Chapter 1
Exam Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use a graphing calculator or computer to determine which of the given viewing windows displays the most appropriate graph of the specified function. 1) f(x) = x3 - 2x2 - 3x + 14 1)
A) [-2, 2] by [-10, 10] C) [-20, 20] by [-100, 100]
B) [-5, 5] by [-5, 25] D) [-5, 25] by [-5, 5]
2) f(x) = 10
2)
x2 - 4
A) [-5, 5] by [-10, 10] C) [-2, 2] by [-10, 10]
B) [-5, 0] by [-10, 10] D) [0, 5] by [-10, 10]
3) f(x) = x2/3(6 - x) A) [-2, 2] by [-15, 15] C) [-4, 0] by [-5, 5]
B) [-4, 9] by [-10, 10] D) [0, 9] by [-10, 10]
3)
1
Match the equation with its graph. 4) y = 4x
4)
A)
B)
C)
D)
Use a graphing calculator or computer to determine which of the given viewing windows displays the most appropriate graph of the specified function. 5) f(x) = x4 - 8x2 + 2x 5)
A) [-10, 15] by [-5, 5] C) [-25, 15] by [-5, 5]
B) [-5, 5] by [-10, 15] D) [-5, 5] by [-25, 15]
2
6) f(x) = x - 4
6)
x2 + 4
A) [-10, 10] by [-10, 10] C) [-10, 10] by [-2, 2]
B) [-5, 5] by [-15, 15] D) [-1, 1] by [-2, 2]
7) f(x) = |x2 - 1| A) [-5, 5] by [-2, 10] C) [0, 5] by [-2, 10]
7) B) [-10, 10] by [-15, 15] D) [-5, 5] by [-15, 15]
2
8) f(x) = x2 + 1 cos 80x
8)
20
A) [-2, 2] by [-1, 1] C) [-0.6, 0.6] by [-0.1, 0.6]
B) [-0.1, 0.1] by [-0.1, 0.1] D) [-10, 10] by [-10, 10]
Match the equation with its graph. 9) y = 5 x
9)
A)
B)
C)
D)
Use a graphing calculator or computer to determine which of the given viewing windows displays the most appropriate graph of the specified function. 10) f(x) = 7 + 6x - x2 10)
A) [-4, 5] by [-15, 25] C) [-10, 20] by [-50, 50]
B) [-4, 5] by [-5, 5] D) [-10, 10] by [-10, 5]
3
Match the equation with its graph. 11) y = x3
11)
A)
B)
C)
D)
Use a graphing calculator or computer to determine which of the given viewing windows displays the most appropriate graph of the specified function. 12) f(x) = 11 + 6x - x3 12)
A) [-4, 5] by [-15, 25] C) [-4, 5] by [-5, 5]
B) [-10, 20] by [-50, 50] D) [-10, 10] by [-10, 5]
13) f(x) = 2 cos 70x A) [-0.2, 0.2] by [-1, 1] C) [-10, 10] by [-10, 10]
13) B) [-1, 1] by [-4, 4] D) [-0.2, 0.2] by [-4, 4]
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the addition formulas to derive the identity.
14) sin x -
2
= -cos x
14)
4
Solve the problem. 15) Graph the functions f(x) = x and g(x) = differences, and two quotients.
3 - x together with their sum, product, two
15)
16) The standard formula for the tangent of the difference of two angles is
16)
17) Use the angle sum formulas to derive sin (A - B) = sin A cos B - cos A sin B.
17)
18) What happens if you set B = -2 in the angle sum formulas for the sine and cosine
18)
tan A - tan B tan (A - B) = . Derive the formula. 1 + tan A tan B
functions? Do the results agree with something you already know?
Use the addition formulas to derive the identity.
19) cos x -
20) sin x +
2
2
= sin x
19)
= cos x
20)
5
Solve the problem.
21) Let f(x) = x - 1 and g(x) = x2 . Graph f and g together with f g and g f.
21)
22) Graph y = cos 2x and y = sec 2x together for - 3
3 . Comment on the behavior of 4
22)
x 2 . Comment on the behavior of csc
23)
x
4
sec 2x in relation to the signs and values of cos 2x.
23) Graph y = sin x and y = csc x together for -2 2
2
x x in relation to the signs and values of sin . 2 2
Use the addition formulas to derive the identity.
24) cos x +
2
= -sin x
24)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Using the graph below, find the domain and range of the given function, and sketch the graph.
6
25) y = f(x + 2) -1
25)
A) D: [-8, 1]; R: [-2.5, 5]
B) D: [-7, 2]; R: [-4, 2]
7
C) D: [-4, 5]; R: [-3, 3]
D) D: [-5, 4]; R: [-3, 3]
Solve the problem.
26) On a circle of radius 6 meters, how long is an arc that subtends a central angle of 4 radians? 3
A) 24 m
B) 2 m
C) 8 m
Graph the function. Determine the symmetry, if any, of the function. 27) y = - 1 x2
8
26)
D) 8 m
27)
A) No symmetry
B) Symmetric about the y-axis
C) Symmetric about the y-axis
9
D) Symmetric about the y-axis
Graph the function. 28) y = 5x - 15
28)
A)
10
B)
C)
D)
11
The problem tells by what factor and direction the graph of the given function is to be stretched or compressed. Give an equation for the stretched or compressed graph. 29) y = 1 + 1 29) compressed horizontally by a factor of 3 x2
A) y = 1 + 1 3
For f(x) = A sin
B) y = 1 + 9
3x2
C) y = 1 + 1
x2
D) y = 3 + 3
9x2
x2
2 (x - C) + D, B
identify either A, B, C, or D as indicated for the sine function. 30) y = 1 cos Find A. -2 2
B) 1
A)
30) C) 2
2
D) 1
Find a formula for the function graphed.
31)
31)
2, A) f(x) = 6, 2, 6,
0 x 2<x 4<x 6<x
2 4 6 8
C) f(x) = 6, 2,
0 x<6 2 x<8
B) f(x) = 2,
0 x<6 2 x<8
2, D) f(x) = 6, 2, 6,
0 2 4 6
6,
x<2 x<4 x<6 x<8
Find the domain and range of the function. 32) g(z) = 16 - z 2
32)
A) D: (- , ), R: (0, 4) C) D: [-4,4], R: [0,4]
B) D: [0, ), R: (- , ) D) D: (-4,4), R: (-4,4)
12
Find a formula for the function graphed.
33)
33)
A) f(x) = 4 + x, x < 2
B) f(x) = 4 + x, x 2
7 x>2 4 x, x 2 C) f(x) = 7 x>2
7 x>2 4 x, x<2 D) f(x) = 7 x 2
Find the formula for the function. 34) Express the area of a circle as a function of its radius r. A) A = r2 B) A = 2 r C) A = r Express the given function as a composite of functions f and g such that y = f(g(x)). 35) y = 1 x2 - 2
A) f(x) = 1 , g(x) = - 1
B) f(x) = 1 , g(x) = x - 2
C) f(x) = 1 , g(x) = x2 - 2
D) f(x) = 1 , g(x) = x2 - 2
x2
2
36) G(x) =
x + 2, 2,
D) A = r3
35)
x2
x
Graph the function.
34)
2
x<0 x 0
36)
13
A)
B)
C)
14
D)
Find the domain and range of the function. 37) g(z) = 6 - z A) D: (- ,6], R: (- , )
37) B) D: (- , ), R: (- ,6] D) D: [0, ), R: (- ,6]
C) D: (- ,0], R: [6, ) Graph the function. 38) y = 1 (x - 2)2
38)
A)
15
B)
C)
D)
16
Determine whether or not the graph is a graph of a function of x.
39)
39)
A) Function
B) Not a function
Graph the function. 40) y - 4 = x2/3
40)
A)
17
B)
C)
D)
18
41) y = 4 - x
41)
A)
B)
19
C)
D)
Find the requested information using the law of cosines and/or the law of sines. Round to three decimal places. 42) A triangle has sides a = 4 and b = 2 and angle C = 40°. Find the sine of B. 42) A) 0.462 B) 0.924 C) 0.115 D) 0.231 The equation of an ellipse is given. Put the equation in standard form and sketch the ellipse. 43) 25x2 + 16y2 = 400
20
43)
2
2
25
16
2
2
16
25
2
2
16
25
A) x + y = 1
B) x + y = 1
C) x + y = 1
21
2
2
25
16
D) x + y = 1
Solve the problem.
44) The accompanying figure shows the graph of y = x2 shifted to a new position. Write the equation
44)
for the new graph.
A) y = x2 - 5
B) y = (x + 5)2
C) y = x2 + 5
D) y = (x - 5)2
Find the domain and range of the function. 45) F(t) = 4 3 t
45)
A) D: (- , ), R: (- , ) C) D: [0, ), R: [0, )
B) D: (0, ), R: (0, ) D) D: (- ,0), R: (- ,0)
Find the exact value of the trigonometric function. Do not use a calculator or tables. 46) cos 5 3
A)
3 2
B) - 1
C) -
2
3 2
46) D) 1 2
Determine an appropriate viewing window for the given function and use it to display its graph.
22
2 47) f(x) = 3x
47)
x2 - 9
A)
B)
23
C)
D)
The problem tells how many units and in what direction the graph of the given equation is to be shifted. Give an equation for the shifted graph. Then sketch the original graph with a dashed line and the shifted graph with a solid line. 48) y = 1 48) Down 6, right 2 x
24
A) y - 6 = 1
x+2
B) y - 6 = 1
x-2
C) y + 6 = 1
x-2
25
D) y + 6 = 1
x+2
Solve the problem. 49) A power plant is located on a river that is 650 feet wide. To lay a new cable from the plant to a location in a city 1 mile downstream on the opposite side costs $200 per foot across the river and $ 125 per foot along the land. Suppose that the cable goes from the plant to a point Q on the opposite side that is x feet from the point P directly opposite the plant. Write a function C(x) that gives the cost of laying the cable in terms of the distance x.
49)
1 mi
650 ft
A) C(x) = 125 x2 + 6502 + 200(5280 - x)
B) C(x) = 200 x2 + 6502 + 125(1 - x)
C) C(x) = 200 x2 + 6502 + 125(5280 - x)
D) C(x) = 200(650 - x) + 125(1 - x)
Graph the function.
50) y = (2 - x)3 + 3
50)
26
A)
B)
C)
27
D)
Solve for the angle , where 0 51) sin2 = 3 4
A) = C) =
6 3
, ,
6 3
, ,
6 3
2
51) , ,
11 6
B) = 0, , 2 D) =
3
4
,
4
,
4
,
4
Determine if the function is even, odd, or neither. 52) g(x) = |3x7 |
A) Even
52)
B) Odd
C) Neither
Find the formula for the function. 53) Express the perimeter of a square as a function of the square's side length x. A) p = 3x B) p = 4x C) p = 6x 2
53) D) p = x3
Find the exact value of the trigonometric function. Do not use a calculator or tables. 54) sec - 3 2
A) -1
B) 0
C) 1
54) D) Undefined
Graph the function. Specify the intervals over which the function is increasing and the intervals where it is decreasing.
28
55) y = -3 x
55)
A) Increasing 0 x <
B) Decreasing 0 x <
29
C) Decreasing - < x <
D) Increasing 0 x <
Use the appropriate addition formula to find the exact value of the expression.
56) cos
56)
12
A)
24
6
B) -
6+ 4
2
C)
6+ 4
2
D)
64
2
Find the exact value of the trigonometric function. Do not use a calculator or tables.
57) sin
57)
4
A)
3 2
B) 1
C) 2
2
D)
2 2
Find the formula for the function.
58) A point P in the fourth quadrant lies on the graph of the function f(x) = -x2 . Express the slope of the line joining P to the origin as a function of x. A) m = 1 B) m = x x
C) m = -x
Graph the function. 30
D) m = -2x
58)
59) y = 1
59)
x+1
A)
B)
31
C)
D)
Find the domain and range of the function. 60) F(t) = t2 + 2
60)
A) D: (- , ), R: (- , ) C) D: [-4, ), R: [2, )
B) D: [0, ), R: (- , 2] D) D: (- , ), R: [2, )
Find the domain and graph the function. 61) F(t) = t - 2 t- 2
61)
32
A) D: (- , 2) (2>, )
B) D: (- , )
C) D: (- , 0) (0, )
33
D) D: (- , 2) (2, )
Determine an appropriate viewing window for the given function and use it to display its graph. 62) f(x) = 6x x2 - 1
A)
34
62)
B)
C)
D)
Find the domain and range for the indicated function. 63) f(x) = 3, g(x) = 3 + x; f/g A) D: x 0 B) D: x 0 R: y 1 R: y 1
63) C) D: x 0 R: y 3
35
D) D: x -3 R: y 0
The problem tells by what factor and direction the graph of the given function is to be stretched or compressed. Give an equation for the stretched or compressed graph. 64) y = 1 + 1 64) stretched vertically by a factor of 5 x2
A) y = 5 + 5
x2
Graph the function.
1
65) g(x) = x - 1 x,
B) y = 1 + 25
C) y = 1 + 1
x2
5
,x<1
5x2
D) y = 1 + 5
x2
65)
x 1
A)
36
B)
C)
D)
Solve the problem.
66) If f(x) = x, g(x) = x , and h(x) = 2x+ 4, find h(g(f(x))).
66)
2
A) 2 x + 4
B) x + 4
C) x + 2
37
D) x + 2
One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. 3 67) cos x = - 1 , x in , 5 2
A) sin x = - 2 6 , tan x = 2 6
B) sin x = - 2 6 , tan x = -2 6
C) sin x = 2 6 , tan x = -2 6
D) sin x = 2 6 , tan x = 2 6
5
67)
5
5
5
Find the domain and graph the function. 68) f(x) = 3x - 2
68)
A) D: (- , )
38
B) D: (- , )
C) D: [0, )
D) D: (- , )
Solve the problem.
69) If f(x) = x, g(x) = x , and h(x) = 4x+ 12, find f(g(h(x))).
69)
4
A) x + 3
B) x + 3
C) x + 12
Graph the function.
39
D) 4 x + 12
70) y = 1 - 4
70)
4x
A)
B)
40
C)
D)
State the period of the function and graph. 71) -sin x 3
71)
41
A) Period 6
B) Period 6
C) Period 6
42
D) Period 6
Graph the function. 72) y = x +2
72)
A)
43
B)
C)
D)
44
73) y = 1 - 2
73)
x2
A)
B)
45
C)
D)
Solve the problem. 74) The kinetic energy K of a mass is proportional to the square of its velocity v. If K = 13,500 joules when v = 15 m/sec, what is K when v = 11 m/sec? A) 8470 joules B) 7260 joules C) 7865 joules D) 6050 joules The equation of an ellipse is given. Put the equation in standard form and sketch the ellipse. 75) 16x2 + 64y2 = 1024
46
74)
75)
2
2
64
16
2
2
16
64
2
2
16
64
A) x + y = 1
B) x + y = 1
C) x + y = 1
47
2
2
64
16
D) x + y = 1
The problem tells how many units and in what direction the graph of the given equation is to be shifted. Give an equation for the shifted graph. Then sketch the original graph with a dashed line and the shifted graph with a solid line. 76) x2 + y2 = 16 76) Up 5, right 3
A) (x + 3)2 + (y - 5)2 = 16
48
B) (x - 3)2 + (y + 5)2 = 16
C) (x - 3)2 + (y - 5)2 = 16
D) (x + 3)2 + (y + 5)2 = 16
Determine if the function is even, odd, or neither. 77) f(x) = (x + 9)(x - 3) A) Even B) Odd
77) C) Neither
49
Express the given quantity in terms of sin x or cos x. 78) sin 3 + x 2
A) sin x - cos x
78)
B) -cos x
C) -cos x - sin x
D) cos x
Solve the problem.
79) If f(x) = 3x - 2 and g(x) = 8x2 - 6x - 7, find g(f(-2)). A) -35 B) -23 C) 553
79) D) 109
Express the given quantity in terms of sin x or cos x. 80) cos 7 - x 2
A) -sin x
80)
B) sin (-x)
C) cos x + sin x
Determine if the function is even, odd, or neither. 81) f(x) = -6 A) Even B) Odd
82) f(x) = 9x5 - 3x3 A) Even
D) sin x
81) C) Neither 82)
B) Odd
C) Neither
Find a formula for the function graphed.
83)
83)
A) f(x) = 6,
-6,
C) f(x) = 6x,
-6x,
x 0 x>0
B) f(x) = -6,
x 0 x>0
D) f(x) = 6,
x<0 x 0
6,
x 0 x>0
-6,
Graph the function.
50
84) g(x) = -2
x + 4,
x 0 x>0
84)
A)
B)
51
C)
D)
Express the given quantity in terms of sin x or cos x. 85) sin (4 - x) A) sin x B) sin (-x)
85) C) cos x - sin x
D) -sin x
Find the function value.
86) sin2
86)
6
A) 3 4
C) 2 - 3
B) 2 - 3
4
D) 1 4
Find the formula for the function.
87) A point P in the first quadrant lies on the graph of the function f(x) = x2 . Express the slope of the line joining P to the origin as a function of x. A) m = x B) m = 1 x
C) m = 2
x
Graph the function. Determine the symmetry, if any, of the function.
52
D) m = 2x
87)
88) y = -5 x
88)
A) Symmetric about the x-axis
B) No symmetry
53
C) No symmetry
D) Symmetric about the x-axis
State the period of the function and graph.
89) sin x +
4
-2
89)
54
A) Period 2
B) Period 2
C) Period 2
55
D) Period 2
Provide an appropriate response. 90) For what values of x is A) 1 x < 2
= 2? B) 2 x < 3
90) C) 2 < x 3
D) 1 < x 2
One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. 91) sin x = - 1 , x in - , 0 2 2
A) cos x = C) cos x =
2 2
3
3
, tan x =
, tan x =
3 3
3
B) cos x = -
3
D) cos x =
2 2
3
3
91)
, tan x = -
, tan x = -
Find the domain and range of the function. 92) f(x) = 7 9+ x
3 3
3
3
92)
A) D: [0, ), R: (- , )
B) D: (- ,0], R:(- ,0]
C) D: (- , ), R: 0, 7
D) D: [0, ), R: 0, 7
9
9
56
Solve the problem.
93) The accompanying figure shows the graph of y = x2 shifted to a new position. Write the equation
93)
for the new graph.
A) y = (x + 3)2
B) y = x2 + 3
C) y = x2 - 3
D) y = (x - 3)2
Determine if the function is even, odd, or neither. 94) f(x) = 3x2 - 4
A) Even
94)
B) Odd
C) Neither
Determine an appropriate viewing window for the given function and use it to display its graph. 95) f(x) = x5 - x3 + x2 + 6
57
95)
A)
B)
C)
58
D)
Use the appropriate addition formula to find the exact value of the expression. 96) tan 13 12
A) 2 + 3 4
C) 2 - 3
B) 2 + 3
4
96) D) 2 - 3
Graph the function. Specify the intervals over which the function is increasing and the intervals where it is decreasing. 97) y = (-x)3/2 97)
59
A) Decreasing - < x 0 Increasing 0 x <
B) Increasing 0 x <
C) Decreasing - < x <
60
D) Decreasing - < x 0
Determine whether or not the graph is a graph of a function of x.
98)
98)
A) Function
B) Not a function
99)
99)
A) Function
B) Not a function
61
Find the exact value of the trigonometric function. Do not use a calculator or tables. 100) tan 5 4
A) -1
B)
2 2
C) 1
100) D) 3
Determine if the function is even, odd, or neither. 101) f(x) = 5 x2 + 2
A) Even
101)
B) Odd
C) Neither
Graph the function in the ts-plane (t-axis horizontal, s-axis vertical). State the period and symmetry of the function. 102) s = csc t 102) 4
A) Period 8, symmetric about the s-axis
62
B) Period 8, symmetric about the origin
C) Period 8, symmetric about the s-axis
D) Period 8, symmetric about the origin
The equation of an ellipse is given. Put the equation in standard form and sketch the ellipse.
63
103) 4x2 + (y + 2)2 = 4
103)
2
A) x2 + (y + 2) = 1 4
2
B) x + (y + 2)2 = 1 4
64
2
C) x + (y + 2)2 = 1 4
2
D) x2 + (y + 2) = 1 4
Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line. State whether the combination of functions (where defined) is even or odd. 104) fg 104) A) Even B) Odd
65
Find a formula for the function graphed.
105)
105)
A) f(x) = 6 - x,
0 x 3 3<x 6
B) f(x) = -x,
0 x 3 3<x 6
C) f(x) = x,
0 x 3 3<x 6
D) f(x) = x + 6,
0 x 3 3<x 6
x,
6 - x,
x + 6,
-x,
Find the formula for the function. 106) Express the volume of a sphere as a function of its radius r. A) V = 4 r3 B) V = 2 r2 C) V = r3 3 3
106) D) V = 3 r3 4
Graph the function. Specify the intervals over which the function is increasing and the intervals where it is decreasing. 107) y = -x 107)
66
A) Decreasing - < x 0
B) Increasing - < x 0
C) Decreasing 0 x <
67
D) Decreasing - < x 0 Increasing 0 x <
Solve the problem. 108) Let g(x) = x + 7. Find a function y = f(x) so that (f g)(x) = 4x + 28 A) f(x) = 4x + 7 B) f(x) = 4x C) f(x) = 4(x + 1)
108) D) f(x) = 4x - 7
Graph the function in the ts-plane (t-axis horizontal, s-axis vertical). State the period and symmetry of the function. 109) s = sec t 109) 3
A) Period 6 , symmetric about the s-axis
68
B) Period 6 , symmetric about the t-axis
C) Period 6 , symmetric about the t-axis
D) Period 6 , symmetric about the s-axis
Express the given quantity in terms of sin x or cos x. 110) sin 3 - x 2
A) -cos (-x)
110)
B) -cos x
C) -cos x - sin x
Graph the function. 69
D) cos x
111) y = 1 - 4
111)
x
A)
B)
70
C)
D)
Determine an appropriate viewing window for the given function and use it to display its graph. 112) f(x) = -0.7x6 - x5 + 5x4 - 3x3 - 6x2 + x - 3
71
112)
A)
B)
C)
72
D)
Solve the problem.
113) The accompanying figure shows the graph of y = -x2 shifted to a new position. Write the equation
113)
for the new graph.
B) y = -(x - 6)2
A) y = -x2 - 6
C) y = -(x + 6)2
D) y = -x2 + 6
One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. 114) sin x = 12 , x in , 13 2
A) cos x = 5 , tan x = - 5 13
B) cos x = - 5 , tan x = - 12
12
13
C) cos x = - 5 , tan x = - 5 13
115) cos x = 5 , 13
x in -
13
115) B) sin x = 12 , tan x = 12
12
13
C) sin x = - 12 , tan x = - 12 13
5
,0
A) sin x = 12 , tan x = - 5 13
5
D) cos x = 5 , tan x = 12
12
2
114)
5
D) sin x = - 12 , tan x = - 5
5
13
Graph the function. Determine the symmetry, if any, of the function. 73
12
116) y = -x
116)
A) No symmetry
B) Symmetric about the y-axis
74
C) Symmetric about the y-axis
D) No symmetry
Determine an appropriate viewing window for the given function and use it to display its graph. 3 117) f(x) = x - 4
75
117)
A)
B)
C)
76
D)
Graph the function.
118) Graph the upper half of the circle defined by the equation x2 + y2 - 10x - 6y + 9 = 0.
A)
77
118)
B)
C)
D)
Use the appropriate addition formula to find the exact value of the expression. 119) cos 19 12
A) 2 - 6
B)
64
2
C)
78
24
6
119) D) - 6 - 2
Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line. State whether the combination of functions (where defined) is even or odd. 120) f f 120)
A) Even
B) Odd
Graph the function. Specify the intervals over which the function is increasing and the intervals where it is decreasing. 121) y = -|x| 121)
A) Increasing - < x <
B) Decreasing - < x 0 Increasing 0 x <
79
C) Decreasing - < x <
D) Increasing - < x 0 Decreasing 0 x <
Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line. State whether the combination of functions (where defined) is even or odd. 122) g g 122)
A) Even
B) Odd
Graph the function.
80
123) Graph two periods of the function f(x) = -cot x + 1. 2
A)
B)
81
123)
C)
D)
Solve the problem.
124) Let f(x) = x - 5. Find a function y = g(x) so that (f g)(x) = x2 - 5. A) g(x) = 2x B) g(x) = x2 + 5 C) g(x) = x2
124) D) g(x) = x2 - 5
Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line. State whether the combination of functions (where defined) is even or odd. 125) g f 125)
A) Even
B) Odd
The problem tells by what factor and direction the graph of the given function is to be stretched or compressed. Give an equation for the stretched or compressed graph. 126) y = x2 - 2 126) stretched horizontally by a factor of 3
A) y = 3x2 - 6
2
B) y = x - 2
C) y = 9x2 - 2
3
2
D) y = x - 2 9
Graph the function in the ts-plane (t-axis horizontal, s-axis vertical). State the period and symmetry of the function.
82
127) s = -cot 4t
A) Period
B) Period
127)
4
4
, symmetric about the s-axis
, symmetric about the origin
83
C) Period , symmetric about the origin
D) Period
4
, symmetric about the origin
Use the appropriate addition formula to find the exact value of the expression. 128) sin 11 12
A)
64
2
B)
24
6
C) -
6+ 4
2
128) D)
6+ 4
Determine an appropriate viewing window for the given function and use it to display its graph.
84
2
129) f(x) = x4 - 4x3 + 12x2 + x - 11
129)
A)
B)
85
C)
D)
Graph the function. 130) y = x + 3 - 4
130)
86
A)
B)
C)
87
D)
Graph the function. Determine the symmetry, if any, of the function. 131) y = -x4/5
A) No symmetry
88
131)
B) Symmetric about the y-axis
C) Symmetric about the y-axis
D) No symmetry
89
Find a formula for the function graphed.
132)
132)
A) f(x) = 2,
x<0 x 0
B) f(x) = 2,
x<3 x>3
C) f(x) = 2,
x<3 x 3
D) f(x) = 2,
x 3 x>3
3 - x,
x - 3,
3 - x,
3 - x,
Find the domain and range of the function. 133) F(t) = 3 t
133)
A) D: (- ,0), R: (- ,0) C) D: (- , ), R: (- , )
B) D: [0, ), R: (- , ) D) D: (0, ), R: (0, )
Find the exact value of the trigonometric function. Do not use a calculator or tables.
134) sec
134)
3
A) 2
B) 2 3
C) 2
3
90
D)
3 2
Solve the problem. 135) A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 inches by 23 inches by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume V of the box as a function of x.
135)
23
12
A) V(x) = (12 - 2x)(23 - 2x) C) V(x) = (12 - x)(23 - x)
B) V(x) = x(12 - 2x)(23 - 2x) D) V(x) = x(12 - x)(23 - x)
The problem tells by what factor and direction the graph of the given function is to be stretched or compressed. Give an equation for the stretched or compressed graph. 136) y = x2 - 3 136) compressed vertically by a factor of 3 2
A) y = x - 1 3
2
B) y = x - 3
C) y = 9x2 - 3
3
D) y = 3x2 - 9
Find the formula for the function. 137) Express the perimeter of an isosceles triangle with side lengths x, 5x, and 5x as a function of the side length. A) p = 10x3 B) p = 25x3 C) p = 11x D) p = 10x
137)
Solve the problem.
138) The accompanying figure shows the graph of y = -x2 shifted to a new position. Write the equation
138)
for the new graph.
A) y = -x2 + 4
B) y = -(x - 4)2
C) y = -x2 - 4
D) y = -(x + 4)2
Graph the function. Specify the intervals over which the function is increasing and the intervals where it is decreasing.
91
139) y = -x2/3
139)
A) Increasing - < x <
B) Increasing - < x < 0 Decreasing 0 < x <
92
C) Decreasing - < x <
D) Decreasing - < x 0 Increasing 0 x <
Determine if the function is even, odd, or neither. 140) f(x) = -2 x-5
A) Even
140)
B) Odd
C) Neither
Determine an appropriate viewing window for the given function and use it to display its graph.
93
141) y = 8
9 + x2 9
141)
A)
B)
94
C)
D)
Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line. State whether the combination of functions (where defined) is even or odd. 142) f2 142)
A) Even
B) Odd
Solve the problem. 143) The variable s is proportional to t, and s = 25 when t = 75. Determine t when s = 50. A) 140 B) 3 C) 150 D) 200
143)
Find the function value.
144) cos2
144)
12
A) 2 + 3
B) 2 + 3
C) 2 - 3
4
4
D) 1 + 3 2
The problem tells how many units and in what direction the graph of the given equation is to be shifted. Give an equation for the shifted graph. Then sketch the original graph with a dashed line and the shifted graph with a solid line.
95
145) y = - x
Left 2
145)
A) y = x - 2
B) y = x - 2
96
C) y = x + 2
D) y = x + 2
Find the domain and range of the function. 146) f(x) = 7 + x A) D: (- , ), R: [7, )
146) B) D: [0, ), R: [7, ) D) D: [0, ), R: (- , )
C) D: (- ,0], R: (- ,7]
Express the given function as a composite of functions f and g such that y = f(g(x)). 147) y = (9x + 13)6
A) f(x) = 9x6, g(x) = x + 13 C) f(x) = x6 , g(x) = 9x + 13
147)
B) f(x) = 9x + 13, g(x) = x6 D) f(x) = (9x)6 , g(x) = 13
Find the exact value of the trigonometric function. Do not use a calculator or tables. 148) sin 11 6
A)
3 2
B) -
3 2
C) - 1 2
97
148) D) 1 2
Solve the problem. 149) Let g(x) =
x. Find a function y = f(x) so that (f g)(x) = x . A) f(x) = 1 B) f(x) = x2 C) f(x) = 1 x x2
149) D) f(x) = x
Use the appropriate addition formula to find the exact value of the expression. 150) sin 17 12
A)
24
6
B)
6+ 4
2
C)
64
150)
2
D) -
6+ 4
2
Provide an appropriate response.
151) Graph the functions f(x) = x and g(x) = 1 + 4 together to identify the values of x for which x > 1 + 2
x
2
151)
4 . x Confirm your findings algebraically. A) (-2, 0) (4, ) B) (- , -2) (0, 4)
C) (4, )
D) (-2, 4)
Solve the problem.
152) Let f(x) = x . Find a function y = g(x) so that (f g)(x) = x.
152)
x-7
A) g(x) = 1
B) g(x) = x - 7
x-7
Solve for the angle , where 0 153) cos2 = 1 4
A) = C) =
4 3
, ,
4 3
, ,
4 3
D) g(x) = 7x
C) g(x) = x(x - 7)
7
x-1
2
153) , ,
B) =
4
6
,
6
,
6
,
11 6
D) = 0, , 2
3
Express the given quantity in terms of sin x or cos x. 154) sin (3 + x) A) -sin x B) cos x - sin x
154) C) cos x + sin x
D) sin x
Use the appropriate addition formula to find the exact value of the expression. 155) sin 19 12
A)
64
2
B)
24
6
C)
98
6+ 4
2
155) D) -
6+ 4
2
Find the requested information using the law of cosines and/or the law of sines. Round to three decimal places.
156) A triangle has side c = 3 and angles A = A) 2.196
4
and B =
6
B) 0.518
Find the domain and range for the indicated function. 157) f(x) = x + 7, g(x) = x - 7; g- f A) D: x -7 B) D: x 7 R: y - 14 R: y 14
. Find the length b of the side opposite B.
C) 0.173
D) 1.553
C) D: x 7
D) D: x 7
156)
157) R: y 0
Graph the function. 158) Graph the function f(x) = sin 4x + cos 2x.
R: y - 14
158)
A)
99
B)
C)
D)
Graph the function. Specify the intervals over which the function is increasing and the intervals where it is decreasing.
100
159) y = - 1
159)
x2
A) Decreasing - < x < 0 Increasing 0 < x <
B) Decreasing - < x < 0 and 0 < x <
101
C) Increasing - < x < 0 and 0 < x <
D) Increasing - < x < 0 Decreasing 0 < x <
State the period of the function and graph. 160) cos 6x
160)
102
A) Period
B) Period
C) Period
3
3
3
103
D) Period
3
Graph the function. Determine the symmetry, if any, of the function. 161) y = 1 x
A) Symmetric about the origin
104
161)
B) Symmetric about the origin
C) Symmetric about the origin
D) No symmetry
105
Solve the problem. 162) The figure shown here shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 6 units long. Express the area A of the rectangle in terms of x.
-3
162)
3
B) A(x) = 2x2 D) A(x) = 2x(x - 3)
A) A(x) = 2x(3 - x) C) A(x) = x(3 - x)
Determine an appropriate viewing window for the given function and use it to display its graph. 3 163) f(x) = x x2 + 9
A)
106
163)
B)
C)
D)
The equation of an ellipse is given. Put the equation in standard form and sketch the ellipse.
107
164) 9(x - 4)2 + 4(y + 2)2 = 36
2
164)
2
A) (x - 4) + (y + 2) = 1 4
9
2
2
B) (x - 4) + (y + 2) = 1 4
9
108
2
2
C) (x - 4) + (y + 2) = 1 9
4
2
2
D) (x - 4) + (y + 2) = 1 9
4
Graph the function.
165) Graph the lower branch of the hyperbola y2 - 4x2 = 1.
109
165)
A)
B)
C)
110
D)
Solve the problem.
166) The accompanying figure shows the graph of y = -x2 shifted to a new position. Write the equation
166)
for the new graph.
A) y = -(x - 4)2
B) y = -(x + 4)2
C) y = -x2 - 4
D) y = -x2 + 4
Use the appropriate addition formula to find the exact value of the expression. 167) tan 7 12
A) 2 + 3
B) 2 - 3
C) -2 - 3
4
Graph the function.
111
167) D) 2 + 3 4
168) y = x2 - 5
168)
A)
B)
112
C)
D)
Solve the problem. 169) If f(x) = x + 9 and g(x) = 8x - 13, find f(g(x)). A) 8 x - 4 B) 2 2x - 1
169) C) 2 2x + 1
113
D) 8 x + 9 - 13
170) The accompanying figure shows the graph of y = x2 shifted to a new position. Write the equation
170)
for the new graph.
A) y = (x + 5)2
C) y = (x - 5)2
B) y = x2 - 5
D) y = x2 + 5
Provide an appropriate response. 171) Graph the equation y2 = x and decide whether or not the graph represents a function of x.
A) Function
B) Not a Function
114
171)
Find a formula for the function graphed.
172)
172)
A) f(x) =
C) f(x) =
1 x + 1, 2
-8 x -2
5, 6 - x,
-2 < x 3 3<x 8
1 x + 1, 2
-8 x -2
5, 6 - x,
-2 < x < 3 3 x 8
B) f(x) =
D) f(x) =
1 x + 1, 2
-8 < x -2
5, 6 - x,
-2 < x 3 3<x<8
-
1 x + 1, 2
5, x - 6,
-8 x -2 -2 < x 3 3<x 8
Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line. State whether the combination of functions (where defined) is even or odd. 173) f/g 173) A) Even B) Odd For f(x) = A sin
2 (x - C) + D, B
identify either A, B, C, or D as indicated for the sine function.
174) y = -2 cos A) - 4 5
+
2
Find C.
B)
174) C) -
8
Graph the function.
115
8
D) - 1
10
175) y = |x + 3| + 4
175)
A)
B)
116
C)
D)
State the period of the function and graph.
176) cos x +
176)
2
117
A) Period 2
B) Period 2
C) Period 2
118
D) Period 2
Solve the problem.
177) The accompanying figure shows the graph of y = x2 shifted to a new position. Write the equation
177)
for the new graph.
A) y = (x - 1)2 - 2
B) y = (x - 1)2 + 2
C) y = (x - 2)2 - 1
178) If f(x) = -6x + 9 and g(x) = 4x + 4, find g(f(x)). A) -24x + 40 B) -24x - 32
D) y = (x + 1)2 + 2 178)
C) 24x + 40
D) -24x + 33
Determine if the function is even, odd, or neither. 179) g(x) = -5x x2 + 7
A) Even
179)
B) Odd
C) Neither
119
Find a formula for the function graphed.
180)
180)
A) f(x) = x,
x 1 x>1
B) f(x) = 2x,
x 1 x>1
C) f(x) = -2x,
x 1 x>1
D) f(x) = -2x,
x 1 x>1
2x + 1,
x + 2,
x + 1,
x + 1,
Determine whether or not the graph is a graph of a function of x.
181)
181)
A) Function
B) Not a function
Find the formula for the function. 182) Express the area of a square as a function of its side length x.
A) A = 4x
B) A = 2x
Find the domain and range for the indicated function. 183) f(x) = x + 3, g(x) = x - 3; f-g A) D: x 3 B) D: x -3 R: y 6 R: y 6
182)
C) A = x2
D) A = x4
C) D: x 3
D) D: x -3
183) R: y 0
120
R: y 0
Determine whether or not the graph is a graph of a function of x.
184)
184)
A) Function
B) Not a function
Find the formula for the function. 185) Express the length d of a square's diagonal as a function of its side length x. A) d = x B) d = x 2 C) d = 2x
185) D) d = x 3
Using the graph below, find the domain and range of the given function, and sketch the graph.
186) y = f(-x - 2) + 1
186)
121
A) D: [-6, 3]; R: [-2, 4]
B) D: [-7, 2]; R: [-2, 4]
C) D: [-7, 2]; R: [-2, 4]
122
D) D: [-3, 6]; R: [-2, 4]
Solve for the angle , where 0 187) sin 2 - cos = 0
A) 0,
2
C) =
, , 2
,
2 2
,
2
187)
,2 6
,
B)
4
D) =
6
,
4 6
,
, 6
6 ,
,
11 6
6
,
11 6
Determine an appropriate viewing window for the given function and use it to display its graph. 188) f(x) = sin 2x x
123
188)
A)
B)
C)
124
D)
Provide an appropriate response.
189) Consider the function y = 1 - 1 . Can x be greater than 0, but less than 1? x
A) Yes
189)
B) No
Using the graph below, find the domain and range of the given function, and sketch the graph.
190) y = -f(x)
190)
125
A) D: [-5, 4]; R: [-3, 3]
B) D: [-7, 2]; R: [-4, 2]
C) D: [-4, 5]; R: [-3, 3]
126
D) D: [-7.5, 1]; R: [-1, 5]
The problem tells how many units and in what direction the graph of the given equation is to be shifted. Give an equation for the shifted graph. Then sketch the original graph with a dashed line and the shifted graph with a solid line. 191) y = x3 191) Down 3, left 1
A) y - 3 = (x - 1)3
127
B) y - 3 = (x + 1)3
C) y + 3 = (x + 1)3
D) y + 3 = (x - 1)3
128
Find a formula for the function graphed.
192)
192)
A) f(x) = 5,
x<0 x 0
B) f(x) = 5,
x<0 x 0
C) f(x) = 5,
x<0 x 0
D) f(x) = 5,
x 0 x>0
-x,
x,
-5x,
-x,
Provide an appropriate response.
193) What is the domain of the function y = 1 - 1 ?
193)
x
A) (0, 1]
B) (- , 0) [1, )
C) (- , 0) (1, )
D) (- , )
Find the domain and range of the function. 194) f(x) = 8 - x2
194)
A) D: (- , ), R: (- , ) C) D: (- , ), R: (- , 8]
B) D: (- , 8], R: (- , ) D) D: (- , ), R: [8, )
The problem tells by what factor and direction the graph of the given function is to be stretched or compressed. Give an equation for the stretched or compressed graph. 195) y = x3 + 1 195) stretched vertically by a factor of 2
A) y = 8x3 + 1
3
B) y = x + 1 2
2
Find the domain and range for the indicated function. 196) f(x) = x + 2, g(x) = x - 2; f ·g A) D: x > 2 B) D: x 2 R: y 0 R: y 0
C) y = 2x3 + 2
D) y = 2x3 + 1
C) D: x 2
D) D: x 2
196) R: - < y <
129
R: y > 0
One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. 3 197) tan x = 3 , x in , 4 2
A) sin x = - 4 , cos x = - 3
B) sin x = 4 , cos x = 3
C) sin x = 3 , cos x = 4
D) sin x = - 3 , cos x = - 4
5
5
5
5
5
197)
5
5
5
Express the given quantity in terms of sin x or cos x. 198) cos 7 + x 2
198)
A) cos x
B) -sin x
C) cos x + sin x
D) sin x
199) cos (3 + x) A) cos x
B) -sin x
C) sin x - cos x
D) -cos x
199)
Graph the function. 200) y = (-3x)2/3 + 2
200)
A)
130
B)
C)
D)
Determine an appropriate viewing window for the given function and use it to display its graph.
131
3 201) f(x) = x
201)
x2 - 4
A)
B)
132
C)
D)
Express the given quantity in terms of sin x or cos x. 202) cos (6 + x) A) -sin x B) cos x
202) C) cos x - sin x
D) -cos x
One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. 3 203) cos x = - 2 , x in ,2 2
A) sin x =
2 , tan x = -1 2
B) sin x =
C) sin x = -
2 , tan x = 1 2
D) sin x = -
Find the domain and range for the indicated function. 204) f(x) = 8, g(x) = 8 + x; g/f A) D: x -8 B) D: x 0 R: y 0 R: y 1
203)
2 , tan x = 1 2 2 , tan x = -1 2
204) C) D: x 0 R: y 8
133
D) D: x 0 R: y 1
One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. 205) tan x = - 5 , x in , 12 2
A) sin x = - 5 , cos x = 12
B) sin x = 5 , cos x = - 12
C) sin x = 12 , cos x = - 5
D) sin x = 5 , cos x = 12
13
13
Graph the function. 3x + 2, 206) f(x) = x, 2x - 1,
13
13
13
13
x < -2 -2 x 3 x>3
205)
13
13
206)
A)
134
B)
C)
D)
135
For f(x) = A sin
2 (x - C) + D, B
identify either A, B, C, or D as indicated for the sine function. 207) y = -3 - 2sin 1 Find D. 2 2
B) 3
A)
207) C) 1
D) -3
Solve the problem.
208) The accompanying figure shows the graph of y = -x2 shifted to a new position. Write the equation
208)
for the new graph.
A) y = -(x + 5)2 - 1 C) y = -(x - 1)2 + 5
B) y = -(x - 5)2 + 1 D) y = -(x - 5)2 - 1
Solve for the angle , where 0 209) cos2 = 3 4
A) =
6
,
6
,
6
2
209) ,
11 6
B) =
C) = 0, , 2
D) =
4 3
, ,
4 3
, ,
4 3
, ,
Find the exact value of the trigonometric function. Do not use a calculator or tables. 210) cot (- ) A) 0 B) -1 C) 1
4 3
210) D) Undefined
The equation of an ellipse is given. Put the equation in standard form and sketch the ellipse.
136
211) (x + 2)2 + 3y2 = 3
211)
2
A) (x + 2)2 + y = 1 3
2
B) (x + 2) + y2 = 1 3
137
2
C) (x + 2) + y2 = 1 3
2
D) (x + 2)2 + y = 1 3
One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. 3 212) tan x = 1, x in , 2
A) sin x = -
2 2 , cos x = 2 2
B) sin x =
2 2 , cos x = 2 2
C) sin x = -
2 2 , cos x = 2 2
D) sin x =
2 2 , cos x = 2 2
212)
Find the requested information using the law of cosines and/or the law of sines. Round to three decimal places. 213) A triangle has sides a = 4 and b = 2 and angle C = 60°. Find the length of side c. 213) A) 4 B) 3.464 C) 2.479 D) 12 Graph the function. Specify the intervals over which the function is increasing and the intervals where it is decreasing.
138
214) y = 1
214)
x
A) Decreasing - < x < 0 and 0 < x <
B) Increasing - < x < 0 Decreasing 0 < x <
139
C) Decreasing - < x < 0 Increasing 0 < x <
D) Increasing - < x < 0 and 0 < x <
State the period of the function and graph.
215) cos x -
4
-2
215)
140
A) Period 2
B) Period 2
C) Period 2
141
D) Period 2
One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. 216) sin x = - 3 , x in - , 0 2 2
A) cos x = - 1 , tan x = - 3
B) cos x = 1 , tan x = 3
C) cos x = 1 , tan x = - 3
D) cos x = -2, tan x =
2
216)
2
2
3 3
Provide an appropriate response. 217) Graph the equation |x + y| = 1 and decide whether or not the graph represents a function of x.
A) Function
B) Not a Function
Use the appropriate addition formula to find the exact value of the expression. 218) cos - 7 12
A) 6 + 2
217)
B) 2 - 6
C)
142
24
6
218) D)
64
2
Solve for the angle , where 0 219) sin2 = 1 4
A) =
3
,
3
,
3
2
219) ,
B) =
3
C) = 0, , 2
D) =
4 6
, ,
4 6
, ,
4 6
, ,
4 11 6
Use the appropriate addition formula to find the exact value of the expression. 220) tan - 7 12
A) 2 - 3 4
B) 2 + 3
C) -2 - 3
220) D) 2 + 3 4
Determine if the function is even, odd, or neither. 221) f(x) = -6x4 - 9x - 6
A) Even
221)
B) Odd
C) Neither
Graph the function.
222) y = (x + 1)2/3
222)
A)
143
B)
C)
D)
Solve the problem. 223) You want to make an angle measuring 135° by marking an arc on the perimeter of a disk with a diameter of 10 inches and drawing lines from the ends of the arc to the disk's center. To the nearest tenth of an inch, how long should the arc be? A) 47.1 in. B) 5.9 in. C) 23.6 in. D) 11.8 in.
144
223)
For f(x) = A sin
2 (x - C) + D, B
identify either A, B, C, or D as indicated for the sine function.
224) y = 2 cos 3x + A)
2
2
Find B.
224)
B) 2
C) 2
3
D)
Using the graph below, find the domain and range of the given function, and sketch the graph.
225) y = f(-x)
225)
145
A) D: [-4, 5]; R: [-3, 3]
B) D: [-5, 4]; R: [-3, 3]
C) D: [-4, 5]; R: [-3, 4]
146
D) D: [-4, 5]; R: [-3, 3]
Solve the problem.
226) The accompanying figure shows the graph of y = -x2 shifted to a new position. Write the equation
226)
for the new graph.
A) y = -(x - 5)2
B) y = -x2 + 5
C) y = -(x + 5)2
D) y = -x2 - 5
Express the given quantity in terms of sin x or cos x. 227) cos (6 - x) A) cos x + sin x B) cos x
C) cos x - sin x
D) -cos x
For f(x) = A sin
227)
2 (x - C) + D, B
identify either A, B, C, or D as indicated for the sine function.
228) y = -5 cos 3x + A)
2
3
Find A.
228) C) 3
B) -5
147
D) -15
Determine if the function is even, odd, or neither. 229) h(t) = t2 + 1
A) Even
229)
B) Odd
C) Neither
Use the appropriate addition formula to find the exact value of the expression. 230) sin - 11 12
A)
64
2
B)
6+ 4
2
Provide an appropriate response. 231) What real numbers x satisfy the equation A) {x|x real numbers}
C) -
=
?
2
D)
24
6
231) B) {x|x = 0} D)
C) {x|x integers} 232) Graph the function f(x) =
6+ 4
230)
.
232)
A)
148
B)
C)
D)
149
Express the given function as a composite of functions f and g such that y = f(g(x)). 233) y = 8 + 2 x2
233)
B) f(x) = 1 , g(x) = 8 + 2
A) f(x) = 8 , g(x) = 2
x
x2
x
D) f(x) = x, g(x) = 8 + 2
C) f(x) = x + 2, g(x) = 8
x
x2
Graph the function. Determine the symmetry, if any, of the function. 234) y = -|x|
A) Symmetric about the origin
150
234)
B) Symmetric about the y-axis
C) Symmetric about the x-axis
D) Symmetric about the y-axis
Solve for the angle , where 0 235) sin 2 + cos = 0
A)
2
235)
11 , , = , 6 6 6 6
C) =
2
,
2
,
6
,
B) D)
6
151
11 , , = , 2 2 6 6 4
,
4
,
6
,
11 6
Express the given function as a composite of functions f and g such that y = f(g(x)). 4 236) y = 9x + 9
A) f(x) = 9x + 9, g(x) = 4
B) f(x) = 4, g(x) = 9 + 9
C) f(x) = 4 , g(x) = 9x + 9
D) f(x) = 4 , g(x) = 9x + 9
236)
x
x
The problem tells by what factor and direction the graph of the given function is to be stretched or compressed. Give an equation for the stretched or compressed graph. 237) y = x + 1 237) compressed vertically by a factor of 3 x+1 A) y = 3 x + 1 B) y = C) y = 3x + 3 D) y = 3x + 1 3
Solve the problem.
238) If f(x) = 7x + 7 and g(x) = 4x2 - 5x + 3, find g(f(-3)). A) 17 B) 49 C) 857
For f(x) = A sin
238) D) 385
2 (x - C) + D, B
identify either A, B, C, or D as indicated for the sine function.
239) y = -3 sin 4x +
2
A) 4
Find B.
B)
239) C)
2
D) 3
Provide an appropriate response.
240) Consider the function y = 1 - 1 . Can x be negative?
240)
x
A) Yes
B) No
Graph the function.
152
241) y = 2 + 3
241)
x2
A)
B)
153
C)
D)
Find the domain and range of the function. 242) g(z) = -9 z+1
242)
A) D: [0, ), R: (- , ) C) D: (- ,-1), R: (0, ) For f(x) = A sin
B) D: (-1, ), R: (- ,0) D) D: [1, ), R: (- , )
2 (x - C) + D, B
identify either A, B, C, or D as indicated for the sine function. 243) y= 1 sin - + Find B. 2
A)
B) 1
243) C) 2
Solve the problem. 244) If f(x) = 5x + 13 and g(x) = 4x - 1, find f(g(x)). A) 20x + 51 B) 20x + 18
D) 2
244) C) 20x + 8
D) 20x + 12
Graph the function. Specify the intervals over which the function is increasing and the intervals where it is decreasing. 154
245) y = 1
245)
x3
A) Increasing - < x <
B) Increasing - < x < 0 and 0 < x <
155
C) Decreasing - < x < 0 and 0 < x <
D) Decreasing - < x < 0; Increasing 0 < x <
Graph the function. 246) F(x) = 5 - x, 1 + 3x,
x 2 x>2
246)
156
A)
B)
C)
157
D)
One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. 247) sin x = - 5 , x in - , 0 3 2
A) cos x = - 2 , tan x = -
5 2
3
C) cos x = 2 , tan x = 3
B) cos x = - 2 , tan x =
5 2
D) cos x = 2 , tan x = -
5 2
3
5 2
3
247)
Provide an appropriate response.
248) Graph the functions f(x) = 3 and g(x) = 2 together to identify the values of x for which x-1
x+1
248)
3 2 < . Confirm your findings algebraically. x-1 x+1
A) (-1, 1) (1, )
B) (-5, )
C) (-5, -1) (1, )
D) (- , -5)
Graph the function.
249) Graph the function f(x) = 2 cos3 x.
249)
158
A)
B)
C)
159
D)
Find the domain and graph the function. 250) G(t) = 1 t+2
250)
A) D: (- , -2) (-2, )
160
B) D: (- , -2) (-2, )
C) D: (- , )
D) D: (- , 0) (0, )
Solve the problem.
251) If f(x) = 4x2 + 6x + 3 and g(x) = 6x - 8, find g(f(x)). A) 24x2 + 36x + 26 B) 24x2 + 36x + 10
161
251) C) 4x2 + 36x + 10
D) 4x2 + 6x - 5
Determine whether or not the graph is a graph of a function of x.
252)
252)
A) Function
B) Not a function
State the period of the function and graph.
253) sin x -
253)
3
A) Period 2
162
B) Period 2
C) Period 2
D) Period 2
163
For f(x) = A sin
2 (x - C) + D, B
identify either A, B, C, or D as indicated for the sine function.
254) y = -5 sin 3x +
3
A) -5
Find A.
254) C) 3
B) -15
D)
3
Solve the problem.
255) The accompanying figure shows the graph of y = x2 shifted to a new position. Write the equation
255)
for the new graph.
A) y = (x + 3)2
C) y = (x - 3)2
B) y = x2 + 3
Graph the function. Determine the symmetry, if any, of the function. 256) y = 1 x3
164
D) y = x2 - 3
256)
A) Symmetric about the origin
B) No symmetry
C) Symmetric about the origin
165
D) Symmetric about the y-axis
Find the domain and graph the function. 257) F(x) = -x
257)
A) D: (- , 0]
166
B) D: (- , )
C) D: [0, )
D) D: (- , 0)
Solve the problem. 258) Boyle's Law says that volume V of a gas at constant temperature increases whenever the pressure P decreases, so that V and P are inversely proportional. If P = 12.8 lbs/in2 when V = 900 in3 , then what is V when P = 21 lbs/in2 ? A) 23625 in3 B) 112 in3 16 375
C) 3840 in3 7
167
D) 375 in3 112
258)
Express the given function as a composite of functions f and g such that y = f(g(x)). 259) y = 10x + 4 A) f(x) = -x , g(x) = 10x - 4 B) f(x) = x , g(x) = 10x + 4
C) f(x) = - x , g(x) = 10x + 4 Graph the function. 260) f(x) = -3 - x, -2,
259)
D) f(x) = x, g(x) = 10x + 4
x<1 x 1
260)
A)
B)
168
C)
D)
Express the given quantity in terms of sin x or cos x. 261) sin (6 + x) A) -sin x B) cos x + sin x
261) C) sin x
D) cos x - sin x
Solve the problem.
262) If f(x) = 1 and g(x) = 9x5, find g(f(x)).
262)
x
A) 9
x
B) 1
C) 1
9x5
x5
Graph the function.
169
D) 9
x5
263) Graph five periods of the function f(x) = tan 4x.
263)
A)
B)
170
C)
D)
Solve the problem.
264) If f(x) = x - 10 and g(x) = 7x + 10, find g(f(x)).
264)
7
A) - 10 7
B) x + 20
C) x
Find the domain and graph the function. 265) G(x) = x
D) 7x + 60
265)
171
A) D: (- , 0)
B) D: (- , 0]
C) D: (- , )
172
D) D: [0, )
266) g(x) = -9 - 3x - x2
266)
A) D: (- , )
173
B) D: (- , )
C) D: (- , - 27 ] 4
D) D: (- , )
Provide an appropriate response.
174
267) Graph the function f(x) =
.
267)
A)
B)
175
C)
D)
Graph the function. Determine the symmetry, if any, of the function. 268) y = (-x)3/2
176
268)
A) Symmetric about the y-axis
B) No symmetry
C) No symmetry
177
D) Symmetric about the y-axis
Provide an appropriate response. 269) For what values of x is A) -3 < x -2
= -2? B) -2 < x -1
269) C) -3 x < -2
D) -2 x < -1
Solve the problem. 270) If you roll a 1-m-diameter wheel forward 71 centimeters over level ground, through what angle (to the nearest degree) will the wheel turn? A) 81° B) 142° C) 41° D) 1°
271) On a circle of radius 3 meters, how long is an arc that subtends a central angle of 25°? A) 5 m 12
B) 5 m
C) 75 m
12
270)
271)
D) 75 m
Find the exact value of the trigonometric function. Do not use a calculator or tables.
272) sec
272)
6
A) 2
C) 2 3
B) 2
3
178
D)
3 2
Answer Key Testname: CHAPTER 1
1) B 2) A 3) B 4) A 5) D 6) C 7) A 8) C 9) D 10) A 11) D 12) A 13) D 14) sin x -
2
= sin x cos -
2
+ cos x sin -
2
= sin x (0) + cos x (-1) = 0 - cos x = -cos x
15)
16) tan (A - B) = sin (A - B) = sin A cos B - sin B cos A = cos (A - B)
cos A cos B + sin A sin B
(cos A cos B)-1 (sin A cos B - sin B cos A) (cos A cos B)-1 (cos A cos B + sin A sin B)
=
tan A - tan B . 1 + tan A tan B
17) sin (A - B)
= sin (A + (-B)) = sin A cos (-B) + cos A sin (-B) = sin A cos B - cos A sin B 18) If B = -2 , then cos (A + B) = cos A and sin (A + B) = sin A. Because the period of both of the sine and cosine functions is 2 , if B is replaced by a multiple of 2 the angle sum formulas must produce the same value as the sine or cosine function.
179
Answer Key Testname: CHAPTER 1
19) cos x -
2
= cos x cos -
2
- sin x sin -
2
= cos x (0) - sin x (-1) = 0 + sin x = sin x
20) sin x +
2
= sin x cos
+ cos x sin
2
2
= sin x (0) + cos x (1) = 0 + cos x = cos x
21)
22) When y = cos 2x is at a maximum point, which is at any multiple of , y = sec 2x is a minimum point. Similarly, when cos (2x) is at a minimum point, which is at any odd multiple of
2
, y = sec 2x is a at maximum point.
23) When y = sin x is at a maximum point, which is at x = (4n + 1) for all integers n, y = csc x is at a minimum point. 2
2
Similarly, when y = sin
x x is at minimum point, , which is at x = (4n - 1) for all integers n, y = csc is at a 2 2
maximum point.
24) cos x +
2
= cos x cos
2
- sin x sin
2
= cos x (0) - sin x (1) = 0 - sin x = -sin x
25) B 26) D 27) C 28) D 29) C 30) B 31) D 32) C 33) C 180
Answer Key Testname: CHAPTER 1
34) A 35) C 36) C 37) D 38) D 39) A 40) B 41) B 42) A 43) B 44) C 45) B 46) D 47) A 48) C 49) C 50) D 51) C 52) A 53) B 54) D 55) B 56) C 57) D 58) C 59) D 60) D 61) D 62) D 63) A 64) A 65) C 66) B 67) A 68) D 69) B 70) A 71) B 72) B 73) A 74) B 75) A 181
Answer Key Testname: CHAPTER 1
76) C 77) C 78) B 79) C 80) A 81) A 82) B 83) A 84) B 85) D 86) D 87) A 88) B 89) D 90) B 91) D 92) D 93) D 94) A 95) C 96) D 97) D 98) A 99) A 100) C 101) A 102) D 103) D 104) B 105) C 106) A 107) A 108) B 109) D 110) B 111) B 112) B 113) C 114) B 115) C 116) A 117) B 182
Answer Key Testname: CHAPTER 1
118) B 119) B 120) A 121) D 122) B 123) C 124) C 125) A 126) D 127) B 128) A 129) C 130) D 131) C 132) D 133) D 134) A 135) B 136) A 137) C 138) A 139) B 140) C 141) A 142) A 143) C 144) B 145) D 146) B 147) C 148) C 149) B 150) D 151) A 152) D 153) C 154) A 155) D 156) D 157) D 158) D 159) A 183
Answer Key Testname: CHAPTER 1
160) D 161) C 162) A 163) B 164) A 165) C 166) A 167) C 168) A 169) B 170) A 171) B 172) A 173) B 174) D 175) D 176) D 177) B 178) A 179) B 180) D 181) A 182) C 183) A 184) B 185) B 186) A 187) C 188) B 189) B 190) A 191) C 192) A 193) B 194) C 195) C 196) B 197) D 198) D 199) D 200) A 201) A 184
Answer Key Testname: CHAPTER 1
202) B 203) A 204) D 205) B 206) C 207) D 208) D 209) A 210) D 211) C 212) A 213) B 214) A 215) B 216) C 217) B 218) C 219) D 220) B 221) C 222) D 223) D 224) B 225) D 226) D 227) B 228) B 229) A 230) D 231) C 232) C 233) C 234) B 235) B 236) C 237) B 238) C 239) B 240) A 241) A 242) B 243) D 185
Answer Key Testname: CHAPTER 1
244) C 245) C 246) D 247) D 248) D 249) B 250) A 251) B 252) B 253) D 254) A 255) D 256) A 257) A 258) C 259) B 260) B 261) C 262) D 263) D 264) C 265) C 266) A 267) B 268) B 269) A 270) A 271) A 272) C
186
Chapter 2
Exam Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the slopes of UQ, UR, US, and UT to estimate the rate of change of y at the specified value of x. 1) x = 5
A) 5 2
C) 25
B) 0
4
D) 5 4
2) x = 5
A) 2
1)
2)
B) 0
C) 5
1
D) 1
3) x = 2.5
A) 0
3)
B) 7.5
C) 1.25
D) 3.75
4) x = 2
A) 6
4)
B) 3
C) 4
2
D) 0
5) x = 3
A) 6
5)
B) 2
C) 0
D) 4
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Sketch the graph of a function y = f(x) that satisfies the given conditions. 6) f(0) = 0, f(1) = 5, f(-1) = 5, lim f(x) = -5. x
6)
Provide an appropriate response. 7) Explain why the following five statements ask for the same information. (a) Find the roots of f(x) = 2x3 - 3x - 5.
7)
(b) Find the x-coordinate of the points where the curve y = 2x3 crosses the line y = 3x + 5. (c) Find all the values of x for which 2x3 - 3x = 5. (d) Find the x-coordinates of the points where the cubic curve y = 2x3 - 3x crosses the line y = 5. (e) Solve the equation 2x3 - 3x - 5 = 0.
3
8) Give an example of a function f(x) that is continuous for all values of x except x = 8, where
8)
it has a nonremovable discontinuity. Explain how you know that f is discontinuous at x = 8 and why the discontinuity is nonremovable.
Prove the limit statement 9) lim (4x - 5) = 11 x 4
9)
Provide an appropriate response.
10) Use the Intermediate Value Theorem to prove that 5x3 - 6x2 - 6x + 5 = 0 has a solution
10)
between 1 and 2.
11) Use the Intermediate Value Theorem to prove that 2x4 + 4x3 + 7x + 4 = 0 has a solution
11)
between -3 and -2.
12) If f(x) = 2x3 - 5x + 5, show that there is at least one value of c for which f(x) equals . Find a function that satisfies the given conditions and sketch its graph. 13) lim f(x) = 0, lim f(x) = , lim f(x) = . x x -3 x -3 +
Prove the limit statement 2 14) lim 4x - 13x- 12 = 19 x-4 x 4
12)
13)
14)
4
Provide an appropriate response. 15) A function y = f(x) is continuous on [-1, 3]. It is known to be positive at x = -1 and negative at x = 3. What, if anything, does this indicate about the equation f(x) = 0? Illustrate with a sketch.
Find a function that satisfies the given conditions and sketch its graph. 16) lim g(x) = -4, lim g(x) = 4, lim g(x) = 4, lim g(x) = -4. x x x 0+ x 0-
15)
16)
Provide an appropriate response.
17) Use the Intermediate Value Theorem to prove that x(x - 7)2 = 7 has a solution between 6
17)
18) Use the formal definitions of limits to prove lim 9 =
18)
and 8.
x 0 x
5
Find a function that satisfies the given conditions and sketch its graph. 19) lim f(x) = 0, lim f(x) = , lim f(x) = . x x 4x 4+
19)
Sketch the graph of a function y = f(x) that satisfies the given conditions. 20) f(0) = 0, lim f(x) = 0, lim f(x) = - , lim f(x) = - , lim f(x) = , lim f(x) = . x x 3x -3 + x 3+ x -3 -
20)
Prove the limit statement 2 21) lim x - 16 = 8 x 4 x-4
21)
6
Find a function that satisfies the given conditions and sketch its graph. 22) lim f(x) = 0, lim f(x) = . x x 0+
22)
Provide an appropriate response. 23) Give an example of a function f(x) that is continuous at all values of x except at x = 3, where it has a removable discontinuity. Explain how you know that f is discontinuous at x = 3 and how you know the discontinuity is removable.
23)
24) Use the formal definitions of limits to prove lim 6 =
24)
Find a function that satisfies the given conditions and sketch its graph. 25) lim f(x) = 1, lim f(x) = -1, lim f(x) = -1, lim f(x) = 1 x x x 0+ x 0-
25)
x 0+
x
Provide an appropriate response.
26) If functions f x and g x are continuous for 0 x 6, could f x possibly be discontinuous gx
26)
at a point of [0,6]? Provide an example.
27) Use the Intermediate Value Theorem to prove that 5 sin x = x has a solution between and .
7
2
27)
Prove the limit statement 28) lim 1 = 1 x 5 x 5
28)
Sketch the graph of a function y = f(x) that satisfies the given conditions. 29) f(0) = 0, f(1) = 4, f(-1) = -4, lim f(x) = -3, lim f(x) = 3. x x
30) f(0) = 5, f(1) = -5, f(-1) = -5, lim f(x) = 0.
29)
30)
x
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the slope of the curve at the given point P and an equation of the tangent line at P. 31) y = x2 + 11x - 15, P(1, -3)
A) slope is 13; y = 13x - 16
B) slope is - 4 ; y = - 4x + 8 25 25 5
C) slope is 1 ; y = x + 1
D) slope is -39; y = -39x - 80
20
20
5
8
31)
Provide an appropriate response. 32) If x3 f(x) x for x in [-1,1], find lim f(x) if it exists. x 0
A) 1
B) does not exist
Find the limit using lim x=0
32) C) 0
D) -1
sinx = 1. x
33) lim sin 3x cot 4x
33)
cot 5x
x 0
A) 15 4
B) does not exist
C) 0
Graph the rational function. Include the graphs and equations of the asymptotes. 2 34) f(x) = 2x 4 - x2
A) asymptotes: x = -2, x = 2, y = -2
9
D) 12 5
34)
B) asymptotes: x = -2, x = 2, y = 0
C) asymptotes: x = -2, x = 2, y = 0
D) asymptotes: x = -2, x = 2, y = 2
10
Provide an appropriate response. 35) If lim f(x) = 1, lim f(x) = -1, and f(x) is an even function, which of the following statements are x 1x 1+ true? I. II.
35)
lim f(x) = -1 x -1 lim f(x) = -1 x -1 +
III. lim f(x) does not exist. x -1
A) I and II only
B) I and III only
C) II and III only
D) I, II, and III
sinx = 1. x
Find the limit using lim x=0 x
36) lim
36)
x 0 sin 3x
A) does not exist
C) 1
B) 1
3
D) 3
A function f(x), a point x 0 , the limit of f(x) as x approaches x 0 , and a positive number is given. Find a number that for all x, 0 < x - x 0 <
f(x) - L < .
37) f(x) = 10 - x, L = 3, x0 = 1, and = 1 A) 5 B) 6 Use the graph to find a
37) C) -7
> 0 such that for all x, 0 < x - x 0 <
D) 8
f(x) - L < .
38)
38) y = 4x - 1
7.2
f(x) = 4x - 1 x0 = 2
7
L=7 = 0.2
6.8
1.95
2
2.05
NOT TO SCALE
A) 5
B) 0.5
C) 0.1
11
D) 0.05
> 0 such
Determine if the given function can be extended to a continuous function at x = 0. If so, approximate the extended function's value at x = 0 (rounded to four decimal places if necessary). If not, determine whether the function can be continuously extended from the left or from the right and provide the values of the extended functions at x = 0. Otherwise write "no continuous extension." 39) f(x) = cos 2x 39) 2x
A) f(0) = 2 C) f(0) = 2 only from the left
B) No continuous extension D) f(0) = 2 only from the right
Answer the question. 40) Does lim f(x) = f(2)? x 2 x3 , f(x) = -2x, 3, 0,
40)
-2 < x 0 0 x<2 2<x 4 x=2
A) Yes Use the graph to find a
B) No > 0 such that for all x, 0 < x - x 0 <
f(x) - L < .
41)
41) f(x) = x - 3 x0 = 4
y=
x-3
L=1 1 = 4
1.25 1 0.75 3.5625 4
4.5625
NOT TO SCALE
A) 1
B) 0.4375
C) 0.5625
12
D) 3
Find the limit, if it exists. 2 42) lim 3h+4 + 2 h 0
A) Does not exist
42) B) 2
C) 1/2
D) 1
For the function f whose graph is given, determine the limit. 43) Find lim f(x). x 0
A) -1
B) 1
43)
C) 0
D) does not exist
Determine if the given function can be extended to a continuous function at x = 0. If so, approximate the extended function's value at x = 0 (rounded to four decimal places if necessary). If not, determine whether the function can be continuously extended from the left or from the right and provide the values of the extended functions at x = 0. Otherwise write "no continuous extension." 44) f(x) = (1 + 2x)1/x 44)
A) No continuous extension C) f(0) = 5.4366
B) f(0) = 2.7183 D) f(0) = 7.3891
Solve the problem. 45) Select the correct statement for the definition of the limit: lim f(x) = L x x0 means that __________________ A) if given a number > 0, there exists a number 0 < x - x0 < implies f(x) - L > .
> 0, such that for all x,
B) if given any number > 0, there exists a number > 0, such that for all x, 0 < x - x0 < implies f(x) - L < .
C) if given any number > 0, there exists a number > 0, such that for all x, 0 < x - x0 < implies f(x) - L > .
D) if given any number > 0, there exists a number > 0, such that for all x, 0 < x - x0 <
implies f(x) - L < .
13
45)
Find the limit.
1
46) lim
46)
2 7 - (x - 7)
x
B) 0
A)
C) -
D) -1
Use a graphing calculator to graph f near x 0 and use Zoom and Trace to estimate the y-value on the graph as x x 0 . 2
47) lim x + 3x + 2
47)
x 2 x2 + 1x - 2
A) 0
B) 1
C) 3
D) 1 3
Find the average rate of change of the function over the given interval.
48) h(t) = sin (2t), 0, A)
B) - 4
4
Use the graph to find a
48)
4
C) 4
> 0 such that for all x, 0 < x - x 0 <
D) 2
f(x) - L < .
49)
49) y = 2x + 3
7.2 7
f(x) = 2x + 3 x0 = 2 L=7 = 0.2
6.8
1.9 2 2.1 NOT TO SCALE
A) 0.1
B) 5
C) 0.4
D) 0.2
Find the limit.
50)
lim tan x x ( /2)+
A) -
50) C) 0
B)
14
D) 1
Find the limit using lim x=0
sinx = 1. x
51) lim sin(sin x) x 0
51)
sin x
A) 0 Find the limit.
52)
x lim x +8 x -1 +
C) 1
B) -1
D) does not exist
1x + 3 x2 + 8x
A) Does not exist
52) B) 2
C) 2
65
49
D) 1 2
Find the limit if it exists. 53) lim 5x + 57 x 5
53)
A) 82
C) - 82
B) -82
D) 82
Find the average rate of change of the function over the given interval.
54) g(t) = 2 + tan t, -
, 4 4
A) - 2
54) B) - 4
C) 4
D) 0
Determine the limit by sketching an appropriate graph. for x -1 55) lim f(x), where f(x) = x2 + 5 0 for x = -1 + x -1
A) -4
55)
B) 0
C) 1
D) 6
Find the limit.
56)
56) 1 1 lim 1/5 (x - 1)3/5 x 1+ x
A) Does not exist
B) 0
C)
15
D)
Use the table of values of f to estimate the limit. 57) Let f(x) = x2 - 5, find lim f(x). x 0 x f(x)
57) 0.001
-0.001
0.01
0.1
-0.1
-0.01
x -0.1 f(x) -1.4970
-0.01 -1.4999
-0.001 -1.5000
0.001 0.01 0.1 ; limit = -15.0 -1.5000 -1.4999 -1.4970
x -0.1 f(x) -1.4970
-0.01 -1.4999
-0.001 -1.5000
0.001 0.01 0.1 ; limit = -1.5000 -1.4999 -1.4970
x -0.1 f(x) -2.9910
-0.01 -2.9999
-0.001 -3.0000
0.001 0.01 0.1 ; limit = -3.0 -3.0000 -2.9999 -2.9910
x -0.1 f(x) -4.9900
-0.01 -4.9999
-0.001 -5.0000
0.001 0.01 0.1 ; limit = -5.0 -5.0000 -4.9999 -4.9900
A) B) C) D)
Provide an appropriate response.
58) It can be shown that the inequalities -x x cos 1
x
x hold for all values of x 0.
58)
1 Find lim x cos if it exists. x x 0
A) does not exist
B) 0.0007
C) 0
D) 1
Find numbers a and b, or k, so that f is continuous at every point.
59)
59) f(x) =
x2 , if x 4
kx, if x > 4 A) k = 1 4
B) k = 4
C) k = 16
D) Impossible
Find all points where the function is discontinuous.
60)
60)
A) x = 1, x = 5 C) None
B) x = 4 D) x = 1, x = 4, x = 5
16
Find the limit, if it exists. 3 61) lim x - 6x + 8 x-2 x 0
A) 4
61) B) Does not exist
C) 0
D) -4
Find the limit.
62) lim
x 3-
3x x - 3 x-3
A) 3
62) B) 0
Use the graph of the greatest integer function y = 63) lim (x ) x 8-
A) 0
C) -3
D) Does not exist
to find the limit.
63)
B) 16
C) 8
D) -16
Find the limit, if it exists. 64) lim 15x + h h 0 x3 (x - h)
A) Does not exist
64) B) 15
C) 15x
x4
D) 15
x3
Find the limit if it exists. 65) lim (10x2 - 7x - 8) x 5
A) 223
65) B) 277
C) 293
Graph the rational function. Include the graphs and equations of the asymptotes. 66) f(x) = x x+2
17
D) 207
66)
A) asymptotes: x = -2, y = 1
B) asymptotes: x = 2, y = 0
C) asymptotes: x = 2, y = 1
18
D) asymptotes: x = -2, y = 0
Answer the question. 67) Does lim f(x) exist? x 0 x3 , f(x) = -3x, 3, 0,
67)
-2 < x 0 0 x<2 2<x 4 x=2
A) No
B) Yes
Give an appropriate answer.
-5f(x) - 9g(x) . 10 + g(x) x -9
68) Let lim f(x) = -8 and lim g(x) = -3. Find lim x -9
A) -9
x -9
B) 67
C) - 5
7
Find the average rate of change of the function over the given interval. 69) y = 2x, [2, 8] A) 7 B) 1 C) 2 3
19
68) D) 13 7
69) D) - 3 10
Find the limit.
70) lim -5 + (2/x)
70)
4 - (1/x2 )
x
B) - 5
A)
C)
4
D) 5 4
Find the limit and determine if the function is continuous at the point being approached. 71) lim tan(sin(- cos(sin ))) -
A) does not exist; no C) 1; yes
71)
B) 0; yes D) 0; no
Find the limit.
2 72) lim x - 7x + 10 x
72)
x3 - 4x
0
A) 10
C) Does not exist
B)
D)
Give an appropriate answer. 73) Let lim f(x) = -5 and lim g(x) = -3. Find lim [f(x) · g(x)]. x 1 x 1 x 1
A) -3
C) 1
B) -8
73) D) 15
Find the limit.
74) lim
x 0
15 + cos2 x
A) 4
74) B) 16
C) 15
D) 15
Determine if the given function can be extended to a continuous function at x = 0. If so, approximate the extended function's value at x = 0 (rounded to four decimal places if necessary). If not, determine whether the function can be continuously extended from the left or from the right and provide the values of the extended functions at x = 0. Otherwise write "no continuous extension." 75) f(x) = tan x 75) x
A) f(0) = 1 only from the left C) No continuous extension
B) f(0) = 1 only from the right D) f(0) = 1
20
Answer the question. 76) Does lim f(x) exist? x 1 -x2 + 1, 5x, f(x) = -3, -5x + 10 1,
76)
-1 x < 0 0<x<1 x=1 1<x<3 3<x<5
A) Yes
B) No
Find the limit.
x 3x +2 x -1
77) lim
77)
A) - 1
B) 1
5
C) does not exist
Answer the question. 78) Is f continuous on (-2, 4]? x3 , f(x) = -2x, 4, 0,
D) 0
78)
-2 < x 0 0 x<2 2<x 4 x=2
A) Yes
B) No
21
Determine if the given function can be extended to a continuous function at x = 0. If so, approximate the extended function's value at x = 0 (rounded to four decimal places if necessary). If not, determine whether the function can be continuously extended from the left or from the right and provide the values of the extended functions at x = 0. Otherwise write "no continuous extension." 2x 79) f(x) = 10 - 1 79) x
A) f(0) = 0 only from the right C) f(0) = 0 only from the left
B) No continuous extension D) f(0) = 0
Use the table to estimate the rate of change of y at the specified value of x. 80) x = 1. x y 0 0 0.2 0.12 0.4 0.48 0.6 1.08 0.8 1.92 1.0 3 1.2 4.32 1.4 5.88 A) 4
B) 6
80)
C) 2
D) 8
A function f(x), a point x 0 , the limit of f(x) as x approaches x 0 , and a positive number is given. Find a number that for all x, 0 < x - x 0 <
f(x) - L < .
81) f(x) = mx + b, m > 0, L = (m/7) + b, x0 = 1/7, and = c > 0 A) = c
m
B) = c
81)
C) = 7
7
D) = 1 + c
m
7
82) f(x) = mx, m > 0, L = 4m, x0 = 4, and = 0.05 A) = 4 - m
m
82) C) = 4 + 0.05
B) = 0.05
m
Graph the rational function. Include the graphs and equations of the asymptotes.
22
D) = 0.05 m
> 0 such
83) f(x) = 2 - x
2
83)
2x + 4
A) asymptotes: x = -2, y = -x + 2
B) asymptotes: x = -2, y = - 1 x + 1 4
2
23
C) asymptotes: x = -4, y = - 1 x + 1 8
2
D) asymptotes: x = -2, y = - 1 x + 1 2
Use the table to estimate the rate of change of y at the specified value of x. 84) x = 2.
84)
x y 0 10 0.5 38 1.0 58 1.5 70 2.0 74 2.5 70 3.0 58 3.5 38 4.0 10
A) 8
B) 0
C) 4
24
D) -8
Find the limit using lim x=0
sinx = 1. x
85) lim sin x cos 4x
85)
x 0 x + x cos 5x
B) 4
A) 0
C) 1
5
D) does not exist
2
86) lim 6x2 (cot 3x)(csc 2x)
86)
x 0
A) does not exist
B) 1
D) 1
C) 1
3
2
For the function f whose graph is given, determine the limit. 87) Find lim f(x). x -1
C) - 4
B) does not exist
A) -1
87)
5
D) 4 5
Find the limit.
2 88) lim -9x + 5x + 9 x -
88)
-13x2 - 9x + 6
A) 1
C) 9
B)
13
D) 3 2
2 89) lim x - 1 x
3 0-
89)
x
A) Does not exist
C) 0
B)
25
D)
Find the intervals on which the function is continuous. 2 90) y = 3 - x x +6 8
90)
A) continuous everywhere B) discontinuous only when x = -6 C) discontinuous only when x = -14 D) discontinuous only when x = -8 or x = -6 Use a graphing calculator to graph f near x 0 and use Zoom and Trace to estimate the y-value on the graph as x x 0 .
91) lim
x -1
x2 - 1 x2 + 3 - 2
A) 4
91) C) 1
B) 1
4
Find the intervals on which the function is continuous. 4 92) y = 2x - 7 A) continuous on the interval 7 , 2
92) B) continuous on the interval 7 , 2
C) continuous on the interval - 7 ,
D) continuous on the interval - , 7
2
Find the limit.
93)
lim x -
A) 2
D) 2
2
2x3 + 4x2 x - 6x2
93) B) - 2
C) -
3
D)
Find the intervals on which the function is continuous. 94) y = 4 - 4x x+4
94)
A) discontinuous only when x = -8 C) discontinuous only when x = 4
B) continuous everywhere D) discontinuous only when x = -4
26
Use the table to estimate the rate of change of y at the specified value of x. 95) x = 1. x y 0 0 0.2 0.02 0.4 0.08 0.6 0.18 0.8 0.32 1.0 0.5 1.2 0.72 1.4 0.98 A) 1.5
B) 2
C) 1
95)
D) 0.5
Find the limit, if it exists. 2 96) lim 3x + 7x - 2 x 1 3x2 - 4x - 2
96) B) - 8
A) 0
Evaluate lim h 0
C) Does not exist
3
f(x 0 + h) - f(x 0 ) h
D) - 7 4
for the given x 0 and function f.
97) f(x) = 4 x for x0 = 9 A) 18
97) B) 2
C) 6
3
D) Does not exist
Find the limit, if it exists. 4 98) lim x - 1 x 1 x-1
A) 4
98) B) Does not exist
C) 0
D) 2
Determine the limit by sketching an appropriate graph. for x < 5 99) lim f(x), where f(x) = -5x - 7 4x 6 for x 5 x 5+
A) -32
99)
B) 14
C) -5
27
D) -6
Find the limit and determine if the function is continuous at the point being approached.
100)
lim cos cos (tan x) 2 x - /2
100)
A) does not exist; no C) 1; yes
B) 1; no D) does not exist; yes
Find the intervals on which the function is continuous. 101) y = 5 x2 - 9
101)
A) discontinuous only when x = 9 C) discontinuous only when x = -3 or x = 3
B) discontinuous only when x = -9 or x = 9 D) discontinuous only when x = -3
102) y = 4 cos
102)
+9
A) discontinuous only when = -9
B) discontinuous only when =
C) continuous everywhere
D) discontinuous only when = 9
Find the limit using lim x=0
2
sinx = 1. x
2
103) lim x - 2x + sin x x 0
103)
x
A) 1
B) 0
C) does not exist
D) -1
Find the intervals on which the function is continuous. 104) y = x2 - 3
104)
A) continuous on the interval [- 3, 3] B) continuous on the intervals (- , - 3] and [ 3, ) C) continuous everywhere D) continuous on the interval [ 3, ) Given the interval (a, b) on the x-axis with the point x 0 inside, find the greatest value for x - x0 <
> 0 such that for all x, 0 <
a < x < b.
105) a = -4, b = 6, x0 = 4 A) 8
105) B) 2
C) 4
D) 1
Find the limit, if it exists. 2 106) lim x + 17x + 72 x+8 x -8
A) 17
106) B) 272
C) 1
28
D) Does not exist
Find the limit.
1 1 2/5 (x - 3)4/5 3- x
107) lim x
107) B) 0
A)
D) Does not exist
C)
Use the table of values of f to estimate the limit. 108) Let f(x) = sin(4x) , find lim f(x). x x 0 x f(x)
-0.1
-0.01 3.99893342
-0.001
108) 0.001
0.01 3.99893342
A) limit = 3.5 C) limit = 0 Use the graph to find a
0.1
B) limit does not exist D) limit = 4
> 0 such that for all x, 0 < x - x 0 <
f(x) - L < .
109)
109) y=
3 x+3 2
6.2
f(x) =
6
3 x+3 2
x0 = 2
5.8
L=6 = 0.2
1.9 2 2.1 NOT TO SCALE
A) 0.2 Find the limit.
C) 4
B) -0.2
D) 0.1
1 +4 2/5 x + 0
110) lim x
A) Does not exist 111) lim
x 4+
A) 0
110) B) 4
C)
D)
2x x - 4 x-4
111) B) 8
C) - 8
29
D) Does not exist
112)
1 lim x +2 x -2 -
112) C) 0
B) -1
A)
D)
Use the graph to estimate the specified limit. 113) Find lim f(x) x 3-
A) 7 3
113)
B) - 7 3
C) -3
3
D) 7 3 3
Use the graph to evaluate the limit. 114) lim f(x) x 0
A) 0
114)
C) does not exist
B) -3
D) 3
Find the limit, if it exists. 3 3 115) lim (x + h) - x h h 0
A) 3x2 + 3xh + h2
115) C) 3x2
B) Does not exist
30
D) 0
Determine the limit by sketching an appropriate graph. for x < 5 116) lim f(x), where f(x) = -5x - 7 3x 6 for x 5 x 5-
117)
A) 9
B) -6
lim f(x), where f(x) = x -6 +
2x 2 0
A) 8
B) Does not exist
116)
C) -32
D) -5
-6 x < 0, or 0 < x 2 x=0 x < -6 or x > 2
C) -0
117)
D) -12
Use the graph to evaluate the limit. 118) lim f(x) x 0
A) does not exist
118)
B) 2
C) -1 31
D) -2
Graph the rational function. Include the graphs and equations of the asymptotes. x 119) f(x) = 2 x +x+3
A) asymptote: y = 0
B) asymptotes: x = 3, x = -3
32
119)
C) asymptote: y = 0
D) asymptote: y = 1
2
120) f(x) = x + 2
120)
x3
33
A) asymptotes: x = 0, y = 0
B) asymptotes: x = 0, y = 0
C) asymptotes: x = 0, y = 0
34
D) asymptote: y = 0
Find numbers a and b, or k, so that f is continuous at every point.
121)
121) 5, x < -5 ax + b, -5 x -4 -2, x > -4 A) a = -7, b = 26
f(x) =
Find the limit.
122) lim x
2
B) a = -7, b = -30
C) a = 5, b = -2
D) Impossible
1 x + -2
A)
122) B) 0
C) -1
D)
Use the table to estimate the rate of change of y at the specified value of x. 123) x = 1. x y 0.900 -0.05263 0.990 -0.00503 0.999 -0.0005 1.000 0.0000 1.001 0.0005 1.010 0.00498 1.100 0.04762 A) 1
Give an appropriate answer. 124) Let lim f(x) = 121. Find lim x 3 x 3
A) 11
C) 0.5
B) -0.5
123)
D) 0
f(x).
124)
B) 3
C) 121
35
D) 3.3166
Find the limit.
125) lim 1
125)
x 0 x2/3
A) -
C) 0
B)
D) 2/3
For the function f whose graph is given, determine the limit. 126) Find lim f(x). x 0
A) 0 127) Find lim f(x) and x 2-
B) does not exist
126)
C) 1
lim f(x). x 2+
D) -1 127)
A) does not exist; does not exist C) 5; -2
B) 1; 1 D) -2; 5
36
Find all points where the function is discontinuous.
128)
128)
A) None
B) x = 3
C) x = 0
D) x = 0, x = 3
Find the slope of the curve at the given point P and an equation of the tangent line at P. 129) y = x3 - 8x, P(1, -7)
A) slope is -5; y = -5x - 2 C) slope is -5; y = -5x Find the limit.
3
129)
B) slope is 3; y = 3x - 6 D) slope is 3; y = 3x - 10
2
130) lim 8x - 5x + 3x x
130)
-x3 - 2x + 6
A) 8
C) 3
B) -8
2
D)
Give an appropriate answer. 131) Let lim f(x) = 27. Find lim log3 f(x). x -2 x -2
A) -2
131)
B) 3
C) 1
D) 27
Use the graph to evaluate the limit. 132) lim f(x) x 0
A) does not exist
132)
C) 2
B) -2
37
D) 0
Provide an appropriate response. 1 1 x+2 2 133) Find lim . x x 0
B) - 1
A) 0 Find the limit. 134) lim x -
A) 0
133) C) 1
4
4
D) Does not exist
x + 6 cos(x + )
134) C) - 6 -
B) 1
Graph the rational function. Include the graphs and equations of the asymptotes. 2 135) f(x) = 2 - 2x - x x
A) asymptotes: x = 0, y = -x
38
D) 6 -
135)
B) asymptotes: x = 0, y = -x + 2
C) asymptotes: x = 0, y = -x - 4
D) asymptotes: x = 0, y = -x - 2
Find numbers a and b, or k, so that f is continuous at every point.
136)
136) x2 ,
x<1 ax + b, 1 x 4 x + 12, x > 4 A) a = -5, b = -4
f(x) =
B) a = 5, b = -4
C) a = 5, b = 4
39
D) Impossible
Find the average rate of change of the function over the given interval. 137) y = 3 , [4, 7] x-2
A) 2
C) 1
B) 7
3
137) D) - 3
10
A function f(x), a point x 0 , the limit of f(x) as x approaches x 0 , and a positive number is given. Find a number that for all x, 0 < x - x 0 <
f(x) - L < .
138) f(x) = 1/x, L = 1/8, x0 = 8, and = 0.5 A) -21.3333 B) 6.4
138) C) -10.6667
D) 0.8
Find the limit, if it exists. 3 2 139) lim x + 12x - 5x 5x x 0
A) 0
> 0 such
139) B) 5
C) Does not exist
D) -1
Solve the problem. 140) When exposed to ethylene gas, green bananas will ripen at an accelerated rate. The number of days for ripening becomes shorter for longer exposure times. Assume that the table below gives average ripening times of bananas for several different ethylene exposure times: Exposure time Ripening Time (minutes) (days) 10 4.2 15 3.5 20 2.6 25 2.1 30 1.1 Plot the data and then find a line approximating the data. With the aid of this line, find the limit of the average ripening time as the exposure time to ethylene approaches 0. Round your answer to the nearest tenth.
40
140)
A)
5.8 days
B)
37.5 minutes
C)
0.1 day
41
D)
2.6 days
Use the graph to find a
> 0 such that for all x, 0 < x - x 0 <
f(x) - L < .
141)
141) y = -x + 3
4.2
f(x) = -x + 3 x0 = -1
4
L=4 = 0.2
3.8
-1.2 -1 -0.8 NOT TO SCALE
A) 0.4
C) 5
B) -0.2
D) 0.2
Provide an appropriate response. 142) Use a calculator to graph the function f to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function's value at x = 0. If the function does not appear to have a continuous extension, can it be extended to be continuous at the origin from the right or from the left? If so, what do you think the extended function's value(s) should be? 7x - 1 f(x) = x
A) continuous extension exists at origin; f(0) = 0 B) continuous extension exists at origin; f(0) 1.9556 C) continuous extension exists from the left; f(0) 1.9556 D) continuous extension exists from the right; f(0) 1.9556 42
142)
Solve the problem. 143) To what new value should f(1) be changed to remove the discontinuity? x2 + 4, x<1 f(x) = 3, x=1 x + 4,
A) 3
143)
x>1
B) 6
C) 5
D) 4
Provide an appropriate response. 144) If lim f(x) = 3, find lim f(x). x 2 x x 2
A) 2
144)
B) 3
C) 6
D) Does not exist
Use the graph to estimate the specified limit. 145) Find lim f(x) x 0
A) does not exist
145)
B) 2
C) -2
D) 0
Provide an appropriate response. 146) Use a calculator to graph the function f to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function's value at x = 0. If the function does not appear to have a continuous extension, can it be extended to be continuous at the origin from the right or from the left? If so, what do you think the extended function's value(s) should be? 7 sin x f(x) = x
A) continuous extension exists at origin; f(0) = 0 B) continuous extension exists from the right; f(0) = 1 continuous extension exists from the left; f(0) = -1
C) continuous extension exists at origin; f(0) = 7 D) continuous extension exists from the right; f(0) = 7 continuous extension exists from the left; f(0) = -7
43
146)
Find the limit.
147)
lim x + 3 x -6 -
x+6 x+6
A) Does not exist
147) B) 3
C) 9
D) -3
Use the graph to estimate the specified limit. 148) Find lim f(x) x 0
A) 3
148)
B) does not exist
C) -3
D) 0
Provide an appropriate response. 149) Given lim f(x) = Ll, lim f(x) = Lr, and Ll Lr, which of the following statements is true? x 0x 0+ I.
149)
lim f(x) = Ll x 0
II. lim f(x) = Lr x 0 III. lim f(x) does not exist. x 0
A) I
B) none
C) II
D) III
150) If lim f(x) = 4, find lim f(x) . x 2 x2
A) 8
150)
x 2 x
B) 4
C) 16
D) 2
Divide numerator and denominator by the highest power of x in the denominator to find the limit. 2 151) lim 81t - 729 t-9 t
A) 9
B) does not exist
C) 81
44
D) 729
151)
Use the graph to evaluate the limit. 152) lim f(x) x 0
A) does not exist
152)
C) 1
B) -1
D)
Find the limit, if it exists. 2 153) lim x - 2x - 15 x+3 x 5
A) 5
153) C) 0
B) -8
D) Does not exist
Use the graph to estimate the specified limit. 154) Find lim f(x) and lim f(x) x /2)x /2)+
A) ;
154)
B) 1; 6
C) 6; 1
45
D)
; 2 2
Find the limit L for the given function f, the point x 0 , and the positive number . Then find a number all x, 0 < |x - x 0 |<
> 0 such that, for
|f(x) - L| < .
155) f(x) = 16 , x0 = 8, = 0.1
155)
x
A) L = 2;
= 0.42
B) L = 2;
= 0.84
C) L = 2;
= 4.21
D) L = 2;
Use the table of values of f to estimate the limit. x+5 156) Let f(x) = , find lim f(x). 2 x + 9x + 20 x -5 x f(x)
-5.1
-5.01
-5.001
= 0.38
156) -4.999
-4.99
-4.9
A) x -5.1 -5.01 -5.001 -4.999 -4.99 -4.9 ; limit = 1 f(x) 0.9091 0.9901 0.9990 1.0010 1.0101 1.1111
B) x -5.1 -5.01 -5.001 -4.999 -4.99 -4.9 ; limit = -1.1 f(x) -1.0091 -1.0901 -1.0990 -1.1010 -1.1101 -1.2111
C) x -5.1 -5.01 -5.001 -4.999 -4.99 -4.9 ; limit = -1 f(x) -0.9091 -0.9901 -0.9990 -1.0010 -1.0101 -1.1111
D) x -5.1 -5.01 -5.001 -4.999 -4.99 -4.9 ; limit = -0.9 f(x) -0.8091 -0.8901 -0.8990 -0.9010 -0.9101 -1.0111
Use the graph to estimate the specified limit. 157) Find lim f(x) and lim f(x) x 2x 2+
157)
A) does not exist; does not exist C) 1; 1
B) -1; 5 D) 5; -1
46
Find the limit using lim x=0
sinx = 1. x
158) lim sin 5x
158)
x
x 0
B) 1
A) 5
C) does not exist
5
D) 1
Find the limit.
159) lim (3x5 - 2x4 - 4x3 + x2 + 5)
159)
x -2
A) -151
B) 57
C) -23
D) -87
160) lim cos 2x
160)
x
x
A) 1
C) 2
B)
D) 0
Provide an appropriate response. 161) If lim f(x) = L, which of the following expressions are true? x 0 I.
lim f(x) does not exist. x 0-
II.
lim f(x) does not exist. x 0+
161)
III. lim f(x) = L x 0IV. lim f(x) = L x 0+
A) I and II only
B) I and IV only
C) II and III only
D) III and IV only
Find all points where the function is discontinuous.
162)
162)
A) x = -2, x = 1
B) None
C) x = -2
D) x = 1
Find the limit.
163) lim
x 1
x2 + 12x + 36
A) 7
163) B) 49
C) ±7
47
D) does not exist
Provide an appropriate response. 164) If lim f(x) - 3 = 2, find lim f(x). x 1 x-1 x 1
A) 2
164)
B) 1
C) 3
D) Does not exist
Divide numerator and denominator by the highest power of x in the denominator to find the limit. -1 -3 165) lim 4x + 2x x 2x-2 + x-5
A)
C) 2
B)
D) 0
Find the limit L for the given function f, the point x 0 , and the positive number . Then find a number all x, 0 < |x - x 0 |<
> 0 such that, for
|f(x) - L| < .
166) f(x) = -4x + 9, x0 = 2, = 0.04 A) L = 1; = 0.01 C) L = 17; = 0.02
166) B) L = 1; = 0.02 D) L = -17; = 0.01
Divide numerator and denominator by the highest power of x in the denominator to find the limit. 167) lim 7x + 5 x 5x2 + 1
A) 7
C) 0
B)
5
167)
D) 7
5
Find the limit L for the given function f, the point x 0 , and the positive number . Then find a number all x, 0 < |x - x 0 |<
165)
> 0 such that, for
|f(x) - L| < .
168) f(x) = 46 - 2x, x0 = 5, = 0.4 A) L = 6; = 2.32 C) L = 7; = 2.32
168) B) L = 6; = 2.48 D) L = j-5; = 1.12
Find all points where the function is discontinuous.
169)
169)
A) x = -2, x = 0 C) x = 2
B) x = -2, x = 0, x = 2 D) x = 0, x = 2
48
For the function f whose graph is given, determine the limit. 170) Find lim f(x). x 0
A) -3
B) 0
170)
C) 3
D) does not exist
Find the intervals on which the function is continuous. x+2 171) y = 2 x - 7x + 12
171)
A) discontinuous only when x = 3 C) discontinuous only when x = -3 or x = 4
B) discontinuous only when x = 3 or x = 4 D) discontinuous only when x = -4 or x = 3
Find the limit if it exists. 172) lim (x + 2)2 (x - 2)3 x -1
A) 1
172) C) 9
B) -27
D) -243
Provide an appropriate response. 173) Which of the following statements defines lim f(x) = ? x x0
173)
I. For every positive real number B there exists a corresponding x0 - < x < x 0 + .
> 0 such that f(x) > B whenever
II. For every positive real number B there exists a corresponding x0 < x < x 0 + .
> 0 such that f(x) > B whenever
III. For every positive real number B there exists a corresponding whenever x0 - < x < x0 .
> 0 such that f(x) > B
A) III
B) II
C) I
49
D) None
Use the table of values of f to estimate the limit. 174) Let f(x) = x2 + 8x - 2, find lim f(x). x 2 x f(x)
1.9
1.99
1.999
174) 2.001
2.01
2.1
A) x 1.9 1.99 1.999 2.001 2.01 2.1 ; limit = 17.70 f(x) 16.692 17.592 17.689 17.710 17.808 18.789
B) x 1.9 1.99 1.999 2.001 2.01 2.1 ; limit = 18.0 f(x) 16.810 17.880 17.988 18.012 18.120 19.210
C) x 1.9 1.99 1.999 2.001 2.01 2.1 ; limit = 5.40 f(x) 5.043 5.364 5.396 5.404 5.436 5.763
D) x 1.9 1.99 1.999 2.001 2.01 2.1 ; limit = f(x) 5.043 5.364 5.396 5.404 5.436 5.763
Provide an appropriate response. 175) Which of the following statements defines lim f(x) = ? x -
175)
I. For every positive real number B there exists a corresponding positive real number N such that f(x) > B whenever x > N. II. For every positive real number B there exists a corresponding negative real number N such that f(x) > B whenever x < N. III. For every negative real number B there exists a corresponding negative real number N such that f(x) < B whenever x < N. IV. For every negative real number B there exists a corresponding positive real number N such that f(x) < B whenever x > N A) III B) I C) IV D) II
Evaluate lim h 0
f(x 0 + h) - f(x 0 ) h
for the given x 0 and function f.
176) f(x) = 3x2 for x0 = 4 A) 24
176) B) 48
C) Does not exist
Find the average rate of change of the function over the given interval. 177) y = x2 + 5x, [5, 9]
A) 19
C) 76
B) 14
9
Graph the rational function. Include the graphs and equations of the asymptotes.
50
D) 12
177) D) 63 2
178) f(x) = 1
178)
x+1
A) asymptotes: x = 1, y = 0
B) asymptotes: x = -1, y = 0
51
C) asymptotes: x = 1, y = 0
D) asymptotes: x = -1, y = 0
Divide numerator and denominator by the highest power of x in the denominator to find the limit. 64x2 + x - 3 179) lim (x - 17)(x + 1) x
A) 8
C) 64
B)
52
D) 0
179)
Answer the question. 180) Is f continuous at f(1)? -x2 + 1, 3x, f(x) = -5, -3x + 6 1,
180)
-1 x < 0 0<x<1 x=1 1<x<3 3<x<5
A) No
B) Yes
Find the limit and determine if the function is continuous at the point being approached. 181) lim sin(x sin2 x + x cos2x + 1) x 2
A) does not exist; no C) sin 3; yes
181)
B) sin -1; yes D) sin 3; no
Answer the question. 182) Does lim f(x) = f(-1)? x -1 + -x2 + 1, 4x, f(x) = -4, -4x + 8 4,
182)
-1 x < 0 0<x<1 x=1 1<x<3 3<x<5
A) Yes
B) No
Divide numerator and denominator by the highest power of x in the denominator to find the limit. 3 x + 5x + 7 183) lim x 3x + x2/3 - 7
A) 3 5
C) 0
B)
53
D) 5 3
183)
Give an appropriate answer. 184) Let lim f(x) = 8 and lim g(x) = -9. Find lim [f(x) - g(x)]. x -8 x -8 x -8
B) 8
A) -1
C) -8
184) D) 17
Find the limit if it exists. 185) lim 5x(x + 3)(x - 6) x -10
A) -1400
185) B) -5600
C) -10,400
D) 5600
Answer the question. 186) Does lim f(x) exist? x (-1)+ -x2 + 1, 3x, f(x) = -5, -3x + 6 2,
186)
-1 x < 0 0<x<1 x=1 1<x<3 3<x<5
A) Yes
B) No
Provide an appropriate response. 187) Which of the following statements defines
lim f(x) = ? x (x0)-
187)
I. For every positive real number B there exists a corresponding x0 - < x < x 0 + .
> 0 such that f(x) > B whenever
II. For every positive real number B there exists a corresponding x0 < x < x 0 + .
> 0 such that f(x) > B whenever
III. For every positive real number B there exists a corresponding whenever x0 - < x < x0 .
> 0 such that f(x) > B
A) II
B) I
C) III
54
D) None
Use the table to estimate the rate of change of y at the specified value of x. 188) x = 1. x y 0 0 0.2 0.01 0.4 0.04 0.6 0.09 0.8 0.16 1.0 0.25 1.2 0.36 1.4 0.49 A) 1.5
B) 2
C) 0.5
Give an appropriate answer.
188)
D) 1
f(x) . g(x) -10
189) Let lim f(x) = -5 and lim g(x) = -4. Find lim x
x
-10
A) 5
x
-10
B) 4
4
C) -1
5
189) D) -10
Find the average rate of change of the function over the given interval. 190) y = 4x2 , 0, 7 4
B) 1
A) 2
C) - 3
3
10
190) D) 7
A function f(x), a point x 0 , the limit of f(x) as x approaches x 0 , and a positive number is given. Find a number that for all x, 0 < x - x 0 <
> 0 such
f(x) - L < .
191) f(x) = -8x - 4, L = -28, x0 = 3, and = 0.01 A) 0.00125 B) 0.000625
191) C) -0.003333
Find the intervals on which the function is continuous. 192) y = 9x + 6 A) continuous on the interval - 2 , 3
D) 0.0025
192) B) continuous on the interval - 2 , 3
C) continuous on the interval 2 ,
D) continuous on the interval - , - 2
3
3
Find the limit, if it exists. 2 193) lim x - 11x + 28 x 4 x2 - 9x + 20
A) Does not exist
193) B) - 11
C) - 3
55
D) 3
194) lim
x+6
194)
x 6 (x - 6)2
A) 6
B) Does not exist
C) -6
D) 0
1 x x 9 -9
195) lim
195)
A) Does not exist
B) 18
C) 0
D) 9
For the function f whose graph is given, determine the limit. 196) Find lim f(x). x 1+
A) 3
196)
C) 3 1
B) does not exist
2
Solve the problem. 197) To what new value should f(2) be changed to remove the discontinuity? 2x + 4, x < 2 f(x) = 10 x=2 x + 6, x > 2 A) 1 B) 0 C) 8 Find the limit.
198) x
lim 3
6
D) 4
197)
D) 7
x2 1 6 x
198)
A)
C) 0
B)
3
D) 2 6
Give an appropriate answer.
199) Let lim f(x) = 9 and lim g(x) = -2. Find lim [f(x) + g(x)]2 . x
A) 7
10
x
10
x
199)
10
B) 49
C) 11
56
D) 85
Find all points where the function is discontinuous.
200)
200)
A) x = -2, x = 2 C) None
B) x = -2, x = 0, x = 2 D) x = 0
Find the limit L for the given function f, the point x 0 , and the positive number . Then find a number all x, 0 < |x - x 0 |<
> 0 such that, for
|f(x) - L| < . 2
201) f(x) = x + -4x + 3 , x0 = 3, = 0.05
201)
x + -3
A) L = -2; = 0.06 C) L = 0; = 0.05
B) L = -4; = 0.06 D) L = 2; = 0.05
Find the limit.
202) lim (3x - 9x2 - 5x + 8)
202)
x
A) 0
C) -6
B)
D) 5 6
1 2 2- x - 4
203) lim x
A) 0
203) B) 1
C)
57
D) -
Use the graph to evaluate the limit. 204) lim f(x) x 0
204)
A) does not exist
C) 1
B)
D) -1
Find numbers a and b, or k, so that f is continuous at every point.
205)
205) f(x) =
x2 ,
if x 5
x + k, if x > 5 A) k = 20
Use the graph to find a
B) k = 30
C) k = -5
> 0 such that for all x, 0 < x - x 0 <
D) Impossible
f(x) - L < .
206)
206) y = 2x2 9
f(x) = 2x2 x0 = 2 L=8 =1
8 7
1.87
2
2.12
NOT TO SCALE
A) 0.25
B) 0.13
C) 6
58
D) 0.12
Find the limit.
207) lim 3x + 1 x
207)
8x - 7
A) - 1 7
C) 0
B)
D) 3 8
Find all points where the function is discontinuous.
208)
208)
A) None
B) x = 0
C) x = 0, x = 1
D) x = 1
Provide an appropriate response. 209) If lim f(x) = 1, find lim f(x) . x 0 x2 x 0 x
A) 1
209)
B) 0
C) 2
Graph the rational function. Include the graphs and equations of the asymptotes. 210) f(x) = 1 (x + 2)2
59
D) Does not exist
210)
A) asymptotes: x = 2, y = 0
B) asymptotes: x = 2, y = 0
C) asymptotes: x = -2, y = 0
60
D) asymptotes: x = -2, y = 0
Find the limit. 211) lim (2x + 2) x 4
A) 4
211) B) 2
C) 10
D) -6
For the function f whose graph is given, determine the limit. 212) Find lim f(x). x 1-
A) -1
212)
C) 1
B) does not exist
2
D) 2
Find the limit.
213)
lim sec x x (- /2)-
A) 1
213) C) 0
B) -
D)
Find the limit if it exists. 214) lim 10 x 7
A) 7
214) B) 7
C) 10 61
D) 10
Find the limit.
215) lim (x2 + 8x - 2)
215)
x 2
A) -18
B) 18
C) 0
D) does not exist
Use a graphing calculator to graph f near x 0 and use Zoom and Trace to estimate the y-value on the graph as x x 0 . 2
216) lim x - 4
216)
x 2 x -2
A) -4
B) 2
C) 0
D) 4
Find numbers a and b, or k, so that f is continuous at every point.
217)
217) f(x) =
6x + 4, if x < -9 kx + 3, if x -9
A) k = 7
B) k = - 1
C) k = 53
3
9
D) k = 1 3
Find the limit and determine if the function is continuous at the point being approached. 218) lim sin(3x - sin 3x) x 0
A) 0; yes C) does not exist; yes
B) 0; no D) does not exist; no
Use the graph to evaluate the limit. 219) lim f(x) x 0
A) 1
219)
B) does not exist
C) -1
D) 0
Find the limit if it exists. 220) lim (25 - 6x) x 15
A) 65
218)
220) B) -65
C) -115 62
D) 115
Use the graph to find a
> 0 such that for all x, 0 < x - x 0 <
f(x) - L < .
221)
221) y=-
4 x+1 3 3.9
4 x+1 3
f(x) = -
3.7
x0 = -2 L = 3.7 = 0.2
3.5
-2.2
-2
-1.9
NOT TO SCALE
A) 0.1
B) 5.7
C) 0.3
D) -0.3
Find the limit and determine if the function is continuous at the point being approached.
222) lim sin x 0
2
cos (tan x)
222)
A) 1; yes C) does not exist; no
B) does not exist; yes D) 1; no
Solve the problem.
223) You are asked to make some circular cylinders, each with a cross-sectional area of 2 cm2 . To do
223)
this, you need to know how much deviation from the ideal cylinder diameter of x0 = 1.72 cm you can allow and still have the area come within 0.1 cm2 of the required 2 cm2 . To find out, let x 2 A= and look for the interval in which you must hold x to make A - 2 < 0.1. What interval 2 do you find? A) 1.5554, 1.6352
B) 2.7568, 2.8983
C) 1.0998, 1.1562
D) 0.5642, 0.5642
224) Ohm's Law for electrical circuits is stated V = RI, where V is a constant voltage, R is the resistance
in ohms and I is the current in amperes. Your firm has been asked to supply the resistors for a circuit in which V will be 11 volts and I is to be 3 ± 0.1 amperes. In what interval does R have to lie for I to be within 0.1 amps of the target value I0 = 3?
A) 110 , 110 29
31
B) 110 , 110 31
C) 31 , 29
29
110 110
63
D) 10 , 10 29 31
224)
Find the limit.
1 x x -2 + 2
225) lim
225)
A) Does not exist
B) -
D) 1/2
C)
Find the limit, if it exists. 2 226) lim x + 4x - 45 x 5 x2 - 25
226)
A) - 2
C) 7
B) Does not exist
5
5
D) 0
Find the intervals on which the function is continuous. 227) y = sin (3 ) 4
A) discontinuous only when =
227) B) discontinuous only when =
2
C) discontinuous only when = 0
D) continuous everywhere
Give an appropriate answer.
228) Let lim f(x) = 64. Find lim x
6
x
A) 6
3
6
f(x).
228)
B) 64
C) 4
D) 3
Find the limit.
2 229) lim x - 6x + 5 x
1+
A)
229)
x3 - x
B) 0
C)
64
D) - 2
Use the graph to evaluate the limit. 230) lim f(x) x 0
A) -2
230)
B) does not exist
C) 2
D) 0
Given the interval (a, b) on the x-axis with the point x 0 inside, find the greatest value for x - x0 <
> 0 such that for all x, 0 <
a < x < b.
231) a = 1.353, b = 2.881, x0 = 1.875 A) = 1.006 B) = 1
231) C) = 0.522
D) = 1.528
Find the limit, if it exists. 232) lim 11 - x x 11 11 - x
A) Does not exist
232) C) 1
B) -1
65
D) 0
Solve the problem. 233) The graph below shows the number of tuberculosis deaths in the United States from 1989 to 1998.
233)
Estimate the average rate of change in tuberculosis deaths from 1991 to 1993. A) About -0.4 deaths per year B) About -45 deaths per year
C) About -80 deaths per year
D) About -30 deaths per year
Find the limit, if it exists. 2 234) lim x - 49 x 7 x-7
A) 1
234) B) 7
C) 14
D) Does not exist
Solve the problem.
235) The cross-sectional area of a cylinder is given by A = D2 /4, where D is the cylinder diameter.
235)
Find the tolerance range of D such that A - 10 < 0.01 as long as Dmin < D < Dmax.
A) Dmin = 3.567, Dmax = 3.570 C) Dmin = 3.567, Dmax = 3.578
B) Dmin = 3.558, Dmax = 3.578 D) Dmin = 3.558, Dmax = 3.570
Find the average rate of change of the function over the given interval. 236) y = -3x2 - x, [5, 6]
A) - 1 6
B) 1 2
C) -2
236) D) -34
Find the limit. 237) lim ( x - 2) x 0
B) 2
A) -2 238) lim x
237) C) 0
D) does not exist
6
238)
8 - (5/x2 )
A) 2
C) 3
B)
4
66
D) 6
A function f(x), a point x 0 , the limit of f(x) as x approaches x 0 , and a positive number is given. Find a number that for all x, 0 < x - x 0 <
> 0 such
f(x) - L < .
239) f(x) = 8x2, L =8, x0 = 1, and = 0.4 A) 0.0247 B) 1.0247
239) C) 0.02532
D) 0.97468
Solve the problem. 240) When exposed to ethylene gas, green bananas will ripen at an accelerated rate. The number of days for ripening becomes shorter for longer exposure times. Assume that the table below gives average ripening times of bananas for several different ethylene exposure times. Exposure time Ripening Time (minutes) (days) 10 4.3 15 3.2 20 2.7 25 2.1 30 1.3 Plot the data and then find a line approximating the data. With the aid of this line, determine the rate of change of ripening time with respect to exposure time. Round your answer to two significant digits.
A)
38 minutes
67
240)
B)
5.6 days
C)
-0.14 day per minute
D)
-6.7 days per minute
68
Find the limit.
2 241) lim x + 7x + 16
241)
x3 + 8x2 + 15
x
B) 16
A) 1
C) 0
15
D)
Find the limit, if it exists. 2 242) lim x + 3x - 18 x-3 x 3
A) Does not exist
B) 9
C) 0
D) 3
x2 - 4
243) lim x
242)
243)
2 x2 - 7x + 10
A) Does not exist
Provide an appropriate response. 244) Given > 0, find an interval I = (7 - , 7), is being verified and what is its value? A) lim 7 - x = 0 x 7-
C) lim
x 7+
C) - 2
B) 0
D) - 4
3
3
> 0, such that if x lies in I, then
B) lim
7-x=0
D) lim
x=7
x 0-
7-x=0
x 7-
7 - x < . What limit
Find the average rate of change of the function over the given interval. 245) y = 8x3 - 8x2 - 3, [-4, 5]
A) 797 5
B) 797 9
C) 160
245) D) 288
Provide an appropriate response. 2 246) The inequality 1- x < sin x < 1 holds when x is measured in radians and x < 1. 2 x Find lim x 0
246)
sin x if it exists. x
A) 0
B) 1
C) 0.0007
D) does not exist
Find the limit if it exists. 247) lim 6x x - 1 7 1 x 3
A) 4
63
244)
247) B) 8
C) 8
21
7
69
D) 20 21
Provide an appropriate response. 248) If lim f(x) = 1, find lim f(x). x 0 x x 0
A) 2 Evaluate lim h 0
248)
B) 1
f(x 0 + h) - f(x 0 ) h
C) 0
D) Does not exist
for the given x 0 and function f.
249) f(x) = 5 for x0 = 6
249)
x
B) - 5
A) -30
C) 5
36
6
D) Does not exist
sinx = 1. x
Find the limit using lim x=0
250) lim sin 5x
250)
x 0 sin 4x
A) does not exist
C) 4
B) 0
5
D) 5 4
A function f(x), a point x 0 , the limit of f(x) as x approaches x 0 , and a positive number is given. Find a number that for all x, 0 < x - x 0 <
f(x) - L < .
251) f(x) = 8x + 3, L = 27, x0 = 3, and = 0.01 A) 0.00625 B) 0.003333 Use the graph to find a
> 0 such
251) C) 0.00125
> 0 such that for all x, 0 < x - x 0 <
D) 0.0025
f(x) - L < .
252)
252) y = -5x - 2
8.2
f(x) = -5x - 2 x0 = -2
8
L=8 = 0.2
7.8
-2.04
-2 -1.96
NOT TO SCALE
A) 0.04
B) 0.4
C) 14
70
D) -0.04
Use the graph to estimate the specified limit. 253) Find lim f(x) and lim f(x) x (-1)x (-1)+
A) -7; -2
253)
B) -5; -2
C) -2; -7
D) -7; -5
Answer the question. 254) Is f continuous at x = 4? x3 , f(x) = -3x, 5, 0,
254)
-2 < x 0 0 x<2 2<x 4 x=2
A) Yes
B) No
Use the table of values of f to estimate the limit. 255) Let f( ) = cos (5 ) , find lim f( ). 0 x -0.1 f( ) -8.7758256
-0.01
-0.001
255) 0.001
A) limit does not exist C) limit = 8.7758256
0.01
B) limit = 0 D) limit = 5
71
0.1 8.7758256
Use the graph to find a
> 0 such that for all x, 0 < x - x 0 <
f(x) - L < .
256)
256) f(x) = 2 x x0 = 2
y=2 x
L=2 2 1 = 4
3.08 2.83 2.58
1.6619
2
2.3694
NOT TO SCALE
A) 0.7075
B) 0.83
Provide an appropriate response. 257) Which of the following statements defines
C) 0.3694
D) 0.3381
lim f(x) = ? x (x0)+
257)
I. For every positive real number B there exists a corresponding x0 - < x < x 0 + .
> 0 such that f(x) > B whenever
II. For every positive real number B there exists a corresponding x0 < x < x 0 + .
> 0 such that f(x) > B whenever
III. For every positive real number B there exists a corresponding whenever x0 - < x < x0 .
> 0 such that f(x) > B
A) III
B) I
C) II
D) None
Find the limit.
258) lim ( 10x2 + 3 - 10x2 - 3)
258)
x
A) 10
B)
1 2 10
C)
72
D) 0
For the function f whose graph is given, determine the limit. 259) Find lim f(x). x 2-
B) 4
A) -1
259)
C) 1.3
Answer the question. 260) Is f continuous at x = 0? x3 , f(x) = -3x, 7, 0,
D) 2.3
260)
-2 < x 0 0 x<2 2<x 4 x=2
A) Yes
B) No
Find the limit if it exists. 261) lim x3/4 x 625
261)
A) 125
B) 625
C) 2.061584302e+15
D) 3
4.398046511e+12
4
73
Provide an appropriate response. 262) Which of the following statements defines lim f(x) = - ? x x0
262)
I. For every negative real number B there exists a corresponding x0 - < x < x 0 + .
> 0 such that f(x) < B whenever
II. For every negative real number B there exists a corresponding whenever x0 < x < x0 + .
> 0 such that f(x) < B
III. For every negative real number B there exists a corresponding whenever x0 - < x < x0 .
> 0 such that f(x) < B
A) III
B) I
C) II
D) None
Find the slope of the curve at the given point P and an equation of the tangent line at P. 263) y = 2 - x3, (-1, 3)
A) slope is 3; y = 3x + 0 C) slope is -3; y = -3x + 0
B) slope is -1; y = -x + 0 D) slope is 0; y = 0
264) y = x3 - 2x2 + 4, P(1, 3) A) slope is 1; y = x + 4 C) slope is -1; y = -1x + 3 Find the limit.
265)
lim x + 4 x -1 +
263)
264) B) slope is 0; y = 4 D) slope is -1; y = -1x + 4
x+1 x+1
A) 3
265) B) 5
C) -3
D) Does not exist
Find the limit, if it exists. 1/3 266) lim (1+h) - 1 h h 0
A) 0
266) B) 1/3
C) 3
D) Does not exist
Find the intervals on which the function is continuous. 4 267) y = (x + 2)2 + 4
267)
A) discontinuous only when x = 8 C) discontinuous only when x = -2
B) continuous everywhere D) discontinuous only when x = -16
74
Provide an appropriate response. 268) Which of the following statements defines
lim f(x) = - ? x (x0 )+
268)
I. For every negative real number B there exists a corresponding x0 - < x < x 0 + .
> 0 such that f(x) < B whenever
II. For every negative real number B there exists a corresponding whenever x0 < x < x0 + .
> 0 such that f(x) < B
III. For every negative real number B there exists a corresponding whenever x0 - < x < x0 .
> 0 such that f(x) < B
A) I
B) III
C) II
D) None
Find all points where the function is discontinuous.
269)
269)
A) x = -2, x = 6
B) x = -2
C) None
D) x = 6
Divide numerator and denominator by the highest power of x in the denominator to find the limit. 2 270) lim 100t - 1000 t - 10 t
A) does not exist
B) 1000
C) 10
D) 100
Solve the problem. 271) The current in a simple electrical circuit is given by I = V/R, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms. When V = 12 volts, what is a 12 resistor's tolerance for the current to be within 1 ± 0.01 amp? A) 0.01% B) 10% C) 0.1% D) 1%
75
270)
271)
Determine the limit by sketching an appropriate graph. 16 - x2 0 x<4 lim f(x), where f(x) = 4 4 x<7 272) x 4 7 x=7
A) 7
272)
B) 4
C) 0
D) Does not exist
A function f(x), a point x 0 , the limit of f(x) as x approaches x 0 , and a positive number is given. Find a number that for all x, 0 < x - x 0 <
> 0 such
f(x) - L < .
273) f(x) = -4x + 6, L = -6, x0 = 3, and = 0.01 A) 0.005 B) 0.01
273) D) 0.0025
C) -0.003333
Find the slope of the curve at the given point P and an equation of the tangent line at P. 274) y = x2 + 5x, P(4, 36)
274)
A) slope is 13; y = 13x - 16
B) slope is -39; y = -39x - 80
C) slope is 1 ; y = x + 1
D) slope is - 4 ; y = - 4x + 8
20
20
5
25
25
5
Divide numerator and denominator by the highest power of x in the denominator to find the limit. 49x2 275) lim x 5 + 25x2
A) 7
B) 6.896136929e+15
C) does not exist
D) 49
5
3.518437209e+15 5
76
275)
For the function f whose graph is given, determine the limit. 276) Find lim f(x). x
A) does not exist
276)
C) 0
B) -2
D)
Find the limit. 277) lim (1 + csc x) x 0+
277)
A) 1
B) 0
D) Does not exist
C)
278) lim (x3 + 5x2 - 7x + 1)
278)
x 2
A) 0 279)
B) 29
C) does not exist
D) 15
1 lim 2 x -4 - x - 16
279)
A) 0 Evaluate lim h 0
C) -
B)
f(x 0 + h) - f(x 0 ) h
D) -1
for the given x 0 and function f.
280) f(x) = 3 x + 6 for x0 = 25 A) 75 2
280) C) 15
B) Does not exist
2
77
D) 3
10
Use the graph to evaluate the limit. 281) lim f(x) x -1
A) - 3
281)
C) 3
B) -1
4
D)
4
Given the interval (a, b) on the x-axis with the point x 0 inside, find the greatest value for x - x0 <
> 0 such that for all x, 0 <
a < x < b.
282) a = 3 , b = 9 , x0 = 4 9
9
A) = 1 9
282)
9
C) = 5
B) = 1
9
D) = 0
Find the limit.
283)
lim x 0.5-
A)
x+9 x+2 17 3
283) B) 19
C)
5
78
19 5
D) Does not exist
Use the graph to evaluate the limit. 284) lim f(x) x 0
A) does not exist
284)
B) 0
C) 6
D) -1
Find the limit if it exists.
285) lim (x - 236)4/5
285)
x -7
A) -27
C) 81
B) -81
D) -243
Find the limit.
286) lim x
x2 + 10x - x
A) 0
286) B) 10
C) 5
D)
Use the graph to estimate the specified limit. 287) Find lim f(x) x 0
A) 0
287)
B) 6
C) does not exist
79
D) -1
Find the limit.
1 +4 4/3 x 0
288) lim x
288)
A) Does not exist Evaluate lim h 0
f(x 0 + h) - f(x 0 ) h
B) 4
C)
D)
for the given x 0 and function f.
289) f(x) = 2x2 + 3 for x0 = -2 A) 8
289) C) Does not exist
B) -5
D) -8
Use the graph to estimate the specified limit. 290) Find lim f(x) and lim f(x) x 0x 0+
A) 1; -1
290)
B) 1; 1
C) -1; 1
Answer the question. 291) Is f continuous at f(3)? -x2 + 1, 3x, f(x) = -2, -3x + 6 3,
D) -1; -1
291)
-1 x < 0 0<x<1 x=1 1<x<3 3<x<5
A) Yes
B) No
80
Find the limit using lim x=0
sinx = 1. x
292) lim sin 4x
292)
x 0 sin 5x
B) 4
A) 0
D) 5
C) does not exist
5
4
A function f(x), a point x 0 , the limit of f(x) as x approaches x 0 , and a positive number is given. Find a number that for all x, 0 < x - x 0 <
f(x) - L < .
293) f(x) = 7x - 10, L = 4, x0 = 2, and = 0.01 A) 0.000714 B) 0.005
293) C) 0.002857
D) 0.001429
Use the table of values of f to estimate the limit. 294) Let f(x) = x - 4 , find lim f(x). x-2 x 4 x f(x)
3.9
> 0 such
3.99
3.999
294) 4.001
4.01
4.1
A) x 3.9 3.99 3.999 f(x) 1.19245 1.19925 1.19993
4.001 4.01 4.1 ; limit = 1.20007 1.20075 1.20745
x 3.9 3.99 3.999 f(x) 1.19245 1.19925 1.19993
4.001 4.01 4.1 ; limit = 1.20 1.20007 1.20075 1.20745
x 3.9 3.99 3.999 f(x) 3.97484 3.99750 3.99975
4.001 4.01 4.1 ; limit = 4.0 4.00025 4.00250 4.02485
x 3.9 3.99 3.999 f(x) 5.07736 5.09775 5.09978
4.001 4.01 4.1 ; limit = 5.10 5.10022 5.10225 5.12236
B) C) D)
Find the limit. 295) lim (3 sin x - 1) x 0
A) -1
295) B) 0
C) 3 - 1
D) 3
Provide an appropriate response. 296) If lim f(x) - 4 = 5, find lim f(x). x 3 x-4 x 3
A) 15
296) C) 3
B) -1
81
D) Does not exist
Find the limit, if it exists. 297) lim 2x - 7 x 1 4x + 5
297)
A) - 1
B) - 7
2
Use the graph to find a
C) Does not exist
5
> 0 such that for all x, 0 < x - x 0 <
D) - 5 9
f(x) - L < .
298)
298) y = x2 - 2
3 2
f(x) = x2 - 2 x0 = 2 L=2 =1
1
1.73
2
2.24
NOT TO SCALE
A) 0
B) 0.51
C) 0.24
D) 0.27
Find the limit and determine if the function is continuous at the point being approached. 299) lim sec(x sec2 x - x tan2 x - 1) x 9
A) sec 8; yes C) csc 1; yes
299)
B) sec 8; no D) does not exist; no
Find the limit.
300) lim x
x2 + 4x -
A) 5
x2 - 6x
300) B) does not exist
C) 10
D) - 1
Use a graphing calculator to graph f near x 0 and use Zoom and Trace to estimate the y-value on the graph as x x 0 .
301) lim
x 3
A) 8
x2 - 9 x2 + 7 - 4
301) B) 1
C) 4
4
82
D) 3
Provide an appropriate response. 302) Which of the following statements defines
lim f(x) = - ? x (x0 )-
302)
I. For every negative real number B there exists a corresponding x0 - < x < x 0 + .
> 0 such that f(x) < B whenever
II. For every negative real number B there exists a corresponding whenever x0 < x < x0 + .
> 0 such that f(x) < B
III. For every negative real number B there exists a corresponding whenever x0 - < x < x0 .
> 0 such that f(x) < B
A) I
B) III
C) II
D) None
Find the limit and determine if the function is continuous at the point being approached. 303) lim tan cos (sin ) -
A) 0; yes C) 0; no
303)
B) does not exist; no D) 1; yes
Find the limit.
304) lim (x2 - 5)
304)
x 0
A) does not exist
B) 0
Provide an appropriate response. 305) Given > 0, find an interval I = (4, 4 + ), is being verified and what is its value? A) lim x = 4 x 4+
C) lim
x 0-
C) 5
D) -5
> 0, such that if x lies in I, then
B) lim
x-4=0
D) lim
x-4=0
x 4-
x-4=0
x 4+
x - 4 < . What limit
305)
Find all points where the function is discontinuous.
306)
306)
A) x = 2
B) x = -2
Use the graph of the greatest integer function y =
307) lim
x 2+
A) 1
C) x = -2, x = 2
D) None
to find the limit.
307)
x
B) 2
C) -2
83
D) 0
For the function f whose graph is given, determine the limit. 308) Find lim f(x) and lim f(x). x -1 x -1 +
A) -5; -2
B) -7; -5
308)
C) -2; -7
D) -7; -2
Find the limit. 309) lim (1 - cot x) x 0
A) 0
309) C) -
B)
D) Does not exist
Divide numerator and denominator by the highest power of x in the denominator to find the limit. -1 310) lim 2 x + x 2x + 4 x
A) 1 2
B) 0
C) 1
310)
D)
Find all points where the function is discontinuous.
311)
311)
A) x = 4, x = 2
B) None
C) x = 2
Graph the rational function. Include the graphs and equations of the asymptotes.
84
D) x = 4
312) f(x) = x - 1
312)
x+1
A) asymptotes: x = 1, y = -1
B) asymptotes: x = -1, y = 1
85
C) asymptotes: x = 1, y = 1
D) asymptotes: x = -1, y = -1
Find the limit and determine if the function is continuous at the point being approached. 313) lim cos(5x - cos 5x) x - /2
A) does not exist; no C) 0; yes
B) 0; no D) does not exist; yes
Provide an appropriate response. 314) If lim f(x) = 1 and f(x) is an odd function, which of the following statements are true? x 0I.
lim f(x) = 1 x 0
II.
lim f(x) = -1 x 0+
III. lim f(x) does not exist. x 0
A) I and III only
313)
B) II and III only
C) I and II only
86
D) I, II, and III
314)
Give an appropriate answer.
315) Let lim f(x) = 2. Find lim (-3)f(x). x
4
x
A) 2
315)
4
B) 9
Find the limit using lim x=0
C) -3
D) 81
sinx = 1. x
316) lim tan 4x B) 1
A) 1 Find the limit.
317) lim x
5
C) does not exist
4
D) 4
1 2 + x - 25
A) 318) lim
316)
x
x 0
317) C) 1
B) 10 -
h 0-
A) -7
2 10
D) 0
8h 2 + 7h + 10 h
B)
318) 7 2 10
C) Does not exist
D) -7
20
A function f(x), a point x 0 , the limit of f(x) as x approaches x 0 , and a positive number is given. Find a number that for all x, 0 < x - x 0 <
> 0 such
f(x) - L < .
319) f(x) = x + 2, L = 3, x0 = 7, and = 1 A) 16 B) 4
319) C) 5
87
D) 7
Give an appropriate answer. 320) Suppose lim f(x) = 1 and lim g(x) = -3. Name the limit rules that are used to accomplish steps x 0 x 0
320)
(a), (b), and (c) of the following calculation. lim (-3f(x) - 2g(x) ) (a) x 0 -3f(x) - 2g(x) = lim x 0 (f(x) + 3)1/2 lim (f(x) + 3)1/2 x 0 (b) =
=
-3 lim f(x) - 2 lim g(x) lim -3f(x) - lim 2g(x) (c) x 0 x 0 x 0 x 0 = ( lim f(x) + 3 )1/2 ( lim f(x) + lim 3)1/2 x 0 x 0 x 0
-3 + 6 3 = 1/2 2 (1 + 3)
A) (a) Quotient Rule
(b) Difference Rule (c) Constant Multiple Rule B) (a) Quotient Rule (b) Difference Rule, Sum Rule (c) Constant Multiple Rule and Power Rule C) (a) Difference Rule (b) Power Rule (c) Sum Rule D) (a) Quotient Rule (b) Difference Rule, Power Rule (c) Constant Multiple Rule and Sum Rule
Find the limit. 1x2 7+x
321) lim
x 2+
321)
A) 4 9
Evaluate lim h 0
4 9
B)
f(x 0 + h) - f(x 0 ) h
C) Does not exist
2 9
D)
for the given x 0 and function f.
322) f(x) = x + 5 for x0 = 7
322)
5
A) 1 5
B) 7
C) Does not exist
5
88
D) 32 5
Solve the problem. 323) Identify the incorrect statements about limits. I. The number L is the limit of f(x) as x approaches x0 if f(x) gets closer to L as x approaches x0. II. The number L is the limit of f(x) as x approaches x0 if, for any > 0, there corresponds a
323)
>0
such that f(x) - L < whenever 0 < x - x0 < .
III. The number L is the limit of f(x) as x approaches x0 if, given any > 0, there exists a value of x for which f(x) - L < . A) I and II
B) II and III
C) I and III
D) I, II, and III
Find the limit.
324) lim
h 0+
h 2 + 3h + 6 h
A) Does not exist f(x 0 + h) - f(x 0 )
Evaluate lim h 0
h
6
324) B) 3
C) 3
2 6
12
D) 1 4
for the given x 0 and function f.
325) f(x) = 3x + 3 for x0 = 7 A) 3
325) B) 24
C) Does not exist
D) 21
Find the limit if it exists. 326) lim (6x - 10) x 10
A) -70
326) B) 70
C) -50
D) 50
Provide an appropriate response. 327) Given lim f(x) = Ll, lim f(x) = Lr , and Ll = Lr, which of the following statements is false? x 0x 0+ I.
327)
lim f(x) = Ll x 0
II. lim f(x) = Lr x 0 III. lim f(x) does not exist. x 0
A) II
B) I
C) none
D) III
Find the limit.
328) lim 1 - 9 x
328)
x
A) -10
B) -9
C) -8
89
D) 9
Evaluate lim h 0
f(x 0 + h) - f(x 0 ) h
for the given x 0 and function f.
329) f(x) = x for x0 = 3 A)
3 3
329) B)
3 6
D) 3
C) Does not exist
2
Use the table of values of f to estimate the limit. 2 330) Let f(x) = x + 4x + 3 , find lim f(x). x2 - 4x - 5 x -1 x f(x)
-1.1
-1.01
-1.001
330) -0.999
-0.99
-0.9
A) x -1.1 -1.01 -1.001 -0.999 -0.99 -0.9 ; limit = -0.2333 f(x) -0.2115 -0.2311 -0.2331 -0.2336 -0.2356 -0.2559
B) x -1.1 -1.01 -1.001 -0.999 -0.99 -0.9 ; limit = -1 f(x) -0.9512 -0.9950 -0.9995 -1.0005 -1.0050 -1.0513
C) x -1.1 -1.01 -1.001 -0.999 -0.99 -0.9 ; limit = -0.4333 f(x) -0.4115 -0.4311 -0.4331 -0.4336 -0.4356 -0.4559
D) x -1.1 -1.01 -1.001 -0.999 -0.99 -0.9 ; limit = -0.3333 f(x) -0.3115 -0.3311 -0.3331 -0.3336 -0.3356 -0.3559
Find the limit, if it exists. 1+x-1 331) lim x x 0
A) 1/2
331) B) Does not exist
C) 1/4
90
D) 0
Use the graph to evaluate the limit. 332) lim f(x) x 0
A) 1
332)
B) does not exist
C) -2
91
D) 0
Answer Key Testname: CHAPTER 2
1) A 2) A 3) D 4) A 5) D 6) Answers may vary. One possible answer:
7) The roots of f(x) are the solutions to the equation f(x) = 0. Statement (b) is asking for the solution to the equation
2x3 = 3x + 5. Statement (d) is asking for the solution to the equation 2x3 - 3x = 5. These three equations are equivalent to the equations in statements (c) and (e). As five equations are equivalent, their solutions are the same. 1 8) Let f(x) = 1 , for all x 8. The function f is continuous for all x 8, and lim = . As f is unbounded as 2 (x - 8) x 8 (x - 8)2 x approaches 8, f is discontinuous at x = 8, and, moreover, this discontinuity is nonremovable.
9) Let > 0 be given. Choose = /4. Then 0 < x - 4 < implies that (4x - 5) - 11 = 4x - 16 = 4(x - 4) =4 x-4 <4 = Thus, 0 < x - 4 < implies that (4x - 5) - 11 < 10) Let f(x) = 5x3 - 6x2 - 6x + 5 and let y0 = 0. f(1) = -2 and f(2) = 9. Since f is continuous on [1, 2] and since y0 = 0 is between f(1) and f(2), by the Intermediate Value Theorem, there exists a c in the interval (1 , 2) with the property that f(c) = 0. Such a c is a solution to the equation 5x3 - 6x2 - 6x + 5 = 0.
11) Let f(x) = 2x4 + 4x3 + 7x + 4 and let y0 = 0. f(-3) = 37 and f(-2) = -10. Since f is continuous on [-3, -2] and since y0 = 0 is between f(-3) and f(-2), by the Intermediate Value Theorem, there exists a c in the interval (-3, -2) with the property that f(c) = 0. Such a c is a solution to the equation 2x4 + 4x3 + 7x + 4 = 0.
12) Notice that f(0) = 5 and f(1) = 2. As f is continuous on [0,1], the Intermediate Value Theorem implies that there is a number c such that f(c) = .
92
Answer Key Testname: CHAPTER 2
13) (Answers may vary.) Possible answer: f(x) =
1 . x+3
14) Let > 0 be given. Choose = /4. Then 0 < x - 4 < implies that 4x2 - 13x- 12 (x - 4)(4x + 3) - 19 = - 19 x-4 x-4
Thus, 0 < x - 4 <
for x 4 = (4x + 3) - 19 = 4x - 16 = 4(x - 4) =4 x -4 < 4 = 4x2 - 13x- 12 - 19 < implies that x-4
15) The Intermediate Value Theorem implies that there is at least one solution to f(x) = 0 on the interval -1, 3 . Possible graph:
16) (Answers may vary.) Possible answer: f(x) =
4, x > 0 -4, x < 0
93
Answer Key Testname: CHAPTER 2
17) Let f(x) = x(x - 7)2 and let y0 = 7. f(6) = 6 and f(8) = 8. Since f is continuous on [6, 8] and since y0 = 7 is between f(6) and f(8), by the Intermediate Value Theorem, there exists a c in the interval (6, 8) with the property that f(c) = 7. Such a c is a solution to the equation x(x - 7)2 = 7.
18) Given B > 0, we want to find > 0 such that 0 < x - 0 < implies 9 > B. x
Now,
9 9 > B if and only if x < . x B
Thus, choosing x <
= 9/B (or any smaller positive number), we see that 9 9 implies B. > x
9 = Therefore, by definition lim x 0 x
19) (Answers may vary.) Possible answer: f(x) =
1 . x-4
20) Answers may vary. One possible answer:
21) Let > 0 be given. Choose = . Then 0 < x - 4 < implies that x2 - 16 (x - 4)(x + 4) -8 = -8 x-4 x-4
for x 4 = (x + 4) - 8 = x -4 < = x2 - 16 Thus, 0 < x - 4 < implies that -8 < x-4
94
Answer Key Testname: CHAPTER 2
22) (Answers may vary.) Possible answer: f(x) = 1 . x
23) Let f(x) = sin (x - 3) be defined for all x 3. The function f is continuous for all x 3. The function is not defined at (x - 3)
x = 3 because division by zero is undefined; hence f is not continuous at x = 3. This discontinuity is removable sin (x - 3) = 1. (We can extend the function to x = 3 by defining its value to be 1.) because lim x-3 x 3
24) Given B > 0, we want to find > 0 such that x0 < x < x0 + implies 6 > B. x
Now,
6 6 > B if and only if x < . x B
We know x0 = 0. Thus, choosing x<
implies
6 6 > x
= 6/B (or any smaller positive number), we see that
B.
6 = Therefore, by definition lim x x 0+ 1, x < 0
25) (Answers may vary.) Possible answer: f(x) = -1, x > 0
26) Yes, if f(x) = 1 and g(x) = x - 3, then h(x) = 1 is discontinuous at x = 3. x-3
95
Answer Key Testname: CHAPTER 2
27) Let f(x) = sin x and let y0 = 1 . f x
f
2
5
2
0.6366 and f( ) = 0. Since f is continuous on
and f( ), by the Intermediate Value Theorem, there exists a c in the interval
f(c) =
1 . Such a c is a solution to the equation 5 sin x = x. 5
28) Let > 0 be given. Choose = min{5/2, 25 /2}. Then 0 < x - 5 < implies that 1 1 5-x = x 5 5x
Thus, 0 < x - 5 <
=
1 1 · · x-5 x 5
<
1 1 25 · · = 5/2 5 2 implies that
1 1 < x 5
29) Answers may vary. One possible answer:
30) Answers may vary. One possible answer:
31) A 32) C 33) C 96
2 2
,
,
and since y0 =
1 is between 5
, with the property that
Answer Key Testname: CHAPTER 2
34) A 35) B 36) C 37) A 38) D 39) B 40) B 41) B 42) C 43) D 44) D 45) D 46) A 47) C 48) C 49) A 50) A 51) C 52) C 53) D 54) C 55) D 56) C 57) D 58) C 59) B 60) C 61) D 62) C 63) A 64) D 65) D 66) A 67) B 68) B 69) B 70) B 71) B 72) C 73) D 74) A 75) D 97
Answer Key Testname: CHAPTER 2
76) A 77) B 78) B 79) D 80) B 81) A 82) D 83) D 84) B 85) C 86) C 87) D 88) C 89) D 90) A 91) A 92) B 93) C 94) D 95) C 96) B 97) B 98) A 99) B 100) A 101) C 102) A 103) D 104) B 105) B 106) C 107) A 108) D 109) D 110) D 111) B 112) A 113) D 114) C 115) C 116) C 117) D 98
Answer Key Testname: CHAPTER 2
118) D 119) C 120) A 121) B 122) D 123) C 124) A 125) B 126) B 127) C 128) B 129) A 130) B 131) B 132) D 133) B 134) D 135) D 136) B 137) D 138) B 139) D 140) A 141) D 142) B 143) C 144) C 145) A 146) D 147) B 148) D 149) D 150) A 151) A 152) A 153) C 154) C 155) A 156) C 157) D 158) A 159) D 99
Answer Key Testname: CHAPTER 2
160) D 161) D 162) D 163) A 164) C 165) A 166) A 167) D 168) A 169) B 170) B 171) B 172) B 173) C 174) B 175) D 176) A 177) A 178) B 179) A 180) A 181) C 182) A 183) D 184) D 185) B 186) A 187) C 188) C 189) A 190) D 191) A 192) B 193) D 194) B 195) A 196) A 197) C 198) C 199) B 200) D 201) D 100
Answer Key Testname: CHAPTER 2
202) D 203) D 204) A 205) A 206) D 207) D 208) A 209) B 210) C 211) C 212) D 213) D 214) D 215) B 216) D 217) C 218) A 219) A 220) B 221) A 222) A 223) A 224) B 225) A 226) C 227) C 228) C 229) D 230) A 231) C 232) A 233) B 234) C 235) A 236) D 237) A 238) C 239) A 240) C 241) C 242) B 243) D 101
Answer Key Testname: CHAPTER 2
244) A 245) C 246) B 247) B 248) C 249) B 250) D 251) C 252) A 253) C 254) A 255) A 256) D 257) C 258) D 259) C 260) A 261) A 262) B 263) C 264) D 265) A 266) B 267) B 268) C 269) D 270) C 271) D 272) C 273) D 274) A 275) A 276) B 277) C 278) D 279) B 280) D 281) C 282) A 283) C 284) B 285) C 102
Answer Key Testname: CHAPTER 2
286) C 287) A 288) D 289) D 290) A 291) B 292) B 293) D 294) C 295) B 296) B 297) D 298) C 299) A 300) A 301) A 302) B 303) A 304) D 305) D 306) C 307) A 308) C 309) D 310) B 311) D 312) B 313) C 314) B 315) B 316) D 317) B 318) A 319) C 320) D 321) B 322) A 323) C 324) B 325) A 326) D 327) D 103
Answer Key Testname: CHAPTER 2
328) B 329) B 330) D 331) A 332) C
104
Chapter 3
Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graph of a function is given. Choose the answer that represents the graph of its derivative.
1)
1)
A)
B)
C)
D)
1
2)
2)
A)
B)
C)
D)
2
3)
3)
A)
B)
C)
D)
3
4)
4)
A)
B)
C)
D)
4
5)
5)
A)
B)
C)
D)
5
6)
6)
A)
B)
C)
D)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. x cos
7) Does the graph of f(x) =
0,
1 , x
x 0 x=0
have a tangent at the origin? Give reasons for
7)
your answer.
8) Rewrite tan x and use the product rule to verify the derivative formula for tan x.
8)
9) Does the curve y = x3 + 4x - 10 have a tangent whose slope is -2? If so, find an equation
9)
for the line and the point of tangency. If not, why not?
6
Find the derivatives of all orders of the function. 7 10) y = x 25,200
10)
Provide an appropriate response. 11) If g(x) = - f(x) - 3, find g (4) given that f (4) = 5.
12) Graph y = -tan x and its derivative together on -
11) , . Does the graph of y = -tan x 2 2
12)
appear to have the smallest slope? If so, what is it? If not, explain.
13) Does the curve y = (x + 3)3 have any horizontal tangents? If so, where? Give reasons for
13)
your answer.
5,
14) Does the graph of f(x) = 0,
-5,
for your answer.
x<0 x = 0 have a vertical tangent at the origin? Give reasons x>0
14)
Solve the problem.
15) For functions of the form y = axn, show that the relative uncertainty dy in the y
dependent variable y is always n times the relative uncertainty
15)
dx in the independent x
variable x.
Provide an appropriate response. 16) Find d998/dx998 (cos x).
16)
17) Find ds/dt when = /4 if s = sin and d /dt = 10.
17)
18) Is there any difference between finding the derivative of f(x) at x = a and finding the slope
18)
19) Find a value of c that will make
19)
of the line tangent to f(x) at x = a? Explain.
f(x) =
sin2 4x , x2
x 0
c,
x=0
continuous at x = 0.
20) Find the tangent to the curve y = 4 cot
x 1 at x = . What is the largest value the slope of 2 2
the curve can ever have on the interval -2 < x < 0?
7
20)
21) Find d997/dx997 (sin x).
21) 5
22) Over what intervals of x-values, if any, does the function y = x decrease as x increases? 5
22)
For what values of x, if any, is y negative? How are your answers related?
23) Graph y = -tan x and its derivative together on -
, . Is the slope of the graph of 2 2
23)
y = -tan x ever positive? Explain.
24) Does the graph of f(x) = 0,
5,
x>0 have a vertical tangent at the point (0, 5)? Give x 0
24)
reasons for your answer. x2 cos
25) Does the graph of f(x) =
1 , x
x 0 have a tangent at the origin? Give reasons for
25)
26) Consider the functions f(x) = x2 and g(x) = x3 and their linearizations at the origin. Over
26)
Provide an appropriate response. 27) Can a tangent line to a graph intersect the graph at more than one point? If not, why not. If so, give an example.
27)
0,
x=0
your answer.
Solve the problem.
some interval x , the approximation error for g(x) is less than the approximation error for f(x) for all x within the interval. Derive a reasonable approximation for the value of . Show your work. (Hint, the absolute value of the second derivative of each function gives a measure of how quickly the slopes of the function and its linear approximation are deviating from one another.)
Find the derivatives of all orders of the function. 28) y = 2 x3 + 7 x2 + 4x - 15 3 2
28)
Provide an appropriate response.
29) Find dy/dt when x = 5 if y = 2x2 - 6x + 7 and dx/dt = 1/2.
29)
30) If g(x) = 2f(x) + 3, find g (4) given that f (4) = 5.
30)
Find the derivatives of all orders of the function. 31) y = (x + 1)(x2 - 4x + 7)
31)
8
Provide an appropriate response.
32) What is the range of values of the slope of the curve y = x3 + 5x - 2?
32)
33) Does the curve y = x ever have a negative slope? If so, where? Give reasons for your
33)
34) Over what intervals of x-values, if any, does the function y = 2x2 increase as x increases?
34)
35) Find equations for the tangents to the curves y = tan 2x and y = -tan(x/2) at the origin.
35)
36) Find d998/dx998 (sin x).
36)
answer.
For what values of x, if any, is y positive? How are your answers related?
How are the tangents related?
37) Graph y = -tan x and its derivative together on -
, . Does the graph of y = -tan x 2 2
37)
appear to have the largest slope? If so, what is it? If not, explain.
Solve the problem.
38) Let Q(x) = bo + b1 (x - a) + b2 (x - a)2 be a quadratic approximation to f(x) at x = a with the properties: i. Q(a) = f(a) ii. Q (a) = f (a) iii. Q (a) = f (a) (a) Find the quadratic approximation to f(x) =
38)
1 at x = 0. 3+x
(b) Do you expect the quadratic approximation to be more or less accurate than the linearization? Give reasons for your answer.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Estimate the slope of the curve at the indicated point.
39)
39)
A) 0
C) 1
B) -1
9
D) Undefined
Provide an appropriate response. 32 40) Evaluate lim x -1 = by first converting it to a derivative at a particular x-value. x 1 x-1
B) 1
A) 31
Use the formula f'(x) = lim z x
C) 33
32
40)
D) 32
f(z) - f(x) to find the derivative of the function. z-x
41) f(x) = 3x2 + 3x + 5 A) 6x2 + 3x
41) B) 6x + 3
C) 3x + 3
D) 6x
Graph the equation and its tangent. 42) Graph y = x2 + 2x - 6 and the tangent to the curve at the point whose x-coordinate is 1.
A)
B)
10
42)
C)
D)
Solve the problem. Round your answer, if appropriate. 43) The radius of a right circular cylinder is increasing at the rate of 3 in./sec, while the height is decreasing at the rate of 9 in./sec. At what rate is the volume of the cylinder changing when the radius is 6 in. and the height is 10 in.? A) -6 in.3/sec B) 36 in.3/sec C) -144 in.3 /sec D) -144 in.3 /sec Find the derivative of the function. 4 6 44) y = 4 - 2x + x x9
44)
A) dy = - 36 + 10 - 3
B) dy = 36 - 10 + 3
C) dy = - 36 + 10 - 3
D) dy = -36x10 + 10x6 - 3x4
dx
x8
dx
x4
x10
x2
x6
dx
x4
x10
x6
x6
dx
Solve the problem. 45) If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, c2 = a 2 + b2 . How is dc/dt related to da/dt and db/dt?
A) dc = a da + b db dt
dt
c
dt
45)
B) dc = a2 da + b2 db
dt
dt
C) dc = 1 a da + b db dt
43)
dt
dt
D) dc = 2a da + 2b db
dt
dt
dt
dt
Determine if the piecewise defined function is differentiable at the origin. if x < 0 46) f(x) = 3x - 6 2 x + 3x + 6 if x 0
A) Differentiable
B) Not differentiable
11
46)
Solve the problem. 47) The position of a particle moving along a coordinate line is s = 5 + 4t with s in meters and t in seconds. Find the particle's acceleration at t = 1 sec. A) 2 m/sec2 B) - 1.125899907e+15 m/sec2 3 7.599824371e+15
C) 1.125899907e+15 m/sec2
47)
D) - 2.814749767e+14 m/sec2
7.599824371e+15
7.599824371e+15
The figure shows the velocity v or position s of a body moving along a coordinate line as a function of time t . Use the figure to answer the question. 48) v (ft/sec) 48)
t (sec)
When does the body reverse direction? A) t = 4, t = 7 sec
B) t = 2, t = 3, t = 5, t = 6, t = 7 sec D) t = 7 sec
C) t = 4 sec
Solve the problem. 49) The graph of y = f(x) in the accompanying figure is made of line segments joined end to end. Graph the derivative of f.
12
49)
A)
B)
C)
13
D)
50) The concentration of a certain drug in the bloodstream x hr after being administered is approximately C(x) =
6x
8 + x2
changes from 1 to 1.3. A) 0.42
50)
. Use the differential to approximate the change in concentration as x
B) 0.41
C) 0.76
D) 0.16
Find the limit.
51) lim x
/3
6 2 + sin( sec x)
51) B) 6 2 + 1
A) 0
C) 1
D) 6
Find the value of (f g) at the given value of x. 52) Assume that f'(3) = 8 , g'(4) = -4, g(4) = 3, and y = f(x). What is y' at x = 4? A) -20 B) -32 C) 24
52) D) 8
Solve the problem. 53) A heat engine is a device that converts thermal energy into other forms. The thermal efficiency, e, of a heat engine is defined by Qh - Qc e= , Qh where Qh is the heat absorbed in one cycle and Qc, the heat released into a reservoir in one cycle, is a constant. Find
d2 e . dQh 2
2 Qc A) d e =
2 Qc B) d e =
2 - Qc C) d e =
2 - 2Qc D) d e =
dQh 2 dQh 2
Qh 3
dQh 2
2Qh 2
dQh 2
14
Qh 2 Qh 3
53)
Solve the problem. Round your answer, if appropriate. 54) One airplane is approaching an airport from the north at 161 km/hr. A second airplane approaches from the east at 242 km/hr. Find the rate at which the distance between the planes changes when the southbound plane is 32 km away from the airport and the westbound plane is 22 km from the airport. A) -540 km/hr B) -270 km/hr C) -405 km/hr D) -135 km/hr Graph the equation and its tangent. 55) Graph y = x3 - 4 and the tangent to the curve at the point whose x-coordinate is 0.
A)
B)
C)
D)
15
54)
55)
Find the value of (f g) at the given value of x. 56) f(u) = 1 - u, u = g(x) = x, x = 8 cos4 u
A) -8
56)
B) 4 -
C) -
D) 8
B) 4 + 24x-1
C) 4x + 8 - 6x-4
D) 4 + 24x-5
Find the second derivative. 57) y = 2x2 + 8x + 2x-3
A) 4 - 24x-5
57)
Solve the problem. 58) The position of a particle moving along a coordinate line is s = 6 + 10t, with s in meters and t in seconds. Find the particle's velocity at t = 1 sec. A) 5 m/sec B) 5 m/sec C) - 1 m/sec D) 1 m/sec 4 2 4 8 Use implicit differentiation to find dy/dx. 59) x7 = cot y 6 A) 7x
59) 2
6 C) - 7x
B) csc y
csc2 y
7x6
D) -
csc2 y
7x6 csc y cot y
Solve the problem. 60) A runner is competing in an 8-mile race. As the runner passes each miles marker (M), a steward records the time elapsed in minutes (t) since the beginning of the race, as shown in the table. What is the runner's average speed over the first 4 miles? Round your answer to four decimal places. M01 2 3 4 5 6 7 8 t 0 15 29 43 55 67 80 93 109 A) 0.0862 miles/min
Find the linearization L(x) of f(x) at x = a. 61) f(x) = x , a = 0 9x + 9 81
60)
B) 13.7552 miles/min D) 0.0727 miles/min
C) 0.0649 miles/min
A) L(x) = 1 x
58)
61)
B) L(x) = - 1 x
C) L(x) = - 1 x
9
D) L(x) = 1 x
81
9
Solve the problem.
62) The kinetic energy K of an object with mass m and velocity v is K = 1 mv2 . How is dm/dt related 2
to dv/dt if K is constant? A) dm = m dv dt v dt
B) dm = - 2mv3 dv
C) dm = - 2m dv
D) dv = - 2m dm
dt
dt
v dt
dt
16
dt
v dt
62)
Find the limit.
63)
lim 4 cos sin x + x - /2
cot
4 csc x
A) 4
+1
63)
B) 1
C) 0
D) - 4
Calculate the derivative of the function. Then find the value of the derivative as specified. 64) g(x) = x3 + 5x; g (1)
A) g (x) = 3x2 + 5; g (1) = 8
B) g (x) = 3x2 + 5x; g (1) = 8
C) g (x) = x2 + 5; g (1) = 6
D) g (x) = 3x2 ; g (1) = 3
64)
Find the derivative of the function. 2 65) g(x) = x + 5 x2 + 6x 3
65)
2
3
A) g (x) = 2x - 5x - 30x x2 (x + 6)2
4
3
2
B) g (x) = 4x + 18x + 10x + 30 x2(x + 6)2
2
2
C) g (x) = x + 6x + 5x + 30x
D) g (x) = 6x - 10x - 30
x2 (x + 6)2
x2 (x + 6)2
Provide an appropriate response.
66) At the two points where the curve x2 + 2xy + y2 = 25 crosses the x-axis, the tangents to the curve
66)
are parallel. What is the common slope of these tangents?
A) -1
C) 3
B) 1
5
D) -5
Find the second derivative. 3 67) y = 11x - 9 6
A) 11x
67) C) 11 x2
B) 11x - 9
2
D) 11 x 6
Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 68) u(1) = 3, u (1) = -7, v(1) = 6, v (1) = -4. 68) d (3v - u) at x = 1 dx
A) 15
C) 21
B) -19
D) -5
Solve the problem. 69) For a motorcycle traveling at speed v (in mph) when the brakes are applied, the distance d (in feet) required to stop the motorcycle may be approximated by the formula d = 0.05v2 + v. Find the instantaneous rate of change of distance with respect to velocity when the speed is 40 mph. A) 10 mph B) 5 mph C) 41 mph D) 4 mph
17
69)
The equation gives the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds). 70) s = 9 + 5 cos t Find the body's acceleration at time t = /3 sec. A) - 5 m/sec2 B) - 5 3 m/sec2 C) 5 m/sec2 D) 5 3 m/sec2 2 2 2 2
70)
Find y .
71) y = ( x - 5)-3
71)
A) 6( x - 5)-5
B) -
C) - 3 ( x - 5)-5 5 - 3 2x
3 2 x
( x - 5)-4
D) 3 ( x - 5)-5 - 5 + 5 4x
x
x
Use implicit differentiation to find dy/dx. 72) y cos 1 = 7x + 7y y
A)
C)
72)
7y2 1 sin - 7y2 y
B)
7 1 1 sin + y cos -7 y y 7 - y sin
7y 1 1 sin + y cos - 7y y y
D) cos
1 y
1 -7 y
Find the second derivative of the function. 73) p = q + 2 q + 6 q q2
73)
2
2
A) d p = - 2 - 48 - 144 dq2
q3
q4
B) d p = 2 + 48 + 144
q5
dq2
2
q2
q3
q2
q3
2
C) d p = - 1 - 16 - 36 dq2
q
D) d p = 2 + 48 + 144
q4
dq2
q3
q4
q5
Find the limit.
74) lim sec -2 t t 0
A) 0
74)
sin t
B) - 1
C) 1
2
18
D) -1
Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 75) u(1) = 4, u (1) = -5, v(1) = 7, v (1) = -3. 75) d (uv) at x = 1 dx
B) 47
A) -47
C) 23
Find an equation of the tangent line at the indicated point on the graph of the function. 76) w = g(z) = 1 + 7 - z, (z, w) = (6, 2) A) w = 1 z - 5 B) w = - 1 z - 5 C) w = 1 z + 5 2 2 2
D) -41
76) D) w = - 1 z + 5 2
Solve the problem. 77) Find an equation for the horizontal tangent to the curve at P.
A) y = 3 2
B) y = 2
C) y = 1
77)
D) y = 0
Provide an appropriate response. 78) The curves y = ax2 + b and y = 2x2 + cx have a common tangent line at the point (-1, 0). Find a, b,
78)
and c.
A) a - 2, b = 1, c = -1 C) a = 1, b = -1, c = 2
B) a = 1, b = 0, c = 2 D) a = -1, b = 1, c = -2
Find the indicated derivative. 79) dy if y = 5x3 dx
A) 15x
79) B) 15x3
C) 3x2
19
D) 15x2
The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? 80) x = 1 80)
A) Differentiable B) Continuous but not differentiable C) Neither continuous nor differentiable Solve the problem.
81) Find equations of all tangents to the curve f(x) = 36 that have slope -1.
81)
x
A) y = x + 12, y = x - 12 C) y = -x + 12 Provide an appropriate response. 82) Find the slope of the tangent to the curve y = A) m = 1 B) m = 1 12 6
B) y = x - 12 D) y = -x + 12, y = -x - 12
x at the point where x = 9. C) m = - 1 12
82) D) m = - 1 6
The equation gives the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds). 83) s = -8 + 3 cos t Find the body's jerk at time t = /3 sec. A) 3 3 m/sec3 B) 3 m/sec3 C) - 3 3 m/sec3 D) - 3 m/sec3 2 2 2 2 Find y .
84) y = 6 + 4
4
84)
x
A) - 48 6 + 4
2
C) 192 6 + 4
2
x2
x4
83)
x
x
+
32 4 3 6+ x x3
B) - 16 6 + 4
+
32 4 3 6+ 3 x x
D) 12 6 + 4
x2
x
x
20
2
3
Use implicit differentiation to find dy/dx. 85) x + y = x2 + y2 x-y 2
85) 2
A) x(x - y) + y
2
B) x(x - y) - y
x - y(x - y)2
2
C) x(x - y) - y
x + y(x - y)2
D) x(x - y) + y
x - y(x - y)2
x + y(x - y)2
Solve the problem. 86) Find all points on the curve y = cos x, 0 x 2 , where the tangent line is parallel to the line y = -x. A) , 0 B) (2 , 1) C) 3 , 0 D) ( , - 1) 2 2
87) Find an equation for the tangent to the curve at P.
A) y = 2x -
4
2
+1+
87)
2
B) y = - 2x +
C) y = - 2x - 1 - 2
4
2
-1-
2
D) y = 2x + 1 + 2
88) Does the graph of the function y = tan x - x have any horizontal tangents in the interval 0 x 2 ? If so, where?
A) Yes, at x =
2
,x=
86)
3 2
88)
B) No
C) Yes, at x = 0, x = , x = 2
D) Yes, at x =
Find the derivative. 89) y = 9 + 8 sec x x
89)
A) y = - 9 + 8 sec x tan x
B) y = - 9 + 8 tan2 x
C) y = - 9 - 8 csc x
D) y = 9 - 8 sec x tan x
x2
x2
x2
x2
21
Find the linearization L(x) of f(x) at x = a. 90) f(x) = x + 1 , a = 3 x
A) L(x) = 10 x + 1 9
2
90)
B) L(x) = 10 x + 2 9
C) L(x) = 8 x + 2
3
9
Solve the problem. 91) Find the tangent to y = 2 - sin x at x = . A) y = - x + - 2 B) y = x - + 2
3
D) L(x) = 8 x + 1 9
2
91) C) y = x - 2
D) y = - x + 2
92) A cube 7 inches on an edge is given a protective coating 0.3 inches thick. About how much coating should a production manager order for 500 cubes? A) About 44,100 in.3
C) About 7350 in.2
92)
B) About 22,050 in.2 D) About 51,450 in.3
Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x. 93) y = csc(cot x) A) y = csc u; u = cot x; dy = csc(cot x) cot(cot x) csc2 x dx
93)
B) y = cot u; u = csc x; dy = csc2 (csc x) csc x cot x dx
C) y = csc u; u = cot x; dy = csc3 x cot x dx
D) y = csc u; u = cot x; dy = - csc(cot x) cot(cot x) dx
Use the linear approximation (1 + x)k 1 + kx, as specified.
94) Find an approximation for the function f(x) = A) f(x)
1 x 110 5
C) f(x)
51-
1 for values of x near zero. 5+x
94)
B) f(x) 5 - x 2
x 10
D) f(x)
1 x 15 2 5
Solve the problem. 95) A rock is thrown vertically upward from the surface of an airless planet. It reaches a height of s = 120t - 10t2 meters in t seconds. How high does the rock go? How long does it take the rock to reach its highest point? A) 720 m, 12 sec
B) 360 m, 6 sec
C) 714 m, 6 sec
D) 1320 m, 12 sec
96) The equation for free fall at the surface of Planet X is s = 7.54t2 m with t in seconds. Assume a
rock is dropped from the top of a 600m cliff. Find the speed of the rock at t = 2 sec. A) 31.16 m/sec B) 30.16 m/sec C) 15.08 m/sec D) 14.08 m/sec
22
95)
96)
Find the derivative of the function. -1
97) y = 1 (9x + 9)3 + 1 - 1 6
97)
x3
A) 9 (9x + 9)2 - 3 1 - 1
-2
C) 3 (9x + 9)2 + 3 1 - 1
-2
2
x4
2
B) 1 (9x + 9)2 - 1 - 1
x3
x4
2
-2
x3
-2 D) 1 (9x)2 - 3
x3
2
x4
Solve the problem. 98) The range R of a projectile is related to the initial velocity v and projection angle by the equation v2 sin 2 R= , where g is a constant. How is dR/dt related to d /dt if v is constant? g 2 A) dR = v cos 2 d
2 B) dR = 2v cos 2 d
2 C) dR = - v cos 2 d
2 D) dR = 2v sin 2 d
dt dt
g
dt
g
Find the indicated derivative. 99) Find y if y = -2 cos x. A) y = -2 sin x
dt
dt
dt
g g
98)
dt
dt
99) B) y = 2 sin x
C) y = 2 cos x
D) y = -2 cos x
Find y .
100) y = (x2 - 4x + 2)(2x3 - x2 + 5) A) 2x4 - 36x3 + 24x2 + 6x - 20 C) 10x4 - 32x3 + 24x2 + 6x - 20
100) B) 10x4 - 36x3 + 24x2 + 6x - 20 D) 2x4 - 32x3 + 24x2 + 6x - 20
The function f(x) changes value when x changes from x 0 to x 0 + dx. Find the approximation error
f - df . Round your
answer, if appropriate. 101) f(x) = x + x2, x0 = 5, dx = 0.06
A) 0.0072
101)
B) 0.0036
C) 0.26544
23
D) 0.13272
The graphs show the position s, velocity v = ds/dt, and acceleration a = d2 s/dt2 of a body moving along a coordinate line as functions of time t. Which graph is which?
102)
102)
A) A = position, C = velocity, B = acceleration B) C = position, A = velocity, B = acceleration C) A = position, B = velocity, C = acceleration D) B = position, A = velocity, C = acceleration Find the slope of the curve at the indicated point. 103) y= x2 + 7x, x = -4
A) m = -8
103)
B) m = -12
C) m = 3
D) m = -1
Find the derivative.
104) y = 11 - 4x2 A) 11 - 4x
104) B) -8x
C) -8
D) 11 - 8x
Solve the problem. 105) A piece of land is shaped like a right triangle. Two people start at the right angle of the triangle at the same time, and walk at the same speed along different legs of the triangle. If the area formed by the positions of the two people and their starting point (the right angle) is changing at 3 m2 /s, then how fast are the people moving when they are 3 m from the right angle? (Round your answer to two decimal places.) A) 0.50 m/s B) 1.00 m/s C) 3.00 m/s D) 2.00 m/s
24
105)
Compare the right-hand and left-hand derivatives to determine whether or not the function is differentiable at the point whose coordinates are given.
106)
106)
y=
1 x
y = -1
A) Since limx -1 + f (x) = -1 while limx -1- f (x) = 0, f(x) is not differentiable at x = -1. B) Since limx -1 + f (x) = 0 while limx -1 - f (x) = -1, f(x) is not differentiable at x = -1. C) Since limx -1 + f (x) = 0 while limx -1 - f (x) = 1, f(x) is not differentiable at x = -1. D) Since limx -1 + f (x) = 0 while limx -1 - f (x) = 0, f(x) is differentiable at x = -1. Provide an appropriate response.
107) Find all points (x, y) on the graph of f(x) = 2x2 -3x with tangent lines parallel to the line y = 9x + 7. A) (0, 0), (3, 9) B) (3, 18) C) (6, 9) D) (3, 9)
Solve the problem. 108) A heat engine is a device that converts thermal energy into other forms. The thermal efficiency, e, of a heat engine is defined by Qh - Qc e= , Qh where Qh is the heat absorbed in one cycle and Qc, the heat released into a reservoir in one cycle, is a constant. Find
de . dQh
Qc A) de = -
B) de = 1
Qc C) de =
D) de = Qh - Qc
dQh dQh
Qh 2
dQh
dQh
Qh 2
25
Qh 2
107)
108)
At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 109) 5x2 y - cos y = 6 , normal at (1, ) 109)
A) y = -2 x + 3
B) y = - 1 x + 1 +
C) y = 1 x - 1 +
D) y = 1 x - 1 + 2
2
Find the derivative of the function. 110) y = (x + 5)(x + 1) (x - 5)(x - 1)
110)
A) y =
12x2 - 60 (x - 5)2 (x - 1)2
B) y =
12x - 60 (x - 5)2 (x - 1)2
C) y =
-x2 + 10 (x - 5)2 (x - 1)2
D) y =
-12x2 + 60 (x - 5)2 (x - 1)2
The function s = f(t) gives the position of a body moving on a coordinate line, with s in meters and t in seconds. 111) s = 10t - t2 , 0 t 10 111) Find the body's speed and acceleration at the end of the time interval. A) 10 m/sec, - 2 m/sec2 B) -10 m/sec, - 2 m/sec2
C) 30 m/sec, -20 m/sec2
D) 10 m/sec, -20 m/sec2
Solve the problem. 112) The driver of a car traveling at 54 ft/sec suddenly applies the brakes. The position of the car is s = 54t - 3t2 , t seconds after the driver applies the brakes. How many seconds after the driver applies the brakes does the car come to a stop? A) 9 sec B) 27 sec
C) 18 sec
D) 54 sec
Find the derivative of the function. 8 6 113) p = q + 4 q + 6 2q q
113)
A) dp = 1 q11 + 2q3 + 3q5 + 24
B) dp = 6q11 + 8q3 + 18q5 - 24
C) dp = 6q11 - 24
D) dp = 8q15 + 16q7 + 30q9 - 24
dq dq
2
112)
q3
dq
q3
dq
q3 q3
Solve the problem.
114) Find equations of all tangents to the curve f(x) = 1
x+1
A) y = -x + 1, y = -x - 3 C) y = -x + 3
that have slope -1.
B) y = -x + 3, y = -x + 1 D) y = -x + 1
26
114)
The equation gives the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds). 115) s = 8 sin t - cos t Find the body's velocity at time t = /4 sec. A) - 7 2 m/sec B) 7 2 m/sec C) 9 2 m/sec D) - 9 2 m/sec 2 2 2 2 Find the derivative. 116) y = 1 + 1 7x2 7x
116)
A) - 1 + 1 7x3
115)
B) - 2 - 1
7x2
7x
C) 2 + 1
7x2
7x3
D) - 2 - 1
7x2
7x3
7x2
Find dy.
117) 6y1/2 - 2xy + x = 0
117)
A)
-1 dx 3y-1/2 - 2x
C)
2y - 1 dx 3y-1/2 - 2x
B)
2y - 1 dx 3y-1/2 + 2x
D) 2y - 1 dx 6y - 2x
Solve the problem.
118) Suppose that the dollar cost of producing x radios is c(x) = 600 + 30x - 0.2x2 . Find the average cost per radio of producing the first 35 radios. A) $1355.00 B) $1405.00
C) $40.14
D) $805.00
119) Does the graph of the function y = 4x + 2 sin x have any horizontal tangents in the interval 0 x 2 ? If so, where? A) Yes, at x = 2 3
C) Yes, at x =
3
,x=
3
2 3
Find the derivative of the function. 2 121) y = (x - 8)(x + 2x) x3
121)
A) dy = 10 - 32
B) dy = 6 + 32
C) dy = x - 32 - 32
D) dy = 32 + 32
x3
x2
120)
B) ( 2, -12 2) D) (-2, 4), ( 2, -4)
C) (- 2, 12 2), (0,0), ( 2, -12 2)
dx
3
D) No
f(x) = x3 - 6x A) (- 2, 4 2), ( 2, -4 2)
x2
119)
B) Yes, at x = 2 , x = 4
120) Find the points where the graph of the function have horizontal tangents.
dx
118)
dx
dx
x3
27
x2
x3
x
Find the derivative. 122) y = 17x-2 - 5x3 + 13x
122)
A) -34x-3 - 15x2 C) -34x-3 - 15x2 + 13
B) -34x-1 - 15x2 + 13 D) -34x-1 - 15x2
Solve the problem. 123) The position(in feet) of an object oscillating up and down at the end of a spring is given by k s = A sin t at time t (in seconds). The value of A is the amplitude of the motion, k is a m
123)
measure of the stiffness of the spring, and m is the mass of the object. Find the object's velocity at time t. A) v = - A k cos k t ft/sec B) v = A m cos k t ft/sec m m k m
C) v = A cos
k t ft/sec m
D) v = A k cos m
k t ft/sec m
Solve the problem. Round your answer, if appropriate. 124) As the zoom lens in a camera moves in and out, the size of the rectangular image changes. Assume that the current image is 7 cm × 6 cm. Find the rate at which the area of the image is changing (dA/dt) if the length of the image is changing at 0.7 cm/s and the width of the image is changing at 0.2 cm/s. A) 5.6 cm2 /sec B) 11.2 cm2 /sec C) 6.1 cm2 /sec D) 12.2 cm2 /sec
124)
Solve the problem.
125) The size of a population of lions after t months is P = 100 (1 + 0.2t + 0.02t2 ). Find the growth rate when P = 2500. A) 180 lions/month
B) 160 lions/month D) 10,020 lions/month
C) 140 lions/month
28
125)
The graphs show the position s, velocity v = ds/dt, and acceleration a = d2 s/dt2 of a body moving along a coordinate line as functions of time t. Which graph is which?
126)
126)
A) A = position, C = velocity, B = acceleration B) A = position, B = velocity, C = acceleration C) B = position, A = velocity, C = acceleration D) C = position, A = velocity, B = acceleration Find the slope of the curve at the indicated point. 127) y = -2x + 1, x = 3 A) m = -6 B) m = 2
127) C) m = 6
D) m = -2
Solve the problem.
128) Find an equation of the tangent to the curve f(x) = 2x2 - 2x + 1 that has slope 2. A) y = 2x B) y = 2x + 2 C) y = 2x + 1 D) y = 2x - 1
Use the formula f'(x) = lim z x
129) g(x) = 2x + x A)
1
2 x
128)
f(z) - f(x) to find the derivative of the function. z-x
129) B) 2 + 1 x
C) 2 -
1 2 x
D) 2 +
1 2 x
Estimate the slope of the curve at the indicated point.
130)
130)
A) -1
B) 1
C) 0
29
D) Undefined
Solve the problem. 131) About how accurately must the interior diameter of a cylindrical storage tank that is 8 m high be measured in order to calculate the tank's volume within 0.5% of its true value? A) Within 0.25% B) Within 0.5 meters
C) Within 0.25 meters
131)
D) Within 0.5%
132) The area A = r2 of a circular oil spill changes with the radius. At what rate does the area change with respect to the radius when r = 9 ft? A) 18 ft2/ft B) 18 ft2 /ft
C) 81 ft2 /ft
132)
D) 9 ft2 /ft
Find dy.
133) y = 7 x - 5
133)
x
A)
7 2 x
+
5 dx x2
B) 7 x + 5 dx 2
C) 7 x - 5 dx
x2
2
x2
D)
7 2 x
-
5 dx x2
Solve the problem.
134) At time t, the position of a body moving along the s-axis is s = t3 - 15t2 + 72t m. Find the body's acceleration each time the velocity is zero. A) a(6) = 0 m/sec2 , a(4) = 0 m/sec2
134)
B) a(6) = 6 m/sec2, a(4) = -6 m/sec2 D) a(12) = 72 m/sec2, a(8) = 12 m/sec2
C) a(6) = -6 m/sec2 , a(4) = 6 m/sec2
You want a linearization that will replace the function over an interval that includes the point x 0 . To make your subsequent work as simple as possible, you want to center the linearization not at x 0 but at a nearby integer x = a at which the function and its derivative are easy to evaluate. What linearization do you use? 135) f(x) = -4x2 - 2x + 7, x0 = 1.1
A) -11 - 10x
B) 11 - 10x
C) 1
135)
D) 12 - 11x
Solve the problem. 136) The elasticity of a particular thermoplastic can be modeled approximately by the relation 2.5x105 , where T is the Kelvin temperature. If the thermometer used to measure T is accurate = T2.3
136)
to 1%, and if the measured temperature is 476 K, how should the elasticity be reported? A) = ± 0.004 B) = 0.174 ± 0.002
C) = 0.174 ± 0.004
D) = 0.174
Find the derivative.
137) w = z -2 - 1
137)
z
A) 2z -3 - 1
z2
B) z-3 + 1
C) -2z -3 - 1
z2
z2
30
D) -2z -3 + 1
z2
138) y = 4x4 + 6x3 + 3 A) 16x3 + 18x2 - 7 C) 4x3 + 3x2
138) B) 16x3 + 18x2 D) 4x3 + 3x2 - 7
Differentiate the function and find the slope of the tangent line at the given value of the independent variable. 139) g(x) = 5 , x = 8 139) 4+x
A) - 1.759218604e+14
B) 1.759218604e+14
C) 5
D) - 5
5.066549581e+15
5.066549581e+15
12
12
Solve the problem. 140) The position (in centimeters) of an object oscillating up and down at the end of a spring is given k by s = A sin t at time t (in seconds). The value of A is the amplitude of the motion, k is a m
140)
measure of the stiffness of the spring, and m is the mass of the object. How fast is the object accelerating when it is accelerating the fastest? A) A2 cm/sec2 B) A k cm/sec2 C) A cm/sec2 D) Ak cm/sec2 m m
The graphs show the position s, velocity v = ds/dt, and acceleration a = d2 s/dt2 of a body moving along a coordinate line as functions of time t. Which graph is which?
141)
141)
A) B = position, A = velocity, C = acceleration B) C = position, A = velocity, B = acceleration C) A = position, B = velocity, C = acceleration D) A = position, C = velocity, B = acceleration Find the value of (f g) at the given value of x. 142) f(u) = u , u = g(x) = 9x2 + x + 3, x = 0 u2 - 1
A) 1
8
142)
B) - 5
C) 5
32
32
31
D) 13 32
Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values of x. Find the derivative with respect to x of the given combination at the given value of x. x f(x) g(x) f (x) g (x) 143) 3 1 16 143) 6 3 -4 4 3 3 2 f2(x) · g(x), x = 3 A) 35
B) 36
C) 99
D) 195
Find an equation for the tangent to the curve at the given point. 144) f(x) = 10 x - x + 1, (100, 1) A) y = 1 B) y = - 1 x + 51 C) y = 1 x - 51 2 2
144) D) y = - 1 x + 1 2
The function f(x) changes value when x changes from x 0 to x 0 + dx. Find the approximation error
f - df . Round your
answer, if appropriate. 145) f(x) = x2 - x, x0 = 2, dx = 0.04
A) 0.0032
145)
B) 0.0016
C) 0.0816
D) 0.08
Write a differential formula that estimates the given change in volume or surface area. 146) The change in the surface area S = 4 r2 of a sphere when the radius changes from r0 to r0 + dx A) dS = 8 r0 dr B) dS = 4 r0 dr C) dS = 4 r0 2 dr D) dS = 2 r0 dr
146)
Provide an appropriate response.
147) The line that is normal to the curve x2 - xy + y2 = 9 at (3, 3) intersects the curve at what other point? A) (0, -3)
B) (-3, 0)
C) (-3, -3)
D) (-6, -6)
Solve the problem. 148) The range R of a projectile is related to the initial velocity v and projection angle by the equation v2 sin 2 R= , where g is a constant. How is dR/dt related to dv/dt and d /dt if neither v nor is g constant? A) dR = v v cos 2 d + 2 sin 2 dv dt g dt dt
B) dR = 2v v cos 2 d + sin 2 dv dt
C) dR = 1 4v cos 2 d dv dt
g
g
dt
dt
D) dR = 1 v cos 2 dv + sin 2 d
dt dt
dt
32
g
147)
dt
dt
148)
Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x. 149) y = tan - 10 x
A) y = tan u; u = - 10 ; dy = 10 sec2
-
B) y = tan u; u = - 10 ; dy = 10 sec
-
x x
dx dx
x2 x2
C) y = tan u; u = - 10 ; dy = sec2 x
-
dx
149)
10 x
10 tan x
-
10 x
10 x
D) y = tan u; u = - 10 ; dy = sec2 10 x
dx
x2
The function f(x) changes value when x changes from x 0 to x 0 + dx. Find the approximation error
f - df . Round your
answer, if appropriate. 150) f(x) = x , x0 = 8, dx = 0.05
A) -0.00885
150) B) 0.00885
C) 0.00001
D) -0.00001
The function s = f(t) gives the position of a body moving on a coordinate line, with s in meters and t in seconds. 151) s = - t3 + 5t2 - 5t, 0 t 5 151) Find the body's speed and acceleration at the end of the time interval. A) 30 m/sec, -20 m/sec2 B) 30 m/sec, -5 m/sec2
C) 5 m/sec, 0 m/sec2
D) -30 m/sec, -20 m/sec2
Given the graph of f, find any values of x at which f is not defined.
152)
152)
A) x = 0 C) x = -1, 1
B) x = -1, 0, 1 D) Defined for all values of x
Graph the curve over the given interval together with its tangent at the given value of x. Graph the tangent with a dashed line.
33
153) y = cos(x), -
x 2 ,x=
153)
2
A)
B)
C)
D)
34
The figure shows the velocity v or position s of a body moving along a coordinate line as a function of time t . Use the figure to answer the question. 154) v (ft/sec) 154)
t (sec)
What is the body's acceleration when t = 4 sec? A) 1 ft/sec2 B) 2 ft/sec2
C) -2 ft/sec2
Find the derivative of the function. 155) f(t) = (6 - t)(6 + t3 )-1 3 2 A) f (t) = 2t - 18t - 6 6 + t3 3
D) 0 ft/sec2
155) 3 2 B) f (t) = - 2t + 18t - 6 2 (6 + t3 )
2
3
C) f (t) = - 4t + 18t - 6
2
D) f (t) = 2t - 18t - 6
2 (6 + t3 )
2 (6 + t3 )
Solve the problem.
2
156) Suppose that the revenue from selling x radios is r(x) = 85x - x dollars. Use the function r (x) to 10
156)
estimate the increase in revenue that will result from increasing production from 110 radios to 111 radios per week. A) $74.00 B) $107.00 C) $62.80 D) $63.00
Find the derivative. 157) p = sec q + csc q csc q
157)
A) dp = - csc q cot q
B) dp = sec2 q + 1
C) dp = sec2 q
D) dp = sec q tan q
dq
dq
dq
dq
35
Solve the problem. 158) Does the graph of the function y = 4x + 8 sin x have any horizontal tangents in the interval 0 x 2 ? If so, where? A) Yes, at x = 2 , x = 4 B) No 3 3
C) Yes, at x = 2
D) Yes, at x =
3
3
,x=
158)
2 3
Estimate the slope of the curve at the indicated point.
159)
159)
B) Undefined
A) -1
C) 1
D) 0
Calculate the derivative of the function. Then find the value of the derivative as specified. 160) f(x) = 8 ; f (-1) x
A) f (x) = - 8 ; f (-1) = -8
B) f (x) = 8; f (-1) = 8
C) f (x) = - 8x2 ; f (-1) = - 8
D) f (x) = 8 ; f (-1) = 8
x2
160)
x2
Solve the problem.
161) Find the tangent to y = cot x at x = A) y = -2x + C) y = -2x +
4
.
161) B) y = 2x + 1
2 2
+1
D) y = 2x -
2
+1
162) The position (in centimeters) of an object oscillating up and down at the end of a spring is given by s = A sin
k t at time t (in seconds). The value of A is the amplitude of the motion, k is a m
measure of the stiffness of the spring, and m is the mass of the object. Find the object's acceleration at time t. A) a = - A sin k t cm/sec2 B) a = - A k sin k t cm/sec2 m m m
C) a = Ak cos m
k t cm/sec2 m
D) a = - Ak sin m
36
k t cm/sec2 m
162)
Find the derivative of the function. 2 163) y = x + 8x + 3 x
163) 2
B) y = 3x + 8x - 3
A) y = 2x + 8 x
x
2
C) y = 3x + 8x - 3
D) y = 2x + 8 2x3/2
2x3/2
Find dy.
164) y = 4x2 + 5x + 2 A) 8x + 10 dx
164) B) 8x + 2 dx
C) (8x + 5) dx
D) 8x dx
Solve the problem. 165) A company knows that the unit cost C and the unit revenue R from the production and sale of x R2 units are related by C = + 10,827. Find the rate of change of unit revenue when the unit 300,000 cost is changing by $10/unit and the unit revenue is $4000. A) $728.85/unit B) $375.00/unit C) $1082.70/unit
165)
D) $200.00/unit
At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 166) y5 + x3 = y2 + 11x, slope at (0, 1) 166)
B) 11
A) - 3
C) 11
3
D) 11
5
7
Find the derivative of the function. 2 167) y = x + 2x - 2 x2 - 2x + 2
167)
A) y =
-4x2 + 8x (x2 - 2x + 2)2
B) y =
-4x2 - 8x (x2 - 2x + 2)2
C) y =
4x2 + 8x (x2 - 2x + 2)2
D) y =
4x2 - 8x (x2 - 2x + 2)2
Use implicit differentiation to find dy/dx. 168) y x + 1 = 4 A) 2y B) y x+1 2(x + 1)
168) C) - y 2(x + 1)
D) - 2y x+1
Solve the problem. 169) The number of gallons of water in a swimming pool t minutes after the pool has started to drain is Q(t) = 50(20 - x)2 . How fast is the water running out at the end of 11 minutes?
A) 450 gal/min
B) 4050 gal/min
C) 2025 gal/min
37
D) 900 gal/min
169)
Solve the problem. Round your answer, if appropriate.
170) The volume of a sphere is increasing at a rate of 2 cm3 /sec. Find the rate of change of its surface area when its volume is
A) 2 cm2 /sec
32 3
cm3 . (Do not round your answer.)
C) 8 cm2 /sec
B) 4 cm2/sec
D) 4 cm2 /sec
3
3
Find the second derivative of the function. 2 171) y = (x - 10)(x + 2x) x3
171)
2
2
A) d y = - 16 - 120 dx2
x
B) d y = 8 + 40
x2
dx2
2
x3
x2
x3
2
C) d y = - 16 - 120 dx2
170)
D) d y = 16 + 120
x4
dx2
x3
x4
Find dy.
172) y = csc(2x2 - 1) A) -4x csc(4x) cot(4x) dx C) -4x csc(2x2 - 1) cot(2x2 - 1) dx
172) B) -2x2 csc(2x2 - 1) cot(2x2 - 1) dx D) 4x csc(2x2 - 1) cot(2x2 - 1)dx
Find the derivative of the function. 173) r = 1 + 6 (6 - ) 6
A) dr = 1 + 1 d
2
173) C) dr = 2 - 1
B) dr = - 1 - 1 d
2
d
D) dr = 1 + 6 d
2
Provide an appropriate response.
174) Find the normal to the curve x2 + y2 = 2x + 2y that is parallel to the line y + x = 0. A) y = x + 2 B) y = -x + 2 C) y = -x - 2 D) y = x - 2
Graph the equation and its tangent. 175) Graph y = -4x2 and the tangent to the curve at the point whose x-coordinate is -2.
38
174)
175)
A)
B)
C)
D)
The graphs show the position s, velocity v = ds/dt, and acceleration a = d2 s/dt2 of a body moving along a coordinate line as functions of time t. Which graph is which?
176)
176)
A) B = position, C = velocity, A = acceleration B) C = position, B = velocity, A = acceleration C) B = position, A = velocity, C = acceleration D) C = position, A = velocity, B = acceleration Find the linearization L(x) of f(x) at x = a. 177) f(x) = tan x, a = A) L(x) = 2x B) L(x) = x +
177) C) L(x) = x - 2 39
D) L(x) = x -
The figure shows the velocity v or position s of a body moving along a coordinate line as a function of time t . Use the figure to answer the question. 178) v (ft/sec) 178)
t (sec)
What is the body's greatest velocity? A) 4 ft/sec B) 3 ft/sec
C) 2 ft/sec
D) 5 ft/sec
Solve the problem.
179) The position of a body moving on a coordinate line is given by s = t2 - 10t + 10, with s in meters
179)
and t in seconds. When, if ever, during the interval 0 t 10 does the body change direction? A) t = 5 sec B) t = 20 sec
C) t = 10 sec
D) no change in direction
Find the derivative of the function. 2 180) y = x - 3x + 2 x7 - 2 8
7
180) 6
8
A) y = -5x + 18x - 13x - 4x + 6 8
7
7
6
B) y = -5x + 18x - 14x - 3x + 6
(x7 - 2)2
(x7 - 2)2
6
8
C) y = -5x + 18x - 14x - 4x + 6
7
6
D) y = -5x + 19x - 14x - 4x + 6
(x7 - 2)2
(x7 - 2)2
Solve the problem. 181) A wheel with radius 3 m rolls at 16 rad/s. How fast is a point on the rim of the wheel rising when the point is /3 radians above the horizontal (and rising)? (Round your answer to one decimal place.) A) 96.0 m/s B) 48.0 m/s C) 24.0 m/s D) 12.0 m/s
40
181)
Find the derivative of the function. -2 182) r = +2
A) r = C) r =
182) B) r = 2
4
+2
2-4
( + 2)
2
D) r = -
( + 2)2
2
( + 2)2
Given the graph of f, find any values of x at which f is not defined.
183)
183)
A) x = -2, 2 C) x = 0
B) x = -2, 0, 2 D) Defined for all values of x
Use implicit differentiation to find dy/dx and d2 y/dx 2 . 184) y2 - x2 = 7
184)
2 2 2 A) dy = x ; d y = y - x dx y dx2 y2 2
2
C) dy = - x ; d y = y - x dx
y dx2
2
A)
5y4
2
D) dy = x ; d y = y - x
y3
Use implicit differentiation to find dy/dx. 185) cos xy + x5 = y5 5x4 - x sin xy
2 2 2 B) dy = x ; d y = y - x dx y dx2 y3
B)
dx
y dx2
2
y2
185) 5x4 + x sin xy
C)
5y4
5x4 + y sin xy 5y4 - x sin xy
D)
5x4 - y sin xy 5y4 + x sin xy
Find the second derivative of the function. 4 186) y = x + 4 x2 2
A) d y = 2 + 24 dx2
x4
186) 2
2
B) d y = 1 + 24 dx2
C) d y = 2x - 8
x4
dx2
41
x3
2
D) d y = 2 - 24 dx2
x4
Given the graph of f, find any values of x at which f is not defined.
187)
187)
A) x = -2, 0, 2
B) x = 2
C) x = -2, 2
D) x = 0
Find the value of (f g) at the given value of x. 188) f(u) = 1 + 14, u = g(x) = 1 , x = 9 u x2 - 14
A) 9
188)
B) 18
C) 81
Use implicit differentiation to find dy/dx and d2 y/dx 2 . 189) x2 + y2 = 3
D) -32
189)
2 2 2 A) dy = - x ; d y = - x + y dx y dx2 y2
2 2 2 B) dy = - x ; d y = - x + y dx y dx2 y3
2
2
y dx2
y2
D) dy = - x ; d y = - x + y
2
C) dy = x ; d y = x - y dx
dx
y dx2
Find the second derivative of the function. 190) r = 1 + 4 (4 - ) 4
y3
190)
2
2
A) d r = 1 -
B) d r = - 1 - 1
d 2
d 2
2
2
2
C) d r = - 2 - 1 d 2
2
D) d r = 2
3
d 2
Find the derivative. 191) s = t5 cos t - 9t sin t - 9 cos t
3
191)
A) ds = - t5 sin t + 5t4 cos t - 9t cos t - 18 sin t dt
B) ds = - t5 sin t + 5t4 cos t - 9t cos t dt
C) ds = - 5t4 sin t - 9 cos t + 9 sin t dt
D) ds = t5 sin t - 5t4 cos t + 9t cos t dt
42
Find the derivative of the function. 3 192) y = x x-1 3
A) y = -2x - 3x
192)
2
3
B) y = 2x - 3x
(x - 1)2 3
C) y = 2x + 3x
2
(x - 1)2
2
3
D) y = -2x + 3x
(x - 1)2
2
(x - 1)2
Calculate the derivative of the function. Then find the value of the derivative as specified. 193) ds t = 4 if s = t2 - t dt
A) ds = 2t + 1; ds t = 4 = 9
B) ds = 2t - 1; ds t = 4 = 7
C) ds = t - 1; ds t = 4 = 3
D) ds = 2 - t; ds t = 4 = -2
dt
dt
dt
dt
dt
193)
dt
dt
dt
Given the graph of f, find any values of x at which f is not defined.
194)
194)
A) x = 5 C) x = 2
B) x = 2, 5 D) Defined for all values of x
Find dy/dt.
195) y = cos7 ( t - 13) A) -7 cos6( t - 13) sin( t - 13) C) -7 sin6 ( t - 13)
195) B) -7 cos6 ( t - 13) sin( t - 13) D) 7 cos6( t - 13)
At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 196) x6 y6 = 64, tangent at (2, 1) 196)
A) y = 1 x
B) y = - 1 x + 2
2
C) y = -32x + 1
2
D) y = 32x - 1
Solve the problem.
197) Find the tangent to y = cos x at x = A) y = x +
2
2
.
197) C) y = - x +
B) y = 1
43
2
D) y = - x -
2
Find the second derivative. 198) y = 1 + 1 7x2 3x
A) 6 + 2 7x4
3x3
198) B) 6 - 2 7x4
C) - 2 - 1
3x3
7x3
3x2
D) - 2 + 1 7x4
3x3
Solve the problem. 199) Assume that the profit generated by a product is given by P(x) = 4 x, where x is the number of units sold. If the profit keeps changing at a rate of $900 per month, then how fast are the sales changing when the number of units sold is 200? (Round your answer to the nearest dollar per month.) A) $6364/month B) $101,823/month
C) $3182/month
D) $32/month
200) Suppose that the radius r and the circumference C = 2 r of a circle are differentiable functions of t. Write an equation that relates dC/dt to dr/dt. A) dr = 2 dC B) dC = 2 r dr dt dt dt dt
C) dC = 2 dr
D) dC = dr
C) 4
D) 0
dt
dt
dt
201) B) 2
202) r = 2 - 5 s3
202)
s
A) 24 + 10 s5
200)
dt
Find the second derivative. 201) y = 2x2 + 9x - 6
A) 4x + 9
199)
s3
B) - 6 + 5
C) 24 - 10
D) 2 - 5
B) 15x + 6 dx 5x + 3
C) 15x - 6 dx 5x + 3
D) 15x - 6 dx 2 5x + 3
s4
s2
s5
s3
s5
s3
Find dy.
203) y = x 5x + 3
A) 15x + 6 dx 2 5x + 3
203)
Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values of x. Find the derivative with respect to x of the given combination at the given value of x. x f(x) g(x) f (x) g (x) 204) 3 1 9 204) 6 5 4 -3 3 2 -5 f(x) + g(x), x = 3 A) 11 2 10
B) -
1 2 10
C) 11
10
44
D)
1 2 10
Solve the problem. 205) A cubic salt crystal expands by accumulation on all sides. As it expands outward find the rate of change of its volume with respect to the length of an edge when the edge is 0.32 mm. A) 3.07 mm3 /mm B) 0.10 mm3 /mm
C) 30.72 mm3 /mm
205)
D) 0.3072 mm3 /mm
Find dy.
206) y = 2 cot 1 x4
206)
4
A) -2x3 csc2 1 x4 dx
B) -2x4 csc2 1 x4 dx
C) -2x3 csc 1 x4 dx
D) 2x3 csc2 1 x4 dx
4
4
4
4
Graph the equation and its tangent. 207) Graph y = -3x2 and the tangent to the curve at the point whose x-coordinate is 1.
A)
B)
45
207)
C)
D)
Find dy.
208) 3x2 y - 6x1/2 - y = 0 A)
3x-1/2 + 6xy 3x2 + 1
C) 3x
-1/2 - 6xy 3x2 + 1
208) 3x-1/2 + 6xy
dx
B)
dx
D) 3x
3x2 - 1
-1/2 - 6xy 3x2 - 1
dx dx
Given the graph of f, find any values of x at which f is not defined.
209)
209)
A) x = 1
B) x = 0
C) x = 2
D) x = 0, 1, 2
The function f(x) changes value when x changes from x 0 to x 0 + dx. Find the approximation error
f - df . Round your
answer, if appropriate. 210) f(x) = 1 , x0 = 2, dx = 0.5 x
A) 0.025
210) B) 0.225
C) 0.55
D) 0.31
Find the value of (f g) at the given value of x. 211) f(u) = 1 , u = g(x) = 8x - x2 , x = 1 u
A) 6
49
211)
B) - 1
C) - 6
6
49
Given y = f(u) and u = g(x), find dy/dx = f (g(x))g (x). 212) y = tan u, u = -17x + 15 A) - sec2 (-17x + 15)
D) 1 6
212) B) -17 sec2(-17x + 15) D) -17 sec (-17x + 15) tan (-17x + 15)
C) sec2 (-17x + 15) 46
Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 213) u(2) = 7, u (2) = 4, v(2) = -1, v (2) = -4. 213) d u at x = 2 dx v
A) - 24
B) 24
D) 3
C) - 32
2
You want a linearization that will replace the function over an interval that includes the point x 0 . To make your subsequent work as simple as possible, you want to center the linearization not at x 0 but at a nearby integer x = a at which the function and its derivative are easy to evaluate. What linearization do you use? 214) f(x) = x , x0 = 2.8 x+1
A) 9 + 1 x 16
16
B) 3 + 1 x 16
C) 1 + 3 x
16
16
16
214)
D) 1 + 9 x 16
16
Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x. 215) y = cot(5x - 9) A) y = cot u; u = 5x - 9; dy = - csc2 (5x - 9) dx
215)
B) y = cot u; u = 5x - 9; dy = - 5 cot(5x - 9) csc(5x - 9) dx
C) y = cot u; u = 5x - 9; dy = - 5 csc2(5x - 9) dx
D) y = 5u - 9; u = cot x; dy = - 5 cot x csc2 x dx
Find the indicated derivative. 216) dv if v = t + 3 dt t
A) 1 + 3
t2
216) B) 1 - 3
C) t - 3
t2
t
D) 1 - 3
t2
At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 217) x6 y6 = 64, slope at (2, 1) 217)
A) - 1 2
C) - 1
B) 2
4
Provide an appropriate response. 218) If xy2 = 4 and dx/dt = -5, then what is dy/dt when x = 4 and y = 1?
A) 8 5
B) 5 8
C) - 8 5
47
D) -32
218) D) - 5 8
The function s = f(t) gives the position of a body moving on a coordinate line, with s in meters and t in seconds. 219) s = 2t2 + 2t + 4, 0 t 2 219) Find the body's displacement and average velocity for the given time interval. A) 20 m, 10 m/sec B) 12 m, 6 m/sec C) 8 m, 10 m/sec D) 12 m, 12 m/sec
Find dy.
220) y = cos(9 x)
220)
A) -9 sin(9 x) dx
B) -9 x sin(9 x) dx
C) 9 x sin(9 x) dx
D) 9 sin(9 x) dx
2
2 x 2
Find the indicated derivative. 221) Find y(4) if y = 2 sin x.
A) y(4) = 2 cos x
2 x
221) B) y(4) = 2 sin x
C) y(4) = - 2 cos x
D) y(4) = - 2 sin x
Find y .
222) y = (3x3 + 3)(3x7 - 7) A) 90x9 + 63x6 - 63x C) 12x9 + 63x6 - 63x
222) B) 90x9 + 63x6 - 63x2 D) 12x9 + 63x6 - 63x2
You want a linearization that will replace the function over an interval that includes the point x 0 . To make your subsequent work as simple as possible, you want to center the linearization not at x 0 but at a nearby integer x = a at which the function and its derivative are easy to evaluate. What linearization do you use? 223) f(x) = x, x0 = 9.1
A) 3 - 1 x 2
6
B) 1 + 1 x 2
C) 3 + 1 x
2
2
6
223)
D) 3 + 1 x 2
3
Solve the problem. Round your answer, if appropriate.
224) Water is discharged from a pipeline at a velocity v (in ft/sec) given by v = 1694p(1/2), where p is
224)
the pressure (in psi). If the water pressure is changing at a rate of 0.463 psi/sec, find the acceleration (dv/dt) of the water when p = 50.0 psi. A) 2770 ft/sec2 B) 120 ft/sec2 C) 55.5 ft/sec2 D) 59.9 ft/sec2
Use implicit differentiation to find dy/dx. 225) xy + x = 2 A) - 1 + x B) 1 + y y x
225) C) 1 + x y
D) - 1 + y x
C) -1
D) -5
Find the value of (f g) at the given value of x. 226) f(u) = sin2 u + u, u = g(x) = -x, x = 5
A) 0
226)
B) 1
48
Find y .
227) y = 2x6 (2x - 9)2 A) 56x7 - 504x6 + 972x5 C) 8x7 - 72x6 + 12x5
227) B) 448x6 - 3024x5 + 4860x4 D) 104x6 - 864x5 + 60x4
The function s = f(t) gives the position of a body moving on a coordinate line, with s in meters and t in seconds. 228) s = 10t - t2 , 0 t 10 228) Find the body's displacement and average velocity for the given time interval. A) -200 m, -20 m/sec B) 200 m, 20 m/sec
C) 0 m, 0 m/sec
D) 200 m, -10 m/sec
Use implicit differentiation to find dy/dx and d2 y/dx 2 . 229) x2 + y2 = 5, at the point (2, 1)
229)
2 A) dy = - 1 ; d y = 0 dx 2 dx2
B) dy = - 2; dx
2
dx2
=1
2
C) dy = 2; d y = 5 dx
d2 y
D) dy = - 2; d y = -5
dx2
dx
dx2
Find the derivative. 230) r = 4 - 9 s3 s
230)
A) 4 - 9 s4
s2
Find the indicated derivative. 231) Find y if y = 7 sin x. A) y = 49 sin x
B) - 12 + 9
C) 12 - 9
D) - 12 + 9
B) y = - 7 sin x
C) y = 7 sin x
D) y = 7 cos x
s4
s2
s4
s2
s2
s2
231)
49
The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? 232) x = 0 232)
A) Differentiable B) Continuous but not differentiable C) Neither continuous nor differentiable The function f(x) changes value when x changes from x 0 to x 0 + dx. Find the approximation error
f - df . Round your
answer, if appropriate. 233) f(x) = x3 , x0 = 5, dx = 0.05
A) 0.037625
233) B) 0.0188125
C) 0.0564375
D) 0.07525
At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 234) y5 + x3 = y2 + 11x, normal at (0, 1) 234)
A) y = - 7 x 11
B) y = 7 x + 1
C) y = - 3 x + 1
11
11
D) y = 11 x + 1 3
The function s = f(t) gives the position of a body moving on a coordinate line, with s in meters and t in seconds. 235) s = 7t2 + 4t + 3, 0 t 2 235) Find the body's speed and acceleration at the end of the time interval. A) 35 m/sec, 14 m/sec2 B) 18 m/sec, 2 m/sec2
C) 32 m/sec, 14 m/sec2
D) 32 m/sec, 28 m/sec2
Solve the problem. Round your answer, if appropriate. 236) Water is being drained from a container which has the shape of an inverted right circular cone. The container has a radius of 5.00 inches at the top and a height of 8.00 inches. At the instant when the water in the container is 6.00 inches deep, the surface level is falling at a rate of 1.9 in./sec. Find the rate at which water is being drained from the container. A) 118 in.3 /s B) 83.9 in.3 /s C) 80.2 in.3 /s D) 97.0 in.3 s
50
236)
Use implicit differentiation to find dy/dx. 237) xy + x + y = x2y2 2
A) 2xy + y
237) 2
2
B) 2xy - y
2x2 y - x
C) 2xy - y - 1
2x2 y + x
-2x2 y + x + 1
2
D) 2xy + y + 1
-2x2 y - x - 1
Use the linear approximation (1 + x)k 1 + kx, as specified.
238) Find an approximation for the function f(x) = (1 - x)4 for values of x near zero. A) f(x) 1 + 4x B) f(x) 4 + 4x C) f(x) 1 - 4x D) f(x) 1 + 5x
Provide an appropriate response. 2/7 239) Evaluate lim x -1 = by first converting it to a derivative at a particular x-value. x -1 x + 1
A) - 2
B) - 1
7
C) 2
7
Use the formula f'(x) = lim z x
239)
D) - 7
7
2
f(z) - f(x) to find the derivative of the function. z-x
240) g(x) = x
240)
x+7
A)
238)
x
(x + 7)2
2 B) x
C)
x+7
7
(x + 7)2
D) -
7
(x + 7)2
Find dy/dt.
241) y = 4t(3t + 2)3 A) 4(3t + 2)3 (6t + 2) C) 4(3t + 2)2
241) B) 4(3t + 2)2 (12t + 2) D) 4(12t + 2)2
Provide an appropriate response.
242) Find the slope of the curve xy3 - x5y2 = -4 at (-1, 2). A) - 3 2
242)
B) - 6
C) - 3
D) 2
B) 4x - 30
C) 4x - 20
D) 20x - 4
5
4
3
Find the second derivative. 243) y = 5x3 - 2x2 + 6
A) 30x - 4
243)
Use the linear approximation (1 + x)k 1 + kx, as specified. 244) Find an approximation for the function f(x) = 3 for values of x near zero. 1-x
A) f(x) 3 - 3x
B) f(x) 3 + 3x
C) f(x) 1 + 3x
51
244) D) f(x) 1 - 3x
Given the graph of f, find any values of x at which f is not defined.
245)
245)
A) x = 1, 3 C) x = 1, 2, 3
B) x = 2 D) Defined for all values of x
Use the linear approximation (1 + x)k 1 + kx, as specified. 3 246) Estimate 1.012. A) 1.04 B) 1.004
246) C) 1.05
D) 1.005
Solve the problem.
247) At time t 0, the velocity of a body moving along the s-axis is v = t2 - 9t + 8. When is the body moving backward? A) 0 t < 1
B) t > 8
C) 0 t < 8
247)
D) 1 < t < 8
At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 248) y4 + x3 = y2 + 12x, tangent at (0, 1) 248)
A) y = 3x + 1
B) y = - 2x
C) y = 6x + 1
D) y = - 3x - 1
Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x. 249) y = cos5 x
249)
A) y = cos u; u = x5 ; dy = - sin(x5 ) dx
B) y = u5 ; u = cos x; dy = 5 cos4 x sin x dx
C) y = cos u; u = x5 ; dy = - 5x4 sin(x5 )
D) y = u5 ; u = cos x; dy = - 5 cos4 x sin x
dx
dx
Find the second derivative of the function. 2 5 250) y = 2 - 2x + x x9
250)
2
2
A) d y = - 18 + 14 - 4 dx2
x10
x8
B) d y = 18 + 14 - 4
x5
dx2
2
x12
x6
A) -
x2 + 3xy x 2 + y2
B) -
x4
D) d y = 180 - 112 + 20
x3
Use implicit differentiation to find dy/dx. 251) x3 + 3x2y + y3 = 8
x7
2
C) d y = 180 + 112 - 20 dx2
x11
dx2
x11
x9
x6
251) x2 + 2xy
C)
x 2 + y2
52
x2 + 2xy x2 + y2
D)
x2 + 3xy x2 + y2
Find dr/d .
252) r
+1=4
A) -
252)
r 2( + 1)
B) - 2r
C)
+1
r 2( + 1)
D) 2r
+1
Solve the problem.
253) The power P (in W) generated by a particular windmill is given by P = 0.015V3 where V is the
253)
velocity of the wind (in mph). Find the instantaneous rate of change of power with respect to velocity when the velocity is 9.8 mph. A) 28.2 W/mph B) 9.6 W/mph C) 4.3 W/mph D) 0.4 W/mph
Given y = f(u) and u = g(x), find dy/dx = f (g(x))g (x). 254) y = u2 , u = 3x - 1
A) 6x - 3
254)
B) 18x
C) 18x - 6
Find the derivative. 255) y = sin x + 10x 10x sin x
255)
A) dy = x cos x + sin x + 10 sin x + 10x cos x dx
10x2
sin2 x
B) dy = cos x + 10 dx
10
cos x
C) dy = x cos x - sin x + 10 sin x - 10x cos x dx
10x2
sin2 x
D) dy = sin x - x cos x + 10x cos x - 10 sin x dx
D) 9x - 3
100x 2
sin2 x
53
The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? 256) x = -1 256)
A) Differentiable B) Continuous but not differentiable C) Neither continuous nor differentiable Find the linearization L(x) of f(x) at x = a. 257) f(x) = 5x + 81, a = 0 A) L(x) = 5 x - 9 B) L(x) = 5 x - 9 9 18
257) C) L(x) = 5 x + 9 9
D) L(x) = 5 x + 9 18
Solve the problem. 258) V = 4 r3, where r is the radius, in centimeters. By approximately how much does the volume of a 3
258)
sphere increase when the radius is increased from 1.0 cm to 1.1 cm? (Use 3.14 for .) A) 1.1 cm3 B) 1.3 cm3 C) 0.1 cm3 D) 1.5 cm3
Find dy/dt.
259) y = (1 + sin 10t)-6 A) -6(1 + sin 10t)-7 cos 10t C) -60(cos 10t)-7
259) B) -6(1 + sin 10t)-7 D) -60(1 + sin 10t)-7 cos 10t
Find the derivative. 260) y = 2 + 1 sin x cot x
260)
A) y = 2 csc x cot x - csc2 x C) y = 2 csc x cot x - sec2 x
B) y = - 2 csc x cot x + sec2 x D) y = 2 cos x - csc2 x
54
Find the derivative of the function. 261) r = (sec + tan )-2
261)
A) -2(sec + tan )-3
B) -2(sec tan
C) -2(sec + tan )-3 (tan2 + sec tan )
D)
-3 + sec2 )
-2 sec (sec
+ tan )2
Find an equation for the tangent to the curve at the given point. 262) y = x2 - 1, (-3, 8)
A) y = -6x - 20
B) y = -6x - 10
C) y = -6x - 19
Use implicit differentiation to find dy/dx and d2 y/dx 2 . 263) xy + 3 = y, at the point (4, -1) 2 A) dy = 1 ; d y = 2 dx 3 dx2 9
263)
2
3 dx2
D) y = -3x - 10
2 B) dy = - 1 ; d y = 0 dx 3 dx2 2
C) dy = 1 ; d y = - 2 dx
262)
D) dy = 3; d y = - 24
9
dx
dx2
The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? 264) x = 0 264)
A) Differentiable B) Continuous but not differentiable C) Neither continuous nor differentiable Given y = f(u) and u = g(x), find dy/dx = f (g(x))g (x). 265) y = 2 , u = 8x - 5 u2
A) - 32
8x - 5
265)
B) 32x
C) -
8x - 5
55
32
(8x - 5)3
D) - 16
8x - 5
Calculate the derivative of the function. Then find the value of the derivative as specified. 266) f(x) = 8 ; f (0) x+2
B) f (x) = -
A) f (x) = 8; f (0) = 8 C) f (x) =
8
(x + 2)2
8
(x + 2)2
266)
; f (0) = -2
D) f (x) = - 8(x + 2)2 ; f (0) = - 32
; f (0) = 2
Solve the problem. 267) Use the following information to graph the function f over the closed interval [-5, 6]. i) The graph of f is made of closed line segments joined end to end. ii) The graph starts at the point (-5, 1). iii) The derivative of f is the step function in the figure shown here.
A)
56
267)
B)
C)
D)
57
Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x. 8 268) y = 3x2 - 8 - x x
A) y = u8 ; u = 3x2 - 8 - x; dy = 8 6x + 8 - 1
7
B) y = u8 ; u = 3x2 - 8 - x; dy = 8 3x2 - 8 - x
7
x
dx
x
268)
x2
dx
x
7
C) y = u8 ; u = 3x2 - 8 - x; dy = 8 3x2 - 8 - x 6x + 8 - 1 x
dx
x
x2
D) y = 3u2 - 8 - u; u = x8; dy = 6x16 - 8 - x8 u
dx
x8
The equation gives the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds). 269) s = 6 sin t - cos t Find the body's acceleration at time t = /4 sec. A) 7 2 m/sec2 B) 5 2 m/sec2 C) - 7 2 m/sec2 D) - 5 2 m/sec2 2 2 2 2 Use implicit differentiation to find dy/dx and d2 y/dx 2 . 270) x2 - y3 = 8 2
3
A) dy = 2x ; d y = 6y - 8x dx
3y2 dx2 2
dx
3y2 dx2
270)
2
2
3
3
B) dy = 2x ; d y = 6y - 8x
9y3
C) dy = 2x ; d y = 6y - 8x
dx
2
3y2 dx2 2
9y6
3
D) dy = 2x ; d y = 5y - 8x
9y5
dx
3y2 dx2
2
2
9y5
Find an equation for the tangent to the curve at the given point. 3 271) y = x , (8, 128) 4
A) y = 16x - 256
269)
B) y = 48x - 256
271) C) y = 16x + 256
D) y = 256x + 48
At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 272) 4x2 y - cos y = 5 , slope at (1, ) 272)
A)
C) -
B) -2
2
Find the derivative. 273) y = 5x2 + 10x + 4x-3
D) 0
273)
A) 10x + 10 + 12x-4
B) 10x + 10 - 12x-4
C) 10x - 12x-4
D) 5x + 4x-4
58
274) y = 8 - 6x3 A) -18x
274) B) 8 - 18x2
C) -18x2
D) -12x2
Provide an appropriate response. 275) The curve y = ax2 + bx + c passes through the point (2, 12) and is tangent to the line y = 4x at the origin. Find a, b, and c. A) a = 2, b = 0, c = 0
275)
B) a = 4, b = 0, c = 1 D) a = 0, b = 1, c = 4
C) a = 1, b = 4, c = 0
276) Find the value of a that makes the following function differentiable for all x-values.
276)
ax, if x < 0 g(x) = 2 x - 3x, if x 0
A) 3
B) 0
C) 9
D) -3
The figure shows the velocity v or position s of a body moving along a coordinate line as a function of time t . Use the figure to answer the question. 277) v (ft/sec) 277)
t (sec)
What is the body's speed when t = 10 sec? A) -5 ft/sec B) 3 ft/sec
C) 0 ft/sec
D) 5 ft/sec
Find y .
278) y = 4 cot x
278)
8
A) -8 csc x
B) 8 csc2 x cot x
8
8
C) 1 csc2 x cot x 8
8
8
D) - 1 csc2 x
8
2
59
8
The figure shows the velocity v or position s of a body moving along a coordinate line as a function of time t . Use the figure to answer the question. 279) v (ft/sec) 279)
t (sec)
When is the body's acceleration equal to zero? A) t = 0, t = 4, t = 7
B) t = 2, t = 3, t = 5, t = 6 D) 0 < t < 2, 6 < t < 7
C) 2 < t < 3, 5 < t < 6 Find dy/dt.
280) y = cos( 10t + 11) A) -
280)
5 sin( 10t + 11) 10t + 11
C) -sin
1 B) sin( 10t + 11) 2 10t + 11
5 10t + 11
D) -sin( 10t + 11)
Calculate the derivative of the function. Then find the value of the derivative as specified. 281) g(x) = 3x2 - 4x; g (3)
A) g (x) = 2x- 4; g (3) = 2 C) g (x) = 6x; g (3) = 18
281)
B) g (x) = 3x - 4; g (3) = 5 D) g (x) = 6x - 4; g (3) = 14
Find y .
282) y = 1 + 3 x2 - 1 + 3 x2
282)
x2
A) - 4 - 6x x5
B) 4 + 6x
C) - 1 + 6x
x3
x5
60
D) 4 + 6x x5
Estimate the slope of the curve at the indicated point.
283)
283)
A) - 1
C) 1
B) 2
20
D) -2
20
Solve the problem. 284) The range R of a projectile is related to the initial velocity v and projection angle by the equation v2 sin 2 R= , where g is a constant. How is dR/dt related to dv/dt if is constant? g
A) dR = 2v dv dt
B) dR = 2v sin 2 dv
g dt
dt
2 C) dR = 2v cos 2 dv dt
g
g
dt
D) dR = 2v cos 2 dv dt
dt
Find the indicated derivative. 285) Find y if y = 4x sin x. A) y = 4x cos x + 12 sin x
C) y
g
dt
285) B) y D) y
= 8 cos x - 4x sin x
Find the derivative. 286) s = t7 tan t -
= - 4x cos x - 12 sin x = - 4x cos x + 12 sin x
t ds 1 A) = 7t6 sec2 t dt 2 t
B) ds = t7 sec2 t + 7t6 tan t - 1 dt 2 t
C) ds = t7 sec t tan t + 7t6 tan t - 1
D) ds = - t7 sec2 t + 7t6 tan t + 1
dt
286)
dt
2 t
2 t
Solve the problem. 287) Find all points on the curve y = sin x, 0 x 2 , where the tangent line is parallel to the line 1 y = x. 2
A) C)
1 2 1 , , , 3 2 3 2 3
,
284)
B)
3 5 3 , ,2 3 2
D)
61
1 11 1 , , ,6 2 6 2 3
,
3 2 3 , , 2 3 2
287)
Solve the problem. Round your answer, if appropriate. 288) Boyle's law states that if the temperature of a gas remains constant, then PV = c, where P = pressure, V = volume, and c is a constant. Given a quantity of gas at constant temperature, if V is decreasing at a rate of 12 in. 3 /sec, at what rate is P increasing when P = 30 lb/in.2 and V = 50 in.3 ? (Do not round your answer.) A) 36 lb/in.2 per sec 5
288)
B) 9 lb/in.2 per sec 25
C) 20 lb/in.2 per sec
D) 125 lb/in.2 per sec
Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 289) u(2) = 6, u (2) = 2, v(2) = -1, v (2) = -4. 289) d v at x = 2 dx u
A) 11
B) - 11
18
C) - 11
18
D) - 13
3
18
At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 290) 2x2 y - cos y = 3 , tangent at (1, ) 290)
A) y = -2 x +
C) y = -
B) y = x
2
x+
3 2
D) y = -2 x + 3
Calculate the derivative of the function. Then find the value of the derivative as specified. 291) g(x) = - 2 ; g (-2) x
291)
A) g (x) = - 2x2; g (- 2) = - 8
B) g (x) = - 2; g (- 2) = - 2
C) g (x) = 2 ; g (-2) = 1
D) g (x) = - 2 ; g (- 2) = - 1
x2
2
x2
62
2
Compare the right-hand and left-hand derivatives to determine whether or not the function is differentiable at the point whose coordinates are given.
292)
292)
y=x
y = 2x
A) Since limx 0 + f (x) = -2 while limx 0 - f (x) = -1, f(x) is not differentiable at x = 0. B) Since limx 0 + f (x) = 1 while limx 0 - f (x) = 2, f(x) is not differentiable at x = 0. C) Since limx 0 + f (x) = 2 while limx 0 - f (x) = 1, f(x) is not differentiable at x = 0. D) Since limx 0 + f (x) = 1 while limx 0 - f (x) = 1, f(x) is differentiable at x = 0. Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values of x. Find the derivative with respect to x of the given combination at the given value of x. x f(x) g(x) f (x) g (x) 293) 3 1 9 293) 6 3 -6 4 -3 3 2 1/f2 (x), x = 4 A) - 1 4
B) 5.629499534e+14 7.599824371e+15
C) - 1.125899907e+15
D) 1.125899907e+15
7.599824371e+15
7.599824371e+15
Solve the problem. 294) The diameter of a tree was 11 in. During the following year, the circumference increased 2 in. About how much did the tree's diameter increase? (Leave your answer in terms of .) A) 13 in. B) in. C) 2 in. D) 11 in. 2
295) Estimate the volume of material in a cylindrical shell with height 30 in., radius 6 in., and shell thickness 0.4 in. (Use 3.14 for .) A) 1130.4 in.3 B) 462.2 in.3
C) 452.2 in.3
63
D) 226.1 in.3
294)
295)
Compare the right-hand and left-hand derivatives to determine whether or not the function is differentiable at the point whose coordinates are given.
296)
296)
y = x2
y=
x
A) Since limx 1 + f (x) = 2 while limx 1 - f (x) = 1 , f(x) is not differentiable at x = 1. 2
B) Since limx 1 + f (x) = 1 while limx 1- f (x) = 2, f(x) is not differentiable at x = 1. 2
C) Since limx 1 + f (x) = 1 while limx 1- f (x) = 1, f(x) is not differentiable at x = 1. 2
D) Since limx 1 + f (x) = 2 while limx 1 - f (x) = 2, f(x) is differentiable at x = 1. Solve the problem. 297) The velocity of water ft/s at the point of discharge is given by v = 11.27 P, where P is the pressure lb/in2 of the water at the point of discharge. Find the rate of change of the velocity with respect to pressure if the pressure is 80.00 lb/in2 . A) 0.0704 ft/s per lb/in2
297)
B) 0.6300 ft/s per lb/in2 D) 1.26 ft/s per lb/in2
C) 50.40 ft/s per lb/in2
Graph the curve over the given interval together with its tangent at the given value of x. Graph the tangent with a dashed line.
298) y = cot(x), -
x 2 , x=
298)
4
64
A)
B)
C)
D)
Use implicit differentiation to find dy/dx and d2 y/dx 2 . 299) xy - x + y = 5 2 A) dy = - 1 + y ; d y = y + 1 dx x + 1 dx2 (x + 1)2
299) 2
B) dy = - 1 + y ; d y = 2y - 2 dx
2
1 + x dx2
(x + 1)2
2
C) dy = 1 - y ; d y = 2y - 2 dx
x + 1 dx2
D) dy = y + 1 ; d y = 2y + 2
(x + 1)2
dx
x + 1 dx2
(x + 1)2
Solve the problem. 300) A rectangular steel plate expands as it is heated. Find the rate of change of area with respect to temperature T when the width is 1.2 cm and the length is 2.2 cm if dl/dt = 1.3 x 10-5 cm/°C and dw/dt = 8.3 x 10-6 cm/°C. A) 3.4 x 10-5 cm2/°C
B) 2.6 x 10-5 cm2/°C D) 1.1 x 10-5 cm2/°C
C) 1.6 x 10-5 cm2/°C
65
300)
Estimate the slope of the curve at the indicated point.
301)
301)
B) - 1
A) 2
C) 1
2
D) -2
2
Find dy/dt.
302) y = t6 (t6 - 4)5
302)
A) t5(t6 - 4)4 (36t6 - 24)
B) 6t5 (t6 - 4)4 (30t6 - 4)
C) 180t34(t6 - 4)4
D) t6(t6 - 4)4 (36t5 - 24)
Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x. 303) y = (2x + 11)4
303)
A) y = u4 ; u = 2x + 11; dy = 8(2x + 11)3
B) y = u4 ; u = 2x + 11; dy = 4(2x + 11)3
C) y = 4u + 11; u = x4 ; dy = 8x3
D) y = u4 ; u = 2x + 11; dy = 2(2x + 11)4
dx
dx
dx
dx
Find the indicated derivative. 304) Find y if y = 5x sin x. A) y = 5 cos x - 10x sin x
304) B) y = - 10 cos x + 5x sin x D) y = - 5x sin x
C) y = 10 cos x - 5x sin x
Solve the problem. 305) The radius of a ball is claimed to be 4.5 inches, with a possible error of 0.05 inch. Use differentials to approximate the maximum possible error in calculating the volume of the sphere and the surface area of the sphere. A) 4.05 in.3 ; 1.8 in.2 B) 1.8 in.3 ; 4.05 in.2
C) 0.9 in.3 ; 18 in.2
D) 8.1 in.3 ; 3.6 in.2
Find the indicated derivative. 306) dt if t = x dx 5x - 6
A) -
6x
(5x - 6)2
305)
306) B) -
6 5x - 6
C) 10x - 6
(5x - 6)2
D) -
6
(5x - 6)2
The equation gives the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds). 307) s = 9 sin t - cos t Find the body's jerk at time t = /4 sec. A) - 4 2 m/sec3 B) - 5 2 m/sec3 C) 4 2 m/sec3 D) 5 2 m/sec3
66
307)
Estimate the slope of the curve at the indicated point.
308)
308)
C) 1
B) 2
A) -2
D) - 1
2
2
Provide an appropriate response. 309) If xy + x = 12 and dx/dt = -3, then what is dy/dt when x = 2 and y = 5? A) 3 B) -9 C) -3
309) D) 9
Calculate the derivative of the function. Then find the value of the derivative as specified. 2 310) dr =3 if r = d 28 -
A) dr = d
C) dr = d
2
dr ; 3/2 d (28 - ) 1
dr (28 - )3/2 d ;
2 =3 = 125
B) dr = -
1 =3 = 125
D) dr = -
d d
2
dr ; 3/2 d (28 - ) 1
dr (28 - )3/2 d ;
310) 2 =3 = - 125 1 =3 = - 125
Find dr/d .
311) 1/3 - r1/3 = 1 A) - r
2/3
311) B)
2/3
C) -
r
Find the linearization L(x) of f(x) at x = a. 312) f(x) = sin x, a = 0 A) L(x) = -x B) L(x) = 0
2/3 r
D) r
2/3
312) C) L(x) = x
D) L(x) = 4x + 1
Calculate the derivative of the function. Then find the value of the derivative as specified. 313) f(x) = 5x + 9; f (2) A) f (x) = 5; f (2) = 5 B) f (x) = 5x; f (2) = 10
C) f (x) = 0; f (2) = 0
313)
D) f (x) = 9; f (2) = 9
Solve the problem.
314) Suppose that the velocity of a falling body is v = ks2 (k a constant) at the instant the body has
314)
315) Suppose that the dollar cost of producing x radios is c(x) = 800 + 40x - 0.2x2 . Find the marginal
315)
fallen s meters from its starting point. Find the body's acceleration as a function of s. A) a = 2ks B) a = 2ks3 C) a = 2k2 s3 D) a = 2ks2
cost when 40 radios are produced. A) $2080 B) -$2080
C) $24 67
D) $56
316) A = r2 , where r is the radius, in centimeters. By approximately how much does the area of a
316)
circle decrease when the radius is decreased from 4.0 cm to 3.8 cm? (Use 3.14 for .) A) 4.8 cm2 B) 5.2 cm2 C) 5.0 cm2 D) 2.5 cm2
Given the graph of f, find any values of x at which f is not defined.
317)
317)
A) x = 0 C) x = 0, 3
B) x = 3 D) Defined for all values of x
Find an equation of the tangent line at the indicated point on the graph of the function. 318) y = f(x) = 6 x - x + 1, (x, y) = (36, 1) A) y = 1 x - 19 B) y = 1 C) y = - 1 x + 19 2 2
318) D) y = - 1 x + 1 2
Find an equation for the tangent to the curve at the given point. 319) y = x2 - x, (-4, 20)
A) y = -9x + 16
B) y = -9x + 12
319) C) y = -9x - 12
Provide an appropriate response. 320) If x3 + y3 = 9 and dx/dt = -3, then what is dy/dt when x = 1 and y = 2?
A) - 4 3
B) 4
C) 3
3
4
D) y = -9x - 16
320) D) - 3 4
Solve the problem. 321) The position (in centimeters) of an object oscillating up and down at the end of a spring is given k by s = A sin t at time t (in seconds). The value of A is the amplitude of the motion, k is a m
321)
measure of the stiffness of the spring, and m is the mass of the object. How fast is the object moving when it is moving fastest? A) A cm/sec B) A k cm/sec C) A m cm/sec D) k cm/sec m k m
The function f(x) changes value when x changes from x 0 to x 0 + dx. Find the approximation error answer, if appropriate. 322) f(x) = x2, x0 = 7, dx = 0.06
A) 0.0018
f - df . Round your
322) B) 0.7836
C) 0.0072
68
D) 0.0036
Find the second derivative. 3 323) s = 5t + 5 3
323)
A) 10t + 5
C) 5t2
B) 5t
D) 10t
Solve the problem. 324) The area of the base B and the height h of a pyramid are related to the pyramid's volume V by the 1 formula V = Bh. How is dV/dt related to dh/dt if B is constant? 3
A) dV = B dh dt
B) dV = B dh
3 dt
dt
C) dV = 1 dh
dt
dt
3 dt
324)
D) dV = dh dt
dt
325) Under standard conditions, molecules of a gas collide billions of times per second. If each
325)
molecule has diameter t, the average distance between collisions is given by 1 L= , 2 t2 n where n, the volume density of the gas, is a constant. Find
A) dL =
2
dt
2 t3 n
C) dL = -
2
dt
B) dL = dt
A) 16x2 - 4
1
2 2 t3 n
D) dL = dt
2 t3 n
Find the second derivative. 326) y = 4x4 - 2x2 + 7
dL . dt
1 2 tn
326) B) 16x2 - 4x
C) 48x2 - 4
Use implicit differentiation to find dy/dx. 327) x = sec(4y) A) 1 sec(4y) tan(4y) 4
D) 48x2 - 4x
327) B) 1 cos(4y) cot(4y) 4
C) 4 sec(4y) tan(4y)
D) cos(4y) cot(4y)
Find y .
328) y = 1 + 5 x - 1 + 5 x
328)
x
A) - 2 - 5 x3
B) 1 + 5
C) 2 + 5
x3
x3
69
D) - 1 - 5 x3
Given the graph of f, find any values of x at which f is not defined.
329)
329)
A) x = 1
B) x = 2
C) x = -1
D) x = 0
The figure shows the velocity v or position s of a body moving along a coordinate line as a function of time t . Use the figure to answer the question. 330) s (m) 330)
t (sec)
When is the body moving forward? A) 0 < t < 3, 3 < t < 5, 5 < t < 8
B) 0 < t < 1, 3 < t < 4, 5 < t < 7, 9 < t < 10 D) 0 < t < 1, 3 < t < 4, 5 < t < 7
C) 0 < t < 8 Find the derivative of the function. 331) y = x6cos x - 6x sin x - 6 cos x
331)
A) -x6 sin x + 6x5 cos x - 6x cos x - 12 sin x C) -6x5 sin x - 6 cos x + 6 sin x
Find dy.
332) y =
B) x6 sin x - 6x5 cos x + 6x cos x D) -x6 sin x + 6x5 cos x - 6x cos x
x 3x - 1
332)
A)
3x - 2 dx 2(3x - 1)3/2
B) 3x + 2 dx
C)
3x + 2 dx 2(3x - 1)3/2
D) 3x - 2 dx
2 3x - 1 2 3x - 1
70
Calculate the derivative of the function. Then find the value of the derivative as specified. 333) f(x) = x2 + 7x - 2; f (0)
A) f (x) = x + 7; f (0) = 7 C) f (x) = 2x; f (0) = 0
333)
B) f (x) = 2x - 2; f (0) = - 2 D) f (x) = 2x + 7; f (0) = 7
Find the derivative. 334) y = (csc x + cot x)(csc x - cot x)
334)
A) y = 1 C) y = - csc x cot x
B) y = 0 D) y = - csc2 x
Find the limit.
335) lim cos 1 - 1 x
x 9
335)
9
B) 1
A) -1
C) 0
2
Given y = f(u) and u = g(x), find dy/dx = f (g(x))g (x). 336) y = u(u - 1), u = x2 + x
A) 2x2 + 4x + 1
A)
-5r4
10r - r5
Find the indicated derivative. 338) Find y(4) if y = -7 cos x.
A) y(4) = -7 sin x
336)
B) 2x2 + 4x
Find the derivative of the function. 337) q = 10r - r5
D) 1
C) 4x3 + 6x2 - 2x
D) 4x3 + 6x2 - 1
337) B)
10 - 5r4
C)
2 10r - r5
1 2 10 - 5r4
D)
1 2 10r - r5
338) B) y(4) = 7 cos x
C) y(4) = 7 sin x
D) y(4) = -7 cos x
Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 339) u(1) = 5, u (1) = -6, v(1) = 7, v (1) = -2. 339) d u at x = 1 dx v
A) - 52 49
C) - 32
B) - 8
49
D) - 32 7
Solve the problem.
340) At time t 0, the velocity of a body moving along the s-axis is v = t2 - 11t + 10. When is the body's velocity increasing? A) t < 10
B) t > 5.5
C) t > 10
71
D) t < 5.5
340)
Solve the problem. Round your answer, if appropriate. 341) Electrical systems are governed by Ohm's law, which states that V = IR, where V = voltage, I = current, and R = resistance. If the current in an electrical system is decreasing at a rate of 9 A/s while the voltage remains constant at 22 V, at what rate is the resistance increasing (in /sec) when the current is 36 A? (Do not round your answer.) A) 11 /sec B) 2 /sec 2 11
C) 4.354066046e+14 2.849934139e+15
D) 99
/sec
341)
/sec
2
Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 342) u(1) = 5, u (1) = -6, v(1) = 6, v (1) = -4. 342) d (2u - 4v) at x = 1 dx
A) 4
C) 34
B) -28
Provide an appropriate response.
343) Find all points (x, y) on the graph of y = y = 5x - 3. A) (0, 0)
D) -14
x with tangent lines perpendicular to the line (x - 5)
B) (0, 0), (5, 2)
C) (10, 2)
343)
D) (0, 0), (10, 2)
Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values of x. Find the derivative with respect to x of the given combination at the given value of x. x f(x) g(x) f (x) g (x) 344) 3 1 9 344) 6 3 -6 4 -3 3 5 f(g(x)), x = 4 A) -30
B) 18
C) -36
D) 6
Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 345) u(1) = 4, u (1) = -6, v(1) = 7, v (1) = -2. 345) d v at x = 1 dx u
A) - 17 8
B) 17
C) - 25
2
8
Find the derivative. 346) s = t5 - csc t + 16
D) 17 8
346)
A) ds = 5t4 + cot2 t
B) ds = 5t4 + csc t cot t
C) ds = 5t4 - csc t cot t
D) ds = t4 - cot2t + 16
dt
dt
dt
dt
72
Estimate the slope of the curve at the indicated point.
347)
347)
A) 1
B) Undefined
D) 0
C) -1
Solve the problem. Round your answer, if appropriate.
348) The volume of a rectangular box with a square base remains constant at 400 cm3 as the area of the
348)
base increases at a rate of 11 cm2 /sec. Find the rate at which the height of the box is decreasing when each side of the base is 15 cm long. (Do not round your answer.) A) 4.837851162e+15 cm/sec B) 1.407374884e+16 cm/sec 3.710851744e+15 7.91648372e+15
C) 3.87028093e+14 cm/sec
D) 1.209462791e+14 cm/sec
7.91648372e+15
1.391569404e+15
Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 349) u(2) = 7, u (2) = 2, v(2) = -1, v (2) = -5. 349) d (3v - u) at x = 2 dx
A) 4
B) -13
C) -17
Use implicit differentiation to find dy/dx and d2 y/dx 2 . 350) 4 y - y = 2x 2 A) dy = 2 - y ; d y = 2 - y dx 4y 2 y dx2 2 C) dy = 2 y ; d y = dx
2-
y dx2
350) B) dy = 2 y ; dx
d2 y
y dx2
2-
=
4 (2 -
y)3
2
2-2 y y(2 -
D) -10
D) dy = 2 - 2; d y = - 2
y)2
dx
dx2
y
y2
Differentiate the function and find the slope of the tangent line at the given value of the independent variable. 351) s = -5t4 + 5t3 , t = -1 351)
A) 5
B) -35
D) 35
C) -5
Find the derivative of the function. 352) s = sin 7 t - cos 7 t 2 2
352)
A) cos 7 t + sin 7 t 2
B) 7 cos 7 t - 7 sin 7 t
2
2
C) - 7 cos 7 t - 7 sin 7 t 2
2
2
2
2
2
D) 7 cos 7 t + 7 sin 7 t
2
2
73
2
2
2
Find y .
353) y = (5x - 4)(5x3 - x2 + 1) A) 100x3 - 25x2 + 75x + 5 C) 25x3 + 25x2 - 75x + 5
353) B) 100x3 - 75x2 + 8x + 5 D) 75x3 + 75x2 - 25x + 5
Solve the problem.
354) Find an equation of the tangent to the curve f(x) = x + 1 that has slope 1 .
354)
4
A) y = 1 x - 5 4
B) y = - 1 x + 5
4
4
C) y = 1 x + 5
4
4
4
D) y = 1 x 4
Differentiate the function and find the slope of the tangent line at the given value of the independent variable. 355) f(x) = 5x + 2 , x = 4 355) x
A) 41
B) 39
2
C) 39
2
D) 41
8
8
Provide an appropriate response.
356) Find an equation for the tangent to the curve y = 8x at the point (1, 4).
356)
x2 + 1
A) y = 4
B) y = 0
C) y = x + 4
D) y = 4x
Find dr/d .
357) cos (r )+ r3 = 3
2 A) 3r + sin r 3 2 - r sin r
357) 2 B) 3r - sin r
2
C) 3r + r sin r
3 2 + r sin r
3 2
Find the derivative. 358) r = 7 - 8 cos
A) dr = 8 7 sin d
2
D) 3r - r sin r 3 2
358) B) dr = 8 7 sin d
- 8 cos
C) dr = - 8 7 cos + 8 sin
D) dr = 8 7 cos - 8 sin
d
d
Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values of x. Find the derivative with respect to x of the given combination at the given value of x. x f(x) g(x) f (x) g (x) 359) 3 1 16 8 5 359) 4 -3 3 2 -6 1/g2 (x), x = 4 A) - 2 27
B) - 4 9
C) 4
D) 3.518437209e+13
9
3.799912186e+15
74
Use implicit differentiation to find dy/dx and d2 y/dx 2 . 360) 2y - x + xy = 3 2 A) dy = - 1 + y ; d y = y + 1 dx x + 2 dx2 (2 + x)2
360) 2 B) dy = 1 - y ; d y = 2y - 2 dx 2 + x dx2 (2 + x)2
2
2
C) dy = y + 1 ; d y = 2y + 2 dx
x + 2 dx2
D) dy = - 1 + y ; d y = 2y - 2
(x + 2)2
dx
Find the linearization L(x) of f(x) at x = a. 3 361) f(x) = x, a = 8 A) L(x) = 1 x + 4 B) L(x) = 1 x + 2 4 12 3
x + 2 dx2
(x + 2)2
361) C) L(x) = 1 x + 4 12
3
D) L(x) = 1 x + 2 4
3
The figure shows the velocity v or position s of a body moving along a coordinate line as a function of time t . Use the figure to answer the question. 362) s (m) 362)
t (sec)
When is the body standing still? A) 2 < t < 3, 4 < t < 5, 7 < t < 9
B) t = 3, t = 5, t = 8 D) 1 < t < 2
C) 8 < t 10 Find the slope of the curve at the indicated point. 363) y = 6 , x = 7 3+x
363)
A) m = - 4.222124651e+14
B) m = - 3
C) m = 3
D) m = 4.222124651e+14
7.036874418e+15
5
5
7.036874418e+15
Use the linear approximation (1 + x)k 1 + kx, as specified. 364) Estimate (1.0005)50.
A) 1.025
364)
B) 1.005
C) 1.05
75
D) 1.01
Use implicit differentiation to find dy/dx. 365) 2xy - y2 = 1
A)
x x-y
B)
365) y y-x
C)
x y-x
D)
y x-y
Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values of x. Find the derivative with respect to x of the given combination at the given value of x. x f(x) g(x) f (x) g (x) 366) 3 1 16 366) 6 3 -5 4 -3 3 2 g(x + f(x)), x = 3 A) -5
B) 21
C) -35
D) -30
5 C) 2 5x + 7
D) - 25 5x + 7 4
Find y .
367) y = 5x + 7 A) -
367) 1
4(5x + 7)3/2
B) -
25
4(5x + 7)3/2
Solve the problem.
368) Suppose that the radius r and volume V = 4 r3 of a sphere are differentiable functions of t. Write 3
an equation that relates dV/dt to dr/dt. A) dV = 4 r2 dr B) dV = 3r2 dr dt dt dt dt
C) dV = 4 r2 dr dt
3
dt
D) dV = 4 dr dt
dt
The equation gives the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds). 369) s = -5 + 7 cos t Find the body's speed at time t = /3 sec. A) - 7 3 m/sec B) - 7 m/sec C) 7 m/sec D) 7 3 m/sec 2 2 2 2 Solve the problem. 370) Under standard conditions, molecules of a gas collide billions of times per second. If each molecule has diameter t, the average distance between collisions is given by 1 L= , 2 t2 n where n, the volume density of the gas, is a constant. Find
2
A) d L = -
2 t4 n
2
6
C) d L = dt2
2
6
dt2
d2 L . dt2
B) d L = -
2 t3 n
2
2
D) d L = -
2 t4 n
dt2
76
2
dt2
2 t2 n
368)
369)
370)
The function f(x) changes value when x changes from x 0 to x 0 + dx. Find the approximation error
f - df . Round your
answer, if appropriate. 371) f(x) = 1 , x0 = 3, dx = 0.8 x2
A) 0.16362
371) B) 0.0174
C) 0.01223
D) 0.20517
Solve the problem.
372) A ball dropped from the top of a building has a height of s = 400 - 16t2 meters after t seconds.
372)
How long does it take the ball to reach the ground? What is the ball's velocity at the moment of impact? A) 10 sec, -80 m/sec B) 5 sec, -160 m/sec
C) 5 sec, 160 m/sec
D) 25 sec, -800 m/sec
Find the value of (f g) at the given value of x. 373) f(u) = tan u , u = g(x) = x2 , x = 10 2
A) 20
373)
B) -10
D) 10
C) -100
Solve the problem. 374) A manufacturer contracts to mint coins for the federal government. How much variation dr in the radius of the coins can be tolerated if the coins are to weigh within 1/10 of their ideal weight? Assume that the thickness does not vary. A) 10% B) 5.0% C) 0.10% D) 0.050%
375) A charged particle of mass m and charge q moving in an electric field E has an acceleration a
374)
375)
given by qE a= , m
where q and E are constants. Find
2
A) d a = qE dm2
2m
d2 a . dm2
2
2
B) d a = - qE dm2
C) d a = qE
m2
dm2
m3
2
D) d a = 2qE dm2
m3
376) Suppose that the velocity of a falling body is v = k (k a constant) at the instant the body has fallen s
s meters from its starting point. Find the body's acceleration as a function of s. 2 2 A) a = - 1 B) a = k C) a = - k s2 s3 s3
376)
D) a = - k
s2
The function s = f(t) gives the position of a body moving on a coordinate line, with s in meters and t in seconds. 377) s = - t3 + 7t2 - 7t, 0 t 7 377) Find the body's displacement and average velocity for the given time interval. A) -49 m, -14 m/sec B) -49 m, -7 m/sec
C) 49 m, 7 m/sec
D) 637 m, 91 m/sec 77
Find the slope of the curve at the indicated point. 378) y = x2 + 9x + 8, x = -1
A) m = -8
378)
B) m = -2
C) m = 15
D) m = 7
Find an equation for the tangent to the curve at the given point. 379) y = x2 + 2, (-3, 11)
A) y = -3x - 7
B) y = -6x - 16
379) C) y = -6x - 7
D) y = -6x - 14
Write a differential formula that estimates the given change in volume or surface area. 380) The change in the volume V = r2 h of a right circular cylinder when the height changes from h
0
380)
to h 0 + dh and the radius does not change
A) dV = r2 h0 dh
B) dV = r0 2 dr
C) dV = 2 rh0 dh
D) dV = r2 dh
Solve the problem. 381) A balloon used in surgical procedures is cylindrical in shape. As it expands outward, assume that the length remains a constant 90.0 mm. Find the rate of change of surface area with respect to radius when the radius is 0.050 mm. (Answer can be left in terms of ). A) 180.0 mm2/mm B) 180.2 mm2/mm
C) 90.2 mm2 /mm Find the derivative. 382) s = 4t2 + 6t - 8
A) 8t + 6
381)
D) 90.1 mm2 /mm
382) B) 4t2 + 6
C) 4t + 6
D) 8t2 + 6
Solve the problem.
383) The area A = r2 of a circular oil spill changes with the radius. To the nearest tenth of a square foot, how much does the area increase when the radius changes from 3 ft to 3.1 ft? A) 1.2 ft2 B) 18.8 ft2 C) 9.4 ft2 D) 1.9 ft2
78
383)
The figure shows the velocity v or position s of a body moving along a coordinate line as a function of time t . Use the figure to answer the question. 384) s (m) 384)
t (sec)
What is the body's velocity when t = 2.5 sec? A) 3 m/sec B) 2 m/sec
C) 1 m/sec
D) -2 m/sec
Solve the problem. 385) The driver of a car traveling at 60 ft/sec suddenly applies the brakes. The position of the car is s = 60t - 3t2 , t seconds after the driver applies the brakes. How far does the car go before coming to a stop? A) 10 ft
B) 1200 ft
C) 300 ft
D) 600 ft
Graph the equation and its tangent. 386) Graph y = x2 + 2x - 5 and the tangent to the curve at the point whose x-coordinate is -2.
79
385)
386)
A)
B)
C)
D)
Find y .
387) y = (6x - 4)(2x + 1) A) 24x - 1
387) B) 12x - 2
C) 24x - 14
D) 24x - 2
Given the graph of f, find any values of x at which f is not defined.
388)
388)
A) x = -2, 2
B) x = -3, 3
C) x = -2, 0, 2
80
D) x = -3, 0, 3
Find the derivative of the function. 5 389) h(x) = cos x 1 + sin x
A) -5 sin x
389)
4
B) 5 cos x
cos x
C) - 4 sin x cos x
4
1 + sin x 4
cos x 4 1 + sin x
D) -5 cos x
(1 + sin x)5
Find the value of (f g) at the given value of x. 390) f(u) = u3 , u = g(x) = x + 4 , x = 0 x-2
A) - 18
B) 18
Use the formula f'(x) = lim z x
391) f(x) =
390) C) - 12
D) 12
f(z) - f(x) to find the derivative of the function. z-x
3 x+8
A) -
391)
3
(x + 8)2
B) -
3 (x + 8)
C) - 3
x2
D)
3
(x + 8)2
Find y .
392) y = 3 + x 3 - x x
392)
x
A) 18 + 2x x3
C) - 18 + 2x
B) - 9 - 2x x3
x
Given y = f(u) and u = g(x), find dy/dx = f (g(x))g (x). 393) y = csc u, u = x3 + 2x
D) - 18 - 2x x3
393)
A) -(3x2 + 2) csc x cot x
B) -(3x2 + 2) csc (x3 + 2x) cot (x3 + 2x)
C) -(3x2 + 2) cot2 (x3 + 2x)
D) - csc (x3 + 2x) cot (x3 + 2x)
Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 394) u(2) = 9, u (2) = 4, v(2) = -2, v (2) = -4. 394) d (2u - 4v) at x = 2 dx
A) 26
C) 10
B) -8
D) 24
Provide an appropriate response. 395) If x + y = x2 + y2 and dx/dt = 12, then what is dy/dt when x = 1 and y = 0? x-y
A) 12
B) - 1
C) 1
12
12
81
395) D) -12
Determine if the piecewise defined function is differentiable at the origin. 5/6 if x 0 396) g(x) = x 1/6 x if x < 0 A) Not differentiable B) Differentiable
396)
Given y = f(u) and u = g(x), find dy/dx = f (g(x))g (x). 397) y = sin u, u = 3x + 19 A) - 3 cos (3x + 19)
397) B) cos (3x + 19) D) - cos (3x + 19)
C) 3 cos (3x + 19) Provide an appropriate response.
398) At the two points where the curve x2 - xy + y2 = 4 crosses the x-axis, the tangents to the curve are parallel. What is the common slope of these tangents? A) -2 B) 2 C) -1
398)
D) 1
Find y .
399) y = x + 1 x - 1 x
399)
x
A) 2x + 1
B) 2x + 2
x3
C) 2x + 1
x3
D) 2x - 1
x2
x2
Solve the problem.
400) The size of a population of mice after t months is P = 100(1 + 0.2t + 0.02t2 ). Find the growth rate at t = 13 months. A) 72 mice/month
400)
B) 36 mice/month D) 144 mice/month
C) 172 mice/month Find the derivative of the function. 401) y = (x + 1)2(x2 + 1)-3
401) 2
2
B) 2(x + 1)(2x - 3x - 1)
A) 2(x + 1)(2x + 3x - 1)
(x2 + 1)4
(x2 + 1)4 2
2
C) -2(x + 1)(2x - 3x - 1)
D) -2(x + 1)(2x + 3x - 1)
(x2 + 1)4
(x2 + 1)4
Find dy.
402) y = sin(2x2) A) 4 cos(2x2 ) dx C) -4 cos(2x2 ) dx
402) B) 4x cos(2x2 ) dx D) -4x cos(2x2) dx
82
Find the value of (f g) at the given value of x. 403) f(u) = u - 1 , u = g(x) = x, x = 64 u+1
A) 1 8
403)
B) 1
C) 1
81
D) 1
64
648
The equation gives the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds). 404) s = 7 + 11 cos t Find the body's velocity at time t = /3 sec. A) - 11 m/sec B) 11 3 m/sec C) - 11 3 m/sec D) 11 m/sec 2 2 2 2
404)
Estimate the slope of the curve at the indicated point.
405)
405)
A) 0
B) Undefined
D) 1
C) -1
Find y .
406) y = 1 tan(8x - 5)
406)
5
A) 2 sec(8x - 5)
B) 2 sec2 (8x - 5) tan(8x - 5)
C) 8 sec2 (8x - 5)
D) 128 sec2(8x - 5) tan(8x - 5)
5
5
5
5
Solve the problem. 407) A charged particle of mass m and charge q moving in an electric field E has an acceleration a given by qE a= , m where q and E are constants. Find
A) da = qEm dm
da . dm
C) da = - m
B) da = - qE dm
m2
dm
qE
D) da = qE dm
m2
Find an equation for the tangent to the curve at the given point. 408) y = x - x2, (-1, -2)
A) y = -3x + 1
407)
B) y = 3x + 1
408) C) y = -x + 1
83
D) y = -x - 1
The figure shows the velocity v or position s of a body moving along a coordinate line as a function of time t . Use the figure to answer the question. 409) v (ft/sec) 409)
t (sec)
When is the body moving backward? A) 7 < t < 10
B) 0 < t < 4 D) 0 < t < 2, 6 < t < 7
C) 4 < t < 7
Solve the problem. 410) Find the points where the graph of the function have horizontal tangents. f(x) = 3x2 + 5x + 1
B) 5 , - 113
A) (0, -1)
6
C) - 5 , - 13
6
6
Use implicit differentiation to find dy/dx and d2 y/dx 2 . 411) x3/5 + y3/5 = 10 2/5 d2 y 2y1/5 - 2x = ; x2/5 dx2 5x7/5y3/5
B) dy = y
2/5 d2 y 2x3/5 + 2y3/5 =; y2/5 dx2 5x1/5y7/5
D) dy = - y
dx
C) dy = x dx
D) (-11, 243)
12
411)
A) dy = - y dx
410)
dx
2/5 d2 y 2x3/5 + 2y3/5 =; 2/5 2 x dx 5x7/5y1/5 2/5 d2 y 2x3/5 + 2y3/5 = ; x2/5 dx2 5x7/5y1/5
Provide an appropriate response. 412) If y x + 1 = 12 and dx/dt = 8, then what is dy/dt when x = 15 and y = 3? A) - 4 B) 3 C) 4 3 4 3
412) D) - 3 4
Find the derivative of the function. 7 413) s = t + 7t + 4 t2
413)
A) ds = 8t9 + 8t2 + 8t
B) ds = 5t4 + 7 + 8
C) ds = 5t4 - 7 - 8
D) ds = t4 - 7 - 4
dt
dt dt
t2
t3
dt
84
t2
t2
t3
t3
At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 414) x6 y6 = 64, normal at (2, 1) 414)
A) y = 1 x 32
C) y = - 1 x + 2
B) y = -2x + 5
2
D) y = 2x - 3
The graphs show the position s, velocity v = ds/dt, and acceleration a = d2 s/dt2 of a body moving along a coordinate line as functions of time t. Which graph is which?
415)
415)
A) B = position, C = velocity, A = acceleration B) C = position, B = velocity, A = acceleration C) B = position, A = velocity, C = acceleration D) A = position, C = velocity, B = acceleration 416)
416)
A) A = position, C = velocity, B = acceleration B) A = position, B = velocity, C = acceleration C) C = position, A = velocity, B = acceleration D) B = position, C = velocity, A = acceleration
85
Solve the problem.
417) Water is falling on a surface, wetting a circular area that is expanding at a rate of 9 mm2 /s. How
417)
fast is the radius of the wetted area expanding when the radius is 161 mm? (Round your answer to four decimal places.) A) 0.0089 mm/s B) 0.0559 mm/s C) 112.3991 mm/s D) 0.0178 mm/s
Find an equation for the tangent to the curve at the given point. 418) h(x) = t3 - 36t - 5, (6, -5)
A) y = 72t - 437
B) y = 67t - 437
418) C) y = -5
D) y = 72t - 5
Find the second derivative of the function. 6 419) s = t + 3t + 4 t2
419)
2
2
A) d s = 12t2 + 6 + 24 dt2
t3
B) d s = 4t2 - 3 - 8
t4
dt2
2
t
t4
2
C) d s = 12t4 + 6 + 24 dt2
t3
D) d s = 4t3 - 3 - 8
t2
dt2
t2
t3
Find the value of (f g) at the given value of x. 420) f(u) = u2 , u = g(x) = x5 + 2, x = 0
A) 4
420)
B) 0
D) 15
C) -30
Find the second derivative. 421) w = z -2 - 1 z
A) -2z -4 - 2
z3
421) B) -2z -3 + 1
C) 6z -4 + 2
z2
z3
D) 6z -4 - 2
z3
Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values of x. Find the derivative with respect to x of the given combination at the given value of x. x f(x) g(x) f (x) g (x) 422) 3 1 4 422) 6 3 4 -3 3 5 -4 g(x), x = 3 A) 3 4
B) 1
C) - 1
4
2 3
D) 1
2 3
Solve the problem. Round your answer, if appropriate. 423) A man 6 ft tall walks at a rate of 2 ft/sec away from a lamppost that is 13 ft high. At what rate is the length of his shadow changing when he is 60 ft away from the lamppost? (Do not round your answer) A) 12 ft/sec B) 6 ft/sec C) 20 ft/sec D) 12 ft/sec 7 19 19
86
423)
Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 424) u(2) = 10, u (2) = 4, v(2) = -3, v (2) = -5. 424) d (uv) at x = 2 dx
B) 62
A) -38 Find the limit.
425) lim 2 sin x
/2
C) -62
D) 55
+ cot x cos x + 6 sin x
A) 4
425) C) 1
B) -1
D) 0
Find the linearization L(x) of f(x) at x = a. 426) f(x) = 2x2 + 5x + 5, a = 3
A) L(x) = 17x + 23
426)
B) L(x) = 7x - 13
C) L(x) = 7x + 23
D) L(x) = 17x - 13
The figure shows the velocity v or position s of a body moving along a coordinate line as a function of time t . Use the figure to answer the question. 427) s (m) 427)
t (sec)
When is the body moving backward? A) 8 < t 10
B) 8 < t < 9 D) 2 < t < 3, 4 < t < 5, 7 < t < 9
C) 2 < t < 3, 4 < t < 5, 7 < t < 8 Solve the problem.
428) Assume that a watermelon dropped from a tall building falls y = 16t2 ft in t sec. Find the
428)
watermelon's average speed during the first 3 sec of fall and the speed at the instant t = 3 sec. A) 24 ft/sec; 48 ft/sec B) 49 ft/sec; 98 ft/sec
C) 48 ft/sec; 96 ft/sec
D) 96 ft/sec; 49 ft/sec
429) At time t, the position of a body moving along the s-axis is s = t3 - 12t2 + 36t m. Find the total distance traveled by the body from t = 0 to t = 3. A) 37 m B) 84 m
C) 27 m
87
D) 59 m
429)
Find the derivative. 430) p = 2 + sec q 2 - sec q
A) dp = dq
430) 4 sin q
B) dp = -
(2 cos q - 1)2
2 C) dp = 4 tan q dq
4 sin q
dq
(2 cos q - 1)2 2
D) dp = - 2 sec q tan q
(2 - sec q)2
dq
Given y = f(u) and u = g(x), find dy/dx = f (g(x))g (x). 431) y = sin u, u = cos x A) - cos x sin x
(2 - sec q)2
431) B) cos x sin x D) sin(cos x) sin x
C) - cos(cos x) sin x Provide an appropriate response.
432) Find an equation for the tangent to the curve y = 18
x2 + 2
A) y = 4x + 2
B) y = -4
at the point (1, 6).
C) y = -2x + 8
432) D) y = -4x + 10
Find the indicated derivative. 433) dp if p = 1 dq q+9
433)
A) -
1 2(q + 9) q + 9
B)
1 2(q + 9) q + 9
C) -
1 (q + 9) q + 9
D) -
1 2 q+9
The function f(x) changes value when x changes from x 0 to x 0 + dx. Find the approximation error
f - df . Round your
answer, if appropriate. 434) f(x) = x - x2 , x0 = 7, dx = 0.04
A) 0.104
434)
B) 0.208
C) 0.0016
D) 0.1056
Solve the problem. 435) The volume of a square pyramid is related to the length of a side of the base s and the height h by 1 the formula V = s2 h. How is dV/dt related to ds/dt if h is constant? 3
A) dV = 2s ds dt
3 dt
2
B) dV = s ds
C) dV = 2hs ds
D) dV = h ds
r 1/3
1/3
r 1/3
dt
dt
3 dt
3 dt
dt
435)
3 dt
Find dr/d .
436) 4/3 + r4/3 = 1 A)
1/3
r
436) B) -
C) -
88
r
D)
Find the linearization L(x) of f(x) at x = a. 437) f(x) = 1 , a = 0 2x - 7
437)
A) L(x) = 2 x + 1 49
B) L(x) = - 2 x - 1
49
49
C) L(x) = - 2 x + 1 49
7
D) L(x) = 2 x - 1
49
49
7
Find y .
438) y = 6 sin(4x + 9) A) -24 sin(4x + 9)
438) B) -96 cos(4x + 9)
C) -96 sin(4x + 9)
89
D) 24 cos(4x + 9)
Answer Key Testname: CHAPTER 3
1) C 2) A 3) B 4) B 5) C 6) B 7) The function f is not differentiable at x = 0 and hence does not have a tangent at the origin. 8) tan x = sin x = sin x sec x cos x
d d sin x · tan x + 1 = tan2 x + 1 = sec2 x (tan x) = (sin x sec x) = sin x sec x tan x + sec x cos x = dx dx cos x
9) The curve has no tangent whose slope is -2. The derivative of the curve, y = 3x2 + 4, is always positive and thus
never equals -2. 6 5 10) y = x , y = x , y 3600 600
=
x4 x3 (5) x2 (6) x (7) 1 (n) , y(4) = ,y = , y = , y = , y = 0 for n 8 120 30 10 5 5
11) g (4) = -5 12)
No, the graph of y = -tan x does not appear to have a smallest slope. The slope of the graph of y = - tan x decreases without bound as x approaches /2 or - /2. 13) The curve y = (x + 3)3 has a horizontal tangent at x = -3. The derivative of the curve is y = 3(x + 3)2 , which equals
zero at x = -3. A curve has a horizontal tangent wherever its derivative equals zero. 14) The function is not differentiable at the x = 0 and hence does not have a tangent at the origin. n-1 dx dx 15) dy = naxn-1 dx, therefore dy = nax =n . y n x ax Therefore,
dy dx = n . Thus, the relative uncertainty in the dependent variable y is always n times the relative y x
uncertainty in the independent variable x.
16) - cos x 17) 5 2
90
Answer Key Testname: CHAPTER 3
18) There is no difference at all. At x = a, the slope of the tangent = lim f(x) - f(a) = f (x). x a
x-a
19) c = 16 20) y = - 4 x + 2 + 4; - 2 21) cos x 5
22) The function y = x decreases (as x increases) over no intervals of x-values. Its derivative y = x4 is negative for no 5
values of x. The function decreases (as x increases) wherever its derivative is negative.
23)
No, the slope of the graph of y = - tan x is never positive. The slope at any point is equal to the derivative, which is y = - sec2 x. Since sec2 x is never negative, y = - sec2 x is never positive, and the slope of the graph is never
positive. 24) The function is not differentiable at x = 0 because it is discontinuous at x = 0. The graph does not have a vertical tangent at (0, 5) since f remains bounded as x approaches zero from either side. 25) The function f is differentiable at x = o and hence has a tangent at the origin. 1 26) . The slopes of g(x) and its linear approximation are deviating from each other more slowly than the slopes of the 3 corresponding curves for f(x) as long as g (x) < f (x) . Since f (x) = 2 and g (x) = 6x, then 1 1 1 1 g (x) < f (x) 6x < 2 or x < or x . Thus, . 3 3 3 3
27) Yes, a tangent line to a graph can intersect the graph at more than one point. For example, the graph y = x3 - 2x2 has a horizontal tangent at x = 0. It intersects the graph at both (0, 0) and (2, 0). 28) y = 2x2 + 7x + 4, y = 4x + 7, y = 4, y(n) = 0 for n 4
29) 7 30) g (4) = 10 31) y' = 3x2 - 6x + 3; y'' = 6x -6; y''' = 6; y(n) = 0 for n 4 32) [5, )
33) The curve y = x never has a negative slope. The derivative of the curve is y = 1 , which is never negative. A 2 x
curve only has a negative slope where its derivative is negative. Since the derivative of y = curve never has a negative slope.
91
x is never negative, the
Answer Key Testname: CHAPTER 3
34) The function y = 2x2 increases (as x increases) over the interval 0 < x < . Its derivative y = 4x is positive for x > 0. The function increases (as x increases) wherever its derivative is positive. 35) y = 2x and y = - 1 x; The tangents are perpendicular. 2
36) - sin x 37)
Yes, the graph of y = -tan x does appear to have a largest slope. It occurs at x = 0, when the slope equals -1. 2 38) (a) Q(x) = 1 - x + x 3 9 27 (b) I expect the quadratic approximation to be more accurate than the linearization. The linearization accounts for how the value of f(x) is changing at the point x = a, but the quadratic approximation also accounts for the change in this change by incorporating the second derivative into the approximation.
39) B 40) D 41) B 42) D 43) B 44) C 45) C 46) B 47) B 48) C 49) C 50) D 51) D 52) B 53) D 54) B 55) A 56) C 92
Answer Key Testname: CHAPTER 3
57) D 58) A 59) C 60) D 61) D 62) C 63) D 64) A 65) D 66) A 67) A 68) D 69) B 70) A 71) D 72) C 73) D 74) C 75) A 76) D 77) B 78) C 79) D 80) C 81) D 82) D 83) A 84) C 85) A 86) A 87) A 88) C 89) A 90) C 91) B 92) A 93) A 94) A 95) B 96) B 97) A 98) B 93
Answer Key Testname: CHAPTER 3
99) C 100) B 101) B 102) C 103) D 104) B 105) B 106) B 107) D 108) C 109) D 110) D 111) A 112) A 113) B 114) A 115) C 116) D 117) C 118) C 119) D 120) A 121) B 122) C 123) D 124) A 125) C 126) C 127) D 128) D 129) D 130) C 131) A 132) B 133) A 134) B 135) B 136) C 137) D 138) B 139) A 140) D 94
Answer Key Testname: CHAPTER 3
141) B 142) B 143) D 144) B 145) B 146) A 147) C 148) B 149) A 150) C 151) A 152) A 153) D 154) B 155) D 156) D 157) C 158) A 159) A 160) A 161) C 162) D 163) C 164) C 165) B 166) B 167) A 168) C 169) D 170) A 171) C 172) C 173) B 174) B 175) C 176) B 177) D 178) D 179) A 180) C 181) C 182) C 95
Answer Key Testname: CHAPTER 3
183) B 184) B 185) D 186) A 187) D 188) B 189) B 190) D 191) B 192) B 193) B 194) C 195) A 196) B 197) C 198) A 199) A 200) C 201) C 202) C 203) A 204) A 205) D 206) A 207) D 208) D 209) B 210) A 211) C 212) B 213) B 214) A 215) C 216) D 217) A 218) B 219) B 220) A 221) B 222) B 223) C 224) C 96
Answer Key Testname: CHAPTER 3
225) D 226) C 227) B 228) C 229) D 230) B 231) B 232) A 233) A 234) C 235) C 236) B 237) C 238) C 239) A 240) C 241) B 242) A 243) A 244) B 245) D 246) B 247) D 248) C 249) D 250) D 251) B 252) A 253) C 254) C 255) C 256) B 257) D 258) B 259) D 260) B 261) D 262) B 263) C 264) C 265) C 266) B 97
Answer Key Testname: CHAPTER 3
267) B 268) C 269) D 270) C 271) B 272) B 273) B 274) C 275) C 276) D 277) D 278) C 279) C 280) A 281) D 282) D 283) B 284) B 285) B 286) B 287) C 288) A 289) B 290) D 291) C 292) C 293) D 294) C 295) C 296) B 297) B 298) D 299) C 300) A 301) C 302) A 303) A 304) C 305) A 306) D 307) B 308) D 98
Answer Key Testname: CHAPTER 3
309) D 310) C 311) D 312) C 313) A 314) A 315) C 316) C 317) C 318) C 319) D 320) C 321) B 322) D 323) D 324) A 325) C 326) C 327) B 328) C 329) D 330) B 331) D 332) A 333) D 334) B 335) D 336) D 337) B 338) D 339) C 340) B 341) C 342) A 343) D 344) C 345) D 346) B 347) A 348) D 349) C 350) B 99
Answer Key Testname: CHAPTER 3
351) D 352) D 353) B 354) C 355) C 356) A 357) B 358) C 359) C 360) B 361) C 362) D 363) A 364) A 365) B 366) C 367) B 368) A 369) D 370) C 371) B 372) B 373) D 374) B 375) D 376) D 377) B 378) D 379) C 380) D 381) B 382) A 383) D 384) D 385) C 386) A 387) D 388) A 389) D 390) A 391) A 392) D 100
Answer Key Testname: CHAPTER 3
393) B 394) D 395) A 396) A 397) C 398) B 399) B 400) A 401) D 402) B 403) D 404) C 405) B 406) D 407) B 408) B 409) B 410) C 411) D 412) D 413) C 414) D 415) A 416) A 417) A 418) A 419) A 420) B 421) D 422) A 423) A 424) C 425) C 426) D 427) D 428) C 429) A 430) A 431) C 432) D 433) A 434) C 101
Answer Key Testname: CHAPTER 3
435) C 436) C 437) B 438) C
102
Chapter 4
Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine from the graph whether the function has any absolute extreme values on the interval [a, b].
1)
1)
A) Absolute maximum only. B) Absolute minimum only. C) No absolute extrema. D) Absolute minimum and absolute maximum.
1
Solve the problem. 2) Find the table that matches the graph below.
2)
A)
B) x f (x) a does not exist b does not exist 8 c 5
x f (x) a does not exist b0 2 c 5
C)
D) x f (x) a0 b0 8 c 5
x f (x) a0 b0 2 c 5
3) Find the graph that matches the given table. x -1 1 3
3)
f (x) 0 does not exist 0
A)
2
B)
C)
D)
3
4) Find the graph that matches the given table.
4)
x f (x) -2 0 1 6
A)
B)
C)
D)
5) The graphs below show the first and second derivatives of a function y = f(x). Select a possible graph f that passes through the point P. f
f
4
5)
A)
B)
[NOTE: Graph vertical scales may vary from graph to graph.]
[NOTE: Graph vertical scales may vary from graph to graph.]
C)
D)
[NOTE: Graph vertical scales may vary from graph to graph.]
[NOTE: Graph vertical scales may vary from graph to graph.]
Determine from the graph whether the function has any absolute extreme values on the interval [a, b].
6)
6)
A) No absolute extrema. B) Absolute minimum and absolute maximum. C) Absolute maximum only. D) Absolute minimum only.
5
Solve the problem. 7) Find the table that matches the given graph.
7)
A)
B) x f (x) a does not exist b does not exist c -1
x f (x) a does not exist b 0 c 1
C)
D) x f (x) a 0 b 0 c -1
x f (x) a does not exist b 0 c -1
Which of the graphs shows the solution of the given initial value problem? 8) dy = -4x, y = 0 when x = 1 dx
A)
6
8)
B)
C)
D)
7
9) dy = 4x, y = 0 when x = -1
9)
dx
A)
B)
C)
8
D)
Solve the problem. 10) The graphs below show the first and second derivatives of a function y = f(x). Select a possible graph f that passes through the point P. f
f
A)
B)
[NOTE: Graph vertical scales may vary from graph to graph.]
[NOTE: Graph vertical scales may vary from graph to graph.]
9
10)
C)
D)
[NOTE: Graph vertical scales may vary from graph to graph.]
[NOTE: Graph vertical scales may vary from graph to graph.]
11) The graphs below show the first and second derivatives of a function y = f(x). Select a possible graph of f that passes through the point P. f
f
A)
B)
[NOTE: Graph vertical scales may vary from graph to graph.]
[NOTE: Graph vertical scales may vary from graph to graph.]
10
11)
C)
D)
[NOTE: Graph vertical scales may vary from graph to graph.]
[NOTE: Graph vertical scales may vary from graph to graph.]
Determine from the graph whether the function has any absolute extreme values on the interval [a, b].
12)
12)
A) No absolute extrema. B) Absolute minimum and absolute maximum. C) Absolute minimum only. D) Absolute maximum only. Solve the problem. 13) The graphs below show the first and second derivatives of a function y = f(x). Select a possible graph f that passes through the point P. f
f
11
13)
A)
B)
[NOTE: Graph vertical scales may vary from graph to graph.]
[NOTE: Graph vertical scales may vary from graph to graph.]
C)
D)
[NOTE: Graph vertical scales may vary from graph to graph.]
[NOTE: Graph vertical scales may vary from graph to graph.]
14) The graphs below show the first and second derivatives of a function y = f(x) . Select a possible graph of f that passes through point P. f
f
12
14)
A)
B)
[NOTE: Graph vertical scales may vary from graph to graph.]
[NOTE: Graph vertical scales may vary from graph to graph.]
C)
D)
[NOTE: Graph vertical scales may vary from graph to graph.]
[NOTE: Graph vertical scales may vary from graph to graph.]
15) The graphs below show the first and second derivatives of a function y = f(x). Select a possible graph f that passes through the point P. f
f
13
15)
A)
B)
[NOTE: Graph vertical scales may vary from graph to graph.]
[NOTE: Graph vertical scales may vary from graph to graph.]
C)
D)
[NOTE: Graph vertical scales may vary from graph to graph.]
[NOTE: Graph vertical scales may vary from graph to graph.]
14
16) Select an appropriate graph of a twice-differentiable function y = f(x) that passes through the points (- 2, 1), -
6 5 , , (0, 0), 3 9
6 5 , and ( 2, 1), and whose first two derivatives have the 3 9
following sign patterns. y :
+
- 2
+ y :
+ 0
-
6 3
2
+ 6 3
A)
B)
C)
D)
15
16)
17) Using the following properties of a twice-differentiable function y = f(x), select a possible graph of f.
x x<2 -2 -2 < x < 0 0 0<x<2 2 x>2
y 9 -7 -23
Derivatives y > 0,y < 0 y = 0,y < 0 y < 0,y < 0 y < 0,y = 0 y < 0,y > 0 y = 0,y > 0 y > 0,y > 0
A)
B)
C)
D)
16
17)
Determine from the graph whether the function has any absolute extreme values on the interval [a, b].
18)
18)
A) Absolute maximum only. B) No absolute extrema. C) Absolute minimum only. D) Absolute minimum and absolute maximum. Provide an appropriate response.
19) Determine the values of constants a and b so that f(x) = ax2 + bx has an absolute maximum at the point (3, 9). A) a = 1, b = 6
B) a = -1, b = 3
C) a = 1, b = 3
19)
D) a = -1, b = 6
Determine from the graph whether the function has any absolute extreme values on the interval [a, b].
20)
20) A) Absolute maximum only. B) Absolute minimum and absolute maximum. C) Absolute minimum only. D) No absolute extrema.
17
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 21) Show that if h > 0, applying Newton's method to x - 8, x 8 f(x) = x<8 - 8 - x,
21)
leads to x2 = h if x0 = h and to x2 = -h if x0 = -h when 0 < 8 < h.
Solve the problem. 22) You are planning to close off a corner of the first quadrant with a line segment 23 units long running from (x, 0) to (0, y). Show that the area of the triangle enclosed by the segment is largest when x = y.
Provide an appropriate response. 23) Imagine there is a function for which f (x) = 0 for all x. Does such a function exist? Is it reasonable to say that all values of x are critical points for such a function? Is it reasonable to say that all values of x are extreme values for such a function. Give reasons for your answer.
22)
23)
Solve the problem.
24) Use Newton's method to estimate the one real solution of 3x3 - 2x - 5 = 0. Start with
24)
x1 = 1.5 and then find x2 .
Answer the question. 25) A trucker handed in a ticket at a toll booth showing that in 3 hours he had covered 225 miles on a toll road with speed limit 65 mph. The trucker was cited for speeding. Why?
25)
Provide an appropriate response. 26) Let f have a derivative on an interval I. f' has successive distinct zeros at x = 1 and x = 5. Prove that there can be at most one zero of f on the interval (1, 5).
26)
18
27) Decide if the statement is true or false. If false, explain.
27)
1 The points (-1, -1) and (1, 1) lie on the graph of f(x) = . Therefore, the Mean Value x Theorem says that there exists some value x = c on (-1, 1) for which f (x) =
1 - (-1) = 1. 1 - (-1)
28) Let f(x) = x3 - 9x
28)
(a) Does f (0) exist? (b) Does f (3) exist? (c) Does f (-3) exist? (d) Determine all extrema of f.
Solve the problem.
29) Use Newton's method to estimate the solutions of the equation 4x - 2x2 + 5 = 0. Start with
29)
x1 = 1.5 for the right-hand solution and with x0 = -1 for the solution on the left. Then, in each case find x2 .
Provide an appropriate response. 30) The accompanying figure shows a portion of the graph of a function that is twice-differentiable at all x except at x = p. At each of the labeled points, classify y and y as positive, negative, or zero.
31) If f(x) is a differentiable function and f (c) = 0 at an interior point c of f's domain, and if
30)
31)
f (x) > 0 for all x in the domain, must f have a local minimum at x = c? Explain.
Answer the question. 32) It took 28 seconds for the temperature to rise from 7° F to 152° F when a thermometer was taken from a freezer and placed in boiling water. Although we do not have detailed knowledge about the rate of temperature increase, we can know for certain that, at some 145 time, the temperature was increasing at a rate of ° F/sec. Explain. 28
19
32)
Provide an appropriate response. 33) Explain why the following four statements ask for the same information. (i) Find the roots of f(x) = 2x3 - 3x - 1
33)
(ii) Find the x-coordinates of the intersections of the curve y = x3 with the line y = 3x + 1. (iii) Find the x-coordinates of the points where the curve y = x3 - 3x crosses the horizontal line y = 1.
(iv) Find the values of x where the derivative of g(x) =
1 4 3 2 x - x - x + 5 equals zero. 2 2
Solve the problem. 34) A team of engineers is testing an experimental high-voltage fuel cell with a potential application as an emergency back-up power supply in cell phone transmission towers. Unfortunately, the voltage of the prototype cell drops with time according to the equation V(t) = -0.0306t3 + 0.373t2 - 2.16t + 15.1, where V is in volts and t is the time of operation in
34)
hours. The cell provides useful power as long as the voltage remains above <v> volts. Use Newton's method to find the useful working time of the cell to the nearest tenth of an hour (that is, solve V(t) = 7.3 volts). Use t = 7 hours as your initial guess and show all your work.
Answer the problem. 35) Use the following function and a graphing calculator to answer the questions. f(x) =
2x + 0.8 sin x, [0, 2 ]
a). Plot the function over the interval to see its general behavior there. Sketch the graph below.
b). Find the interior points where f = 0 (you may need to use the numerical equation solver to approximate a solution). You may wish to plot f as well. List the points as ordered pairs (x, y).
c). Find the interior points where f does not exist. List the points as ordered pairs (x, y).
20
35)
d). Evaluate the function at the endpoints and list these points as ordered pairs (x, y).
e). Find the function's absolute extreme values on the interval and identify where they occur.
Show that the function has exactly one zero in the given interval. 36) r( ) = 3 cot + 1 + 5, (0, ). 2
36)
Provide an appropriate response. 37) Let f(x) = (x - 3)2/3
37)
(a) Does f (3) exist? (b) Show that the only local extreme value of f occurs at x = 3. (c) Does the result of (b) contradict the Extreme Value Theorem? (d) Repeat parts (a) and (b) for f(x) = (x - c)2/3.
Give reasons for your answers.
38) Suppose that g(0) = -4 and that g (t) = -2 for all t. Must g(t) = -2t - 4 for all t?
38)
Solve the problem.
39) The curve y = tan x crosses the line y = 2x between x = 0 and x =
2
. Use Newton's
39)
method to find where the line and the curve cross. (Round your answer to two decimal places.)
Provide an appropriate response.
40) Consider the quartic function f(x) = ax4 + bx3 + cx2 + dx + e, a 0. Must this function
40)
41) Sketch a smooth curve through the origin with the following properties:
41)
Answer the problem. 42) Use the following function and a graphing calculator to answer the questions.
42)
have at least one critical point? Give reasons for your answer. (Hint: Must f (x) = 0 for some x?) How many local extreme values can f have?
f (x) > 0 for x < 0; f (x) < 0 for x > 0; f (x) approaches 0 as x approaches - ; and f (x) approaches 0 as x approaches .
f(x) = x4 - 4x2 + 3x + 3, [-0.5, 1.8] a). Plot the function over the interval to see its general behavior there. Sketch the graph below.
21
b). Find the interior points where f = 0 (you may need to use the numerical equation solver to approximate a solution). You may wish to plot f as well. List the points as ordered pairs (x, y).
c). Find the interior points where f does not exist. List the points as ordered pairs (x, y).
d). Evaluate the function at the endpoints and list these points as ordered pairs (x, y).
e). Find the function's absolute extreme values on the interval and identify where they occur.
Solve the problem. 43) Use Newton's method to estimate the solution of the equation 2sin x - 4x + 1 = 0. Start with x1 = 1.5. Then, in each case find x2 . Provide an appropriate response. 44) Suppose Newton's Method is used with an initial guess xo that lies at a critical point (a, b), b 0. What happens to x 1 and later approximations? Give reasons for your answer.
22
43)
44)
Solve the problem.
45) Use Newton's method to estimate the solutions of the equation 3x4 + 2x - 4 = 0. Start with
45)
x1 = 1 for the right-hand solution and with x0 = -1.5 for the solution on the left. Then, in each case find x2 .
46) Find the inflection points (if any) on the graph of the function and the coordinates of the
46)
points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function's first and second derivatives. y = x3 - 15x2
47) Use Newton's method to find the four real zeros of the function f(x) = 3x4 - 6x2 + 2 = 0.
47)
48) How many solutions does the equation cos 4x = 0.95 - x2 have?
48)
Provide an appropriate response. 49) What can you say about the inflection points of the quartic curve y = ax4 + b x3 + cx2 + dx + e, a 0? Give reasons for your answer.
49)
Solve the problem.
50) Use Newton's method to estimate the one real solution of the equation 2x5 - 3x - 5 = 0.
50)
Answer the question. 51) A marathoner ran the 26.2 mile New York City Marathon in 2.5 hrs. Did the runner ever exceed a speed of 9 miles per hour?
51)
Start with x1 = 1. Then, in each case find x2 .
Provide an appropriate response.
52) Apply Newton's method to f(x) =
a
x , a > 0, and write an expression for xn + 1. If the
initial guess xo is greater than or equal to 1, what happens to xn + 1 as n
52)
?
Solve the problem.
53) Let c(x) = t(p0 - p)p3 where t and p0 are constants. Show that c(x) is greatest when p=
53)
3 p . 4 0
Provide an appropriate response. 54) Can anything be said about the graph of a function y = f(x) that has a second derivative that is always equal to zero? Give reasons for your answer.
23
54)
Solve the problem. 55) Write down the first four approximations to the solution of the equation sin 3x = x using Newton's method with an initial estimate of x0 = 1.
55)
56) Use Newton's method to find the two real solutions of the equation
56)
57) Use Newton's method to find the positive fourth root of 5 by solving the equation
57)
Provide an appropriate response. 58) If the derivative of an odd function g(x) is zero at x = c, can anything be said about the value of g at x = -c? Give reasons for you answer.
58)
x4 - 3x3 - 3x2 - 3x + 4 = 0.
x4 - 5 = 0. Start with x1 = 1 and find x2 .
Solve the problem. 59) A manufacturer uses raw materials to produce p products each day. Suppose that each delivery of a particular material is $d, whereas the storage of that material is x dollars per unit stored per day. (One unit is the amount required to produce one product). How much should be delivered every x days to minimize the average daily cost in the production cycle between deliveries? Show that the function has exactly one zero in the given interval. 60) f(x) = x3 + 4 + 6, (- , 0). x2
59)
60)
Provide an appropriate response. 61) If the derivative of an even function f(x) is zero at x = c, can anything be said about the value of f at x = -c? Give reasons for your answer.
62) As x moves from left to right though the point c = 8, is the graph of f(x) = x + 1 rising, or x
61)
62)
is it falling? Give reasons for your answer.
63) The function P(x) = 2x + 200 , 0 < x < models the perimeter of a rectangle of dimensions x
x by
63)
100 . x
(a) Find the extreme values for P. (b) Give an interpretation in terms of perimeter of the rectangle for any values found in part (a).
Solve the problem.
64) The function y = cot x - 2 3 csc x has an absolute maximum value on the interval 0 < x < 3
. Find it.
24
64)
65) Use Newton's method to find the negative fourth root of 2 by solving the equation
65)
66) Find the approximate values of r1 through r4 in the factorization
66)
x4 - 2 = 0. Start with x1 = -1 and find x2 .
6x4 - 12x3 - 7x2 + 13x - 1 = 6(x - r1 )(x - r2 )(x - r3)(x - r4 ).
67) Use Newton's method to estimate the solutions of the equation -3x2 - 2x + 4 = 0. Start
67)
with x1 = 0.5 for the right-hand solution and with x0 = -2 for the solution on the left. Then, in each case find x2 .
Provide an appropriate response. 68) The function f(x) = 2x 0 x < 1 is zero at x = 0 and x = 1 and differentiable on (0, 1), 0 x=1 but its derivative on (0,1) is never zero. Does this example contradict Rolle's Theorem? Solve the problem. 69) Marcus Tool and Die Company produces a specialized milling tool designed specifically for machining ceramic components. Each milling tool sells for $4, so the company's revenue in dollars for x units sold is R(x) = 4x. The company's cost in dollars to produce x tools can be modeled as C(x) = 299 + 30x5/8. Use Newton's method to find the break-even
68)
69)
point for the company (that is, find x such that C(x) = R(x)). Use x = 370 as your initial guess and show all your work.
Provide an appropriate response. 70) Sketch a continuous curve y = f(x) with the following properties: f(2) = 3; f (x) > 0 for x > 4; and f (x) < 0 for x < 4 .
71) The graph below shows the position s = f(t) of a body moving back and forth on a
coordinate line. (a) When is the body moving away from the origin? Toward the origin? At approximately what times is the (b) velocity equal to zero? (c) Acceleration equal to zero? (d) When is the acceleration positive? Negative?
25
70)
71)
Solve the problem.
72) Use Newton's method to estimate the solutions of the equation 3x2 + 2x - 1 = 0. Start with
72)
x1 = 0.5 for the right-hand solution and with x0 = -2 for the solution on the left. Then, in each case find x2 .
Provide an appropriate response. 73) Assume that f(x) and g(x) are two functions with the following properties: g(x) and f(x) are everywhere continuous, differentiable, and positive; f(x) is everywhere increasing and g(x) is everywhere decreasing. Which of the following functions are everywhere decreasing? Prove your assertions. i). h(x) = f(x) + g(x) ii). j(x) = f(x) · g(x) g(x) iii). k(x) = f(x)
73)
iv). p(x) = f(x) g(x) v). r(x) = f(g(x)) = (f g)(x)
Solve the problem. 74) Find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function's first and second derivatives.
74)
y = x5 - 4x4 - 200
75) Show that g(x) =
a+x b2 + (a + x)2
is an increasing function of x.
76) Use Newton's method to estimate the one real solution of -2x3 - 3x - 4 = 0. Start with
75)
76)
x1 = -0.5and then find x2 .
Provide an appropriate response. 77) You plan to estimate to five decimal places by using Newton's method to solve the equation cos x = 0. Does it matter what your starting value is? Give reasons for your answer.
78) For x > 0, sketch a curve y = f(x) that has f(1) = 0 and f (x) = - 1 . Can anything be said x
about the concavity of such a curve? Give reasons for your answer.
26
77)
78)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Identify the function's extreme values in the given domain, and say where they are assumed. Tell which of the extreme values, if any, are absolute. 79) f(x) = 16 - x2 , -4 x < 4 79)
A) local and absolute maximum: 0 at x = -4; local and absolute minimum: 4 at x = 0
B) local and absolute minimum: 0 at x = -4 and x = 4; local and absolute maximum: 4 at x = 0
C) no local extrema; no absolute extrema D) local and absolute minimum: 0 at x = -4; local and absolute maximum: 4 at x = 0
The graph below shows solution curves of a differential equation. Find an equation for the curve through the given point.
80)
80)
dy 6 = 1 - x 1/5 dx 5
A) y = x + x6/5 + 3
B) y = 1 - x6/5 - 3
C) y = x - x6/5 + 3
D) y = x - x6/5 + 5
5
5
5
3
Solve the problem. 81) Find the number of units that must be produced and sold in order to yield the maximum profit, given the following equations for revenue and cost: R(x) = 70x - 0.5x2 C(x) = 2x + 8. A) 68 units
B) 69 units
C) 76 units
27
D) 72 units
81)
Find the largest open interval where the function is changing as requested. 82) Increasing f(x) = 1 x2 + 1
A) (1, )
B) (- , 0)
C) (- , 1)
For the given expression y , find y'' and sketch the general shape of the graph of y = f(x). 83) y' = cos x, 0 x 2
A)
B)
28
82) D) (0, )
83)
C)
D)
Find the largest open interval where the function is changing as requested. 84) Decreasing f(x) = x3 - 4x
A)
2 3 , 3
C) - , - 2 3 3
B) - ,
84) D) - 2 3 , 2 3 3 3
Solve the problem. 85) At about what velocity do you enter the water is you jump from a 15 meter cliff? (Use g = 9.8 m/ sec2 .)
A) 2 m/sec
B) -8.5 m/sec
C) -17 m/sec
85)
D) 17 m/sec
Use Newton's method to estimate the requested solution of the equation. Start with given value of x 0 and then give x 2 as the estimated solution. 86) x4 - 4x + 2 = 0; x0 = 0; Find the left-hand solution.
A) 0.5
B) 0.57
86) C) 0.51
29
D) 0.52
Find all possible functions with the given derivative. 87) y = 3 2 t
A) 3t + C 2
87)
B) 3 t + C
C) t + C
2
D) 3t + C 2
Solve the problem. 88) Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's position at time t. v = -13t + 8, s(0) = 8 A) s= 13 t2 + 8t - 8 B) s = -13t2 + 8t + 8 2
C) s = - 13 t2 + 8t + 8
D) s = - 13 t2 + 8t - 8
2
2
Find the location of the indicated absolute extremum for the function. 89) Minimum
A) x = -1
88)
B) x = 2
C) No minimum
30
89)
D) x = 1
Find the extrema of the function on the given interval, and say where they occur. 90) sin x + cos x, 0 x 2
A) local maxima: 1 at x = 2 and 2 at x = local minima: 1 at x = 0 and - 2 at x =
B) local maxima: 1 at x = 2 and 2 at x = local minima: 1 at x = 0 and - 2 at x =
C) local maxima: 1 at x = 0 and - 2 at x = local minima: 1 at x = 2 and
2 at x =
D) local maxima: 1 at x = 0 and - 2 at x = local minima: 1 at x = 2 and
2 at x =
4
90)
;
4 4
;
4 4
;
4 4
;
4
Find the absolute extreme values of the function on the interval. 7 91) f( ) = sin + , 0 2 4
91)
A) absolute maximum is 1 at = 0; absolute minimum is -1 at = B) absolute maximum is 1 at = 1 ; absolute minimum is -1 at = 7 , 8
8
C) absolute maximum is 1 at = 7 ; absolute minimum is -1 at = 1 8
8
D) absolute maximum is 1 at = 9 ; absolute minimum is -1 at = 7 , 8
8
Find all possible functions with the given derivative. 92) r = 5 + 1 3
A) r = 5 - 4
92) C) r = 5 - 1
B) r = 5 + 3
2 2
31
D) r = 5 + 1
4 4
Find the location of the indicated absolute extremum for the function. 93) Minimum
A) x = 2
B) x = 1
C) x = -2
Graph the rational function. 2 94) y = x x2 + 5
93)
D) x = -1
94)
A)
B)
32
C)
D)
Solve the problem. 95) Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's position at time t. v = cos
2
95)
t, s(0) = 1
A) s = 2 sin C) s = 2 sin
2 2
t+
B) s = sin t +1
t
D) s = 2 sin
2
t +1
Find the extreme values of the function and where they occur. 96) y = 7x x2 + 1
96)
A) The maximum value is 0 at x = 0. B) The minimum value is - 7 at x = -1. The maximum value is 7 at x = 1. 2
2
C) The minimum value is 0 at x = 1. The maximum value is 0 at x = -1. D) The minimum value is 0 at x = 0. Answer each question appropriately.
97) Suppose the velocity of a body moving along the s-axis is ds = 9.8t - 4. dt
Is it necessary to know the initial position of the body to find the body's displacement over some time interval? Justify your answer. A) No, displacement has nothing to do with the position of the body.
B) No, the initial position is necessary to find the curve s= f(t) but not necessary to find the
displacement. The initial position determines the integration constant. When finding the displacement the integration constant is subtracted out. C) Yes, knowing the initial position is the only way to find the exact positions at the beginning and end of the time interval. Those positions are needed to find the displacement. D) Yes, integration is not possible without knowing the initial position.
33
97)
Find the largest open interval where the function is changing as requested. 98) Increasing y = 7x - 5 A) (- , 7) B) (-5, 7) C) (-5, )
98) D) (- , )
Graph the function, then find the extreme values of the function on the interval and indicate where they occur. 99) y = x + 4 + x - 3 on the interval - < x < 99)
A) Absolute minimum is 7 on the interval (-5, 4]
B) Absolute minimum is 6 on the interval (-5, 2]
34
C) Absolute minimum is 7 on the interval (-4, 3]
D) Absolute minimum is 8 on the interval (-4, 3]
Solve the problem.
100) On our moon, the acceleration of gravity is 1.6 m/sec2 . If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 45 seconds later? A) 72 m/sec B) -72 m/sec C) 3240 m/sec
D) -36 m/sec
Use a computer algebra system (CAS) to solve the given initial value problem. 2 101) y = 9(1 - x ) , y(0) = 1 1+ x2
A) y = 10 tan-1 x - x + 1
B) y = 18 tan-1 x - 9x
C) y = 9 ln x + 1 - 9x + 1
D) y = 18 tan-1 x - 9x + 1
x-1
100)
101)
Solve the problem.
102) The graphs of y = x and y = 4 - x2 intersect at one point x = r. Use Newton's method to estimate the value of r to four decimal places. A) r 2.4681 B) r 1.6481
C) r 2.6481 35
D) r 1.4681
102)
Find an antiderivative of the given function. 103) 27x2 - 18x - 8
103)
A) 10x3 - 9x2 - 8x
B) 9x3 - 8x2 - 8x
C) 9x3 - 9x2 - 7x
D) 9x3 - 9x2 - 8x
Find the largest open interval where the function is changing as requested. 104) Decreasing f(x) = x - 8 A) (- , -8) B) (8, ) C) (- , 8)
104) D) (-8, )
Determine all critical points for the function. 105) f(x) = x2 + 16x + 64
A) x = -16
105)
B) x = 8
C) x = 0
D) x = -8
Find the curve y = f(x) in the xy-plane that has the given properties. 2 106) d y = 36x and the graph of y passes through the point (0, 10) and has a horizontal tangent there. dx2
A) y = 18x3 + 10
B) y = 6x3 + 10
C) y = 6x3 - 10
106)
D) y = 18x2 + 10
Find the most general antiderivative.
107)
sin (cot
+ csc ) d
107)
A) csc + cos + C C) sin + C
B) cos + C D) sin + + C
Find the function with the given derivative whose graph passes through the point P. 108) g (x) = 4 + 6x, P(-1, 2) x2
A) g(x) = 4x2 + 3x2 - 5 C) g(x) = -4x-1 + 3x2
108)
B) g(x) = 4x-2 + 6x - 5 D) g(x) = -4x-1 + 3x2 - 5
Use Newton's method to estimate the requested solution of the equation. Start with given value of x 0 and then give x 2 as the estimated solution. 109) x3 + 5x + 2 = 0; x0 = -1; Find the one real solution.
A) -0.39
109)
B) -0.38
C) -0.44
36
D) -0.64
Find the extrema of the function on the given interval, and say where they occur.
110) sin 4x, 0 x
110)
2
A) local maxima: 1 at x =
4
and 0 at x =
local minima: 0 at x = 0 and -1 at x =
B) local maxima: 1 at x =
8
and 0 at x =
local minimum: -1 at x =
C) local maxima: 1 at x =
8
8
;
8 2
;
8 and 0 at x =
local minima: 0 at x = 0 and -1 at x =
D) local maxima: 1 at x =
2
and 0 at x =
2
;
8
4
;
local minimum: 0 at x = 0
Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. 111) f(x) = x4 - 4x3 - 53x2 - 86x + 67 111)
A) Approximate local maximum at 0.958; approximate local minima at -3.245 and 7.205 B) Approximate local maximum at 0.866; approximate local minima at -3.122 and 7.22 C) Approximate local maximum at 0.93; approximate local minima at -3.277 and 7.164 D) Approximate local maximum at -0.944; approximate local minima at -3.192 and 7.136 Provide an appropriate response. 112) An approximation to the total profit (in thousands of dollars) from the sale of x hundred thousand tires is given by p = -x3 + 15x2 - 48x + 450, x 3. Find the number of hundred thousands of tires that must be sold to maximize profit. A) 5 hundred thousand
112)
B) 3 hundred thousand D) 8 hundred thousand
C) 10 hundred thousand
Solve the problem. 113) The stiffness of a rectangular beam is proportional to its width times the cube of its depth. Find the dimensions of the stiffest beam than can be cut from a 14-in.-diameter cylindrical log. (Round answers to the nearest tenth.)
14"
A) w = 8.0 in.; d = 13.1 in. C) w = 6.0 in.; d = 13.1 in.
B) w = 8.0 in.; d = 11.1 in. D) w = 7.0 in.; d = 12.1 in. 37
113)
Find the function with the given derivative whose graph passes through the point P. 114) f (x) = x2 + 9, P(3, 65)
A) f(x) = x3 + 9x + 1.548112372e+15
114)
B) f(x) = x3 + 9x2 + 29
1.407374884e+14
3
3
C) f(x) = x + 9x
D) f(x) = x + 9x + 29
3
3
Solve the problem.
115) On our moon, the acceleration of gravity is 1.6 m/sec2 . If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 45 seconds later? A) -72 m/sec B) 72 m/sec C) -36 m/sec
115)
D) 3240 m/sec
Find the extreme values of the function and where they occur. 116) y = x3 - 12x + 2
116)
A) Local maximum at (0, 0). B) None C) Local maximum at (-2, 18), local minimum at (2, -14). D) Local maximum at (2, -14), local minimum at (-2, 18). Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down.
117)
117)
A) Local minimum at x = 2; local maximum at x = 0; concave down on (0, ); concave up on (, 0) B) Local minimum at x = 2; local maximum at x = 0; concave up on (0, ); concave down on (, 0) C) Local minimum at x = 0; local maximum at x = 2; concave up on (0, ); concave down on (, 0) D) Local minimum at x = 0; local maximum at x = 2; concave down on (0, ); concave up on (, 0)
38
Solve the problem.
118) How close does the curve y = x come to the point 7 , 0 ? (Hint: If you minimize the square of the 3
118)
distance, you can avoid square roots.)
A) The distance is minimized when x = 11 ; the minimum distance is 7 units. 3
3
B) The distance is minimized when x = - 1 ; the minimum distance is
25 units. 12
7
C) The distance is minimized when x = 4 ; the minimum distance is 7 units. 3
3
D) The distance is minimized when x = 11 ; the minimum distance is 6
25 units. 12
Find the largest open interval where the function is changing as requested. 119) Increasing y = (x2 - 9)2
A) (-3, 0)
B) (3, )
C) (- , 0)
119) D) (-3, 3)
Answer each question appropriately. 120) Find the standard equation for the position s of a body moving with a constant acceleration a along a coordinate line. The following properties are known: d2 s = a, i. dt2 ii.
120)
ds = v0 when t = 0, and dt
iii. s = s0 when t = 0, where t is time, s0 is the initial position, and v0 is the initial velocity.
A) s = at + s0 2
2
B) s = at - v0 t - s0 2
2
C) s = at2 + v0 t + s0
D) s = at + v0 t + s0 2
2
Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval. 121) g(x) = x3/4, 0,4
A) No
B) Yes
Find the largest open interval where the function is changing as requested. 122) Decreasing y = 1 + 7 x2
A) (-7, 0)
121)
B) (7, )
C) (-7, 7)
39
122) D) (0, )
Find the absolute extreme values of the function on the interval. 123) F(x) = - 2 , 0.5 x 5 x2
123)
A) absolute maximum is 2 at x = 1 ; absolute minimum is -8 at x =5 25
2
B) absolute maximum is - 2 at x = 5; absolute minimum is -8 at x = - 1 25
2
C) absolute maximum is - 2 at x = 1 ; absolute minimum is -8 at x = -5 25
2
D) absolute maximum is - 2 at x = 5; absolute minimum is -8 at x = 1 25
2
Find an antiderivative of the given function. 124) 6 csc2 x 7 7
124)
A) 6 cot x
B) -6 cot x
C) 6 cot x 49 7
D) 12 csc 2 x cot x 7 7 7
7
7
Find the absolute extreme values of the function on the interval. 125) f(x) = |x - 7|, 4 x 11 A) absolute maximum is -3 at x = 4; absolute minimum is 4 at x = 11
125)
B) absolute maximum is 4 at x = 11; absolute minimum is 0 at x = 7 C) absolute maximum is 4 at x = 11; absolute minimum is 3 at x = 4 D) absolute maximum is 3 at x = 4; absolute minimum is 0 at x = 7 Find an antiderivative of the given function. 126) cos x + 5 sin x 5
126)
A) - sin x + cos x
B) 1 sin x - cos x
5
5
C) 1 sin x - 25 cos x
D) -sin x - 25 cos x
5
5
Find the most general antiderivative.
127)
(-9 sec2 x) dx
127)
A) -9 tan x + C
128)
sec
sec - cos
A) cot + C
B) tan x + C
C) 9 cot x + C
9
D) -9 cot x + C
d
128) B) cos2 + C
C) -cot + C
40
D)
+ tan
+C
129)
(3x3 + 5x + 5) dx
129)
A) 3x4 + 5x2 + 5x + C
B) 9x2 + 5 + C
C) 9x4 + 10x2 + 5x + C
D) 3 x4 + 5 x2 + 5x + C 4
2
Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down.
130)
130)
A) Local minimum at x = 1; local maximum at x = -1; concave down on (- , ) B) Local minimum at x = 1; local maximum at x = -1; concave down on (0, ); concave up on (- , 0) C) Local minimum at x = 1; local maximum at x = -1; concave up on (- , )
D) Local minimum at x = 1; local maximum at x = -1; concave up on (0, ); concave down on (- , 0)
Solve the problem.
131) A rocket lifts off the surface of Earth with a constant acceleration of 30 m/sec2. How fast will the rocket be going 2.5 minutes later? A) 187.5 m/sec B) 37.5 m/sec
C) -75 m/sec
Determine all critical points for the function. 2 132) f(x) = 3x x-7
131)
D) 75 m/sec
132)
A) x = 7 C) x = 14 and x = 0
B) x = -7 D) x = 0 and x = 7
41
Solve the problem.
133) Suppose that c(x) = 4x3 - 33x2 + 7331x is the cost of manufacturing x items. Find a production
133)
level that will minimize the average cost of making x items. A) -66 items
B) 83 items C) 7331 items D) There is not a production level that will minimize average cost. Use differentiation to determine whether the integral formula is correct. 134) 4x(7x + 6)3 dx = 1 x2 (7x + 6)4 + C 2
A) Yes
134)
B) No
Find the extreme values of the function and where they occur. 135) y = x3 - 3x2 + 4x - 4
A) None C) The maximum is 0 at x = 1.
135) B) The minimum is 0 at x = -1. D) The maximum is 0 at x = 2.
Solve the problem. 136) A private shipping company will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. What dimensions will give a box with a square end the largest possible volume?
A) 36 in. × 36 in. × 36 in. C) 18 in. × 18 in. × 90 in.
B) 18 in. × 18 in. × 36 in. D) 18 in. × 36 in. × 36 in.
42
136)
Find the absolute extreme values of the function on the interval. 137) g(x) = 10 - 8x2 , -3 x 4
137)
A) absolute maximum is 8 at x = 0; absolute minimum is -138 at x = 4 B) absolute maximum is 10 at x = 0; absolute minimum is -118 at x = 4 C) absolute maximum is 80 at x = 0; absolute minimum is -62 at x = -3 D) absolute maximum is 20 at x = 0; absolute minimum is -62 at x = 4 Use differentiation to determine whether the integral formula is correct. 2 138) x cos x dx = x sin x + C 2
A) No
138)
B) Yes
For the given expression y , find y'' and sketch the general shape of the graph of y = f(x). 139) y' = 2|x| = -2x, x 0 2x, x>0
A)
43
139)
B)
C)
D)
44
Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down.
140)
140)
A) Local minimum at x = 3 ; local maximum at x = -3 ; concave up on (0, ); concave down on
(- , 0) B) Local minimum at x = 3 ; local maximum at x = -3 ; concave down on (- , -3) and (3, ); concave up on (-3, 3) C) Local minimum at x = 3 ; local maximum at x = -3 ; concave up on (- , -3) and (3, ); concave down on (-3, 3) D) Local minimum at x = 3 ; local maximum at x = -3 ; concave down on (0, ); concave up on (- , 0)
Find the location of the indicated absolute extremum for the function. 141) Minimum
A) x = -3
B) x = 2
C) x = -4
Find the absolute extreme values of the function on the interval. 3 142) f(x) = csc x, x 2 2
A) absolute maximum is -1 at x = ; absolute minimum is 1 at x = 0 B) absolute maximum is 1 at x = ; absolute minimum is -1 at x = C) absolute maximum does not exist; absolute minimum does not exist D) absolute maximum is 0 at x = - ; absolute minimum is -1 at x =
45
141)
D) x = 0
142)
143) g(x) = -x2 + 10x - 21, 3 x 7
143)
A) absolute maximum is 3.236962232e+15 at x = 5; absolute minimum is 0 at 7 and 0 at x = 3 7.036874418e+13
B) absolute maximum is 1.407374884e+15 at x = 6; absolute minimum is 0 at 7 and 0 at x = 3 2.814749767e+14
C) absolute maximum is 1.125899907e+15 at x = 6; absolute minimum is 0 at 7 and 0 at x = 3 2.814749767e+14
D) absolute maximum is 1.125899907e+15 at x = 5; absolute minimum is 0 at 7 and 0 at x = 3 2.814749767e+14
Solve the initial value problem. 144) dy = 1 + 10, y(9) = -4 dx 2 x
144)
A) y = x + 10x - 97
B) y = -1 x + 10x - 373
C) y = x + 10x + 89
D) y = 1 + 10x - 283
4
4
x
3
Graph the equation. Include the coordinates of any local and absolute extreme points and inflection points. 145) y = x + cos 2x, 0 x
A) local minimum: (1.444, -0.246)
local maximum: (0.126, 1.031) inflection points: (0.785, 0.393), (2.356, 1.178)
46
145)
B) local minimum: local maximum: inflection point:
4
, -1
3 ,3 4 2
,1
C) no local extrema inflection point:
, 2 2
47
D) local minimum: 5 , 5 - 6 3 12
local maximum: inflection points:
12
12
,
+6 3 12
3 3 , , , 4 4 4 4
Solve the problem. 146) Given the acceleration, initial velocity, and initial position of a body moving along a coordinate line at time t, find the body's position at time t. a = 16, v(0) = 15, s(0) = -10 A) s = 8t2 + 15t - 10 B) s = 8t2 + 15t
C) s = -8t2 - 15t - 10
146)
D) s = 16t2 + 15t - 10
147) Find the optimum number of batches (to the nearest whole number) of an item that should be
147)
produced annually (in order to minimize cost) if 100,000 units are to be made, it costs $2 to store a unit for one year, and it costs $380 to set up the factory to produce each batch. Assume that units of this item will be sold off throughout the year, so the cost equation will use the average cost. A) 11 batches B) 13 batches C) 16 batches D) 18 batches
Find the extreme values of the function and where they occur. 1 148) y = 1 - 5x2
A) The minimum is 0 at x = 1. C) The maximum is 1 at x = 2.
148) B) The maximum is 1 at x = -2. D) The minimum is 1 at x = 0.
Provide an appropriate response. 149) Suppose that the second derivative of the function y = f(x) is y'' = (x - 3)(x + 6). For what x-values does the graph of f have an inflection point? A) -3, -6 B) -3, 6 C) 3, 6 D) 3, -6
48
149)
Find the absolute extreme values of the function on the interval. 150) f(x) = 2x - 4, -3 x 4 A) absolute maximum is 12 at x = 4; absolute minimum is - 2 at x = -3
150)
B) absolute maximum is 12 at x = -4; absolute minimum is - 10 at x = 3 C) absolute maximum is 4 at x = 4; absolute minimum is - 10 at x = -3 D) absolute maximum is 4 at x = -3; absolute minimum is - 2 at x = 4 Identify the function's extreme values in the given domain, and say where they are assumed. Tell which of the extreme values, if any, are absolute. 151) f(x) = x2 - 4x, - < x 4 151)
A) local minimum: 0 at x = 4; local and absolute maximum:-4 at x = 2 B) local and absolute minimum: -4 at x = 2; local maximum: 0 at x = 4 C) local minimum: -4 at x = 2; local and absolute maximum: 0 at x = 4 D) local and absolute minimum: -4 at x = 2; local and absolute maximum: 0 at x = 4 Use differentiation to determine whether the integral formula is correct.
152)
36(9x + 2)3 dx = (9x + 2)4 + C
152)
A) No
B) Yes
Find the most general antiderivative. t 153) 6t2 + dt 10
A) 2t3 + t + C
153) 2
B) 2t3 + t + C
C) 12t + 1 + C 10
20
Find all possible functions with the given derivative. 154) y = 5x7
A) 5 x8 + C
5
154)
B) 1 x8 + C
8
D) 18t3 + 1 t2 + C
C) 7 x8 + C
4
D) 5 x8 + C
5
7
Find an antiderivative of the given function. 155) x-5 + 1 9 x
A) - 1 + 2 x1/2 4x4
9
155)
B) - 1 + 2 x1/2 4x5
C) - 1 + 2 x1/2
9
5x4
Use differentiation to determine whether the integral formula is correct. 156) sec (2x - 5) tan (2x - 5) dx = sec (2x - 5) + C 2
A) Yes
B) No
49
9
D) - 1 + 2 x1/2 5x5
9
156)
Solve the problem. 157) The 8 ft wall shown here stands 27 feet from the building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall.
157)
8' wall
27'
A) 47.9 ft
B) 35 ft
C) 46.9 ft
D) 45.9 ft
158) A company is constructing an open-top, square-based, rectangular metal tank that will have a
158)
volume of 52.5 ft3. What dimensions yield the minimum surface area? Round to the nearest tenth, if necessary. A) 4.7 ft × 4.7 ft × 2.4 ft B) 5.4 ft × 5.4 ft × 1.8 ft
C) 10.2 ft × 10.2 ft × 0.5 ft
D) 3.7 ft × 3.7 ft × 3.7 ft
159) How close does the semicircle y = 16 - x2 come to the point (1, 4)? A) x = 0.24 B) x = 2.24 C) x = 1.76
159) D) x = 2.27
160) Find the number of units that must be produced and sold in order to yield the maximum profit, given the following equations for revenue and cost: R x = 2x C x = 0.001x2 + 0.5x + 70.
A) 1500 units
B) 1250 units
C) 750 units
D) 2500 units
Sketch the graph of the function and determine whether it has any absolute extreme values on its domain.
50
160)
161) y = 4 cos x, 0 < x < 2
161)
A) no absolute maximum;
B) absolute maximum at x = 0 and x = 2 ;
C) no absolute maximum;
D) absolute maximum at x = 0;
no absolute minimum
absolute minimum at x =
absolute minimum at x =
absolute minimum at x =
Solve the initial value problem. 2 162) d y = 3 - 4x, y (0) = 8, y(0) = 2 dx2
162)
A) y = 3x2 + 4x3 + 8x + 2
B) y = 2
C) y = 3 x2 - 2 x3 + 8x + 2
D) y = 3 x2 + 2 x3 - 8x - 2
2
3
2
51
3
Identify the function's local and absolute extreme values, if any, saying where they occur. 163) f(x) = x2 + 8x + 32
163)
A) absolute minimum: 4 at x = -4 B) absolute maximum: 4 at x = -4 C) no local extrema D) relative minimum: 4 at x = -4; relative maximum: -4 at x = 4 Find the largest open interval where the function is changing as requested. 164) Increasing f(x) = x2 - 2x + 1
A) (0, )
B) (1, )
C) (- , 0)
Find the derivative at each critical point and determine the local extreme values. 2 165) y = -x - 5x + 10, x 1 -x2 + 15x - 10, x > 1
A) Critical Pt. derivative Extremum Value 5 65 x=0 local max 4 2 x=1 x=
15 2
undefined local min 0
local max
4 185 4
B) Critical Pt. derivative Extremum Value 5 65 x= 0 local max 4 2 x=1
x=
15 2
undefined local min
undefined local max
6
185 4
C) Critical Pt. derivative Extremum Value 5 65 x=0 local min 4 2 x=1
x=
15 2
undefined local max
0
local min
4
185 4
52
164) D) (- , 1)
165)
D) Critical Pt. derivative Extremum Value 5 65 x=0 local max 4 2 x=1 x=-
undefined local min 15 2
0
local max
6 185 4
Use Newton's method to estimate the requested solution of the equation. Start with given value of x 0 and then give x 2 as the estimated solution. 166) 3x2 + 2x - 1 = 0; x0 = 1; Find the right-hand solution.
A) 0.33
B) 0.35
166) C) 0.85
D) 0.50
Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down.
167)
167)
A) Local minimum at x = 1; local maximum at x =-1; concave down on (0, ); concave up on (, 0)
B) Local minimum at x = 1; local maximum at x =-1; concave up on (0, ); concave down on (, 0)
C) Local maximum at x = 1; local minimum at x =-1; concave up on (- , ) D) Local maximum at x = 1; local minimum at x =-1; concave up on (0, ); concave down on (, 0)
53
Find the location of the indicated absolute extremum for the function. 168) Maximum
A) No maximum
B) x = 1
C) x = 4
168)
D) x = -1
Identify the function's local and absolute extreme values, if any, saying where they occur. 169) f(x) = -x3+ 3x2 + 9x + 3
169)
A) local maximum at x = -1; local minimum at x = 3 B) local maximum at x = -3; local minimum at x = 1 C) local maximum at x = 1; local minimum at x = -3 D) local maximum at x = 3; local minimum at x = -1 Find the extrema of the function on the given interval, and say where they occur. 170) csc2 x + 2 cot x, 0 < x <
A) local minimum: 0 at x = C) local maximum: 0 at x =
B) local minimum: 0 at x =
4
D) local maximum: 0 at x =
4
54
170) 4 4
Plot the zeros of the given polynomial on the number line together with the zeros of the first derivative. 171) y = x2 + 9x + 18
171)
A)
B)
C)
D)
Solve the problem. 172) A trough is to be made with an end of the dimensions shown. The length of the trough is to be 16 feet long. Only the angle can be varied. What value of will maximize the trough's volume?
A) 46°
B) 14°
Find the value or values of c that satisfy the equation
C) 32°
D) 30°
f(b) - f(a) = f (c) in the conclusion of the Mean Value Theorem for b-a
the function and interval. 173) f(x) = x2 + 4x + 4, [-2, 1]
A) 0, - 1 2
173) C) - 1 , 1 2 2
B) -2, 1
D) - 1 2
Solve the problem. 174) An object is dropped from 15 ft above the surface of the moon. How long will it take the object to hit the surface of the moon if d2 s/dt2 = -5.2 ft/sec2 ?
A) 1.70 sec
172)
B) 5.77 sec
C) 2.40 sec 55
D) 1.44 sec
174)
Find the function with the given derivative whose graph passes through the point P.
175) r ( ) = 6 - csc2 , P
4
,0
175)
A) r( ) = 6 - cot - 3 + 2
B) r( ) = 6 + cot - 3 + 1
C) r( ) = 6 + cot - 3 - 1
D) r( ) = 6 - cot - 3 - 2
2
2
2
2
Determine all critical points for the function. 176) f(x) = 5x3 - 3x5
176)
A) x = -1 and x = 1 C) x = -1
B) x = 0, x = -1, and x = 1 D) x = 1
Find the absolute extreme values of the function on the interval. 177) h(x) = 1 x + 5, -3 x 4 2
177)
A) absolute maximum is - 3 at x = -3; absolute minimum is 7 at x = 4 2
B) absolute maximum is 7 at x = 4; absolute minimum is 7 at x = -3 2
C) absolute maximum is - 3 at x = -4; absolute minimum is -4 at x = 3 D) absolute maximum is - 3 at x = 4; absolute minimum is 7 at x = -3 2
Solve the initial value problem. 178) dr = - cos , r(0) = -9 d 3 3
A) r = cos C) r = -sin
3 3
178)
- 10
B) r = sin
-9
D) r = -
3
3 sin
-9 3
-9
For the given expression y , find y'' and sketch the general shape of the graph of y = f(x). 179) y = x-2/3(x - 2)
56
179)
A)
B)
C)
57
D)
Find the location of the indicated absolute extremum for the function. 180) Minimum
A) x = 5
B) x = -5
C) x = 3
180)
D) x = -3
Find the extreme values of the function and where they occur. 181) y = 1 x2 - 1
181)
A) Local maximum at (1, 0), local minimum at (-1, 0). B) Local maximum at (0, -1). C) None D) Local maximum at (-1, 0), local minimum at (1,0). Determine all critical points for the function. 182) f(x) = x3 - 12x + 5
182)
A) x = -2 C) x = 2
B) x = -2 and x = 2 D) x = -2, x = 0, and x = 2
58
183) f(x) = x3 - 6x2 + 10 A) x = -2 and x = 2 C) x = 0
183) B) x = 0 and x = 2 D) x = 0 and x = 4
Sketch the graph and show all local extrema and inflection points. 184) y = x - sin x, 0 x 2
184)
A) Local minimum: (0, 0)
B) Local minimum: (0, 0)
C) Local minimum: (0, 0)
D) Local minimum: (0, 0)
Local maximum: (2 , 2 ) No inflection points
Local maximum: (2 , 2 ) Inflection point: ( , )
Local maximum: (2 , 2 ) No inflection points
Local maximum: (2 , 2 ) Inflection point: ( , )
59
Find the curve y = f(x) in the xy-plane that has the given properties. 185) f(x) has a slope at each point given by - 1 and passes through the point 1 , 11 8 x2
A) y = - 1 + 13 x
B) y = 2 + 3
C) y = 1 + 3
x
x
185) D) y = 3 + 3 x3
Sketch the graph and show all local extrema and inflection points. 4 186) y = x x - 4 2
186)
B) Absolute maximum: 8 , 524,288
A) Absolute minimum: (8, 0)
5
No inflection points
3125
Absolute minimum: (8, 0) 16 331,776 Inflection points: , 5 3125
60
D) Local maximum: 8 , 524,288
C) Local maximum: 8 , 524,288 5
5
3125
3125
Local minimum: (8, 0) 16 331,776 Inflection points: , , (8, 0) 5 3125
Local minimum: (8, 0) 16 331,776 Inflection point: , 5 3125
Find the derivative at each critical point and determine the local extreme values. 187) y = x(4 - x2 )
187)
A) Critical Pt. derivative Extremum Value x = 1.15 0 local max 3.08 x = -1.15 0 local min -3.08
B) Critical Pt. derivative Extremum Value x = -1.15 0 local max 3.08 x = 1.15 0 local min -6.16
C) Critical Pt. derivative Extremum Value x = 1.15 0 local max -6.16 x = -1.15 0 local min 3.08
D) Critical Pt. derivative Extremum Value x = -1.15 0 local max -6.16 x = 1.15 0 local min 3.08
Provide an appropriate response. 188) Suppose that f'(x) = 2x for all x. Find f(6) if f(-2) = 2. A) 32 B) 36 Graph the rational function.
61
188) C) 38
D) 34
189) y = x - 3
189)
x2 - 9
A)
B)
C)
D)
62
Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down.
190)
190)
A) Local maximum at x = 3; local minimum at x = -3 ; concave up on (0, -3) and (3, ); concave down on (-3, 3) B) Local minimum at x = 3; local maximum at x = -3 ; concave down on (0, ); concave up on (- , 0) C) Local minimum at x = 3; local maximum at x = -3 ; concave up on (0, -3) and (3, ); concave down on (-3, 3) D) Local minimum at x = 3; local maximum at x = -3 ; concave up on (0, ); concave down on (- , 0)
Graph the rational function. 191) y = 2x x2 + 1
191)
63
A)
B)
C)
D)
Answer each question appropriately.
192) Suppose the velocity of a body moving along the s-axis is ds = 9.8t - 2. dt
192)
Find the body's displacement over the time interval from t = 2 to t = 3 given that s=s0 when t=0.
B) 14.5 D) Not enough information is given.
A) -6.9 C) 22.5
Find the absolute extreme values of the function on the interval. 193) f(x) = x8/3, -1 x 8
A) absolute maximum is 512 at x = 8; absolute minimum is 0 at x = 01 B) absolute maximum is 256 at x = 8; absolute minimum is 0 at x = 01 C) absolute maximum is 256 at x = 8; absolute minimum does not exist D) absolute maximum is 256 at x = 8; absolute minimum is 1 at x = -1
64
193)
Solve the problem. 194) Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's position at time t. v = cos
2
194)
t, s(0) = 1
A) s = 2 sin C) s = 2 sin
2 2
t+
B) s = 2 sin
t
D) s = sin t
2
t
Find the most general antiderivative. 195) x x + x dx x2
195)
A) C
B) 2 x - 2 + C
C) 2 - 2 x + C
D) -
x
x
x 3 x +C 2 2
Find the absolute extreme values of the function on the interval. 3 196) F(x) = x, -3 x 27 A) absolute maximum is 3 at x = -27; absolute minimum is 0 at x =0
196)
B) absolute maximum is 0 at x = 0; absolute minimum is 3 at x = 27 C) absolute maximum is 3 at x = 27; absolute minimum is -3 at x = -27 D) absolute maximum is 3 at x = 27; absolute minimum is 0 at x =0 Find the extreme values of the function and where they occur. 197) y = 1 x2 + 1
197)
A) The maximum value is 1 at x = 0.5, the minimum value is -1 at x = 0.5. B) The maximum value is 1 at x = 0.5. C) The minimum value is -1 at x = 0.5. D) The maximum value is 1 at x = 0. Solve the initial value problem. 198) ds = cos t - sin t, s =3 dt 2
198)
A) s = 2sin t + 1 C) s = sin t + cos t + 4
B) s = sin t + cos t + 2 D) s = sin t - cos t + 2
65
Solve the problem. 199) A rectangular sheet of perimeter 30 cm and dimensions x cm by y cm is to be rolled into a cylinder as shown in part (a) of the figure. What values of x and y give the largest volume?
A) x = 11 cm; y = 4 cm C) x = 12 cm; y = 3 cm
199)
B) x = 9 cm; y = 6 cm D) x = 10 cm; y = 5 cm
Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. 200) f(x) = 0.1x3 -15x2 - 89x - 32 200)
A) Approximate local minimum at -2.884; approximate local maximum at 102.884 B) Approximate local minimum at -102.884; approximate local maximum at 2.884 C) Approximate local maximum at -102.884; approximate local minimum at 2.884 D) Approximate local maximum at -2.884; approximate local minimum at 102.884 Find all possible functions with the given derivative. 201) y = 6x2 - 8x
201)
A) - 2x3 - 4x2 + C C) 2x3 + C
B) 2x2 + 4x + C D) 2x3 - 4x2 + C
Solve the initial value problem. 202) dy = 4x-3/4, y(1) = 5 dx
202)
A) y = 16x1/4 - 11
B) y = 4x1/4 + 1
C) y =- 3 x-7/4 - 11 4
D) y = 16x1/4 + 80
4
66
Find the location of the indicated absolute extremum for the function. 203) Maximum
A) x = 0
B) x = 2
C) x = -1
203)
D) No maximum
Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval. 204) f(x) = x1/3, -4,4
A) Yes
204)
B) No
Use differentiation to determine whether the integral formula is correct. -1 205) (4x + 6)-2 dx = - (4x + 6) + C 4
A) No
205)
B) Yes
Answer each question appropriately. 206) If differentiable functions y = F(x) and y = G(x) both solve the initial value problem dy = f(x) , y(x0 ) = y0 , on an interval I, must F(x) = G(x) for every x in I? Justify the answer. dx
206)
A) F(x) and G(x) are not unique. There are infinitely many functions that solve the initial value problem. When solving the problem there is an integration constant C that can be any value. F(x) and G(x) could each have a different constant term. B) F(x) = G(x) for every x in I because integrating f(x) results in one unique function.
C) F(x) = G(x) for every x in I because when given an initial condition, we can find the
integration constant when integrating f(x). Therefore, the particular solution to the initial value problem is unique. D) There is not enough information given to determine if F(x) = G(x).
Find all possible functions with the given derivative. 207) y = 6x2 + 1
A) 2x3 + x + C
207)
B) x + C
C) 12x + C
67
D) 2x3 + C
Answer each question appropriately.
208) Suppose the velocity of a body moving along the s-axis is ds = 9.8t - 3.
208)
dt
Find the body's displacement over the time interval from t = 3 to t = 7 given that s = 9 when t = 0. A) 184 B) 208 C) 39.2 D) 4
Solve the problem. 209) Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's position at time t. 8 4t v = sin , s( 2 ) = 2
A) s = -2 cos 4t + 3.3073
B) s = -2 cos 4t + 4
C) s = 2 cos 4t + 4
D) s = -2 cos 4t + 8.2134
209)
Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. 210) f(x) = x4 - 3x3 - 21x2 + 74x - 24 210)
A) Approximate local maximum at 1.694; approximate local minima at -3.051 and 3.694 B) Approximate local maximum at 1.517; approximate local minima at -3.123 and 3.713 C) Approximate local maximum at 1.651; approximate local minima at -3.021 and 3.785 D) Approximate local maximum at 1.604; approximate local minima at -3.089 and 3.735 Solve the initial value problem. 211) dv = 1 csc t cot t, v 3 dt 4 2
=-
3 4
211)
A) v = - sec t - 3
B) v = csc t - 7
C) v = - csc t - 1
D) v = - csc t - 1
4
4
4
4
2
4
Find the largest open interval where the function is changing as requested. 212) Decreasing f(x) = - x + 3 A) (-3, ) B) (3, ) C) (- , 3)
212) D) (- , -3)
Graph the equation. Include the coordinates of any local and absolute extreme points and inflection points.
68
213) y = 2x3 + 15x2 + 24x
213)
A) local minimum: (3, 6)
B) no extrema
C) local minimum: (-8, 512)
D) local minimum: (-1, -11)
no inflection point
inflection point: (0, 0)
local maximum: (0, 0) inflection point: (-4, 256)
local maximum: (-4, 16) 5 8.796093022e+13 inflection point: - , 2 3.518437209e+13
69
214) y = 3x2 + 30x
214)
A) absolute minimum: (-10,-30)
B) absolute minimum: (5,-75)
C) absolute minimum: (10,-30)
D) absolute minimum: (-5,-75)
no inflection points
no inflection points
no inflection points
no inflection points
Solve the problem. 215) From a thin piece of cardboard 50 in. by 50 in., square corners are cut out so that the sides can be folded up to make a box. What dimensions will yield a box of maximum volume? What is the maximum volume? Round to the nearest tenth, if necessary. A) 33.3 in. × 33.3 in. × 16.7 in.; 18,518.5 in3 B) 25 in. × 25 in. × 12.5 in.; 7812.5 in3
C) 33.3 in. × 33.3 in. × 8.3 in.; 9259.3 in3
D) 16.7 in. × 16.7 in. × 16.7 in.; 4629.6 in3
70
215)
Find the derivative at each critical point and determine the local extreme values. 216) y = x2 1 - x
216)
A) Critical Pt. derivative Extremum Value 0 0 min x=0 0 undefined min x=1 x=
4 5
0
local max
16 5 125
B) Critical Pt. derivative Extremum Value x=
4 5
0
local max
16 5 125
C) Critical Pt. derivative Extremum Value 0 0 min x=0
x=1 x=
4 5
0
min
0
local max
0
16 5 125
D) Critical Pt. derivative Extremum Value 0 undefined min x=1 x=
4 5
0
local max
16 5 125
Graph the function, then find the extreme values of the function on the interval and indicate where they occur. 217) y = x - 3 + x + 3 on the interval -5 < x < 5 217)
71
A) Absolute minimum is 6 on the interval [-3, 3]
B) Absolute minimum is 7 on the interval [-3, 3]
C) Absolute minimum is 7 on the interval [-3, 3]
72
D) Absolute minimum is 6 on the interval [-3, 3]
Use Newton's method to estimate the requested solution of the equation. Start with given value of x 0 and then give x 2 as the estimated solution. 218) x4 - 3 = 0; x0 = 1; Find the negative solution.
A) 1.32
218)
B) 1.29
C) 1.33
For the given expression y , find y'' and sketch the general shape of the graph of y = f(x). 219) y = x2(1 - x)
73
D) 1.31
219)
A)
B)
C)
74
D)
Graph the rational function. x+3 220) y = 2 x + 7x + 12
220)
A)
B)
75
C)
D)
Find all possible functions with the given derivative. 221) y = 4t - 2 t
A) 4t2 - 4 t + C
221) C) 4t2 - 2 + C
B) 2t2 - 4 t + C
D) t2 - 2 t + C
t
Find the extreme values of the function and where they occur. 222) y = x2 + 2x - 3
A) The minimum is 1 at x = 4. C) The minimum is 1 at x = -4.
222) B) The minimum is -4 at x = -1. D) The minimum is -1 at x = 4.
Solve the problem. 223) At noon, ship A was 12 nautical miles due north of ship B. Ship A was sailing south at 12 knots (nautical miles per hour; a nautical mile is 2000 yards) and continued to do so all day. Ship B was sailing east at 7 knots and continued to do so all day. The visibility was 5 nautical miles. Did the ships ever sight each other? A) Yes. They were within 3 nautical miles of each other.
223)
B) No. The closest they ever got to each other was 7.0 nautical miles. C) Yes. They were within 4 nautical miles of each other. D) No. The closest they ever got to each other was 6.0 nautical miles. 224) The positions of two particles on the s-axis are s1 = sin t and s2 = sin t +
6
, with s1 and s2 in
meters and t in seconds. At what time(s) in the interval 0 t 2 do the particles meet? A) t = 5 sec and t = 17 sec B) t = 1 6 6 2
C) t = 5
12
sec and t =
17 12
sec
sec and t =
3 2
sec
D) t = sec and t = 2 sec
76
224)
Solve the initial value problem. 225) dr = 8t + sec2 t, r(- ) = 2 dt
225)
A) r = 4t2 + tan t + 2 - 4 2 C) r = 8 + tan t - 6
B) r = 8t2 + tan t + 2 - 8 2 D) r = 4t2 + cot t + 2 - 4 2
Solve the problem. 226) A small frictionless cart, attached to the wall by a spring, is pulled 10 cm back from its rest position and released at time t = 0 to roll back and forth for 4 sec. Its position at time t is s = 1 - 10 cos t. What is the cart's maximum speed? When is the cart moving that fast? What is the magnitude of of the acceleration then? A) 10 31.42 cm/sec; t = 0 sec, 1 sec, 2 sec, 3 sec; acceleration is 0 cm/sec2
226)
B) 10 31.42 cm/sec; t = 0.5 sec, 2.5 sec; acceleration is 1 cm/sec2 C) 10 31.42 cm/sec; t = 0.5 sec, 1.5 sec, 2.5 sec, 3.5 sec; acceleration is 0 cm/sec2 D) 3.14 cm/sec; t = 0.5 sec, 1.5 sec, 2.5 sec, 3.5 sec; acceleration is 0 cm/sec2 227) Suppose a business can sell x gadgets for p = 250 - 0.01x dollars apiece, and it costs the business
227)
c(x) = 1000 + 25x dollars to produce the x gadgets. Determine the production level and cost per gadget required to maximize profit. A) 13,750 gadgets at $112.50 each B) 10,000 gadgets at $150.00 each
C) 111 gadgets at $248.89 each
D) 11,250 gadgets at $137.50 each
Use a computer algebra system (CAS) to solve the given initial value problem. 228) y = 4x2 sin x, y(0) = 1
A) y = -4(x2 - 2) cos x + 8x sin x - 7 C) y = -4x cos x + 4 sin x + 1
228)
B) y = -4x sin x cos x + 4 sin2 x + 4x2 + 1 D) y = -4 cos x2 + 5
Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. 229) f(x) = 0.01x5 - x4 + x3 + 8x2 - 7x - 26 229)
A) Approximate local maxima at -1.765 and 2.198; approximate local minima at 0.352 and 79.096 B) Approximate local maxima at -1.861 and 2.247; approximate local minimum at 0.423
C) Approximate local maxima at -1.861 and 2.247; approximate local minima at 0.423 and 79.192
D) Approximate local maxima at -1.894 and 2.302; approximate local minima at 0.476 and 79.247
Sketch the graph of the function and determine whether it has any absolute extreme values on its domain.
77
230) y = x - 2 , 0 < x 10
230)
A) absolute maximum at x = 10;
B) absolute maximum at x = 10;
C) absolute maximum at x = 10;
D) absolute maximum at x = 10;
absolute minimum at x = 2
no absolute minimum
absolute minimum at x = 2
no absolute minimum
Solve the problem. 231) A long strip of sheet metal 12 inches wide is to be made into a small trough by turning up two sides at right angles to the base. If the trough is to have maximum capacity, how many inches should be turned up on each side? A) 6 in. B) 4 in. on one side, 5 in. on the other
C) 3 in.
D) 4 in.
78
231)
Find the most general antiderivative.
232)
( t-
4
t) dt
232)
A) 3 t3/2 - 5 t5/4 + C
B) -1 t1/2 - 1 t-3/4 + C
C) 2 t3/2 - 4 t5/4 + C
D) t -
2
4
3
2
5
4
3
t+C
Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval. cos , <0
233) f( ) =
0,
233)
=0
A) No
B) Yes
Use Newton's method to estimate the requested solution of the equation. Start with given value of x 0 and then give x 2 as the estimated solution. 234) -x2 + 4x -1 = 0; x0 = 0; Find the left-hand solution.
A) 0.14
B) 0.23
234) C) 0.25
D) -0.33
Find the function with the given derivative whose graph passes through the point P.
235) r ( ) = csc cot - 3, P
4
A) r( ) = -csc - 3 +
,0
4
235) +
2
B) r( ) = csc - 3 +
9
C) r( ) = -csc - t
4
+
2
D) r( ) = -csc - 3
3
Answer each question appropriately. 236) The position of an object in free fall near the surface of the plane where the acceleration due to gravity has a constant magnitude of g (length-units)/sec2 is given by the equation:
1 s = - gt2 + v0 t + s0 , where s is the height above the earth, v0 is the initial velocity, and s0 is 2
the initial height. Give the initial value problem for this situation. Solve it to check its validity. Remember the positive direction is the upward direction. 2 2 A) d s = -g B) d s = g , s (0) = v0 , s(0) = s0 dt2 dt2
2
2
C) d s = -gt , s(0) = s0
D) d s = -g , s (0) = v0 , s(0) = s0
dt2
dt2
79
236)
Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. 237) f(x) = 0.1x4 - x3- 15x2 + 59x + 8 237)
A) Approximate local maximum at 1.672; approximate local minima at -6.85 and 12.457 B) Approximate local maximum at 1.704; approximate local minima at -6.847 and 12.446 C) Approximate local maximum at 1.77; approximate local minima at -6.852 and 12.572 D) Approximate local maximum at 1.735; approximate local minima at -6.777 and 12.542 Find the open intervals on which the function is increasing and decreasing. Identify the function's local and absolute extreme values, if any, saying where they occur.
238)
238)
A) increasing on (-3, 3); decreasing on (-6, -3) and (3, 6);
absolute maximum at (3, 1); absolute minimum at (-3, -1)
B) increasing on (-3, 3); decreasing on (0, 6);
absolute maximum at (3, 1); absolute minimum at (-3, -1) C) increasing on (-3, 3); decreasing on (-6, -3) and (3, 6); no absolute maximum; no absolute minimum D) increasing on (-3, 3); decreasing on (-6, 0); absolute maximum at (3, 1); absolute minimum at (-3, -1) Identify the function's extreme values in the given domain, and say where they are assumed. Tell which of the extreme values, if any, are absolute. 239) f(x) = (x + 5)2 , < x 0 239)
A) local and absolute maximum: 25 at x = 0; local and absolute minimum: 0 at x = -5
B) no local extrema; no absolute extrema C) local maximum: 25 at x = 0;
local and absolute absolute minimum: 0 at x = -5 D) local and absolute minimum: 0 at x = -5
Find an antiderivative of the given function. 240) 4 x - 4 A) 8 x3/2 - 4x B) 4x3/2 - 4 3
240) C) 8 x3/2 - 4 3
80
D) 4x3/2 - 4x
Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. 241) f(x) = 0.1x5 + 5x4 - 8x3 - 15x2 - 6x - 57 241)
A) Approximate local maxima at -41.132 and -0.273; approximate local minima at -0.547 and 1.952 B) Approximate local maxima at -41.183 and -0.202; approximate local minima at -0.459 and 2.023 C) Approximate local maxima at -41.193 and -0.284; approximate local minima at -0.554 and 2.013 D) Approximate local maxima at -41.185 and -0.322; approximate local minima at -0.588 and 1.993
Graph the equation. Include the coordinates of any local and absolute extreme points and inflection points. 242) y = x1/3(x2 - 112)
B) local minimum: ± 48, - 24
A) no extrema
inflection point: (0, 0)
local maximum: (0, 0)
inflection points: ±4, -
81
20 3
242)
3
C) local minimum: 4, -96 4 local maximum: -4, 96 inflection point: (0, 0)
3
D) local minimum: (0, 0) no inflection points
4
Solve the problem. 243) Given the acceleration, initial velocity, and initial position of a body moving along a coordinate line at time t, find the body's position at time t. a = 20 cos 4t, v(0) = -10, s(0) = -6 A) s = - 5 sin 4t - 10t - 6 B) s = 5 sin 4t - 10t - 6 4 4
C) s = - 5 cos 4t - 10t - 6
243)
D) s = 5 cos 4t + 10t - 6
4
4
Graph the function, then find the extreme values of the function on the interval and indicate where they occur. 244) y = x - 3 - x - 4 on the interval -2 < x < 7 244)
82
A) Absolute maximum is -1 on the interval [4, 7); Absolute minimum is 1 on the interval (-2, 3]
B) Absolute maximum is 2 on the interval [4, 7); Absolute minimum is 0 on the interval (-2, 3]
C) Absolute maximum is 1 on the interval [4, 7); Absolute minimum is 0 on the interval (-2, 4]
83
D) Absolute maximum is 1
on the interval [4, 7); Absolute minimum is -1 on the interval (-2, 3]
Solve the problem. 245) At about what velocity do you enter the water if you jump from a 15 meter cliff? (Use g = 9.8 m/sec2 .)
A) 17 m/sec
B) -17 m/sec
C) -8.5 m/sec
D) 2 m/sec
Find the location of the indicated absolute extremum for the function. 246) Maximum
A) x = 5
B) No maximum
C) x = 3
Find the most general antiderivative. y 6 247) dy + 2 y
A) 1 y 4
4
12
246)
D) x = 0
247) B) 1 y3/2 - 12 y + C
1 +C 12 y
C) 3 y3/2 + 1
245)
3
D) 1 y3/2 + 12 y + C
y+C
3
84
Find the function with the given derivative whose graph passes through the point P. 248) g (x) = 1 + 2x , P(-4, 4) x2
248)
A) g(x) = 1 + x2 + 49
B) g(x) = - 1 + x2 - 49
C) g(x) = 1 + x2 - 49
D) g(x) = - 1 - x2 - 49
x x
4
x
4
x
4 4
Graph the equation. Include the coordinates of any local and absolute extreme points and inflection points. 249) y = x 5 - x2
A) local maximum: (0, 5) no inflection points.
85
249)
B) local minimum: local maximum:
10 5 ,2 2 10 5 , 2 2
inflection point: (0, 0)
C) local minimum: local maximum:
15 2 · 5 3/2 · 3 ,3 9 15 2 · 5 3/2 · , 3 9
3
inflection point: (0, 0)
86
D) local maximum: 10 , 2 · 5 3
3/2 · 3 9
no inflection point.
Answer each question appropriately. 250) Find a curve y = f(x) with the following properties: d2 y i. = 4x dx2
250)
ii. The graph passes through the point (0,1) and has a horizontal tangent at that point. A) y = 4x + 1 B) y = 2 x3 + x + 1 C) y = 2 x3 + 1 D) y = 2x2 + 1 3 3
For the given expression y , find y'' and sketch the general shape of the graph of y = f(x). 2 251) y' = x - 2 4
87
251)
A)
B)
C)
D)
Find all possible functions with the given derivative. 252) y = 5 - 1 x2
A) 5x - 1 + C x
252)
B) 5x + 1 + C
Find an antiderivative of the given function. 253) 4 csc 6x cot 6x
A) -4 csc 6x
C) 5 + 1 + C
D) 5 - 1 + C
C) - 2 cot 6x 3
D) 2 csc 6x 3
x2
x
x
253)
B) - 2 csc 6x 3
254) - 63
254)
x8
A) 9
x7
B) 7
C) - 7
x10
x9
88
D) 9
x8
Find the extreme values of the function and where they occur. 255) y = x + 1 x2 + 2x + 2
255)
A) The maximum is 2 at x = 0; the minimum is 1 at x = -2. 2
B) None C) The maximum is - 1 at x = 0; the minimum is 1 at x = -2. 2
2
D) The maximum is 1 at x = 0; the minimum is - 1 at x = -2. 2
2
Identify the function's local and absolute extreme values, if any, saying where they occur. 256) f(x) = x3 + 3x2 + 3x + 2
256)
A) no local extrema B) local maximum at x = -1; local minimum at x = 1 C) local maximum at x = -1 D) local minimum at x = -1 Graph the rational function. 2 257) y = x + 2x - 15 x-4
257)
A)
B)
89
C)
D)
Find the location of the indicated absolute extremum for the function. 258) Maximum
A) x = -4
B) x = 0
C) No maximum
258)
D) x = 11 4
Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing. 259) f (x) = (5 - x)(7 - x) A) Decreasing on (5, 7); increasing on (- , 5) (7, )
259)
B) Decreasing on (- , 5) (7, ); increasing on (5, 7) C) Decreasing on (- , 5); increasing on (7, ) D) Decreasing on (- , -5) (-7, ); increasing on (-5, -7) Find the largest open interval where the function is changing as requested. 260) Decreasing f(x) = 4 - x A) (- , 4) B) (- , -4) C) (-4, )
90
260) D) (4, )
Plot the zeros of the given polynomial on the number line together with the zeros of the first derivative. 261) y = (x - 5)(x + 1)2
261)
A)
B)
C)
D)
Graph the rational function. 262) y = x x2 - 36
262)
A)
B)
91
C)
D)
Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval. 263) s(t) = t(4 - t), -1,5 A) No B) Yes Find the most general antiderivative. 1 1 264) dx - x5 5 3 x
264)
6 A) 1 - x - 1 + C 5x6
6
4 B) 1 - x + 1 + C
3x
6x6
6
C) -1 - x - x + C 4x4
6
4
9
D) -5x4 - 5x5 + C
3
Find all possible functions with the given derivative. 265) y = x3 - 7x
A) 3x2 - 7 + C
263)
3
265)
2
4
B) x - 7x + C 3
C) x + 7x2 + C
2
4
4
2
D) x - 7x + C 4
2
Graph the equation. Include the coordinates of any local and absolute extreme points and inflection points. 266) y = 6x x2 + 1
92
266)
B) local minimum: -1, - 3
A) local minimum: (1, -3)
2
local maximum: (-1, 3) inflection point: (0, 0)
local maximum: 1,
3 2
inflection point: (0,0)
C) absolute maximum: 0, 6
D) local minimum: (-1, -3)
no inflection point
local maximum: (1, 3)
inflection points: (0, 0), (-1 3, (1 3,
Use differentiation to determine whether the integral formula is correct. 267) sec2 x - 3 dx = -6 cot x - 3 + C 6 6
A) No
B) Yes
93
3 2
3 2
3),
3)
267)
Find the derivative at each critical point and determine the local extreme values. 1 1 15 , x 1 - x2 - x + 4 2 4 268) y = x3 - 6x2 + 8x, x>1
268)
A) Critical Pt. derivative Extremum Value x = -1 0 local max 4 x = 3.15 0 local min -3.08
B) Critical Pt. derivative Extremum Value x=1 0 local max 4 x = 3.15 undefined local max -3.08
C) Critical Pt. derivative Extremum Value x = -1 undefined local min 4 x = 3.15 0 local max -3.08
D) Critical Pt. derivative Extremum Value x = -1 0 local min 4 x = 3.15 0 local max -3.08
Solve the initial value problem. 2 269) d r = 6 ; dr t=1 = 5, r(1) = 5 dt2 t3 dt
269)
A) r = 3t + 8t + 16
B) r = 3 + 8t + 16
C) r = 3 + 8t - 6
D) r = 6 + 31 t - 6
t
-5t5
t
5
Use differentiation to determine whether the integral formula is correct. 4 1 270) dx = +C 5 (x + 5) (x + 5)4
A) No
270)
B) Yes
Identify the function's local and absolute extreme values, if any, saying where they occur. 271) f(x) = x3 + 0.5x2 - 2x + 2
271)
A) local maximum at x = -2; local minimum at x = 1 3
B) local maximum at x = -2 ; local minimum at x = 1 3
C) local maximum at x = -1; local minimum at x = 2 3
D) local maximum at x = - 1 ; local minimum at x = 2 3
Graph the function, then find the extreme values of the function on the interval and indicate where they occur. 94
272) y = x + 2 - x - 5 on the interval - < x <
272)
A) Absolute maximum is 7
on the interval [5, ); Absolute minimum is -7 on the interval (- , 2]
B) Absolute maximum is 8
on the interval [5, ); Absolute minimum is -8 on the interval (- ,2]
95
C) Absolute maximum is 7
on the interval [6, ); Absolute minimum is -7 on the interval (- ,1]
D) Absolute maximum is 6
on the interval [5, ); Absolute minimum is -6 on the interval (- ,2]
Identify the function's local and absolute extreme values, if any, saying where they occur. 273) f(r) = (r - 7) 3
A) local minimum: x = 0 B) local minimum: x = 0; local maximum: x = 7 C) no local extrema D) local minimum: x = 7
96
273)
Answer each question appropriately. 274) How many curves y = f(x) have the following properties? d2 y i. = 5x dx2
274)
ii. The graph passes through the point (0,3) and has a horizontal tangent at that point. A) 2 B) Infinitely many
C) 0
D) 1
Solve the problem. 275) You are driving along a highway at a steady 72 ft/sec when you see a deer ahead and slam on the brakes. What constant deceleration is required to stop your car in 296 ft? A) 8.76 ft/sec2 B) 0.11 ft/sec2 C) 4.38 ft/sec2 D) 17.51 ft/sec2
275)
Use Newton's method to estimate the requested solution of the equation. Start with given value of x 0 and then give x 2 as the estimated solution. 276) x4 - 6x + 3 = 0; x0 = 2; Find the right-hand solution.
A) 1.604
B) 1.602
276) C) 1.607
Use differentiation to determine whether the integral formula is correct. 4x + 5 5 277) 4x + 5 4 dx = +C 20
A) No
D) 1.600
277)
B) Yes
Sketch the graph and show all local extrema and inflection points. 278) y = |x2 - 5x|
97
278)
B) Local minimum: 5 , 25
A) Local minima: (0, 0), (5, 0) Local maximum:
2
5 25 , 2 4
4
No inflection points
Inflection points: (0, 0), (5, 0)
D) Local maximum: 5 , 25
C) Local minima: (0, 0), (5, 0) Local maximum:
2
5 25 , 2 4
4
Inflection points: (0, 0), (5, 0)
No inflection points
Find the value or values of c that satisfy the equation
f(b) - f(a) = f (c) in the conclusion of the Mean Value Theorem for b-a
the function and interval. 279) f(x) = x + 32 , [2, 16] x
A) 0, 4 2
279) B) -4 2, 4 2
C) 2, 16
98
D) 4 2
Identify the function's extreme values in the given domain, and say where they are assumed. Tell which of the extreme values, if any, are absolute. 280) h(x) = x3 + 2x2 + 6x + 3, < x 0 280)
A) local and absolute maximum: 3 at x = 0;
local and absolute minimum: 2 at x = -1
B) no local extrema; no absolute extrema C) local and absolute maximum: 3 at x = 0; D) local and absolute minimum: 2 at x = -1; Find the absolute extreme values of the function on the interval. 281) f(x) = 7x8/3, -27 x 8
281)
A) absolute maximum is 45,927 at x = -27 ; absolute minimum is 0 at x = 0 B) absolute maximum is 1792 at x = 8 ; absolute minimum is 0 at x = 0 C) absolute maximum is 6561 at x = -27 ; absolute minimum is 0 at x = 0 D) absolute maximum is 45,927 at x = -27 ; absolute minimum is 1792 at x = 8 Solve the problem.
282) The acceleration of gravity near the surface of Mars is 3.72 m/sec2 . If a rock is blasted straight up
282)
from the surface with an initial velocity of 85 m/sec (about 190 mph), how high does it go? (Hint: When is velocity zero?) A) Approximately 1942.25 meters B) Approximately 22.85 meters
C) Approximately 170 meters
D) Approximately 971.1 meters
Find the extreme values of the function and where they occur. 283) y = x3 - 3x2 + 1
283)
A) Local maximum at (0, 1), local minimum at (2, -3). B) None C) Local maximum at (0, 1). D) Local minimum at (2, -3). Solve the problem. 284) Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's position at time t. v = -11t + 6, s(0) = 7 A) s = -11t2 + 6t + 7 B) s = - 11 t2 + 6t + 7 2
C) s = 11 t2 + 6t - 7
284)
D) s = - 11 t2 + 6t - 7
2
2
Using the derivative of f(x) given below, determine the critical points of f(x). 285) f (x) = (x + 6)(x + 8) A) -8, -6 B) -14 C) 6, 8
99
285) D) 0, -8, -6
Solve the problem. 286) Determine the dimensions of the rectangle of largest area that can be inscribed in a semicircle of radius 3. A) h = 3 2 , w = 3 2 B) h = 3 2, w = 2 2
C) h = 3 2, w = 3 2 ,
286)
D) h = 2, w = 3 2
2
Find all possible functions with the given derivative. 287) y = csc2 3
287)
A) - 1 csc3 3 + C 9
B) 1 csc 3 + C 6
C) -3 cot 3 + C
D) - 1 cot 3 + C 3
Solve the problem.
288) A particle moves on a coordinate line with acceleration a = d2s/dt2 = 5/ t + 9 t, subject to the
288)
conditions that ds/dt = 3 and s = 1 when t = 1. Find the velocity v = ds/dt in terms of t and the position s in terms of t. 3 5 3 A) v = 20 t + 12 t - 13t + 74 ; s = 10 t + 6 t - 13 3 5 15 3
3
5
B) v = 10 t - 6 t + 13; s = 20 t - 12 t + 13t + 74 3
3
3
5
5
15
3
C) v = 10 t + 6 t - 13; s = 20 t + 12 t - 13t + 74 3
3
5
3
15
5
D) v = 10 t + 6 t - 13; s = 20 t + 12 t - 13t + 74 3
5
15
Identify the function's local and absolute extreme values, if any, saying where they occur. 4 289) g(x) = x - 2x3 + 5 x2 + 12x - 4 4 2
A) local maxima at x = 1 and x = -4; local minimum at x = 3 B) local maximum at x = -1; local minimum at x = 4 C) local maxima at x = -1 and x = 4; local minimum at x = 3 D) local maximum at x = 3; local minima at x = -1 and x = 4
100
289)
Find the open intervals on which the function is increasing and decreasing. Identify the function's local and absolute extreme values, if any, saying where they occur.
290)
290)
A) increasing on (-3, 0); decreasing on (-5, -3) and (2, 5)
absolute maximum at (-5, 4); local maximum at (0, -2) and (2, -2); absolute minimum at (5, -5) B) increasing on (-3, 0); decreasing on (-5, -3) and (2, 5) absolute maximum at (-5, 4); local minimum at (-3, -5) and (5, -5) C) increasing on (-3, 1); decreasing on (-5, -3) and (0, 5) absolute maximum at (-5, 4); no absolute minimum D) increasing on (-3, 0); decreasing on [-5, -3) and (2, 5] absolute maximum at (-5, 4); absolute minimum at (5, -5)
Solve the problem. 291) Given the acceleration, initial velocity, and initial position of a body moving along a coordinate line at time t, find the body's position at time t. a = 12, v(0) = 3, s(0) = -13 A) s = 12t2 + 3t - 13 B) s = 6t2 + 3t
C) s = -6t2 - 3t - 13
291)
D) s = 6t2 + 3t - 13
Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. 292) f(x) = x5 - 15x4 - 3x3 - 172x2 + 135x - 0.099 292)
A) Approximate local maximum at 0.379; approximate local minima at -0.472 and 12.565 B) Approximate local maximum at 0.379; approximate local minimum at 12.565 C) Approximate local maximum at 0.361; approximate local minima at -0.531 and -12.542 D) Approximate local maximum at 0.347; approximate local minimum at -12.496 Solve the problem.
293) Suppose c(x) = x3 - 18x2 + 30,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items. A) 11 items B) 10 items C) 8 items
D) 9 items
Find the extreme values of the function and where they occur. 294) y = (x + 4)2/3
A) The minimum value is 0 at x = -4. C) There are no definable extrema.
294) B) The maximum value is 0 at x = 4. D) The minimum value is 0 at x = 4.
101
293)
The graph below shows solution curves of a differential equation. Find an equation for the curve through the given point.
295)
295)
dy = sin x + cos x dx
A) y = sin x - cos x -2 C) y = sin x - cos x -1
B) y = sin x - cos x D) y = sin x - cos x +1
Graph the rational function. 296) y = 5 x2
296)
A)
B)
102
C)
D)
Find the function with the given derivative whose graph passes through the point P.
297) r ( ) = 4 + sec2 , P(
4
, 0)
297)
A) r( ) = 2 2 + tan - 3 + 1
B) r ( ) = 2 2 + tan + 3
C) r( ) = 4 + 1 sec3
D) r( ) = 4 + tan - 3 + 1
3
Solve the initial value problem. 3 298) d y = 7; y (0) = 4, y (0) = -2, y(0) = 5 dx3
298)
A) y = 5
B) y = 7 x3 + 2x2 - 2x + 5
C) y = 7 x3 - 2x2 - 2
D) y = 7 x3 + 2x2 - 2x + 5
3
6
6
Using the derivative of f(x) given below, determine the critical points of f(x). 299) f (x) = (x - 3)2 (x + 10)
A) -3, 0, 10
B) -10, 3
C) -3, 10
299) D) -10, -3, 3
Use differentiation to determine whether the integral formula is correct.
300)
x sin x dx = -x cos x + sin x + C
300)
A) Yes
B) No
Find an antiderivative of the given function. 301) 6 cos 4x
A) sin 4x
301)
B) 6 sin 4x
C) - 24 sin 4x
103
D) 3 sin 4x 2
Solve the problem. 302) A private shipping company will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 96 in. Suppose you want to mail a box with square sides so that its dimensions are h by h by w and it's girth is 2h + 2w. What dimensions will give the box its largest volume? A) 64 in. × 64 in. × 16 in. B) 16 in. × 16 in. × 32 in. 3 3
C) 16 in. × 16 in. × 80 in.
302)
D) 32 in. × 16 in. × 32 in.
303) A rocket lifts off the surface of Earth with a constant acceleration of 30 m/sec2. How fast will the rocket be going 2.5 minutes later? A) 2250 m/sec B) -4500 m/sec
C) 3375 m/sec
Find the function with the given derivative whose graph passes through the point P. 304) f (x) = x - 8, P(2, -7) 2 A) f(x) = x2 - 8x + 5 B) f(x) = x - 8x + 8 2
303)
D) 4500 m/sec
304)
2
D) f(x) = x - 8x + 7
C) f(x) = x2 - 8x
2
Find the derivative at each critical point and determine the local extreme values. 305) y = x2/3(x2 - 9); x 0
A) Critical Pt. derivative Extremum Value x=0 Undefined local max 3 x = 1.5 0 minimum -8.845
B) Critical Pt. derivative Extremum Value x=0 Undefined local max 0 x = 1.5 0 minimum -8.845
C) Critical Pt. derivative Extremum Value x=0 0 maximum 0 x = 1.5 0 minimum -8.845
D) Critical Pt. derivative Extremum Value x=0 Undefined local max 0 x = 1.5 0 minimum 14.742
Sketch the graph and show all local extrema and inflection points.
104
305)
306) y = -x4 + 4x2 - 8
306)
A) Absolute maxima: (- 2, -4), ( 2, -4) Local minimum: (0, -8) 2 4 Inflection points: , , 3 9
B) Absolute minima: (- 2, 4), ( 2, 4) Local maximum: (0, 8) 2 52 Inflection point: , , 3 9
2 4 , 3 9
D) Absolute maxima: (- 2, -4), ( 2, -4)
C) Absolute maxima: (- 2, -4), ( 2, -4) Inflection points: -
2 4 , , 3 9
2 52 , 3 9
Local minimum: (0, -8) No inflection points
2 4 , 3 9
105
Find all possible functions with the given derivative. 307) y = x11
307)
B) 1 x10 + C 11
A) 11x10 + C
C) 1 x12 + C 12
D) 12x12 + C
Solve the problem. 308) A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, whereas the semicircle is of tinted glass that transmits only one-third as much light per unit area as clear glass does. The total perimeter is fixed. Find the proportions of the window that will admit the most light. Neglect the thickness of the frame.
A) width = 12 height
6+
B) width =
12 6+2
C) width =
12 3+2
D) width =
3 6+2
height
height height
Use differentiation to determine whether the integral formula is correct. 5 309) (3x - 3)4 dx = (3x - 3) + C 15
A) No
308)
309)
B) Yes
Solve the problem. 310) A rectangular field is to be enclosed on four sides with a fence. Fencing costs $6 per foot for two opposite sides, and $4 per foot for the other two sides. Find the dimensions of the field of area 620 ft2 that would be the cheapest to enclose.
A) 37.3 ft @ $6 by 16.6 ft @ $4 C) 30.5 ft @ $6 by 20.3 ft @ $4
B) 16.6 ft @ $6 by 37.3 ft @ $4 D) 20.3 ft @ $6 by 30.5 ft @ $4
106
310)
311) The strength S of a rectangular wooden beam is proportional to its width times the square of its
311)
depth. Find the dimensions of the strongest beam that can be cut from a 12-in.-diameter cylindrical log. (Round answers to the nearest tenth.)
12"
A) w = 5.9 in.; d = 10.8 in. C) w = 7.9 in.; d = 8.8 in.
B) w = 7.9 in.; d = 10.8 in. D) w = 6.9 in.; d = 9.8 in.
Find the derivative at each critical point and determine the local extreme values. 312) y = 5 - 2x, x 1 x + 2, x > 1
312)
A) Critical Pt. derivative Extremum Value x=0 0 minimum 4
B) Critical Pt. derivative Extremum Value x=2 undefined minimum 4
C) Critical Pt. derivative Extremum Value x=1 0 minimum 3
D) Critical Pt. derivative Extremum Value x=1 undefined minimum 3
Find an antiderivative of the given function. 313) 2 x5/3 3
A) 2 x8/5 5
313)
B) 2 x8/3
C) 1 x8/3
3
4
D) 1 x8/5
Identify the function's local and absolute extreme values, if any, saying where they occur. 314) h(x) = x - 2 x2 + 3x + 6
A) local minimum at x = -2; no local maxima B) local minimum at x = -5; local maximum at x = 6 C) local minimum at x = -2; local maximum at x = 6 D) no local extrema
107
4
314)
Find an antiderivative of the given function. 315) x8 - 1 x8 9 A) x - 1 8
8x7
315) 9 C) x + 1
B) 8x7 + 1
8x7
9
7x7
9 D) x - 1 9
9x9
Provide an appropriate response.
316) Suppose the derivative of the function y = f(x) is y' = (x - 9)2 (x + 3). At what points, if any, does
316)
the graph of f have a local minimum or local maximum? A) local minimum at x = -3 B) no local minimum or local maximum
C) local maximum at x = -3
D) local minimum at x = 9
Graph the rational function. 3 317) y = x x2 - 4
317)
A)
B)
108
C)
D)
Find the open intervals on which the function is increasing and decreasing. Identify the function's local and absolute extreme values, if any, saying where they occur.
318)
318)
A) increasing on (-2, 0) and (2, 4); decreasing on (0, 2);
absolute maximum at (4, 4); local maximum at (0, 2); absolute minimum at (-2, 0) and (2, 0)
B) increasing on (2, 4); decreasing on (0, 2);
absolute maximum at (4, 4); local maximum at (0, 2); absolute minimum at (-2, 0) and (2, 0)
C) increasing on (-2, 0) and (2, 4); decreasing on (0, 2);
absolute maximum at (4, 4) and(0,2); absolute minimum at (-2, 0) and (2, 0) D) increasing on (-2, 0) and (2, 4); decreasing on (0, 2); absolute maximum at (4, 4); absolute minimum at (-2, 0) and (2, 0)
Find the function with the given derivative whose graph passes through the point P. 319) r (t) = sec2 t - 4, P(0, 0)
A) r(t) = sec t tan t - 4t -1 C) r(t) = sec t - 4t - 4
B) r(t) = tan t - 4t D) r(t) = sec t - t - 6
109
319)
Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing. 320) f (x) = x1/3(x - 3)
320)
A) Decreasing on (0, 3); increasing on (- , 0) (3, ) B) Decreasing on (- , 0) (3, ); increasing on (0, 3) C) Decreasing on (0, 3); increasing on (3, ) D) Increasing on (0, ) Find the absolute extreme values of the function on the interval.
321) f(x) = tan x, -
4
x
321)
3
A) absolute maximum is 3 at x = B) absolute maximum is 3 at x = C) absolute maximum is -1 at x = D) absolute maximum is 3 at x =
9 3 3
; absolute minimum is -1 at x = ; absolute minimum is -1 at x = -
; absolute minimum is
3
and -
4
3 at x = -
A) Critical Pt. derivative Extremum Value x=0 2 undefined local min 1 2
0
local max
9 4
B) Critical Pt. derivative Extremum Value x=0 2 undefined local min x=
3 2
0
local max
17 4
C) Critical Pt. derivative Extremum Value 2 x=2 undefined local min x=0
0
local max
9 4
D) Critical Pt. derivative Extremum Value -2 x=0 undefined local min x=
1 2
0
local max
4 4
; absolute minimum does not exist
Find the derivative at each critical point and determine the local extreme values. x<0 322) y = 2 - x, 2 2 + 1x - x , x 0
x=
8
7 4
Graph the rational function. 110
322)
2
323) y = x + x - 20
323)
x2 - x - 12
A)
B)
C)
D)
Find the most general antiderivative.
324)
(-5 cos t) dt
A) - 5 + C sin t
324) C) - sin t + C
B) -5sin t + C
5
111
D) -5 cos t + C
Determine all critical points for the function. 325) y = 2x2 - 64 x
325)
A) x = 4 C) x = 0
B) x = 0, x = 4, and x = -4 D) x = 0 and x = 4
Find the largest open interval where the function is changing as requested. 326) Increasing f(x) = 1 x2 - 1 x 4 2
A) (1, )
B) (- , )
326)
C) (- , -1)
D) (-1, 1)
Solve the problem. 327) Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's position at time t. 8 4t v = sin , s( 2 ) = 2
A) s = -2 cos 4t + 3.3073
B) s = 2 cos 4t + 4
C) s = -2 cos 4t + 8.2134
D) s = -2 cos 4t + 4
327)
Identify the function's extreme values in the given domain, and say where they are assumed. Tell which of the extreme values, if any, are absolute. 3 328) g(t) = t - 17 t2 + 72t, 0 t < 328) 3 2
A) local minimum: 608 at x =8; local maximum: 0 at x = 0; absolute maximum: 405 at x = 9 3
2
B) local and absolute minimum: 405 at x = 9; local maximum: 608 at x =8 2
3
C) local minimum: 608 at x =8; local and absolute maximum: 405 at x = 9 3
2
D) local minimum: 0 at x = 0; local and absolute minimum: 405 at x = 9; local maximum: 608 2
3
at x =8
Graph the equation. Include the coordinates of any local and absolute extreme points and inflection points.
112
329) y =
x2 2 x + 13
329)
B) local minimum: 0, - 1
A) local minimum: (0, 0)
13
no inflection points
no inflection points
C) local minimum: 0, 1
D) local minimum: (0, 0)
no inflection points
inflection points: -
13
113
39 1 , , 3 4
39 1 , 3 4
Find the location of the indicated absolute extremum for the function. 330) Maximum
A) x = 4
B) No maximum
330)
C) x = 1
D) x = -4
Solve the problem. 331) If the price charged for a candy bar is p(x) cents, then x thousand candy bars will be sold in a x certain city, where p(x) = 121 . How many candy bars must be sold to maximize revenue? 34
A) 2057 candy bars C) 2057 thousand candy bars
B) 4114 candy bars D) 4114 thousand candy bars
Solve the initial value problem. 332) dy = 1 + x, x > 0; y(3) = 0 dx x3
A) y = -1 + x 2x2
332)
2
B) y = - 1 + 1 2x2
2
2
2
18
2 D) y = - 1 + x - 2.814749767e+15
C) y = 4 + x - 1.296544111e+16 x4
331)
2.849934139e+15
2x2
2
Determine all critical points for the function. 333) f(x) = (x - 1)5
6.333186976e+14
333)
A) x = 0, x = 1, and x = 5 C) x = 1
B) x = 1 and x = 5 D) x = 0 and x = 1
114
Answer Key Testname: CHAPTER 4
1) D 2) D 3) D 4) B 5) C 6) A 7) D 8) D 9) A 10) D 11) B 12) D 13) B 14) C 15) C 16) A 17) C 18) A 19) D 20) D
115
Answer Key Testname: CHAPTER 4
21) f(x) =
x - 8, - 8 - x,
f (x) =
x 8 x<8
1 , 2 x-8
x 8
1 , 2 8-x
x<8
Let x0 = h. Then, x1 = h -
h-8 = h - 2(h - 8) = 16 - h 1 2 h-8
x2 = 16 - h +
8 - (16 - h) = 16 - h + 2(h - 8) = h 1 2 8 - (16 - h)
Likewise, let x0 = - h. Then, x1 = - h +
8+h = - h + 2(8 + h) = 16 + h 1 2 8+h
x2 = 16 + h -
(16 + h) - 8 = 16 + h - 2(h + 8) = - h 1 2 (16 + h) - 8
22) If x , y represent the legs of the triangle, then x2 + y2 = 232 . 529 - x2 A(x) = xy = x 529 - x2 x2 529 - x2 A'(x) = + 2 2 529 - x2 Solving for y, y =
Solving A'(x) = 0, x = ±
23 2 2
Substitute and solve for y: (
23 2 2 23 2 ) + y2 = 529 ; y = 2 2
x = y.
23) Yes. the function is a flat line f(x) = C. One may say that every value of x is a critical point since critical points are by definition points where f (x) = 0. But none of these points correspond to extreme values. 24) x2 = 1.3836
25) As the trucker's average speed was 75 mph, the Mean Value Theorem implies that the trucker must have been going that speed at least once during the trip. 26) Assume there are 2 (or more) zeros a and b on the interval (1, 5) where 1 < a < b < 5. f(a) = f(b) = 0. By Rolle's theorem there is at least one number c in (a, b) such that f'(c) = 0 and 1 < a < c < b < 5. This implies there is another zero between 1 and 5 which is a contradiction. Thus, there cannot be more that 1 zero of f on (1, 5). 27) False. The function has a non-removable discontinuity at x = 0. The mean value theorem does not apply.
116
Answer Key Testname: CHAPTER 4
28) (a) No
(b) No (c) No (d) Minimum: 0 at x = ± 3 and x = 0; local maximum: 10.3923048 at x = ± 1.73205081 29) f(x) = 4x - 2x2 + 5, f '(x) = 4 - 4x Right-hand solution: x1 = 1.5 xn + 1 = x n -
4x - 2x2 + 5 -2x2 - 5 = 4 - 4x 4 - 4x
therefore x2 = 4.7500 Left-hand solution: x1 = -1 xn + 1 = x n -
4x - 4x2 + 5 -2x2 - 5 = 4 - 4x 4 - 4x
therefore x2 = -0.8750
30) a: both y and y are undefined.
b: y =0 and y > 0 c: y > 0 and y = 0 d: y = 0 and y = 0 e: y > 0 and y = 0 f: y = 0 and y < 0 g: y < 0 and y = 0 31) Yes. The point x = c is either a local maximum, a local minimum, or an inflection point. But, since f (x) > 0 for all x in the domain, there are no inflection points and the curve is everywhere concave up and thus cannot have a local maximum. Hence, there is a local minimum at x = c. 32) The average rate of temperature change is 145 ° F/sec during the 28 seconds. Therefore, the Mean Value Theorem 28
implies that at sometime during this time period, the temperature was changing at a rate of
145 ° F/sec. 28
33) (i) The roots of the function y = 2x3 - 3x - 1 can be calculated by using Newton's method. Resulting in three roots: 1.77777064, -0.36602451, and 1.36602545.
(ii) The x-coordinate of the intersection of y = x3 and y= 3x + 1 is the root of y = 2x3 - 3x - 1: 1.77777064. (iii) The curve y = x3 - 3x crosses the horizontal line y = 1 at the solution of y = 2x3 - 3x - 1 which is the same as the functions in part (i) and (ii). (iv) The values of x where the derivative of g(x) equals zero are the same as the functions in part (i) and (ii)
117
Answer Key Testname: CHAPTER 4
34) Find the root of f(x) = -0.0306t3 + 0.373t2 - 2.16t + 15.1 - 7.3. f (x) = -0.0918t2 + 0.746t - 2.16 x1 = 7 x2 = 7 -
f(7) -0.0306(7)3 + 0.373(7)2 - 2.16(7) + 15.1 =7= 12.40 f (7) -0.0918(7)2 + 0.746(7) - 2.16
x3 = 12.40 -
-0.0306(12.40)3 + 0.373(12.40)2 - 2.16(12.40) + 15.1 f(12.40) = 12.40 = 10.60 f (12.40) -0.0918(12.40)2 + 0.746(12.40) - 2.16
The useful working time is t = 10.60 hours.
35) a).
Solid line: f(x); dashed line: f (x) b). See figure above. f (x) = 0 at x = 4.2704 and x = 2.2079. Critical points of f(x) are 4.2704, 2.1993 and 2.2079, 2.7444 . c). f (x) is undefined at the endpoint x = 0. d). Endpoints are (0, 0) and (2 , 3.5449). e). Absolute minimum: (0, 0); absolute maximum (2 , 3.5449). 36) The function r( ) is continuous on the open interval (0, ). Also, r( ) approaches as approaches 0 from the right, and r( ) approaches - as approaches from the left. Since r( ) is continuous and changes sign along the interval, it must have at least one root on the interval. The first derivative of r( ) is r ( ) = -3 csc2
-
2 , which is everywhere negative on (0, ). Thus, r( ) has a single 3
root on (0, ).
37) (a) No, since f (x) = 2 (x - 3)-1/3, which is undefined at x = 3. 3
(b) The derivative is defined and nonzero for all x 3. (c) No, f(x) need not have a global maximum because its domain is all real numbers. Any restriction of f to a closed interval of the form [a, b] would have both a maximum value and a minimum value on the interval. (d) The answers are the same as (a) and (b) with 3 replaced by c.
118
Answer Key Testname: CHAPTER 4
38) Yes, all antiderivatives of g are of the form G(t) = -2t + C, where C is a constant. The only such function to satisfy the
initial condition g(0) = -4 is g(t) = -2t - 4. 39) The curves cross at x = 1.17. 40) Yes. The derivative is cubic: 4ax3 + 3bx2 + 2cx + d. The derivative approaches - as x approaches - and it approaches as x approaches . By the Intermediate Value Theorem, f (x) must equal zero at at least one point on the interval x . 41) Answers will vary. A general shape is indicated below:
42) a).
Solid line: f(x); dashed line: f (x) b). See figure above. f (x) = 0 at x = 0.4093 and x = 1.1644. Critical points of f(x) are 0.4093, 3.5859 and 1.1644, 2.9082 . c). f (x) is defined on the entire interval. d). Endpoints are (-0.5, 0.5625) and (1.8, 5.9376). e). Absolute minimum: (-0.5, 0.5625); absolute maximum (1.8, 5.9376).
119
Answer Key Testname: CHAPTER 4
43) f(x) = 2sin x - 4x + 1 f (x) = 2 cos x- 4 x1 = 1.5 xn + 1 =
2x cos x - 2sin x - 1 2cos x - 4
therefore x2 must be 0.72119422
44) At the critical point f = 0.This puts x1 and all later approximations out at - or . 45) f(x) = 3x4 + 2x - 4 = 0, f (x) = 12x3 + 2 Right-hand solution: x1 = 1 xn + 1 = x n -
3x4 + 2x - 4 9x4 + 4 = 12x3 + 2 12x3 + 2
therefore x2 = 0.9286 Left-hand solution: x1 = -1.5 xn + 1 = x n -
3x4 + 2x - 4 9x4 + 4 = 12x3 + 2 12x3 + 2
therefore x2 = -1.2873
46) The zeros of y = 0 and y = 0 are extrema and points of inflection, respectively. Inflection at x = 5, local maximum at x = 0, local minimum at x = 10.
120
Answer Key Testname: CHAPTER 4
47) f(x) = 3x4 - 6x2 + 2, f (x) = 12x3 - 12x
xn+1 = xn -
4 2 3x n - 6x n - 2 3 12 x n - 12xn
1st zero x0 = -1.5
2nd zero x0 = -0.5
3rd zero x0 = 0.5
4th zero x0 = 1.5
x1 = -1.3361
x1 = -0.6528
x1 =0 .6528
x1 = 1.3361
x2 = -1.2686
x2 = -0.6501
x2 =0 .6501
x2 = 1.2686
x3 = -1.2563
x3 = -0.6501
x3 = 0.6501
x3 = 1.2563
48) 4 solutions 49) The curve can have 0 or 2 inflection points. The second derivative is quadratic, y = 12ax2 + 6bx + 2c, and quadratics
may have 0, 1, or 2 roots, depending on the value of the discriminant. If 36b2 - 96ac < 0, then y has no real roots and y has no inflection points. If 36b2 - 96ac = 0, then y has exactly one real root and y has a single inflection point.
Finally, if 36b2 - 96ac > 0, then y has two real roots and y has exactly two inflection points. 50) f(x) = 2x5 - 3x - 5, f (x) = 10x4 - 3
x1 = 1 2x5 - 3x - 5 8x5 + 5 = 10x4 - 3 10x4 - 3
xn + 1 = x n -
therefore x2 = 1.8571
51) Yes, the Mean Value Theorem implies that the runner attained a speed of 10.5 mph, which was her average speed throughout the marathon. 1 -1 a 1 a 52) f(x) = x ; f (x) = x a a xn+1 = xn -
xn
1 1 a -1 x a n
= (1-a)xn
x0 = 1 x1 = 1 - a x2 = 1 - a 2 x3 = 1 - a 3 xn+1 = as n
1 - a n+1
, xn
121
Answer Key Testname: CHAPTER 4
53) We calculate c (x) = 3tp0 p2 - 4tp3 , and find that the only critical point is p=
3 3 3 3 p0. As c (x) < 0 for p > p0 and c (x) > 0 for p < p0 , the absolute maximum of c(x) occurs at p = p0 . 4 4 4 4
54) The graph will be a straight line. Since y = 0, this means there is no change in y , which is the slope of y. A constant slope implies a straight line. x0 = 1
55)
x1 = 0.7837 x2 = 0.7602 x3 = 0.7596
56) f(x) = x4 - 3x3 - 3x2 - 3x + 4 , f '(x) = 4x3 - 9x2 - 6x - 3
xn+1 = xn -
4 3 2 x n - 3 x n - 3 x n - 3 xn + 4 3 2 4 x n - 9 x n - 6xn - 3
Left-hand solution Right-hand solution x0 = 1 x0 = 3.5 x1 = 0.7143
x1 = 4.0856
x2 = 0.6657
x2 = 3.9204
x3 = 0.6641
x3 = 3.8995
57) f(x) = x4 - 5, f '(x) = 4x3 x1 = 1 xn + 1 = x n -
x4 - 5 3x4 + 5 = 4x3 4x3
therefore x2 = 2.0000
58) Yes. The value of g at x = -c is also zero since odd functions are symmetric with respect to the origin.
122
Answer Key Testname: CHAPTER 4
59) If he asks for a delivery every x days, then he must order (px) to have enough material for that delivery cycle. The average amount in storage is approximately one-half of the delivery amount, or
px . Thus, the cost of delivery and 2
storage for each cycle is approximately Cost per cycle = delivery costs + storage costs px Cost per cycle = d + ·x 2 We compute the average daily cost c(x) by dividing the cost per cycle by the number of days x in the cycle. d px c(x) = + x 2 We find the critical points by determining where the derivative is equal to zero. d p c (x) = + =0 2 2 x 2d p
x=±
Therefore, an absolute minimum occurs at
2d days. p
60) The function f(x) is continuous on the open interval (- , 0). Also, f(x) approaches - as x approaches - , and f(x)
approaches as x approaches 0 from the left. Since f(x) is continuous and changes sign along the interval, it must have at least one root on the interval. The first derivative of f(x) is f (x) = 3x2 -
8 , which is everywhere positive on (- , 0). Thus, f(x) has a single root on (x3
, 0).
61) Yes. The value of f at x = -c must also be zero since even functions are symmetrical with respect to the y-axis.
62) The derivative of the function is f (x) = 1 - 1 , which is positive for all x > 1. By the first derivative test, the function is x2
increasing at c = 8. 63) (a) The only critical point in the interval (0, ) at x = 10.0. The minimum value of P(x) is 40 at x = 10.0. (b) The smallest possible perimeter of the rectangle is 40 units and it occurs at x = 10.0, which makes the rectangle a 10.0 by 10.0 square. 64) y = - csc2 (x) + 2 3 csc(x)cot(x) = csc(x) 2 3 cot(x) - csc(x) = 0 2 3 cot(x) - csc(x) = 0 1 - 2 3 cos(x) = 0 3 3 3 3 cos(x) =
1 2 3 3
x=
6
65) f(x) = x4 - 2, f '(x) = 4x3 x1 = -1 xn + 1 = x n -
x4 - 2 3x4 + 2 = 4x3 4x3
therefore x2 = -1.2500
123
Answer Key Testname: CHAPTER 4
66) r1 = 2.0783 r2 = 0.0809 r3 = 0.9191 r4 = -1.0783 67) f(x) = -3x2 - 2x + 4, f (x) = -6x - 2 Right-hand solution: x1 = 0.5 xn + 1 = x n -
-3x2 - 2x + 4 -3x2 - 4 = -6x - 2 -6x - 2
therefore x2 = 0.9500 Left-hand solution: x1 = -2 xn + 1 = x n -
-3x2 - 2x + 4 -3x2 - 4 = -6x - 2 -6x - 2
therefore x2 = -1.6000
68) This example does not contradict Rolle's Theorem because the function f is not continuous on the closed interval [0, 1]. In particular, f is not continuous at the right end point x = 1. 69) Find the root of f(x) = C(x) - R(x) = 299 + 30x5/8 - 4x. f (x) =
75 -3/8 -4 x 4
x1 = 370 x2 = 370 -
f(370) 299 + 30·3705/8-4(370) = 370 = 384.04 f (370) 75 ·370-3/8 - 4 4
x3 = 384.04 -
f(384.04) 299 + 30·384.045/8-4(384.04) = 384.04 = 383.94 f (384.04) 75 ·384.04-3/8 - 4 4
The break-even point is x = 383.94 tools.
124
Answer Key Testname: CHAPTER 4
70) Answers will vary. A general shape is indicated below:
71) (a). Moving towards to origin on (1, 2) and (5.7, 7); moving away from the origin on (0, 1), (2, 5.7), and (7, 10).
(b). Velocity is zero at the extrema. These occur at t 1 sec and t 5.7 sec. (c). Acceleration is zero at the inflection points. These occur at t 2.3 sec, t 4 sec, t 5.1 sec, t 7 sec, and t 8.5 sec. (d). Acceleration is positive where f(t) is concave up and negative where it is concave down. Acceleration is positive on (2.3, 4), (5.1, 7), and (8.5, 10). Acceleration is negative on (0, 2.3), (4, 5.1), and (7, 8.5). 72) f(x) = 3x2 + 2x - 1, f (x) = 6x + 2 Right-hand solution: x1 = 0.5 xn + 1 = x n -
3x2 + 2x - 1 3x2 + 1 = 6x + 2 6x + 2
therefore x2 = 0.3500 Left-hand solution: x1 = -2 xn + 1 = x n -
3x2 + 2x - 1 3x2 + 1 = 6x + 2 6x + 2
therefore x2 = -1.3000
125
Answer Key Testname: CHAPTER 4
73) i). h (x) = f (x) + g (x) . The signs of the terms are (+) + (-), therefore h (x) may be positive or negative. Without more information, we cannot determine where h(x) is increasing or decreasing. ii). j (x) = f (x)g(x) + f(x)g (x) . The signs of the terms are (+)(+) + (+)(-) , therefore j (x) may be positive or negative. Without more information, we cannot determine where j(x) is increasing or decreasing. g (x)f(x) - g(x)f (x) (-)(+) - (+)(+) g(x) = (-). The quotient iii). k (x) = . The signs of the terms are is continuous and 2 (+) f(x) (f(x))
differentiable for all x, and k (x) is everywhere negative. So, according to the first derivative test, k(x) is everywhere decreasing. iv). p (x) = (g(x))( f(x) g(x)-1 )(g (x) ). The signs of the factors are (+)(+)(-) = (-). The function f(x) g(x) is everywhere
continuous and differentiable, and p (x) is everywhere negative. So, according to the first derivative test, p (x) is everywhere decreasing. v). r (x) = f (g(x))g (x). The signs of the factors are (+)(-) = (-). The function (f g)(x) is everywhere continuous and differentiable, and r (x) is everywhere negative. So, according to the first derivative test, r(x) is everywhere decreasing. 74) The zeros of y = 0 and y = 0 are extrema and points of inflection, respectively. Inflection at x = 12 , local maximum 5 at x = 0, local minimum at x =
75) Notice that g (x) =
16 . 5
b2
3/2 (b2 +(a + x)2 )
is positive for all values of x. Therefore g is increasing everywhere.
76) f(x) = -2x3 - 3x - 4, f (x) = -6(x2 ) - 3 x1 = -0.5 xn + 1 = x n -
-2x3 - 3x - 4 -4x3 + 4 = -6(x2 ) - 3 -6x2 - 3
therefore x2 = -1.0000
77) Yes. Since cos x = 0 at all points of to a different solution.
+ k , if you choosing your starting value too large or too small, it will converge
126
Answer Key Testname: CHAPTER 4
78)
Since f (x) =
1 > 0 for all x > 0, then the function is everywhere concave up. x2
79) D 80) C 81) A 82) B 83) D 84) D 85) C 86) D 87) B 88) C 89) C 90) B 91) A 92) C 93) C 94) B 95) D 96) B 97) B 98) D 99) C 100) B 101) D 102) B 103) D 104) C 105) D 127
Answer Key Testname: CHAPTER 4
106) B 107) D 108) D 109) C 110) C 111) D 112) D 113) D 114) D 115) A 116) C 117) B 118) D 119) B 120) D 121) B 122) D 123) D 124) B 125) B 126) C 127) A 128) C 129) D 130) D 131) D 132) C 133) D 134) B 135) A 136) B 137) B 138) A 139) C 140) A 141) A 142) C 143) D 144) A 145) D 146) A 147) C 128
Answer Key Testname: CHAPTER 4
148) D 149) D 150) C 151) B 152) B 153) B 154) A 155) A 156) A 157) C 158) A 159) C 160) C 161) C 162) C 163) A 164) B 165) A 166) B 167) B 168) D 169) D 170) B 171) D 172) D 173) D 174) C 175) C 176) B 177) B 178) C 179) D 180) C 181) B 182) B 183) D 184) B 185) C 186) C 187) A 188) D 189) A 129
Answer Key Testname: CHAPTER 4
190) D 191) D 192) C 193) B 194) B 195) B 196) D 197) D 198) B 199) D 200) D 201) D 202) A 203) B 204) B 205) B 206) C 207) A 208) A 209) B 210) D 211) D 212) A 213) D 214) D 215) C 216) A 217) A 218) C 219) C 220) D 221) B 222) B 223) D 224) C 225) A 226) C 227) D 228) A 229) C 230) C 231) C 130
Answer Key Testname: CHAPTER 4
232) C 233) A 234) B 235) A 236) D 237) D 238) A 239) C 240) A 241) A 242) C 243) C 244) D 245) B 246) D 247) D 248) B 249) B 250) C 251) B 252) B 253) B 254) A 255) D 256) A 257) B 258) C 259) A 260) A 261) A 262) B 263) A 264) C 265) D 266) D 267) A 268) A 269) C 270) B 271) C 272) A 273) C 131
Answer Key Testname: CHAPTER 4
274) D 275) A 276) D 277) B 278) C 279) D 280) C 281) A 282) D 283) A 284) B 285) A 286) A 287) D 288) D 289) D 290) D 291) D 292) B 293) D 294) A 295) C 296) C 297) D 298) D 299) B 300) A 301) D 302) A 303) D 304) D 305) B 306) A 307) C 308) B 309) B 310) D 311) D 312) D 313) C 314) C 315) C 132
Answer Key Testname: CHAPTER 4
316) A 317) A 318) A 319) B 320) A 321) B 322) A 323) B 324) B 325) D 326) A 327) D 328) D 329) D 330) C 331) C 332) D 333) C
133
Chapter 5
Exam Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer each question appropriately. 1) Which of the following integrals, if any, calculates the area of the shaded region?
A)
0
4x dx
B)
-2
2
4x dx
C)
-2
4
-4x dx
D)
-4
1)
0
-4x dx
-2
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response.
b
2) What values of a and b minimize the value of
(x6 - 4x4 ) dx?
2)
a
3) A plane averaged 300 mph on a 600 mile trip and then returned over the same 600 miles
3)
at the rate of 400 mph. What was the plane's average speed for the entire trip? Give reasons for your answer. 1
4) Use the max-min inequality to show that 1
1 + x4 dx
2.
4)
0 b
5) Explain why the rule
dx = b - a holds for any constants a and b.
5)
a
6) Integrate
csc 2 x cot x dx using the substitution u = cot x and using the substitution
u = csc x. Show the results are the same.
1
6)
2
7) Use the max-min inequality to find upper and lower bounds for 3 2
1 dx . Add these to arrive at an estimate of x
3 1
b
8) What values of a and b maximize the value of
1
1 dx and x
7)
1 dx . x
(6x - 3x2 ) dx?
8)
a 3
9) Use the max-min inequality to find upper and lower bounds for the value of 1
1 dx. x
10) Use the max-min inequality to show that if f is integrable and f(x) 0 on [a, b], then b
9)
10)
f(x) dx 0.
a
11) Derive a formula for the area of a circumscribed regular n-sided polygon for a circle of
11)
radius r.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the average value of the function over the given interval. 12) y = 2 - x2; [-2, 5]
A) - 33 7
B) - 13 3
12) C) - 17 3
D) - 25 7
Answer each question appropriately.
13) Suppose the velocity of a body moving along the s-axis is ds = 9.8t - 1. dt
Is it necessary to know the initial position of the body to find the body's displacement over some time interval? Justify your answer. A) No, displacement has nothing to do with the position of the body.
B) Yes, integration is not possible without knowing the initial position. C) No, the initial position is necessary to find the curve s= f(t) but not necessary to find the
displacement. The initial position determines the integration constant. When finding the displacement the integration constant is subtracted out. D) Yes, knowing the initial position is the only way to find the exact positions at the beginning and end of the time interval. Those positions are needed to find the displacement.
2
13)
Evaluate the integral. /2
14)
0
d
14)
2
A)
2
B)
8
2
C)
8
D) 2
Find the area of the shaded region.
f(x) = x3 + x2 - 6x
15)
15)
g(x) = 6x
A) 343 12
B) 81
C) 768
12
12
D) 937 12
Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. 16) f(x) = x2 between x = 1 and x = 5 using a lower sum with four rectangles of equal width. 16)
A) 30
B) 41
C) 69
D) 54
Evaluate the integral using the given substitution.
17)
3 18(y6 + 2y3 + 1) (2y5 + 2y2 ) dy , u = y6 + 2y3 + 1
17)
A) 9 (y6 + 2y3 + 1)4 (10y4 + 4y) + C
B) 3 (y6 + 2y3 + 1)4 + C
C) 18(y6 + 2y3 + 1)2 + C
D) 9 (y6 + 2y3 + 1)4 + C
2
2 2
Graph the function f(x) over the given interval. Partition the interval into 4 subintervals of equal length. Then add to 4 your sketch the rectangles associated with the Riemann sum f(ck) x k , using the indicated point in the kth k=1 subinterval for ck.
3
18) f(x) = -2x2 , [0, 4], left-hand endpoint
18)
A)
B)
C)
D)
4
Provide an appropriate response. 19) Suppose that f has a positive derivative for all values of x and that f(2) = 0. Which of the following x statements must be true of the function g(x) = f(t) dt ? 0 A) The graph of g has an inflection point at x = 2.
19)
B) The function g has a local maximum at x = 2. C) The graph of g crosses the x-axis at x = 2. D) The function g has a local minimum at x = 2. Find the average value of the function over the given interval. 20) f(x) = 3 - x on [0, 3] A) 27 B) 3 2
20) C) 9 2
D) 3 2
Solve the problem. 21) Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's position at time t. v = cos
2
21)
t, s(0) = 1
A) s = 2 sin
2
B) s = 2 sin
t+
D) s = 2 sin
C) s = sin t Find the derivative. tan x 22) y = 0
2 2
t+ 1 t+ 1
t dt
A) sec2 x tan x
22) B) 2 tan3/2 x
C) tan x
3
5
D) sec x tan3/2 x
Find the area of the shaded region. 23) y = x4 - 32
23)
y = -x4 B) 516 5
A) 256 5
C) 2816 5
D) 512 5
Find the area enclosed by the given curves.
24) y = sin x, y = csc 2 x, A)
3 1 + 3 2
3
x
24)
2
B) 1 -
3 2
C)
3 1 2 2
D)
3 1 3 2
Use a finite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. 25) f(x) = x on [3,7] divided into 4 subintervals (Round to the nearest hundredth.) 25) 10
A) 8.89 Evaluate the integral. 0 26) 3x2 + x + 5 dx 9 A) - 1629 2
B) 0.22
C) 0.28
26) B) - 1629
C) 1629
Find the area enclosed by the given curves. 27) y = - 4sin x, y = sin 2x, 0 x
A) 16
D) 0.89
D) 1629 2
27) C) 1
B) 8
2
6
D) 4
Evaluate the integral. 16 28) 3 x dx 0 A) 24 3
29)
28) B) 288
C) 192
D) 128
(x + 4)3 dx
29)
0
A) 2145
B) 99
C) 4.716904883e+15
D) 5.279854837e+15
8.796093022e+12
Find the derivative. cot 30) d d /4
8.796093022e+12
csc2 y dy
30)
A) - csc3 cot C) csc2 cot
B) - csc2 csc2 (cot ) D) csc2 (cot )
Solve the initial value problem. 7 31) dy = , y(0) = 4 dx (4 + x) 2
31)
A) y = - 7 + 23
B) y = 7 + 9
C) y = - 7 + 4
D) y = 7 + 4
4+x
4
4+x
4+x
4+x
Solve the problem.
32) Suppose that h is continuous and that
3
-1
8
h(x) dx = 2 and
h(x) dx = -10. Find
3
-1 and
4
8
h(t) dt
32)
-1
h(t) dt.
8
A) 8; -8
B) -12; 12
C) 12; -12
D) -8; 8
Solve the initial value problem. 2 33) d s = -49sin 7t - 3 , s (0) = 15, s(0) = -2 2 dt2
33)
A) s = sin 7t - 3
+ 14t - 2
B) s = 7cos 7t - 3
+ 15
C) s = sin 7t - 3
-3
D) s = sin 7t - 3
+ 15t - 3
2 2
2
2
7
Express the limit as a definite integral. n 34) lim ck2 + 16 xk, where P is a partition of [-6, 7] P 0 k=1 7 7 A) B) x + 16 dx -6 -6 n -6 C) D) x2 + 16 dx 7 1 Solve the problem.
35) Suppose that f is continuous and that
3 -3
A) -2
5
f(z) dz = 0 and
34) x2 + 16 dx x2 + 16 dx
-3
C) 2
B) -4
3
f(z) dz = 2. Find
f(x) dx .
35)
5
D) 4
Find the area of the shaded region.
36)
36)
A) 18
B) 27
C) 36
Use the substitution formula to evaluate the integral. 4 3- x 37) dx x 1 A) 6 B) 3 2
D) 45
37) C) - 3 2
Solve the initial value problem. 38) dy = 12 sin 2 x cos x, y(0) = 5 dx
D) 3
38)
A) y = 6 cos 2 x + 5 C) y = 24 sin x cos x + 5
B) y = - 4 sin 3 x - 5 D) y = 4 sin 3 x + 5
8
Evaluate the integral by using multiple substitutions. 2 2 39) x cos 21x sin 21x dx 3 3/2 21x2
A) sin C)
39) 3/2 x2
+C
B) sin
sin 21x2 +C 126
D) sin
189
9
3/2 x
189
+C
+C
Evaluate the integral using the given substitution.
40)
5 x4 (x5 - 1) dx , u = x5 - 1
40)
A) 1 (x5 - 1)6 + C
B) 1 (x5 - 1)4 + C
C) 1 x30 - 1 + C
D) 1 (x5 - 1)6 + C
6
20
30
30
Provide an appropriate response. x 41) Suppose that f(t) dt = 5x2 + 7x - 4. Find f(x). 1
41)
A) 5x2 + 7x - 4
B) 5 x3 + 5 x2 - 4x - 8
C) 5 x3 + 7 x2 - 4x
D) 10x + 7
3
2
Find the area of the shaded region.
42)
3
3
y = x2 - 4x + 3
42)
y=x- 1
A) 9 2
B) 41
C) 25
6
6
9
D) 3
Solve the problem. 43) After a new firm starts in business, it finds that its rate of profits (in hundreds of dollars per year) dP after t years of operation is given by = 3t2 + 6t + 5. Find the profit in year 8 of the operation. dt
A) $52,500
B) $69,600
C) $63,450
Compute the definite integral as the limit of Riemann sums. 2 44) (4x - 1) dx -1 A) 3 B) - 3 2 Provide an appropriate response.
45) What definite integral is represented by lim n
evaluate the integral. 4 A) x3 dx = 64 0 1 4 C) 4x2 dx = 3 0
n i=1
D) $21,900
44) C) - 1
D) 10
2
4i 2 4 ? Use the Fundamental Theorem to n n
4
B) 0
D)
4
x2 dx =
45)
64 3
x2 dx = 21
1
Evaluate the integral. dx 46) xln x3
A) ln ln x3 + C
43)
46) C) 1 ln x3 + C
B) ln x3 + C
3
D) 1 ln ln x3 + C 3
Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. 47) f(x) = 1 between x = 2 and x = 9 using a lower sum with two rectangles of equal width. 47) x
A) 105 44
B) 35
C) 203
66
44
D) 203 198
48) f(x) = 16 - x2 between x = -4 and x = 4 using the "midpoint rule" with two rectangles of equal width. A) 96
B) 48
C) 8
D) 32
Answer each question appropriately. 49) Suppose the area of the region between the graph of a positive continuous function f and the x-axis from x = a to x = b is 9 square units. Find the area between the curves y = f(x) and y = 3f(x). A) -18 square units B) 36 square units
C) 27 square units
D) 18 square units
10
48)
49)
Find the value of the specified finite sum. n n n 50) Given a k = -5 and bk = 6, find a k + bk . k=1 k=1 k=1 A) -1 B) -30 C) 1
50) D) 30
Estimate the value of the quantity. 51) A piece of tissue paper is picked up in gusty wind. The table shows the velocity of the paper at 2 second intervals. Estimate the distance the paper travelled using left-endpoints. Time Velocity (sec) (ft/sec) 0 0 2 8 4 12 6 6 8 25 10 30 12 20 14 10 16 2 A) 113 ft
B) 226 ft
C) 206 ft
D) 203 ft
Solve the problem. 52) Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's position at time t. v = -5t + 3, s(0) = 14 A) s= 5 t2 + 3t - 14 B) s = -5t2 + 3t + 14 2
C) s = - 5 t2 + 3t + 14
51)
52)
D) s = - 5 t2 + 3t - 14
2
2
Express the limit as a definite integral. n ck 53) lim cos xk, where P is a partition of [0, 4 ] 4 P 0 k=1 4 4 1 x x A) B) dx cos dx - sin 4 4 4 0 0 n 0 x C) D) cos x dx cos dx 4 1 4
11
53)
Evaluate the integral. x3
54)
x4 + 3 dx
54)
A) 8 x4 + 3 3/2 + C
B) 2 x4 + 3 3/2 + C
C) 1 x4 + 3 3/2 + C
D) - 1 x4 + 3 -1/2 + C
3
3
6
3
55)
13
2
x2 dx
55)
0
B) 13 13
A) 13 3
3
C) 169
3
D)
13 3
Solve the problem. 56) Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's position at time t. 8 4t v = sin , s( 2 ) = 2
A) s = -2 cos 4t + 8
B) s = 2 cos 4t + 4
C) s = -2 cos 4t + 3
D) s = -2 cos 4t + 4
Find the total area of the region between the curve and the x-axis. 57) y = 2x - x2 ; 0 x 2
A) 5 3
B) 7 3
57)
C) 2 3
D) 4 3
Solve the initial value problem. 58) dy = x4(x5 - 8)5 , y(1) = 5 dx
58)
A) y = 1 (x5 - 8)6 - 8.074469797e+15
B) y = 1 (x5 - 8)4 - 5.059952511e+15
C) y = 1 x30 - 8 + 33
D) y = 1 (x5 - 8)6 - 1.73796138e+15
30 30
56)
2.061584302e+12
20
10
6
12
4.398046511e+13
2.199023256e+13
Answer each question appropriately. 59) If differentiable functions y = F(x) and y = G(x) both solve the initial value problem dy = f(x) , y(x0 ) = y0 , on an interval I, must F(x) = G(x) for every x in I? Justify the answer. dx
59)
A) F(x) and G(x) are not unique. There are infinitely many functions that solve the initial value problem. When solving the problem there is an integration constant C that can be any value. F(x) and G(x) could each have a different constant term. B) There is not enough information given to determine if F(x) = G(x).
C) F(x) = G(x) for every x in I because integrating f(x) results in one unique function. D) F(x) = G(x) for every x in I because when given an initial condition, we can find the
integration constant when integrating f(x). Therefore, the particular solution to the initial value problem is unique.
Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. 60) f(x) = x2 between x = 3 and x = 7 using the "midpoint rule" with four rectangles of equal width. 60)
A) 126
B) 117
C) 105
D) 86
Solve the problem.
61) Suppose that
-3
g(t) dt = -4. Find
-6
A) 1; -4
-3 -6
g(x) dx and -4
B) -1; 4
-6
- g(t) dt .
61)
-3
C) 1; 4
D) 0; -4
Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. 62) f(x) = x2 between x = 0 and x = 3 using an upper sum with two rectangles of equal width. 62)
A) 8.4375 Evaluate the integral. 6 63) 4x3 dx -1 A) 35
B) 12.5
C) 3.375
63) C) 5180
B) -1295
Graph the integrand and use areas to evaluate the integral. 5 64) 6 dx -1 A) 6 B) 24 Solve the problem.
11
D) 1295
64) C) 36
11
65) Suppose that f and g are continuous and that Find
D) 16.875
D) 18
11
f(x) dx = -3 and
7
g(x) dx = 9.
7
f(x) - 4g(x) dx .
7
A) -48
C) 33
B) -39 13
D) -12
65)
Find the area of the shaded region.
66)
66)
y = 1 - sin (x - /2)
B) 2
A)
C) 2 + 2
D) 2
Find the total area of the region between the curve and the x-axis. 67) y = (x + 4) x; 1 x 16 A) 5871 B) 2886 C) 1275 2 5
67) D) 591 2
Solve the initial value problem. 68) dy = 9(3x - 5)-6 , y(0) = 1 dx
68)
A) y = - 3 (3x - 5)-5 + 1
B) y = - 9 (3x - 5)-5 + 1
C) y = - 3 (3x - 5)-7 + 1
D) y = (3x - 5)-5 - 5
5
5
7
Express the sum in sigma notation. 69) 1 + 1 + 1 + 1 5 25 125 625
A)
4 k=1
1 k- 1 5
69) 5
B)
k=1
1 k- 1 5
4
C)
k=1
1 k 5
D)
4 k=0
1 k+ 1 5
Evaluate the integral using the given substitution.
70)
csc2 7 cot 7
d
, u = cot 7
70)
A) 1 cot2 + C
B) 1 csc3 7 cot2 7 + C
C) - 1 cot2 7 + C
D) - 1 tan2 7 + C
14
6
14
14
14
Find the area of the shaded region.
71)
71) y=x- 4 y=
B) 32
A) 32
A)
2
C) 128
3
Find the area enclosed by the given curves.
72) y = csc 2 x, y = cot 2 x, x =
2x
4
, and x =
D) 64
3
3
3 4
72)
B) 3
C)
4
D)
4
Express the limit as a definite integral. n 73) lim (3ck2 - 10ck + 2) xk, where P is a partition of [-10, 3] P 0 k=1 3 n A) B) (3x2 - 10x + 2) dx (6x - 10) dx 1 -10 3 -10 C) D) (3x - 10) dx (3x2 - 10x + 2) dx 3 -10
73)
Use a definite integral to find an expression that represents the area of the region between the given curve and the x-axis on the interval [0, b]. 74) y = 12x2 74)
A) 4b3
Express the sum in sigma notation. 75) 3 + 6 + 9 + 12 + 15 5 A) 3(k + 1) k=1
D) -4b3
C) 24b
B) -24b
75) B)
6
3k
C)
k=1
4 k=0
15
3(k + 1)
D)
5 k=2
3(k - 1)
Graph the function f(x) over the given interval. Partition the interval into 4 subintervals of equal length. Then add to 4 your sketch the rectangles associated with the Riemann sum f(ck) x k , using the indicated point in the kth k=1 subinterval for ck.
76) f(x) = 2x + 3, [0, 2], left-hand endpoint
76)
A)
B)
C)
D)
16
Write the sum without sigma notation and evaluate it. 4 77) 2 cos k k=1
A) 2 cos + 2 cos B) 2 cos + 2 cos C) 2 cos + 2 cos D) 2 cos + 2 cos
4 2 2 2
= -2 + + 2 cos + 2 cos + 2 cos
77)
2 3 3 3
+ 2 cos + 2 cos + 2 cos
4 4 4
2
= -1 + =3+ = -2 +
2 3+
2
Find the area of the shaded region.
78)
78)
A) 22 3
B) 29
C) 16
3
3
D) 38 3
Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. 79) f(x) = 1 between x = 2 and x = 5 using an upper sum with two rectangles of equal width. 79) x
A) 33 70
B) 51
C) 51
70
28
D) 33 28
Graph the function f(x) over the given interval. Partition the interval into 4 subintervals of equal length. Then add to 4 your sketch the rectangles associated with the Riemann sum f(ck) x k , using the indicated point in the kth k=1 subinterval for ck.
17
80) f(x) = x2 - 3, [0, 8], midpoint
80)
A)
B)
C)
D)
Find the value of the specified finite sum. n n ak 81) Given a k = 4, find . 4 k=1 k=1 A) 1 B) 16
81) C) -1
18
D) 0
Find the average value of the function over the given interval. 82) f(x) = x + 2 on [-3, 3] A) 7 B) 7 6
82) C) 7 2
D) 21
Graph the function f(x) over the given interval. Partition the interval into 4 subintervals of equal length. Then add to 4 your sketch the rectangles associated with the Riemann sum f(ck) x k , using the indicated point in the kth k=1 subinterval for ck.
83) f(x) = -4x2 , [0, 4], midpoint
83)
A)
B)
19
C)
D)
Provide an appropriate response. 84) Which of the following express 1 + 3 + 9 + 27 + 81 in sigma notation? 5 4 3 I. 3k - 1 II. 3k III. 3k + 1 k=1 k=0 k = -1 A) I, II, and III B) II and III C) I and II
84)
D) II only
Find the formula and limit as requested. 85) For the function f(x) = 10x + 2, find a formula for the upper sum obtained by dividing the interval [0, 3] into n equal subintervals. Then take the limit as n to calculate the area under the curve over [0, 3]. 2 2 A) 6 + 90n + 90n ; Area = 51 B) 6 + 90n + 90n ; Area = 102 5 2n 2 2n 2 2
2
C) 6 - 90n + 90n ; Area = - 39
D) 6 + 87n + 91n ; Area = 99
2n 2
2n 2
2
Estimate the value of the quantity. 86) The table shows the velocity of a remote controlled race car moving along a dirt path for 8 seconds. Estimate the distance traveled by the car using 8 subintervals of length 1 with right-end point values. Time Velocity (sec) (in./sec) 0 0 1 8 2 21 3 28 4 27 5 25 6 28 7 23 8 4 A) 160 in.
85)
B) 164 in.
C) 154 in.
20
D) 170 in.
86)
Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. 87) f(x) = x2 between x = 4 and x = 8 using an upper sum with four rectangles of equal width. 87)
A) 165
B) 149
C) 174
D) 126
Graph the function f(x) over the given interval. Partition the interval into 4 subintervals of equal length. Then add to 4 your sketch the rectangles associated with the Riemann sum f(ck) x k , using the indicated point in the kth k=1 subinterval for ck.
88) f(x) = -2x - 1, [0, 2], left-hand endpoint
88)
A)
B)
C)
D)
Evaluate the integral.
89)
x3 (x4 - 3)4 dx 4
A) x - 3 12
3
+C
89) 4
B) x - 3 4
5
4
C) x - 3
+C
20
21
5
+C
D) x4 - 3 5 + C
Graph the integrand and use areas to evaluate the integral. 17 90) x dx 13 A) 60 B) 15
90) C) 30
Use the substitution formula to evaluate the integral. 2 91) 3 cos2 x sin x dx /3 A) - 129 B) - 21 1024 8
D) 120
91) C) - 7
D) 7
C) 4
D) 2
8
Find the average value of the function over the given interval. 92) f(x) = 2x + 4 on [-2, 2] A) 8 B) 16
22
8
92)
Graph the function f(x) over the given interval. Partition the interval into 4 subintervals of equal length. Then add to 4 your sketch the rectangles associated with the Riemann sum f(ck) x k , using the indicated point in the kth k=1 subinterval for ck.
93) f(x) = cos x + 2, [0, 2 ], right-hand endpoint
93)
A)
B)
C)
D)
Find the value of the specified finite sum. n n n 94) Given a k = 2 and bk = 6, find a k - 2bk . k=1 k=1 k=1 A) 14 B) -14 C) -10
23
94) D) 10
Find the area enclosed by the given curves. 95) y = 2x - x2 , y = 2x - 4
A) 34
95)
B) 31
3
C) 37
3
D) 32
3
3
Evaluate the integral by using multiple substitutions. sin t 96) dt t cos3 t 4 +C cos t
A) -
B)
4 sin
t
96) C) - 2 + C
+C
t3/2
D)
4 cos
t
+C
Use a finite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. 97) f(x) = 4x sin x on [0, 4] divided into 4 subintervals 97) A) 0 B) -8 C) - 2 D) - 1
Express the limit as a definite integral. n 5 98) lim xk, where P is a partition of [6, 7] P 0 k = 1 ck7 7
A) 6
5 dx x
n
B) 1
5 dx x
98) 7
C) 6
5 dx x7
6
D) 7
5 dx x7
Graph the function f(x) over the given interval. Partition the interval into 4 subintervals of equal length. Then add to 4 your sketch the rectangles associated with the Riemann sum f(ck) x k , using the indicated point in the kth k=1 subinterval for ck.
99) f(x) = x2 - 2, [0, 8], right-hand endpoint
99)
24
A)
B)
C)
D)
Find the area of the shaded region.
100)
100)
y=2 y = csc2 x
A)
B)
-2
C)
+2
D) 2
Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. 101) f(x) = x2 between x = 0 and x = 1 using the "midpoint rule" with two rectangles of equal width. 101)
A) .625
B) .75
C) .125
25
D) .3145
Evaluate the sum. 13 102) k k=1
102)
A) 91
B) 182
C) 13
D) 91 2
Find the derivative. x 103) y = 7t cos (t10) dt
103)
0
A) 10 x cos (x5 )
B) 7 cos (x5 ) - 7 x sin (x5)
C) 7 cos (x5 )
D) 7 x cos (x5 )
2
Use a definite integral to find an expression that represents the area of the region between the given curve and the x-axis on the interval [0, b]. 104) y = x + 9 104) 3
A) - b - 9 3
2
B) b + 9b
C) b + 9 3
6
2
D) - b - 9b 6
Use a finite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. 105) f(x) = 6 2 x on [1, 7] divided into 3 subintervals 105)
A) 168
B) 84
C) 336
Compute the definite integral as the limit of Riemann sums. 3 106) x3 dx -3
D) 504
106)
A) 27
B) 2.849934139e+15
C) 2.849934139e+15
D) 0
7.036874418e+13
1.407374884e+14
Use a definite integral to find an expression that represents the area of the region between the given curve and the x-axis on the interval [0, b]. 107) y = 9 x2 107)
A) -3 b3
Evaluate the sum. 23 108) 5 k=2 A) 105
B) 3 b3
C) 18 b
D) -18 b
108) B) 110
C) 115 26
D) 113
Evaluate the integral by using multiple substitutions.
109)
5(3x2 - 8) sin5 (x3 - 8x) cos (x3 - 8x) dx
109)
A) 25 sin4 (x3 - 8x) + C
B) 6 sin6 (x3 - 8x) + C
C) 5 sin6 (x3 - 8x)+ C
D) 5 cos6 (3x2) + C
5
6
6
Find the area enclosed by the given curves. 110) y = 1 x2 , y = -x2 + 6 2
A) 16
110)
B) 4
C) 8
D) 32
111) Find the area of the region in the first quadrant bounded on the left by the y-axis, below by the
111)
1 line y = x, above left by y = x + 4, and above right by y = - x2 + 10. 3
A) 39 2
112) y = x3, y = 4x A) 2
B) 73
C) 15
6
D) 39 4
112) B) 8
C) 16
D) 4
Evaluate the integral. 4 113) 3x2 6 + 3x3 dx
113)
A) 12 6 + 3x3 5/4 + C
B) 4 6 + 3x3 5/4 + C
C) 3 6 + 3x3 5/4 + C
D) - 2 6 + 3x3 -3/4 + C
5
15
Use a finite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. 114) f(x) = x2 - 9 on [-3, 7] divided into 5 subintervals 114)
A) 3
C) 51
B) 12
5
27
D) 15
Find the area of the shaded region.
115)
115)
A) 5
B) 4
3
C) 2
3
D) 1
3
3
Express the limit as a definite integral. n 116) lim (sec2 ck) xk, where P is a partition of [-3 , 3 ] P 0 k=1 3 3 A) B) sec2 x dx tan x dx -3 -3 n -3 C) D) sec2 x dx sec x dx 3 1
116)
Estimate the value of the quantity. 117) The velocity of a projectile fired straight into the air is given every half second. Use right endpoints to estimate the distance the projectile travelled in four seconds. Time Velocity (sec) (m/sec) 0 138 0.5 133.1 1.0 128.2 1.5 123.3 2.0 118.4 2.5 113.5 3.0 108.6 3.5 103.7 4.0 98.8 A) 483.4 m
B) 966.8 m
C) 463.8 m
Write the sum without sigma notation and evaluate it. 3 7 118) (-1)k sin 2 k=1 A) -sin 7 - sin 7 = 2 2 2 2
D) 927.6 m
118) B) -sin 7 + sin 7 - sin 7 = 0 2
C) -sin 7 + sin 7 - sin 7 = -1 2
117)
2
2
D) -sin 7 + sin 7 - sin 7 = 1
2
2
28
2
2
Solve the problem.
10
119) Suppose that f and g are continuous and that 10
Find
10
f(x) dx = -4 and
6
g(x) dx = 9.
119)
6
5f(x) + g(x) dx .
6
A) 14
B) 25
C) -11
D) 41
Use a definite integral to find an expression that represents the area of the region between the given curve and the x-axis on the interval [0, b]. 120) y = 18x 120) 2 2 A) -9b B) -18b C) 9b D) 18b
Answer each question appropriately. 121) Suppose the area of the region between the graph of a positive continuous function f and the x-axis from x = a to x = b is 11 square units. Find the area between the curves y = f(x) and y = -5f(x). A) -66 square units B) 44 square units
C) 66 square units
D) -44 square units
Graph the integrand and use areas to evaluate the integral. 7 122) (2x + 14) dx -7 A) 98 B) 392
122) C) 196
Provide an appropriate response. x 123) Find f( ) if f(t) dt = x sin 8x. 0 A) B) 8
3
D) 28
123) C) 0
Find the total area of the region between the curve and the x-axis. 124) y = x2 - 6x + 9; 2 x 4
A) 4
121)
B) 7
124)
C) 1
3
3
Express the limit as a definite integral. n 8 125) lim xk, where P is a partition of [10, 12] P 0 k = 1 7 - 12ck2 n 12 8 8 A) B) dx dx 7 - 12x 2 7 12x 1 10 10 12 8 8 C) D) dx dx 2 7 - 12x 12 7 - 12x 10 29
D) 8
D) 2 3
125)
Evaluate the integral.
126) 0
3 - sin 2x dx 2
126)
A) 3
B) 3 - 1
2
127)
3 /4
2
2 sec
tan
C) - 3
2
D) 3 + 1
2
2
2
d
127)
- /4
A) -2 2
C) 2 2
B) 0
D) -4 2
Evaluate the integral using the given substitution.
128)
15(5x - 2)-5 dx , u = 5x - 2
128)
A) (5x - 2)-4 + C
B) - 3 (5x - 2)-4 + C
C) - 3 (5x - 2)-4 + C
D) - 1 (5x - 2)-6 + C
2
4
2
Solve the initial value problem. 129) dr = csc2 13 cot 13 , r =3 d 4
129)
A) r = - 1 cot2 13 + 79
B) r = - 1 tan2 13 + 79
C) r = 1 csc3 13 cot2 13 + 3
D) r = 1 cot2 + 77
26
26
26
6
26
26
26
Find the area of the shaded region.
130)
130)
y = cos2 x
y = -cos x
A) 2 +
C) 2 -
B) 2
30
2
D) 2 +
2
Evaluate the sum. 13 131) k3 k=1 A) 8281
131) B) 2548
C) 2197
Find the average value of the function over the given interval. 132) f(x) = 10x on [5, 7] A) 120 B) 240 Compute the definite integral as the limit of Riemann sums. d 133) 3x2 , a < b c 3 3 A) 9(d - c) B) 3 d - c 3 3 Evaluate the integral. 1/5 134) t2 dt 0 A) - 1 5
B) - 1
C) 30
D) 60
133) C) d2 - c2
3
3
3
3
D) 27 d - c
D) 1
C) 375
375
375
135) III.
6
(k + 1)2
k=4
B) All are equivalent. D) III
C) II
137)
132)
134)
Provide an appropriate response. 135) Which formula is not equivalent to the other two? 7 -3 I. (k - 2)2 II. (k - 1)2 k = -5 k=5 A) I
Evaluate the integral. 9 136) 4 dx 6 A) 12
D) 819
136) B) 0
C) -30
csc2 (9 + 2) d
D) 36 137)
A) 18 csc (9 + 2) cot (9 + 2) + C
B) - 1 cot (9 + 2) + C
C) 9 cot (9 + 2) + C
D) -cot (9 + 2) + C
9
31
Answer each question appropriately. 2 0 138) Suppose that f(x) dx = 10. Find f(x) dx , if f is even. 0 -2 A) 20 B) 10 C) -20 Express the limit as a definite integral. n 139) lim ck5 xk, where P is a partition of [-3, 1] P 0 k=1 n 1 1 A) B) C) x dx x5 dx 5x4 dx 1 -3 -3 Use the substitution formula to evaluate the integral. 3 /4 140) 3 + cot csc 2 d /4 A) -3 B) 6
C) 12
139) -3
D)
x5 dx
1
D) 3
141)
B) 1
2
D) -10
140)
Find the area enclosed by the given curves. 141) y = x, y = x2
A) 1
138)
C) 1
6
D) 1
3
12
Evaluate the integral. x dx 142) (7x2 + 3)5
142)
A) - 7 (7x2 + 3)-4 + C
B) - 1 (7x2 + 3)-6 + C
C) - 7 (7x2 + 3)-6 + C
D) - 1 (7x2 + 3)-4 + C
3
14
3
56
32
Estimate the value of the quantity. 143) Joe wants to find out how far it is across the lake. His boat has a speedometer but no odometer. The table shows the boats velocity at 10 second intervals. Estimate the distance across the lake using right-end point values. Time Velocity (sec) (ft/sec) 0 0 10 12 20 30 30 56 40 53 50 58 60 55 70 58 80 48 90 15 100 0 A) 385 ft
B) 5800 ft
C) 3950 ft
D) 3850 ft
Use the substitution formula to evaluate the integral. 1 10 r dr 144) 9 + 5r2 0
A) 14 - 3 Evaluate the integral. 7 145) z2
A) - 11 2
144)
B) - 2 14 + 6
C) 2 14 - 6
D)
14 3 2 2
7 dz
7
143)
145) C) - 11 + 2 7
B) - 7
2
Evaluate the integral by using multiple substitutions. 3 2 2 146) 12y sin 6y + 1 cos 6y + 1 dy 6y2 + 1
D) - 7 2
7
146)
A) 2 sin4 6y2 + 1 + C
B) 1 cos4 12y + C
C) 1 sin4 6y2 + 1 + C
D) 6 sin2 6y2 + 1 + C
2
2
33
Find the area of the shaded region.
147)
147)
A) 10
B) 5
C) 12.5
D) 7.5
Solve the initial value problem. 148) ds = (18t + 3) sin (9t2 + 3t), s(0) = -8 dt
148)
A) s = sin (9t2 + 3t) - 8
B) s = -cos (9t2 + 3t) - 7
C) s = cos (9t2 + 3t) - 9
D) s = -cos 18t + 3 - 13 4
Find the area of the shaded region.
149)
149)
A) 33 4
B) 41
C) 17
4
D) 9
4
Use the substitution formula to evaluate the integral. /2 150) cot x csc6 x dx /6 A) 127 B) 21 7 2
4
150) C) - 21 2
D) - 127 7
Use a finite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. 151) f(t) = 1e-t cos 2 t on [-2, 3] divided into 5 subintervals (Round to the nearest hundredth.) 151)
A) -7.41
B) 1.41
C) -1.41
34
D) -19.14
Provide an appropriate response. 152) Which formula is not equivalent to the other two? 6 4 (-1)k+1 (-1)k+3 I. II. k+ 1 k+ 3 k=4 k=2 A) I
152) III.
k=3
(-1)k-1 k-1
B) III D) All are equivalent.
C) II Find the average value of the function over the given interval. 153) y = x2 - 4x + 2; [0, 9]
A) 11
5
B) 47
35
153) C) 17 4
D) 27
Graph the function f(x) over the given interval. Partition the interval into 4 subintervals of equal length. Then add to 4 your sketch the rectangles associated with the Riemann sum f(ck) x k , using the indicated point in the kth k=1 subinterval for ck.
154) f(x) = cos x + 3, [0, 2 ], left-hand endpoint
154)
A)
B)
C)
D)
36
Find the area of the shaded region. 155) y = 2x2 + x - 6
A) 9 2
y = x2 - 4
155)
B) 8
C) 19
3
3
Express the limit as a definite integral. n 156) lim (sin ck) xk, where P is a partition of [- /4, 0] P 0 k=1 0 n 0 A) B) C) sin x dx sin x dx cos x dx 1 - /4 - /4 Find the value of the specified finite sum. n n n 157) Given a k = -4 and bk = 4, find a k - bk . k=1 k=1 k=1 A) 16 B) -8 C) 8 Provide an appropriate response.
158) Find the linearization of f(x) = 4 + A) x + 4
x+1
1 B) 4x + 1
tan
D) 11 6
156) /4
D)
sin x dx
0
157) D) -10
t dt at x = 0. 4
C) 4
Graph the integrand and use areas to evaluate the integral. 3 159) (-2x + 6) dx -6 A) 162 B) 81
158) D) 2x + 4
159) C) 108
37
D) 27
Write the sum without sigma notation and evaluate it. 3 160) 5 k sin 6 k=1
160) B) 5 sin
A) 5 + 25 + 125 = 155 C) 5 sin
6
+ 25 sin
6
+ 125 sin
6
=
155 2
6
D) 5 sin
6
+ 125 sin + 25 sin
6
Solve the initial value problem. 161) dy = x cos (2x2 ), y(0) = 8 dx
6
= 65
+ 125 sin
6
=
155 3 2
161)
2
A) y = x sin(2x2 ) + 8
B) y = 1 sin (u)
C) y = sin (2x2) + 8
D) y = 1 sin(2x2 ) + 8
u
2
4
38
Estimate the value of the quantity. 162) The table gives dye concentrations for a cardiac-output determination. The amount of dye injected was 4.7 mg. Plot the data and connect the data points with a smooth curve. Find the area under the curve using rectangles. Use this area to estimate the cardiac output.
162)
Seconds Dye Conc. after (adjusted for injection recirculation) 0 0 2 0.5 4 1.2 6 2.3 8 3.8 10 4.6 12 4.5 14 3.2 16 1.8 18 0.5 20 0 Dye concentration (mg/L)
Time (sec)
A) 0.1 L/min
B) 12.9 L/min
C) 6.4 L/min
Graph the integrand and use areas to evaluate the integral. 6 163) (6 - x ) dx -6 A) 72 B) 108
D) 279.6 L/min
163) C) 18
D) 36
Evaluate the integral using the given substitution.
164)
x cos (6x2 ) dx, u = 6x2
164)
A) 1 sin (u) + C
B) sin(6x2 ) + C
C) 1 sin (6x2 ) + C
D) x sin (6x2) + C
u
2
12
2
39
dx , u = 4x + 3 4x + 3
165)
A) 2
165)
1 +C 4x + 3
B) 2 4x + 3 + C
C) 1 4x + 3 + C
D)
2
1
+C
4(4x + 3)3/2
Evaluate the integral by using multiple substitutions.
166)
5 + sin2 (x - 3) sin (x - 3) cos (x - 3) dx
166)
A) 1 (5 + sin2 x)3/2 + C
B) 1 (5 + sin2 (x - 3))3/2 + C
C) (5 + cos2 (x - 3))3/2 + C
D) 3 5 + sin2 (x - 3) + C
3
3 4
Solve the problem.
5
167) Suppose that g is continuous and that
g(x) dx = 8 and
4
A) 26
168)
(9 + cos t)6
A) C)
g(x) dx = 18. Find
4 C) -26
B) -10
Evaluate the integral. sin t
9
5
g(x) dx .
dt
1
168) +C
B)
+C
D)
7(9 + cos t)7 1
(9 + cos t)5
167)
9 D) 10
1
5(9 + cos t)5 5
(9 + cos t)5
+C
+C
Evaluate the sum. 5 169) k2 - 6
169)
k=1
A) 25
B) 49
Find the average value of the function over the given interval. 170) f(x) = 4 - x on [-4, 4] A) 16 B) 1 Find the derivative. x3 171) d sin t dt dx 0
A) 3x2 sin (x3)
C) 19
D) 55
C) 2
D) 4
170)
171) C) 1 x4 sin (x3)
B) -cos (x3) - 1
4
40
D) sin (x3 )
Use a finite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. 2 172) f(t) = 1 - cos t 172) on [0, 4] divided into 4 subintervals 2
A) 1 - 2
C) 1
B) 1
2
Express the sum in sigma notation. 173) 4 + 5 + 6 + 7 + 8 + 9 5 A) k k=0 5 C) k+4 k=0 Evaluate the integral. 3 /4 174) -6 csc 2 x dx /4 A) 0
D) - 1
2
4
173) 2
B) D)
k = -1 5
(-1)2k k
k+4
k=4
174) C) 12
B) -6
D) -12
Solve the problem. 175) A certain company has found that its expenditure rate per day (in hundreds of dollars) on a dE certain type of job is given by = 4x + 7, where x is the number of days since the start of the job. dx Find the expenditure if the job takes 6 days. A) $11,400 B) $3100
C) $114
175)
D) $31
Evaluate the integral.
176)
csc z +
6
A) csc z + C) -
6
cot z +
6
dz
176)
+C
6
csc z +
6
B) -cot z + +C
D) -csc z +
6 6
+C +C
Evaluate the integral using the given substitution. t 2 t t 177) 4 - sin cos dt, u = 4 - sin 2 2 2
A) 2 4 - sin t
2
3
177) B) - 2 4 - sin t
+C
3
3
C) 1 4 - sin t sin t + C 3
2
2
D) 2 4 - cos t
2
3
41
2
3
3
+C
+C
Express the sum in sigma notation. 178) 1 - 2 + 4 - 8 + 16 4 A) (-1)k 2 k
C)
k=0 3
178) 2
B)
(-1)k + 1 2 k + 1
k = -2 5 D) (-2)k k=1
(-1)k + 1 2 k
k = -1
Evaluate the integral. 2d
179) 0
A) 19
179) 3
B) 27
24
3
3
C)
24
D)
3
3 3
Solve the initial value problem. 180) dy = cot x, y(-1) = 7 dx -1
A) y = x
C) y =
x
180)
cot t dt + 7
x
B) y = 7
cot t dt + 7
D) y = -
-1
x
csc 2 t dt + 7
-1
Find the average value of the function over the given interval. 181) f(x) = x on [-8, 8] A) 4 B) 8 Evaluate the integral. /2 182) 9 sin x dx 0 A) 0
cot t dt + -1
181) C) 2
D) 64
182) B) 1
C) 9
D) -9
Use a finite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. 183) f(x) = 4 x on [-3, 3] divided into 4 subintervals 183) A) 8 B) 24 C) 6 D) 9 3
Evaluate the integral. 2 1 2 184) t+ dx t 1 A) 37 6
184) B) 15
C) 5
2
6
42
D) 29 6
Estimate the value of the quantity. 185) A swimming pool has a leak. The leak is getting worse. The following table gives the leakage rate every 6 hours. Time Leakage (hr) (gal/hour) 0 0 6 0.6 12 1.3 18 1.9 24 3.0 30 4.5 36 5.9 42 7.0 48 8.1 Give the upper estimate for the number of gallons lost. A) 32.3 gallons B) 145.2 gallons C) 253.8 gallons D) 193.8 gallons Solve the initial value problem. 186) ds = 8t(2t2 - 1)3, s(1) = 5 dt
186) B) s = 1 (2t2 - 1)4
A) s = (2t2 - 1)4 + 4
2
C) s = 1 (2t2 - 1)4 + 9 2
185)
D) s = 1 (2t2 - 1)4 + 5
2
2
Find the area of the shaded region.
187)
187)
y = x2 - 2x
A) 7
15
y = -x4
C) 76
B) 2
15
43
D) 22 15
Write the sum without sigma notation and evaluate it. 4 188) 2 sin k k=1
A) 2 sin + 2 sin B) 2 sin + 2 sin C) 2 sin + 2 sin D) 2 sin + 2 sin
2 2
2 4
+ 2 sin + 2 sin
+ 2 sin =
3 3
3
+ 2 sin + 2 sin
+ 2 sin
4 4
4
188)
=6+
2
=1+
3+ 2
2
=2+
3+
2
2
Solve the initial value problem. 189) dy = csc x, y(3) = -6 dx x
A) y =
-6
C) y = -
189)
csc t dt + 3 x
3
B) y = x
csc t cot t dt - 6
D) y =
3
x
csc t dt - 6
csc t dt - 6
3
Find the area enclosed by the given curves.
190) Find the area of the region in the first quadrant bounded on the left by the line x =
6
and on the
190)
right by the curves y = tan 2 x and y = cot 2 x. (Round to four decimal places.) A) 0.5858 B) 0.4126 C) 0.3094 D) 4.3094
Evaluate the integral by using multiple substitutions. 2 191) 2 sin 2y + 9 cos 2y + 9 dy 2 2y + 9 3
A) sin
191)
2y + 9 +C 12
B)
2y + 9 3 +C 3 3
D) sin
C) 2 sin 2y + 9 + C
44
3
2y + 9
+C
Solve the problem. 192) In a certain memory experiment, subject A is able to memorize words at a rate given by dm = -0.003t2 + 0.6t dt
192)
(words per minute).
In the same memory experiment, subject B is able to memorize at the rate given by dM = -0.006t2 + 0.6t dt
(words per minute).
How many more words does subject B memorize from t = 0 to t = 19 (during the first 19 minutes)? A) -21 B) 101 C) 95 D) -7
Find the derivative. x d
193)
dx
18t9 dt
193)
1
A) 18x9/2
C) 9 x6 - 9
B) 12x6
5
Use the substitution formula to evaluate the integral. /2 cos x 194) dx 3 + (3 4 sin x) 0 A) 20 B) - 20 441 441
D) 9x4
5
194) C) - 1
D) 5
9
441
Solve the problem. 195) Given the acceleration, initial velocity, and initial position of a body moving along a coordinate line at time t, find the body's position at time t. a = 20 cos 5t, v(0) = -9, s(0) = 6 A) s = 4 cos 5t + 9t + 6 B) s = - 4 cos 5t - 9t + 6 5 5
C) s = - 4 sin 5t - 9t + 6
D) s = 4 sin 5t - 9t + 6
5
5
Find the formula and limit as requested. 196) For the function f(x) = 2x2 + 5, find a formula for the upper sum obtained by dividing the interval [0, 3] into n equal subintervals. Then take the limit as n to calculate the area under the curve over [0, 3]. 3 2 3 2 A) 15 + 108n + 162n + 54n ; Area = 33 B) 15 + 108n + 162n + 54n ; Area = 15 6n 4 6n 4 3
195)
2
3
C) 15 + 108n + 162n + 54n ; Area = 33
2
D) 15 + 108n + 162n + 54n ; Area = 18
6n 3
6n 3
45
196)
Evaluate the integral. 9 197) 3x2 + x + 5 dx 0
A) 95
197) B) 1629
C) 771
2
2
Graph the integrand and use areas to evaluate the integral. 3 198) 9x dx 0 A) 9 B) 81 2
D) 1614
198) C) 27
D) 81 2
Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. 199) f(x) = x2 between x = 0 and x = 1 using a lower sum with two rectangles of equal width. 199)
A) .625
B) .75
C) .3145
Find the total area of the region between the curve and the x-axis. 200) y = -x2 + 9; 0 x 5
A) 10 9
B) 10 3
C)
200)
C) 5 9
Write the sum without sigma notation and evaluate it. 2 16k 201) k + 33 k=1 A) 16 + 32 = 824 1 + 33 2 + 33 595
D) .125
D) 98 3
201)
16 32 16 + = 1 + 33 2 + 33 23
B)
16 32 256 + = 1 + 33 2 + 33 595
D)
16 16 552 + = 1 + 33 2 + 33 595
Find the area of the shaded region.
202)
202)
A) 3
B) 23
C) 5
3
3
46
D) 5
Use a finite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. 203) f(x) = 5 on 1 , 11 divided into 5 subintervals 203) x 2 2
A) 137
B) 137
12
C) 5
60
D) 15
2
4
Express the sum in sigma notation. 204) - 1 + 2 - 3 + 4 - 5 7 7 7 7 7 5
A) C)
(-1)k - 1
k=0 5 k=1
(-1)k
204)
k 7
4
B)
k 7
D)
k=1 5
(-1)k
k+ 1 7
(-1)k + 1
k=1
k 7
Solve the problem. 205) Given the acceleration, initial velocity, and initial position of a body moving along a coordinate line at time t, find the body's position at time t. a = 18, v(0) = -3, s(0) = -9 A) s = 9t2 - 3t B) s = 18t2 - 3t - 9
C) s = -9t2 + 3t - 9
205)
D) s = 9t2 - 3t - 9
Use the substitution formula to evaluate the integral. 0 2t 206) dt 23 + 4 t -1 A) - 1.266637395e+15 2.814749767e+16
206) B) 1.266637395e+15 1.125899907e+17
C) - 1.266637395e+15
D) - 1.266637395e+15
5.629499534e+16
1.125899907e+17
Use a finite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. 207) f(x) = 3x5 on [1, 3] divided into 4 subintervals 207)
A) 103.8
B) 58.6
C) 175.8
Solve the problem.
11
208) Suppose that f and g are continuous and that Find
7
D) 703.2
11
f(x) dx = -4 and
7
g(x) dx = 10.
7
g(x) - f(x) dx .
11
A) -6
C) 6
B) -14
47
D) 14
208)
Provide an appropriate response. 209) Which of the following express -3 + 9 - 27 + 81 - 243 in sigma notation? 5 6 5 I. (-1)k3 k II. (-3)k-1 III. (-1)k-1 3 k k=1
k=2
k= 1
A) III only
B) I and II
C) II only
Evaluate the integral. 3 /2 210) 22 cos x dx /2 A) -22
D) I, II, and III
210) B) 22
C) 44
Graph the integrand and use areas to evaluate the integral. 3 211) ( x + 10) dx -3 A) 39 B) 129
D) -44
211) C) 69
D) 78
Evaluate the integral using the given substitution. 3 3 3 212) sin2 dx , u = 2 x x x
212)
A) - 3 + sin3 6 + C
B) 3 + 1 sin 3 + C
C) - 3 + 1 sin 6 + C
D) - 3 + 1 sin 6 + C
x x
x
2
209)
2x
x
2x
2
x
4
x
Use a finite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. 213) f(x) = 4 + sin x on [0, 2] divided into 4 subintervals 213) A) 4 B) 8 + 2 C) 16 + 2 D) 0 2 4
48
Estimate the value of the quantity. 214) The table gives dye concentrations for a cardiac-output determination. The amount of dye injected was 5.8 mg. Plot the data and connect the data points with a smooth curve. Find the area under the curve using rectangles. Use this area to estimate the cardiac output.
214)
Seconds Dye Conc. after (adjusted for injection recirculation) 0 0 2 0.2 4 0.7 6 1.9 8 4.3 10 4.1 12 2.5 14 1.2 16 0.8 18 0.2 20 0 Dye concentration (mg/L)
Time (sec)
A) 0.2 L/min
B) 10.9 L/min
C) 329.0 L/min
Evaluate the integral using the given substitution. 32s3 ds 215) , u = 8 - s4 4 8-s
A) C)
-8 2 8 - s4
D) 21.9 L/min
215) B) -16s3 8 - s4 + C
+C
16s4
D) -16 8 - s4 + C
8 - s4
49
Find the area of the shaded region.
216)
216)
A) 1.50
B) 1.39
C) 1.69
D) 1.25
y = sec2 x
217)
217)
y = cos x
A)
2 2
B) 1 -
2 2
C) 2 - 2
Find the total area of the region between the curve and the x-axis. 218) y = x2 + 1; 0 x 1
A) 2 3
Find the derivative. 0 219) y = cos 8 x
B) 5 3
C) 4 3
D) 1 + 2
218) D) 1 3
t dt
A) -sin (x4 )
219) B) -8x7 cos (x4)
C) 8x7 cos (x4 )
Answer each question appropriately. 3 0 220) Suppose that f(x) dx = 8. Find f(x) dx , if f is odd. 0 -3 A) -24 B) 24 C) -8
50
D) 1 - cos (x4 )
220) D) 8
Solve the initial value problem. 221) dy = x 4 + x2 6 , y(0) = 0 dx
221)
A) y = 1 4 + x2 7
B) y = 1 4 + x2 7 - 16384
7
7
C) y = 1 4 + x2 7 - 8192 14
7
D) y = 1 4 + x2 7
7
14
Use the substitution formula to evaluate the integral. (1 + cos 9t) 2 sin 9t dt
222) 0
A) 8
27
222) B) 1
C) 8
27
D) 1
3
9
Evaluate the integral by using multiple substitutions. 2 223) (2r - 1) cos 3(2r - 1) + 5 dr 3(2r - 1)2 + 5
223)
A) 1 sin 3(2r - 1)2 + 5 + C
2 B) cos 3r + 6 + C
C) 1 sin 3(2r - 1)2 + C
D) -sin 3(2r - 1)2 + 5 + C
6
12
12
3r2 + 6
Find the area of the shaded region.
224)
224)
A) 25 3
Find the derivative. x dt 225) y = 3t + 5 0 A) 1 3x + 5
B) 23
C) 26
3
3
D) 22 3
225) B)
-3
C)
(3x + 5)2
51
1 1 3x + 5 5
D)
-3
(3x + 5)2
+
3 25
Write the sum without sigma notation and evaluate it. 3 k+ 6 226) k k=1 A) 1 + 6 · 2 + 6 · 3 + 6 = 84 1 2 3
226) B) 1 + 6 + 3 + 6 = 10 1
C) 1 + 6 + 2 + 6 + 3 + 6 = 24 1
227)
3
2
3
D) 1 + 6 + 2 + 6 + 3 + 6 = 14
3
1
2
3
(-1)k (k - 4)2
227)
k=1
A) (1 - 4)2 - (3 - 4)2 = -10 C) -(1 - 4)2 + (2 - 4)2 - (3 - 4)2 = 6
B) -(1 - 4)2 + (2 - 4)2 - (3 - 4)2 = -6 D) -(1 - 4)2 -2(2 - 4)2 -3(3 - 4)2 = -20
Estimate the value of the quantity. 228) A swimming pool has a leak. The leak is getting worse. The following table gives the leak rate every 6 hours. Time Leakage (hr) (gal/hour) 0 0 6 0.5 12 0.7 18 1.1 24 2.4 30 4.5 36 5.6 42 6.0 48 6.3 Suppose it keeps leaking 6.3 gallons every 6 hours. After losing 3100 gallons the leak is fixed. Approximately how long did the leak last? Use the right endpoints to estimate the first 48 hours. A) 27.11 hours B) 55.22 hours C) -83.60 hours D) 33.11 hours Evaluate the integral. 1 4 229) sin + 6 dt 2 t t
228)
229)
A) 4 cos 4 + 6 + C
B) - 1 cos 4 + 6 + C
C) -cos 4 + 6 + C
D) 1 cos 4 + 6 + C
t
4
t
4
Write the sum without sigma notation and evaluate it. 3 230) (-1)k+1 cos 10k k=1 A) cos 10 - cos 20 + cos 30 = - 1
t
t
230) B) cos 10 + cos 30 = 2 D) -cos 10 + cos 20 - cos 30 = -1
C) cos 10 - cos 20 + cos 30 = 1
52
Solve the problem.
5
231) Suppose that 3
A) 0; 1
5
f(x) dx = -1. Find
3
f(x) dx and
5
f(x) dx .
231)
5
B) 5; -1
C) -1; 1
Graph the integrand and use areas to evaluate the integral. 6 232) 36 - x2 dx -6 A) 36 B) 36
D) 0; -1
232) C) 6
D) 18
Evaluate the integral. 11 233) x dx
233)
1
A) 10
C) 11 - 1
B) - 5
Use the substitution formula to evaluate the integral. 1 -1/3 234) (8y2 - y + 1) (64y - 4) dy 0
A) 8
234)
B) 24
C) 18
Graph the integrand and use areas to evaluate the integral. 9 235) x dx -2 A) 77 B) 85 2 2 Evaluate the integral. 9 236) 10 dx 1 A) 100
D) 5
D) 9 2
235) C) 85
D) 11
236) B) 10
C) 80
Write the sum without sigma notation and evaluate it. 3 237) 6 k cos k k=1 A) 6 cos + 36 cos 2 + 216 cos 3 = -186
D) -10
237) B) 6 cos + 36 cos 2 + 216 cos 3 = 186 D) 6 cos + 36 cos + 216 cos = -258
C) 6 cos + 216 cos 3 = -222
53
Find the value of the specified finite sum. n n 238) Given a k = -4, find 4 ak . k=1 k=1 A) 16 B) -1
238) D) 1
C) -16
Evaluate the integral by using multiple substitutions. 2 2 239) 9 tan x sec x dx 2 (7 + tan3 x)
A) C)
9x2
2 (7 + x3 )
3
7 + cot3 x
239) -3
+C
+C
B)
+C
D) 3 tan x sec x + C
7 + tan3 x 3
3
21 + tan3 x
Estimate the value of the quantity. 240) A hat is dropped from a hot air balloon. The hat falls faster and faster but its acceleration decreases over time due to air resistance. The acceleration is measured every second after the drop for 6 seconds. Find an upper estimate for the speed when t = 8 seconds. Time Acceleration (sec) (ft/sec2 ) 0 32.00 1 16.64 2 8.65 3 4.50 4 2.34 5 1.22 6 0.63 A) 65.35 ft/sec
B) 192 ft/sec
C) 32.04 ft/sec
Find the total area of the region between the curve and the x-axis. 241) y = 2x + 7; 1 x 5 A) 18 B) 52 C) 26 Compute the definite integral as the limit of Riemann sums. w 242) u dx v 2 2 A) u w - v B) u(w - v) 2 2
54
240)
D) 33.98 ft/sec
241) D) 9
242) C) u
D) w - v
Express the limit as a definite integral. n 243) lim 4ck5 xk, where P is a partition of [6, 11] P 0 k=1 n 11 11 A) B) C) 4x dx 4x5 dx 20x4 dx 1 6 6
243) D)
6
4x5 dx
11
Evaluate the integral using the given substitution.
244)
x cos2 (x3/2 - 6) dx , u = x3/2 - 6
A) 1 3
244) B) 2 sin3 (x3/2 - 6) + C
x sin (x3/2 - 6) + C
9
C) x3/2 - 6 + 1 sin 2(x3/2 - 6)+ C
D) 1 (x3/2 - 6) + 1 sin 2(x3/2 - 6) + C
2
3
6
Graph the function f(x) over the given interval. Partition the interval into 4 subintervals of equal length. Then add to 4 your sketch the rectangles associated with the Riemann sum f(ck) x k , using the indicated point in the kth k=1 subinterval for ck.
245) f(x) = x2 - 1, [0, 8], left-hand endpoint
245)
A)
B)
55
C)
D)
Find the area of the shaded region.
246)
246) y=2 y = 2 sin( x)
A) 4
C) 4 + 4
B) 4
D) 8
Find the total area of the region between the curve and the x-axis. 247) y = 1 ; 1 x 4 x
A) 4
B) 1
247)
C) 1
4
2
D) 2
Find the area enclosed by the given curves. 248) Find the area of the region in the first quadrant bounded by the line y = 8x, the line x = 1, the 1 curve y = , and the x-axis. x
A) 3 4
C) 5
B) 6
4
56
D) 3 2
248)
Evaluate the integral.
249)
sin (4x - 5) dx
249)
A) - 1 cos (4x - 5) + C
B) 4 cos (4x - 5) + C
C) -cos (4x - 5) + C
D) 1 cos (4x - 5) + C
4
4
Graph the integrand and use areas to evaluate the integral. 3 250) 10x dx 1 A) 4 B) 40 Find the derivative. x 251) y = 8t + 3 dt 0
A) 8x + 3
250) C) 80
D) 20
251) B) 1 (8x + 3)3/2
4 8x + 3
C)
12
D) 8x + 3 - 3
Find the total area of the region between the curve and the x-axis. 252) y = (x + 1)3; 0 x 1
A) 4
B) 9
Solve the problem.
7
253) Suppose that A) -35; 5
7
f(x) dx = -5. Find
6
252)
C) 15 4
7
7f(u) du and
6
D) 15
- f(u) du .
253)
6
C) -35; - 1
B) 7; 5
5
D) 2; -5
Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. 254) f(x) = 1 between x = 3 and x = 7 using the "midpoint rule" with two rectangles of equal width. 254) x
A) 5
B) 4.573968372e+14
C) 5
D) 2.286984186e+14
6
5.066549581e+15
12
3.799912186e+15
57
Find the derivative. sin t 1 255) d du dt 16 - u2 0 A) - cos t 16 - sin2 t
C)
255) B)
1
D)
16 - sin2 t
Use the substitution formula to evaluate the integral. 1 256) x + 4 dx 0 A) 10 5 B) 15 5 - 15 3 2 2
1
cos t (16 - sin2 t) cos t
16 - sin2 t
256) C) 5 5 - 8
D) 10 5 - 16 3
Solve the initial value problem. 2 257) d s = -162cos (9t), s (0) = 10, s(0) = 8 dt2
3
257)
A) s = 2 cos (9t) + 10t + 6 C) s = 18 cos (9t) + 10t - 10
B) s = 2 cos (9t) + 6 D) s = -18 sin (9t) + 10
258) dy = sin (4x + ), y(0) = 2
258)
dx
A) y = - 1 cos (4x + ) + 2
B) y = 4 cos (4x + ) + 2
C) y = - cos (4x + ) + 1
D) y = - 1 cos (4x + ) + 7
4
4
Express the limit as a definite integral. n 259) lim (sec ck tan ck) xk, where P is a partition of [0, 5 ] P 0 k=1 n 5 A) B) sec x dx (sec x tan x) dx 1 0 5 0 C) D) sec x dx (sec x tan x) dx 0 5
58
4
259)
Estimate the value of the quantity. 260) The table shows the velocity of a remote controlled race car moving along a dirt path for 8 seconds. Estimate the distance traveled by the car using 8 subintervals of length 1 with left-end point values. Time Velocity (sec) (in./sec) 0 0 1 10 2 18 3 14 4 24 5 27 6 29 7 12 8 5 A) 134 in.
B) 124 in.
C) 268 in.
Write the sum without sigma notation and evaluate it. 4 k2 261) 3 k=1 2 2 2 2 A) 1 · 2 · 3 · 4 = 7 3 3 3 3 2
2
2
2
3
3
3
3
D) 139 in.
261) 2
2
3
3
3
2
2
2
2
3
3
3
3
B) 1 + 4 = 17
C) 1 + 2 + 3 + 4 = 10
D) 1 + 2 + 3 + 4 = 10 3
Find the total area of the region between the curve and the x-axis. 262) y = 3 ; 1 x 3 x3
A) 4 3
B) 1
262)
C) 3
2
Evaluate the integral. 9 2 t +1 263) dt t 4
D) 1 3
263)
A) 212
B) 1.96592679e+15
C) 1.899956093e+15
D) 2.075877953e+15
2.199023256e+13
2.199023256e+13
2.199023256e+13
Find the total area of the region between the curve and the x-axis. 264) y = x2(x - 2)2 ; 0 x 2
A) 17 15
260)
B) 16 15
C) 15 17
59
264) D) 15 16
Find the area of the shaded region. 265) f(x) = -x3 + x2 + 16x
265)
g(x) = 4x
A) 937
B) 343
12
C) - 343
12
D) 1153
12
12
Find the formula and limit as requested. 266) For the function f(x) = 11 - 5x2 , find a formula for the lower sum obtained by dividing the
266)
interval [0, 1] into n equal subintervals. Then take the limit as n to calculate the area under the curve over [0, 1]. 3 2 3 2 A) 11 - 10n + 15n + 5n ; Area = 28 B) 11 - 10n + 15n + 5n ; Area = 28 3 3 6n 3 3n 3 3
2
3
C) 11 + 10n + 15n + 5 ; Area = 38 6n 3
2
D) 10n + 15n + 5n ; Area = 5
3
6n 3
Find the average value of the function over the given interval. 267) f(x) = -2x + 12 on [-12, 6] A) 6 B) 18
60
3
267) C) 36
D) 324
Graph the function f(x) over the given interval. Partition the interval into 4 subintervals of equal length. Then add to 4 your sketch the rectangles associated with the Riemann sum f(ck) x k , using the indicated point in the kth k=1 subinterval for ck.
268) f(x) = cos x + 2, [0, 2 ], midpoint
268)
A)
B)
C)
D)
Solve the problem.
269) Suppose that f is continuous and that
4 -4
A) 28
7
f(z) dz = 0 and
-4
B) -4
C) -7
61
7
f(z) dz = 7. Find -
4f(x) dx .
4
D) -28
269)
Find the derivative. x8 270) y = cos 0
A) sin (x4 )
t dt
270) B) cos (x4 ) - 1
C) 8x7 cos (x4 )
62
D) cos (x4 )
Answer Key Testname: CHAPTER 5
1) D 2) a = -2 and b = 2
3) Since the plane averaged 300 mph on the first 600 miles, the first part of the trip took 600 = 2 hours. The return trip 300
took
600 1200 = 1.5 hours. The entire trip of 1200 miles took 3.5 hours. So the average speed was = 343 mph. 400 3.5
4) If f(x) = 1 + x4 on [0, 1], then max f = 2 and min f = 1. 1
So min f · (1 - 0) b
5)
dx =
1 + x4 dx
0
b
1
max f · (1 - 0) or 1
1 + x4 dx
b 1 dx = x a = b - a. The integral represents the area of the rectangle under y = 1 from x = a to x = b,
a a which equals 1(b-a) = b - a.
6) If u = cot x and du = - csc 2 x dx, then du = - csc x cot x, then =-
2 2.
0
csc 2 x cot x dx =
csc 2 x cot x dx =
- u du = -
- u du = -
u2 cot 2 x +C=+ C. If u = csc x and 2 2
u2 csc 2 x 1 + cot2 x +C=+C=+C 2 2 2
1 cot 2 x cot 2 x +C= + C1 . 2 2 2
7) If f(x) = 1 on [1, 2], then max f = 1 and min f = 1 . x
2
2
So min f · (2 - 1) 1
If f(x) =
1
1 1 1 on [2, 3], then max f = and min f = . x 2 3 3
So min f · (3 - 2)
So
2
1 1 dx max f · (2 - 1) or x 2
1 1 + 2 3
3 1
2
3
1 1 dx max f · (3 - 2) or x 3
1 1 5 dx 1 + or x 2 6
3 1
1 dx x
2
1 dx 1. x
1 dx x
1 . 2
3 . 2
8) a = 0 and b = 2
9) If f(x) = 1 on [1, 3], then max f = 1 and min f = 1 . x
3
So min f · (3 - 1)
3
1 1 dx max f · (3 - 1) and · (3 - 1) x 3
1 2 bound is , and the upper bound is 2. 3
b
10) If f(x) 0 on [a, b], then max f 0. So
f(x) dx
1
1 2 dx 1 · (3 - 1) and x 3
max f · (b - a) 0.
a
11) A = nr2 tan
3
n
63
3 1
1 dx 2. So the lower x
Answer Key Testname: CHAPTER 5
12) B 13) C 14) C 15) D 16) A 17) B 18) B 19) D 20) D 21) B 22) A 23) D 24) D 25) B 26) A 27) B 28) D 29) C 30) B 31) A 32) D 33) D 34) B 35) A 36) C 37) D 38) D 39) A 40) D 41) D 42) A 43) D 44) A 45) B 46) D 47) D 48) A 49) D 50) C 51) B 52) C 53) B 64
Answer Key Testname: CHAPTER 5
54) C 55) A 56) D 57) D 58) A 59) D 60) C 61) A 62) D 63) D 64) C 65) B 66) D 67) B 68) A 69) C 70) C 71) D 72) A 73) A 74) A 75) C 76) C 77) B 78) D 79) D 80) D 81) A 82) C 83) D 84) A 85) A 86) B 87) C 88) B 89) C 90) A 91) C 92) C 93) D 94) C 95) D 65
Answer Key Testname: CHAPTER 5
96) D 97) C 98) C 99) D 100) B 101) D 102) A 103) C 104) B 105) A 106) D 107) B 108) B 109) C 110) A 111) B 112) B 113) B 114) A 115) C 116) A 117) C 118) D 119) C 120) C 121) C 122) C 123) B 124) D 125) B 126) A 127) D 128) C 129) A 130) D 131) A 132) D 133) B 134) D 135) C 136) A 137) B 66
Answer Key Testname: CHAPTER 5
138) B 139) B 140) B 141) B 142) D 143) D 144) C 145) C 146) C 147) A 148) B 149) B 150) B 151) C 152) B 153) A 154) D 155) C 156) A 157) B 158) A 159) B 160) C 161) D 162) C 163) D 164) C 165) C 166) B 167) B 168) B 169) A 170) C 171) A 172) C 173) C 174) D 175) A 176) D 177) B 178) A 179) D 67
Answer Key Testname: CHAPTER 5
180) C 181) A 182) C 183) C 184) D 185) D 186) C 187) A 188) C 189) D 190) C 191) D 192) D 193) D 194) D 195) B 196) C 197) B 198) D 199) D 200) D 201) A 202) B 203) B 204) C 205) D 206) D 207) C 208) B 209) B 210) D 211) C 212) D 213) B 214) B 215) D 216) B 217) B 218) C 219) B 220) C 221) C 68
Answer Key Testname: CHAPTER 5
222) A 223) A 224) C 225) A 226) D 227) B 228) A 229) D 230) C 231) A 232) D 233) D 234) C 235) B 236) C 237) A 238) C 239) B 240) A 241) B 242) B 243) B 244) D 245) C 246) B 247) D 248) C 249) A 250) B 251) A 252) C 253) A 254) A 255) D 256) D 257) A 258) D 259) B 260) A 261) C 262) A 263) C 69
Answer Key Testname: CHAPTER 5
264) B 265) A 266) A 267) B 268) D 269) D 270) C
70
Chapter 6
Exam
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) The region shown here is to be revolved about the y-axis to generate a solid. Which of the methods (disk, washer, shell) could you use to find the volume of the solid? How many integrals would be required in each case?
1)
x = 2y - y2
2) The region bounded by the lines x = 2, x = 6, y = -2, and y = 1 is revolved about the y-axis
2)
3) The region shown here is to be revolved about the x-axis to generate a solid. Which of
3)
to form a solid. Explain how you could use elementary geometry formulas to verify the volume of this solid.
the methods (disk, washer, shell) could you use to find the volume of the solid? How many integrals would be required in each case? x = - y2
x = -4y2 + 3
4) The first-quadrant region bounded by y = 4 - x2 , the x-axis, and the y-axis is revolved about the y-axis to form a solid. Explain how you could use elementary geometry formulas to verify the volume of the solid.
1
4)
5) The region shown here is to be revolved about the line x = 3 to generate a solid. Which of
5)
the methods (disk, washer, shell) could you use to find the volume of the solid? How many integrals would be required in each case? x = - y2
x = -4y2 + 3
6) The region shown here is to be revolved about the y-axis to generate a solid. Which of
6)
the methods (disk, washer, shell) could you use to find the volume of the solid? How many integrals would be required in each case? y=x
y = 4x- x2
7) The region shown here is to be revolved about the x-axis to generate a solid. Which of the methods (disk, washer, shell) could you use to find the volume of the solid? How many integrals would be required in each case?
x = 2y - y2
2
7)
8) The region shown here is to be revolved about the y-axis to generate a solid. Which of
8)
the methods (disk, washer, shell) could you use to find the volume of the solid? How many integrals would be required in each case?
y = -2x2
y = -x
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Set up an integral for the length of the curve. 9) x = sin 2y, y 0 0 A) 1 + 4 cos2 2y dy 0 C) 1 + 4 sin2 2y dy -
9) 0
B) -
D)
0
1 + cos2 2y dy
1 + 2 cos 2y dy
-
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. 10) y = 6 , y = 0, x = 1, x = 36 10) x
A)
2.83674e+15 3.298534883e+12
B) 2.849934139e+15
C)
2.83674e+15 1.649267442e+12
D) 2.863128279e+15
1.649267442e+12 1.649267442e+12
3
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. 11) About the y-axis 11) x=5
y = 2 + x2 /25
A) 125
B) 75
4
C) 50
D) 125 2
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line. 12) About the line y = -1 12) x=5
3
y (solid)
x = 5y2 (dashed)
A) 250 21
B) 50
C) 125
21
21
D) 25 14
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. 13) x = 2y2 , x = - 2y, y = 2 13)
A) 160 3
B) 80
C) 40
3
3
D) 20 3
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 3 14) y = 5csc x, y = 5 2, 14) x 4 4
A) 25 2 - 50
B) 2 + 10
C) 25 2 + 50
4
D) 5 2 - 25
Solve the problem. 15) An auxiliary fuel tank for a helicopter is shaped like the surface generated by revolving the curve x2 y=1, - 2 x 2, about the x-axis (dimensions are in feet). How many cubic feet of fuel will 4 the tank hold to the nearest cubic foot? A) 2 B) 8
C) 3
15)
D) 7
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. 16) x = 3y - y2 , x = 0 16)
A) 9
B) 2.849934139e+15
C) 2.849934139e+15
D) 2.849934139e+15
2
4.222124651e+14
2.111062325e+14
1.055531163e+14
Find the volume of the described solid. 17) The base of a solid is the region between the curve y = 4cos x and the x-axis from x = 0 to x = /2. The cross sections perpendicular to the x-axis are squares with bases running from the x-axis to the curve. A) 8 B) 15 C) 2 D) 4 4 Find the length of the curve. 4 18) x = y + 1 from y = 1 to y = 3 8 4y2
17)
18)
A) 6.473924464e+15
B) 2.152110759e+15
C) 6.49151665e+15
D) 6.473924464e+15
6.333186976e+14
2.111062325e+14
6.333186976e+14
3.166593488e+14
Find the volume of the solid generated by revolving the region about the given line. 19) The region in the first quadrant bounded above by the line 2x + y = 4, below by the x-axis, and on the left by the y-axis, about the line x = -2 A) 128 B) 64 C) 32 D) 10 3 3 3 3 Solve the problem. 20) A bead is formed from a sphere of radius 2 by drilling through a diameter of the sphere with a drill bit of radius 1. Find the volume of the bead. A) 8 B) 5 C) 32 D) 10 3 3 3 3
19)
20)
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 21) y = 2x + 3, y = 0, x = 0, x = 1 21) A) 4 B) 2 C) 3 D) 2 5
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. 22) About the y-axis 22) x=
5
y=
A) 18
x2 + 4
C) 19
B) 19
3
D) 38 3
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 23) y = 2cos ( x), y = 2, x = -0.5, x = 0.5 23) 4 A) 8 B) C) 2 D) 4 3 Use your grapher to find the surface's area numerically. 24) y = sinx, 0 x /4; x-axis A) 2.42 B) 14.32
24) C) 8.18
D) 5.46
Find the volume of the solid generated by revolving the region about the given line. 25) The region in the first quadrant bounded above by the line y = 1, below by the curve y = sin 4x, and on the left by the y-axis, about the line y = 1 2 2 A) B) + 4 C) D) - 1 + 4 8 4 8 4 8 4 Solve the problem. 26) A dome is in the form of a partial sphere, with a hemisphere of radius 10 feet on top and the remaining part of the sphere extending 5 feet to the ground from the center of the sphere. Find the surface area of the dome to the nearest square foot. A) 9425 ft2 B) 94 ft2 C) 9 ft2 D) 942 ft2
27) A frustum of a right circular cone has a height of 10 m, a base of radius 4m, and a top of radius 3m. Find its volume. A) 37 3
B) 370
C) 37
26)
27)
D) 370 3
28) It takes a force of 12,000 lb to compress a spring from its free height of 12 in. to its fully
compressed height of 7 in. How much work does it take to compress the spring the first inch? A) 120,000 in.-lb B) 1200 in.-lb C) 600 in.-lb D) 2400 in.-lb
6
25)
28)
A variable force of magnitude F(x) moves a body of mass m along the x-axis from x 1 to x2 . The net work done by the x2 force in moving the body from x 1 to x2 is W =
x1
F(x) dx =
1 1 mv2 2 - mv1 2 , where v1 and v2 are the body's velocities 2 2
at x 1 and x 2 . Knowing that the work done by the force equals the change in the body's kinetic energy, solve the problem.
29) How many foot-pounds of work does it take to throw a baseball 90 mph? A baseball weighs 5 oz, or 0.3125 lb. A) 85 ft-lb
B) 1266 ft-lb
C) 2723 ft-lb
29)
D) 40 ft-lb
Find the fluid force exerted against the vertically submerged flat surface depicted in the diagram. Assume arbitrary units, and call the weight-density of the fluid w.
30)
30) 2 2 2
A) 4w
B) 12w
C) 48w
D) 24w
Find the volume of the solid generated by revolving the region about the y-axis. 31) The region in the first quadrant bounded on the left by the circle x2 + y2 = 25, on the right by the line x = 5, and above by the line y = 5 A) 125 B) 250 3
C) 125 3
31)
D) 125 4
Solve the problem. 32) A construction crane lifts a bucket of sand originally weighing 130 lb at a constant rate. Sand is lost from the bucket at a constant rate of 0.5 lb/ft. How much work is done in lifting the sand 90 ft? (Neglect the weight of the bucket.) A) 11,700 ft · lb B) 7650 ft · lb C) 13,725 ft · lb D) 9675 ft · lb
32)
33) The gravitational force (in lb) of attraction between two objects is given by F = k/x2, where x is the
33)
34) At lift-off, a rocket weighs 40.3 tons, including the weight of fuel. It is fired vertically, and the
34)
distance between the objects. If the objects are 10 ft apart, find the work required to separate them until they are 75 ft apart. Express the result in terms of k. A) 2 k B) 13 k C) 1 k D) 1 k 15 150 65 750
fuel is consumed at the rate of 2.63 tons per 1,000 ft of ascent. How much work is done in lifting the rocket to an altitude of 14,000 ft? A) 3.06 x 105 ft·ton B) 4.35 x 105 ft·ton
C) 8.22 x 105 ft·ton
D) 3.92 x 105 ft·ton
7
Find the center of mass of a thin plate of constant density covering the given region. 35) The region enclosed by the parabolas y = - x2 + 72 and y = x2
A) x = 0, y = 36
B) x = 0, y = 144 5
C) x = 36, y = 0
35) D) x = 0, y = 72
Solve the problem. 36) An isosceles triangular plate is submerged vertically in seawater, with its base on the bottom. The base is 8 ft long, and the height of the triangle is 8 ft. Find the force exerted on one face of the plate if the water level is 2 ft above the base of the triangle. Seawater weighs 64 lb/ft3 . Round your answer to one decimal place if necessary. A) 256 lb B) 938.7 lb
C) 2816 lb
36)
D) 1408 lb
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. 37) y = 6x3 , y = 6x, for x 0 37)
A) 8
B) 4
5
C) 2
5
D) 12
5
5
Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. 38) The region bounded by y = 4x - x2 and y = x about the line x = 3 38)
A) 27
B) 27
2
C) 81
4
D) 81
8
4
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 39) y = 8x, y = 8, x = 0 39) A) 4 B) 128 C) 64 D) 24 3 3 Set up an integral for the length of the curve. 40) x = y2 + 5y, 0 y 5 5
A) 0
C)
5
40)
4y2 + 26 dy
5
B) 0
4y2 + 10y + 25 dy
D)
0
5
2y + 6 dy 4y2 + 20y + 26 dy
0
Find the volume of the solid generated by revolving the region about the y-axis. 41) The region enclosed by x = y1/3, x = 0, y = 27
A) 243
B) 8.549802418e+15 1.407374884e+14
C) 81
D) 3.206175907e+15 2.199023256e+13
8
41)
Solve.
42) Find the lateral surface area of the cone generated by revolving the line segment y = x/6, 0 x 3 about the y -axis. A) 18 37
B) 54
39
C) 54
37
D) 18
39
Find the volume of the described solid. 43) The solid lies between planes perpendicular to the x-axis at x = -4 and x = 4. The cross sections perpendicular to the x-axis between these planes are squares whose bases run from the semicircle y = - 16 - x2 to the semicircle y = 16 - x2 .
A) 256 3
B) 512
C) 128
3
3
B) 12,480 ft · lb
C) 224,640 ft · lb
3
B) x = 0, y = 144
C) x = 36, y = 0
5
Find the volume of the solid generated by revolving the region about the y-axis. 47) The region in the first quadrant bounded on the left by y = 4 , on the right by the line x = 4, and x
B) 24
C) 16
45)
D) x = 0, y = 36
Solve the problem. 46) The line segment joining the origin to the point (5, 2) is revolved about the x-axis to generate a cone of height 5 and base radius 2. Find the cone's surface area using the parametrization x = 5t, y = 2t, 0 t 1. A) 2 29 B) 5 29 C) 10 29 D) 2 29
above by the line y = 2 A) 16 3
44)
D) 24,960 ft · lb
Find the center of mass of a thin plate covering the given region with the given density function. 45) The region enclosed by the parabolas y = 72 - x2 and y = x2 , with density (x) = x2
A) x = 0, y = 72
43)
D) 1024
Solve the problem. 44) A swimming pool has a rectangular base 10 ft long and 20 ft wide. The sides are 6 ft high, and the pool is full of water. How much work will it take to lower the water level 2 feet by pumping the water out over the top of the pool? Assume that the water weighs 62.4 lb/ft3 . Give your answer to the nearest ft · lb. A) 449,280 ft · lb
42)
46)
47)
D) 8
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. 48) y = x2 - 5, y = 4x, x = 0, for x 0 48)
A) 7.696581394e+15
B) 7.696581394e+15
C) 2.199023256e+15
D) 7.696581394e+15
1.055531163e+14
5.277655813e+13
1.759218604e+13
2.638827907e+13
9
49) y = 4x, y = 8x, x = 4
49)
A) 256 3
B) 128 3
C) 512 3
D) 384
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line. 50) About the line y = -1 50) x=
y (solid)
x = y/2 (dashed)
A) 52
B) 64
15
C) 104
15
D) 224
15
15
Solve the problem.
51) A bathroom scale is compressed 1 in. when a 170 lb person stands on it. Assuming that the scale 5
51)
behaves like a spring that obeys Hooke's law, how much does someone who compresses the scale 1 in. weigh? 10
A) 127.5 lb
B) 42.5 lb
C) 85 lb
Set up an integral for the length of the curve. 52) x = y1/5 , 0 y 2 2
A) 0
C) 0
2
D) 340 lb
52)
2 dy 25y8/5
B)
5y4/5 + 1 dy 5y4/5
D)
2 0
0
2
25y8/5 + 1 dy 25y8/5 1 dy 25y8/5
Find the volume of the solid generated by revolving the region about the y-axis. 53) The region enclosed by the triangle with vertices (0, 0), (2, 0), (2, 2) A) 16 B) 16 C) 8 3 3
10
53) D) 4 3
Find the volume of the solid generated by revolving the shaded region about the given axis. 54) About the x-axis
54)
y = 4 - x2
A) 416 5
B) 16
C) 416
3
15
D) 256 15
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line. 55) About the line y = 3 55)
x = 3y - y2
A) 2.849934139e+15
B) 2.849934139e+15
C) 9
D) 2.849934139e+15
4.222124651e+14
2.111062325e+14
2
1.055531163e+14
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 56) y = x + 3, y = 0, x = -3, x = 6 56) 9 A) 243 B) 36 C) 81 D) 2 Solve the problem. 57) A rescue cable attached to a helicopter weighs 2 lb/ft. A 180-lb man grabs the end of the rope and is pulled from the ocean into the helicopter. How much work is done in lifting the man if the helicopter is 50 ft above the water? A) 14,000 ft · lb B) 9100 ft · lb C) 11,500 ft · lb D) 2680 ft · lb
11
57)
Find the volume of the solid generated by revolving the shaded region about the given axis. 58) About the y-axis
y=
58)
5x
A) 50
B) 25
C) 3.435973837e+15
D) 25
3
5.497558139e+12
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. 59) y = 2x, y = - x , x = 1 59) 2
A) 5
C) 5
B) 5
3
D) 3
6
2
Find the moment or center of mass of the wire, as indicated. 60) Find the center of mass of a wire of constant density that lies along the line y = x from x = 0 to x = 1. A) x = 1 , y = 1 B) x = 1 , y = 1 C) x = 1, y = 0 D) x = 1, y = 1 3 3 2 2 Set up an integral for the length of the curve. 61) y = 3 cos x, 0 x
61)
1 + 9 sin2 x dx
A)
1 + 9 cos2 x dx
B)
0
0 1 + 3 sin x dx
C)
60)
1 - 3 sin x dx
D)
0
0
Solve the problem. 62) Find the work done in winding up a 150-ft cable that weighs 4.00 lb/ft. A) 45,000 ft·lb B) 135,000 ft·lb C) 11,300 ft·lb
12
62) D) 1200 ft·lb
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. 63) x = 6 y, x = - 6y, y = 1 63) A) 22 B) 44 C) 16 D) 12 5 5
Find the length of the curve. 64) x = 2 (y - 1) 3/2 from y = 1 to y = 4 3
A) 14
C) 21
B) 7
3
64) D) 10
2
3
E) 1
Solve the problem. 65) A swimming pool has a rectangular base 12 ft long and 24 ft wide. The sides are 7 ft high, and the pool is half full of water. How much work will it take to empty the pool by pumping the water out over the top of the pool? Assume that the water weighs 62.4 lb/ft3 . Give your answer to the nearest ft · lb. A) 440,294 ft · lb
B) 220,147 ft · lb
C) 330,221 ft · lb
D) 165,110 ft · lb
Find the volume of the solid generated by revolving the region about the y-axis. 66) The region enclosed by x = 2tan y , x = 0, y = - 7 7 4
A) 14 - 7 2 2
C) 7 + 7 2
B) 28 - 7
65)
66) D) 28 - 7 2
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. 67) y = x, y = 0, y = x - 6 67) 63 225 63 A) B) C) D) 27 2 2 4
13
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. 68) About the y-axis 68) x=3
y=
A) 54
3x
C) 108
B) 12
5
D) 27
5
5
Find the center of mass of a thin plate of constant density covering the given region. 69) The region bounded by y = x4 , x = 3, and the x-axis
69)
A) x = 5 , y = 3.562417674e+15 4 3.166593488e+14
B) x = 1 , y = 2.849934139e+15 2 6.333186976e+14
C) x = 7 , y = 2.374945116e+15
D) x = 5 , y = 3.562417674e+15
4
3.166593488e+14
2
1.583296744e+14
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 70) y = 1 , y = 0, x = 1, x = 6 70) x
A) 1
C) 5
B) ln 6
2
D) 5
12
6
Solve.
71) Locate the centroid of a semicircular region. A) y = V = 2
(4/3) a 2
2 (1/2) a 2
=
71) 3 B) y = V = (4/3) a = 2 a 2 3 2 a2
4 a 3
3 C) y = V = (4/3) a = 4 a 2
2 (1/2) a 2
D) y = V =
3
2
a2
2 (1/2) a 2
=
1
a
Find the moment or center of mass of the wire, as indicated. 72) Find the moment about the x-axis of a wire of constant density that lies along the curve y = 2 x from x = 0 to x = 3. A) 28 B) 21 C) 14 D) 14 3 3
14
72)
Solve the problem. 73) A force of 2 N will stretch a rubber band 5 cm. Assuming Hooke's law applies, how much work is done on the rubber band by a 6 N force? A) 4500 J B) 0.05 J C) 0.15 J D) 0.45 J
74) The line segment joining the points (0, 2) and (16, 6) is revolved about the x-axis to generate a
73)
74)
frustum of a cone. Find the surface area of the frustum using the parametrization x = 4t, y = t + 2, 0 t 4. A) 32 17 B) 16 17 C) 32 5 D) 32 17
Set up an integral for the length of the curve. x 75) y = cot t dt , x 6 3 0 /3 A) 1 + csc4 x dx /6 /3 C) csc x dx /6
75) /3
B)
/6 /3
D)
1 + cot x dx
csc x dx
/6
Find the center of mass of a thin plate of constant density covering the given region. 76) The region cut from the first quadrant by the circle x2 + y2 = 4
A) x = 8 , y = 8 3 3
B) x = 0, y = 8 3
C) x = 1, y = 1
D) x = 2 , y = 2 3
76)
3
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. 77) About the x-axis 77) y=2
x=2
x=
A) 16 3
4 - y2
C) 4
B) 8
3
15
D) 8 3
Find the center of mass of a thin plate of constant density covering the given region. 78) The region bounded by y = 8 - x and the axes A) x = 32, y = 32 B) x = 8 , y = 8 3 3
C) x = 256 , y = 256 3
D) x = 8, y = 8
3
Find the centroid of the thin plate bounded by the graphs of the given functions. Use covered by the plate. 79) g(x) = x2 and f(x) = x + 20
A) x = 1 , y = 62 2
5
78)
B) x = 3 , y = 62 4
C) x = 20, y = 62
5
5
= 1 and M = area of the region
79) D) x = 1 , y = 5 2
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. 80) x = 32 - y2 , x = y2 , y = 0 80)
A) 256
B) 128
C) 512
D) 64
Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. 81) y = tanx, 0 x /4; x-axis 81) /4 /4 A) 2 B) tanx 1 + sec4 x dx secx 1 + tanx dx 0 0 /4 /4 C) D) 2 tanx 1 + sec4 x dx secx 1 + tanx dx 0 0 Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. 82) The region bounded by y = 7x - x2 and y = x about the y-axis 82)
A) 324
B) 108
C) 216
16
D) 162
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line. 83) About the line y = 2 83) x=5
3
y (solid)
x = 5y2 (dashed)
A) 275 21
B) 50
C) 275
21
42
D) 5
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. 84) About the y-axis 84)
y =4sin(x2 )
A) 12
B) 4
C) 16
D) 8
Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. 85) The triangle with vertices (0, 0), (0, 2), and (4, 2) about the line x = 4 85) A) 80 B) 16 C) 32 D) 64 3 3 3 3
86) The region bounded by y = 7 x, y = 7, and x = 0 about the line x = 1 A) 56 15
B) 49 15
C) 49 30
17
86) D) 119 15
Find the area of the surface generated by revolving the curve about the indicated axis. 87) y = x, 3/2 x 9/2; x-axis A) 7 7 B) 19 19 - 7 7 2 6 6
C) 7 7 2
3 6
87)
D) 7 7 6
Find the moment or center of mass of the wire, as indicated.
88) Find the moment about the x-axis of a wire of constant density that lies along the curve y = x
3
3
from x = 0 to x = A) 7 12
4
3.
B) 1
C) 7
12
D) 3
18
4
Solve the problem. 89) A tank truck hauls oil in a 10-ft-diameter horizontal right circular cylindrical tank. If the density of the oil is 60 lb/ft3 , how much force does the oil exert on each end of the tank when the tank is half full? A) 5000 lb
B) 7500 lb
88)
C) 2500 lb
89)
D) 11,250 lb
Find the volume of the described solid.
90) The base of the solid is the disk x2 + y2 25. The cross sections by planes perpendicular to the
90)
y-axis between y = - 5 and y = 5 are isosceles right triangles with one leg in the disk. A) 1000 B) 1250 C) 2000 D) 250 3 3 3 3
Find the volume of the solid generated by revolving the region about the given line. 91) The region bounded above by the line y = 9, below by the curve y = 9 - x2, and on the right by the line x = 3, about the line y = 9 A) 2.493692372e+15 2.199023256e+13
91)
B) 2.849934139e+15 2.199023256e+13
C) 8.549802418e+15
D) 9
1.759218604e+14
Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. 92) The region bounded by y = 6 x, y = 6, and x = 0 about the line y = 6 92) A) 3 B) 6 C) 12 D) 18 Solve.
93) Find the surface area of the cone frustum generated by revolving the line segment y = (x/3) + (1/3), 1 x 4, about the x-axis. A) 7 10 3
B) 22 9
C) 22
10
3
18
10
D) 7
10
93)
Solve the problem. 94) A right triangular plate of base 8 m and height 4 m is submerged vertically, as shown below. Find the force on one side of the plate. (w = 9800 N/m 3 )
4m
94)
4m
8m
A) 420,000 N
B) 310,000 N
C) 100,000 N
D) 630,000 N
95) It took 1950 J of work to stretch a spring from its natural length of 3 m to a length of 5 m. Find the spring's force constant. A) 1462.5 N/m
B) 3900 N/m
C) 487.5 N/m
95)
D) 975 N/m
Find the moment or center of mass of the wire, as indicated.
96) Find the moment about the y-axis of a wire of constant density that lies along the curve y = x2 from x = 0 to x = A) 5 2
2.
B) 9
C) 13
4
D) 2
6
Find the center of mass of a thin plate covering the given region with the given density function. 2 97) The region between the x-axis and the curve y = 3 , 1 x 2, with density (x) = x 2 3 x
A) x = 3 , y = 3 2
4
B) x = 5 , y = 3 4
C) x = 3 , y = 3
4
96)
2
2
97)
D) x = 3 , y = 9 2
4
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. 98) About the x-axis 98)
y = 9 - x2
A) 81 2
B) 9
C) 18
19
D) 27
Set up an integral for the length of the curve. 99) y = 1 - x6 , - 1 x 1 4 4
A) C)
1/4 -1/4 1/4 -1/4
99)
4 - 4x6 + 36x10 dx 4(1 - x6 ) 5 - 4x6
4(1 - x6 )
B)
dx
D)
1/4 -1/4 1/4 -1/4
4 - 4x6 + 6x5 dx 4(1 - x6 ) 4 + 36x10 dx 4
Find the center of mass of a thin plate of constant density covering the given region. 100) The region bounded by the parabola y = 36 - x2 and the x-axis
A) x = 0, y = 2.849934139e+15 6.871947674e+11
B) x = 72 , y = 0 5
C) x = 0, y = 5
D) x = 0, y = 72
72
100)
5
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 101) y = x, y = 0, x = 1, x = 3 101) A) 2 B) 5 3
D) 7.318349394e+15
C) 4
8.444249301e+14
Find the moment or center of mass of the wire, as indicated. 102) Find the center of mass of a wire of constant density that lies along the first-quadrant portion of the circle x2 + y2 = 4.
A) x = 4 , y = 4
B) x = 6 , y = 6
C) x = 2 , y = 2
Find the volume of the solid generated by revolving the region about the y-axis. 103) The region enclosed by the triangle with vertices (4, 0), (4, 2), (6, 2) A) 152 B) 28 C) 40 3 3 3
20
102)
D) x = 0, y = 4
103) D) 56 3
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. 104) About the x-axis 104)
x = y2 /2
A) 4
B) 2
D) 8
C) 8
3
Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. 105) xy = 3, 1 y 2; y-axis 105) 2 2 1 1 A) 3 B) 6 1 + 9y-4 dy 1 + 9y-4 dy y y 1 1 2 2 1 1 C) 6 D) 3 1 + 3y-4 dy 1 + 3y-4 dy y y 1 1 Solve the problem.
1
106) Find a curve through the point (0, 3) whose length integral, 0 x 1, is L = A) y = x
B) y = x2
C) y = x2 + 3
21
1 + 4x2 dx.
0
D) y = 2x + 3
106)
Find the volume of the solid generated by revolving the shaded region about the given axis. 107) About the y-axis
x=
A) 25
107)
y2 5
B) 250
C) 100
3
D) 625 6
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 108) y = x2 + 3, y = 5x + 3 108)
A) 3.332894622e+15
B) 1.803886264e+15
C) 4.466765988e+15
D) 125
8.246337208e+12
2.061584302e+12
8.246337208e+12
Solve the problem. 109) A vertical right circular cylindrical tank measures 22 ft high and 12 ft in diameter. It is full of oil weighing 60 lb/ft3 . How long will it take a (1/2)-horsepower (hp) motor (work output
109)
275 ft · lb/sec) to pump the oil to the level of the top of the tank? Give your answer to the nearest minute. A) 100 min B) 5972 min C) 32 min D) 199 min
Find the length of the curve. x 110) y = 4 sin2 t - 1 dt , 0 x 0
A) 2
110)
2
B) 4
C) 1
D) 2 3
Find the volume of the solid generated by revolving the region about the given line. 111) The region in the first quadrant bounded above by the line y = 7, below by the curve y = on the left by the y-axis, about the line y = 7 A) 343 B) 343 C) 343 D) 49 2 6 5 3
22
7x, and
111)
Find the moment or center of mass of the wire, as indicated.
112) Find the center of mass of a wire that lies along the first-quadrant portion of the circle x2 + y2 = 36 if the density of the wire is A) x = 12 , y = 12
= cos .
C) x = 3 , y = 3
B) x = 2, y = 2
2
2
112)
D) x = 3, y = 3
Solve the problem. 113) A rectangular swimming pool has a parabolic drain plate at the bottom of the pool. The drain 1 1 plate is shaped like the region between y = x2 and the line y = from x = -1 to x = 1. The pool 2 2
113)
is 10 ft by 20 ft and 8 ft deep. If the pool is being filled at a rate of 200 ft3 /hr, what is the force on the drain plate after 2 hours of filling? Round your answer to two decimal places if necessary. A) 237.12 lb B) 35.36 lb C) 83.2 lb D) 70.72 lb
114) A company applies a clear glaze finish on the outside of the ceramic bowls it produces. The bowl
114)
corresponds to the bottom half of a sphere which is created by rotating the circle x2 + y2 = 36 around the x-axis. The finish is to be 0.2 cm thick, and the company wants to create 3000 bowls. Use the fact that 1 L = 1000 cm3 to calculate how many liters of finish are required. Assume that all specifications for the bowl are in cm. A) 226.19 L of finish
B) 135.72 L of finish D) 67.86 L of finish
C) 2.26 L of finish
Find the volume of the solid generated by revolving the shaded region about the given axis. 115) About the y-axis
x = 2tan
A) 20
115)
y 5
B) 5 2 - 10
C) 10 2 + 5
D) 10 2 - 20
Solve the problem. 116) A water noodle is formed from a cylinder of radius 5 and height 10 by drilling through the diameter of the cylinder with a drill bit of radius 1. Find the volume of the water noodle. A) 250 B) 240 C) 10 D) 120
23
116)
Find the moment or center of mass of the wire, as indicated. 117) Find the center of mass of a wire that lies along the first-quadrant portion of the circle x2 + y2 = 4 if the density of the wire is = 2sin .
A) x = 2 , y = 2
B) x = 1 , y = 1
C) x = 0, y = 1
D) x = 1 , y = 1
4
4
117)
4
2
2
Solve.
118) Find the volume of the torus generated by revolving the circle (x - 7)2 + y2 = 1 about the y-axis. A) 7 2 B) 21 2 C) 28 2 D) 14 2
Find the volume of the described solid. 119) The base of a solid is the region between the curve y = 3cos x and the x-axis from x = 0 to x = /2. The cross sections perpendicular to the x-axis are squares with diagonals running from the x-axis to the curve. A) 3 B) 9 C) 2 D) 9 2 8 4
118)
119)
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 120) y = - 4x + 8, y = 4x, x = 0 120) A) 16 B) 8 C) 32 D) 96 Find the volume of the solid generated by revolving the region about the given line. 121) The region in the first quadrant bounded above by the line y = 6, below by the curve y = on the left by the y-axis, about the line x = -1 A) 336 B) 246 C) 216 D) 192 5 5 5 5
6x, and
121)
122) A water tank is formed by revolving the curve y = 2x4 about the y-axis. Find the volume of water
122)
Solve the problem.
in the tank as a function of the water depth, y. A) V(y) = 3 y3/2 2 2
C) V(y) =
9
B) V(y) = 2
y3/2
D) V(y) =
y1/2
3 2
y9
2 2
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves about the given lines. 123) y = 25 - x2 , 123) y = 25, x = 5; revolve about the line y = 25
A) 3.435973837e+15
B) 3.435973837e+15
C) 125
D) 4.008636143e+14
2.061584302e+12
5.497558139e+12
3
2.748779069e+11
24
Find the length of the curve. 124) y = 1 x3/2 - x1/2 from x = 4 to x = 9 3
A) 19
124)
B) 22
3
C) 6
3
D) 11
Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. 125) The region bounded by y = 7 x, y = 7, and x = 0 about the y-axis 125) A) 14 B) 7 C) 7 D) 7 3 5 3 10 Find the area of the surface generated by revolving the curve about the indicated axis. 126) y = 6x - x2 , 0.5 x 1.5; x-axis
A) 5
C) 7
B)
126) D) 6
Find the volume of the solid generated by revolving the region about the given line. 127) The region in the first quadrant bounded above by the line y = 4x3 , below by x-axis, and on the right by the line x = 1, about the line y = - 1 A) 18 B) 30 7 7
C) 60
2
A) 1
C)
7
128)
y6 + 2y3 + 2 dy
2
B) 1
2
D) 23
7
Set up an integral for the length of the curve. 128) y4 + 4y = 4x - 1, 1 y 2
y6 + y3 + 1 dy
D)
1
2
y6 + 2 dy y3 + 2 dy
1
Find the area of the surface generated by revolving the curve about the indicated axis. 129) y = x3/14, 0 x 4; x-axis
A) 263 9
B) 1132 49
C) 4528
Find the center of mass of a thin plate of constant density covering the given region. 130) The region bounded by the x-axis and the curve y = 5sin x, 0 x A) x = , y = 25 B) x = , y = 5 2 4 2 8
C) x =
127)
2
,y=
5 2
D) x = , y = 5
4
25
129) D) 15551 756
130)
Set up an integral for the length of the curve.
131) y = 6 cot x,
4
/2
A)
/4 /2
C)
x
131)
2
1 - 36 csc2 x dx
/2
B)
1 + 6 csc2 x dx
/4 /2
D)
/4
/4 0
C)
/4
1 + 36 csc4 x dx
/4
132) x = 4 tan y, 0 y A)
1 + 36 csc2 x dx
132)
4
1 - 16 sec2 y dy
/4
B) 0
1 + 16 sec2 y dy
D)
0
/4
1 + 4 sec4 y dy 1 + 16 sec4 y dy
0
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. 3 133) x = 6y2 , x = 6 y 133) 15 15 A) B) 9 C) 2 D) 14 7 Find the centroid of the thin plate bounded by the graphs of the given functions. Use covered by the plate. 134) g(x) = x2, f(x) = 9, and x = 0
A) x = 1 , y = 27 2 5
B) x = 9 , y = 9 8
C) x = 9 , y = 27 8 5
= 1 and M = area of the region
134) D) x = 9, y = 27 5
Find the center of mass of a thin plate of constant density covering the given region. 135) The region bounded by the x-axis and the semicircle y = 16 - x2
A) x = 0, y = 16
B) x = 0, y = 4
C) x = 16 , y = 16
D) x = 4 , y = 0
3
3
135)
3
3
3
Find the volume of the described solid. 136) The solid lies between planes perpendicular to the x-axis at x = - 1 and x = 1. The cross sections perpendicular to the x-axis are semicircles whose diameters run from y = - 1 - x2 to 1 - x2 . A) 1 3
y=
B) 4
C) 2
3
3
26
D) 8 3
136)
Find the center of mass of a thin plate of constant density covering the given region. 137) The region bounded by the parabola x = y2 and the line x = 4
A) x = 4, y = 0
B) x = 0, y = 12 5
C) x = 12 , y = 0 5
137) D) x = 32 , y = 0 3
Solve the problem. 138) A construction crane lifts a 100-lb bucket originally containing 140 lb of sand at a constant rate. The sand leaks out at a constant rate so that there is only 70 lb of sand left when the crane reaches a height of 60 feet. How much work is done by the crane? A) 12,300 ft · lb B) 14,400 ft · lb C) 2700 ft · lb D) 6300 ft · lb Find the volume of the solid generated by revolving the region about the y-axis. 139) The region enclosed by x = 3 , x = 0, y = 1, y = 4 y
A) 27 4
B) 27
C) 9
16
139) D) 45
4
4
2
140) The region enclosed by x = y , x = 0, y = - 4, y = 4
140)
4
A) 32 3
B) 2048
C) 128
5
138)
D) 64
5
5
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 141) y = x, y = 0, x = 0, x = 5 141) 25 25 5 A) 5 B) C) D) 3 2 2 Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. 142) y = 4x3 , y = 4x, for x 0 142)
A) 16 15
B) 64
C) 32
3
D) 64
21
21
Solve.
143) Find the surface area of the cone frustum generated by revolving the line segment y = (x/6) + (1/6), 1 x 5, about the y-axis. A) 136 37
B) 137
37
C) 141
37
D) 134
143)
37
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. 144) y = 2 x , y = 2 144) 8 16 64 2 A) B) C) D) 3 3 3 3
27
Solve the problem. 145) The spring of a spring balance is 6.0 in. long when there is no weight on the balance, and it is 9.8 in. long with 8.0 lb hung from the balance. How much work is done in stretching it from 6.0 in. to a length of 13.0 in.? A) 140 in.-lb B) 7.4 in.-lb C) 52 in.-lb D) 12 in.-lb
145)
Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. 146) y = x4, 0 x 2; x-axis 146) 2
A) 0
x4 1 + 16x6 dx 2
C) 2
2
B) 0
x4 1 + 16x6 dx
2
D) 2
0
x3 1 + 16x6 dx x3 1 + 16x6 dx
0
Solve the problem. 147) A semicircular plate 14 ft in diameter sticks straight down into fresh water with the diameter along the surface. Find the force exerted by the water on one side of the plate. A) 14,268.8 lb B) 10,701.6 lb C) 21,403.2 lb D) 32,104.8 lb Find the volume of the solid generated by revolving the region about the y-axis. 148) The region in the first quadrant bounded on the left by y = x3, on the right by the line x = 2, and below by the x-axis A) 32 5
C) 64
B) 4
5
147)
148)
D) 96 5
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line. 149) About the line y = 8 149)
x = 6y - y2
A) 2.374945116e+15
B) 2.374945116e+15
C) 2.849934139e+15
D) 2.374945116e+15
6.597069767e+12
2.199023256e+12
1.319413953e+13
1.319413953e+13
28
Solve the problem. 150) A spherical tank of water has a radius of 14 ft, with the center of the tank 50 ft above the ground. How much work will it take to fill the tank by pumping water up from ground level? Assume the water weighs 62.4 lb/ft3 . Give your answer to the nearest ft · lb.
A) 17,930,703 ft · lb C) 35,861,406 ft · lb
150)
B) 11,415,040 ft · lb D) 44,826,757 ft · lb
151) A vertical right circular cylindrical tank measures 26 ft high and 16 ft in diameter. It is full of oil
151)
weighing 60 lb/ft3 . How much work does it take to pump the oil to the level of the top of the tank? Give your answer to the nearest ft · lb. A) 4,077,536 ft · lb B) 16,310,144 ft · lb
C) 8,155,072 ft · lb
D) 67,959 ft · lb
Find the area of the surface generated by revolving the curve about the indicated axis. 152) x = y3/9, 0 y 2; y-axis
A) 256
B) 2.909307767e+15
C) 2.909307767e+15
D) 1163
27
152)
2.968681395e+13
2.40463193e+15
2187
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. 153) y = 9x2 , y = 9 x 153)
A) 27
B) 243
10
C) 243
2
D) 243
10
20
Find the center of mass of a thin plate covering the given region with the given density function. 154) The region bounded by the parabola y = 4 - x2 and the x-axis, with density (x) = 2x2
A) x = 0, y = 8 5
B) x = 0, y = 16
C) x = 8 , y = 0
7
5
154)
D) x = 0, y = 8 7
Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. 155) The region bounded by x = 5 y, x = - 5y, and y = 1 about the line y = 1 155) 22 13 11 A) B) C) 5 D) 3 3 3 Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves about the given lines. 156) y = 3x, 156) y = 0, x = 2; revolve about the line x = -1 A) 14 B) 28 C) 4 D) 14
29
Find the moment or center of mass of the wire, as indicated.
157) Find the center of mass of a wire that lies along the semicircle y = 4 - x2 if the density of the wire is = cos
.
A) x = 1, y = 1
C) x = 0, y = 2
B) x = 0, y = 1
3
157)
D) x = 2 , y = 2 3
3
Solve the problem.
158) How much work is done by a 160-lb woman as she walks up 14 steps, each with a 1 -ft rise? 2
A) 560 ft · lb
B) 1120 ft · lb
C) 2240 ft · lb
158)
D) 4480 ft · lb
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 159) y = 4csc x, y = 0, x = , x = 3 159) 4 4
A) 48
B) 16
C) 32
D) 8
Find the length of the curve. 160) y = 1 x3 + 1 from x = 1 to x = 5 6 2x
160)
A) 5.55913079e+15
B) 5.585519069e+15
C) 5.55913079e+15
D) 5.55913079e+15
2.638827907e+14
2.638827907e+14
3.518437209e+14
1.319413953e+14
Find the center of mass of a thin plate covering the given region with the given density function. 161) The triangular region cut from the first quadrant by the line y = -x + 2, with density (x) = x A) x = 1 , y = 1 B) x = 1, y = 1 C) x = 1 , y = 1 D) x = 1, y = 1 2 2 2 2
161)
Solve.
162) Find the lateral (side) surface area of the cone generated by revolving the line segment y = x/2, 0 x 5, about the x-axis. A) 25 7 4
B) 5 2
C) 25
5
4
5
D) 5 2
7
Solve the problem. 163) A rectangular swimming pool has a parabolic drain plate at the bottom of the pool. The drain 1 1 plate is shaped like the region between y = x2 and the line y = from x = -1 to x = 1. The pool 2 2 is 10 ft by 20 ft and 8 ft deep. If the drain plate is designed to withstand a fluid force of 200 lb, how high can the pool be filled without exceeding this limitation? A) 5 ft 6 in. B) 5 ft 1 in. C) 5 ft 9 in. D) 4 ft 11 in.
30
162)
163)
164) The hemispherical bowl of radius 7 contains water to a depth 3. Find the volume of water in the bowl.
A) 27
C) 505
B) 54
3
164)
D) 848 3
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 165) y = 3x, y = 3, x = 0 165) 27 27 A) B) 9 C) D) 18 4 2 Find the length of the curve.
166) y = 2x3/2 from x = 0 to x = 5
166)
4
A) 335 72
B) 335
C) 335
3
108
D) 9 4
Solve the problem. 167) A spring has a natural length of 24 in. A force of 1400 lb stretches the spring to 34 in. How far beyond its natural length will a 350 lb force stretch the spring? A) 5 in. B) 2.5 in. C) 40 in. D) 4 in.
168) A right circular cylinder is obtained by revolving the region enclosed by the line x = r, the x-axis, and the line y = h, about the y-axis. Find the volume of the cylinder. A) r2 h B) rh C) rh2
167)
168)
D) 2 r2 h
169) A fisherman is about to reel in a 8-lb fish located 10 ft directly below him. If the fishing line
169)
weighs 1 oz per foot, how much work will it take to reel in the fish? Round your answer to the nearest tenth, if necessary. A) 90 ft · lb B) 130 ft · lb C) 83.1 ft · lb D) 86.3 ft · lb
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line. 170) About the line y = 3 170) x=y+6 x = y2
A) 99 2
B) 225
C) 99
2
31
D) 63 2
Solve the problem.
171) A tank is designed by revolving the parabola y = 5x2 , 0 x 2, about the y-axis. The tank, with
171)
dimensions in meters, is filled with water weighing 9800 N/m 3 . How much work will it take to empty the tank by pumping the water to the tank's top? Give your answer to the nearest J. A) 8,210,029 J B) 256,563 J C) 24,630,086 J D) 1,642,006 J
The centroid of a triangle lies at the intersection of the triangle's medians, because it lies one-third of the way from each side towards the opposite vertex. Use this result to find the centroid of the triangle whose vertices appear as following. 172) (0, 0), (1, 0), ( 1 , 8) 172) 2
A) (0, 8 ) 3
B) ( 1 , 16 ) 2
C) ( 1 , 8 )
3
D) ( 1 , 4)
2 3
3
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 173) y = x2, y = 25, x = 0 173)
A) 3.435973837e+15
B) 250
C) 3.435973837e+15
D) 1.288490189e+15
5.497558139e+12
3
1.374389535e+12
3.435973837e+11
Solve the problem. 174) A conical tank is resting on its apex. The height of the tank is 14 ft, and the radius of its top is 7 ft. The tank is full of gasoline weighing 45 lb/ft3 . How much work will it take to pump the gasoline to a level 14 ft above the cone's top? Give your answer to the nearest ft · lb. A) 1,056,015 ft · lb B) 565,722 ft · lb C) 226,289 ft · lb
D) 113,144 ft · lb
Find the volume of the solid generated by revolving the shaded region about the given axis. 175) About the y-axis
x = 2 tan
A) 20 - 5
y 5
C) 10 - 5 2
B) 5 + 5 2
2
32
174)
D) 20 - 5 2
175)
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 176) y = 49 - x2 , y = 0, x = 0, x = 7 176)
A) 14
B) 196
C) 1.005686636e+15
D) 1.005686636e+15
4.398046511e+12
2.199023256e+12
Solve the problem. 177) Find the volume that remains after a hole of radius 1 is bored through the center of a solid sphere of radius 4. Round to the nearest tenth. A) 77.5 B) 60.8 C) 730 D) 243.3
177)
Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. y 178) x = 178) sin t dt, 0 y /3; y-axis 0 /3 y /3 y A) B) 2 cos t dt sin y dy sin t dt cos y dy 0 0 0 0 /3 y /3 y C) 2 D) cos t dt sin y dy sin t dt cos y dy 0 0 0 0 Find the center of mass of a thin plate covering the given region with the given density function. 179) The region between the curve y = 7 and the x-axis from x = 1 to x = 9, with density (x) = x
A) x = 13 , y = 98 3
3
B) x = 5, y = 7
C) x = 13 , y = 7
3
3
5
x
179)
D) x = 13 , y = 7 3
6
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 180) y = 7 , y = - x + 8 180) x
A) 72
B) 512
C) 54
3
33
D) 42
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. 181) About the y-axis 181)
y = 3x - x2
A) 2.849934139e+15
B) 2.849934139e+15
C) 2.849934139e+15
D) 2.849934139e+15
1.407374884e+14
2.814749767e+14
4.222124651e+14
2.111062325e+14
Find the volume of the solid generated by revolving the shaded region about the given axis. 182) About the x-axis
182)
y = 3sec x
B) 9
A) 3
C) 9
2
D) 21 2
Solve the problem. 183) A conical tank is resting on its apex. The height of the tank is 12 ft, and the radius of its top is 4 ft. The tank is full of gasoline weighing 45 lb/ft3 . How much work will it take to pump the gasoline to the top? Give your answer to the nearest ft · lb. A) 4524 ft · lb B) 2,198,612 ft · lb
C) 81,430 ft · lb
D) 27,143 ft · lb
184) An auxiliary fuel tank for a helicopter is shaped like the surface generated by revolving the curve y=1-
x2 25
, - 5 x 5, about the x-axis (dimensions are in feet). How many cubic feet of fuel will
the tank hold to the nearest cubic foot? A) 17 B) 21
C) 8
34
183)
D) 5
184)
185) The base of a rectangular tank measures 9 ft by 18 ft. The tank is 16 ft tall, and its top is 10 ft below
185)
ground level. The tank is full of water weighing 62.4 lb/ft3 . How much work does it take to empty the tank by pumping the water to ground level? Give your answer to the nearest ft · lb. A) 5,499,187 ft · lb B) 1,294,086 ft · lb C) 2,911,334 ft · lb D) 1,293,926 ft · lb
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
186) y = sin 3x, y = 1, x = 0 to x = A)
2 6
-
3
186)
6
B)
2 6
-
C)
2 6
+
D)
-
6
1 3
Find the volume of the described solid. 187) The base of a solid is the region between the curve y = 5cos x and the x-axis from x = 0 to x = /2. The cross sections perpendicular to the x-axis are isosceles right triangles with one leg on the base of the solid. A) 6 B) 5 C) 25 D) 25 2 8 4
187)
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 188) y = x2, y = 0, x = 0, x = 4 188)
A) 256
B) 64
C) 64
3
D) 1024 5
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. 189) y = 4x, y = 8x, y = 4 189) 16 8 A) 8 B) C) D) 16 3 3
Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. 190) The region in the first quadrant bounded by x = 6y - y2 and the y-axis about the x-axis 190)
A) 2.849934139e+15
B) 2.849934139e+15
C) 2.849934139e+15
D) 2.849934139e+15
1.759218604e+13
2.638827907e+13
1.319413953e+13
8.796093022e+12
Find the length of the curve. 191) y = (9 - x2/3) 3/2 from x = 1 to x = 27
A) 36
191)
B) 72
C) 24
35
D) 81 2
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line. 192) About the line y = - 6 192) x=y+6 x = y2
A) 387
C) 387
B) 192
2
D) 63
4
2
Find the area of the surface generated by revolving the curve about the indicated axis. 193) x = 3 4 - y, 0 y 15/4; y-axis A) 125 - 5 10 B) 5 10 2
C) 125
193)
D) 125 + 5 10
2
2
Find the moment or center of mass of the wire, as indicated.
194) Find the center of mass of a wire that lies along the semicircle y = 9 - x2 if the density of the wire is = 3sin .
A) x = 0, y = 3
B) x = 3 , y = 3
C) x = 0, y = 3
D) x = 3 , y = 3
8
8
4
4
8 4
Find the volume of the solid generated by revolving the region about the given line. 195) The region bounded above by the line y = 7, below by the curve y = 7cos ( x), on the left by the line x = -0.5, and on the right by the line x = 0.5, about the line y = 7 A) 147 B) 147 - 98 C) 147 - 196 D) 49 2 2 2 Solve the problem. 196) An auxiliary fuel tank for a helicopter is shaped like the surface generated by revolving the curve x2 y=1, - 3 x 3, about the x-axis (dimensions are in feet). If a cubic foot holds 7.481 gallons 9 and the helicopter gets 3 miles to the gallon, how many additional miles will the helicopter be able to fly once the tank is installed (to the nearest mile)? A) 226 B) 56 C) 113 D) 75
36
194)
195)
196)
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. 197) y = x2, y = 6 + 5x, for x 0 197)
A) 3.799912186e+15
B) 3.799912186e+15
C) 3.799912186e+15
D) 3.799912186e+15
2.638827907e+13
4.398046511e+12
8.796093022e+12
1.319413953e+13
Find the center of mass of a thin plate covering the given region with the given density function. 198) The region bounded below by the parabola y = x2 and above by the line y = x + 2, with density (x) = 2x2
A) x = 8 , y = 118
B) x = 9 , y = 131
C) x = 531 , y = 18
D) x = 18 , y = 531
7
49
70
8
5
64
5
70
199) The region bounded by the curves y = ± 6 and the lines x = 1 and x = 9, with density (x) = 6
x
x
A) x = 5, y = 0
B) x = 4, y = 0
C) x = 3, y = 0
B) 8192
C) 16384
5
199)
D) x = 2, y = 0
Find the volume of the described solid. 200) The solid lies between planes perpendicular to the x-axis at x = - 4 and x = 4. The cross sections perpendicular to the x-axis are circular disks whose diameters run from the parabola y = x2 to the parabola y = 32 - x2. A) 256 3
198)
200)
D) 8192
15
15
Find the volume of the solid generated by revolving the region about the given line.
201) The region in the first quadrant bounded above by the line y = 5, below by the line y = 5x , and on 6
the left by the y-axis, about the line y = 5 A) 25 B) 60
C) 350
D) 50
Solve the problem. 202) A rectangular sea aquarium observation window is 14 ft wide and 6 ft high. What is the force on this window if the upper edge is 5 ft below the surface of the water. The density of seawater is 64 lb/ft3 .
A) 43,000 lb
B) 86,000 lb
C) 54,200 lb
2
,y=2
B) x =
2
,y=3
C) x =
37
2
,y=
202)
D) 38,100 lb
Find the center of mass of a thin plate of constant density covering the given region. 3 203) The region between the x-axis and the curve y = 3csc2 x, x 4 4
A) x =
201)
9 2
203) D) x = 3 , y = 4 8
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
204) y = sec x, y = tan x, x = 0, x = A)
2
B)
4
204)
4 2
C)
2
D)
2
Solve the problem.
2
205) Find a curve through the point (-8, 1) whose length integral, 1 y 2, is L = A) x = -8 y
C) x = 4 y
B) x = -8y5/2
4
1+
1
16 dy. y3
205)
D) x = -8 y
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. 206) About the y-axis 206) x=4
y = 3 - x2 /16
A) 20
B) 40
C) 48
D) 32
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line. 207) About the line y = 1 207) x=5
3
y (solid)
x = 5y2 (dashed)
A) 25 28
B) 25
C) 50
21
21
38
D) 25 14
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
208) y = sin 8x, y = 0, 0 x A) 8
208)
8
C) 1
B) 2
4
D) 16
Find the fluid force exerted against the vertically submerged flat surface depicted in the diagram. Assume arbitrary units, and call the weight-density of the fluid w.
209)
209) 7 7 7
A) 343w
B) 3.771324883e+15 w
C) 686 w 3
D) 343 w
1.319413953e+13 2
Find the center of mass of a thin plate covering the given region with the given density function. 210) The region bounded by x = y2 and the line x = 9, with density (x) = y2
210)
Find the volume of the solid generated by revolving the shaded region about the given axis. 211) About the y-axis
211)
A) x = 27 , y = 0 7
B) x = 27 , y = 0 5
C) x = 45 , y = 0 7
D) x = 0, y = 27 5
x = 6y/5
A) 120
B) 50
C) 15
39
D) 60
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves about the given lines. 212) y = 5x, 212) y = 0, x = 3; revolve about the x-axis A) 5.93736279e+15 B) 5.93736279e+15 5.277655813e+13 5.277655813e+13
C) 5.93736279e+15
D) 5.93736279e+15
2.638827907e+13
1.319413953e+13
Find the volume of the solid generated by revolving the shaded region about the given axis. 213) About the x-axis
213)
y = - 4x + 8
B) 128
A) 24
C) 896
3
D) 256
3
3
Find the moment or center of mass of the wire, as indicated. 214) Find the moment about the x-axis of a wire of constant density that lies along the first-quadrant portion of the circle x2 + y2 = 16.
B) 8
A) 16
C) 4
D) 4
Solve the problem. 215) Find the force on one side of a cubical container 5 cm on an edge if the container is filled with mercury. The density of mercury is 133 kN/m 3 .
A) 1.7 N
B) 8.3 N
C) 330 N
Set up an integral for the length of the curve. 216) y = x5, 0 x 1 1
A) 0
C)
1
214)
215)
D) 8300 N
216)
1 + 5x8 dx
1
B) 0
1 + 25x8 dx
D)
0
1
1 + 5x4 dx 1 + 25x10 dx
0
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 217) y = 9cos x, y = 0, x = -0.5, x = 0.5 217) 81 A) 27 B) 162 C) D) 81 2
40
Find the volume of the solid generated by revolving the region about the given line. 218) The region in the first quadrant bounded above by the line y = 1, below by the curve y = sin 7x, and on the left by the y-axis, about the line y = - 1 A) 3 2 - 1 B) 1 2 + 1 C) 1 2 - 1 D) 3 - 1 14 21 14 7 14 7 14 21
218)
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. 219) y = 5 , y = 0, x = 6, x = 8 219) x
A) 15
B) 30
C) 10
D) 20
Solve the problem. 220) One end of a pool is a vertical wall 15 ft wide. What is the force exerted on this wall by the water if it is 6 ft deep? The density of water is 62.4 lb/ft3 .
A) 16,800 lb
B) 33,700 lb
C) 2810 lb
D) 8420 lb
Use your grapher to find the surface's area numerically. 221) x1/2 + y1/3 = 3, 1 x 4; x-axis
A) 234.59
221)
B) 210.41
C) 342.94
D) 324.54
Solve the problem.
2
222) Find a curve through the point 1, 6 whose length integral, 1 x 2, is L = 7
A) y = 6 x6 7
220)
1
B) y = 6x6
C) y = 6x7
1 + 36x12 dx.
222)
D) y = 6 x7 7
Find the volume of the solid generated by revolving the region about the y-axis.
223) The region enclosed by x = sin 6y, 0 y A) 12
B)
12
,x=0
223) C)
6
12
D) 6
Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. 224) The region in the first quadrant bounded by x = 5y - y2 and the y-axis about the line x = -1 224)
A) 4.810363372e+15
B) 4.810363372e+15
C) 4.810363372e+15
D) 4.810363372e+15
1.099511628e+13
4.398046511e+13
1.649267442e+13
3.298534883e+13
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. 225) y = 9 x, y = 0, x = 1 225) 18 36 A) 18 B) 45 C) D) 5 5
41
Find the length of the curve. 1 226) x = t3 - 1 dt , 1 y 16 y A) 2046 B) 42 5
226) C) 2043 5
D) 255 4
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line. 227) About the line y = -1 227)
x = 5y - y2
A) 7.696581394e+15
B) 7.696581394e+15
C) 7.696581394e+15
D) 5.497558139e+15
1.759218604e+13
1.055531163e+14
5.277655813e+13
5.277655813e+13
Solve the problem.
228) The disk (x - 4)2 + y2 4 is revolved about the y-axis to generate a torus. Find its volume. (Hint: 2
-2
4 - y2 dy = 2 , since it is the area of a semicircle of radius 2.)
A) 16 2
B) 32 2
C) 8 2
42
D) 16
228)
Find the fluid force exerted against the vertically submerged flat surface depicted in the diagram. Assume arbitrary units, and call the weight-density of the fluid w.
229)
229) 10 10 20
A) 2000w
B) 8.246337208e+14 w
C) 4000w
D) 8.246337208e+14 w
5.497558139e+11 2.748779069e+11
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. 230) y = 50 - x2 , y = x2, x = 0 230)
A) 5.497558139e+15
B) 1250
C) 625
D) 5.497558139e+15
1.759218604e+13
3.518437209e+13
Solve the problem. 231) A force of 1000 lb compresses a spring from its natural length of 15 in. to a length of 13 in. How much work is done in compressing it from 13 in. to 5 in.? A) 0.096 in.-lb B) 24,000 in.-lb C) 48,000 in.-lb D) 16,000 in.-lb
231)
Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. 232) The region in the first quadrant bounded by x = 4y - y2 and the y-axis about the y-axis 232)
A) 1024 15
B) 512
C) 512
15
5
43
D) 128 5
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line. 233) About the line y = 9 233) x=
y (solid) x = y/3 (dashed)
A) 8.549802418e+15
B) 2.493692372e+15
C) 8.549802418e+15
D) 2.849934139e+15
3.518437209e+14
2.199023256e+13
1.759218604e+14
8.796093022e+13
Find the fluid force exerted against the vertically submerged flat surface depicted in the diagram. Assume arbitrary units, and call the weight-density of the fluid w.
234)
234) 5 5 5
A) 625 w 6
B) 250 w
C) 125 w
3
3
D) 125w
Find the volume of the solid generated by revolving the region about the given line. 235) The region in the second quadrant bounded above by the curve y = 4 - x2 , below by the x-axis, and on the right by the y-axis, about the line x = 1 A) 256 B) 8 15
44
C) 56 3
D) 32 3
235)
Find the volume of the solid generated by revolving the shaded region about the given axis. 236) About the x-axis
236)
y = 4 - x2
A) 8 3
B) 224
C) 256
15
15
D) 64 15
Solve the problem.
237) A water tank is formed by revolving the curve y = 7x4 about the y-axis. Water drains from the
237)
238) A vertical right circular cylindrical tank measures 24 ft high and 10 ft in diameter. It is full of oil
238)
tank through a small hole in the bottom of the tank. At what constant rate does the water level, y, fall? (Use Torricelli's Law: dV/dt = -m y.) A) dy = - 7 B) dy = - m 7 C) dy = D) dy = - m dt dt dt m dt m 7 7
weighing 60 lb/ft3 . How much work does it take to pump the oil to a level 2 ft above the top of the tank? Give your answer to the nearest ft · lb. A) 1,357,168.03 ft · lb B) 3,166,725.4 ft · lb
C) 1,319,468.91 ft · lb
D) 1,583,362.7 ft · lb
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves about the given lines. 239) y = 4x, 239) y = x2 ; revolve about the y-axis
A) 64 3
B) - 128
C) 128
3
3
Find the length of the curve. x 240) y = t2 - 1 dt , 3 x 6 1 A) 27 B) 27 2
3
240) C) 18
Find the center of mass of a thin plate of constant density covering the given region. 241) The region bounded by y = x2 and y = 5
A) x = 0, y = 15
D) 896
B) x = 0, y = 3
C) x = 0, y = 6
45
D) 9
241) D) x = 0, y = 25 3
The centroid of a triangle lies at the intersection of the triangle's medians, because it lies one-third of the way from each side towards the opposite vertex. Use this result to find the centroid of the triangle whose vertices appear as following. 242) (-4, 0), (4, 0), (0, 8) 242) A) (0, 8 ) B) (0, 16 ) C) (0, 8) D) (0, 4) 3 3 Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. 243) y = 8x2 , y = 8 x 243)
A) 12
C) 24
B) 6
5
5
D) 6 5
Find the volume of the described solid. 244) The solid lies between planes perpendicular to the x-axis at x = 0 and x = 5. The cross sections perpendicular to the x-axis between these planes are squares whose bases run from the parabola y = - 2 x to the parabola y = 2 x. A) 50 B) 192 C) 200 D) 100
244)
A variable force of magnitude F(x) moves a body of mass m along the x-axis from x 1 to x2 . The net work done by the x2 force in moving the body from x 1 to x2 is W =
x1
F(x) dx =
1 1 mv2 2 - mv1 2 , where v1 and v2 are the body's velocities 2 2
at x 1 and x 2 . Knowing that the work done by the force equals the change in the body's kinetic energy, solve the problem.
245) A 2-oz tennis ball was served at 140 ft/sec about (95 mph). How much work was done on the ball
245)
to make it go this fast? (To find the ball's mass from its weight, express the weight in pounds and divide by 32 ft/sec2 , the acceleration of gravity.)
A) 77 ft-lb
B) 38 ft-lb
C) 36 ft-lb
D) 18 ft-lb
Solve the problem. 246) A right triangular plate of base 6 m and height 3 m is submerged vertically, as shown below. Find the force on one side of the plate if the top vertex is 1 m below the surface. (w = 9800 N/m 3 )
246)
1m
3m
6m
A) 210,000 N
B) 150,000 N
C) 410,000 N
D) 180,000 N
247) Find the volume that remains after a hole of radius 1 is bored through the center of a solid cylinder of radius 4 and height 8. A) 8 B) 128
C) 60
46
D) 120
247)
Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. 248) About the y-axis 248)
y=
A) 20
4sin (x) ;0<x x
B) 8
C) 12
D) 16
Find the volume of the solid generated by revolving the shaded region about the given axis. 249) About the x-axis
249)
y = 3 sin x
A) 9 2 2
B) 9 2 - 3
C) 9 2 + 9
2
2
Find the length of the curve. 250) y = 3 (x4/3 - 2x2/3) from x = 1 to x = 27 8
D) 9 2 - 9 2
250)
A) 2.53327479e+15
B) 93
C) 3.061040372e+15
D) 2.691604465e+15
7.036874418e+13 7.036874418e+13
7.036874418e+13
47
Find the volume of the described solid. 251) The solid lies between planes perpendicular to the x-axis at x = /6 to x = /2. The cross sections perpendicular to the x-axis are circular disks with diameters running from the curve y = cot x to the curve y = csc x. 2 2 A) ( 3 - 1) B) (2 3 - 2) 2 12 3
C) ( 3 + 1) 2
-
2
D) ( 3 - 1) +
6
48
2 6
251)
Answer Key Testname: CHAPTER 6
1) The volume can be found using the disk method with 1 integral. Since x = 2y - y2 cannot be solved for y, the shell method cannot be used.
2) The resulting solid is a cylindrical shell whose volume is given by V = the inner radius, and h is the height. Since r1 = 6, r2 = 2 and h = 3, V =
r1 2 - r2 2 h, where r1 is the outer radius, r2 is 6 2 - 2 2 3 = 96 .
3) The volume can be found using the washer method with 2 integrals or using the shell method with 1 integral. 4) The resulting solid is a hemisphere of radius 2. The volume is given by V = 2 r3 = 16 3
3
5) The volume can be found using the washer method with 1 integral or using the shell method with 2 integrals. 6) The volume can only be found using the shell method. Only 1 integral is required. Since y = 4x- x2 cannot be solved for x, the washer method cannot be used.
7) The volume can be found using the shell method with 1 integral. Since x = 2y - y2 cannot be solved for y, the washer method cannot be used.
8) The volume can be found using the washer method with 2 integrals or using the shell method with 1 integral. 9) A 10) C 11) D 12) C 13) B 14) A 15) D 16) C 17) D 18) A 19) B 20) D 21) A 22) D 23) C 24) A 25) A 26) D 27) D 28) B 29) A 30) B 31) C 32) D 33) B 34) A 35) A 36) B 37) A 49
Answer Key Testname: CHAPTER 6
38) A 39) B 40) D 41) D 42) C 43) D 44) D 45) D 46) D 47) D 48) B 49) C 50) C 51) C 52) B 53) A 54) D 55) B 56) A 57) C 58) D 59) A 60) B 61) A 62) A 63) B 64) A 65) C 66) D 67) A 68) C 69) D 70) D 71) C 72) A 73) D 74) D 75) D 76) A 77) D 78) B 79) A 50
Answer Key Testname: CHAPTER 6
80) A 81) A 82) C 83) C 84) D 85) D 86) B 87) B 88) C 89) A 90) A 91) C 92) B 93) A 94) A 95) D 96) C 97) A 98) C 99) A 100) D 101) D 102) A 103) D 104) A 105) B 106) C 107) C 108) C 109) A 110) A 111) B 112) D 113) D 114) B 115) D 116) B 117) D 118) D 119) B 120) C 121) A 51
Answer Key Testname: CHAPTER 6
122) B 123) B 124) B 125) B 126) D 127) B 128) A 129) B 130) B 131) D 132) D 133) D 134) C 135) A 136) C 137) C 138) A 139) A 140) C 141) C 142) D 143) A 144) B 145) C 146) C 147) A 148) C 149) A 150) C 151) A 152) C 153) C 154) D 155) B 156) B 157) B 158) B 159) C 160) A 161) B 162) C 163) B 52
Answer Key Testname: CHAPTER 6
164) B 165) C 166) C 167) B 168) A 169) C 170) A 171) A 172) C 173) C 174) B 175) D 176) C 177) D 178) B 179) D 180) A 181) D 182) C 183) D 184) A 185) C 186) A 187) C 188) D 189) B 190) C 191) A 192) A 193) A 194) C 195) C 196) A 197) D 198) A 199) C 200) C 201) D 202) A 203) A 204) A 205) A 53
Answer Key Testname: CHAPTER 6
206) B 207) C 208) C 209) B 210) C 211) D 212) C 213) B 214) A 215) B 216) C 217) C 218) A 219) D 220) A 221) B 222) D 223) B 224) D 225) D 226) A 227) C 228) B 229) D 230) C 231) B 232) B 233) C 234) B 235) C 236) B 237) B 238) D 239) C 240) A 241) B 242) A 243) A 244) C 245) B 246) C 247) D 54
Answer Key Testname: CHAPTER 6
248) D 249) D 250) A 251) A
55
Chapter 7
Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use logarithmic differentiation to find the derivative of y. 1) y = x(x - 2)
1)
A) x(x - 2)(2x - 2)
B) ln x + ln(x - 2)
C) 1 1 + 1
D)
2 x
2
x-2
x(x - 2) 1 1 + 2 x x-2
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 2) Show that lim ln(x + 1) = lim ln(x + 9955) . ln x ln x x x
2)
Explain why this is the case.
3) Which one is correct, and which one is wrong? Give reasons for your answers.
3)
x+2 1 1 (a) lim = lim =2 2x 4 x -2 x -4 x -2 (b)
x+2 0 = =0 lim 2 8 x -2 x -4
Estimate the limit by graphing the function for an appropriate domain. Confirm your estimate by using l'Hopital's rule. Show each step of your calculation. 4) lim 1 - cos x 4) x x 0
Provide an appropriate response.
5) Show that
ln ax dx = x ln ax - x + C.
5)
Estimate the limit by graphing the function for an appropriate domain. Confirm your estimate by using l'Hopital's rule. Show each step of your calculation. 6) lim x1/x 6) x
1
Provide an appropriate response. 7) Which one is correct, and which one is wrong? Give reasons for your answers. x-4 1 1 (a) lim = lim = x 4 x2 - 4 x 4 2x 8
7)
x-4 0 = =0 (b) lim 12 x 4 x2 - 4
8) A polynomial f(x) has a degree smaller than or equal to another polynomial g(x). Does
8)
f = O(g) and does g = O(f)?
Estimate the limit by graphing the function for an appropriate domain. Confirm your estimate by using l'Hopital's rule. Show each step of your calculation. 9) lim cos x - 1 9) x 0 ex - x - 1
10) lim x x
10)
2x
Solve the problem. 11) You are sitting in a classroom next to the wall looking at the blackboard at the front of the room. The blackboard is 10 ft long and starts 5 ft from the wall you are sitting next to. x x Show that your viewing angle is = cot-1 - cot-1 . 10 5 Provide an appropriate response. 12) Which of the following items is undefined and why? 1 1 csc-1 or cos-1 5 5
11)
12)
13) Suppose you are looking for an item in an ordered list one million items long. Which
13)
14) Which of the following items is undefined and why?
14)
would be better, a sequential search or a binary search? Why?
tan-1 3 or cos-1 3
15) Find the error in the following incorrect application of L'Hôpital's Rule. lim x -2
x3 - 2x2 + 1 3x2 - 6x
3x2 - 4x
= lim = lim x -2 6x - 6 x -2
15)
6x - 4 -16 . = 6 6
16) Graph y = sin-1 (sin x). Explain why the graph looks like it does.
2
16)
17) Graph f(x) = cos-1 x together with its first derivative. Comment on the behavior of f and
17)
18) Derive the identity sec-1 (-x) = - sec-1 x by combining the following two equations:
18)
the shape of its graph in relation to the signs and values of f .
cos-1 (-x) = - cos-1 x sec-1 x = cos-1 (1/x)
Estimate the limit by graphing the function for an appropriate domain. Confirm your estimate by using l'Hopital's rule. Show each step of your calculation. 19) lim x x 19) + x 0 1 1 sin x x x 0
20) lim
20)
Provide an appropriate response. 21) Give an example of two differentiable functions f and g with lim f(x) = lim g(x) = x x
21)
f(x) that satisfy lim = 4. g(x) x
22) Graph f(x) = cos-1
x 1 and g(x) = tan-1 . Explain why the graph looks like it does. x 2 x +1
22)
Estimate the limit by graphing the function for an appropriate domain. Confirm your estimate by using l'Hopital's rule. Show each step of your calculation. 23) lim x x - 2 x + 3x - 6 23) x-2 x 2
Provide an appropriate response.
24) Consider the graphs of y = cos-1 x and y = sin-1 x. Does it make sense that the derivatives of these functions are opposites? Explain.
3
24)
25) Show that y = x6 + x and y = x6 + x4 grow at the same rate as x they both grow at the same rate as y = x3 as x
by showing that
25)
.
26) Give an example of two differentiable functions f and g with lim f(x) = lim g(x) = 0 that x
x
26)
f(x) satisfy lim = . g(x) x
27) Find the error in the following incorrect application of L'Hôpital's Rule.
27)
28) A student attempted to use l'Hôpital's Rule as follows. Identify the student's error.
28)
sinx cosx - sinx lim = lim = lim = 0. x 0 x + x2 x 0 1 + 2x x 0 2
lim x
sin (1/x) = lim e1/x x
-x-2 cos (1/x)
= lim x
-x-2 e1/x
cos (1/x) 1 = =1 1 e1/x
29) If f(x) = (x - 3)2 and g(x) =
1
(x - 3)2
, show that lim f(x)g(x) = 0 x 3
29)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Graph the inverse of the function plotted, on the same set of axes. Use a dashed curve for the inverse.
30)
30)
4
A)
B)
C)
D)
Evaluate the integral. 2 dx 31) 3 + 6x
31)
A) 2 ln 3 + 6x + C
B) -2 ln -3 - 6x + C
C) 1 ln -3 - 6x + C
D) ln -3 - 6x + C
3 3
Verify the integration formula. -1 32) tanh (lnx) dx = ln x tanh-1 (ln x) + 1 ln (1 - (ln x)2)+ C x 2
A) Yes
32)
B) No
Express the value of the inverse hyperbolic function in terms of natural logarithms. 33) csch-1 9 4
A) ln 3 2
C) ln 4- 97
B) ln 97
9
5
33) D) ln 4+ 97 9
Find the limit.
34) lim cos-1 x
34)
x 1-
A) 1
B) 0
C) -1
D)
L'Hopital's rule does not help with the given limit. Find the limit some other way. 35) lim sec x tan x x 0+
A) 0
D) 1
C) -1
B)
Evaluate the integral. 6 ln x 5 36) dx x 1
35)
36)
A) 15,620
C) 5
B) 29
ln 5
ln 5
ln 6
D) 5
ln 5
ln 6 - 1 ln 5
Solve the initial value problem. 2 37) d y = 2 - e-t, y(1) = -1 , y (0) = -6 e dt2
37)
A) y = 2t2 + e-t - 7t + 5 - 2
B) y = t2 - e-t - 7t + 6
C) y = t2 - e-t
D) y = t2 - e-t - 6t
e
Solve for t.
38) ex2 e5x+8 = et A) x2 - 5x - 8
38) B) 5x3 + 8x2
C) x2 + 5x + 8
D) ln (x2 + 5x + 8)
Solve the problem.
5 , 2x + 1
39) Find the volume of the solid that is generated by revolving the area bounded by y = x = 0, x = 6, and y = 0 about the x-axis. A) 5 ln (2) B) 25 ln (13) 2 2
C) 5 2 ln (13) 2
D) 25 ln (2) 2
Find the inverse of the function. 40) f(x) = x3 - 6
A) f-1 (x) = C) f-1 (x) =
3 3
39)
40)
x+6
B) Not a one-to-one function
x-6
D) f-1 (x) =
6
3
x+6
Evaluate the integral. /8 41) 4 cot (4 ) d /16
A) ln 2
41) B) ln 8
C) ln 2
2
D) - ln 2
2
2
Find the domain and range of the inverse of the given function. 42) f(x) = x3 - 4
A) Domain: all real numbers; range: [-4, ) C) Domain and range: all real numbers
42) B) Domain: [-4, ); range: all real numbers D) Domain: [0, ); range: [0, )
Use l'Hopital's Rule to evaluate the limit. 2 43) lim 11 + 5x - 10x x 4 - 2x - 2x2
A) 1
43) C) 11
B) 5
D)
4
Find the derivative of y with respect to x. 44) y = -cot-1 2x 3
A)
6 2 4x + 9
B)
44) 9 2 4x + 9
C)
2 9 - 4x2
Find the derivative of y with respect to x, t, or , as appropriate. 45) y = ln ln 2x A) 1 B) 1 C) 1 ln 2x x ln 2x x
D)
-6 4x2 + 9
45) D) 1 2x
Solve the differential equation. 46) dy = 7x6 e-y dx
46)
A) y = x7 + C C) y = ln (x7 + C)
B) y = ln (7x7 + C) D) y = C ln (x7 )
Find the formula for df-1 /dx. 47) f(x) = 1024x5
A)
1 20x4/5
47) B) x
1/5
C) 20x4
4
7
D) 5120x4
Solve the initial value problem. 48) dy = 6 , y(0) = 1 dx 16 + x2
48)
A) y= tan-1 x + 1
B) y = 3 cot-1 x + 1
C) y = tan-1 x
D) y = 3 tan-1 x + 1
4
2
4
2
Find the derivative of y with respect to x. 49) y = 2 sin-1 (4x4) 32x3
A)
4
B)
1 - 16x8
4
49) 32x3
C)
1 - 16x4
32x3
1 - 16x8
Rewrite the expression in terms of exponentials and simplify the results. 50) cosh 3x + sinh 3x A) 3x B) e3x - e-3x C) e3x Find the formula for df-1 /dx. 51) f(x) = x3/7
D)
2 1 - 16x8
50) D) 2e3x
51)
A) 3 x-4/7
C) 7 x4/3
B) x-4/3
7
D) x7/3
3
Evaluate the integral. 1/x 52) e dx 2x2 1/x
A) e
2
52) 1/x
B) - e
+C
2
-1/x
C) e
+C
Find the slowest growing and the fastest growing functions as x 53) y = ln 6x y = 5 ln x 1 y= x y=
2
D) -2 e1/x + C
.
x
A) Slowest: y = 1
x
Fastest: y =
x 1 B) Slowest: y = x
Fastest: y = 5 ln x C) Slowest: y = x Fastest: y = ln 6x and y = 5 ln x grow at the same rate. D) Slowest: y = ln 6x and y = 5 ln x grow at the same rate. Fastest: y = x
8
+C
53)
Solve the problem. 54) The barometric pressure p at an altitude of h miles above sea level satisfies the differential dp equation = -0.2 p. If the pressure at sea level is 29.92 inches of mercury, find the barometric dh pressure at 12,000 ft. A) 18.99 in.
B) 9.5 in.
C) 47.14 in.
D) 2.71 in.
Determine whether the integration formula is correct. 3 x5 dx 55) x4 sin-1 8x dx = x sin-1 8x + 8 +C 3 5 1 - 64x2
A) No
55) B) Yes
Evaluate the integral. dx
56)
54)
56)
64 - x2
A) 1 sin-1 1 x + C
B) 2 cos-1 1 x + C
C) cos-1 1 x + C
D) sin-1 1 x + C
2
8
8
8
8
Find the limit.
57) lim sec-1 x
57)
x
B) 0
A)
C)
2
D) -
2
Evaluate the integral. dx 58) x 3 + 4 ln x
58)
A) 1 ln 4 + 3 ln x + C
B) 1 ln 3 + 4 ln x + C
C) 1 ln 3 + 4 ln x + C 3
D) 3 ln 3 + 4 ln x + C
4
4 4
Find the domain and range of the inverse of the given function. 59) f(x) = - 1 x
A) Domain: (0, ); range: (- , 0) C) Domain and range: all real numbers Solve the problem.
59) B) Domain: (- , 0) (0, ); range: (- , 0) D) Domain and range: (- , 0) (0, )
2
60) Find the length of the curve x = y - 4 ln y , 8 y 16. 32
A) 6 + 4 ln 2
60)
5
B) 8 + 4 ln 2
C) 8 + 4 ln 2
5
9
D) 6 + 4 ln 2 5
Evaluate the integral in terms of natural logarithms. 5/2 dx 61) 2 6/5 1 - x A) ln 2 B) 1 ln 7 2 33
61) C) 1 ln - 7 2
33
D) ln 5
11
^
Use l'Hopital's rule to find the limit. 4x + 7 62) lim x 7x2 + 7x - 6
A) 4
62) C) 2
B) 1
7
7
D) 0
Find a value of a so that f is continuous at c, or indicate this is impossible. 4x - 20 , x 5 63) f(x) = x - 5 a,
63)
x = 5; c = 5
A) -4
C) Impossible
B) -20
Answer the question appropriately. 64) Find the equation for the line through the origin and tangent to y = ln 2x. A) y = 2x B) y = e C) y = (ln 2) x e
D) 4
64) D) y = - ex 2
Evaluate the integral. 7 x6 e-x dx
65)
A) - 1 e-x8 + C 7
65) C) -7e-x8 + C
B) - 1 e-x7 + C 7
D) e-x7 + C
Express the value of the inverse hyperbolic function in terms of natural logarithms. 66) tanh-1 10 11
A) 1 ln 231 2
B) 1 ln -21
C) 1 ln 1
2
2
66) D) 1 ln 21 2
Find the limit.
67) lim tan-1 x
67)
x
A)
2
B) -
C) 0
2
10
D)
Evaluate the integral in terms of natural logarithms. 11 -sin x dx 68) 1+ cos2 x 0 A) 0 B) ln -1 + 2 1+ 2
68) D) ln 2
C) -2
Find the limit.
69) lim csc-1 x
69)
x
A)
C) -
B)
2
D) 0
2
Find the derivative of y.
70) y = -2t3 tanh 1
70)
t2
A) -6t2 tanh 1 - 4 sech2 1
B) -6t2 tanh 1 + 2 sech2 1
C) -6t2 tanh 1 + 4 sech2 1
D) -6t2 tanh 1 + 4 sech 1
t2
t2
t2
t2
t2
t2
t2
t2
Evaluate the integral. /2 2 sin 2 d 71) 2 /4 1 + cos 2
A)
72)
ln 4
B)
8
A) 1 ln 1025 2
32
D)
4
2
72) B) 9
C) ln 1025
16
32
D) 1 ln 2 2
t 5-1 dt
A) 1
74)
C)
coth 2x dx
ln 2
73)
71)
73) B) t
5-1 ln t
C) t
+C
5 5
+C
dx 2 x 1+x
D) t
5-2 5-2
+C
74)
A) tan-1 x + C
B) 1 tan-1 x + C
C) 1 ln x + C
D) 1 sin-1 x + C
2
2
2
11
Determine if the statement is true or false as x 75) 5x + ln x = O(x) A) True
.
75) B) False
Solve the problem. 76) The intensity L(x) of light x ft beneath the surface of a lake satisfies the differential equation dL = - 0.03L. At what depth, to the nearest foot, is the intensity one tenth the intensity at the dx surface? A) 51 ft
B) 38 ft
C) 77 ft
D) 115 ft
Express the value of the inverse hyperbolic function in terms of natural logarithms. 77) cosh-1 5
A) ln (5 + 26)
A) 4
77)
B) ln (5 + 24)
C) ln (5 - 24)
D) ln (10)
B) 10
C) 1 10
D) 1 4
B) -
C) 3 4
Find the value of df-1 /dx at x = f(a). 78) f(x) = 4x + 10, a = 2
76)
78)
Find the angle. 79) cot-1 (-1)
79)
A) - 3 4
4
D)
4
Find the derivative of y with respect to the appropriate variable.
80) y = csch-1 1
80)
6
A)
ln 6 1 6
1+ ln
C) 1+
1 6
ln 6
1+
1 2 6
B)
1 6
1 6
ln 6
D)
1 2
1+
6
1 2 6
Determine if the given function y = f(x) is a solution of the accompanying differential equation. 81) Differential equation: y = e- 5x - 5y Initial condition: y(0) = 0
Solution candidate: y = 5xe- 5x A) Yes
B) No
12
81)
Find the derivative of y with respect to x, t, or , as appropriate. 82) y = esin t (ln t4 + 10) sin t cos t A) 4e t
82) B) esin t ln t4 + 10 + 4 t
sin t
C) ecos t (cos t)(ln t4 + 10) + 4e
D) esin t (cos t)(ln t4 + 10) + 4 t
t
Verify the integration formula.
83)
x4 1 2x3 sech-1 x2 dx = sech-1 x + 2 2
1 - x4+ C
83)
A) No
B) Yes
Answer the question appropriately. 84) Find the area of the "triangular" region in the first quadrant that is bounded above by the curve y = e4x, below by the curve y = ex, and on the right by the line x = ln 3.
A) 2.427721674e+15
B) 3 ln 3
C) 2.53327479e+15
D) 3.448068465e+15
1.407374884e+14 1.407374884e+14
1.407374884e+14
Find the derivative of y with respect to x, t, or , as appropriate. 85) y = (x2 - 2x + 9) ex
A)
x3 3
+ 7x + 9 ex
86)
1 + e4
D) (2x - 2) ex
d
86)
A) ln (1 + 4e ) + C
B) 4 ln (1 + e4 ) + C
4 C) ln (1 + e ) + C
D) ln (1 + e4 ) + C
4 4
87)
85) B) (x2 + 4x + 7) ex
C) (x2 + 7) ex Evaluate the integral. e4
21e 3x dx 2 x
A) 21 e 3x + C 2
84)
87) B) 3 e 3x + C
C) 7 3 e 3x + C
13
D) 21 e 3x + C
Find the derivative of y. 88) y = csch 6x 9
88)
A) 6 csch 6x coth 6x
B) -csch 6x coth 6x
C) csch 6x coth 6x
D) - 6 csch 6x coth 6x
9
9
9
9
9
9
9
9
9
9
Evaluate the integral. 89) sec x tan x dx 3 + sec x
89)
A) 3 ln 3 + sec x + C C) ln 3 + sec x + C
B) -ln 3 + sec x + C D) 3 ln sec x + C
Answer the question appropriately.
90) Find the absolute maximum value of f(x) = ex - 2.6x on [0, 2]. A) 1 B) e2 - 5.2 C) 2.6 - 2.6 ln 2.6
90) D) 2.6 - ln 2.6
Find the derivative of y with respect to the independent variable. 91) y = 5 x
B) 5x ln x
A) x ln 5 Evaluate the integral. 6 - 3x
92)
25 - 36x2
92)
5
C) 1
12
D) 5x
dx
A) 1 tan-1 6 x + 1 5
91)
C) 5x ln 5
12
B) sin-1 6 x + 1 ln ( 25 - 36x2 ) + C
25 - 36x2 + C
5
12
D) sin-1 6 x + 1
25 - 36x2 + C
5
Find the value of df-1 /dx at x = f(a). 93) f(x) = x3 - 6x2 - 6, x 4, a = 6
12
93)
A) 36
B) 1 36
C) - 1
D) - 1.759218604e+13
6
1.055531163e+14
Find the inverse of the function. 94) f(x) = 4x3 - 8
94) B) f-1 (x) =
A) Not a one-to-one function C) f-1 (x) =
25 - 36x2 + C
3 x 4
D) f-1 (x) =
+8
14
3 x-8 4
3 x+8 4
Determine if the statement is true or false as x 95) x = O(x + 7) A) True
.
95) B) False
Is the function graphed below one-to-one?
96)
96)
A) Yes
B) No
Determine if the given function y = f(x) is a solution of the accompanying differential equation. 97) Differential equation: 5xy + 5y = cos x Initial condition: y( ) = 0 sin x Solution candidate: y = 5x
A) Yes
97)
B) No
Find the angle.
98) cos-1 1
98)
2
A)
3
B)
C)
6
2
Find the derivative of y. 99) y = sinh2 8x
D) 3
99)
A) 16 sinh 8x cosh 8x C) 2 cosh 8x
B) 16 cosh 8x D) 2 sinh 8x cosh 8x
15
Find the slowest growing and the fastest growing functions as x 100) y = 2x2 + 7x
.
100)
y = ex y = ex/2
y = log2 x
A) Slowest: y = log2 x Fastest: y = ex B) Slowest: y = ex/2
Fastest: 2x2 + 7x C) Slowest: y = log2 x Fastest: y = ex and y = ex/2 grow at the same rate. D) Slowest: 2x2 + 7x Fastest: y = ex
Solve the initial value problem. 101) dy = 10 , y(0) = -5 dx 1 - x2
101)
A) y = 10 cos-1 x + -10 - 10
B) y = 10 sin-1 x - 5
C) y = 10 cos-1 x - 15
D) y = 10 sin-1 x
2
Verify the integration formula.
102)
6 sech x dx = sin-1 (1 - x2 ) + C
102)
A) Yes
B) No
Solve the problem. 103) A certain radioactive isotope decays at a rate of 2% per 100 years. If t represents time in years and y represents the amount of the isotope left then the equation for the situation is y = y0 e-0.0002t. In how many years will there be 93% of the isotope left? A) 350 years B) 700 years C) 253 years D) 363 years Rewrite the expression in terms of exponentials and simplify the results. 104) ln (cosh 9x - sinh 9x) A) ln (e9x - e-9x) B) -18x C) -9
104) D) -9x
Solve the problem. 105) Find the half-life of the radioactive element radium, assuming that its decay constant is k = 4.332 x 10-4 , with time measured in years.
A) 1400 years
B) 1600 years
C) 800 years
16
103)
D) 2308 years
105)
Use logarithmic differentiation to find the derivative of y. 106) y = x(x + 8)(x - 8) A) x(x + 8)(x - 8) 1 + 1 + 1 x x+8 x-8
106) B) x(x + 8)(x - 8)(ln x + ln(x + 8) + ln(x - 8))
C) 1 + 1 + 1 x
x+8
D) 1
x-8
Evaluate the integral.
107)
4x sech x2 tanh x2 dx 2
A) sech x + C 2x
107) B) - 2 sech x2 + C
C) 2 sech x2 + C
D) 4 csch x2 + C
Use l'Hopital's Rule to evaluate the limit. 1 cos x 2 108) lim x /3 x 3
A) - 3
B)
108) 3 2
C) -
3 2
D)
2 2
Solve the differential equation. 109) dy = 6x5 sec y dx
109)
A) y = cos-1 (x6 + C) C) y = sin-1 (x6 + C)
B) y = x6 + C D) y = sin (x6 + C)
Use logarithmic differentiation to find the derivative of y. 110) y = cos x 3x + 4 A) cos x 3x + 4 ln cos x + 1 ln(3x + 4) 2
C) cos x 3x + 4 -tan x +
110) 1 3 B) cos x 3x + 4 + sin x · cos x 3x + 4
3 2(3x + 4)
D) -6-tan x - 5-tan x 3x + 8
Find the derivative of y with respect to x. 111) y = cos-1 (9x2 + 4)
111)
A)
-18x 1 - (9x2 + 4)2
18x B) 1 + (9x2 + 4)2
C)
18x 1 - (9x2 + 4)2
D)
17
9 1 + (9x2 + 4)2
Evaluate the integral.
112)
3 2 -7 2 -7
A) - 5
6
-dx 2 -x - 14x - 40
112)
B) -
C)
4
4
D)
2 2 5
Graph the inverse of the function plotted, on the same set of axes. Use a dashed curve for the inverse.
113)
113)
A)
B)
C)
D)
18
Use logarithmic differentiation to find the derivative of y with respect to the independent variable. 114) y = (ln x)ln x
A) ln (ln x) + 1 (ln x)
ln x
B) ln (ln x) + 1 x
x
C) (ln x)
114)
ln x
D) ln x ln (ln x)
x
Evaluate the integral. ln 9 115) tanh x dx ln 2 A) ln 164 45
115) B) ln 119
C) ln 2
18
D) 119 18
A value of sinh x or cosh x is given. Use the definitions and the identity cosh2 x - sinh2 x = 1 to find the value of the other indicated hyperbolic function. 116) cosh x = 5 , x > 0, sinh x = 116) 3
A) 4
B) - 4
3
C) 16
3
9
D) 3 5
Evaluate the integral. 2
117) 1
2 dx -1 x(sec x) 1 - x2
117)
A) ln 4
C) 2 ln 4
B) 0
D) 2 1 -
Find the derivative of y with respect to the independent variable. 118) y = t5 - e 6-e B) t 6-e
A) (5 - e)t4 - e
118)
C) (4 - e)t5 - e
D) t5 - e
Find the derivative of y with respect to the appropriate variable. 119) y = (1 - 10t) coth-1 10t
A)
10 - 10 tanh-1 2 t
10t
B)
C) (1 - 10t) coth-1 10t
119) 10 - 10 coth-1 2 t
D) - 5t
19
4
10t
Evaluate the integral. ln 10 120) cosh x dx 0 A) - 89 20
120) B) 99
C) 89
20
Determine if the statement is true or false as x 121) 4x3 + cos x = O(4x2 )
D) 99
20
10
.
121)
A) True
B) False
Use l'Hopital's Rule to evaluate the limit. 2 122) lim x - 7x + 12 x-3 x 3
A) 6
122) C) 13
B) -4
D) -1
Answer the question appropriately.
123) Find the absolute minimum value of f(x) = ex - 5x on [0, 2]. A) 1 B) 5 - 5 ln 5 C) 5 - ln 5
Find a value of a so that f is continuous at c, or indicate this is impossible. 5 , x<0 2 x 124) f(x) = a, x = 0; c = 0 3x, x > 0 A) 25 B) 5 C) -5
123) D) e2 - 10
124) D) Impossible
Find the limit.
125) lim (e4/x - 7x)x/2
125)
x 0+
A) 1
C) e2
B) 0
D)
E) 2
Evaluate exactly.
126) sin cos-1
3 2
A) 1 2
126) B) -1
C) 1
2
D) - 3 2
Answer the question appropriately.
127) Find the linearization of f(x) = log 5 x at x = 5. Round the coefficients to 2 decimal places. A) L(x) = 0.32x - 0.61 B) L(x) = x - 5 C) L(x) = 0.12x + 0.38 D) L(x) = 0.12x + 1
20
127)
Evaluate the integral. e4 5 128) dt t 1
128)
A) 4
/2
129)
C) 2 - 1
B) 5 ln 4
8 cosh (sin ) cos
2e8
D) 20
2
d
129)
0
B) 4 e + 1
A) 4
e
C) 4 e - 1
D) 8(e /2 - e- /2 - 1)
e
Solve the problem. 130) A region in the first quadrant is bounded above by the curve y = cosh x, below by the curve y = sinh x, on the left by the y-axis, and on the right by the line x = 10. Find the volume of the solid generated by revolving the region about the x-axis.
A) 2
B) 10
C) 0
D)
2
e-20 + 1
Solve the differential equation. 131) dy = 8x7 y - 1 dx
131)
A) y = 1 x16 + C
B) y = 1 x8 + C
C) y = (2x8 + C) 2 + 1
D) y = (x8 + C) 2
4
2
2
+1
Find the domain and range of the inverse of the given function. 132) f(x) = 3 , x 0 x2 + 1
A) Domain and range: [0, ) C) Domain: [0, ); range: (0, 3]
132) B) Domain: (0, 3]; range: [0, ) D) Domain: (- , 0}; range: [-3, 0)
Use l'Hopital's Rule to evaluate the limit. 133) lim sin 5x x 0 sin x
A) 0
130)
133)
B) 1
C) -5
21
D) 5
Find the derivative of y with respect to x, t, or , as appropriate.
134) y = 7e (sin - cos ) A) 14e (sin - cos ) C) 7e (sin - cos ) + 7e
134) B) 0 D) 14e sin
Find the derivative of y with respect to x, t, or , as appropriate. 135) y = ln (6x) A) 1 B) 1 C) - 1 6x x 6x Rewrite the expression in terms of exponentials and simplify the results. 136) 4 cosh (ln x) + 12 sinh (ln x)
A) 8x
C) 8 x + 1 x
B) 0
135) D) - 1
x
136) D) 8x - 4 x
Express as a single logarithm and, if possible, simplify. 137) 1 ln 16t12 - ln 4 2
A) ln 4t6
137)
B) ln 4 t6 - 1
C) ln t6
D) ln 2t6
Find the value of df-1 /dx at x = f(a). 138) f(x) = 1 x + 6, a = 1 3
A) 6
Find
138) B) 1
C) 1
6
D) 3
3
dy . dx
139) e3x = sin (x + 5y) A)
139)
e3x 5 cos (x + 5y)
3x - cos (x + 5y)
B) 3e
C) ln sin (x + 5y)
D)
5 cos (x + 5y)
3e3x -1 5 cos (x + 5y)
Determine if the given function y = f(x) is a solution of the accompanying differential equation. 4x t e 140) y = 1 dt 4x t 1 1 2 x y + xy = e4x 4
A) Yes
B) No
22
140)
Find the limit. 2 141) lim 4 x - 1
141)
-1 x 1 + sec x
C) 1
B) 4
A) -4
D) 1
4
Evaluate the integral. 8 142) ( 6 + 1)x 6 dx 0
142) 6
B) 8
A) 8 6 + 1
C) x 6 + 1 + C
ln 8
D) 8 6 + 1 - 1
Evaluate the integral in terms of natural logarithms. 2/2
143)
16dx
143)
1 + 16x2
0
B) 4 ln (2 2 + 7) D) 16 ln (2 2 + 3)
A) ln 11 C) 4 ln (2 2 + 3) Evaluate the integral.
144)
2x 3+4 dx
A) 2x
3+4
ln x
144) +C
B)
C) 2x
2 +C 3+5
3+5 3+5
+C
D) 2x
3+3 3+3
+C
Find the limit.
145) lim x-2/ln x
145)
x 0+
A) 1 e
C) 1
B) -2
e2
Find the slowest growing and the fastest growing functions as x 146) y = x + 6 y = ex
D) e2
.
146)
y = x3 + cos2 x y = 7x
A) Slowest: y = ex
B) Slowest: y = x + 6
C) Slowest: y = x + 6
D) Slowest: y = x + 6
Fastest: y = x3 + cos2 x
Fastest: y = x3 + cos2 x
Fastest: y = ex
Fastest: y = 7 x
23
Determine if the statement is true or false as x 147) ex + 9x = O(ex)
.
147)
A) True
B) False
Solve the problem. 148) Locate and identify the absolute extreme values of sin (ln x) on 4, 5 A) Absolute maximum at (e /2, 1); absolute minimum at 5, sin ln 5
148)
B) Absolute maximum at (5, sin (ln 4)); absolute minimum at (4, sin (ln 4)) C) Absolute maximum at(5, sin (ln 4)); absolute minimum at (e /2, -1) D) Absolute maximum at (e /2, 1); absolute minimum at (4, sin (ln 4)) Find the domain and range of the inverse of the given function. 149) f(x) = (7x - 4)3
A) Domain and range: all real numbers C) Domain: [7, ); range: [0, )
149) B) Domain: [4, ); range: [0, ) D) Domain: [0, ); range: all real numbers
Express the value of the inverse hyperbolic function in terms of natural logarithms. 150) sech-1 12 13
A) Undefined
B) ln 18
C) ln 2
13
3
150) D) ln 3 2
Evaluate the integral.
151)
sinh 3x dx
151)
A) cosh 3x + C
B) cosh-1 3x + C
C) - 1 cosh 3x + C
D) 1 cosh 3x + C
3
152)
3
2 -(cot-1 t) dt 1 + t2
A) (cot
-1 t)3 3
152) B) 2(cot-1 t)1 + C
+C
D) -2(cot-1 t)3 + C
C) cot-1 t+ C
153)
0 -ln 3
x 5 sinh2 dx 2
A) 5 4 + ln 3 2 3
153) B) 5 (e6 - e-6 )
C) - 320
3
81
24
D) 5 4 - ln 3 2 3
Evaluate exactly.
154) cos-1 cos -
154)
3
A) 4
B) -
3
C) - 3
3
D)
3
Find a value of a so that f is continuous at c, or indicate this is impossible. x+5 , x -5 155) f(x) = 2 x + 5 a,
155)
x = -5; c = -5
A) 2
B) 5
D) Impossible
C) -5
Find the length of the curve. 156) y = ln(sec x) from x = 0 to x = /4
A) ln( 2 + 1)
156) 2+1 2
B) ln
C) ln(1 - 2)
1 2+1
D) ln
Find the derivative of y with respect to the independent variable. ln 7 157) y =log7 x + 8 x-8
A) -2
B)
x+8
C)
1 x + 8 ln 7 ln 7 x - 8
D)
157) 1
(x - 8)2
-16 (x + 8)(x - 8)
Evaluate the integral. (ln
158)
3)/4 4 e4x dx
158)
1 + e8x
0
A)
B) -
6
C)
12
Find the derivative of y with respect to the appropriate variable. 159) y = 5 sinh-1 (ln x)
A)
5
1 + (ln x)2
B)
5
C)
x 1 + (ln x)2
25
D) -
12
6
159) 5 x
(ln x)2 - 1
D)
5 1+
1 2 x
Evaluate the integral. ln 3 160) 6etcosh t dt ln 2 A) 15 + 3 ln 3 2 2
160) B) 3 + 3 ln 3
C) 15 + ln 1
2
Determine if the statement is true or false as x 161) ln x = o(ln(x2 + 7))
D) 39 + 3 ln 6
2
2
.
161)
A) True
B) False
Solve the problem. 162) The solid lies between planes perpendicular to the x-axis at x = - 3 and x =
3. The cross 4 sections are squares whose diagonals stretch from the x-axis to the curve y = 4/ 4 - x2 . Find the volume of the solid. A) 16 B) 8 C) 24 D) 32 3 3 3
Find the formula for df-1 /dx. 163) f(x) = 1 x + 5 3 6
162)
163)
A) 3x - 5
C) 1
B) 3
2
D) x - 5
3
2
Use l'Hopital's Rule to evaluate the limit. 2 164) lim x - 9 x -3 x + 3
A) 3
164)
B) 6
C) -6
D) -3
Evaluate exactly.
165) cos sin-1 -12
165)
13
A)
25 13
B) 12
D) - 25
C) -12
13
13
25
Solve the initial value problem. 166) dy = e8x cos e8x, y(0) = 0 dx
166)
A) y = 1 sin e8x - 1 sin 1
B) y = 1 sin e8x - 1
C) y = 1 sin x
D) y = - 1 sin e8x + 1 sin 1
8
8
8
8
8
8
26
8
Find the derivative of y with respect to x, t, or , as appropriate. 167) y = 8xex - 8ex
A) 8xex + 16ex
Evaluate the integral. ln 5 x 168) 8 cosh2 dx 2 0 A) 4 13 - ln 5 5
B) 8x
167) C) 8xex
D) 8ex
168) B) 17576
C) 4 12 + ln 5
125
D) 8 (e15 - e-15)
5
3
Verify the integration formula. coth-1
169)
x dx = (x - 1)coth-1
x+
x+ C
169)
A) Yes
B) No
Solve the differential equation. 2 170) dy = 6y dx x
A) y = - 6ln x + C
170) B) y =
-1 6ln x + C
C) y =
1 6ln x + C
D) y = 18ln x + C
Express as a single logarithm and, if possible, simplify. 171) ln 5x2 - 15x + ln 1 5x
171)
A) ln (x - 3)
B) ln (x - 15)
C) ln 25x2 (x - 3)
D) ln 5x2 - 15x + 1
5x
Evaluate the integral in terms of natural logarithms. 4 3
172)
dx
172)
16 + x2
0
A) ln ( 3 + 2)
B) ln
Find the derivative of y with respect to x. 173) y = sin-1 (e3t)
A)
3 e3t 1 - e9t
B)
3+2 4
C) ln ( 2 + 3)
D) ln ( 2 + 3)
173) -3 e3t 1 - e6t
C)
27
e3t 1 - e6t
D)
3 e3t 1 - e6t
Solve the problem.
174) Find the area bounded by y = A) 1 sin-1 9 7
7
3 49 - 9x2
, x = 0, y = 0, and x = 3.
B) 3 tan-1 3 7
C) sin-1 9
7
7
174) D) 1 tan-1 9 7
7
Determine if the given function y = f(x) is a solution of the accompanying differential equation. 175) Differential equation: xy + y = 2x 7
175)
Initial condition: y(7) = 2 7 x Solution candidate: y = + x 7
A) Yes Evaluate the integral. e 176) 4xln 5 - 1 dx 1 A) 4 - e ln 4 /2
177)
B) No
176) B)
4 +C 4 ln x
C) 16
ln 5
D) 16
8 cos t sin t dt
0
A) 8
/2-1
ln 8
177) C) 7
B) -7
ln 8
ln 8
D) 7
Use logarithmic differentiation to find the derivative of y with respect to the independent variable. 178) y = xln x
A) ln x 2
B) xln x - 1ln x
C) 2xln x - 1ln x
178)
D) 2 ln x x
Graph the inverse of the function plotted, on the same set of axes. Use a dashed curve for the inverse.
179)
179)
28
A)
B)
C)
D)
Solve the problem. 180) A certain radioactive isotope decays at a rate of 2% per 400 years. If t represents time in years and y represents the amount of the isotope left, use the condition that y = 0.98y0 to find the value of k
180)
in the equation y = y0 ekt.
A) -0.02020
B) 0.00666
C) 0.00005
Find the derivative of y with respect to x, t, or , as appropriate. 181) y = e(6 x + x6)
D) -0.00005
181)
A) (6 x + 6x5) ln (6 x + x6 )
B) 6 x + 6x5) e(6 x + x6 )
C) e(3 x + 6x5 )
D) 3 + 6x5 e(6 x + x6 ) x
29
Provide an appropriate response.
182) Find the inverse of the function f(x) = x + 3. How is the graph of f-1 related to the graph of f?
182)
A) f(x) = 1 + 3. The graph of f-1 is a curve intersecting the graph of f at two points equidistant x from the y-axis.
B) f(x) = -x - 3. The graph of f-1 is a line perpendicular to the graph of f at x = 3. C) f(x) = x - 3. The graph of f-1 is a line parallel to the graph of f. The graphs of f and f-1 lie on
opposite sides of the line y = x and are equidistant from that line. D) f(x) = x + 1 . The graph of f-1 is a line parallel to the graph of f. The graphs of f and f-1 lie on 3 the same side of the line y = x.
Find the inverse of the function. 183) f(x) = (x - 6)2 , x 6
183)
A) f-1 (x) = x + 6, x 0 C) f-1 (x) = x - 6, x 6
B) Not a one-to-one function D) f-1 (x) = - x + 6, x 0
Find the derivative of y with respect to x. 184) y = tan-1 (ln 4x)
A)
4
x(1 + (ln 4x)2 )
B)
184) 1
C)
1 + (ln 4x)2
Express the following logarithm as specified. 185) ln 4.5 in terms of ln 2 and ln 3 A) 2 ln 3 - 2 ln 2 B) 2 ln 3 + ln 2
1
x(1 + (ln 4x)2 )
D)
1 x 1 + (ln 4x)2
185) C) 2 ln 3 - ln 2
D) ln 3 + 2 ln 2
Solve the initial value problem. 7 186) dy = 2 + , y(0) = -4 dx 25 + x2 1 - x2
186)
A) y = 2 tan-1 x + 7 sin-1 x
B) y = 2 tan-1 x + 7 sin-1 x - 4
C) y = 2 cot-1 x - 7 sin-1 x
D) y = 2 tan-1 x - 7 sin-1 x - 4
5
5
5
5
Provide an appropriate response.
187) Consider the graph of f(x) = 4 - x2, 0 x 1. What symmetry does the graph have? Is f its own
inverse? A) The graph of f has no symmetry. The function f is not its own inverse because there is no symmetry. B) The graph of f is symmetric with respect to the y-axis. The function f is its own inverse because (f f)(x) = x. C) The graph of f is symmetric with respect to the y-axis. The function f is not its own inverse because (f f)(x) = x . D) The graph of f is symmetric with respect to the line y = x. The function f is its own inverse because (f f)(x) = x.
30
187)
Find the angle. 188) sin-1 0
188)
A)
B)
C) -
2
D) 0
2
Solve the initial value problem. 3 189) dy = , x > 1, y(2) = 4 dx x x2 - 1
189)
A) y = 3 sin-1 x + 4
B) y = 3 sec-1 x + 3
C) y = 3 csc-1 x + 2
D) y = 3 sec-1 x
3
Find the derivative of y. 190) y = sech (-5 ) (1 - ln sech(-5 )) A) -5 sech (-5 ) tanh (-5 ) ln (-5 )
190)
B) -5 sech (-5 ) tanh (-5 ) ln sech (-5 ) C) 1 + 5 sech (-5 ) tanh (-5 ) ln sech (-5 ) D) sech (-5 ) tanh (-5 ) ln sech (-5 ) Find the derivative of y with respect to x, t, or , as appropriate. 5x ln t2 dt
191) y =
191)
x
A) ln 5x2 C)
B) ln x
5x ln 5x ln x 2x 2 x
D) -ln 5
Solve the problem.
192) Find the area bounded by the x-axis, the curve y = 1 , x = 0, y = 0, and x = 3.
192)
x+2
A) ln 5
B) ln 5
C) 2 ln 5
2
Solve the differential equation. 193) dy = e4x - 4y dx
D) 1 ln 5 2
2
193)
A) y = 4ln (e4x + C)
B) y = 4e4x + C
C) y = ln (e4x + C)
D) y = 1 ln (e4x + C) 4
31
Use logarithmic differentiation to find the derivative of y with respect to the independent variable. 194) y = (9x + 1)x
A) ln (9x + 1) + 9x 9x + 1
B) x ln (9x + 1)
C) (9x + 1)x ln (9x + 1) + 1
D) (9x + 1)x ln (9x + 1) + 9x
1
9x + 1
Solve the initial value problem. 7 195) dy = , x > 3, y(6) = -2 dx x x2 - 9
195)
A) y = 7 sin-1 x - 2
B) y = 7 sec-1 x - 2 - 7
C) y = 7 sec-1 x - 2
D) y = sec-1 x - 2
3 3
3
3
3
Evaluate the integral. 2 3 196) 9x2 2 x dx 1
A) 762
194)
3
9
3
196) B) 18
C) 762
ln 2
ln 2
D) 3 + C ln x
Determine whether the integration formula is correct. 197) 1 + 3 cot-1 3 x dx = ln x + ln (4 + 9x2 ) + 3x cot-1 3 x + C x 2 2
A) No
197)
B) Yes
Solve for t.
198) e t = x8 A) (ln x16)
198) B) x16
C) 64(ln x)2
D) 8 ln x
Determine whether the integration formula is correct.
199)
2x ln(1 + x2 ) dx = (1 + x2 ) ln (1 + x2) - (1 + x2 ) + C
A) Yes
199) B) No
Answer the question appropriately.
200) Find the area of the region between the curve y = 43-x and the interval 0 x 2 on the x-axis. A) 64 ln 4
C) 60 ln 4
B) 64
32
D) 60 ln 4
200)
^
Use l'Hopital's rule to find the limit. 7 201) lim sin 0
A) -
201) C) 0
B)
Evaluate the integral. ln 2 202) ex dx ln 5 A) -3
D) 1
202) B) 7
C) 4
Express as a single logarithm and, if possible, simplify. 203) ln (8 sec ) + ln (2 cos ) A) ln (16 cot )
D) 3
203) B) ln (16) D) ln 4
C) ln (8 sec + 2 cos )
Answer the question appropriately. 204) Find a curve through the origin in the xy-plane whose length from x=0 to x=1 is 1 1 x L= 1+ e dx. 16 0 A) y = 1 ex/2 - 1 B) y = x2 C) y = ex -1 D) y = 1 ex/2 2 2 2 Evaluate the integral. log5 x
205)
x
dx
204)
205) B) ln x + C
A) 5x ln 5 + C
ln 5
2
2
C) ln 5 (ln x) + C
D) (ln x) + C
2
2 ln 5
^
Use l'Hopital's rule to find the limit. 206) lim x2 + 8x - x x
A) 4
206) B) - 4
Determine if the statement is true or false as x 207) 4 + cos x = O(4) A) True
C) 8
D) 0
.
207) B) False
33
Determine if the given function y = f(x) is a solution of the accompanying differential equation. 208) y + y = e2x
208)
1 y = 4e- x + e2x 3
A) Yes
B) No
Solve the problem. 209) Find the area bounded by xy = 7, x = 1, x = 5, and y = 0. A) 7 ln 1 B) ln 5 C) 7 ln 1 2 5 5 Verify the integration formula. 7 csch 7x dx = ln tanh
210)
209) D) 7 ln 5
7 x +C 2
210)
A) No
B) Yes
Evaluate the integral. 1/3 5x dx
211)
0
211)
36 - x4
A) 5 sin-1 1 2
54
B)
C)
8
Evaluate the integral in terms of natural logarithms. 1/2 dx 212) 1 - x2 0
A) ln 5 Find the value of df-1 /dx at x = f(a). 213) f(x) = x2 - 6x + 4; a = 4
A) 1 8
D)
2
212)
B) ln 0
C) 1 ln 1
B) 1 2
C) 1 14
2
3
D) 1 ln 3 2
213) D) 2
Determine if the given function y = f(x) is a solution of the accompanying differential equation. 214) Differential equation: 8xy + 8y = cos x Initial condition: y( ) = 0 8sin x Solution candidate: y = x
A) Yes
B) No
34
214)
Evaluate the integral.
215)
sech2 (10x - 4) dx
215)
A) 10 sech3 (10x - 4) + C
B) 1 coth (10x - 4) + C
C) tanh (10x - 4) + C
D) 1 tanh (10x - 4) + C
3
10 10
Solve the initial value problem. 216) dy = et sin (et - 6), y(ln 6) = 0 dt
216)
A) y = cos (et - 6) - 1 C) y = et cos (et - 6) - 6
B) y = sin et - sin 2 D) y = -cos (et - 6) + 1
L'Hopital's rule does not help with the given limit. Find the limit some other way. 217) lim csc cot /2-
A) 1
B) -1
217) D) 0
C)
Determine if the given function y = f(x) is a solution of the accompanying differential equation. 218) y = - y2
218)
1 y= x+5
A) Yes
B) No
Find the derivative of y with respect to the appropriate variable. 219) y = sinh-1 5x 1 A) 2 5x(1 + 5x)
5 B) 2 5x(5x - 1)
219)
5 C) 2 5x(1 + 5x)
Evaluate the integral. dx 220) -x2 - 8x - 7
D)
1 1 + 5x
220)
A) -sin-1 x + 4 + C
B) sin-1 x + 4 + C
C) cos-1 x + 4 + C
D) 1
3
3
3
2
-x2 - 8x - 7+ C
Determine whether the integration formula is correct.
221)
2 2 (sin-1 x) - 1 dx = x(sin-1 x) - 3x + 2 1 - x2 sin-1 x + C
A) Yes
B) No
35
221)
Find the derivative of y with respect to the appropriate variable. 222) y = 8 ln x + 1 - x2 sech-1 x -1 A) 8 - sech x x 1 - x2
C) 7 - x sech x
222)
B) 7 x
-1 x
x sech-1 x 1 - x2
D) 8 ln x - sech-1 x
2 1 - x2
^
Use l'Hopital's rule to find the limit. 223) lim sin 2x x 0 5x
A) 1
223) B) 2
5
C) 1
5
D) 0
Is the function graphed below one-to-one?
224)
224)
A) Yes Evaluate the integral. 225) -ecos-1 x
B) No
dx
225)
1 - x2
A) -esin-1 x + C
B) -ecos-1 x + C
C) ecos-1 x + C
D) e
sin-1 x
sin-1 x
+C
Rewrite the expression in terms of exponentials and simplify the results. 226) (sinh x + cosh x)2
A) ex2
B) e2x
C) e2x - e-2x
36
226) 2x
D) e
4
Answer the question appropriately. 227) Find the linearization of f(x) = sin x at x = 2. Round the coefficients to 2 decimal places. A) L(x) = -0.42x +1.74 B) L(x) = 0.42x + 0.49
C) L(x) = cos x
227)
D) L(x) = -0.42x + 0.07
Provide an appropriate response. 228) Find the average value of f(x) = 1/x over [5, 9] A) ln 9 - ln 5 B) ln 9 + ln 5 4 4
228) C) 4 45
D) ln 5 - ln 9 4
Is the function graphed below one-to-one?
229)
229)
A) Yes
B) No
Solve the differential equation. 230) dy = 8 cos x sec y dx
230)
A) y = sin (8 sin x + C) C) y = sin-1 (8 cos x + C)
B) y = 8 sin x + C D) y = sin-1 (8 sin x + C)
Evaluate the integral in terms of natural logarithms. 6 dx 231) 2 3 x x +9 A) 1 ln 2(1 + 2) 3 1+ 5
231) B) - 1 ln 1 + 2 3
1+
5
D) 1 ln 1 + 2
C) -ln 1 + 5 2
3
37
1+
3
Is the function graphed below one-to-one?
232)
232)
A) Yes
B) No
Verify the integration formula.
233)
7 tanh 7x dx = ln cosh-1 7x + C
233)
A) No
B) Yes
Find the inverse of the function. 234) f(x) = 4x + 1 A) f-1 (x) = x + 1 4
234) B) f-1 (x) = x - 1 4 D) f-1 (x) = x - 1
C) Not a one-to-one function
4
Determine whether the integration formula is correct. 8 235) x7tan-1 x dx = 1 x8 tan-1 x - x dx + C 8 1 + x2
235)
A) Yes
B) No
Evaluate the integral.
236)
2 sinh (4x- ln 7) dx
236)
A) 8 cosh (4x - ln 7) + C
B) 1 cosh (4x - ln 7) + C
C) 1 cosh 4x + C
D) 2 cosh (4x - ln 7) + C
2
14
38
Find
dy . dx
237) tan y = ex + ln 5x x A) xe + 1 x sec2 y
237) B) ex + 5 - sec2 y x
Determine if the statement is true or false as x 238) ln x = o(ln 10x) A) True Find the angle. 239) sec-1
A) C)
x C) xe + 5 x sec2 y
x D) e + 5 sin2 y
.
238) B) False
2
239) B)
4
D)
4
4
± 2 n,
4
±2 n
4
Find the derivative of y with respect to the independent variable.
240) y = (ln 2 ) A)
(ln 2 ) -1
240) B)
(ln 2 ) -1
2
C) (ln 2 ) -1
Evaluate the integral. 1/2 -1 10 dx 241) esin x 1 - x2 0
A) 10e /6 - 10
241) B) e1/2 - 1
C) 10e /6
Rewrite the expression in terms of exponentials and simplify the results. 242) 4 cosh (ln x)
A) 2(ex+ e-x)
D) (2 ) ln
B) 2x
C) 0
D) 5 3
242) D) 2 x + 1 x
A value of sinh x or cosh x is given. Use the definitions and the identity cosh2 x - sinh2 x = 1 to find the value of the other indicated hyperbolic function. 243) cosh x = 17 , x < 0, sech x = 243) 8
A) 15 17
B) - 8
C) - 289
15
64
D) 8
17
Solve for t.
244) et/970 = k A) 970ek
244) B) ln k 970
C) ln 970k
39
D) 970 ln k
Determine if the given function y = f(x) is a solution of the accompanying differential equation. 245) Differential equation: y = e- 7x - 7y
245)
Initial condition: y(- 7) = 0
Solution candidate: y = (x + 7)e- 7x A) Yes
B) No
Solve the initial value problem. 8 246) dy = + x2 , x >1, y(2) = -5 dx x x2 - 1
246)
3
A) y = 8 sec-1 x + x - 31 3
B) y = 8 sin-1 x + x
3
3
3
C) y = 8 csc-1 x + x - 31 3
Find the limit.
247) lim tan x 0
3
D) y = 8 sec-1 x + 2x - 35 3
6
-1 7x2
247)
2x2
C) 7
B) 1
A) -7
D) 1
2
Find the length of the curve. 248) y = ln(sin x) from x = /6 to x = /4 A) ln( 2 - 1) B) ln 2 - 3 2-1
2
248) C) ln
2-1 2- 3
D) ln(2 - 3)
Provide an appropriate response. 249) If f(x) is one-to-one, is g(x) = f(-x) also one-to-one? Explain. A) g(x) is a reflection of f(x) across the line y = x. It will not be one-to-one.
249)
B) g(x) is a reflection of f(x) across the y-axis. It will be one-to-one. C) g(x) is a reflection of f(x) across the x-axis. It will be one-to-one. D) There is not enough information to determine whether g(x) is one-to-one. Find the derivative of y with respect to x, t, or , as appropriate. 250) y = ln 1 - x (x + 2)4 4
A) (x + 2) 1-x
B) 3x - 6
250) C) ln 5x - 6
(x + 2)5
(x + 2)5
40
D)
3x - 6 (x + 2)(1 - x)
A value of sinh x or cosh x is given. Use the definitions and the identity cosh2 x - sinh2 x = 1 to find the value of the other indicated hyperbolic function. 251) sinh x = - 12 , csch x = 251) 5
A) 13
B) 169
5
C) - 5
25
12
D) 5
12
Solve the problem.
252) Find the length of the curve y = 4 - x2 between x = 0 and x = 1. A) 1 6
B) 1 3
C) 1 2
252) D) 2 3
253) A region in the first quadrant is bounded above by the curve y = tanh x, below by the x-axis, on
253)
the left by the y-axis, and on the right by the line x = ln 7. Find the volume of the solid generated by revolving the region about the x-axis. A) 2 B) - 24 C) ln 7 - 24 D) 0 25 25
Use l'Hopital's Rule to evaluate the limit. 2 254) lim 13x - 3x + 4 x 7x2 + 4x + 15
A) - 13 7
254)
B) 13
C) 1
7
D) 7
13
Evaluate the integral. cos x dx 255) 1 + 7 sin x
255)
A) ln 1 + 7 sin x + C
B) 7 sin x + C
C) 7 ln 1 + 7 sin x + C
D) 1 ln 1 + 7 sin x + C 7
Express the following logarithm as specified. 256) ln 3 3 in terms of ln 2 and ln 3 A) 3 ln 3 B) ln 2 + ln 3 2 2
256) C) ln 3 2
D) ln 3 + ln 3 2
^
Use l'Hopital's rule to find the limit. 257) lim x sin 14 x x
A) 1
14
257) B) 1
C) 14
41
D) 0
258) lim sin 5x
258)
x 0 tan 3x
A) 5 3
B) - 5
C) 3
3
D) 0
5
Find the limit.
259)
lim sin-1 x x -1 +
A) -1
259) B) 1
C)
D) -
2
2
Evaluate the integral.
260)
coth (6x) dx
260)
A) 1 ln sinh 6x + C
B) 6 ln sinh x + C
C) 1 csch2 6x + C
D) ln sinh 6x + C
6
6
6
Solve the problem. 261) A loaf of bread is removed from an oven at 350° F and cooled in a room whose temperature is 70° F. If the bread cools to 210° F in 20 minutes, how much longer will it take the bread to cool to 145° F. A) 26 min B) 18 min C) 19 min D) 38 min Find the slowest growing and the fastest growing functions as x 262) y = 5x10 y = ex y = ex-6 y = xex
A) Slowest: y = xex Fastest: y = ex
B) Slowest: y = 5x10
Fastest: y = ex and y = ex-6 grow at the same rate. C) Slowest: y = ex-6
Fastest: y = xex D) Slowest: y = 5x10 Fastest: y = xex
42
261)
.
262)
Solve the initial value problem. 263) dy = e-t sec2 ( e-t ), y(-ln 8) = 2 dt
263)
-t A) y = -tan ( e ) + 2
-t -t B) y = -e cot ( e ) + 1
-t C) y = tan ( e ) + 10
D) y = cot ( e-t ) + 2
Evaluate the integral. 3/2
264)
7 dt
264)
24 - 2t - t2
-1
A)
2 2
D) 7
B) ln 9
C) ln 9 2
9 D) e 2
B) y
x+y - y C) xye x - xyex+y
x+y D) e e2x
B) 7
6
C)
3
6
Solve for t.
265) e2t = 9
265)
A) ln 3
Find
dy . dx
266) ln 2xy = ex+y
x+y A) 2xye x+y
266) x
Find the derivative of y with respect to x, t, or , as appropriate. 267) y = ln 1 + x x5
A) -10 - 9 x 2x
B) 10 - 9 x 2x(1 +
C) -10 - 9 x
x)
2x(1 +
Evaluate the integral. 3 2 x +1 268) dx 3 + 3x x 2 A) 1 ln 2 3 3
x)
D) -10 - 9 x 2(1 +
x)
268) B) 1 ln 2.53327479e+15 3
C) 2 ln 3 3
267)
9.851624185e+14
D) 1 ln 9.851624185e+14
2
3
43
2.322168558e+15
A value of sinh x or cosh x is given. Use the definitions and the identity cosh2 x - sinh2 x = 1 to find the value of the other indicated hyperbolic function. 269) cosh x = 13 , x > 0, tanh x = 269) 12
A) 13
B) 5
5
C) 5
13
D) - 25
12
Find the inverse of the function. 270) f(x) = x + 8 A) f-1 (x) = x - 8
144
270) B) f-1 (x) = x2 - 8, x 0 D) f-1 (x) = (x + 8)2
C) Not a one-to-one function
Find the derivative of y with respect to the independent variable. 271) y = 3 ln 6t
A) 6 ln 3 3ln 6t t
B) ln 3 3 ln 6t t
271)
C) 6 ln 3 t
D) 3ln 6t
Find the derivative of y with respect to the appropriate variable. 272) y = ( 2 + 9 ) tanh-1 ( + 8)
A) -
272)
B) (2 + 9) -
+7
C) (2 + 9) tanh-1 ( + 8) -
1 + 63
D) (2 + 9) tanh-1 ( + 8) -
+7
2+9
1 + ( + 8)2
L'Hopital's rule does not help with the given limit. Find the limit some other way. 273) lim x cot x x 0 cos x
A) 1
C) 0
B) -1
273) D)
Solve the problem.
274) Find the length of the segment of the curve y = 1 cosh 2x from x=0 to x = ln 2.
274)
2
A) 1 4
2-
1 2
C) 3
B) 2
8
D) 5 8
Express the following logarithm as specified.
275) ln 8 in terms of ln 2 A) 6 ln 2
275) B) 2 ln 2
C) 3 ln 2
3
2
44
D) 1 ln 8 2
Find the derivative of y with respect to the independent variable. 276) y = 7 t
B) ln 7
A) 7 t ln 7
t
276)
C) ln 7 7 t
2 t
D) 1 7 t
2 t
2 t
Use logarithmic differentiation to find the derivative of y. 277) y = 5 x(x - 1) x3 + 5
277)
A) 1 (ln x + ln(x - 1) - ln(x3 + 5))
2 B) 1 5 x(x - 1) 1 + 1 - 3x
2 C) 1 + 1 - 3x
2 D) 5 5 x(x - 1) 1 + 1 - 3x
5
x
x-1
5
x3 + 5
x3 + 5 x
x3 + 5 x
x-1
x-1
x3 + 5
x3 + 5
Evaluate the integral. 3/2 (sin-1 x)6
278)
1 - x2
0
A) ln
ln
279)
dx
7 6
278)
B)
7
C)
2187
7
D)
15,309
7 1,959,552
2 2 2x ex sin ex dx
279)
0
A) 1 - cos 1
B) 1 + cos 1
D) 1
C) -1
Determine if the given function y = f(x) is a solution of the accompanying differential equation. 280) y - y = 2ex y = e2x + 2xex A) Yes
280)
B) No
Find the derivative of y with respect to the appropriate variable. 281) y = 2 tanh-1 (cos x)
281)
A) -2 sin x 1+ cos2 x
B) ln
C) -2
D) -2
cos x
sin x
45
1 1 - x2
sin x
Express the following logarithm as specified. 282) ln 35 + ln (1/7) in terms of ln 5 and ln 7 ln 125
A) ln 7
282)
B) ln 5
3
D) 1
C) 1
2
3
Use logarithmic differentiation to find the derivative of y. x 283) y = x-6
283)
A) - 3
x x-6
B) 1 (lnx - ln(x - 6))
C) 1
x 1 1 x-6 x x-6
D) 1 1 - 1
2
2
2 x
x-6
Evaluate the integral. csch (ln x) coth (ln x) 284) dx 7x
284)
A) 7 sech (ln x) + C
B) 1 csch (ln x) + C
C) x csch (ln x) + C
D) -1 csch (ln x) + C
7
7
^
Use l'Hopital's rule to find the limit. 2 285) lim 6x - 5x + 1 x 4x2 + 3x - 8
A) 6
285) B) 1
C)
Find the derivative of y with respect to x, t, or , as appropriate. 286) y = sin e- 5 A) 5 4 cos e- 5 B) cos (-5 4 e- 5 ) C) (-5 4 e- 5 ) cos e- 5 D) cos e- 5 Determine if the statement is true or false as x 287) ex = o(e3x)
D) 3 2
286)
.
287)
A) True
B) False
46
Evaluate the integral.
288)
- 2/3 -2/3
A)
dt t 9t2 - 1
288)
B) -
6
C) -
6
Find the derivative of y with respect to the appropriate variable. 289) y = (8 - 8 ) tanh-1
A) -8 1+ C)
8 1+
B) - 8 tanh-1
D)
D)
12
12
289) 8+8
- 8 tanh -1
8 1-
- 8 tanh-1
1+ 2
Use logarithmic differentiation to find the derivative of y. 290) y = x sin x x+3
290)
A) x sin x ln x + ln sin x - 1 ln(x + 3)
B) 1 + cot x -
C) x sin x 1 + cot x -
D) 1 1 + 1 + 1
x+3
x+3 x
2
x
1 2x + 6
2 x
1 2x + 6
sin x
x+3
Answer the question appropriately.
291) Find the linearization of f(x) = 7 x at x = 1. Round the coefficients to 2 decimal places. A) f (x) = 3.60x + 3.40 B) f (x) = 13.62x - 6.62 C) f (x) = 1.95x + 5.05 D) f (x) = 13.62x + 1
Solve the initial value problem. 2 292) d y = -6e-x, y(0) = -6, y (0) = 0 dx2
291)
292)
A) y = -6e-x + 6x - 12 C) y = 6e-x + C
B) y = -6e-x - 6 D) y = -6e-x - 6x + 0
Evaluate exactly.
293) tan(sec-1 1) + cos(tan-1 (- 3)) A) -1 2
293) C) - 3
B) 1 2
2
47
D) Undefined
Solve the problem.
294) Consider the area of the region in the first quadrant enclosed by the curve y = 1 cosh 6x, the 6
294)
coordinate axes, and the line x = 10. This area is the same as the area of a rectangle of a length s, where s is the length of the curve from x = 0 to x = 10. What is the height of the rectangle? A) sinh 60 B) 1 sinh 60 C) 6 D) 1 36 6
Express as a single logarithm and, if possible, simplify. 295) ln (8x + 4) - 2 ln 2 A) ln (16(2x + 1)) B) ln (2x + 2)
295) C) ln (2x + 1)
D) ln (8x)
A value of sinh x or cosh x is given. Use the definitions and the identity cosh2 x - sinh2 x = 1 to find the value of the other indicated hyperbolic function. 296) cosh x = 17 , x < 0, csch x = 296) 15
A) - 8
B) - 15
15
C) - 15
17
Solve the problem.
297) Find the average value of the function y = A) 1
8
9 25 - 9x2
B) 1
5
1 1 - x2
4x3
C)
1 - x2
Evaluate the integral. 5e(5 sin 3x)
299)
sec 3x
5 . 6
297)
D) 1
5
Find the derivative of y with respect to x. 298) y = 4x3 sin-1 x
A)
8
over the interval from x = 0 to x =
C) 3
3
D) 15
2
298) 4x3
+ 12x2
B)
+ 12x2 sin-1 x
3 D) 4x + 12x2 sin-1 x
1 - x2 1 + x2
dx
299)
A) 1 ln sec 3x + C
B) 5 ln sec 3x + C
C) 1 e(5 sin 3x) + C
D) e(5 sin 3x) + C
3 3
48
Find
dy . dx
300) 4x + y = yln 4
300)
1 ln 4 -1 y
B)
C) ln 4 ln y
D)
A)
xy
Find the derivative of y with respect to x. 301) y = tan-1 3x
A)
1 1 - 3x
4x + y
-4 x + y + yln 4 - 1 4x + y x y + 4 + ln 4 yln 4 - 1
301)
1 B) 6 3x(1 + 3x)
3 C) 2(1 + 3x) 3x
D) 1 1 + 3x
Find the limit.
302) lim sin x 0
-1 9x
302)
x
A) 1
C) 1
B) 9
9
D)
Find the domain and range of the inverse of the given function. 303) f(x) = x - 2 A) Domain and range: all real numbers B) Domain: [2, ); range: [2, )
C) Domain: [0, ); range: [2, )
D) Domain: [2, ); range: [0, )
304) f(x) = x4 + 6, x 0 A) Domain: (- , 0]; range: (- , 6] C) Domain and range: all real numbers
B) Domain: [0, ); range: [6, ) D) Domain: [6, ); range: [0, )
303)
304)
Solve the problem. 305) The solid lies between planes perpendicular to the x-axis at x = -2 and x = 2. The cross sections perpendicular to the x-axis are circles whose diameters stretch from the curve y = - 5/ 4 + x2 to the curve y = 5/ 4 + x2 . Find the volume of the solid. A) 25 2 B) 25 C) 25 2 4
Determine whether the integration formula is correct. -1 -1 306) 3tan x dx = 3 ln x - 3 ln (1 + x2 ) - 3tan x + C 2 2 x x
A) No
D) 5 2 4
306) B) Yes
49
305)
Evaluate the integral.
307)
2 (sin-1 x) dx 1 - x2
307)
A) ln (sin x) + C
-1 x)3
C) (sin
B) 2(sin-1 x)1 + C
1 - x2
3
D) (cos-1 x)3 + C
+C
Evaluate exactly.
308) sec cos-1
3 2
308)
A) 2 3
C) - 3
B) -1
3
D) - 2 3
2
3
Express the value of the inverse hyperbolic function in terms of natural logarithms. 309) coth-1 11 10
A) 1 ln -21
C) 1 ln 231
B) 0
2
309) D) 1 ln 21
2
2
Express as a single logarithm and, if possible, simplify. 310) ln (x2 - 4) - ln (x + 2)
310)
B) ln (x2 - 2)
A) ln (x + 2)
C) ln (x - 2)
D) ln (x - 4)
Evaluate the integral. 311) 3 + 16x dx 4 + 9x2
311)
A) 1 tan-1 3 x + 1 sin-1 3 x + C
B) sin-1 3 x + 8 ln 4 + 9x2 + C
C) 18x + 8 ln 4 + 9x2 + C
D) 1 tan-1 3 x + 8 ln 4 + 9x2 + C
2
2
2
2
2
9
2
9
2
9
Find the derivative of y with respect to the independent variable. 312) y =log6 sin cos e 8
312)
A) e6 (cos - sin - e 8 )
B) 1 (sec csc - ln 8 -1)
C) 1 (cot - tan - ln 8 -1)
D) 1
ln 6
e 8 ln 6 sin cos
ln 6
50
Evaluate the integral. 3 /4 x 313) tan dx 3 0 A) -3 2 2
313) C) 3 2
B) 3 ln 2 2
D) -3 ln 2 2
2
Solve the problem. 314) The charcoal from a tree killed in a volcanic eruption contained 63.3% of the carbon-14 found in living matter. How old is the tree, to the nearest year? Use 5700 years for the half-life of carbon-14. A) 3760 years B) 2607 years C) 1807 years D) 5700 years Determine whether the integration formula is correct. 2 2 315) 25x sin-1 5x dx = 50x -1 sin-1 5x - 5x 1 - 25x + C 4 4
A) No
315)
B) Yes
Solve the initial value problem. 3 316) dy = 3 , y(0) = -1 dx 1 + x2 1 - x2
316)
A) y = 3 tan-1 x - 3 sin-1 x - 1 C) y = 3 tan-1 x - 3 sin-1 x + 1 Evaluate the integral.
317)
9 cosh
B) y = 3 tan-1 x - 1 D) y = 3 cot-1 x - 3 sin-1 x
x - ln 3 dx 2
317)
A) 9 sinh x + C
B) 18 sinh x - ln 3 + C
C) 9 sinh x - ln 3 + C
D) 6 sinh x - ln 3 + C
2
2
2
2
2
Use l'Hopital's Rule to evaluate the limit. 318) lim cos 7x - 1 x 0 x2
A) 0
314)
318)
B) - 49
C) 49
2
2
D) 7 2
Solve the problem. 319) In a chemical reaction, the rate at which the amount of a reactant changes with time is dy proportional to the amount present, such that = -0.7y, when t is measured in hours. If there dt are 77 g of reactant present when t = 0, how many grams will be left after 4 hours? Give your answer to the nearest tenth of a gram. A) 4.7 g B) 0.1 g C) 7 g D) 2.3 g
51
319)
Verify the integration formula.
320)
x2 1 x csch-1 x dx = csch-1 x + 2 2
1 + x2 + C
320)
A) No 321)
B) Yes
3 sech x dx = tan-1 (sinh 3x) + C
321)
A) Yes
B) No
Evaluate the integral. 4 dt 322) 2 2 t - 4t + 0
A) Undefined
322)
B)
C)
12
10
+
3 2
D)
6
Use l'Hopital's Rule to evaluate the limit. 2 323) lim x + 7x + 13 x x3 + 8x2 + 5
323)
B) 1
A)
C) 0
D) -1
Rewrite the expression in terms of exponentials and simplify the results. 324) sinh (2 ln 7x) A) 14x B) 1 49x2 - 1 2 49x2
D) 7 x - 1
C) 1 49x2 + 1 2
49x2
Evaluate the integral. /12 sec2 3x 325) dx 4 + tan 3x 0 A) 1 ln 1 3 4
324)
2
x
325) C) ln 5
B) e5/4
4
52
D) 1 ln 5 3
4
Solve the initial value problem. 3 5 326) dy = , y(2) = -1 + dx 2 8-x x x2 - 1
326)
2x + 5 sec-1 x + -1 4
A) y = sin-1
B) y = sin-1 x + 5 sin-1 x 8
C) y = sin-1
2x 5 + 5 sec-1 x + -1 4 3
D) y = sin-1
2x + sec-1 x + -3 4
-
3 4
Find the angle. 327) tan-1 -1
327)
A) 1
B)
D) 4
C) 0
4
Solve the problem. 328) Find the equation that satisfies the following conditions: dy 1 = 5 + , y(1) = 13 dx x
A) y = 5x + ln x + 7 C) y = x + ln x + 12
B) y = ln x + 13 D) y = 5x + ln x + 8
Find the derivative of y with respect to x, t, or , as appropriate. 329) y = ln (5 e- )
A) ln (5e- (1- ))
328)
B)
1
329) C)
5 e
1
-1
D) e
1
+1
Rewrite the expression in terms of exponentials and simplify the results. 330) sinh 4x - cosh 4x 2
B) e-4x2
A) e-8x
330)
C) e8x - e-8x
D) -e-8x
Evaluate the integral.
331)
x csch2 1 dx 10
331)
A) 10 tanh 1 - x + C
B) 10 coth 1 - x + C
C) 10 csch3 1 - x + C
D) -coth 1 - x + C
10
3
10
10
10
53
/16
332)
16 tan 4x dx
332)
0
A) 2 ln 3
B) 2 ln 2
C) 4 ln 2
D) -2 ln 2
Provide an appropriate response. 333) If f(x) is one-to-one, can anything be said about h(x) = 3f(x) + 2? Is it also one-to-one? Give reasons for your answer. A) Yes, h(x) will be one-to-one. For every distinct value of f(x) there is one distinct value of h(x). B) No, h(x) will not be one-to-one. The function h(x) does not pass the horizontal line test.
333)
C) No, h(x) will not be one-to-one. The function h(x) assumes the same value for at least two different f(x)-values.
D) Yes, h(x) will be one-to-one. The inverse of f(x) is h(x) and is therefore one-to-one. 334) Given that x > 0, find the maximum value, if any, of x1/x3 . A) 1 B) e1/(3e) C) Rewrite the expression in terms of exponentials and simplify the results. 335) sinh (6 ln x) A) 1 x6 + 1 B) 3 x - 1 C) 1 x6 - 1 2 2 x x6 x6
334) D) e3e
335) D) 2x
Graph the inverse of the function plotted, on the same set of axes. Use a dashed curve for the inverse.
336)
336)
54
A)
B)
C)
D)
Find the derivative of y. 337) y = ln(sech (4x + 10)) 4 A) sech (4x + 10)
337) B) -4 tanh (4x + 10)
C) -tanh (4x + 10)
D) tanh (4x + 10)
Find the angle.
338) sin-1 A)
2 2
4
338) B)
C)
4
3
D)
3
Solve for t.
339) 100e3t = 300 A) ln 3 3
339) 3 C) e 3
B) ln 200 3
55
D) ln 1
Determine if the given function y = f(x) is a solution of the accompanying differential equation. 340) y + y = e2x y = 7e- x + e2x A) Yes
340)
B) No
Solve the problem. 341) The velocity of a body of mass m falling from rest under the action of gravity is given by the mg gk equation v = tanh t , where k is a constant that depends on the body's aerodynamic k m
341)
properties and the density of the air, g is the gravitational constant, and t is the number of seconds lim into the fall. Find the limiting velocity, v, of a 220 lb. skydiver (mg = 220) when k = 0.006. t
A) 0.01 ft/sec C) There is no limiting speed.
B) 191.49 ft/sec D) 60.55 ft/sec
^
Use l'Hopital's rule to find the limit. 342) lim 7 - 7cos 0 sin 3
A) 7
342) B) 1
3
C) 0
D)
Use logarithmic differentiation to find the derivative of y with respect to the independent variable. 343) y = x9 sin x
A) 9 cos x ln x + sin x
B) 9 x9 sin x cos x ln x + sin x
C) 9 sin x ln x
D) x sin x cos x ln x + sin x
x
343)
x
x
Find the derivative of y with respect to the appropriate variable. 344) y = sinh-1 (cos x)
A) - sin x
B)
1 + x2
1
344)
C) - sin x
1 + cos2 x
D)
- sin x 1 + cos2 x
Use logarithmic differentiation to find the derivative of y with respect to the independent variable. 345) y = (x + 2)x
A) x + (2)x-1
345)
B) x ln(x + 2)
C) ln(x + 2) + x
D) (x + 2)x ln(x + 2) + x
x+2
x+2
Solve the problem. 346) Suppose that the amount of oil pumped from a well decreases at the continuous rate of 11% per year. When, to the nearest year, will the well's output fall to one-eighth of its present value? A) 2 years B) 28 years C) 13 years D) 19 years
56
346)
Find the angle.
347) cos-1
3 2
347)
A) 11
B)
6
C)
6
4
D)
4
Find the derivative of y with respect to x, t, or , as appropriate. 348) y = x4ln (x) - 1 x3 3
A) 4x3 - x2 C) 5x3 - x2
348) B) x3 - x2 + 4x3ln x D) x4 ln x - x2 + 4x3
Evaluate exactly.
349) csc(tan-1 1 + csc-1 1) A) - 2
349) C) 2
B) 0
D)
2 2
Find the derivative of y with respect to x, t, or , as appropriate. e2x
350) y =
ln t dt
350)
e10 x A) 4e2x - 10 xe10 x
B) ln t
C) e2x ( 2x - 1) - e10 x (10 x - 1)
D) 4xe2x - 50e10 x
Rewrite the expression in terms of exponentials and simplify the results. 351) ln cosh 4x - sinh 4x + ln (cosh 9x + sinh 9x)
A) 5
B) 13x
C) 5x
Solve the initial value problem. 352) dy = -7e-x sec e-x tan e-x, y(0) = 7 sec 1 + 6 dx
351) D) ln (e9x - e-4x)
352)
A) y = 7 tan e-x + 6 C) y = -7 sec x + 1
B) y = 7 sec e-x + 6 D) y = -7 sec e-x + 1
Verify the integration formula.
353)
tanh-1 x dx = x tanh-1 x +
1 ln (1 - x2 )+ C 2
353)
A) Yes
B) No
57
Provide an appropriate response. 354) Consider a linear function that is perpendicular to the line y = x. Will this function be its own inverse? Explain. A) Yes it will be its own inverse. All perpendicular lines are their own inverses.
354)
B) Yes it will be its own inverse. If it is perpendicular to y = x it is symmetric with respect to y
= x. Therefore it is its own inverse. C) No it won't be its own inverse. The slope will be the same but the y-intercept will be different. D) No it won't be its own inverse. Its inverse will be some other line that is perpendicular to it.
Find a value of a so that f is continuous at c, or indicate this is impossible. x2 - 4, x<0 355) f(x) = a, x = 0; c = 0
355)
3(x - 4) + 8, x > 0
A) 3 356) f(x) =
B) 4
C) -4
D) Impossible
-2x - 1, x < 0 a, x = 0; c = 0 4x - 2, x > 0
A) 2
356) C) Impossible
B) -2
D) -1
Provide an appropriate response. 357) Find the derivative of the inverse of the function f(x) = mx, where m is a nonzero constant. 2 A) 1 B) mx C) m D) 1 m 2 L'Hopital's rule does not help with the given limit. Find the limit some other way. 358) lim 16x + 1 x+5 x
A) 4 Find the value of df-1 /dx at x = f(a). 359) f(x) = 5x2, x 0, a = 5
A) 50 Evaluate the integral. 5 /3 t 360) 2 cot dt 5 5 /6 A) 10 ln 3
357)
358)
B) 16
C) 0
D)
B) 3 625
C) 1 10
D) 1 50
359)
360) B) -5 ln 3
C) 5 ln 3
58
D) -10 ln 3
Find the slowest growing and the fastest growing functions as x 361) y = x2 + 3x
.
361)
y = x2
x4 + x 2 y = 4x2 y=
A) Slowest: y = x4 + x2
Fastest: y = x2 + 3x B) Slowest: y = x2 and y = 4x2 grow at the same rate. Fastest: y = x4 + x2 C) They all grow at the same rate.
D) Slowest: y = x4 + x2 Fastest: y = 4x2
A value of sinh x or cosh x is given. Use the definitions and the identity cosh2 x - sinh2 x = 1 to find the value of the other indicated hyperbolic function. 362) cosh x = 13 , x < 0, coth x = 362) 5
A) - 5
B) - 12
13
C) 12
5
D) - 13
13
Find the inverse of the function. 363) f(x) = x + 2, x -2 A) f-1 (x) = x2 - 2, x 0
12
363) B) f-1 (x) = -x2 + 2, x 0 D) f-1 (x) = x2 - 4, x 0
C) Not a one-to-one function
Find the derivative of y with respect to x, t, or , as appropriate. 364) y = ln (cos (ln )) A) tan (ln ) B) - tan (ln ) C) tan (ln )
364) D) -
Evaluate the integral. 9 dx
365)
365)
16 - 81x2
A) 1 sin-1 9 x + C
B) 1 tan-1 9 x + C
C) tan-1 9 x + C
D) sin-1 9 x + C
4
4
4
4
366)
tanh
tan (ln )
4
4
x dx 10
366)
A) ln coth x + C
B) 10 ln cosh x + C
C) 10 ln sinh x + C
D) 10 sech2 x + C
10
10
10
10
59
Find the derivative of y with respect to the independent variable. 367) y = 2 cos
A) -2 cos
2 cos
C) Find
367)
ln 2 sin
B) 2cos
ln 2
D) - 2cos
ln 2 sin
dy . dx
368) sin y = 2x + 3y
368)
1 A) sin y - 3
B) 2 + 3 cos y
C) 2 + 3 - cos y
2 D) cos y - 3
Use logarithmic differentiation to find the derivative of y. 4 369) y = sin x cos x sec x x1/3
369)
4
A) sin x cos x sec x cot x - 3tan x - 1 x1/3
3x
4
B) sin x cos x sec x cotx - 4tan x + x1/3
1 1 sin x cos x 3x
4
C) sin x cos x sec x cot x + 5tan x - 1 x1/3
3x
4
D) sin x cos x sec x 2cot x - 3tan x - 1 x1/3
3x
Find the limit.
370) lim (ln x)4/x
370)
x
A) 4
C) e4
B) 1
D) 0
Find the derivative of y with respect to the independent variable. 2 371) y =log3 x 4 x+1
B) e3 4 x+1
A) 1
2 1 ln 3 x 2(x+1)
C) 1
ln 3
371) x2
4 x+1 x2
D) 1
2 1 ln 3 x2 2 x+1
Express the value of the inverse hyperbolic function in terms of natural logarithms. 372) sinh-1 (7)
A) ln (7 - 50)
B) ln (7 + 50)
C) ln (7 + 48)
60
372) D) ln 14
Find the derivative of y with respect to x. 373) y = sin-1 6x + 5 11
373)
66
A)
B)
1 + (6x + 5)2
C) -
6
D)
121 - (6x + 5)2
6 121 - (6x + 5)2
6
1 + (6x + 5)2
Determine whether the integration formula is correct. 6 x6 dx 374) x5 cos-1 3x dx = x cos-1 3x + 3 +C 6 6 1 - 9x2
A) No
374) B) Yes
Find the derivative of y with respect to x, t, or , as appropriate. 375) y = ln (5x2)
A) 10
B)
x
2x
375) C)
x2 + 5
1 2x + 5
D) 2
x
Find the derivative of y with respect to x. 376) y = sin-1 1 x4
A)
-4 x 1 - x8
B)
376) -4 x x8 - 1
Solve the problem.
C)
-4
1 + x8
4 D) -4x
1 - x8
2
377) Find the length of the curve y = x - 1 ln x, 2 x 4. 4
A) 2 + ln 3
377)
2
B) 3 + ln 2
C) 3 + ln 2
2
D) ln 2 2
L'Hopital's rule does not help with the given limit. Find the limit some other way. 378) lim tan x sec x x 0+
A) 1
B) 0
C) -1
Evaluate the integral in terms of natural logarithms. e4 6 dx 379) x 1+ (lnx)2 1 A) 6 ln (1 + 2 ) B) ln (4 + 17 )
378) D)
379) C) 6 ln (4 + 17 )
61
D) 4
Is the function graphed below one-to-one?
380)
380)
A) Yes
B) No
Evaluate the integral. 2
381)
2 x9x dx
381)
1
A) 36
C) 9
B) 36
ln 9
2-9
D) 9
ln 9
2 ln 9
Solve the initial value problem. 2 382) d y = e2t + 9 sin t, y(0) = 0, y (0) = 4 dt2
382) 2t
B) y = e
A) y = e2t - 9 sin t + 12t - 1 4
2t
C) y = e
4
4 2t
D) y = e
- 9 sin t
4
- 9 sin t + 4t - 9 sin t +
1 4
25 1 t2 4
Express as a single logarithm and, if possible, simplify. 383) ln cos - ln cos 5 2
A) ln cos
5
383)
B) ln 1
C) ln cos
5
D) ln 5
Use l'Hopital's Rule to evaluate the limit. 3 2 384) lim x - 10x + 9 x-1 x 1
A) 13
384)
B) 23
C) 20
62
D) -17
Solve the initial value problem. 2 385) d y = 5e-x, y(0) = 1, y (0) = 0 dx2
385)
A) y = -5e-x + C C) y = 5e-x + 1
B) y = 5e-x + 5x - 4 D) y = 5e-x - 5x + 6
Evaluate the integral. 2/2
386)
-1 -ecos x
0
A)
dx
386)
1 - x2
B) e /4
2
Express the following logarithm as specified. 387) ln 13.5 in terms of ln 3 and ln 2 A) 3 ln 3 - ln 2 B) 3 ln 3 2 2
C) e /4 - e /2
D) 1
C) 3 ln 3
D) 3 ln 3 + ln 2 2
387)
Find the derivative of y with respect to x, t, or , as appropriate. 388) y = ln (x - 5) A) 1 B) - 1 C) 1 5-x x+5 x+5
388) D) 1 x-5
Express the value of the inverse hyperbolic function in terms of natural logarithms. 389) sinh-1 -3 4
A) ln -3 + 10 4
B) ln 1
C) ln (4)
2
389) D) ln 2
Evaluate the integral. dt 390) 2 t + 14t + 53
390) B) 1 tan-1 t + 7 + C
A) -7t+ C
2
C) 2 tan-1 t + 7 + C
2
D) tan-1 (t - 7) + C
2
63
Find the domain and range of the inverse of the given function. 391) f(x) = 1 x - 6 7
391)
A) Domain: (- , 6) (6, ); range: (- , 7) (7, ) B) Domain: (- , 7) (7, ); range: all real numbers C) Domain and range: (- , 7) (7, ) D) Domain and range: all real numbers Find the length of the curve. 392) y = 1 (ex + e-x) from x = 0 to x = 1 8 2
A) e - 1 8e
392)
2
2
B) e + e - 2
C) e - 2e + 1
8
Determine if the statement is true or false as x 393) x = O(4x) A) True
8e
D) e
8
.
393) B) False
Solve the problem. 394) Locate and identify the absolute extreme values of ln (sin x) on
/6, 3 /4 A) Absolute maximum at ( /2, 0); absolute minimum at 3 /4, - ln 2 2
394)
B) Absolute maximum at ( /2, 0); absolute minimum at ( /6, - ln 2) C) Absolute maximum at 3 /4, ln 2 ; absolute minimum at ( /2, 0) 2
D) Absolute maximum at ( /6, ln 2); absolute minimum at ( /2, 0) Express the value of the inverse hyperbolic function in terms of natural logarithms. 395) cosh-1 13 5
A) ln 5 Evaluate the integral. ln 4 396) e-t sinh t dt 0 A) ln 4 - 15 2 64
B) ln 13 + 170
5
396) B) ln 4 + 1 2
C) ln 2 - 15
64
64
Find the derivative of y with respect to x, t, or , as appropriate. 397) y = e8 - 3x
A) e-3
D) ln 1
C) ln 4
5
395)
B) 8e8 - 3x
64
D) ln 4 + 15 2
4
397) C) -3e8 - 3x
D) -3 ln (8 - 3x)
Find the length of the curve. 398) y = ln(ex - 1) - ln(ex + 1) from x = ln 4 to x = ln 5
A) ln 20 21
Find
398)
B) ln 21 20
C) ln 32 25
D) ln 19 15
dy . dx
399) e3y = cos (8x +y)
399)
A) -8 sin (8x + y) + 8
B) sin (8x + y)
3e3y - 1
C)
3e3y
-8 sin (8x + y) 3y 3e + sin (8x + y)
D) ln cos (8x + y)
Express the following logarithm as specified. 400) ln 36 in terms of ln 2 and ln 3 A) 4 ln 2 B) 2 ln 2 + 2 ln 3
400) C) -2 ln 2 - 2 ln 3
D) 2 ln 2 - 2 ln 3
Answer the question appropriately.
401) Where does the periodic function f(x) = 2esin(x/2) take on its extreme values? A) x is an odd integer B) x = ±k where k is an even integer C) x = ±k /2 where k is an even integer D) x = ±k where k is an odd integer
Evaluate the integral. /20 402) (1 + etan 5x) sec2 5x dx 0
A) 5e
Find
401)
402)
B) - e
C) e
5
D) e
5
dy . dx
403) ln y = ey cos 6x
403)
y A) -6ye sin 6x 1 - yey cos 6x
y B) ye sin 6x 1 - ey cos 6x
C) -6yey sin 6x
D) ey cos 6x - 6ey sin 6x
Graph the inverse of the function plotted, on the same set of axes. Use a dashed curve for the inverse.
65
404)
404)
A)
B)
C)
D)
Find the derivative of y with respect to the appropriate variable. 405) y = cosh-1 2 x + 2
405)
A)
1 (4x + 9)(x + 2)
B)
1 (2x + 3)(x + 2)
C)
1 (4x + 7)(x + 2)
D)
1 (2x + 3)
66
Solve the differential equation. 406) x2 dy = 6y dx
A) y = Ce6/x
406) B) y = Ce- 6/x
C) y = Ce- 6x
D) y = - 6 + C x
Graph the inverse of the function plotted, on the same set of axes. Use a dashed curve for the inverse.
407)
407)
A)
B)
C)
D)
67
Evaluate the integral. 408) cosh x dx 3
408)
A) sin-1 x + C
B) -3 sinh x + C
3
C) sinh x + C
3
3
D) 3 sinh x + C 3
Find the derivative of y with respect to x, t, or , as appropriate. 409) y = ln x x6
A) 6ln x - 1
B) 1 - 6ln x
x7
409) C) 1 - 6ln x
x12
x7
D) 1 + 6ln x x12
Determine whether the integration formula is correct. 2 2 410) 25x cos-1 5x dx = 50x -1 cos-1 5x - 5x 1 - 25x + C 4 4
A) No
410)
B) Yes
Find the inverse of the function. 411) f(x) = x - 2, x 2 A) f-1 (x) = x2 + 2, x 0 C) f-1 (x) = x2 - 2, x 0
411) B) Not a one-to-one function D) f-1 (x) = x + 2, x 0
Use logarithmic differentiation to find the derivative of y. 5 412) y = x x + 4 (x - 3)2/3
412)
5 A) x x + 4 ln x + 1 ln(x5 + 4) - 2 ln(x - 3) (x - 3)2/3
2
3
B) ln x + 1 ln(x5 + 4) - 2 ln(x - 3) 2
3
5 4 C) x x + 4 1 + 5x (x - 3)2/3 x
2x5 + 8
4 D) 1 + 5x -
2 3x - 9
x
2x5 + 8
2 3x - 9
Use logarithmic differentiation to find the derivative of y with respect to the independent variable. 413) y = (x + 4) sin x
A) sin x ln (x + 4)
B)
C) cos x ln (x + 4) + sin x
-cos x (x + 4) sin x x+4
D) (x + 4) sin x cos x ln (x + 4) + sin x
x+4
x+4
68
413)
Determine if the given function y = f(x) is a solution of the accompanying differential equation. 414) y - y = 4ex y = ex + 4xex A) Yes
414)
B) No
Find the limit.
415) lim cot-1 x
415)
x
A)
B)
2
Evaluate the integral. /4 416) 9 sinh (tan ) sec2 - /4 A) 9 e - 1 e
D) 0
C)
d
416) B) 0 D) 9(e /4 - e- /4 )
C) 9 Solve the differential equation. 417) dy = 8x7 cos2 y dx
417)
A) y = tan-1 (x7 + C) C) y = tan (x8 + C)
B) y = x8 + C D) y = tan-1 (x8 + C)
Use logarithmic differentiation to find the derivative of y. 2 418) y = 5 (4x + 1)(x + 2) (x3 + 6)(x + 7)
418)
A) 1 (ln(4x + 1) + 2ln(x + 2) - ln(x3 + 6) - ln(x + 7)) 5
2
4 2 3x2 1 + (x3 + 6)(x + 7) 4x + 1 x + 2 x3 + 6 x + 7
B) 1 5 (4x + 1)(x + 2) 5
2
4 2 3x2 1 + (x3 + 6)(x + 7) 4x + 1 x + 2 x3 + 6 x + 7
C) 5 5 (4x + 1)(x + 2) D)
4 2 3x2 1 + 4x + 1 x + 2 x3 + 6 x + 7
Find the limit.
419) lim x tan-1 3
419)
x
x
A) -3
C) 1
B) 3
3
69
D)
Solve the differential equation. 420) dy = 4x 4 - y2 dx
420)
A) y = sin (2x2 + C) C) y = 2 sin (2x2 + C) Find
B) y = sin-1 (2x2 + C) D) y = 2 sin-1 (2x2 + C)
dy . dx
421) exy = sin x
421)
xy A) cos x - ye exy
xy B) sin x - ye xexy
xy C) cos x - ye xexy
D) cos x exy
Find the inverse of the function. 422) f(x) = 7 x+3
422)
x 3 + 7x
B) f-1 (x) = -3x + 7
C) Not a one-to-one function
D) f-1 (x) = 3 + 7x
A) f-1 (x) =
x
Determine if the statement is true or false as x 423) x = o(x + 6) A) True
x
.
423) B) False
Evaluate exactly.
424) sec tan-1 -4
424)
3
A) - 5 3
B) 1
C) - 5
3
D) 5
4
3
Use logarithmic differentiation to find the derivative of y with respect to the independent variable. 425) y = 9t t t + 1(ln 9t) A) 9t t+ 1
B) 9t 1 ln 9t + 1
C) 1 1 ln 9t + 1
D) 9t
2
t 2
Express the following logarithm as specified. 426) ln (1/25) in terms of ln 5
A) 2 ln 25
425)
t 1 t
2
ln 9t + 1
426) C) - 1 ln 5
B) -2 ln 5
2
70
D) 1 ln 5 2
Find the formula for df-1 /dx. 427) f(x) = (7 - x)3
427)
A) x2/3
B) -3(7 - x)2
C) -
1 3x2/3
D) 7 - x1/3
Find a value of a so that f is continuous at c, or indicate this is impossible. 25x - 5 sin 5x , x 0 428) f(x) = 3x3 c,
x=0
A) 0
B) 125
428)
C) 625
18
D) 25
18
A value of sinh x or cosh x is given. Use the definitions and the identity cosh2 x - sinh2 x = 1 to find the value of the other indicated hyperbolic function. 429) sinh x = 5 , cosh x = 429) 12
A) 13
B) 169
12
C) 12
144
D) - 13
13
12
Use logarithmic differentiation to find the derivative of y. 430) y = (x4 + 1)2 (x - 1)4 x2 A) (x4 + 1)2 (x - 1)4x2 10 + 4 x x-1
430)
3 B) (x4 + 1)2 (x - 1)4x2 8x + 4 + 2 x4 + 1
x-1
x
3 C) 8x + 4 + 2 x4 + 1
x-1
x
D) (x4 + 1)2 (x - 1)4x2 (2ln(x4 + 1) + 4ln(x - 1) + 2ln x) Find the inverse of the function. 431) f(x) = x - 2, x 0 A) f-1 (x) = (x + 2)2 , x 2 C) f-1 (x) = x + 2, x 2
431) B) f-1 (x) = (x - 2)2 D) f-1 (x) = -(x + 2)2 , x 2
Find a value of a so that f is continuous at c, or indicate this is impossible. 432) Let f(x) = (sin x)x, x 0. Extend the definition of f to x = 0 so that the extended function is continuous there.
x A) f(x) = (sin x) , x 0
x B) f(x) = (sin x) , x 0
x C) f(x) = (sin x) , x 0
x D) f(x) = (sin x) , x 0
e,
0,
x=0
1,
x=0
-1,
71
x=0 x=0
432)
Solve the differential equation. 433) dy = 9 xy dx
433)
A) y = 9x3 + C
B) y = 3x3/2 + C
C) y = 3x3 + x3/2 + C
D) y = (3x3/2 + C)2
Solve the problem. 434) An oil storage tank can be described as the volume generated by revolving the area bounded by 24 y= , x = 0, y = 0, x = 2 about the x-axis. Find the volume (in m 3 ) of the tank. 36 + x2
A) 1.01 m 3
B) 24.3 m 3
C) 97.0 m 3
D) 615 m 3
Use l'Hopital's Rule to evaluate the limit. 435) lim x x 0 sin x
A) -1
434)
435)
B) 1
C) 0
2
D) 1
Use logarithmic differentiation to find the derivative of y with respect to the independent variable. 436) y = (sin x)cos x
A) cos x cot x - sin x ln(sin x) C) cos x cot x - ln (sin x)
436)
B) cos x ln ( sin x) D) (sin x)cos x(cos x cot x - sin x ln (sin x))
Verify the integration formula. 437) 9x tanh x2 dx = 9 ln (cosh x2) + C 2
437)
A) Yes
B) No
Find the derivative of y with respect to the independent variable. 438) y = (cos ) 6
A) - 6(cos ) 6-1 sin
438)
B) - 6 cos sin D) -(cos ) 6-1 sin
C) 6(cos ) 6-1 Solve the problem.
439) The region between the curve y = 1 and the x-axis from x = 1 to x = 6 is revolved about the x2
6
y-axis to generate a solid. Find the volume of the solid. A) 2 ln 6 B) 2 ln 6 C)
72
ln 6 -
D) 4 ln 6
439)
Find the derivative of y with respect to x, t, or , as appropriate.
440) y = ln
e
440)
9+e
A) 9 + 2e 9+e
B) ln
C) 9 + e
9
e
9+e
D)
9 9+e
Is the function graphed below one-to-one?
441)
441)
A) Yes
B) No
Graph the inverse of the function plotted, on the same set of axes. Use a dashed curve for the inverse.
442)
442)
73
A)
B)
Function is its own inverse.
C)
D)
A value of sinh x or cosh x is given. Use the definitions and the identity cosh2 x - sinh2 x = 1 to find the value of the other indicated hyperbolic function. 443) sinh x = - 4 , tanh x = 443) 3
A) 4 5
B) - 5
C) - 4
4
D) 5
5
3
Solve the problem. 444) Find the volume of the solid that is generated by revolving the area bounded by the x-axis, the 3x curve y = , x = 1, and x = 3 about the x-axis. 2 x +1
A) 3 ln 5 2
B) 3 ln 5
C) 3 ln 1
2
5
74
D) 3 ln 1 4
5
444)
Evaluate the integral.
445)
dx (x + 4) x2 + 8x + 15
-1 (x + 4)
A) sin
4
445) B) sec-1 (x + 4) + C
+C
D) sec
C) csc-1 (x + 4) + C
446)
-1 (x + 4) 4
+C
2 ex + e-x dx
446)
A) 1 e2x - e-2x + 2x + C
B) 1 e2x + e-2x + C
C) 1 e2x + e-2x + 2x + C
D) 1 e2x - e-2x + C
2
2
2
2
Evaluate the integral in terms of natural logarithms. 10 dx 447) x2 - 9 4
447) B) ln 10 + 91 - 1
A) ln 7 C) 1 ln 2
23
91 4+
D) ln 10 + 91
7
4+
7
Use logarithmic differentiation to find the derivative of y with respect to the independent variable. 448) y = 7xx2 A) 7xx2 + 1(2 ln x + 1) B) 7xx2 + 1 ln x C) 7x(2 ln x +1) D) 7xx2 + 1(2 ln x)
448)
Evaluate the integral.
449)
4e9x dx
A) 2 e9x+1 + C 5
449) B) 2 e9x2 + C
C) 4 e9x + C
9
9
D) 4e9x + C
L'Hopital's rule does not help with the given limit. Find the limit some other way. 1 450) lim cot x sinx x 0+
A)
B) 0
C) 1
B) coth 3x
C) 1 sinh 3x
Find the derivative of y. 451) y = ln (sinh 3x)
A) 3 coth 3x
450) D) -1
451)
75
D) 3 csch 3x
Evaluate the integral.
452)
453)
dt -1 3(tan t)(1 + t2 )
452)
A) 3 cot-1 t + C
B) 1 ln tan-1 t + C
C) ln 3 tan-1 t + C
D)
3
1
2 3(tan-1 t)
+C
dx
453)
-x2 - 4x - 3
A) 1 2
-x2 - 4x - 3+ C
B) cos-1 (x + 2) + C
C) -sin-1 (x + 2) + C
D) sin-1 (x + 2) + C
Find the domain and range of the inverse of the given function. 454) f(x) = 3.1 - 0.52x A) Domain: [3.1, ); range: all real numbers
454)
B) Domain: all real numbers; range: [3.1, ) C) Domain and range: all real numbers D) Domain: all real numbers; range: (- , 3.1] Find the derivative of y. 455) y = cosh x4
A) -4x3 sinh x4
455) B) 4x3 sinh x4
C) sinh x4
D) -sinh x4
A value of sinh x or cosh x is given. Use the definitions and the identity cosh2 x - sinh2 x = 1 to find the value of the other indicated hyperbolic function. 456) sinh x = 3 , coth x = 456) 4
A) 5
16
B) 3
C) 5
5
D) 5
3
Evaluate the integral. 9 sinh x 457) 4 dx x 1
4
457)
A) -4
B) 4 e3 + e-3 -e - 1
C) 4(e3 - e)
D) 8 e3 + e-3 -e - 1
e e
76
A value of sinh x or cosh x is given. Use the definitions and the identity cosh2 x - sinh2 x = 1 to find the value of the other indicated hyperbolic function. 458) sinh x = 8 , sech x = 458) 15
A) 15
B) 17
8
C) 15
15
Find the limit.
459) lim 1 + 3
17
D) 64
289
x
459)
x5
x
B) 0
A)
C) 3
D) 1
Determine whether the integration formula is correct. 3 460) x2cos-1 x dx = 1 x3 cos-1 x - x dx + C 3 1 - x2
460)
A) No
B) Yes
Find the slowest growing and the fastest growing functions as x 461) y = ex
.
461)
y = ex/2 y = xx y = 4x
A) Slowest: y = ex/2 and y = ex grow at the same rate. Fastest: y = xx B) Slowest: y = xx
Fastest: y = 4 x C) Slowest: y = ex/2 and y = ex grow at the same rate.
Fastest: y = 4 x D) Slowest: y = ex/2 Fastest: y = xx
Evaluate the integral. -9/2
462)
462)
- x2 - 10x - 24
-5
A) -
- dx
6
B) - 3 +
C)
6
77
6
D)
3
Solve for t.
463) e(ln 0.3)t = 0.8
463)
0.8 A) e ln 0.3
B) 9.007199255e+15
C) ln 9.007199255e+15
D) ln 0.8
3.377699721e+15
3.377699721e+15
ln 0.3
Solve the initial value problem. 464) dy = 4 , y(0) = 1 dx 1 + x2
464)
A) y = 4 sin-1 x + 1 C) y = 4 tan-1 x + 1
B) y = 4 cot-1 x - 3 D) y = 4 tan-1 x
Solve the problem. 465) The amount of alcohol in the bloodstream, A, declines at a rate proportional to the amount, that is, dA = - kA. If k = 0.6 for a particular person, how long will it take for his alcohol concentration to dt
465)
decrease from 0.10% to 0.05%? Give your answer to the nearest tenth of an hour. A) 1.2 hr B) 1.7 hr C) 2.3 hr D) 0.4 hr
Use logarithmic differentiation to find the derivative of y with respect to the independent variable. 466) y = (cos x)x
A) (cos x)x (ln cos x + x cot x) C) (cos x)x (ln cos x - x tan x)
466)
B) ln x(cos x)x - 1 D) ln cos x - x tan x
Rewrite the expression in terms of exponentials and simplify the results. 467) cosh 7x - sinh 7x A) e-7x B) e7x - e-7x C) -7x
467) D) 2e-7x
Is the function graphed below one-to-one?
468)
468)
A) Yes
B) No
78
L'Hopital's rule does not help with the given limit. Find the limit some other way. 469) lim sec x x 0 csc x
B) 1
A) -1
469) D) 0
C)
Find the derivative of y with respect to x. 470) y = sec-1 2x + 1 9
A) C)
470)
18 2x + 1
B)
(2x + 1)2 - 81 -18
2x + 1
D)
(2x + 1)2 - 1
18 (2x + 1)2 - 1
-18
1 + (2x + 1)2
Solve the initial value problem. 471) dy = -5 , y(1) = 3 dx 1 - x2
471)
A) y = 5 cos-1 x
B) y = -5 cos-1 x + 3
C) y = 5 sin-1 x - 2
D) y = 5 sin-1 x + 6 - 5 2
Solve the problem. 472) The velocity of a body of mass m falling from rest under the action of gravity is given by the mg gk equation v = tanh t , where k is a constant that depends on the body's aerodynamic k m
472)
properties and the density of the air, g is the gravitational constant, and t is the number of seconds lim into the fall. Find the limiting velocity, v, of a 300 lb. skydiver (mg = 300) when k = 0.006. t
A) There is no limiting speed. C) 223.61 ft/sec
B) 70.71 ft/sec D) 0.01 ft/sec
473) Find the angle given x = 50, y = 22, = 65°.
A) 41.25°
473)
B) 1.25°
C) 115° 79
D) 23.75°
Answer Key Testname: CHAPTER 7
1) D 2) lim ln(x+1) = lim 1/(x + 1) = lim ln x
lim x
ln(x+9955) 1/(x + 9955) x = lim = lim =1 ln x 1/x x + 9955 x x
x
1/x
x =1 x+1
x
x
y = ln x, y = ln(x+1), and y = ln (x+9955) all grow at the same rate. 3) Choice (a) is correct. L'Hôpital's rule can be applied to lim x + 2 since it corresponds to the indeterminate form x - 2 x2 - 4 0 . Choice (b) is incorrect because (-2)2 = 4 not -4. 0
4)
Using the graph, students should estimate the limit to be 0. Using l'Hopital's rule: 1 - cos x sin x = lim lim x x 0 x 0 1 =
sin 0 0 = = 0 1 1
5) Let y = x ln ax - x + C and take its derivative. dy = (1)ln ax + ax 1 - 1 = ln ax dx
ax
80
Answer Key Testname: CHAPTER 7
6)
Using the graph, students should estimate the limit to be 1. Using l'Hopital's rule: the limit leads to the indeterminate form 0 so let f(x) = ln x ln f(x) = (1/x) ln x = x lim ln f(x) x
= lim x = lim x =
So
lim x
ln
x
x1/x and take logarithms of both sides
x
(1/ x ) · (1/2 x) 1
differentiate
1 =0 lim 2x x
x1/x = lim eln f(x) = e0 = 1 x
7) Choice (a) is incorrect. L'Hôpital's rule cannot be applied to lim x - 4 because it corresponds to 0 which is not an x 4 x2 - 4
indeterminate form. Choice (b) is correct.
8) f = O(g) but g O(f) except in the case where their degrees are equal.
81
12
Answer Key Testname: CHAPTER 7
9)
Using the graph, students should estimate the limit to be -1. Using l'Hopital's rule: cos x - 1 - sin x lim = lim x x 0 e -x-1 x 0 ex - 1 = lim x 0
- cos x -1 = = -1 x 1 e
10)
Using the graph, students should estimate the limit to be 0. Using l'Hopital's rule: x 1 lim = lim x 2x x ln2 · 2 x
=0
11) The angle made between the ground and the top of the blackboard is equal to cot-1 x . The angle - made 10
between the ground and the bottom of the blackboard is equal to cot-1 x cot-1 5
12) csc-1 1 , There is no angle whose cosecant is 1 . 5
5
82
x . Therefore 5
-( - )=
x = cot-1 10
Answer Key Testname: CHAPTER 7
13) The binary search. The sequential search could take up to a million steps. The binary search would take at most 20 steps. 14) cos-1 3, There is no angle whose cosine is 3.
3 2 15) L'Hôpital's Rule cannot be applied to lim x - 2x + 1 because it corresponds to - 15 which is not an indeterminate x -2
3x2 - 6x
24
form.
16) When plugging in angles such that y = sin-1 x is -
2
y
2
2
x
2
the output is the same angle. However, the range of
. Therefore, when plugging in angles outside of that interval the output will be different.
Instead of getting back the same angle you are getting back the first or fourth quadrant angle whose sine is the same value. The overall result is a function going back and forth between 1 and -1 in a linear fashion. 17) The graph of f(x) = cos-1 x has negative slope on its domain [-1, 1]. Therefore all values of the first derivative are
below the x-axis. The graph of f(x) becomes gradually less steep as the graph of f approaches its vertex at x = 0. To the right of the y-axis, the graph of f quickly descends and f(x) becomes more steep. 18) sec-1(-x) = cos-1(-1/x) = - cos-1 (1/x) = - sec-1 x
19)
Using the graph, students should estimate the limit to be 1. Using l'Hopital's rule: the limit leads to the indeterminate form 0 0 so let f(x) = x x and take logarithms of both sides ln x ln f(x) = x ln x = 1/ x lim ln f(x) x 0+
=
ln x lim 1/ x x 0+
=
1/x lim 3/2 x 0 + -1/2x
=
So
-
differentiate
lim -2 x = 0 x 0+
lim x x = lim eln f(x) = e0 = 1 x 0+ x 0+
83
Answer Key Testname: CHAPTER 7
20)
Using the graph, students should find the limit to be . Using l'Hopital's rule: the limit leads to the indeterminate form 1 1 lim sin x x x 0
= lim x 0
-
so combine the fractions and apply l'Hopital's rule:
x - sin x x sin x
(1/2 x ) - cos x = lim x 0 (1/2 x ) sin x + x cos x 1 - 2 x cos x = lim sin x + 2x cos x x 0 =
1 = 0
21) Two such functions are f(x) = 4x2 + 4 and g(x) = x2 + 1. 22) When x is positive these graphs are identical because they are both giving the same angle. cos
=
x x2 + 1
tan
=
1 . When x is negative both functions are still referring to the same angle. However, x
inverse cosine gives values between /2 and
while inverse tangent gives values between - /2 and 0.
84
Answer Key Testname: CHAPTER 7
23)
Using the graph, students should estimate the limit to be approximately 4.4. Using l'Hopital's rule: x x - 2 x + 3x - 6 = lim lim x-2 x 2 x 2 =
x + (x/2 x) - (1/ x ) + 3 1
2 + (2/2 2) - (1/ 2 ) + 3 = 1
2+3
24) Yes, They both have domains -1 x 1. They have the same basic shape with opposite slopes. Since the slopes are opposites the derivatives will be opposites. 6 6 25) lim x + x = lim x + x = lim 1 + 1 = 1 x x3 x x x5 x6 lim x
x6 + x4 = lim x3 x
x6 + x4 = lim x x6
1+
1 =1 x2
Therefore both functions grow at the same rate as x 26) Two such functions are f(x) = 1 and g(x) = 1 . x x2
.
27) L'Hôpital's Rule cannot be applied to lim cosx because it corresponds to 1 which is not an indeterminate form. x 0 1 + 2x
1
28) L'Hôpital's Rule cannot be applied to lim sin (1/x) because it corresponds to 0 , which is not an indeterminate form. x
e1/x
1
29) lim f(x)g(x) = lim (x - 3) 2/(x - 3)2 = 0 x 3
x 3
30) D 31) C 32) A 33) D 34) B 35) B 36) D 37) B 85
Answer Key Testname: CHAPTER 7
38) C 39) B 40) A 41) C 42) C 43) B 44) A 45) B 46) C 47) A 48) D 49) A 50) C 51) C 52) B 53) A 54) A 55) A 56) D 57) C 58) B 59) D 60) A 61) B 62) D 63) D 64) A 65) B 66) D 67) A 68) B 69) D 70) C 71) C 72) A 73) C 74) A 75) A 76) C 77) B 78) D 79) B 86
Answer Key Testname: CHAPTER 7
80) D 81) B 82) D 83) A 84) C 85) C 86) C 87) C 88) D 89) C 90) B 91) C 92) D 93) B 94) D 95) A 96) A 97) A 98) A 99) A 100) C 101) B 102) B 103) D 104) D 105) B 106) A 107) B 108) C 109) C 110) C 111) A 112) B 113) B 114) A 115) A 116) A 117) C 118) A 119) B 120) B 121) B 87
Answer Key Testname: CHAPTER 7
122) D 123) B 124) D 125) C 126) A 127) C 128) D 129) C 130) B 131) B 132) B 133) D 134) D 135) B 136) D 137) C 138) D 139) B 140) A 141) B 142) A 143) C 144) C 145) C 146) D 147) A 148) D 149) A 150) D 151) D 152) A 153) D 154) D 155) D 156) A 157) D 158) C 159) B 160) A 161) B 162) A 163) B 88
Answer Key Testname: CHAPTER 7
164) C 165) A 166) A 167) C 168) C 169) A 170) B 171) A 172) A 173) D 174) C 175) A 176) C 177) C 178) C 179) C 180) D 181) D 182) C 183) A 184) C 185) C 186) B 187) D 188) D 189) B 190) B 191) C 192) B 193) D 194) D 195) B 196) C 197) B 198) C 199) A 200) C 201) C 202) A 203) B 204) A 205) D 89
Answer Key Testname: CHAPTER 7
206) A 207) A 208) A 209) D 210) B 211) A 212) D 213) B 214) B 215) D 216) D 217) C 218) A 219) C 220) B 221) A 222) B 223) B 224) B 225) C 226) B 227) A 228) A 229) B 230) D 231) A 232) B 233) A 234) B 235) A 236) B 237) A 238) B 239) D 240) A 241) A 242) D 243) D 244) D 245) A 246) A 247) C 90
Answer Key Testname: CHAPTER 7
248) B 249) B 250) D 251) C 252) B 253) C 254) B 255) D 256) A 257) C 258) A 259) D 260) A 261) B 262) D 263) A 264) D 265) A 266) C 267) C 268) B 269) B 270) B 271) B 272) C 273) A 274) C 275) C 276) C 277) B 278) C 279) B 280) B 281) D 282) D 283) C 284) D 285) D 286) C 287) A 288) C 289) C 91
Answer Key Testname: CHAPTER 7
290) C 291) B 292) D 293) B 294) D 295) C 296) C 297) C 298) C 299) C 300) B 301) C 302) B 303) C 304) D 305) A 306) B 307) C 308) A 309) D 310) C 311) D 312) C 313) B 314) A 315) A 316) A 317) B 318) B 319) A 320) B 321) A 322) D 323) C 324) B 325) D 326) C 327) D 328) D 329) C 330) A 331) B 92
Answer Key Testname: CHAPTER 7
332) B 333) A 334) B 335) C 336) D 337) B 338) B 339) A 340) B 341) B 342) C 343) B 344) D 345) D 346) D 347) B 348) B 349) C 350) D 351) C 352) B 353) A 354) B 355) C 356) C 357) A 358) A 359) D 360) C 361) C 362) D 363) A 364) D 365) D 366) B 367) D 368) D 369) A 370) B 371) A 372) B 373) B 93
Answer Key Testname: CHAPTER 7
374) B 375) D 376) B 377) B 378) B 379) C 380) B 381) A 382) D 383) D 384) D 385) B 386) C 387) A 388) D 389) B 390) B 391) D 392) A 393) A 394) B 395) A 396) A 397) C 398) C 399) C 400) B 401) D 402) D 403) A 404) A 405) C 406) B 407) B 408) D 409) C 410) B 411) A 412) C 413) D 414) A 415) B 94
Answer Key Testname: CHAPTER 7
416) B 417) D 418) B 419) B 420) C 421) C 422) B 423) B 424) D 425) D 426) B 427) C 428) C 429) A 430) B 431) A 432) B 433) D 434) C 435) D 436) D 437) A 438) A 439) D 440) D 441) A 442) A 443) C 444) A 445) B 446) A 447) D 448) A 449) C 450) C 451) A 452) B 453) D 454) C 455) B 456) C 457) B 95
Answer Key Testname: CHAPTER 7
458) C 459) D 460) A 461) D 462) A 463) D 464) C 465) A 466) C 467) A 468) A 469) D 470) A 471) B 472) C 473) A
96
Chapter 8
Exam Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response.
1) i) Show that 0
4x3 dx diverges, and hence x4 + 1 b
ii) Show that lim b
-b
2) A student claims that
-
4x3 dx diverges. x4 + 1
1)
4x3 dx = 0. x4 + 1
b
f(x) dx always exists, as long as a and b are both positive.
2)
a Refute this by giving an example of a function for which this is not true.
3) Here is an argument that ln 2 = - . Where does the argument go wrong? 1 ln 2 = ln 1 + ln 2 = ln 1 - ln 2 = lim ln b
b-1 1 - ln b 2
= lim b
ln
b x-1 x 2
= lim b
ln (x - 1) - ln x
= lim b = 2 = 2 = lim b =
b 2
b 2
1 1 dx x-1 x
1 1 dx x-1 x 1 dx x-1 ln(x - 1)
2 b
1 dx x
- lim 2 b
ln x
b 2
-
1
3)
4) (a) Express
3x - 1 as a sum of partial fractions. 2 x - 3x - 10
(b) Evaluate
3x - 1 dx . 2 x - 3x - 10
(c) Evaluate
2x2 - 3x - 21 dx . x2 - 3x - 10
4)
(d) Find a solution to the initial value problem: dy 3xy - y , y(4) = 12. = dx x2 - 3x - 10
5) A student wishes to find the integral
+
f(x) dx of a function that has the property limit
5)
0 lim f(x) = 1. Why can this not be done? x +
6) The standard normal probability density function is defined by f(x) = 1 e- x2 /2. 2
(a) Use the fact that
1 2
2 e- x /2 dx = 1 to show that
1 2
6)
2 1 x2 e- x /2 dx = . 2
0 (b) Use the result in part (a) to show that the standard normal probability density function has variance 1.
7) Let f(x) = 1 , x 1.
7)
x2
(a) Find the area of the region between the graph of y = f(x) and the x-axis. (b) If the region described in part (a) is revolved about the x-axis, what is the volume of the solid that is generated? (c) A surface is generated by revolving the graph of y = f(x) about the x-axis. Write an integral expression for the surface area and show that the integral converges. (d) Use numerical techniques to estimate the area of the region in part (c), to an accuracy of at least two decimal places.
8) A student wishes to take the integral over all real numbers of f(x) = 1 , if x < 0 1 , if x > x
0 and claims this is zero because - +
9) The Cauchy density function, f(x) = 1
-
(1 + x2 )
x2
8)
equals zero. What is wrong with this thinking? 1
(1 + x2 )
, occurs in probability theory. Show that
dx = 1.
2
9)
1
10) (a) Find the values of p for which 0 (b) Find the values of p for which
1
0
1 dx converges. xp
10)
1 dx diverges. xp
1 e-2x dx = e-4 < 0.0092 and hence that 2
11) (a) Show that 2
(b) Explain why this means that
2 e-x dx < 0.0092. 2
2
2 e-x dx can be replaced by
0 introducing an error of magnitude greater than 0.0092.
2 e-x dx without
0
12) The standard normal probability density function is defined by f(x) = 1 e- x2 /2. 2
1 2
(a) Show that
2 x e- x /2 dx =
11)
12)
1 . 2
0 (b) Use the result in part (a) to show that the standard normal probability density function has mean 0.
0
dx converges. 3 x +1
50
dx x3 + 1
13) (a) Show that (b) Show that
13)
0.0002.
dx is approximated by 3 x +1
(c) Suppose
50
dx . Based on your 3 x +1
0 0 answer to part (b), what is the maximum possible error? (d) Use a numerical method to estimate the value of 0 (e) Determine whether
dx . 3 x +1
dx converges or diverges, and justify 3 -1 x + 1
your answer. If it converges, estimate its value to an accuracy of at least two decimal places.
+
14) A student needs
3
15) Show that -
dx
1 + x2
e-|x| dx , and if so,
14)
0
-
why?
+
e-|x| dx . Is this integral the same as 2
dx
+ 3
1 + x2
5
= -
dx
1 + x2
dx
+ 5
3
1 + x2
.
15)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the improper integral converges or diverges. /2 16) sec d 0 A) Converges B) Diverges
16)
Use a trigonometric substitution to evaluate the integral. dx 17) x 64 + ln2 x
17)
A) tan-1 ln x + C
B) cosh-1 ln x + C
C) sinh-1 ln x + C
D) sin-1 ln x + C
8
8
8
8
Evaluate the improper integral or state that it is divergent. 0 25 18) dx 2 (x 1) A) 25 B) 50
18) C) Divergent
D) -25
Determine whether the improper integral converges or diverges. -3 19) (e100x + x3 )dx A) Converges B) Diverges Solve the problem by integration.
20) Find the x-coordinate of the centroid of the first-quadrant area bounded by y =
19)
8 , x = 2, and 3 x +x
20)
x = 5.
A) 2.895
C) 1.448
B) -0.727
D) -0.364
Integrate the function. 1 dx
21)
0
21)
64 - x2
A) 8 cos-1 1 8
B) cos-1 1
C) sin-1 1
8
8
4
D) 1 sin-1 1 8
8
dx 3/2 2 (x + 64)
22)
A) C)
8 x 64 - x2
22) +C
x 64 64 - x2
B) 64 - x2 +C x
+
D)
x 64 64 + x2 x 8 64 + x2
+C
+C
Express the integrand as a sum of partial fractions and evaluate the integral. 4 2 23) 5x + 40x + 75 dx 2 x(x2 + 5)
A) 3 ln x + ln x2 + 5 + C C) 3 ln x + ln x2 + 5 -
23)
B) 7 ln x + ln x2 + 5 + C
4 +C 2 x +5
D) 3 ln x -
4 +C 2 x +5
Evaluate the improper integral or state that it is divergent.
24) 1
1 dx 2 x(x + 4)
24)
A) ln 5
B) ln 5
8
C) Divergent
D) ln 3
Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions. 4 2 25) 5y + 3y - 5y dy 25) y3 - 1
A) 5 y2 + ln y - 1 + ln y2 + y + 1 + C 2
B) 5 y2 + 5 ln y - 1 + ln y2 - y + 1 + C 2
C) 5 y2 - ln y - 1 + 5 ln y2 + y + 1 + C 2
D) 5 y2 + ln y - 1 + (2y + 1) ln y2 + y + 1 + C 2
Evaluate the integral.
26)
x csc2 3x dx
26)
A) - 1 x cot 3x + 1 ln sin 3x + C
B) - x cot 3x + ln sin 3x + C
C) - 3x cot 3x + 9 ln sin 3x + C
D) 1 x cot 3x - 1 ln sin 3x + C
3
9
3
5
9
Solve the problem. 27) Suppose that the accompanying table shows the velocity of a car every second for 8 seconds. Use Simpson's Rule to approximate the distance traveled by the car in the 8 seconds. Round your answer to the nearest hundredth if necessary. Time (sec) Velocity (ft/sec) 0 20 1 21 2 22 3 24 4 23 5 25 6 22 7 20 8 21 A) 176.33 ft B) 177.5 ft
C) 132 ft
27)
D) 178.33 ft
28) The amount of time that goes by between a new driver getting a license and the moment the
28)
driver is involved in an accident is exponentially distributed. An insurance company observes a sample of new drivers and finds that 70% are involved in an accident during the first 2 years after they get their driver's license. In a group of 200 new drivers, how many should the insurance company expect to be involved in an accident during the first 3.0 years after receiving their license? A) 176 drivers B) 165 drivers C) 24 drivers D) 190 drivers
Use a trigonometric substitution to evaluate the integral. 2 6e-t 29) dt -2t + 1 36e 0 A) tan-1 6 - tan-1 6 e2
29) B) tan-1 2 - tan-1 6
C) tan-1 6 - tan-1 6
D) 1 tan-1 6 - 1 tan-1 6
e2
6
6
e2
Provide an appropriate response.
30) A student knows that
+
+
f(x) dx diverges, but needs to investigate
a g(x) = 16f(x). Does this integral necessarily also diverge? A) Yes B) No
Use the Trapezoidal Rule with n = 4 steps to estimate the integral. 4 31) x dx 0 A) 16 B) 10 C) 8 Solve the problem.
6
g(x) dx , where
30)
a
31) D) 4
32) A spinner marked off in five equal-sized regions, numbered 1, 2, 3, 4, 5, is spun twice. Let the random variable X be the sum of the numbers of the regions where the pointer stops. (a) Find the set of possible outcomes (b) Create a probability bar graph for X. A) (a) {2, 3, 4, 5, 6, 7, 8, 9, 10} (b)
1
1/5 2
3
4
5
6
7
8
9
10
B) (a) {2, 3, 4, 5, 6} (b)
1/5
2 3 4 C) (a) {2, 3, 4, 5, 6, 7, 8, 9, 10} (b)
5
6
9/25
5/25
1/25 2
3
4
5
6
7
8
9
10
7
32)
D) (a) {2, 3, 4, 5, 6, 7, 8, 9, 10} (b)
5/25
1/25 2
3
4
Evaluate the integral. 3 33) x4 ln 8x dx 1 A) 144.36
5
6
7
8
9
10
33) B) 163.80
C) 145.98
D) -45.80
Solve the problem by integration. 34) The current i (in A) as a function of the time t (in s) in a certain electric circuit is given by 8t + 2 i= . Find the total charge that passes a given point in the circuit during the first 2 4t + 2t + 1 three seconds. A) 15.045 C
B) 3.761 C
C) 3.738 C
Solve the problem.
35) Find an upper bound for ES in estimating
3
D) 0.940 C
(8x4 - 7x) dx with n = 8 steps.
1
A) 1
B) 3.518437209e+13
C) 1.759218604e+13
D) 1.125899907e+15
34)
35)
4.222124651e+15
4.222124651e+15
2.849934139e+15
Use the Trapezoidal Rule with n = 4 steps to estimate the integral. 2 36) (x4 + 8) dx 0 A) 369 B) 369 C) 269 16 8 12
8
36) D) 497 16
Evaluate the integral. y2 sin 4y dy
37)
37)
A) - 1 y2 sin 4y + 1 y cos 4y + 1 sin 4y + C
B) - 1 y2 cos 4y + 1 y sin 4y + 1 cos 4y + C
C) - 1 y2 cos 4y + 1 y sin 4y + 1 cos 4y + C
D) 1 y2 cos 4y - 1 y sin 4y - 1 cos 4y + C
4
8
4
32
2
4
2
8
4
32
8
32
Solve the problem.
38) The charge q (in coulombs) delivered by a current i (in amperes) is given by q =
i dt , where t is
38)
the time (in seconds). A damped-out periodic wave form has current given by i = e-3t cos 5t. Find a formula for the charge delivered over time t. -3t -3t A) e (-3 cos 5t + 5 sin 5t) + C B) e (-3 cos 5t + sin 5) + C 25 34
-3t(-3 cos 5t + 5 sin 5t)
D) e
C) -3 cos 5t + 5 sin 5t + C 34
34
+C
Express the integrand as a sum of partial fractions and evaluate the integral. -2x2 + 8x + 8 39) dx (x2 + 4)(x - 2)3
A) tan-1 x - 2 2
x-2
(x - 2)2
C) 1 tan-1 x + 1 2
2
1
x-2
1
B) 1 tan-1 x -
+C
(x - 2)2
39)
2
2
1
+
(x - 2)2
D) 1 tan-1 x + ln x - 2 -
+C
2
2
1
(x - 2)3 1
(x - 2)2
+C +C
Determine whether the improper integral converges or diverges.
40) 1
x dx x4 + 1
40)
A) Diverges Evaluate the integral. 3e2t + 4et
41)
e2t - 12et + 36
B) Converges
dt
41)
A) 3 ln t - 6 - 22 + C
B) 3 ln et - 6 + 22 + C
C) 4 ln et - 6 - 18 + C
D) 3 ln et - 6 - 22 + C
et - 6
t- 6
et - 6
et - 6
Express the integrand as a sum of partial fractions and evaluate the integral. 5 2x2 + x + 50 42) dx 2 1 ( x + 25) (x + 2) A) 1.43 B) 1.76 C) 0.88
9
42) D) 3.52
Integrate the function. 36 dx 43) 2 x x2 + 16
43)
2 A) 9 x + 16 + C
2 B) - 9 x + 16 + C
2 C) 4 x + 16 + C
D) -
4x
4x
9x
x2 + 16 + C 9x
Evaluate the integral. /2 44) cos2 5x sin3 5x dx 0
A) 2
44) B) 4
75
C) 1
75
D) 8
75
75
Expand the quotient by partial fractions. 45) 7x + 2 x2 - 6x + 8
A) 15 + x-4
45) B) 15 + -8
-8 (x - 4)(x - 2)
x-4
C) 15 + 8 x-4
x-2
D) 30 + 16
x-2
x-4
x-2
Evaluate the integral. et dt
46)
46)
e2t - 7et + 10
A) 1 et ln et - 5 - 1 et ln et - 2 + C
B) 1 ln et - 5 - 1 ln et - 2 + C
C) 1 ln et - 5 + 1 ln et - 2 + C
D) 1 ln t - 5 - 1 ln t - 2 + C
3
3
3
47)
3
3
3
3
sin 6t sin 2t dt
47)
A) 1 sin 4t - 1 cos 8t + C
B) 1 sin 4t - 1 sin 8t + C
C) 1 sin 6t - 1 sin 2t + C
D) 1 sin 4t + 1 sin 8t + C
8 8
48)
3
16
8
16
8
16 16
4x sin x dx
48)
A) 4 sin x - 4 cos x + C C) 4 sin x - 4x cos x + C
B) 4 sin x - x cos x + C D) 4 sin x + 4x cos x + C
10
e-2x sin x dx
49)
49)
0
B) 1
A) 1
C) 2
5
D) Diverges
5
Express the integrand as a sum of partial fractions and evaluate the integral. 13x + 64 50) dx 3 x + 8x2 + 16x
50)
A) 4 ln
x 3 + +C x+4 x+4
B) 4 ln
x 5 + +C x+4 x+4
C) 4 ln
x 6 +C x+4 x+4
D) 2 ln
x 3 +C x+4 x+4
Evaluate the improper integral or state that it is divergent. 14
51)
8x(x + 1)2
1
dx
51)
A) 0.337
C) Divergent
B) -1.569
Find the area or volume.
52) Find the area between the graph of y = A) 9
18
(x - 1)2
D) 1.569
and the x-axis, for - < x 0.
B) 36
C) 18
52) D) 1
Determine whether the improper integral converges or diverges. 5 dx 53) 1/3 -5 (x + 1) A) Converges B) Diverges
53)
Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions. 3 2 54) 9x + 8x - 3x - 5 dx 54) x3 - x 2
A) 9x + 8ln x2 - 5 + 9ln x - 1 + C
B) 9x + 8ln x - 5 + 9ln x - 1 + C
C) 9x + 9ln x + 5 + 9ln x - 1 + C
D) 9 + 8ln x - 5 + 9ln x - 1 + C
x2
x
x
x
x2
Determine whether the improper integral converges or diverges.
55) 15
dx dx ex - x5
55)
A) Diverges
B) Converges
11
Provide an appropriate response. 56) Given that we know the Fundamental Theorem of Calculus, why would we want to develop numerical methods for definite integrals?
56)
i) Antiderivatives are not always expressible in closed form. ii) Numerical methods are a good excuse to use our graphics calculator. iii) The function f(x) may not be continuous on [a,b]. A) Both i and iii are correct.
C) Only i is correct.
B) Only ii is correct. D) Only iii is correct.
Solve the problem. 57) The lifespan for a pair of walking shoes is normally distributed with a mean of µ = 790,000 steps and standard deviation of = 16,000 steps. If a manufacture produces 8000 pairs of these shoes, how many can be expected to last for at least 802,800 steps? A) 1695 pairs B) 1855 pairs C) 1751 pairs D) 1607 pairs
58) Two dice are rolled, and the random variable X assigns to each outcome the sum of the number of dots showing on each face. What is the probability that X > 9? A) 1 B) 6 C) 1 12 6
Evaluate the integral. 4 59) 7x ln x dx 2 A) 64.9
60)
/4
A)
8
-
2 3
58)
D) 1 4
59) B) 46.9
C) 6.70
D) 11.06
cot4 2t dt
/8
57)
60) B)
4
-
1 3
C) - 1 3
Determine whether the improper integral converges or diverges. 0 61) 25e5x dx A) Diverges B) Converges
12
D)
8
-
1 3
61)
Evaluate the integral by making a substitution (possibly trigonometric) and then applying a reduction formula. 1 62) 62) 5 x2 + 1 dx 0
A) 5[ 2 - ln( 2 + 1)]
B) 2 + ln( 2 + 1)
C) 5 [ 2 + ln( 2 + 1)]
D) 5[ 2 + ln( 2 + 1)]
2
Evaluate the improper integral or state that it is divergent. 22xe-x dx
63)
63)
-
B) 22
A) -22
C) 0
Solve the problem. 64) Find the area between y = ln x and the x-axis from x = 1 to x = 3. A) 2 B) 3 ln 3 - 3 C) 3 ln 3 - 2 3
D) Divergent
64) D) ln 3
Determine whether the improper integral converges or diverges. 3 ex dx ln x
65) 1
65)
A) Converges 3
66) 0
B) Diverges
dx dx x-2
66)
A) Converges Evaluate the integral. 4x2 - 25
67)
x
B) Diverges
dx
67) 4x2 - 25 +C x
A) ln x + 4x2 - 25 -
B) 4x2 - 25 - 5 sec -1 2x + C 5
C) x2 - 25 - 5 sec -1 x + C
D) ln x + x2 - 25 -
5
x2 - 25 +C x
Solve the problem.
68) Find the area between y = (x - 2)ex and the x-axis from x = 2 to x = 7. A) e7 + e2 B) e7 - e2 C) 4e7 + e2
13
68) D) 4e7
69) A rectangular swimming pool is being constructed, 18 feet long and 100 feet wide. The depth of
69)
the pool is measured at 3-foot intervals across the length of the pool. Estimate the volume of water in the pool using the Trapezoidal Rule. Width (ft) Depth (ft) 0 3 3 3.5 6 4 9 5 12 5.5 15 6 18 7 A) 8700 ft3
B) 7700 ft3
C) 10,200 ft3
D) 5800 ft3
Evaluate the integral by using a substitution prior to integration by parts.
70)
ln (5x + 5x2 ) dx
70)
A) ln (5x + 5x2) - 2x + C C) xln (5x + 5x2 ) + C
B) ln (5x + 5x2) + ln (x +1) + C D) xln (5x + 5x2 ) + ln (x +1) - 2x + C
Integrate the function. dx 71) ,x>5 2 x x2 - 25
71)
A) 125 + C
B) ln x +
C) ln x + x2 - 25 + C
D) 1 + C
x
5
x2 - 25 x
+C
25x
Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions. 4 2 72) 6x + 36x + 6 dx 72) x3 + 6x
A) 3x2 + ln x - 1 ln x2 + 6 + C
B) 3x2 + ln x - x ln x2 + 6 + C
C) 3x2 + ln x + 1 ln x2 + 6 + C
D) 3x2 - 1 ln x2 + 6 + C
2 2
2
Solve the problem.
73) Find an upper bound for ES in estimating
5x cos x dx with n = 8 steps. 0 Give your answer as a decimal rounded to five decimal places. A) 0.00830 B) 0.00872 C) 0.01038
14
73) D) 0.00652
Use the Trapezoidal Rule with n = 4 steps to estimate the integral. 3 7 74) dx 2 1 x A) 987 B) 987 C) 987 400 200 100
74) D) 497 100
Expand the quotient by partial fractions. x2 - 6x + 30
75)
75)
x2 - 10x + 24
A) 15 + -11
B) 1 + 15 + 11
C) x + 15 + -x - 11
D) 1 + 15 + -11
x-6
x-4
x-6
x-6
x-4
x-6
x-4 x-4
Solve the initial value problem for y as a function of x. 76) x2 - 36 dy = 1, x > 6, y(12) = ln (2 + 6 3) dx
76)
A) y = ln (x + x2 - 36) + ln (2 + 6 3) B) y = ln (x + x2 - 36) + ln (2 + 6 3) - ln (12 + 6 3) C) y = 1 ln x + 6 + ln 6 12
x-6
D) y = ln sec x + tan x + ln (2 + 6 3) Integrate the function. dx 11 77) ,x> 2 2 4x - 121 4x2 - 121 +C 11
A) 1 ln 2 x + 2
11
C) 1 ln sec-1 11 x + 2
77)
2
B) 1 ln 1 x + 2
4x2 - 121 +C 11
2
D) 1 ln 11 x + 11
2
11 +C 4x2 - 121 4x2 - 121 +C 2
Solve the problem. 78) Estimate the minimum number of subintervals needed to approximate the integral 6 1 dx 2 (x 1) 3 with an error of magnitude less than 10-4 using the Trapezoidal Rule.
A) 368
B) 92
C) 46
D) 15
79) Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = ex, and the line x = ln 8 about the line x = ln 8. A) 2 (7 + ln 8) B) 2 (8 - ln 9) C) 2 (7 - ln 8) D) 2 (8 - ln 8)
15
78)
79)
Determine whether the improper integral converges or diverges. 9
80) 2
(x + 1)2
dx
80)
A) Converges
B) Diverges
Find the area or volume.
81) Find the area under y =
3
1 + x2
B) 3
A) 6
10/ 3 5
A)
81) D) 3
C) 3
4
Evaluate the improper integral. 49 dx 82) 49 - x 0 A) 7
83)
in the first quadrant.
2
82) B) 0
D) 14
C) -14
dt t t2 - 25
83) B)
10
C) 1
3
D)
5
30
Evaluate the integral. The integral may not require integration by parts. 84) ln x dx x3
84)
A) - 1 ln x - 1 + C
B) - 1 ln x - 1 + C
C) 4 + C
D) - 1 ln x - 1 + C
2x2
4x2
2
x5
x
2x2 4x2
Evaluate the integral. cot4 9x dx
85)
85)
A) - 1 cot39x + 1 cot 9x + C
B) - 1 cot39x + 1 cot2 9x + x + C
C) - 1 cot3 9x + cot 9x + x + C
D) - 1 cot39x + 1 cot 9x + x + C
27
9
27
3
27
9 9
Determine whether the improper integral converges or diverges.
86) 1
4x + 9 x2
86)
A) Diverges
B) Converges
16
Use a trigonometric substitution to evaluate the integral. 1 dy 87) 2 1/e 16y - y(ln y) A) 0.245 B) -0.511 Evaluate the integral. /2 88) 1 + cos x dx /3 A) 2 + 2
87) C) 0.064
D) 0.061
88) C) 2 - 2
B) 1
D) 2
Solve the problem. 89) Estimate the minimum number of subintervals needed to approximate the integral
89)
4 cos x dx 0 with an error of magnitude less than 10-4 using Simpson's Rule. A) 16 B) 20 C) 18
D) 19
Evaluate the integral.
90)
sin 8x cos 6x dx
90)
A) - cos 2x + cos 16x + C
B) - cos 2x - cos 14x + C
C) sin 2x - sin 8x + C
D) sin 2x + sin 14x + C
4
4
14
4
28
4
28
28
Find the value of the constant k so that the given function in a probability density function for a random variable over the specified interval. 91) f(x) = k over [1, 13] 91) x
A) 1 - ln 13
B)
1 ln 13
C)
Use Simpson's Rule with n = 4 steps to estimate the integral. 1 92) (x2 + 3) dx -1 A) 20 B) 27 3 4
17
2 ln 13
D) ln 13
92) C) 10 3
D) 11 2
Evaluate the improper integral or state that it is divergent. 0 dx 93) (9 + x) x -
A)
3
93) C) 0
B) -3
D) -
3
Expand the quotient by partial fractions. 94) 4x + 46 x2 + 8x + 7
A)
3 -7 + x+1 x+7
B)
94) 7 -3 + x+1 x+7
C)
7 3 + x+1 x+7
D)
7 -3 + x-1 x-7
Solve the problem. 95) The rate of water usage for a business, in gallons per hour, is given by W(t) = 25te-t, where t is the number of hours since midnight. Find the average rate of water usage over the interval 0 t 5. A) 0.20 gallons per hour
95)
B) 4.80 gallons per hour D) 4.87 gallons per hour
C) 5.20 gallons per hour Evaluate the improper integral or state that it is divergent. 0 96) 16 ex sin x dx A) - 8 B) 8
96) C) 0
D) -16
Determine whether the improper integral converges or diverges.
97) -
dx 5x6 + 1
97)
A) Diverges Evaluate the improper integral. 5 98) x ln4x dx 0 A) 25 ln20 - 25 2 2
B) Converges
98) B) 25 ln20 - 25 2
C) - 1 ln20 + 25
4
2
18
4
D) 25 ln4 - 25 2
4
Evaluate the integral. 4e2t - 7et
99)
e3t - 3e2t + et - 3
dt
99)
A) 1 ln et - 3 - 1 ln e2t + 1 + 5 tan-1(et) + C 2
4
2
B) 1 ln et - 3 - 1 ln e2t + 1 + C 2
4
C) 1 ln et - 3 + 1 ln e2t + 1 + 3 tan-1(et) + C 4
2
2
D) 1 ln t - 3 - 1 ln t + 1 + 5 tan-1 t + C 2
4
2
Evaluate the integral by using a substitution prior to integration by parts.
100)
x2
x + 25 dx
100) B) (15x - 300x + 5000)(x + 25) 105
105
3/2
2
C) (30x - 600x + 400)(x + 25) 105
3/2
2
2
A) (30x - 600x + 10,000) (x + 25) + C
2
3/2
D) (30x - 600x + 10,000)(x + 25)
+C
+C
105
+C
Solve the problem. 101) The average life (in months) of an automobile battery is 4 months. If the battery life is exponentially distributed, find the probability that the life of a randomly selected battery is greater than 5 years. A) 0.1784 B) 0.2865 C) 0.0716 D) 0.7135 Evaluate the improper integral. 5 dx 102) 25 - x2 0
A)
Integrate the function. 9 - x2
103)
x4
A)
102) B)
10
C) 10
2
D) 1
dx, x < 3
9
1/2 (9 - x2 ) 27x3
103) 2 3/2
B) (9 - x )
+C
2 3/2
C) - (9 - x )
101)
x3
+C
D) -
19
+C
27x3 +C 3/2 (9 - x2 )
Express the integrand as a sum of partial fractions and evaluate the integral. 3 2 104) 7x + 54x + 134x + 104 dx (x + 4)(x + 2)3
A) ln (x + 4)2 (x + 2)5 B) ln (x + 4)2 (x + 2)5 + C) ln (x + 4)2 (x + 2)5 D) ln (x + 4)2 (x + 2)5 -
104)
2 1 + +C (x + 2) (x + 2)2 1
(x + 2)2 3
(x + 2)2
+C +C
1 2 + +C (x + 2) (x + 2)2
Solve the problem.
105) Find the length of the curve y = 64 - x2 between x = 0 and x = 4. A) 1 6
B) 8 3
105) D) 4 3
C) 2
106) Estimate the minimum number of subintervals needed to approximate the integral /2
106)
5 sin x dx
- /2 with an error of magnitude less than 10-4 using the Trapezoidal Rule. A) 65 B) 360 C) 128
D) 255
Use a trigonometric substitution to evaluate the integral. ex dx
107)
107)
1 - e2x
A) -2 1 - e2x + C C) ex sin-1 (ex) + C
B) sec-1 (ex) + C D) sin-1 (ex) + C
Evaluate the improper integral or state that it is divergent.
108) 0
25(1 + tan-1 x) dx 1 + x2
A) 25 ln 1 +
2
108) B) 25 1 + 2
C) 25 1 +
4
2
20
2 2
D) 25 2
Evaluate the integral by making a substitution and then using a table of integrals. x dx 109) 2 25x + 60x + 36
A) 1 ln(5x + 6) + 25
6 +C 5x + 6
B) 1 ln(5x2 + 6x) 5
C) x - 6 ln(5x + 6) + C 5
109)
D) 1 ln(6x + 5) +
25
36
5 +C 6x + 5
Solve the problem. 110) A surveyor measured the length of a piece of land at 100-ft intervals (x), as shown in the table. Use Simpson's Rule to estimate the area of the piece of land in square feet. x Length (ft) 0 50 100 60 200 80 300 55 400 50 A) 24,500 ft2
B) 24,000 ft2
Evaluate the integral. /5 111) sin3 5x dx 0 A) 4 15
C) 29,500 ft2
110)
D) 25,000 ft2
111) C) 1
B) 0
D) 2
5
15
Express the integrand as a sum of partial fractions and evaluate the integral. 2 112) 3x - 33x + 36 dx x3 - 9x2 + 18x
A) 2 ln x - 3 ln x - 6 + 4 ln x - 3 + C C) 2 ln x + 4 ln x - 6 - 3 ln x - 3 + C
112)
B) ln x - ln x - 6 + ln x - 3 + C D) -3 ln x - 6 + 4 ln x - 3 + C
Evaluate the integral. f-1 (x) dx = xf-1 (x) -
113) Use the formula
f(y) dy , y = f-1 (x) to evaluate the integral.
113)
cos-1 x dx
A) x cos-1 x + sin(cos-1 x) + C C) x cos-1 x - x + C 114)
B) x cos-1 x - sin x + C D) x cos-1 x - sin(cos-1 x) + C
tan4 5t dt
114)
A) 1 tan3 5t - 1 tan 5t + t + C
B) 1 tan3 5t - 1 tan2 5t + 1 tan 5t + t + C
C) - 1 tan3 5t + 1 tan 5t + C
D) 1 tan3 5t - tan 5t + t + C
15
15
5
15
5
3
21
25
5
Express the integrand as a sum of partial fractions and evaluate the integral. 5x + 34 115) dx 2 x + 10x + 24 2 A) ln (x + 4) + C
3 B) ln (x + 4) + C
6 C) ln (x + 4) + C
7 D) ln (x + 4) + C
(x + 6)7
115)
(x + 6)7
(x + 6)7
(x + 6)2
Solve the problem. 116) Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 12.22 ounces and a standard deviation of 0.04 ounce. Find the probability that the bottle contains between 12.12 and 12.18 ounces. A) 0.1649 B) 0.1525 C) 0.8351 D) 0.8475
117) Find the volume of the solid generated by revolving the region in the first quadrant bounded by y = ex and the x-axis, from x = 0 to x = ln 7, about the y-axis. A) 14 ln 7 B) 2 (7ln 7 - 6) C) 7 ln 7
118)
B) Yes
Solve the problem.
24
119) Find the average value of the function y = A) 4
49 - 36x2
B) 1
21
over the interval from x = 0 to x =
C) 8
3
7 . 12
3
2
120) The slope of a curve is given by dy = 98x + 20 . Find the equation of the curve if it passes dx
16x4 + 4x2
1 ,4 . 2
A) y = - 5 + 9 tan-1 2x + 14 - 3
B) y = 5 - 9 tan-1 2x - 14 + 3
C) y = - 5 + 9 tan-1 2x + 14 - 9
D) y = 5 + 9 tan-1 2x - 14 - 9
x x
4 4
8
x
16
x
22
119)
D) 2
7
Solve the problem by integration.
through
117)
D) 2 (7ln 7 - 7)
Determine whether the function is a probability density function over the given interval. 118) f(x) = 1 x, 0 x 6 18
A) No
116)
4 4
8 16
120)
Use Simpson's Rule with n = 4 steps to estimate the integral. 2 121) (x4 + 2) dx 0 A) 125 B) 67 24 8 1
122) 0
3
1 + x2
121) C) 177
D) 125
16
12
dx
122)
A) 15,969
B) 8011
3400
C) 15,969
1700
D) 8011
6800
3400
Determine whether the improper integral converges or diverges. ln 6 2 123) x-3 e1/x dx 0 A) Converges B) Diverges Evaluate the integral. 1
124)
x 64 + x2
dx
124)
2 A) - 1 ln 8 - 64 + x + C
2 B) - 1 ln 8 + 64 + x + C
2 C) 1 ln 8 + 64 + x + C
2 D) - 1 ln 8 + 64 + x + C
8
8
125)
123)
x
8
x
8
x2 x
cos t dt 2 sin t - 13 sin t + 40
125)
A) 1 ln t - 8 - 1 ln t - 5 + C
B) ln sin t - 8 - ln sin t - 5 + C
C) 1 ln sin t - 8 - 1 ln sin t - 5 + C
D) 1 ln sin t - 8 + 1 ln sin t - 5 + C
3
3
3
3
3
3
Evaluate the integral by making a substitution and then using a table of integrals.
126)
sin-1
x + 2 dx
A) 1 (2x+3) sin-1 x + 2 + 2
126) x+2 2
-x - 1 + C
B) 1 (x + 2) sin-1 x + 2 - x + 2 + C 2
C) 1 (x + 3) tan-1 x + 2 -
x+2 +C 2
D) 1 (2x + 3) sin-1 x + 2 +
x+2 4
2 4
-x - 1 + C
23
Solve the problem.
127) Find an upper bound for ET in estimating A) 8
B) 1
49
2
(6x2 + 7) dx with n = 7 steps.
1
C) 1
588
128) Find the first-quadrant area bounded by y =
D) 1
98
Solve the problem by integration.
1
x3 + 5x2 + 4x
127) 49
, x = 1, and x = 4.
128)
A) 1 ln 4.503599627e+15
B) 1 ln 1,717,986,918,399,999
C) 12 ln 1,717,986,918,399,999
D) 1 ln 2.251799814e+15
12
1.717986918e+15
12
175,921,860,444,159,904
6
175,921,860,444,159,904
5.629499534e+15
Evaluate the improper integral or state that it is divergent. 25e-25x dx
129)
129)
0
A) 1
B) 0
Evaluate the integral. 6e4t + 36e2t + 6
130)
e2t + 6
C) Divergent
D) -1
dt
130)
A) e2t + ln et + 6 - 1 ln e2t + C
B) 3e2t - t + ln e2t + 6 + C
C) e2t + t - 1 ln et + 6 + C
D) 3e2t + t - 1 ln e2t + 6 + C
2
2
2
Integrate the function. dx
131)
131)
2 (49x2 + 1)
A) tan-1 7x C) ln
7x
49x2 + 1
+C
B)
7x
49x2 + 1
+C
D) 1 tan-1 7x +
49x2 + 1 + 7x + C
14
x +C 98x2 + 2
Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions. 3 2 132) 4x + 4x + 7 dx 132) x2 + x
B) 4x2 + 7 ln x + 1 - 7 ln x + C D) 2x2 + 7 ln x - 1 - 7 ln x + C
A) 7 ln x + 1 - 7 ln x + C C) 2x2 - 7 ln x + 1 + 7 ln x + C
24
Expand the quotient by partial fractions. 5x + 2 133) (x - 5)(x - 2)
133)
A)
9 4 + x-5 x-2
B) 27 + 12
C)
9 -4 + x - 5 (x - 5)(x - 2)
D)
x-5
x-2
9 -4 + x-5 x-2
Evaluate the integral.
134)
e2x cos 7x dx 2x
A) e
53 2x
C) e
2
134)
[7 sin 7x + 2 cos 7x] + C
B) 1 [7 e2x sin 7x + 2 cos 7x] + C
[ sin 7x + cos 7x] + C
D) e
53
2x
53
Use a trigonometric substitution to evaluate the integral. 1 ex dx 135) 2x 0 36 - e A) 0.641 B) 0.043
[7 sin 7x - 2 cos 7x] + C
135) C) 0.053
D) 0.260
Find the area or volume.
136) Find the volume of the solid generated by revolving the area under y = 7e-x in the first quadrant about the y-axis. A) 7
B) 28
C) 14
D) 1
Solve the problem. 137) Find the volume of the solid generated by revolving the region bounded by the curve y = ln x, the x-axis, and the vertical line x = e2 about the x-axis.
B) 2 (e2 - 1)
e
A)
136)
C) (e2 - 1)
137)
D) (e - 1)
Evaluate the integral by making a substitution (possibly trigonometric) and then applying a reduction formula. dx 138) 138) 2 (4 - x2 )
A) 1
x
8 4 - x2
C)
x
8(4 - x2 )
-
1 x+2 ln 4 x-2
B) 1 ln x + 2 + C
+C
4
D) 1
x-2
x 1 x+2 + ln 8 4 - x2 4 x-2
+C
25
+C
Find the surface area or volume.
139) The region between the curve y =
1 , -1.0 x 1.3, and the x-axis is revolved about the 1 + cos x
139)
x-axis to generate a solid. Use a table of integrals to find the volume of the solid generated to two decimal places. A) 4.31 B) 4.10 C) 4.83 D) 5.23
Determine whether the improper integral converges or diverges. 9 dx 140) 2 0 81 - x A) Diverges B) Converges
140)
Use any method to evaluate the integral.
141)
5 csc3 x dx tan x
141)
A) - 5 csc3 x + C
B) - 5 csc4 x + C
C) 5 csc4 x cot x + C
D) - 5 cot3 x + C
3
3
4
3
Express the integrand as a sum of partial fractions and evaluate the integral. 4x3 - 5x2 + 8x - 10 142) dx (x2 + 2)(x - 2)3
A) - 4 x-2
C)
3
2(x - 2)2
B) 2 ln x - 2 - 2 -
+C
x-2
4 5 +C x - 2 2(x - 2)3
Evaluate the improper integral. 1 dx 143) 2/3 x -27 A) 0
142)
D)
3
2(x - 2)2
+C
1 2 3 +C 2 x 2 x +2 2(x - 2)2
143) C) 4
B) -6
Solve the problem. 144) Find the length of the curve y = ln(sin x), /6 x A) ln( 3) B) ln( 3 + 1)
D) 12
/2
C) ln( 3 + 2)
26
144) D) 1 - ln( 3 + 2)
Use integration by parts to establish a reduction formula for the integral. cosn x dx
145)
145)
A)
1 n-1 cosn x dx = cosn-1 x sin x n n
cosn-1 x dx
B)
1 n-1 cosn x dx = cosn-1 x sin x + n n
cosn-2 x dx
C)
cosn x dx = cosn-1 x sin x - (n - 1)
sin x cosn-2 x dx
D)
cosn x dx = - cosn-1 x sin x + (n - 1)
cosn-2 x dx
Find the area or volume.
146) Find the volume of the solid generated by revolving the region under the curve y = 10 , from x = 1 x
to x = , about the x-axis.
A) 10
B) 1
C) 20
10
146)
D) 10
Solve the problem.
147) Estimate the area of the surface generated by revolving the curve y = cos 2x, 0 x x-axis. Use the Trapezoidal Rule with n = 6. A) 5.108 B) 7.091
C) 4.606
about the
4
D) 4.652
Evaluate the integral. dx
148)
148)
2 (25 - x2)
A)
x
50(25 - x2 )
147)
B) 1
x 1 x+5 ln +C 50 25 - x2 10 x-5
+C
D) 1 ln x + 5 + C
C) 1
x 1 x+5 ln + +C 50 25 - x2 10 x-5
10
x-5
Evaluate the improper integral or state that it is divergent.
149) 6
dx 2 x - 25
A) 1 ln 11 10
149) B) 1 ln 6
C) 1 ln 1
10
10
Use the Trapezoidal Rule with n = 4 steps to estimate the integral. 1 150) (x2 + 5) dx -1 A) 43 B) 55 C) 32 2 8 3 27
6
D) - 1 ln 11 5
150) D) 43 4
Use Simpson's Rule with n = 4 steps to estimate the integral. 0 151) sin t dt -1 A) -1 - 2 2 B) - 2 + 2 6
151) C) - 1 + 2 2
D) - 1 + 2
6
4
Evaluate the integral. dx 152) 2 x + 4x + 13
152)
A) 1 sin-1 x + 2 + C
B) 3 tan-1 x + 2 + C
C) (2x + 4) ln x2 + 4x + 13 + C
D) 1 tan-1 x + 2 + C
3
3
3
3
3
Use reduction formulas to evaluate the integral.
153)
sin3 2x sec5 2x dx
153)
4
6
A) tan 2x sec 2x + C
B) tan 2x + C
8
6
6
4
C) sec 2x + C
D) tan 2x + C
6
8
Solve the problem. 154) A data-recording thermometer recorded the soil temperature in a field every 2 hours from noon to midnight, as shown in the following table. Use the Trapezoidal Rule to estimate the average temperature for the 12-hour period. Round your answer to the nearest hundredth if necessary. Time Temp (°F) Noon 62 2 63 4 65 6 65 8 64 10 64 Midnight 63
A) 63.92°F
B) 63.94°F
C) 76.7°F
28
D) 74.33°F
154)
Evaluate the integral by making a substitution (possibly trigonometric) and then applying a reduction formula.
155)
et csc3 (et - 4) dt
155)
A) 1 [csc(et - 4) cot(et - 4) + ln csc(et - 4) + cot(et - 4) ] + C 2
B) - 1 [csc(et - 4) cot(et - 4) - ln csc(et - 4) + cot(et - 4) ] + C 2
C) 1 [csc(et - 4) cot(et - 4) - ln csc(et - 4) + cot(et - 4) ] + C 2
D) - 1 [csc(et - 4) cot(et - 4) + ln csc(et - 4) + cot(et - 4) ] + C 2
Solve the problem.
156) Find the area bounded by y(4 + 9x2) = 3, x = 0, y = 0, and x = 3. A) 3 sin-1 9 2
B) sin-1 9 2
C) 1 tan-1 3 2 2
156) D) 1 tan-1 9 2 2
Use integration by parts to establish a reduction formula for the integral.
157)
xn ex dx
157)
A)
xn ex dx = xn ex - n
xn+1 ex dx
B)
xn ex dx = xn ex -
C)
xn ex dx = xn ex + n
xn-1 ex dx
D)
xn ex dx = xn ex - n
xn-1 ex dx
1 n+1
xn-1 ex dx
Express the integrand as a sum of partial fractions and evaluate the integral. 8 5x3 - 9x 158) dx 4 - 81 x 4 A) -3.67 B) 4.16 C) 7.34
158) D) 3.67
Evaluate the integral by using a substitution prior to integration by parts.
159)
ln 3x 2 dx
159)
A) x ln 3x 2 - x ln 3x + x + C C) x ln 3x 2 - 2x ln 3x + C
B) x ln 3x 2 - 2x ln 3x + 2x + C D) x ln 3x 2 + 2x ln 3x - 2x + C
29
Evaluate the integral.
160)
sin 6x cos 3x dx
160)
A) - 1 cos 9x - 1 cos 3x + C
B) - 1 cos 9x - 1 sin 9 x + C
C) 1 sin 3x - 1 sin 9x+ C
D) 1 sin 3x + 1 sin 9x+ C
18
6
6
18
18
6
18
18
Find the surface area or volume. 161) Use numerical integration with a programmable calculator or a CAS to find, to two decimal places, the area of the surface generated by revolving the curve y = sin 2x, 0 x x-axis. A) 7.21
B) 1.48
C) 14.42
2
, about the
D) 9.29
Solve the problem by integration. 162) Under certain conditions, the velocity v (in m/s) of an object moving along a straight line as a 2t2 + 14t + 11 function of the time t (in s) is given by v = . Find the distance traveled by the object (2t + 1)(t + 2)2 during the first 3 s. A) 1.046 m
B) 1.747 m
C) 4.792 m
163)
A) 1 tan-1
6 + 5x +C 6
6 6
162)
D) 2.846 m
Evaluate the integral. dx 163) x 6 + 5x
C) 1 ln
161)
6 + 5x 6 + 5x +
B) 2 6 + 5x + 2 6 tan-1
6 +C 6
D) 1 ln 6
6 + 5x + 6 + 5x -
6 + 5x +C 6
6 +C 6
Solve the problem. 164) There are 3 balls in a hat; one with the number 2 on it, one with the number 6 on it, and one with the number 8 on it. You pick a ball from the hat at random and then you flip a coin to obtain heads (H) or tails (T). Determine the set of possible outcomes. A) {2 H, 2 T, 6 H, 6 T, 8 H, 8 T} B) {2 6 8 H, 2 6 8 T}
C) {2 H, 6 H, 8 H}
D) {2 H T, 6 H T, 8 H T}
Evaluate the integral.
165) Use the formula
f-1 (x) dx = xf-1 (x) -
164)
x
d -1 f (x) dx to evaluate the integral. dx
cot-1 x dx
A) x cot-1 x - ln x2 + 1 + C C) x cot-1 x + x + C
B) x cot-1 x + ln x + C D) x cot-1 x + ln x2 + 1 + C
30
165)
166)
x3 cos 6x dx
166)
A) 1 x3 sin 6x + 1 x2 cos 6x - 1 x sin 6x - 1 cos 6x + C 6
12
36
216
B) 1 x3 sin 6x - 1 x2 cos 6x + 1 x sin 6x + 1 cos 6x + C 6
12
36
216
C) 1 x3 sin 6x + 1 x2 cos 6x - 1x sin 6x - 1 cos 6x + C 6
2
D) 1 x3 cos 6x + 1 x2 sin 6x - 1 x cos 6x - 1 sin 6x + C 6
12
36
216
Use Simpson's Rule with n = 4 steps to estimate the integral. 2 167) 7x2 dx 0 A) 49 B) 77 3 4
167) C) 56
D) 28
3
3
Integrate the function. dx 168) x 64x2 - 16
168) B) 1 sin-1 2x + C
A) 2 sin-1 2x + C
C) 2 sec-1 2x + C
8
D) 1 sec-1 2x + C 4
Evaluate the integral.
169)
tan5 3x dx
169)
A) 1 tan4 3x - 1 tan2 3x + 1 ln cos x + C 12
6
3
B) 1 tan4 3x - 1 tan2 3x - ln cos x + C 4
2
C) 1 tan4 3x - 1 tan2 3x - 1 ln cos x + C 12
6
3
D) - 1 tan4 3x + 1 tan2 3x - 1 ln cos x + C 12
170)
6
3
25 - x2 dx
170)
A) x
25 - x2 +
25 x sin-1 +C 2 25
B) sin-1 x + C
C) x
25 - x2 +
25 x sin-1 + C 2 5
D) x
2 2
5
2
31
25 - x2 -
25 ln x + 2
25 - x2 + C
Evaluate the improper integral. 13 dx 171) x-9 0 A) -2
171) B) 5
C) 10
D) 4
Determine whether the improper integral converges or diverges.
172) 1
dx x3/8 + 3
172)
A) Converges
B) Diverges
Evaluate the integral. 2 cos3 5x dx
173)
173)
A) 2 sin 5x + 2 sin3 5x + C
B) 2 sin 5x - 2 sin3 5x + C
C) 2 sin 5x - 2 cos3 5x + C
D) 2 sin 5x - 2 sin3 5x + C
5
15
5
5
15
15
3
Use reduction formulas to evaluate the integral. sin2 3x cos2 3x dx
174)
174)
A) - 1 sin 3x cos3 3x + x + 1 cos 6x + C
B) - 1 sin 3x cos3 3x + x + 1 sin 6x + C
C) - 1 sin 3x cos2 3x + x + 1 sin 6x + C
D) - 1 sin 3x cos3 3x + x + 1 sin 6x + C
12 12
8 8
48
12
48
3
8
8
48
3
Evaluate the integral.
175) -
1 + cos x dx 2
A) 2
175) B) 0
C) 4
D) 1
Solve the problem. 176) The pulse rate for a species of small mammal follows a normal distribution with a mean µ = 215 beats per minute and standard deviation = 30 beats per minute. In a population of 900 of these mammals, how many would you expect to have a pulse rate between 203 and 251? A) 621 of these mammals B) 486 of these mammals
C) 432 of these mammals
D) 450 of these mammals
32
176)
Evaluate the integral. 177) -sin t (3 cos t + 4) dt cos2 t - 14cos t + 49
177)
A) 3 ln cos t - 7 -
25 +C cos t - 7
B) 3 ln t - 7 - 25 + C
C) 4 ln cos t - 7 -
21 +C cos t - 7
D) 3 ln cos t - 7 +
t- 7
25 +C cos t - 7
Use a trigonometric substitution to evaluate the integral. dx 178) 2 x 1+x
178)
A) 1 tan-1 x + C
B) 1 sin-1 x + C
C) 1 ln x + C
D) tan-1 x + C
2
2
2
Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions. x3 179) 179) dx x2 + 2x + 1 2 A) x - 2x + 3ln x + 1 + 1 + C
2 B) x - 2x + 3ln x + 1 - 1 + C
C) 3ln x - 2 + 3 -
D) x - 2x - 3ln x + 1 +
2
x+1
x+1
1
(x + 1)2
+C
2
x+1
2
1
2
Use a trigonometric substitution to evaluate the integral. ln 3 et dt 180) 49 + e2t 0 A) 0.022 B) 0.038
(x + 1)2
+C
180) D) 0.263
C) -0.038
Determine whether the improper integral converges or diverges.
181) 5
dx x5/2
181)
A) Diverges
B) Converges
Evaluate the integral.
182)
(2x-1) ln(21x) dx
182) 2
2
A) x2 - x ln 21x - x + x + C
B) x2 - x ln 21x - x + 2x + C
2
2
2
2
4
2
C) x - x ln 21x - x + x + C
D) x2 - x ln 21x - x2 + x + C
33
Find the area or volume.
183) Find the volume of the solid generated by revolving the region under the curve y = 2e-2x in the first quadrant about the y-axis. A) 2 3 B) 1
C) 4
D) 2
Express the integrand as a sum of partial fractions and evaluate the integral. 2 184) 48x + 32x + 3 dx 2 (16x2 + 1)
A) 3 tan-1(4x) -
1 16x2 + 1
C) ln 16x2 + 1 -
1 +C 16x2 + 1
4
1 (16x2 + 1)
2
B) 3 tan-1 (4x) -
+C
4
184) 1 +C 16x2 + 1
D) 3 tan-1 (16x) + 4
1 +C 16x2 + 1
Evaluate the integral. 6x - 1 185) dx 6x - 1
185)
A) 3 x x + 6x + C /20
C) 2 x 6x + x + C
B) 4x2 + x + C
2
186)
183)
3
D) 9x 6x + x + C
tan4 5t dt
186)
- /20
A) - 4
B)
15
10
-
4 15
C)
D)
10
15
-
2 15
Find the area or volume.
187) Find the area of the region bounded by the curve y = 8x-2 , the x-axis, and on the left by x = 1. A) 64 B) 4 C) 16 D) 8 Integrate the function. x2 + 4
188)
dx
7x2
C) ln
188) x2 + 4
+
x2 + 4 + C 7x
B) 1 ln
x2 + 4 + x
x - sin-1 + C 2
x2 + 4 + x
+
x2 + 4 + C 7x
D) 1 ln
x2 + 4 + x
-
A) x + ln
7 7
x2 + 4 + C 7x
Use the Trapezoidal Rule with n = 4 steps to estimate the integral. 3 189) (12x + 2) dx 1
A) 52
187)
189)
C) 65
B) 104
2
34
D) 26
Use Simpson's Rule with n = 4 steps to estimate the integral. 3 6 190) dx 2 1 x A) 423 B) 83 100 25
190) C) 1813
D) 1813
450
900
Evaluate the integral.
191)
x4 ln 8x dx
191)
A) 1 x5 ln 8x - 1 x6 + C
B) 1 x5 ln 8x + 1 x5 + C
C) 1 x5 ln 8x - 1 x5 + C
D) ln 8x - 1 x5 + C
5
30
5
5
25
25
5
Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions. 3 2 192) 27x + 54x + 30x + 5 dx 192) 9x2 + 18x + 9
A) 3 x2 + 1 ln 3x + 3 - 2
1 +C 3 3x + 3
B) 3 x2 + ln 3x + 3 - 2
C) 3 x2 + ln 3x + 3 -
2 +C 3x + 3
D) 3 x2 - 3x + 5 + C
2
3
2
1 +C 3 3x + 3
2 2
3x + 3
Use integration by parts to establish a reduction formula for the integral.
193)
secn x dx , n 1
193)
A)
secn x dx = secn-2 x tan x + (n - 2)
secn-2 x dx
B)
secn x dx =
1 n-1 secn x tan x n n
secn-1 x dx
C)
secn x dx =
1 n-2 secn-2 x tan x + n-1 n-1
D)
secn x dx = secn-2 x tan x - (n - 2)
secn-2 x dx
secn-2 x tan x dx
Determine whether the improper integral converges or diverges. dx
194) 10
6
194)
x-4
A) Converges
B) Diverges
35
Solve the problem. 195) Estimate the minimum number of subintervals needed to approximate the integral 4 (5x + 1)dx 0 with an error of magnitude less than 10-4 using the Trapezoidal Rule.
A) 0
B) 1
C) 2
195)
D) 3
Find the indicated probability. 196) f(x) = e-x; [0, ), P(x 3) A) 0.0498
196)
B) 0 C) 0.9502 D) The function f(x) is not a probability density function. Evaluate the integral.
197)
(5x + 7) e-3x dx
197)
A) 5 x e-3x + 26 e-3x + C
B) - 5 x e-3x - 26 e-3x + C
C) -15x e-3x - 66 e-3x + C
D) - 5 x e-3x - e-3x + C
3
9
3
9
3
Solve the problem.
198) The voltage v (in volts) induced in a tape head is given by v = t2 e3t, where t is the time (in
198)
seconds). Find the average value of v over the interval from t = 0 to t = 2. Round to the nearest volt. A) 1564 volts B) 40 volts C) 194 volts D) 6 volts
Use Simpson's Rule with n = 4 steps to estimate the integral. 1 3 199) dx 1+x 0 A) 1171 B) 1747 560 1680 Evaluate the integral. 4+x
200)
16 - x2
199) C) 1747
D) 1171
840
840
dx
200)
A) sin-1 x - 2 16 - x2 + C
B) 1 tan-1 x + 1 16 - x2 + C
C) 4 sin-1 x - 16 - x2 + C
D) 4 tan-1 x - 16 - x2 + C
16
4
4
4
4
36
2
Use various trigonometric identities to simplify the expression then integrate.
201)
cos6
sin 2 d
201)
A) 2 cos6 + C
B) 1 cos7 + C
7
C) - 1 cos8 + C
8
4
D) - 2 cos9 + C 9
Find the surface area or volume. 202) Use an integral table and a calculator to find to two decimal places the area of the surface generated by revolving the curve y = x2 /2 , -3 x 3, about the x-axis.
A) 160.59
B) 196.27
C) 140.14
202)
D) 114.20
Solve the problem. 203) Estimate the minimum number of subintervals needed to approximate the integral 5 1 dx x-1 2 with an error of magnitude less than 10-4 using Simpson's Rule.
A) 24
B) 570
C) 48
203)
D) 6
Provide an appropriate response. 204) The error formula for the Trapezoidal Rule depends upon i) f(x). ii) f'(x). iii) f'' (x). iv) the number of steps A) iii and iv B) i and iii C) i, iii, and iv
204)
D) ii and iv
Evaluate the integral.
205)
y3 e-5y dy
A) -e-5y 1 y3 + 3 y2 + 6 y + 6
+C
B) - 1 y4 e-5y + C
C) e-5y 1 y3 - 3 y2 + 6 y - 6
+C
D) - 1 e-5y y3 + y2 + y + 6 + C
5
5
206)
205) 25
25
125
125
625
625
20 5
dx 5 + 13 sin 2x
206)
A) - 1 ln 5 + 13 sin 2x + 12 cos 2x 12
5 + 13 sin 2x
B) - 1 ln 13 + 5 sin 2x + 12 cos 2x
+C
24
C) 1 ln 5 + 13 sin x + 12 cos x + C 24
5 + 13 sin 2x
+C
D) - 1 ln 13 + 5 cos 2x + 12 sin 2x + C
5 + 13 sin 2x
24
37
5 + 13 sin 2x
csc3 4t dt
207)
207)
A) - 1 csc 4t cot 4t + 1 ln csc 4t + cot 4t + C 2
2
B) - 1 csc 4t cot 4t - 1 ln csc 4t + cot 4t + t + C 2
2
4
C) - 1 csc 4t cot 4t - 1 ln csc 4t + cot 4t + C 8
8
D) 1 csc 4t cot2 4t - 1 ln csc 4t + cot 4t + C 8
8
Solve the problem.
208) Find an upper bound for ET in estimating A) 0
B) 4 75
4
(8x + 4) dx with n = 10 steps.
0
C) 8
75
Use Simpson's Rule with n = 4 steps to estimate the integral. 9 209) x dx 0 A) 81 B) 81 4
208) D) 32 75
209) C) 81 2
D) 135 4
Evaluate the improper integral or state that it is divergent. 12xe2x dx
210)
210)
0
A) 1.6667
B) 1.3333
C) 2.6667
D) Divergent
Solve the problem. 211) The following table shows the rate of water flow (in gal/min) from a stream into a pond during a 30-minute period after a thunderstorm. Use the Trapezoidal Rule to estimate the total amount of water flowing into the pond during this period. Round your answer to the nearest gallon. Time (min) Rate (gal/min) 0 200 5 250 10 300 15 250 20 220 25 200 30 150 A) 6983 gal
B) 6975 gal
C) 6383 gal
38
D) 7850 gal
211)
For the given probability density function, over the stated interval, find the requested value. 212) f(x) = 1 x2, over [-2, 3]; Find the mean. 7
A) 2.849934139e+15
B) 2.568459162e+15
C) 2.498090418e+15
D) 2.286984186e+15
9.851624185e+14
212)
9.851624185e+14
9.851624185e+14
9.851624185e+14
Evaluate the improper integral or state that it is divergent.
213) 1
9 dx (1 + x2 )tan-1 x
A) 9 ln 1 +
2
213) B) 9 1 + 2
2
C) 9 ln
2
Use Simpson's Rule with n = 4 steps to estimate the integral. 0 214) sin x dx A) - 1 + 2 B) - 1 + 2 2 4 6
2
D) 9 ln 2
214) C) -(1 + 2 2)
D) - 2 + 2 6
Solve the problem by integration. 215) The force F (in N) applied by a stamping machine in making a certain computer part is 3x F= , where x is the distance (in cm) through which the force acts. Find the work done 2 x + 7x + 9 by the force from x = 0 to x = 0.4 cm. A) 0.0846 N· cm B) 0.0173 N· cm
C) 6.7653 N· cm
Solve the initial value problem for y as a function of x. 216) x dy = x2 - 4, x 2, y(2) = 0 dx
A) y = 2
215)
D) 0.0058 N· cm
216)
x2 - 4 x - sec-1 2 2
B) y =
C) y = 2 ln x
D) y =
2
39
x2 - 4 2 sec-1 (x/2) x2 - 4 -x+2 2
Evaluate the integral. /15 217) sec3 5x dx - /15
217)
Give your answer in exact form. A) 2 3 + 1 ln(2 3) 5 10
C)
B) 2 3 + 1 ln(7 + 4 3) 5
3 1 ln(7 + 4 3) + 5 10
10
D) 2 + 1 ln 4 5
10
sin 8t cos 7t dt
218) 0
218)
A) 8
B) 16
15
C) 1
15
15
Use the Trapezoidal Rule with n = 4 steps to estimate the integral. 1 9 219) dx 1+x 0 A) 3513 B) 1747 C) 2229 560 560 280
D) 17 15
219) D) 3513 280
Express the integrand as a sum of partial fractions and evaluate the integral. x3 220) dx x2 + 4x + 4
A) 12 ln x - 4 + 12 x+2
8
(x + 2)2
2
C) x - 4x - 12 ln x + 2 + 2
8
2 B) x - 4x + 12 ln x + 2 + 8 + C
+C
(x + 2)2
220)
2
x+2
2 D) x - 4x + 6 ln x + 2 - 4 + C
+C
2
x+2
Evaluate the integral. 9x 221) dx x-5
221)
A) 6(x - 5)3/2 + C
B) 6 x + 5(x - 10) + C
C) 27 (x - 5)3/2 + C
D) 6 x - 5(x + 10) + C
2
Evaluate the integral by making a substitution and then using a table of integrals. ln x 222) dx x(9 + ln x)
A) - 9 ln ln x + 9 + C C) ln x - 9 ln x + 9 + C
B) ln x + x - 9 ln ln x + 9 + C D) ln x - 9 ln ln x + 9 + C
40
222)
Solve the problem. 223) An oil storage tank can be described as the volume generated by revolving the area bounded by 24 y= , x = 0, y = 0, x = 2 about the x-axis. Find the volume (in m 3 ) of the tank. 2 64 + x
A) 18.5 m 3
B) 0.770 m 3
C) 457 m 3
223)
D) 55.4 m 3
Use integration by parts to establish a reduction formula for the integral.
224)
cotn x dx , n 1
224)
A)
cotn x dx =
-1 cotn-2 x n-1
cotn-1 x dx
B)
cotn x dx =
1 cotn-1 x + n-1
cotn-1 x dx
C)
-1 cotn x dx = cotn-1 x n-1
cotn-2 x dx
D)
cotn x dx = - cotn-1 x +
1 n-1
cotn-2 x dx
For the given probability density function, over the stated interval, find the requested value. 9 9 1 x 8 225) f(x) = x2 ; Find the median. 0 1 A) 2
Otherwise
B) 9 ln 9
C) 18
8
225)
D) 9
17
16
Determine whether the improper integral converges or diverges. 1 1 226) dx x ln x -1 A) Converges B) Diverges
226)
Evaluate the integral.
227)
4 cos4 3x dx
227)
A) 3x + 1 sin 3x + 1 sin 12x + C
B) 3x + 1 sin 3x + 1 sin 6x + C
C) 3 x + 2 sin 6x + 1 sin 12x + C
D) 3 x + 1 sin 6x + 1 sin 12x + C
3
2
3
6
3
8
2
3
Determine whether the improper integral converges or diverges. 2 228) x - 1 -4/3 dx 0 A) Diverges B) Converges
41
24
24
228)
Solve the problem by integration.
229) Find the volume generated by revolving the first-quadrant area bounded by y = and x = 2 about the y-axis. A) 5 ln 720 B) 10 ln 9 2 5
20 4 x + 8x2 + 12
229)
D) 5 ln 9
C) 5 ln 720
5
Determine whether the improper integral converges or diverges.
230) 1
1 dx 6 x +2
230)
A) Converges
B) Diverges
Solve the problem. 231) Estimate the minimum number of subintervals needed to approximate the integral 2 (5x4 - 9x)dx 1 with an error of magnitude less than 10-4 using Simpson's Rule.
A) 22
B) 12
C) 10
231)
D) 14
232) A rectangular swimming pool is being constructed, 18 feet long and 100 feet wide. The depth of
232)
the pool is measured at 3-foot intervals across the length of the pool. Estimate the volume of water in the pool using Simpson's Rule. Width (ft) Depth (ft) 0 4 3 4.5 6 5 9 6 12 6.5 15 7 18 8 A) 9300 ft3
B) 7000 ft3
C) 10,500 ft3
D) 12,300 ft3
233) Find the area of the region enclosed by the curve y = x sin x and the x-axis for 8 x 9 . A) 16 B) 17 C) 0 D) 16
233)
Evaluate the integral.
234)
9 - x2 dx x
234)
2 A) 9 - x2 - 3 ln 3 + 9 - x + C
B) -sin-1 x -
C) 9 - x2 - sin-1 x + C
2 D) 9 - x2 + 3 ln 3 + 9 - x + C
x
3
3
9 - x2 +C x
x2
42
Evaluate the integral by making a substitution and then using a table of integrals. dx 235) x(16 + (ln x)2 )
A) 1 sin-1 ln x 4
4
B) 1 tan-1 ln x + C
+C
4
C) 1 sin-1 x + C 4
4
D) 1 tan-1 x
4
4
Evaluate the integral. f-1 (x) dx = xf-1 (x) -
236) Use the formula
235)
x
4
+C
d -1 f (x) dx to evaluate the integral. dx
236)
sin-1 x dx 1
A) x sin-1 x +
1 - x2
B) x sin-1 x + x + C
+C
C) x sin-1 x - 1 - x2 + C
D) x sin-1 x + 1 - x2 + C
Solve the problem. 237) There are 3 balls in a hat; one with the number 1 on it, one with the number 5 on it, and one with the number 8 on it. You pick a ball from the hat at random and then you flip a coin to obtain heads (H) or tails (T). Determine the set of possible outcomes, then find the probability that the number on the ball is greater than 6. A) 2 B) 1 C) 1 D) 1 3 4 3 6
237)
Evaluate the integral.
238)
cos
2
cos
5
A) 5 sin 3 3
10
d
238) +
5 7 sin 7 10
+C
+
5 7 cos +C 7 10
D) 5 sin 3
-
5 7 sin 7 10
3
C) 5 sin 3 + 5 sin 7 + C 3
B) 5 cos 3
7
3
10 10
+C
Evaluate the integral by using a substitution prior to integration by parts. 239) 1 e 6x + 3 dx 2
A)
6x + 3 e 6x + 3 + C 6
B) 1 e 6x + 3 [ 6x + 3 - 1] + C 6
D) 1 e 6x + 3 [ 6x + 3 - 6] + C
C) (6x + 3) e 6x + 3 + C
6
43
239)
Solve the problem. 240) Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 14.14 onces and a standard deviation of 0.04 ounce. Find the probability that the bottle contains more than 14.14 ounces of beer. A) 0 B) 0.4 C) 0.5 D) 1
240)
Determine whether the improper integral converges or diverges. x6 dx ex - 1
241) 1
241)
A) Diverges
B) Converges
Expand the quotient by partial fractions. 242) x + 7 (x + 2)2
242)
A)
5 1 + x + 2 (x + 2)2
B)
1 5 + x + 2 (x + 2)2
C)
1 6 + x+2 x+7
D)
1 -5 + x + 2 (x + 2)2
Provide an appropriate response. 243) The "trapezoidal" sum can be calculated in terms of the left and right-hand sums as ? . A) left-hand sum + right-hand sum B) left-hand sum - right-hand sum 2
C) left-hand sum + right-hand sum
D) None of the above is correct.
Use the Trapezoidal Rule with n = 4 steps to estimate the integral. 2 244) 2x2 dx 0 A) 11 B) 15 C) 16 2 2 3
244) D) 11
Solve the problem. 245) Estimate the minimum number of subintervals needed to approximate the integral 3 (3x3 - 2x)dx 0 with an error of magnitude less than 10-4 using Simpson's Rule.
A) 2
243)
B) 16
C) 1
44
D) 0
245)
246) Find an upper bound for ET in estimating A) 768 49
B) 3000 49
4
(6x3 + 7x) dx with n = 7 steps.
-1
C) 375
246) D) 1500
49
49
Evaluate the integral. f-1 (x) dx = xf-1 (x) -
247) Use the formula
f(y) dy , y = f-1 (x) to evaluate the integral.
247)
cot-1 x dx
A) x cot-1 x + ln sin(cot-1 x) + C C) x cot-1 x - x + C 248)
B) x cot-1 x - ln sin(cot-1 x) + C D) x cot-1 x - ln sin x + C
(x + 6)2 tan-1 x + (3x - 19) (x + 6) dx 2 (x2 + 1) (x + 6)
248)
-1 2 A) (tan x) - ln x + 6 - 5 tan-1 x + ln(x2 + 1) + C 2
-1 2 B) (tan x) - ln x + 6 + 1 ln(x2 + 1) + C 2
2
-1 2 C) (tan x) - ln x + 6 - 3 tan-1 x + 1 ln(x2 + 1) + C 2
2
-1 2 D) (tan x) - 3 tan-1 x + 1 ln(x2 + 1) + C 2
249)
2
sec3 3x dx
249)
A) 1 sec23x tan 3x + 1 ln sec 3x + tan 3x + C 6
6
B) 1 sec 3x tan 3x - 1 ln sec 3x + tan 3x + C 2
2
C) 1 sec 3x tan 3x + 1 ln sec 3x + tan 3x + C 6
6
D) 1 sec 3x tan 3x + x + C 6
6
Use reduction formulas to evaluate the integral.
250)
8 cos3 5x dx
250)
A) 8 sin 5x - 8 sin3 5x + C
B) 8 sin 5x - 8 sin3 5x + C
C) 8 sin 5x - 8 cos3 5x + C
D) 8 sin 5x + 8 sin3 5x + C
5 5
15
3
15
5
45
15
Integrate the function. 4 dx 251) 6 + x2
A) 4 ln
251) 6 + x2
+ C
C) x + ln 4 + 6 + x2
B)
2 + C 2 x +6
D) 4 ln x + 6 + x2
+ C
+ C
Evaluate the integral. -8x cos 3x dx
252)
252)
A) - 8 cos 3x - 8 x sin 8x + C
B) - 8 cos 3x - 8x sin 3x + C
C) - 8 cos 3x - 8 x sin 3x + C
D) - 8 cos 3x - 8 sin 3x + C
9
3
9
3
3
9
3
dx
253)
253)
x 16 - x2 2 A) - 1 ln 4 + 16 - x + C
B) -
2 C) 16 - x2 - 4 ln 4 + 16 - x
2 D) - 1 ln x + 16 - x + C
4
x
x
16 - x2 +C 16x 4
x
Use any method to evaluate the integral. 2 254) tan x dx csc x
254)
2
A) cos x + C
B) cos x + C
C) sin x - sec x + C
D) cos x + sec x + C
2
Evaluate the improper integral or state that it is divergent. 8 x7 e-x dx
255) -
255)
A) 1
B) - 1
8
C) 0
4
D) Divergent
Provide an appropriate response. +
256) A student knows that a
-1
dx = 72. Can
f(x) dx be found, and if so, what is it?
-
A) Yes, -72
B) No
46
256)
Use a trigonometric substitution to evaluate the integral. dx 257) x (1 + 81 ln2 x)
257)
A) 1 tan-1 (9 ln x) + C
B) 1 tan-1 (9 ln x) + C
C) 1 tan-1 (81 ln2 x) + C
D) 1 ln ( 1 + 81 ln2x) + C
9x
9
9
162
Solve the initial value problem for y as a function of x. 258) x2 - 49 dy = x, x > 7, y(14) = 0 dx
258)
A) y =
x2 - 49 x
B) y = x2 - 49
C) y =
x2 - 49 3 x 2
D) y = x2 - 49 - 7 3
Use any method to evaluate the integral.
259)
6x sin2 x dx
259) 3
A) 6x2 - 3 sin 2x - 3 cos 2x + C
B) 3x2 + sin x + C
C) 3 x2 - 3 sin 2x - 3 cos 2x + C
D) 3x2 - 3 sin 2x - 3 cos 2x + C
2
2
2
3
4
2
Solve the problem by integration.
260) Find the volume generated by rotating the area bounded by y = y = 0 about the y-axis. A) 1 ln 14 6 13
B) 1 ln 7 3
3
Solve the problem.
A)
1 51,200
1 , x = 5, x = 6, and 3 x + 8x2 + 7x
C) 1 ln 14
13
261) Find an upper bound for ET in estimating
4
6 5
1
(x - 1)2
C) 3,799,912,185,593,855
900,719,925,474,099,200
13
D) 3 ln 13
dx with n = 10 steps.
B)
68,719,476,736 8,589,934,591,999,995
D)
1 153,600
260)
14
261)
Evaluate the integral by making a substitution and then using a table of integrals.
262)
tan t · 1 - cos2 t dt
262)
A) ln sec t + cos t + C C) ln sec t + tan t - sin t + C
B) ln csc t - cot t - sin t + C D) ln sec t + tan t + sin t + C
47
Solve the problem. 263) Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 12.38 ounces and a standard deviation of 0.04 ounce. Find the probability that the bottle contains fewer than 12.28 ounces of beer. A) 0.4938 B) 0.0062 C) 0.5062 D) 0.9938 Evaluate the integral by making a substitution and then using a table of integrals. x 264) dx 4-x x +x 4-x +C 2
B)
C) 4 sin-1
x 2
D) 4 sin-1 x - 4 - x + C
4x - x2 + C
2+
264)
A) 4 sin-1
4 - x - 2 ln
263)
4-x +C x
2
Solve the problem.
265) Find an upper bound for ET in estimating
6x sin x dx with n = 11 steps.
0 Give your answer as a decimal rounded to four decimal places. A) 0.0427 B) 0.4025 C) 0.2563
265) D) 0.1281
Evaluate the integral.
266)
cos 9x cos 4x dx
266)
A) 1 sin 5x - 1 sin 13x + C
B) 1 cos 5x + 1 cos 13x + C
C) 1 sin9x + 1 sin 4x + C
D) 1 sin 5x + 1 sin 13x + C
10
26
10
10
26
10
Express the integrand as a sum of partial fractions and evaluate the integral. 3x2 + x + 72 267) dx x3 + 36x
A) 2 ln x + 1 ln x2 + 36 + sin-1 x + C 2
6
B) ln x + 1 ln x2 + 36 + tan-1 x + C 2
6
C) 2 ln x - 1 ln x2 + 36 - tan-1 x + C 2
D) 2 ln x + 1 ln x2 + 36 + 1 tan-1 x + C 2
6
6
48
26
26
267)
Evaluate the integral.
268)
2 sec4 x dx
268)
A) 2 tan x + 2 tan3 x + C
B) 2 tan3 x + C
C) - 2 tan3 x + C
D) 2(sec x + tan x)5 + C
3
3
3
For the given probability density function, over the stated interval, find the requested value. 269) f(x) = 1 x, over [1, 3]; Find the mean. 3
A) 5
B) 7.318349394e+15
C) 7.036874418e+15 2.53327479e+15
D) 2.53327479e+15
6
269)
2.53327479e+15
Evaluate the integral. 1/8 270) y tan-1 8y dy 0 1 A) 256 128
8.444249301e+14
(Give your answer in exact form.)
B) 1
C)
256
270) 4
-
1 2
D)
512
-
1 128
Find the surface area or volume. 271) Use an integral table and a calculator to find to two decimal places the area of the surface generated by revolving the curve y = x2 , -3 x 2, about the x-axis.
A) 384.47
B) 224.27
C) 448.55
271)
D) 314.54
Use integration by parts to establish a reduction formula for the integral.
272)
cscn x dx , n 1
272) -1 n-2 cscn-2 x cot x + n-1 n-1
A)
cscn x dx =
cscn-2 x dx
B)
cscn x dx = cscn-2 x cot x + (n - 2)
C)
cscn x dx =
-1 n-1 cscn-2 x cot x n-1 n
cscn-1 x dx
D)
cscn x dx =
-1 n-2 cscn-2 x cot x n-1 n-1
cscn-2 x cot x dx
cscn-2 x dx
Solve the problem. 273) Find the volume generated by revolving the curve y = cos 4x about the x-axis, 0 x /48 2 2 2 1 A) B) C) D) + + + 96 32 96 32 96 96 48
49
273)
Evaluate the integral. /4 274) sin7 y dy 0 A) 128 - 119 2 560
274) C) - 177 2
B) 16 35
Solve the problem.
275) Find an upper bound for ES in estimating A)
17,179,869,184 3,221,225,471,999,993
D) 256 - 177 2
560
6 2
560
1 dx with n = 8 steps. x-1
275)
B) 70,368,744,177,664
4,222,124,650,659,835
C) 140,737,488,355,328
D) 8,549,802,417,586,173
4,222,124,650,659,835
33,776,997,205,278,682
Express the integrand as a sum of partial fractions and evaluate the integral. 5x2 + x + 1 276) dx (x2 + 2)(x - 9)
A) 5 ln x - 9 + tan-1 x 2 + C
B) ln x - 9 +
C) 5 ln x - 9 + 1 tan-1 x + C
D) 5 ln x - 9 +
2
2
Integrate the function. x2 - 16
277)
x
2
276) 2 x 2 tan-1 +C 2 2 2 x 2 tan-1 +C 2 2
dx
277)
A) 4
x2 - 16 x - sin-1 4 4
+C
B) 4 ln x2 - 16 - x
C)
x2 - 16 x - sec-1 16 4
+C
D) 4
Evaluate the integral. 3 278) ln 4x dx 1 A) 8.07
4
+C
x2 - 16 x - sec-1 4 4
+C
278) B) 4.07
C) 11.1
D) -1.93
Solve the problem. 279) Find the volume of the solid generated by revolving the region in the first quadrant bounded by the x-axis and the curve y = sin 6x, 0 x /6 about the line x = /6. 2 A) 1 B) 1 2 C) D) 1 2 18 18 18 36
50
279)
Integrate the function. dx
280)
3/2 (x2 - 25)
,x>5
280)
x +C 2 x - 25
A) -
25x +C x2 - 25
C)
B) -
x +C 25 x2 - 25
D)
x2 - 25 +C x
Find the value of the constant k so that the given function in a probability density function for a random variable over the specified interval. 281) f(x) = kex over [0, 14] 281)
A) 2
B)
e14
1 14 e +1
C) e14 - 1
D)
1 14 e -1
Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions. x4 282) 282) dx x2 - 25 3
A) x + 25x + 125 ln x - 5 - 125 ln x + 5 + C 3
B)
x3 3
2
+ 25x -
2
125 ln x - 5 2
+
125 ln x + 5 2
+C
3
C) x + 25 ln x - 5 - 25 ln x + 5 + C 3
2
2
3
D) x + 25x + 125 ln x - 25 - 125 ln x + 25 + C 3
Integrate the function. x3
283)
x2 + 5
2
2
dx
283)
A) 1 (x2 + 5)3/2 - x2 + 5 + C
B) 1 x2 + 5 -
C) 1 (x2 + 5)3/2 - 5 x2 + 5 + C
D) 1 (x2 + 5)3/2 + tan-1 x + C
3
5 3
3
Use Simpson's Rule with n = 4 steps to estimate the integral. 3 284) (8x + 5) dx 1 A) 84 B) 35
51
5 +C x2 + 5
5
284) C) 21
D) 42
Integrate the function. x2
285)
5/2 (x2 - 4)
A) C) -
dt
285)
x3
3/2 12(x2 - 4) x3
1/2 12(x2 - 4)
+C
B) -
+C
D) -
x2
+C
3/2 6(x2 - 4) x
+C
5/2 6(x2 - 4)
Expand the quotient by partial fractions. t4 + t2 - 9t - 12
286)
286)
t4 + 3t2
A) 1 + 3t + 2 + -3 + -4 t2 + 3
C) 1 +
t
B) 1 + 3t + 2 + -4
t2
t2 + 3
2 -3 -4 + + 2 t t +3 t2
t2
D) 1 + 3t + 2 + -3 + -4 t2 + 3
Solve the problem by integration.
t
t2
2
287) Find the first-quadrant area bounded by y = 12x + 32x + 28 , and x = 3.
287)
(x2 + 7)(x + 4)
A) 4 ln 4
C) 4 ln 1
B) 4 ln 112
4
D) 2 ln 4
Use integration by parts to establish a reduction formula for the integral.
288)
tann x dx , n 1
288)
A)
tann x dx =
1 tann-2 x n-1
tann-1 x dx
B)
tann x dx =
1 tann-1 x + n-1
tann-1 x dx
C)
tann x dx = tann-1 x -
1 n-1
tann-2 x dx
D)
tann x dx =
1 tann-1 x n-1
tann-2 x dx
Solve the problem. 289) Find the volume of the solid generated by revolving the region in the first quadrant bounded by the x-axis and the curve y = x cos x, 0 x /2 about the y-axis. 3 2 3 3 A) B) C) D) -8 -4 -4 +2 2-4 2 2 2 2
52
289)
Find the area or volume.
290) Find the area under the curve y = A) 1
1
(x + 1)3/2
bounded on the left by x = 3.
B) 1
2
290) D) 2
C) 4
3
Use a trigonometric substitution to evaluate the integral. dx 291) x x2 - 9
291)
A) 1 sec-1 x + C
B) sin-1 x + C
C) 1 sin-1 x + C
D) 3sec-1 x + C
3
3
3
3
3
Evaluate the improper integral or state that it is divergent. 23x
292) -
(x2 - 1)
2
dx
A) 23
292) B) 0
C) 46
Solve the problem. 293) Find the area of the region enclosed by y = 7x sin x and the x-axis for 0 x A) 7 B) C) 7 2 Express the integrand as a sum of partial fractions and evaluate the integral. 7 4x dx 294) 3 2 (x - 5) A) - 85 B) 85 C) - 77 72 18 18
D) Divergent
.
293) D) 14
294) D) - 85 18
Solve the problem. 295) The height of a vase is 5 inches. The table shows the circumference of the vase (in inches) at half-inch intervals starting from the top down. Estimate the volume of the vase by using the Trapezoidal rule with n = 10. Round your answer to the nearest thousandth. [Hint: you will first need to find the areas of the cross-sections that correspond to the given circumferences.] Circumferences 4.7 8.1 4.2 9.4 4.1 10.1 4.8 8.5 5.6 6.4 6.8 A) 105.479 in.3
B) 101.552 in.3
C) 61.856 in.3
53
D) 62.876 in.3
295)
Determine whether the improper integral converges or diverges.
296) 1
9x + 2 3 9x + 8x2 + 1
296)
A) Converges
B) Diverges
Evaluate the integral.
297)
cos-1 x dx
297) 1
+C
B) x cos-1 x - 1 - x2 + C
C) x cos-1 x - 2 1 - x2 + C
D) x cos-1 x + 1 - x2 + C
A) x cos-1 x -
1 - x2
Solve the initial value problem for y as a function of x. 298) 25 - x2 dy = 1, x < 5, y(0) = 14 dx
298)
A) y = sin-1 x
B) y = sin-1 x + 14
C) y = ln sec x + tan x
D) y = ln sec x + tan x + 14
5
5
Evaluate the integral by making a substitution and then using a table of integrals. cos 299) d sin 36 + sin2 2 A) - 1 ln 6 + 36 + sin 6
sin
+C
cos
2 C) - 1 ln 6 cos + 36 + sin 6
sin
2 B) - 1 ln 6 - 36 + sin
+C
2 D) - 1 ln 6 + 36 + sin
+C
6
+C
299)
sin
6
sin
Find the value of the constant k so that the given function in a probability density function for a random variable over the specified interval. 300) f(x) = 1 x over [2, k] 300) 4
B) 10
A) 12
C) 2 2
D) 2 3
Use various trigonometric identities to simplify the expression then integrate.
301)
sin2
cos 4 d
301)
A) 1 sin 2 - 1 sin 4 - 1 sin 6 + C
B) 1 sin 4 - 1 sin 2 - 1 sin 6 + C
C) 1 sin 4 - 1 sin 6 - 1 sin 2 + C
D) 1 sin 4 - 1 sin 2 - 1 sin 6 + C
8 8
4 4
24
2
24
8
54
4 4
6
24
Solve the initial value problem for y as a function of x. 302) (x2 + 64) dy = 1, y(8) = 0 dx
302)
A) y = 1 tan-1 x 8
C) y = 1 sin-1 x 8
B) y = 1 tan-1 x -
8
8
8
8
32
D) y = x - 1
16
8
Evaluate the integral. 5x - 7 303) dx x2
303)
A) -
5x - 7 5 7 ln + x 7
5x - 7 5x - 7 +
C) -
5x - 7 5 7 tan-1 + x 7
7 +C 7
5x - 7 +C x2
B) -
5x - 7 5 7 tan-1 + x 7
D)
5x - 7 + 5tan-1 x
5x - 7 +C 7
5x - 7 +C 7
Determine whether the improper integral converges or diverges. /2 sin t 304) dt t 0 A) Diverges B) Converges
304)
Express the integrand as a sum of partial fractions and evaluate the integral. 40 305) dt 3 t + 3t2 - 4t
305)
A) -2 ln t + 8ln t - 1 - 2ln t + 4 + C
B) - 10 + 8ln t - 1 + 2ln t + 4 + C
C) -10 ln t + 8ln t2 - 1 + C
D) -10 ln t + 8ln t - 1 + 2ln t + 4 + C
t
Provide an appropriate response. 306) The "Simpson" sum is based upon the area under a ? . A) parabola B) rectangle C) triangle Solve the problem. 307) The length of one arch of the curve y = 3 sin 2x is given by /2 L= 1 + 36cos2 2x dx 0 Estimate L by the Trapezoidal Rule with n = 6. A) 6.4115 B) 6.2807 C) 5.8169
55
306) D) trapezoid
307)
D) 6.1995
Use various trigonometric identities to simplify the expression then integrate. sin
308)
cos
cos 8
d
308)
A) 1 cos 6 - 1 cos 10 + C
B) 1 cos 4 + C
C) 1 cos 10 + C
D) 1 cos 6 - 1 cos 10 + C
8
16
40
10
24
40
Evaluate the integral. e2x x2 dx
309)
A) 1 x2 e2x - 1 xe2x + 1 e2x + C
B) 1 x2 e2x - xe2x + 1 e2x + C
C) 1 x2 e2x - 1 xe2x + C
D) 1 x2 e2x - 1 xe2x + 1 e2x + C
2
4
2
/2
310)
309) 4
2
2
4
2
4
cos 4t cos 3t dt
0
310) B) 3
A) 1
311)
2
C) 4
5
7
D) 3 7
9 + 5x dx x
311)
A) 2 9 + 5x + C C) 2 9 + 5x + 3 tan-1
9 + 5x +C 3
B) 2 9 + 5x - 3 ln
9 + 5x - 3 +C 9 + 5x + 3
D) 2 9 + 5x + 3 ln
9 + 5x - 3 +C 9 + 5x + 3
Expand the quotient by partial fractions. 6z 312) 3 z - 2z 2 - 8z
312)
A) 1 + -1 + 1
B)
1 -1 + z-4 z+2
C) 1 + 1 + -1
D)
1 1 + z-4 z+2
z z
z-4 z-4
z+2 z+2
Solve the problem.
313) Find the area of the region enclosed by the curve y = x cos x and the x-axis for 11 2
A) 12
B) 13
C) 11
56
D) 11
x
13 . 2
313)
Use the Trapezoidal Rule with n = 4 steps to estimate the integral. 1 1 314) dx 2 0 1+x A) 3299 B) 5323 C) 9403 3400 6800 6800
314) D) 5323 3400
Evaluate the integral.
2x2
315)
16 - x2
dx
16 - x2 - 4 ln
A)
315) 4+
16 - x2 +C x
B) -4sin-1 x - x2 16 - x2 + C 4
C) 16 sin-1 x - x 16 - x2 + C
D) 16 sin-1 x - 16 - x2 + C
4
4
Expand the quotient by partial fractions. 316) y + 7 y2 (y + 1)
316)
A) -6 + 7 + 7
B) 7 + 6
C) 6 + 7 + 6
D) -6 + 7 + 6
y y
y2
y+ 1
y2
y2
y+ 1
y
y+ 1
y2
y+ 1
Evaluate the integral. 1 317) dx x ln x4
317)
A) ln x4 + C
C) 1 ln(ln x4) + C
B) ln(ln x4 ) + C
D) 1 ln x4 + C
4
4
Solve the problem.
318) During each cycle, the velocity v (in ft/s) of a robotic welding device is given by v = 2t - 20 , 9 + t2
318)
where t is the time (in s). Find the expression for the displacement s (in ft) as a function of t if s = 0 for t = 0. A) s = t2 - 20 tan-1 t B) s = t2 - 20 sin-1 t 3 3
C) s = t2 - 20 tan-1 20 t 3
D) s = t2 - 20 tan-1 t
3
3
3
Express the integrand as a sum of partial fractions and evaluate the integral. 5x - 11 319) dx 2 x - 6x - 7
A) ln 3(x - 7) + 2(x + 1) + C C) 3ln x - 7 + 2ln x + 1 + C
B) 4ln x - 7 - 2ln x + 1 + C D) 3ln x + 7 + 2ln x - 1 + C 57
319)
Find the surface area or volume. 320) The region between the curve y = sinx, 0 x 1.6, and the x-axis is revolved about the x-axis to generate a solid. Use a table of integrals to find, to two decimal places, the volume of the solid generated. A) 2.44 B) 2.70 C) 2.61 D) 2.56 Solve the problem. 321) The lifetime of an appliance that costs $500 is exponentially distributed with a mean of 4 years. The manufacturer gives a full refund if the appliance fails in the first year following its purchase. If the manufacturer sells 200 printers, how much should it expect to pay in refunds? A) $19,908 B) $22,120 C) $77,880 D) $221 Evaluate the improper integral or state that it is divergent. -5 6 322) dx x4 A) - 2.576980378e+10 5.0331648e+15
B) Divergent D) 2
125
125
Evaluate the integral. 3 323) cot x dx 7
323)
A) 1 cot2 x + ln sin x + C
B) - 1 cot2 x - 1 ln sin x + C
C) 1 cot4 x + C
D) 1 cot4 x sec x + C
14
14
28
/2 0
7
28
sin 3t sin 2t dt
A) 4
11
321)
322)
C) - 6
324)
320)
324) B) 3
C) 2
5
5
Expand the quotient by partial fractions. 325) x + 6 x2 + 6x + 9
D) 1 2
325)
A)
3 1 + x + 3 (x + 3)2
B)
1 3 + x + 3 (x + 3)2
C)
1 -3 + x + 3 (x + 3)2
D)
1 4 + x+3 x+6
58
Solve the problem. 326) The length of time it takes college students to find a parking spot in the library parking lot follows a normal distribution with a mean of 7.0 minutes and a standard deviation of 1 minute. Find the probability that a randomly selected college student will find a parking spot in the library parking lot in less than 6.5 minutes. A) 0.1915 B) 0.2674 C) 0.3551 D) 0.3085
326)
Evaluate the integral. 17x cos
327)
1 x dx 2
327)
A) 17 sin 1 x + 34x cos 1 x + C
B) 34x sin 1 x + 68 cos 1 x + C
C) 17x sin 1 x - 34 cos 1 x + C
D) 68 sin 1 x - 34x cos 1 x + C
2
2
2
2
2
2
2
2
e-x cos 3 x dx
328) 0
328)
A) 1
B) 1
10
D) 3
C) Diverges
10
Use reduction formulas to evaluate the integral.
329)
9 cot3 x sin2 x dx
329)
A) 9 ln sin x - 9 cos2 x + C
B) 9 ln sin x - 9 sin x + C
C) 9 ln sin x + 9 cos2 x + C
D) 9 ln sin x + 9 sin x + C
2
2
2
Solve the problem. 330) Suppose that the accompanying table shows the velocity of a car every second for 8 seconds. Use the Trapezoidal Rule to approximate the distance traveled by the car in the 8 seconds. Round your answer to the nearest tenth if necessary. Time (sec) Velocity (ft/sec) 0 17 1 18 2 19 3 21 4 20 5 22 6 19 7 17 8 18 A) 307 ft B) 233.5 ft
C) 153.5 ft
59
D) 171 ft
330)
Evaluate the integral. (x2 - 9x) ex dx
331)
A) 1 x3 ex - 9 x2ex + C
B) ex[x2 - 11x + 11] + C
C) ex[x2 - 11x - 11] + C
D) ex[x2 - 9x + 9] + C
3
1/4
332) 0
331)
2
7 sin4 2 x dx
332)
A) 21 - 7 16
B) 21
8
C) 21 - 7
32
D) 21
32
16
Solve the initial value problem for x as a function of t. 333) (t2 - 5t + 6) dx = 1 (t > 3), x(4) = 0 dt
333)
A) x = ln t - 2 - ln t - 3 + ln 2
B) x = -ln t - 2 + ln t - 3 + 2
C) x = -ln t - 2 + 1 + ln 2
D) x = -ln t - 2 + ln t - 3 + ln 2
t-3
Solve the initial value problem for y as a function of x. 334) (49 - x2 ) dy = 1, y(0) = 3 dx
334)
A) y = 1 ln x + 7 + 3
B) y = 1 ln x + 7
C) y = 1 ln sec x + tan x + 3
D) y =
14
x-7
14
7
x-7
x 49 49 - x2
+3
Determine whether the improper integral converges or diverges. e-x sin x dx
335)
335)
1
A) Converges
B) Diverges
Use reduction formulas to evaluate the integral.
336)
3 cos3 x sin5 x dx
336)
A) 3 sin4 x - 3 cos8 x + C
B) 1 (sin6 x - sin8 x) + C
C) 3 cos4 x - 1 cos6 x + C 4 2
D) 1 sin6 x - 3 sin8 x + C
4
8
2 2
60
8
Solve the problem. 337) Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e-2x, and the line x = 8 about the y-axis.
A) 1 (1 - 16 e-16)
B) 1 (1 - 15 e-16)
C) - 1 (1 + 17 e-16)
D) 1 (1 - 17 e-16)
2
337)
2
2
2
Evaluate the improper integral or state that it is divergent. dx (x - 4)( x - 3)
338) 5
338) B) - 1 ln 2
A) ln 4
D) 1 ln 5
C) ln 2
4
2
Solve the problem by integration.
339) Find the x-coordinate of the centroid of the area bounded by y(x2 - 16 ) = 1, y = 0, x = 5, and x = 8. A) -1.81 B) 1.81 C) 1.52 D) 6.09
Solve the problem. 340) The following table shows the rate of water flow (in gal/min) from a stream into a pond during a 30-minute period after a thunderstorm. Use Simpson's Rule to estimate the total amount of water flowing into the pond during this period. Round your answer to the nearest gallon. Time (min) Rate (gal/min) 0 225 5 275 10 325 15 275 20 245 25 225 30 175 A) 7050 gal
B) 7733 gal
C) 7725 gal
339)
340)
D) 8850 gal
Evaluate the integral by making a substitution and then using a table of integrals.
341)
ex
4 - e2x dx x
A) 2 sin-1 e
2
x
C) e
2
341) x
B) e
+C
4 - e2x + 2 ln ex +
2
e2x - 4
x
D) e
+C
2
ex 4 - e2x + 2 sin-1 2
+C
4 - e2x + 2 ln x +
x2 - 4 + C
Solve the problem. 342) Two dice are tossed, and the random variable X assigns to each outcome the sum of the number of dots showing on each face. What is the probability that X = 5? A) 1 B) 5 C) 4 D) 8 9 6 9
61
342)
Expand the quotient by partial fractions. 343) 5x + 45 (x + 2)(x + 7)
A)
7 -2 + x-2 x-7
B)
343) 7 -2 + x+2 x+7
C)
2 -7 + x+2 x+7
D)
7 2 + x+2 x+7
Use reduction formulas to evaluate the integral. cot4 4x dx
344)
344)
A) - 1 cot3 4x + 1 cot 4x + C
B) - 1 cot3 4x + 1 cot 4x + x + C
C) - 1 cot2 4x + 1 cot 4x - x + C
D) - 1 cot3 4x - 1 cot 4x + x + C
12
4
12
12
4
4
3
4
sec3 5x dx
345)
345)
A) 1 sec2 5x tan 5x + 1 ln sec 5x + tan 5x + C 10
10
B) 1 sec 5x tan 5x - 1 ln sec 5x + tan 5x + C 10
5
C) 1 sec 5x tan 5x + 1 ln sec 5x + tan 5x + C 10
10
D) 1 sec 5x tan 5x + 1 ln sec 5x + cot 5x + C 10
10
Solve the problem by integration.
346) The general expression for the slope of a curve is 3x + 4 . Find the equation of the curve if it x2 + 4x
passes through (1, 0). 2 A) y = ln x(x - 4) 25
2 B) y = ln x( x + 4)
2 C) y = ln x(x + 4)
D) y = ln x(x + 4)2
346)
5
25
Evaluate the integral. 1 - cos 2x dx
347)
347)
0
A) 2
B) 2
C) 2 2
62
D)
2 2
Evaluate the integral. The integral may not require integration by parts.
348)
x7 sec x8 dx
348)
A) 1 ln sec x8 + tan x8 + C
B) x9 sec x9 + C
C) 8 ln sec x8 - tan x8 + C
D) 1 ln sec x8 + tan x8 + C
8
7
Solve the problem. 349) Find the volume of the solid generated by revolving the region bounded by the curve y = 4cos x 3 and the x-axis, x , about the x-axis. 2 2
A) 8 2
B) 8 3
C) 16 2
349)
D) 16
Evaluate the integral by using a substitution prior to integration by parts.
350)
x 3 - x dx
350)
A) - 2 x(3 - x)3/2 + 4 (3 - x)5/2 + C
B) - 2 x(3 - x)3/2 - 4 (3 - x)5/2 + C
C) 2 x(3 - x)3/2 + 4 (3 - x)5/2 + C
D) - 2 x(3 - x)3/2 - 2 (3 - x)5/2 + C
3
3
15
3
15
3
15 5
Solve the problem. 351) The number of new mini-vans sold by a particular salesperson during the month of March is exponentially distributed with a mean of 10. What is the probability that the salesperson will sell between 2 and 7 mini-vans in March? A) 0.322 B) 0.424 C) 0.422 D) 0.200 Provide an appropriate response. +
352) A student knows that
a
f(x) dx converges. Does
a
f(x) dx also necessarily converge?
351)
352)
-
A) No
B) Yes
Find the value of the constant k so that the given function in a probability density function for a random variable over the specified interval. 353) f(x) = k(9 - x) over [0, 9] 353) 2 1 A) B) 81 C) D) 9 81 81
63
Integrate the function. 2 3/2 354) (36 - t ) dt t6
354)
5 36 - t2 +C t
A) - 1 5
B) - 1
5 36 - t2 t + sec-1 +C t 6
D) - 1
36 - t2 t
180
2 5/2
C) - (36 - t )
180t7
180
5
+C
Solve the problem by integration. 355) By a computer analysis, the electric current i (in A) in a certain circuit is given by 0.0050(10t2 + 10t + 90 ) i= , where t is the time (in s). Find the total charge that passes a point in ( t + 2)( t2 + 18) the circuit in the first 0.75 s. A) 0.0399 C B) 0.0437 C
C) 0.0017 C
D) 0.0083 C
For the given probability density function, over the stated interval, find the requested value. 356) f(x) = 1 x over [0, 2]; Find the median. 2
C) 1
B) 2
A) 2
356)
D) 1
4
2
Evaluate the integral. 357) x + 8 dx x+5
358)
357)
A) x + 3 ln x + 5 + C
B) x + 13 ln x + 5 + C
C) (x2 /2 + 8x) ln x + 5 + C
D) -
sin 4t sin
3
(x + 5)2
+C
t dt 3
358)
A) 3 sin t - 3 sin 7t + C
B) 22 sin 11 t - 26 sin 13 t + C
C) 3 sin 11 t - 3 sin 13 t + C
D) 3 cos 11 t + 3 cos 13 t + C
22 22
355)
26
3
26
3
3
22
Solve the problem. 359) Find the length of the curve y = ln(csc x), /6 x A) ln( 3 + 1) B) ln( 3 + 2)
3
3
3
26
3
3
/2
C) 1 - ln( 3 + 2)
64
359) D) ln( 3)
Use integration by parts to establish a reduction formula for the integral.
360)
xn e-ax dx
360)
A)
xn e-ax dx = -
xn e-ax n + a a
xn-1 e-ax dx
B)
xn e-ax dx = - axn e-ax + na
xn-1 e-ax dx
C)
xn e-ax n xn e-ax dx = a a
xn-1 e-ax dx
D)
xn e-ax dx = -
xn e-ax n + a a
xn-2 e-ax dx
Integrate the function.
361)
36 - x2 dx
361)
2 A) 18 sin-1 x + x 36 - x + C 6
C)
x 36 36 - x2
+
36x
B)
2
36 - x2 +C x
36 - x2
x +C 2
+
2 D) 18x - x 36 - x + C 2
Use a CAS to perform the integration. 362) ln x x6
A) -5 ln x - 1
362) B) -4 ln x - 1
25x5
C) -ln x - 1
16x4
D) -5 ln x - 1
25x5
5x5
Use reduction formulas to evaluate the integral.
363)
sin5 2x dx
363)
A) - 1 sin4 2x cos 2x - 4 sin2 2x cos 2x - 2 cos 2x + C 5
15
3
B) - 1 sin4 2x cos 2x - 2 sin2 2x cos 2x - 1 sin 4x + x + C 10
15
3
2
C) - 1 cos4 2x sin 2x - 4 cos2 2x sin 2x - 1 cos 2x + C 10
15
15
D) - 1 sin4 2x cos 2x - 2 sin2 2x cos 2x - 4 cos 2x + C 10
15
15
Use a CAS to perform the integration.
364)
x8 ln x dx
364)
9
9
9
81
9
9
9
9
A) x ln x + x + C
9
10
9
90
9
9
9
81
B) x ln x - x
C) x ln x - x + C
+C
D) x ln x - x + C
65
Determine whether the improper integral converges or diverges. 8 x2 dx 365) 64 - x2 0 A) Converges B) Diverges
365)
Determine whether the function is a probability density function over the given interval. 366) f(x) = 1 x, 4 x 9 2
A) No
366)
B) Yes
Evaluate the integral. e-x cos 2 x dx
367) 0
367)
A) 2
C) 1
B) Diverges
5
D) 1
5
Find the indicated probability. 368) f(x) = 1 e-x/3 ; [0, ), P(1 x 5) 3
A) 0.0879
368)
B) 0.01173
C) 0.1759
D) 0.5277
Evaluate the integral by using a substitution prior to integration by parts. x2 369) dx x2 + 25
369)
A) 3x x2 + 25 - 25 ln (x + x2 + 25) + C
B) x x2 + 25 + 25 ln (x + x2 + 25) + C
C) x x2 + 25 - 25 ln (x + x2 + 25) + C
D) 3x x2 + 25 + 25 ln (x + x2 + 25) + C
2
2
2
2
2
2
2
2
Determine whether the improper integral converges or diverges.
370) 1
ln x dx x
370)
A) Converges
B) Diverges
Determine whether the function is a probability density function over the given interval. 5 x 0 371) f(x) = (x2 + 9) 0 A) No
x<0
B) Yes
66
371)
Express the integrand as a sum of partial fractions and evaluate the integral. 10 6 372) dx 2 x -9 5 A) 2.847 B) -0.767 C) 3.767 1
373) 0
x3
x2 + 12x + 36
372) D) 0.767
dx
373)
A) 108 ln 7 - 233
B) 18 ln 7 - 251
C) 144 ln 7 - 125
D) 108 ln 7 - 18ln 6 + 233
6 6
14
14
14
14
Find the area or volume. 374) Find the volume of the solid generated by revolving the region in the first quadrant under the 10 curve y = , bounded on the left by x = 1, about the x-axis. x2
A) 10
B) 2
C) 100
Evaluate the improper integral or state that it is divergent. e 375) 14e-x dx A) 28 B) Divergent
374)
D) 100 3
375) C) 14
D) -14
Evaluate the integral by using a substitution prior to integration by parts.
376)
cos (ln x) dx
376)
A) x cos (ln x) + sin (ln x)+ C
B) x[cos (ln x) + sin (ln x)] + C
C) x [cos (ln x) - sin (ln x)] + C
D) x [cos (ln x) + sin (ln x)] + C
2
2
Evaluate the integral.
377)
4xex dx
A) xex - 4ex + C
377) B) 4ex - ex + C
C) 4ex - 4xex + C
D) 4xex - 4ex + C
Provide an appropriate response. 378) The error formula for Simpson's Rule depends upon i) f(x). ii) f''(x). iii) f(4) (x) iv) the number of steps A) i, iii, and iv
B) i and iii
378)
C) iii and iv 67
D) ii and iv
Solve the problem.
379) Find an upper bound for ES in estimating A) 1.288490189e+15
5
(3x2 - 1) dx with n = 12 steps.
0
379)
B) 3.435973837e+15
2.564940725e+17
2.05195258e+18
C) 1.288490189e+15
D) 0
5.129881451e+17
Use a trigonometric substitution to evaluate the integral. ln 3 ex dx 380) e2x + 1 0
380)
A) ln 6 - ln(1 + 2 )
B) ln (e3 + 10 )
C) ln(3 + 10 ) - ln(1 + 2 )
D) ln 3 2
Use integration by parts to establish a reduction formula for the integral.
381)
(ln ax) n dx
381)
A)
(ln ax) n dx =
x(ln ax) n n + n a
(ln ax) n-1 dx
B)
(ln ax) n dx =
x(ln ax) n n n a
(ln ax) n-2 dx
C)
(ln ax) n dx = x(ln ax) n - n
(ln ax) n-1 dx
D)
(ln ax) n dx = ax(ln ax) n - an
(ln ax) n-1 dx
Evaluate the integral.
382)
x sin-1 x dx
382)
2
2
A) x sin-1 x - 1 cos-1 x + 1 1 - x2 + C 2
4
B) x sin-1 x - 1 x sin-1 x - 1 1 - x2 + C
4
2
2
4
4
2
C) x sin-1 x - 1 sin-1 x + 1 x 1 - x2 + C 2
4
D) x sin-1 x + 1 sin-1 x - 1 x 1 - x2 + C
4
2
2
2
Solve the problem. 383) Estimate the minimum number of subintervals needed to approximate the integral 3 (4x3 + 9x)dx 2 with an error of magnitude less than 10-4 using the Trapezoidal Rule.
A) 142
B) 245
C) 200
68
D) 123
383)
384) The time between major earthquakes in the Alaska panhandle region can be modeled with an
384)
exponential distribution having a mean of 680 days. Find the probability that the time between a major earthquake and the next one is less than 200 days. A) 0.0004 B) 0.2548 C) 0.7452 D) 0.0011
Express the integrand as a sum of partial fractions and evaluate the integral. 3 2 385) 2x + 5x + 14x + 7 dx 2 (x2 + 2x + 5)
A) - 1 tan-1 x +1 2
2
1
x2 + 2x + 5
+C
B) ln x2 + 2x + 5 - 1 tan-1 x +1 2
C) ln x2 + 2x + 5 -
385)
2
1
x2 + 2x + 5
+C
1 +C 2 x + 2x + 5
D) ln x2 + 2x + 5 - 1 tan-1 x +1 + C 2
2
Find the area or volume.
386) Find the area of the region in the first quadrant between the curve y = e-4x and the x-axis. B) 1 e 4
A) 4
C) 1
387) Find the volume of the solid generated by revolving the area under y = 6e-x in the first quadrant about the x-axis A) 72
Integrate the function. 8 81 dx
388)
0
(81 - x2 )
B) 18
C) 36
388) B) 17 - 17
17
C)
17 17
D) 173/2
Use reduction formulas to evaluate the integral. 389) csc4 t dt 5
389)
A) - 5 csc3 t cot t + 10 cot t + C
B) - 5 csc2 t cot t - 10 csc t + C
C) - 1 csc2 t cot t - 2 cot t + C
D) - 5 csc2 t cot t - 10 cot t + C
15
5
5
387)
D) 6
3/2
A) 8 17
3
386)
D) 1 4
5
5
3
15
5
3
5
3
69
5 5
5 5
3 3
5 5
Evaluate the integral by using a substitution prior to integration by parts. 1 x 390) dx x+1 0 A) -1.33 B) -0.94 C) 0.39 Integrate the function. 1 2 391) dt 2 -1 1 + 4t
390) D) -2.27
391)
A) 2sin-1 2
B)
C) 2tan-1 1
2
2
D) 2tan-1 2
Determine whether the improper integral converges or diverges. 6 - x-5 e 392) dx x-5 5 A) Converges B) Diverges
392)
Evaluate the integral. (-sin t - 7) cos t dt 393) 3 sin t + 2 sin2 t + sin t + 2
393)
A) -ln sin t + 2 + 1 ln sin2 t + 1 + C 2
B) -ln sin t + 2 + 1 ln sin2 t + 1 - 3 tan-1 (sin t) + C 2
C) ln sin t + 2 - ln sin2 t + 1 - 5 tan-1(sin t) + C D) -ln t + 2 + 1 ln t2 + 1 - 3 tan-1 t + C 2
Solve the problem. 394) The length of the ellipse x = a cos t, y = b sin t, 0 t 2 is /2 Length = 4a 1 - e2 cos2 t dt 0 where e is the ellipse's eccentricity.
394)
Use Simpson's Rule with n = 6 to estimate the length of the ellipse when a = 2 and e =
A) 11.8956
B) 11.2546
C) 12.1751
70
1 . 3
D) 12.2097
Evaluate the improper integral or state that it is divergent.
395) 3
dt 2 t - 2t
395)
A) 1 ln 2
B) 1 ln 3
3
C) - 1 ln 3
2
D) 2 ln 3
2
Use the Trapezoidal Rule with n = 4 steps to estimate the integral. 0 396) sin t dt -1 A) - 1 + 2 B) -1 - 2 C) - 1 + 2 4 8
396) D) - 1 + 2 2
Evaluate the integral. 1 - cos2 x dx
397)
397)
0
A) 0
398)
csc
-csc - cot
B)
2 2
C) 2
D) 2
d
398) B) csc2 - cot + C D) cot + csc + C
A) cot - csc + C C) cot2 + csc + C
Find the surface area or volume. 399) Use substitution and a table of integrals to find, to two decimal places, the area of the surface generated by revolving the curve y = ex, 0 x 4, about the x-axis.
A) 11,342.93
B) 12,238.78
C) 9376.82
D) 10,148.47
Express the integrand as a sum of partial fractions and evaluate the integral. 90 dx 400) x3 - 9x
400)
A) 10 ln x - 5ln x - 3 - 5ln x + 3 + C
B) -10 ln x + 1 tan-1 x + C
C) -10 ln x + 5ln x - 3 + 5ln x + 3 + C
D) - 10 + 5ln x - 3 + 5ln x + 3 + C
3
3
x
Find the indicated probability. 401) f(x) = 3x 4 - x over [0, 1], P(0.4 < x) 3
A) 0.744
399)
401)
B) 0.384
C) 0.644
71
D) 0.904
Evaluate the integral. (x + 5)2 tan-1 (2x) + 4x3 + x
402)
2 (4x2 + 1) (x + 5)
A) (tan
-1 2x)2 2
+ ln x + 5 -
dx
402)
5 +C x+5
B) (tan D) (tan
C) (x + 5)2 tan-1 2x + ln x + 5 + 5 + C x+5
-1 2x)2 4
-1 2x)2 4
+
5 +C x+5
+ ln x + 5 +
5 +C x+5
Determine whether the improper integral converges or diverges.
403) 1
sin x dx x2
403)
A) Diverges
B) Converges
Solve the initial value problem for x as a function of t. 404) (2t3 - 2t2 + t - 1) dx = 3, x(2) = 0 dt
404)
A) x = ln t - 1 - 2 tan-1 2 t + 2 tan-1 2 2 B) x = ln t - 1 - tan-1 2 t - ln t2 + 1 + tan-1 2 2 + ln 4.5 2
C) x = ln t - 1 - 2 tan-1 2 t - 1 ln t2 + 1 + 2 tan-1 2 2 + 1 ln 4.5 2
2
2
D) x = ln t - 1 - 1 ln t2 + 1 + 1 ln 4.5 2 2 2 Solve the problem. 405) The length of time it takes college students to find a parking spot in the library parking lot follows a normal distribution with a mean of 6.0 minutes and a standard deviation of 1 minute. Find the probability that a randomly selected college student will take between 4.5 and 7.0 minutes to find a parking spot in the library lot. A) 0.4938 B) 0.2255 C) 0.0919 D) 0.7745
406) Estimate the area of the surface generated by revolving the curve y = 2x2 , 0 x 3 about the x-axis. Use Simpson's Rule with n = 6. A) 996.028 B) 1007.254
407) Find the area bounded by y = A) sin-1 9 5
C) 1024.885
3 25 - 9x2
5
406)
D) 1021.107
, x = 0, y = 0, and x = 3.
B) 3 tan-1 3
407)
C) 1 sin-1 9
5
5
72
405)
5
D) 1 tan-1 9 5
5
Use the Trapezoidal Rule with n = 4 steps to estimate the integral. 0 408) sin x dx A) - 1 + 2 B) - 1 + 2 C) -(1 + 8 4
408) D) - 1 + 2
2)
2
Solve the problem by integration. 409) Under specified conditions, the time t (in min) required to form x grams of a substance during a dx chemical reaction is given by t = . Find the equation relating t and x if x = 0 g (6 - x)(3 - x) when t = 0 min. A) t = 1 ln 3 - x - 1 ln 1 3 6-x 3 2
B) t = 1 ln 6 - x + 1 ln 2
C) t = 1 ln 6 - x - 1 ln 2
D) t = 1 ln 3 - x + 1 ln 1
3
3-x
3
3
3-x
3
6-x
409)
3 3
2
Find the value of the constant k so that the given function in a probability density function for a random variable over the specified interval. 410) f(x) = 8e-7x over [0, k] 410)
A) 1 ln 8
B) - 1 ln 15
7
7
C) 1 ln 7
8
7
8
Express the integrand as a sum of partial fractions and evaluate the integral. 6x3 + 22x2 + 56x + 13 411) dx x2 ( x2 + 4x + 13)
D) e-7x ln 8
411)
A) 4 ln x - 1 + ln x2 + 4x + 13 + 1 tan-1 x + 2 + C x
3
3
B) 4 ln x - 1 + ln x2 + 4x + 13 + 1 tan-1 x + 2 + C x
3
3
C) 7 ln x + ln x2 + 4x + 13 + 1 tan-1 x + 2 + C 3
3
D) 5 ln x + 1 - ln x2 + 4x + 13 + sin-1 x + 2 + C x
3
Determine whether the improper integral converges or diverges.
412) 1
4 x2 + 6
412)
A) Converges
B) Diverges
73
Use reduction formulas to evaluate the integral. tan5 3x dx
413)
413)
A) - 1 tan4 3x + 1 tan2 3x - 1 ln cos 3x + C 12
6
3
B) 1 tan4 3x - 1 tan2 3x + 1 ln cos 3x + C 12
6
3
C) 1 tan4 3x - 1 tan2 3x - ln cos 3x + C 4
2
D) 1 tan4 3x - 1 tan2 3x - 1 ln cos 3x + C 12
6
3
Evaluate the integral. 2 sin 8x sin 5x dx
414)
414)
A) 2 sin 3x + sin 13x + C
B) 2 sin 3x - sin 13x + C
C) 2 cos 3x - cos 13x + C
D) 2 sin 3x - sin 13x + C
6
26
6
6
26
26
t2 ln (t3 + 9) dt t3 + 9
415)
415)
2
3
3
A) (ln t) + C
B) t ln (t + 9) + C
C) [ln (t3 + 9)]2 + C
D) [ln (t + 9)] + C
6
t3 + 9 3
2
6
Evaluate the improper integral or state that it is divergent. 0 416) 10xe3x dx A) Divergent B) -4.667
416) C) -1.1111
D) 0.3333
Solve the problem.
417) The growth rate of a certain tree (in feet) is given by y = 2 + e-t2 /2 , where t is time in years. t+ 1
417)
Estimate the total growth of the tree through the end of the second year by using Simpson's rule. Use 2 subintervals. Round your answer to the nearest hundredth. A) 5.11 ft B) 2.34 ft C) 3.68 ft D) 3.41 ft
Determine whether the improper integral converges or diverges. ex
418) 1
1 + x2
dx
418)
A) Diverges
B) Converges
74
Evaluate the integral by making a substitution (possibly trigonometric) and then applying a reduction formula. sec3
419)
419)
A) -sec B) sec C) sec D) -sec
tan
+ ln sec
+ tan
+C
tan
+ ln sec
+ tan
+C
tan
- ln sec
+ tan
+C
tan
+ ln sec
- tan
+C
Evaluate the integral. dx 420) x2 - 6x - 16
420)
A) 1 tan-1 x - 3 + C
B) sin-1 x + 3 + C
C) ln x - 3 + x2 - 6x - 16 + C
D) sin-1 x - 6 + C
5
5
5 5
Solve the problem. 421) Find the volume generated by revolving the curve y = cos 5x about the x-axis, 0 x /12 2 2 2 1 A) B) C) D) + + + 24 8 24 24 8 24 12 Find the surface area or volume. 422) Use numerical integration with a programmable calculator or a CAS to find, to two decimal places, the area of the surface generated by revolving the curve y = cos x, 0 x x-axis. A) 14.42
B) 2.29
C) 1.15
2
421)
422)
, about the
D) 7.21
Use any method to evaluate the integral.
423)
cos x dx sin2 x
A) -sec x + C
423) B) csc x + C
C) sec x + C
Integrate the function. 1 424) dt t2 7 - t2
D) -csc x + C
424)
A) -
7 - t2 +C 7t
B)
7 - t2 +C t
C) -
7 - t2 + sin-1 7t + C 7t
D) -
7 - t2 +C 7t2
75
Solve the problem. 425) Find the length of the curve y = ln(sin x), A) ln( 2 + 1) B) ln(2 2)
/4 x
/2
425)
C) ln( 2)
D) 1 - ln( 2 + 1)
Solve the initial value problem for y as a function of x. 426) x x2 - 49 dy = 1, x > 7, y(14) = 0 dx
A) y = sec-1 x 7
B) y = 1 sin-1 x +
3
C) y = 1 sec-1 x 7
426)
7
7
7
21
D) y = 1 sec-1 x
21
7
7
Evaluate the integral. dx 427) 2 x 5x - 7
427)
A)
5x - 7 5 tan-1 + 7x 7 7
C)
5x - 7 5 ln + 7x 7 7
5x - 7 +C 7
5x - 7 5x - 7 +
5x - 7 5 tan-1 7x 7 7
B) -
7 +C 7
D)
5x - 7 5 tan-1 7 7 7
5x - 7 +C 7 5x - 7 +C 7x
Determine whether the function is a probability density function over the given interval.
428) f(x) = 10 sin 10x over
, 20 10
428)
A) Yes
B) No
Integrate the function. y2
429)
3/2 (64 - y2)
dy
429)
A) 64 - y2 - sin-1 y + C
B)
8
8y
C)
64 - y2
- sin-1 y + C
D)
y 64 - y2 y 64 - y2
+C - sin-1
y +C 8
Evaluate the integral.
430)
cos6
sin 2 d
A) 2 cos6 + C 7
431)
/2
430) B) 1 cos7 + C
C) - 2 cos9 + C
8
9
D) - 1 cos8 + C 4
cos5 7x dx
- /2
A) 2 7
431) B) 8
C) - 16
105
105
76
D) 0
Determine whether the improper integral converges or diverges. 2 3xe-x dx
432)
432)
-
A) Converges
B) Diverges
Express the integrand as a sum of partial fractions and evaluate the integral. x+8 433) dx x2 + 3x
A) 1 ln x8(x + 3)5 + C
B) 1 ln
C) 8 ln x8(x + 3)5 + C
D) ln
3
3
3
x8
433)
(x + 3)5 x8
(x + 3)5
+C
+C
Solve the initial value problem for x as a function of t. 434) (t2 + 5t) dx = 5x + 10, x(1) = 1 dt
434)
t -2 t+5
A) x = 18 ln
B) x = 17t - 2 t+5
D) x = 18t - 2
C) x = 18t + ln t + 5 - 2
t+5
Evaluate the integral. dx 435) x 49x2 - 64
435)
A) 7 sin-1 7 x + C
B) 7 sec-1 7 x + C
C) 1 sin-1 7 x + C
D) 1 sec-1 7 x + C
8 7
8
8
8
8
8 8
Solve the problem by integration.
436) Find the volume generated by revolving the first-quadrant area bounded by y = about the x-axis. A) 1 144
Evaluate the improper integral. 9 x 437) dx 2 81 x 0 A) 81
B) 1
C) 1
4
72
x
(x + 6)2
and x = 6
436)
D) 1
288
437) B) 9
C) -9
77
D) -81
Solve the initial value problem for y as a function of x. 438) (x2 + 4) 2 dy = x2 + 4, y(0) = 2 dx
438)
A) y = x + 2
B) y = x + 2
2
C) y =
4
4
x +2 x2 + 4
D) y =
4
x x2 + 4
Solve the initial value problem for x as a function of t. 439) (t + 3) dx = x2 + 1 , t > -3 , x(1) = tan 1 dt
439)
A) x = tan [ ln t + 3 - ln 4 + 1]
B) x = tan-1 [ ln t + 3 - ln 4 + 1]
C) x = 1 - 1
D) x = tan [ ln t + 3 - ln 4]
t+3
t- 4
Evaluate the integral. /2 440) (1 - cos 4x) cos 2x dx /12 A) 13 B) - 11 24 24 Solve the problem by integration.
441) Find the area bounded by y = A) 8.01
440) C) - 1
24
D) 13 48
x - 12 , y = 0, x = 1, and x = 3. Round to the nearest hundredth. 2 x - 7x - 18
B) 6.29
C) 0.73
D) 0.57
Provide an appropriate response. 442) When we use Simpson's rule to approximate a definite integral, it is necessary that the number of partitions be ? . A) a multiple of 4 B) an even number
C) an odd number
A) 1.759218604e+13 8.549802418e+15
442)
D) either an even or odd number
Solve the problem.
443) Find an upper bound for ES in estimating
441)
4
(4x5 - 2x) dx with n = 6 steps.
3
B) 1.801439851e+16 8.549802418e+15
C) 7.036874418e+13
D) 1.407374884e+14
8.549802418e+15
8.549802418e+15
78
443)
Evaluate the integral by making a substitution and then using a table of integrals. ex 444) dx e2x - 9 x A) 1 ln e + 3 + C
B) 1 ln 3 - x + C
x C) 1 ln 3 - e + C
2x D) 1 ln 3 - e + C
6 6
445)
6
ex - 3
ex + 3
444)
x+3
6
e2x + 3
e2x dx 6ex + 4
445)
x
A) e - 1 ln 6ex + 4 + C 6
C)
B) x - 1 ln 6x + 4 + C 6
9
4 + ln 6ex + 4 6ex + 4
9
x
D) e + 1 sin-1 6ex + 4 + C
+C
6
9
Solve the problem. 446) A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 470 seconds and a standard deviation of 60 seconds. Find the probability that a randomly selected boy in secondary school can run the mile in less than 332 seconds. A) 0.9893 B) 0.0107 C) 0.5107 D) 0.4893
447) A physical fitness association is including the mile run in its secondary-school fitness test. The
446)
447)
time for this event for boys in secondary school is known to possess a normal distribution with a mean of 450 seconds and a standard deviation of 50 seconds. Find the probability that a randomly selected boy in secondary school will take longer than 335 seconds to run the mile. A) 0.0107 B) 0.5107 C) 0.9893 D) 0.4893
Find the indicated probability.
448) f(x) = cos x over 0, A) -0.092
2
,P
7
x
448)
5
B) 0.588
C) 0.566
D) 0.154
Evaluate the integral. The integral may not require integration by parts.
449)
7 x6 ex dx
A) 1 ex6 + C 6
449) 7
B) x ex7 + C
C) 1 ex7 + C 7
7
Solve the problem. 450) Find the length of the curve y = ln(csc x), A) 1 - ln( 3) B) ln(2 3)
/3 x
D) 7ex7 + C
/2
C) ln( 3)
79
450) D) ln( 3 + 1)
Evaluate the improper integral or state that it is divergent. 2dx
451) 0
451)
81 + x2
A) 0
B)
+9
C)
D)
81
9
Use integration by parts to establish a reduction formula for the integral. 2 xn e- x dx
452)
452)
A)
2 2 xn e- x dx = - 2xn-1 e- x - 2(n - 1)
2 xn-2 e- x dx
B)
2 1 2 n-1 xn e- x dx = - xn-1 e- x + 2 2
2 xn-2 e- x dx
C)
2 2 xn e- x dx = n xn-1 e- x + 2n
2 xn-1 e- x dx
D)
2 1 2 n xn e- x dx = - xn e- x + 2 2
2 xn-1 e- x dx
Evaluate the integral. e3 ln3 (x4 ) 453) dx x 1
453)
A) 3e12
B) 2.849934139e+15
C) 2.849934139e+15 e4
D) 2.849934139e+15
1.374389535e+11
2.199023256e+12
2.199023256e+12
Express the integrand as a sum of partial fractions and evaluate the integral. dx 454) 2 x (x2 - 4)
454)
A) 1 + 1 ln x - 2 + C
B) 1 + 1 ln x - 2 + C
C) 1 + 1 ln x - 2 + C
D) 1 + 1 ln x + 2 + C
4x 8x
16 16
x+2
4x
x+2
4x
8
16
x+2
x-2
Determine whether the improper integral converges or diverges. sin
455) 0
d ( - )4/7
455)
A) Diverges
B) Converges
80
Solve the problem. 456) Estimate the minimum number of subintervals needed to approximate the integral 5 x + 4 dx 0 with an error of magnitude less than 10-4 using Simpson's Rule.
A) 7
B) 8
C) 4
456)
D) 6
Evaluate the improper integral or state that it is divergent.
457) 1
dx 2.757 x
A)
1 2.757
457) B)
1 1.757
C) Divergent
D)
1 3.757
Determine whether the function is a probability density function over the given interval. 458) f(x) = 5 x over 0, ln(1 + ln5) ln5
A) No
458)
B) Yes
Use integration by parts to establish a reduction formula for the integral.
459)
sinn x dx
459)
A)
1 n-1 sinn x dx = - sinn-1 x cos x n n
sinn-1 x dx
B)
sinn x dx = sinn-1 x cos x - (n - 1)
cos x sinn-2 x dx
C)
1 n-1 sinn x dx = - sinn-1 x cos x + n n
sinn-2 x dx
D)
sinn x dx = sinn-1 x cos x + (n - 1)
sinn-2 x dx
Solve the problem. 460) Estimate the minimum number of subintervals needed to approximate the integral 4 (5x2 + 5)dx 1 with an error of magnitude less than 10-4 using the Trapezoidal Rule.
A) 92
B) 949
C) 238
81
D) 475
460)
Answer Key Testname: CHAPTER 8
1) i) 0
4x3 dx = lim x4 + 1 b b
ii) lim b
-b
b 0
4x3 dx = lim ln (x4 + 1) x4 + 1 b
4x3 dx = lim ln (x4 + 1) 4 x +1 b
2) Answers will vary, but f(x) =
1
(x - c)d
b 4 4 -b = lim (ln (b +1) - ln (b +1)) = lim 0 = 0 b b
where a < c < b and d a positive integer is a family of examples.
3) The mistake is in the second to last step: lim ln(x - 1) b
4) (a)
b 4 0 = lim (ln (b +1) - ln 1) = b
b 2
- lim b
ln x
b 2
= ln
- ln
+ ln 2 = ln 2
1 2 + x+2 x-5
(b) ln x + 2 + 2ln x - 5 + C (c) 2x + ln x + 2 + 2ln x - 5 + C (d) y = 2(x + 2)(x - 5)2
5) The only way the limit of the integral can exist is if the limit of the function is zero. 1 2
6) (a) Since 1 2
0
0
2 e- x /2 dx = 1 and f(x) is an even function,
2 x2 e- x /2 dx = lim b
b 0
0 1 2
2 x2 e- x /2 dx = lim b
1 2
1 2
2 1 e- x /2 dx = . Then 2
2 b -x e-x /2 0 +
b
2 e- x /2 dx =
0
1 2
2 1 e- x /2 dx = . 2
(b) The variance of the distribution is equal to 1 2
0
1 2
0
2 x2 e- x /2 dx = -
1 2
2 x2 e- x /2 dx +
2 1 1 x2 e- x /2 dx = + = 1. 2 2
7) (a) 1 (b)
3
(c) 2 1
1 x5
x6 + 4 dx converges by using the limit comparison
test with the function y = x-2 . (d) 7.61 8) Infinity cannot be added like this. b 1 1 1 1 1 1 9) dx = lim dx = lim tan-1 b - tan-1 0 = = . Since f(x) is an even function, 2 2 2 2 (1 + x ) b b 0 0 1+x 0 1 1 1 1 1 dx = and dx = + = 1. 2 2 2 2 2 (1 + x ) (1 + x ) -
82
Answer Key Testname: CHAPTER 8
10) (a) The integral converges if p <1.
(b) The integral diverges if p 1. b 1 1 1 11) (a) e-2x dx = lim e-2x dx = lim - e-2b + e-4 = e-4 < 0.0092 2 2 2 b b 2 2 2 (b) Since 0 < e-x e-2x for x 2, 2
2 e-x dx +
2
2 e-x dx
0
2 1 2
12) (a) 0
2 e-x dx 2
1 2
2
0
b
1 2 x e- x /2 dx = lim 2 b
0
1 2
-
1 2
+
0
1 2
1 2
-
2 b e- x /2 0 =
2 x e- x /2 dx = 0
2 x e- x /2 dx = -
1 2
0 1 2
1 . 2 2 x e- x /2 dx = -
2 x e- x /2 dx + 0
1 2
1 . The 2
2 x e- x /2 dx = -
= 0.
13) (a) Possible answer: Note that 1
Since 0
0
dx = x3 + 1
1 0
dx
dx
+
x3 + 1
1
x3 + 1
dx is a proper integral and 3 x +1 1
by direct comparison to 1 (b)
0
2 x e- x /2 is an odd function,
mean of the distribution is equal to 1 2
2
2 e-x dx =
2 e-x dx, with an error of magnitude no greater than 0.0092.
2 x e- x /2 dx = lim b
(b) Since y =
2 e-x dx < 0.0092. Therefore,
e-2x dx . So
dx x3 + 1
dx , x3
0
.
dx converges 3 x +1
dx converges. 3 x +1
dx = 0.0002. x3
50 50 (c) 0.0002 (d) 1.209 (e) diverges 14) Yes, the function is symmetric about the y-axis. 3 5 5 5 dx dx dx dx dx 15) + = + + 2 2 2 2 2 3 1+x 3 1+x 3 1+x 5 - 1+x - 1+x
16) B 17) C 18) A 19) A 20) A 21) C 83
dx
1 + x2
5
= -
dx
1 + x2
dx
+ 5
1 + x2
Answer Key Testname: CHAPTER 8
22) B 23) A 24) A 25) A 26) A 27) D 28) A 29) A 30) A 31) C 32) D 33) A 34) B 35) B 36) A 37) B 38) D 39) C 40) B 41) D 42) B 43) B 44) A 45) B 46) B 47) B 48) C 49) B 50) A 51) A 52) C 53) A 54) B 55) B 56) A 57) A 58) C 59) B 60) D 61) B 62) C 63) D 84
Answer Key Testname: CHAPTER 8
64) C 65) B 66) B 67) B 68) C 69) A 70) D 71) D 72) A 73) B 74) B 75) D 76) B 77) A 78) B 79) C 80) A 81) D 82) D 83) D 84) A 85) D 86) B 87) C 88) C 89) C 90) B 91) B 92) A 93) D 94) B 95) B 96) A 97) B 98) B 99) A 100) D 101) B 102) B 103) C 104) A 105) D 85
Answer Key Testname: CHAPTER 8
106) B 107) D 108) B 109) A 110) B 111) A 112) A 113) D 114) A 115) D 116) B 117) B 118) B 119) C 120) C 121) D 122) D 123) B 124) D 125) C 126) A 127) D 128) A 129) A 130) D 131) D 132) C 133) D 134) A 135) C 136) C 137) B 138) D 139) B 140) A 141) A 142) A 143) D 144) C 145) B 146) D 147) C 86
Answer Key Testname: CHAPTER 8
148) C 149) A 150) D 151) C 152) D 153) D 154) A 155) D 156) D 157) D 158) D 159) B 160) A 161) D 162) D 163) C 164) A 165) D 166) A 167) C 168) D 169) C 170) C 171) C 172) B 173) B 174) B 175) C 176) B 177) A 178) D 179) A 180) B 181) B 182) A 183) D 184) B 185) C 186) B 187) D 188) D 189) A 87
Answer Key Testname: CHAPTER 8
190) C 191) C 192) A 193) C 194) B 195) B 196) A 197) B 198) C 199) C 200) C 201) C 202) C 203) A 204) A 205) A 206) B 207) C 208) A 209) C 210) D 211) B 212) D 213) D 214) B 215) B 216) A 217) B 218) B 219) A 220) B 221) D 222) D 223) D 224) C 225) C 226) B 227) D 228) A 229) D 230) A 231) C 88
Answer Key Testname: CHAPTER 8
232) C 233) B 234) A 235) B 236) D 237) C 238) A 239) B 240) C 241) B 242) B 243) A 244) A 245) A 246) D 247) B 248) C 249) C 250) A 251) D 252) C 253) A 254) D 255) C 256) B 257) B 258) D 259) C 260) C 261) A 262) C 263) B 264) C 265) C 266) D 267) D 268) A 269) B 270) A 271) D 272) A 273) A 89
Answer Key Testname: CHAPTER 8
274) D 275) C 276) D 277) D 278) B 279) D 280) B 281) D 282) A 283) C 284) D 285) A 286) A 287) A 288) D 289) C 290) B 291) A 292) B 293) A 294) D 295) C 296) A 297) B 298) B 299) D 300) D 301) D 302) B 303) B 304) B 305) D 306) A 307) B 308) D 309) D 310) D 311) D 312) B 313) A 314) B 315) C 90
Answer Key Testname: CHAPTER 8
316) D 317) C 318) D 319) C 320) D 321) B 322) D 323) B 324) C 325) B 326) D 327) B 328) A 329) C 330) C 331) B 332) B 333) D 334) A 335) A 336) D 337) D 338) C 339) D 340) B 341) B 342) A 343) B 344) B 345) C 346) C 347) C 348) A 349) A 350) B 351) A 352) A 353) A 354) D 355) D 356) B 357) A 91
Answer Key Testname: CHAPTER 8
358) C 359) B 360) A 361) A 362) A 363) D 364) D 365) A 366) A 367) C 368) D 369) C 370) B 371) A 372) D 373) A 374) D 375) B 376) D 377) D 378) C 379) D 380) C 381) C 382) C 383) B 384) B 385) B 386) D 387) B 388) A 389) D 390) C 391) D 392) A 393) B 394) D 395) B 396) A 397) D 398) D 399) C 92
Answer Key Testname: CHAPTER 8
400) C 401) A 402) D 403) B 404) C 405) D 406) C 407) A 408) B 409) A 410) A 411) B 412) B 413) D 414) B 415) D 416) C 417) D 418) A 419) B 420) C 421) B 422) D 423) D 424) A 425) A 426) C 427) A 428) A 429) D 430) D 431) C 432) A 433) B 434) D 435) D 436) A 437) B 438) C 439) A 440) C 441) C 93
Answer Key Testname: CHAPTER 8
442) B 443) C 444) C 445) A 446) B 447) C 448) D 449) C 450) C 451) D 452) B 453) D 454) A 455) B 456) D 457) B 458) B 459) C 460) D
94
Chapter 9
Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Match the differential equation with the appropriate slope field. 1) y = x2 - y2
A)
B)
1
1)
C)
D)
Determine which of the following equations is correct. 2) 1 tan x dx = sin x
2)
A) 1/sin x + C C) -ln(cos x)/sin x + C
B) cos x + C D) sin x + C
Match the differential equation with the appropriate slope field.
2
3) y = x - y A)
3)
B)
C)
3
D)
4) y = y + 2 A)
4)
B)
4
C)
D)
5) y = (y + 3)(y - 3) A)
5)
5
B)
C)
D)
6
6) y = y(y + 2)( y - 2) A)
6)
B)
C)
7
D)
7) y = x
7)
y
A)
B)
8
C)
D)
Determine which of the following equations is correct. 1 8) x dx = 6x
A) 6x ln x + C
8)
B) x + C
C) x ln x + C
6
6
2
D) x ln x + C 12
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the orthogonal trajectories of the family of curves. Sketch several members of each family. 9) kx2 + y2 = 1
9
9)
Solve the problem. 10) Using the given conditions, obtain a slope field and graph the particular solution over the specified interval. Then find the general solution of the differential equation. y = y(2 - lny), y(0) =
10)
1 ; 4
0 x 4, 0 y 3
Obtain a slope field and add to its graphs of the solution curves passing through the given points. 11) y = 2y with (-2, 0) x
11)
Show that the curves are orthogonal. 12) x2 + y2 = 5 and y2 = x3
12)
Find the orthogonal trajectories of the family of curves. Sketch several members of each family. 13) y = -mx
13)
The autonomous differential equation represents a model for population growth. Use phase line analysis to sketch solution curves for P(t), selecting different starting values P(0). Which equilibria are stable, and which are unstable? 14) dP = P(P - 7) 14) dt
Solve the problem. 15) Using the given conditions, obtain a slope field and graph the particular solution over the specified interval. Then find the general solution of the differential equation. y = y(3 - y), y(0) =
15)
1 ; 3
0 x 5, 0 y 4
Sketch several solution curves. 16) dy = y3 - 3y dx
16)
Obtain a slope field and add to its graphs of the solution curves passing through the given points. 17) y = -y with (0, 2)
17)
Show that the curves are orthogonal. 18) y2 + x2 = 1 and xy = 2
18)
The autonomous differential equation represents a model for population growth. Use phase line analysis to sketch solution curves for P(t), selecting different starting values P(0). Which equilibria are stable, and which are unstable? 19) dP = 1 - 3P 19) dt
10
Obtain a slope field and add to its graphs of the solution curves passing through the given points. 20) y = xy with (0, -1) x2 + 2
21) y' = 3(y - 1) with (2, 0)
20)
21)
Sketch several solution curves. 22) y = y4 - y3
22)
23) dy = y2 - 1
23)
dx
Obtain a slope field and add to its graphs of the solution curves passing through the given points. 24) y = y(1 - y) with (0, -1)
24)
Sketch several solution curves. 25) y = y - 2 y
25)
Obtain a slope field and add to its graphs of the solution curves passing through the given points. 26) y = -y2 with (0, 1)
26)
Solve the problem. 27) Using the given conditions, obtain a slope field, solve for the general solution, and plot solution curves for the arbitrary constant values C = -2, 2, and 4. y =
27)
4x2 + 3x + 1 ; 2(y - 1)
-3 x 3, -3 y 3
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve.
28) A 55-kg skateboarder on a 1-kg board starts coasting on level ground at 4 m/sec. Let k = 3.2 kg/sec. How long will it take the skater's speed to drop to 1 m/sec? A) 24.26 sec B) 0.43 sec C) -24.26 sec
D) 23.83 sec
Provide an appropriate response. 29) If a body of mass m falling from rest under the action of gravity encounters an air resistance proportional to two times the square root of velocity, then the body's velocity t seconds into the fall satisfies the equation: dv m = mg - 2kv2 dt where k is a constant that depends on the body's aerodynamic properties and the density of the air. Determine the equilibrium, velocity curve, and the terminal velocity for a 140 lb skydiver (mg =
11
28)
29)
140) with k = 0.005.
A) Equilibrium: v =
mg 2k
mg 2k
vterminal = 118.32 ft/sec
B) Equilibrium: v =
mg 2k
mg 2k
vterminal = 118.32 ft/sec
C) Equilibrium: v =
2k mg
2k mg
vterminal = 118.32 ft/sec
12
D) Equilibrium: v =
2k mg
2k mg
vterminal = 118.32 ft/sec
Use Euler's method with the specified step size to estimate the value of the solution at the given point x *. Find the value of the exact solution at x*.
30) y = 6x5 ex6 , y(0) = 3, dx = 0.1, x* = 1 A) Euler's method gives y 4.7183; the exact solution is 5.2619 B) Euler's method gives y 6.9279; the exact solution is 4.9901 C) Euler's method gives y 6.2981; the exact solution is 4.7183 D) Euler's method gives y 5.6683; the exact solution is 4.4465
Solve the differential equation. 31) 4y = ex/4 + y
31) x/4 + Cex/4
A) y = xex/4 + Cex/4
B) y = -xe
x/4 + Cex/4
D) y = xe
C) y = xe
30)
4
x/4 + C
4
4
32) x dy = y + (x2 - 1)2
32)
dx
A) y = x4 - x2 - 1 + Cx
B) y = 1 x4 - 2x ln x+ Cx
C) y = 1 x3 - 2x - 1 + C
D) y = 1 x4 - 2x2 - 1 + Cx
3
3
x
3
13
Provide an appropriate response. 33) A catamaran is running along a course with the wind providing a constant force of 75 lb. The only other force acting on the boat is resistance as the boat moves through the water. The resisting force is numerically equal to three times the boat's speed, and the initial velocity is 1 ft/sec. What is the maximum velocity in feet per second of the boat under this wind? A) The maximum velocity occurs when dv = 0 or v = 20 ft/sec dt
33)
B) The maximum velocity occurs when dv = 0 or v = 25 ft/sec dt
C) The maximum velocity occurs when dv = 25 or v = 0 ft/sec dt
D) The maximum velocity occurs when dv = 20 or v = 0 ft/sec dt
Solve the differential equation. 34) 3x2 y - 2xy = y-3
34)
A) y2 = -4 + Cx8/3 11x
B) y4 = -4 + Cx8/3 11x
C) y4 = -4 + Cx11/3
D) y2 = -4 + Cx11/3
11x
11x
Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Round your results to four decimal places. 35) y = -x(1 - y), y(1) = 2, dx = 0.2 35) A) y1 = 1.2000, y2 = 4.9760, y3 = 5.8093 B) y1 = 3.0000, y2 = 24.8800, y3 = 29.0464
C) y1 = 0.3000, y2 = 1.2440, y3 = 1.4523
D) y1 = 2.2000, y2 = 2.4880, y3 = 2.9046
Solve the problem.
36) An office contains 1000 ft3 of air initially free of carbon monoxide. Starting at time = 0, cigarette
36)
smoke containing 4% carbon monoxide is blown into the room at the rate of 0.5 ft3 /min. A ceiling fan keeps the air in the room well circulated and the air leaves the room at the same rate of 0.5 ft3 /min. Find the time when the concentration of carbon monoxide reaches 0.01%.
A) 8.01 min
B) 5.01 min
C) 6.01 min
D) 7.01 min
Identify equilibrium values and determine which are stable and which are unstable. 37) y = (y - 3)(y - 5)(y - 6) A) y = 6 is a stable equilibrium value and y = 5 and y = 3 are unstable equilibria.
37)
B) y = 3 is a stable equilibrium value and y = 5 and y = 6 are unstable equilibria. C) y = 5, y = 3 and y = 6 are unstable equilibria. D) y = 5 is a stable equilibrium value and y = 3 and y = 6 are unstable equilibria. Solve the differential equation. 38) y + y = y2
A)
1
1 + Ce-x
38) B) 1 + Cex
C) 1 - Ce-x
14
D)
1
1 + Cex
Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Round your results to four decimal places. 39) y = 5xex5, y(1) = 3, dx = 0.1 39) A) y1 = 4.3591, y2 = 7.1121, y3 = 14.3367 B) y1 = 3.4873, y2 = 6.2403, y3 = 13.4649
C) y1 = 3.9232, y2 = 6.6762, y3 = 13.9008 Solve the initial value problem. 40) 2 dy - 3 y = 5 sec d
A) y =
3
+ 3,
cos
C) y = -
2 cos
tan ;
D) y1 = 4.7951, y2 = 7.5480, y3 = 14.7726
> 0, y( ) = 0
40)
>0
- 2,
B) y = -
>0
D) y =
3
- 3,
cos 2
cos
+ 2,
>0 >0
Solve. Round your results to four decimal places.
41) Use the Euler method with dx = 0.5 to estimate y(2) if y = y2 / 2x and y(1) = -1. What is the exact value of y(2)? A) y -0.6750, exact value is -0.2557
41)
B) y -0.5258, exact value is -0.2612 D) y -0.7496, exact value is -0.2335
C) y -0.5631, exact value is -0.2723 Solve the problem.
42) The system of equations dx = (-3 + 4y)x and dy = (-3 + x)y describes the growth rates of two dt
dt
42)
symbiotic (dependent) species of animals (such as the rhinoceros and a type of bird which eats insects from its back). What is necessarily true of the two populations at the equilibrium points? A) Both populations equal zero. B) They both remain constant over all time.
C) They are both at a maximum.
D) They are both at a minimum.
Solve the initial value problem. 43) t dy + 4y = t3; t > 0, y(2) = 1 dt
43)
3
3
A) y = t - 2t-4 , t > 0
B) y = t + 144 t-4, t > 0
7
7
3
3
C) y = t + 16 t-4 , t > 0 7
7
D) y = t - 16 t-4 , t > 0
7
7
7
Solve the problem. 44) A 200 gal tank is half full of distilled water. At time = 0, a solution containing 1 lb/gal of concentrate enters the tank at the rate of 4 gal/min, and the well-stirred mixture is withdrawn at the rate of 2 gal/min. When the tank is full, how many pounds of concentrate will it contain? A) 120 pounds B) 200 pounds C) 150 pounds D) 100 pounds
15
44)
45) dy/dt = ky + f(t) is a population model where y is the population at time t and f(t) is some function
45)
to describe the net effect on the population. Assume k = .02 and y = 10,000 when t = 0. Solve the differential equation of y when f(t) = -5t. A) y = 250t + 12,500 - 2500e-.02t B) y = 250t + 12,500 - 2500e.02t
C) y = -250t - 12,500 - 2500e.02t
D) y = -250t + 12,500 - 2500e-.02t
Solve the differential equation. 46) y - y = -xy2
46)
A) x - 1 + C
ex C) (x + 1) + C
B) x + 1 + C
D)
ex
ex(x - 1) + C
Solve. Round your results to four decimal places. 47) Use the Euler method with dx = 0.2 to estimate y(1) if y = -y and y(0) = 1. What is the exact value of y(1)? A) y 0.2277, exact value is e-2 B) y 0.0277, exact value is -2e
C) y 0.3277, exact value is e-1
D) y 0.1277, exact value is -e
Solve the initial value problem. 48) dy + y = cos ; > 0, y( ) = 1 d
A) y = sin + 2
C) y = -sin + 2
, ,
48)
>0
B) y = sin + , > 0
>0
D) y = -sin + , > 0
Solve the differential equation. 49) x dy + 2y = 9 - 1 , x > 0 dx x
49)
2
2
A) y = 9x + 2x + C , x > 0
B) y = x - 9x + C , x > 0
2x2
2x2
2
2
C) y = 9x - 2x + C , x > 0
D) y = 9x - x + C , x > 0
2x2
2x2
50) -x3 y + 2x2 y = y2 A) y =
x2
lnx + C
47)
50) B) y =
x lnx + C
C) y =
16
x2
lnx - C
D) y =
x lnx - C
Identify equilibrium values and determine which are stable and which are unstable. 51) dy = (y + 5)(y + 4) dx
51)
A) y = 5 is a stable equilibrium value and y = 4 is an unstable equilibrium. B) y = -4 is a stable equilibrium value and y = 5 is an unstable equilibrium. C) y = 4 is a stable equilibrium value and y = -5 is an unstable equilibrium. D) y = -5 is a stable equilibrium value and y = 4 is an unstable equilibrium. Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Round your results to four decimal places. 52) y = 2xy - 2y, y(1) = 6, dx = 0.2 52) A) y1 = 3.0000, y2 = 7.2900, y3 = 17.4960 B) y1 = 6.0000, y2 = 6.4800, y3 = 7.5168
C) y1 = 1.8000, y2 = 4.0500, y3 = 26.2440
D) y1 = 0.6000, y2 = 6.4800, y3 = 13.9968
Solve the problem. 53) A tank contains 100 gal of fresh water. A solution containing 2 lb/gal of soluble lawn fertilizer runs into the tank at the rate of 1 gal/min, and the mixture is pumped out of the tank at the rate of 2 gal/min. Find the maximum amount of fertilizer in the tank and the time required to reach the maximum. A) 60 pounds, 40 minutes B) 50 pounds, 50 minutes
C) 48 pounds, 40 minutes
53)
D) 48 pounds, 60 minutes
Solve the initial value problem. 54) 2 dy - 4xy = 8x; y(0) = 22 dx
54)
A) y = 2 + 22ex2
B) y = -1 + 23ex2
C) y = -2 + 24ex2
D) y = -2 + 24e-x2
Solve the differential equation. 55) xy + 4y = cos x , x > 0 x3
55)
A) y = cos x + C , x > 0
B) y = sin x + C , x > 0
C) y = cos x + C , x > 0
D) y = sin x + C , x > 0
x4
x3
x3
x4
Solve the problem. 56) If the switch is thrown open after the current in an RL circuit has built up to its steady-state value, di the decaying current obeys the equation L + Ri = 0. How long after the switch is thrown open dt will it take the current to fall to 20% of its original value? A) -4.60 L/R seconds B) 1.61 L/R seconds
C) 1.81 L/R seconds
D) -4.40 L/R seconds
17
56)
Use Euler's method with the specified step size to estimate the value of the solution at the given point x *. Find the value of the exact solution at x*.
57) y = 3y2 / 3x, y(1) = -1, dx = 0.5, x* = 5 A) Euler's method gives y -0.1949; the exact solution is -0.2594 B) Euler's method gives y -0.1482; the exact solution is -0.2224 C) Euler's method gives y -0.0879; the exact solution is -0.1893 D) Euler's method gives y 0.0821; the exact solution is -0.1601
Construct a phase line. Identify signs of y and y . 58) dy = (y + 2)(y - 3) dx
58)
A) y >0
y <0
y <0
y >0
y <0
y >0
y >0
y >0
1 2
-2
3
B) y >0
y <0
y <0
y >0
y <0
y <0
y <0
y >0
1 2
-3
2
C) y >0
y <0
y <0
y >0
y <0
y >0
y >0
y >0
1 2
-3
2
D) y >0
y <0
y <0
y >0
y <0
y <0
y <0
y >0
-2
1 2
57)
3
18
Use Euler's method with the specified step size to estimate the value of the solution at the given point x *. Find the value of the exact solution at x*.
59) y = y - e-2x, y(0) = 2, dx = 1/3, x* = 2 A) Euler's method gives y 8.2740; the exact solution is 11.1513 B) Euler's method gives y 9.2983; the exact solution is 12.9315 C) Euler's method gives y 8.9605; the exact solution is 12.3212 D) Euler's method gives y 9.4866; the exact solution is 13.3004
59)
Identify equilibrium values and determine which are stable and which are unstable. 60) dy = y2 - 25 dx
60)
A) y = -5 and y = 5 are stable equilibrium values. B) y = -5 is a stable equilibrium value and y = 5 is an unstable equilibrium. C) y = 5 is a stable equilibrium value and y = -5 is an unstable equilibrium. D) There are no equilibrium values. Solve the problem. 61) dy/dt = ky + f(t) is a population model where y is the population at time t and f(t) is some function to describe the net effect on the population. Assume k = .02 and y = 10,000 when t = 0. Solve the differential equation of y when f(t) = 19t. A) y = 950t - 47,500 + 57,500e-.02t B) y = 950t + 47,500 + 57,500e-.02t
61)
D) y = -950t - 47,500 + 57,500e.02t
C) y = -950t - 47,500 + 57,500e-.02t Solve the differential equation. 62) x dy = cos x - 4y, x > 0 dx x3
62)
A) y = x4(cos x + C), x > 0 C) y = x-4 (cos x + C), x > 0
B) y = x4(sin x + C), x > 0 D) y = x-4 (sin x + C), x > 0
Solve.
63) A local pond can only hold up to 31 geese. Six geese are introduced into the pond. Assume that the rate of growth of the population is dP = (0.0013)(31 - P)P dt
where t is time in weeks. Find a formula for the goose population in terms of t. 31 31 A) P(t) = B) P(t) = -0.040t 1 + 4.2e 1 + 25e-31t
C) P(t) =
31 1 + 25e-0.040t
D) P(t) =
19
31 1 + 4.2e-31t
63)
Construct a phase line. Identify signs of y and y . 64) y = 3y, y > 0
64)
A) y >0
y <0
y <0
y >0 3 2
B) y >0 y >0
3 2
C) y <0
y >0
y >0
y <0
3 2
D) y <0 y >0
3 2
Identify equilibrium values and determine which are stable and which are unstable. 65) y = 2y, y > 0 A) y = 0 is an unstable equilibrium value. B) y = 0 is a stable equilibrium value.
C) y = 2 is an unstable equilibrium value.
D) There are no equilibrium values.
20
65)
Solve the problem. 66) A tank initially contains 100 gal of brine in which 40 lb of salt are dissolved. A brine containing 2 lb/gal of salt runs into the tank at the rate of 4 gal/min. The mixture is kept uniform by stirring and flows out of the tank at the rate of 3 gal/min. Find the solution to the differential equation that models the mixing process.
A) y = 4(50 + t) C) y = 4(50 + t) -
C
B) y = 2(100 + t) -
(100 + t)3 108
D) y = 2(100 + t) -
(100 + t)3
66)
C
(100 + t3 ) 108
(100 + t)3
Solve the differential equation. 67) dy - y = ( ln x)4 dx x
67)
A) y = x (ln x) 5 + Cx
B) y = 1 x5 + Cx
C) y = 1 x (ln x) 5 + Cx
D) y = 1 (ln x) 5 + Cx
5
5
5
Solve the problem. 68) A 100 gal tank is half full of distilled water. At time = 0, a solution containing 2 lb/gal of concentrate enters the tank at the rate of 4 gal/min, and the well-stirred mixture is withdrawn at the rate of 3 gal/min. When the tank is full, how many pounds of concentrate will it contain? A) 150 pounds B) 175 pounds C) 187.5 pounds D) 200 pounds Solve the initial value problem. 69) y + y = 2ex; y(0) = 15
68)
69)
A) y = 2ex + 12e-x C) y = 15ex
B) y = 4e2 + 20e-x D) y = ex + 14e-x
Solve. Round your results to four decimal places. 70) Use the Euler method with dx = 0.2 to estimate y(2) if y = -y/x and y(1) = 4. What is the exact value of y(2)? A) y 1.4286, exact value is 8.0000 B) y 0.8571, exact value is 4.0000
C) y 1.7778, exact value is 1.3333
70)
D) y 0.5714, exact value is 0.7500
Solve the differential equation. 71) cos x dy + y sin x = sin x cos x dx
71)
A) y = cos x ln sec x + tan x + C cos x C) y = cos x ln sec x + C cos x
B) y = cot x + C cos x D) y = sin x ln sec x + C sin x
72) y + y tan x = cos x, - /2 < x < /2 A) y = x cos x + C cos x, - /2 < x < /2 C) y = x sin x + C cos x, - /2 < x < /2
72) B) y = x cos x + C sin x, - /2 < x < /2 D) y = x sin x + C sin x, - /2 < x < /2
21
Solve the problem.
73) The system of equations dx = (-6 + 7y)x and dy = (-6 + x)y describes the growth rates of two dt
73)
dt
symbiotic (dependent) species of animals (such as the rhinoceros and a type of bird which eats insects from its back). What happens to the rhinoceros population when the bird population decreases? A) The rhinoceros population decreases.
B) The rhinoceros population could either increase or decrease. C) The rhinoceros population stays the same. D) The rhinoceros population increases. Solve the differential equation. 74) ex dy + 2ex y = 1, x > 0 dx
74)
A) y = e-2x + Ce-x, x > 0 C) y = e-x + e-2x, x > 0
B) y = e-x + Ce-2x, x > 0 D) y = ex + Ce-2x, x > 0
Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Round your results to four decimal places. 75) y = y2(1 - 2x), y(-1) = -1, dx = 0.5 75)
A) y1 = 0.5, y2 = 0.75, y3 = 1.03125 C) y1 = 0.7, y2 = 0.99, y3 = 1.9136
B) y1 = 0.7, y2 = 0.99, y3 = 1.2656 D) y1 = 0.4, y2 = 0.63, y3 = 0.7472
Solve the initial value problem. 76) dy + 8y = 3; y(0) = 1 dt
76)
A) y = 5 e8t + 3 8
B) y = 3 e-8t + 5
8
8
x2
77) (x + 2) dy - 2(x2 + 2x)y = e dx
A) y = ex2
x+2
C) y = 5 e-8t + 3
8
8
D) y = 3 e8t + 5 8
; x > -2, y(0) = 0
8
77)
x , x > -2 2x + 4
C) y = -e-x2
8
x , x > -2 2x + 4
B) y = e-x2
x , x > -2 2x + 4
D) y = -ex2
x , x > -2 2x + 4
Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Round your results to four decimal places. 78) y = y - ex - 1, y(-1) = 2, dx = 0.5 78)
A) y1 = 1.8161, y2 = 1.3708, y3 = 2.6562 C) y1 = 2.3161, y2 = 2.6708, y3 = 3.0062
B) y1 = 2.8161, y2 = 3.9708, y3 = 3.3562 D) y1 = 3.3161, y2 = 5.2708, y3 = 3.7062
22
Solve the problem. 79) How many seconds after the switch in an RL circuit is closed will it take the current i to reach 20% of its steady state value? Express answer in terms of R and L and round coefficient to the nearest hundredth. A) 0.22 L/R seconds B) 0.42 L/R seconds
C) 1.61 L/R seconds
79)
D) 1.81 L/R seconds
Identify equilibrium values and determine which are stable and which are unstable. 80) dy = y2 - 5y dx
80)
A) y = 0 and y = 5 are unstable equilibrium values. B) y = 5 is a stable equilibrium value and y = 0 is an unstable equilibrium. C) y = 0 is a stable equilibrium value and y = 5 is an unstable equilibrium. D) There are no equilibrium values. Solve.
81) A 51-kg skateboarder on a 2-kg board starts coasting on level ground at 8 m/sec. Let k = 3.2 kg/sec. About how far will the skater coast before reaching a complete stop? A) 2611.20 m B) 40.80 m C) 132.50 m
81)
D) 255.00 m
82) Solve the initial value problem
82)
dP = 4kP2, P(0) = P0 dt
A) P(t) = C) P(t) =
P0
B) P(t) =
4 - 4kP0 t 4P0 1 - 4kP0 t
D) P(t) =
P0 1 - kP0 t P0 1 - 4kP0 t
Solve the initial value problem. 83) dy + xy = 4x; y(0) = -4 dx
83)
A) y = 4e-x2/2 - 8
B) y = -8ex2/2 + 4
C) y = -8e-x2 /2 + 4
D) y = 4ex2 /2 - 8
Solve the problem. 84) A tank initially contains 120 gal of brine in which 40 lb of salt are dissolved. A brine containing 1 lb/gal of salt runs into the tank at the rate of 6 gal/min. The mixture is kept uniform by stirring and flows out of the tank at the rate of 8 gal/min. Write, in standard form, the differential equation that models the mixing process. A) dy + 6 y = 8 B) dy + 8 y = 6 dt 120 - 2t dt 120 - 2t
C) dy = 6 dt
8 y 120 - 2t
D) dy + dt
23
8 y=6 120 + 2t
84)
Solve the differential equation. 85) -xy - y = y-2
85)
A) y3 = 1 - Cx-3
B) y2 = -1 + Cx-3
C) y2 = 1 - Cx-3
86) y e3x + 3ye3x = 4x A) y = 4x2 e-3x + Ce-3x C) y = 4x2 e3x + Ce3x
D) y3 = -1 + Cx-3 86)
B) y = 2x2 e-3x + Ce-3x D) y = 2x2 e3x + Ce3x
Solve the problem.
87) The system of equations dx = (-3 + 4y)x and dy = (-3 + x)y describes the growth rates of two dt
dt
symbiotic (dependent) species of animals (such as the rhinoceros and a type of bird which eats insects from its back). Find the equilibrium points. A) (x, y) = (0, 0) and (x, y) = -3, - 3 B) (x, y) = 3, 3 4 4
C) (x, y) = (0, 0) and (x, y) = 3, 4
D) (x, y) = (0, 0) and (x, y) = 3, 3
3
4
24
87)
Construct a phase line. Identify signs of y and y . 88) dy = y2 - 36 dx
88)
A) y >0
y <0
y <0
y >0
y >0
y <0
y >0
y >0
0
-6
6
B) y <0
y >0
y >0
y <0
y <0
y >0
y <0
y >0
0
-6
6
C) y >0
y <0
y <0
y >0
y <0
y >0
y <0
y >0
0
-6
6
D) y <0
y <0
y <0
y >0
y <0
y <0
y <0
y >0
-6
0
6
Solve.
89) A local pond can only hold up to 58 geese. Eight geese are introduced into the pond. Assume that the rate of growth of the population is dP = (0.0011)(58 - P)P dt
where t is time in weeks. How long will it take for the goose population to be 23? A) 22.14 weeks B) 14.23 weeks C) 34.65 weeks
25
D) 54.74 weeks
89)
Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Round your results to four decimal places. 90) y = 1 + y , y(3) = -3, dx = 0.5 90) x
A) y1 = -4.5000, y2 = -3.5143, y3 = -3.3536 B) y1 = -6.0000, y2 = -5.8571, y3 = -11.1786 C) y1 = -3.0000, y2 = -2.9286, y3 = -2.7946 D) y1 = -6.0000, y2 = -4.3929, y3 = -5.5893
26
Answer Key Testname: CHAPTER 9
1) C 2) C 3) B 4) A 5) D 6) B 7) B 8) C 9) y = Cx1/k
10)
y=
5 2 1 2 1 y - y lny + 4 2 4
27
Answer Key Testname: CHAPTER 9
11)
12)
13) y = 1 x + C m
28
Answer Key Testname: CHAPTER 9
14)
7 7 2
0 -
7 2
dP d2 P = P(P - 7) has a stable equilibrium at P = 0 and an unstable equilibrium at P = 7; = P(P - 7)(2P - 7) dt dt2
15)
y=
3y2 y3 1 + 2 3 3
29
Answer Key Testname: CHAPTER 9
16)
17)
18)
30
Answer Key Testname: CHAPTER 9
2
19) dP = 1- 3P has stable equilibrium at P = 1 , d P = -3(1 - 3P) dt
3 dt2
1 3
-
1 3
20)
21)
31
Answer Key Testname: CHAPTER 9
22)
23)
24)
32
Answer Key Testname: CHAPTER 9
25)
26)
27) y=
x(8x2 + 9x + 6) +C 12(y - 1)
28) A 29) A 30) C 31) C 33
Answer Key Testname: CHAPTER 9
32) D 33) B 34) B 35) D 36) B 37) D 38) D 39) A 40) B 41) B 42) B 43) D 44) C 45) B 46) D 47) C 48) B 49) C 50) A 51) D 52) B 53) B 54) C 55) D 56) B 57) C 58) D 59) C 60) B 61) D 62) D 63) A 64) B 65) D 66) D 67) C 68) C 69) D 70) C 71) C 72) A 73) A 34
Answer Key Testname: CHAPTER 9
74) B 75) A 76) C 77) A 78) C 79) A 80) C 81) C 82) D 83) C 84) B 85) D 86) B 87) D 88) C 89) A 90) C
35
Chapter 10
Exam Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Which of the following statements is false?
1)
A) If an and f(n) satisfy the requirements of the Integral Test, and if
n=1
B) n=1
an =
f(x)dx converges, then 1
f(x) dx. 1
1 converges if p >1 and diverges if p 1. np
C) The integral test does not apply to divergent sequences. D)
1 n=2 n(ln n)
p
converges if p > 1.
2) For which of the following is the corresponding Taylor series a finite polynomial of degree 3? A) 5x3 + 2x2 - 12
B) e-2x3
C) x2 sin x
D) 3ln(x)
3) Which of the following statements is false?
3)
A) If {an} and {bn} meet the conditions of the Limit Comparison test, then, if lim n
bn converges, then
2)
an bn
= 0 and
a n converges.
B) The series an must have no negative terms in order for the Direct Comparison test to be applicable.
C) The sequences {a n} and {bn} must be positive for all n to apply the Limit Comparison Test. D) All of these are true. For what values of x does the series converge conditionally?
4) n=1
(-1)n (x + 4)n n
A) x = -5
4) B) x = -3
C) x = -5, x = -3
D) None
Find the sum of the series. 5) 1 + 5 + 25 + 125 + 6 + ... 2! 3! 4!
A) ln 6
5) C) e5
B) e-5
1
D) sin 1 5
Find the first four nonzero terms in the Maclaurin series for the function. 6) e5x
A) 1 + 5x +
25x2 2!
125x3
+
3!
2
3
2!
3!
6)
B) 5x + 125x3 + 3125x5 + 78,125x7 + . . .
+...
C) 1 + 5x + 5x + 5x + . . .
2
3
2!
3!
D) x + 5x + 5x + 5x + . . .
Find the sum of the geometric series for those x for which the series converges. (-1)n
7) n=0
A)
x-5 n 9
9 -4 - x
7) B)
9 4+x
C)
9 4-x
D)
9 -4 + x
Find the sum of the series.
8) n=0
7 8n
8) B) 7
A) 1
D) 56
C) 8
9
9
For what values of x does the series converge conditionally?
9) n=1
(x + 1)n n
9)
A) x = 0
B) x = -2
C) x = -2, x = 0
D) None
Determine if the series converges or diverges. If the series converges, find its sum.
10) n=1
9n (2n - 1)2 (2n + 1)2
A) converges; 15 2
10) B) converges; 27
C) converges; 9
8
8
D) converges; 21 2
Provide an appropriate response.
11) If the series
(-1)n (x + 6)n is integrated twice term by term, for what value(s) of x (if any) does
n=0 the new series converge and for which the given series does not converge? A) x = -7 B) none C) x = -7, x = -5
2
D) x = -5
11)
Find the Taylor polynomial of order 3 generated by f at a. 12) f(x) = 1 , a = 0 x+8
12)
2 3 A) P3(x) = 1 - x + x - x
4096
2 3 4 B) P3(x) = x + x + x + x
2 3 C) P3(x) = 1 + x + x + x
2 3 4 D) P3(x) = x - x + x - x
8 8
64 64
512 512
8
4096
8
64 64
512 512
4096 4096
Determine if the series converges absolutely, converges, or diverges. (-1)n
13) n=1
(n!)2 2 n (5n + 1)!
13)
A) converges conditionally B) diverges C) converges absolutely Find the first four nonzero terms in the Maclaurin series for the function. 14) ln(1 + x3 )
14)
A) x3 - 1 x6 + 1 x9 - 1 x12 + . . . 2 6 24
B) 1 x3 - 1 x6 + 1 x9 - 1 x12 + . . . 3 6 9 12
C) x3 + 1 x6 - 1 x9 + 1 x12 + . . .
D) x3 - 1 x6 + 1 x9 - 1 x12 + . . .
2
6
24
2
3
4
Use series to evaluate the limit. 15) lim sin 7x - tan 7x x 0 x3
15)
A) - 7
B) - 3.017059907e+15
C) - 49
D) -7
2
1.759218604e+13
2
Find the sum of the series as a function of x. (x - 4)n
16) n=0
A)
1 x+5
16) B)
1 x-5
C) - 1
x+5
D) - 1
x-5
Use the Comparison Test to determine if the series converges or diverges.
17) n=1
1 2 n ln n - 10
17)
A) converges
B) diverges
3
Use the nth-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. cos
18) n=1
9 n
18)
A) diverges
B) converges, 9
C) inconclusive
D) converges, 1
Use the Root Test to determine if the series converges or diverges.
19)
n
19)
n n=1 ln n + 8 A) Converges
B) Diverges
Find the Maclaurin series for the given function. 20) e-10x
A) n=1
C) n=0
20)
(-1)n 10n xn n!
B)
(-1)n 10n xn n!
D)
n=1
n=0
10n xn n! 10n xn n!
Provide an appropriate response.
21) A sequence of rational numbers {rn } is defined by r1 = 2 , and if rn = a then rn+1 = a + b . Find 1
b
a-b
21)
r50.
A) 50
B) 2
C) 49
D) 3
Determine if the sequence is monotonic and if it is bounded. 22) an = (n + 8)! (8n + 1)!
A) decreasing; bounded C) not monotonic; bounded
22) B) increasing; unbounded D) nondecreasing; bounded
Find the polynomial of lowest degree that will approximate F(x) throughout the given interval with an error of magnitude less than than 10-3 . x
23) F(x) = A)
0 x3 3
-
2 (tan-1 t) dt , [0, 0.5] 2x15 15
23) 3
B) x - 2x 3
9
3
9
3
27
C) x - 2x
9
Determine whether the nonincreasing sequence converges or diverges. n+1 + 2 n 24) an = 8 n · 8n
A) Diverges
B) Converges
4
3
15
3
15
D) x - x
24)
Use the Comparison Test to determine if the series converges or diverges.
25)
1
25)
n n=1 (ln 5n) A) diverges
B) converges
Use power series operations to find the Taylor series at x = 0 for the given function. 26) x7 tan-1 (9x)
A) n=0
C) n=0
(-1)n+1 9 2n+1 x2n+8 2n + 1
B)
(-1)n 9 2n+1 x2n+7 2n + 1
D)
n=0
n=0
26)
(-1)n 9 2n+1 x2n+8 2n + 1 (-1)n+1 9 2n+1 x2n+7 2n + 1
Solve the problem.
27) Using the Maclaurin series for ln(1 + x), obtain a series for ln 1 + x .
27)
1-x
A) n=0
C) n=0
x2n+1 2n + 1
B) 2
(-1)n x2n+1 2n + 1
D) 2
n=0
n=0
x2n+1 2n + 1 (-1)n x2n+1 2n + 1
Provide an appropriate response. (-1)n (x - 2)n is integrated term by term, for what value(s) of x does the new
28) If the series n=0 series converge? A) 1 x < 3
B) 1 x 3
C) 1 < x < 3
28)
D) 1 < x 3
Use power series operations to find the Taylor series at x = 0 for the given function. 29) x10e6x
A) n=0
6 n xn+10 n!
B) n=0
6 n+10xn+10
C)
D)
n!
n=0
n=0
29)
6 n xn+10 (n + 10)! 6 n+10xn+10 (n + 10)!
Find the polynomial of lowest degree that will approximate F(x) throughout the given interval with an error of magnitude less than than 10-3 . x
30) F(x) = 0
sin t dt , [0, 1] t 3
5
18
600
A) x - x + x
30) 3
5
18
90
B) x - x + x
3
5
18
90
C) x + x + x
5
3
5
18
600
D) x + x + x
Solve the problem. 31) Find the sum of the infinite series 1 + 3r + r2 + 3r3 + r4 + 3r5 + r6 + ...
31)
for those values of r for which it converges. A) 1 - 3r B) 1 + 3r 2 1+r 1 + r2
C) 1 + 3r 1 - r2
D) 1 - 3r 1 - r2
Use the limit comparison test to determine if the series converges or diverges.
32) n=1
5 - 2 sin n 9n 5/4 + 8 cos n
32)
A) Converges
B) Diverges
Use the Comparison Test to determine if the series converges or diverges. 2 e-3n 33) 2 n=1 n A) diverges B) converges
33)
Find the Taylor series generated by f at x = a. 34) f(x) = e5x, a = 6
A) n=0
C) n=0
34)
e30 5 n+1 (x - 6)n (n + 1)!
B)
e30 5 n (x - 6)n n!
D)
n=0
n=0
e30 5 n+1 (x - 6)n n! e30 5 n (x - 6)n (n + 1)!
Use substitution to find the Taylor series at x = 0 of the given function. 35) ln(1 + x2 )
A) n=1
C)
(-1)n x2n n
B)
(-1)n-1 x2n
D)
n=1
n=1
n=1
35) (-1)n-1 x2n (2n)! (-1)n-1 x2n n
Use series to evaluate the limit. 2 36) lim 1 + ln(1 + 4x ) - cos 4x x 0 x2
A) 18
36) B) 2
C) 12
6
D) 8
Find the Taylor series generated by f at x = a. 37) f(x) = 1 , a = 6 x2
37) B)
n=0
(-1)n n(x - 6)n 6 n+2
D)
n=0
(-1)n (n + 1)(x - 6)n 6 n+2
A) C)
n=0
(-1)n (n + 1)(x - 6)n 6 n+1
n=0
(-1)n n(x - 6)n 6 n+1
Use series to evaluate the limit. 4x 38) lim e - 1 x x 0
A) -4
38) B) 0
C) 1
D) 4
Solve the problem. 39) Use the Alternating Series Estimation Theorem to estimate the error that results from replacing x2 e-x by 1 - x + when 0 < x < 0.2. 2
A) 0.00027
B) 0.00267
C) -0.00133
39)
D) 0.00133
Determine convergence or divergence of the series.
40) n=1
5n 1/2 2n 3/2 + 6
40)
A) Converges
B) Diverges
Provide an appropriate response.
41) For an alternating series
(-1)n+1 un where it is not true that un un+1 for sufficiently large
41)
n=1 n, what can be said about the convergence or divergence of the series? A) The series always converges.
B) The series always diverges. C) The series may or may not converge. Find the limit of the sequence if it converges; otherwise indicate divergence. 42) an = n! 7n · 2n
A) e14
B) 0
C) 1
42) D) Diverges
Find the interval of convergence of the series. (x - 6)n
43)
43)
n=0
A) x < 7
B) 5 x < 7
C) 5 < x < 7 7
D) -7 < x < 7
Answer the question.
1
2 4
44) Estimate the error if e-2x2 is approximated by 1 - 2x2 + 2 x in the integral of 2!
A) 2
5
6 B) 2
6(3!)
C) 2
6
7(3!)
7(3!)2
2 e-2x dx.
44)
0
4 D) 2
5(2!)2
By calculating an appropriate number of terms, determine if the series converges or diverges. If it converges, find the limit L and the smallest integer N such that a n - L < 0.01 for n N; otherwise indicate divergence.
45) an = 10551/n A) L = 1, N = 1065
45) B) L = 1, N = 700
C) L = 0, N = 1065
Find a formula for the nth term of the sequence. 46) 0, 2, 2, 2, 0, 2, 2, 2 (0, 2, 2, 2 repeated) A) an = 1 + (-1)n(n+1)/2 C) a n = 1 + (-1)n(n+1)(n+2)/6
D) diverges
46) B) an = 1 + (-1)(n+1)(n + 2)/2 D) an = 1 + (-1)n(n-1)/2
For what values of x does the series converge conditionally? (-1)n (9x + 6)n
47) n=0
A) x = - 5 9
47) B) x = - 7
C) x = -6
9
D) None
Use the limit comparison test to determine if the series converges or diverges.
48)
6 n
48)
3/2 - 7n + 10 n=1 4n
A) Diverges
B) Converges
Find the sum of the geometric series for those x for which the series converges. 2 n xn
49) n=0
A)
1 1 - 2x
49) B)
2 1 - 2x
C)
1 1 + 2x
D)
2 1 + 2x
Find the limit of the sequence if it converges; otherwise indicate divergence. 50) an = (-1)n 3 n
A) ±3
B) 0
C) 3
8
50) D) Diverges
For what values of x does the series converge absolutely?
51) n=1
xn n2 + 3
51)
A) - 1 < x < 1
B) - 1 < x < 1
C) -1 < x < 1
D) -3 < x < 3
3
3
3
3
Use the Root Test to determine if the series converges or diverges. n
52) n=1
52)
n 4n 1/n - 1
A) Converges
B) Diverges
Provide an appropriate response.
3
5
3
5
53) For approximately what values of x can tan-1 x be replaced by x - x + x with an error of magnitude no greater than 5 × 10-3 ? A) |x| < 0.60596 B) |x| < 0.61945
C) |x| < 0.55743
D) |x| < 0.57193
Find the smallest value of N that will make the inequality hold for all n > N. n 54) 5 < 10-2 n!
A) 16 Determine if the series
B) 13
C) 15
53)
54) D) 14
a n defined by the formula converges or diverges.
n=1 55) a1 = 3, a n+1 = 5 + sin n a n n
55)
A) Converges
B) Diverges
Determine if the geometric series converges or diverges. If it converges, find its sum. 56) 1 + (-9) + (-9)2 + (-9)3 + (-9)4 + . . .
A) converges, 90
B) converges, 9
C) diverges
56) D) converges, 81
Solve the problem. 57) A child on a swing sweeps out a distance of 24 ft on the first pass. If she is allowed to continue 3 swinging until she stops, and if on each pass she sweeps out a distance of the previous pass, 4 how far does the child travel? A) 96 ft B) 120 ft
C) 48 ft
9
D) 72 ft
57)
Provide an appropriate response. 58) Which of the following statements is false? I. For a function f(x), the Taylor polynomial approximation can always be improved by increasing the degree of the polynomial. II. Of all polynomials of degree less than or equal to n, the Taylor polynomial of order n gives the best approximation of f(x). III. The Taylor series at x = a can be obtained by substituting x - a for x in the corresponding Maclaurin series. A) III only B) I, II, and III C) I and III D) I and II
59) If an and bn both converge conditionally, what can be said about A) The series always diverges. B) The series may converge or diverge. C) The series always converges. Determine if the series
max {a n , bn } ?
58)
59)
a n defined by the formula converges or diverges.
n=1 1 60) a1 = , a n+1 = (an)n+1 9
60)
A) Converges
B) Diverges
A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. an 61) a1 = 1, a n+1 = 61) n+5
A) 1, 1 , 1 , 1 , 1
B) 1, 1 , 6 , 7 , 8
C) 1, 1 , 1 , 1 , 1
D) 1, 1 , 42, 1 , 3024
6 7 8 9
6 7 8 9
6 42 336 3024
6
Express the number as the ratio of two integers. 62) 0.812812 . . . A) 8120 B) 812 999 999
336
62) C) 8120 99
D) 812 99
Find the first four nonzero terms in the Maclaurin series for the function. 63) sin x cos x A) x + 1 x3 - 2 x5 + 4 x7 + . . . B) x - 1 x3 + 1 x5 - 2 x7 + . . . 3 15 315 6 30 315
C) x - 2 x3 + 2 x5 - 4 x7 + . . . 3
15
D) 1 + x - 1 x2 - 1 x3 + . . .
315
2
10
6
63)
Use series to evaluate the limit. -1 64) lim sin 4x - sin 4x x 0 tan-1 4x - tan 4x
A) - 1
64) B) - 1
3
C) - 2
2
D) 1
3
3
Find the limit of the sequence if it converges; otherwise indicate divergence. 65) an = ln(7n - 1) - ln(6n + 2)
B) ln 6
A) ln 1
C) ln 7
7
65) D) Diverges
6
Find the values of x for which the geometric series converges. (-1)n (9x)2n
66)
66)
n=0
A) |x| < 18
B) |x| < 1
C) |x| < 9
D) |x| < 1 9
Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series. (-1)n+1
67) n=1
1 9n
67)
A) Error < 1.69 × 10-5
B) Error < 1.88 × 10-6
C) Error < 1
D) Error < 1.52 × 10-4
n
Find the sum of the series as a function of x.
68) n=0
(x + 7)2n 9n
A) -
9
(x + 7)2 - 9
68) B)
9
C) -
(x + 7)2 - 9
9
(x + 7)2 + 9
D)
9
(x + 7)2 + 9
A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. 69) a1 = 1, a n+1 = an + 3 69)
A) 1, 4, 7, 10, 13 C) 4, 7, 10, 13, 16
B) 1, 3, 9, 27, 81, 243 D) 1, 4, 7, 10, 13, 16
Find the limit of the sequence if it converges; otherwise indicate divergence. 7/n 70) an = 7 n
A) 1
B) ln 7
C) 0
11
70) D) Diverges
Determine convergence or divergence of the alternating series. (-1)n
71) n=1
3n 2 + 6 6n 7 + 3
71)
A) Converges
B) Diverges
For what values of x does the series converge absolutely? (4x)n
72)
72)
n=0
A) 0 < x < 1
B) - 1 < x < 1
4
4
Find the sum of the series. 73) x7 + x8 + x9 + x10 + ...
A) 7x
D) None
4
73) B)
1-x
C) x > 1
4
x6
C)
1-x
x7
D)
1+x
x7
1-x
Use the Comparison Test to determine if the series converges or diverges.
74) n=1
n n 2n + 3
74)
A) diverges
B) converges
Provide an appropriate response.
75) Obtain the first nonzero term of the Maclaurin series for sin-1 x - tan-1 x. A) -
x3
B)
3
x3
C)
6
x3
75) D)
3
x3 2
Determine if the series converges or diverges. If the series converges, find its sum.
76) n=1
1 1 1/(n+1) 1/n 3 3
A) converges; 2 3
76) C) converges; - 1
B) diverges
3
D) converges;1
Solve the problem.
77) You plan to estimate e by evaluating the Maclaurin series for f(x) = ex at x = 1. How many terms
of the series would you have to add to be sure the estimate is off by no more than one in the fourth decimal place? Use the remainder estimation theorem. A) 9 B) 12 C) 11 D) 10
12
77)
Use the integral test to determine whether the series converges.
78) n=1
7 n
78)
A) converges
B) diverges
Use the limit comparison test to determine if the series converges or diverges.
79)
1 n=1 8 n + 3(ln n) A) Diverges
79)
2
B) Converges
Provide an appropriate response. 80) Which of the following is not a condition for applying the integral test to the sequence {a n}, where
80)
a n = f(n)? I. f(x) is everywhere positive II. f(x) is decreasing for x N III. f(x) is continuous for x N A) I only
B) III only C) II only D) All of these are conditions for applying the integral test. Determine if the series
a n defined by the formula converges or diverges.
n=1 81) a1 = 8, a n+1 = n a n n+1
81)
A) Diverges
B) Converges
Determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001. (-1)n+1
82) n=1
27 (n +
82)
n)3
A) n 26
B) n 24
C) n 27
Find the quadratic approximation of f at x = 0. 83) f(x) = eln 3x
D) n 25
83) B) Q(x) = 1 + 3x + 9 x2 2
A) Q(x) = 3x C) Q(x) = 3x + 9 x2
D) Q(x) = 3x - 9 x2
2
2
13
Provide an appropriate response.
3
5
3
5
84) If tan-1 x is replaced by x - x + x and |x| < 0.5, what estimate can be made of the error? A) |E| < 0.0011161 C) |E| < 0.0013021
84)
B) |E| < 0.0022321 D) |E| < 0.0026042
Solve the problem. 85) Find the value of b for which 1 1 - eb + e2b - e3b + ... = . 10
85)
B) ln 10
A) ln 10
C) ln 9
11
D) ln 11
Find the first four terms of the binomial series for the given function. 86) (1 + 9x)1/3
86)
A) 1 + 3x - 2.849934139e+15 x2 + 1.335906628e+15 x3 3.166593488e+14
2.968681395e+13
B) 1 + 3x + 2.849934139e+15 x2 + 6.412351813e+15 x3 3.166593488e+14
2.374945116e+14
C) 1 - 3x + 2.849934139e+15 x2 - 1.335906628e+15 x3 1.055531163e+14
2.968681395e+13
D) 1 + 3x - 2.849934139e+15 x2 + 6.412351813e+15 x3 3.166593488e+14
2.374945116e+14
A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. n+1 87) a1 = 5, a n+1 = (-1) 87) an
A) 5, 1 , -5, - 1 , 5
B) -5, 1 , 5, - 1 , -5
C) 5, - 1 , -5, 1 , 5
D) 5, - 1 , 5, - 1 , 5
5
5
5
5
5
5
5
5
Find the polynomial of lowest degree that will approximate F(x) throughout the given interval with an error of magnitude less than than 10-3 . x
88) F(x) = A)
0 2 x 2
-
ln(1 + t2 ) dt , [0, 0.6] t x 4 x6 + 4 6
88) 2
4
6
2
16
128
B) x - x + x
2
4
6
2
8
32
C) x - x + x
Solve the problem. 89) If an is a convergent series of nonnegative terms, what can be said about
A) Always converges B) Always diverges C) May converge or diverge 14
2
4
6
2
8
18
D) x - x + x
na n ?
89)
Provide an appropriate response.
90) Obtain the first nonzero term of the Maclaurin series for 2 sin x - sin 2x - x2 tan x. A)
7x5
B) -
18
7x5
C) -
12
7x5
90) 7x5
D)
18
12
Find the Taylor polynomial of order 3 generated by f at a. 91) f(x) = 1 , a = 0 5-x 2
3
4
25
125
625
2
3
125
625
A) P3(x) = x + x + x + x 5
C) P3(x) = 1 - x + x - x 5
25
91) 2
3
4
25
125
625
2
3
125
625
B) P3(x) = x - x + x - x 5
D) P3(x) = 1 + x + x + x 5
25
Solve the problem.
92) If an is a convergent series of nonnegative terms, what can be said about
k
a n , where k is a
92)
positive integer? A) Always converges
B) Always diverges C) May converge or diverge Use the Ratio Test to determine if the series converges or diverges.
93) n=1
10n n!
93)
A) Diverges
B) Converges
Assume that the sequence converges and find its limit. 94) a1 = 3, a n + 1 = 3an
94)
B) 1
A) -3
C) 9
D) 3
Find the limit of the sequence if it converges; otherwise indicate divergence. 95) an = n - n2 - 2n
A) 2
B) 0
95)
C) 1
D) Diverges
Find the Maclaurin series for the given function. 96) e10x
96)
(-1)n 10n xn
A) n=0
C) n=1
(-1)n 10n xn
B)
n!
n=1
10n xn n!
D) n=0
15
n!
10n xn n!
Determine if the sequence is monotonic and if it is bounded. 97) an = 2n + 1 n+1
A) not monotonic; bounded C) decreasing; bounded
97) B) nondecreasing; bounded D) decreasing; unbounded
Find the values of x for which the geometric series converges. (3x + 1)n
98)
98)
n=0
A) 0 < x < 2
B) - 1 < x < 1
3
3
C) - 2 < x < 0
3
D) 0 < x < 1
3
3
Find the limit of the sequence if it converges; otherwise indicate divergence. 6 99) an = (ln n) n
B) e6
A) ln 6
99)
C) 0
D) Diverges
Provide an appropriate response. 100) Which of the following sequences do not meet the conditions of the Integral Test? I. an = n(sin n + 1) II. a n = III. a n =
1 np + p 1 n n
A) I only
B) I, II, and III
C) I and III
D) II and III
Find the first four terms of the binomial series for the given function. 101) (1 - 7x2 )-1/2 A) 1 + 7 x2 + 147 x4 + 3.394192395e+15 x6 B) 1 + 7 x2 + 147 x4 + 3.771324883e+15 x6 2 4 8.796093022e+12 2 4 1.759218604e+13
C) 1 + 7 x2 + 147 x4 + 3.771324883e+15 x6 2
100)
8
D) 1 + 7 x2 + 147 x4 + 3.394192395e+15 x6
3.518437209e+13
Provide an appropriate response. 102) If an converges, what can be said about
101)
2
a n an+1 ?
A) The series may converge or diverge. B) The series always converges. C) The series always diverges.
16
8
1.759218604e+13
102)
Find the quadratic approximation of f at x = 0. 103) f(x) = ln(1 + sin 9x) A) Q(x) = 1 - 9x + 81 x2 2
103) B) Q(x) = 1 + 9x + 81 x2 2
C) Q(x) = 9x + 81 x2
D) Q(x) = 9x - 81 x2
2
Provide an appropriate response.
2
2
104) If cos x is replaced by 1 - x and |x| < 0.4, what estimate can be made of the error? 2
A) |E| < 0.0042667 C) |E| < 0.0010667
B) |E| < 0.0026667 D) |E| < 0.010667
Determine whether the nonincreasing sequence converges or diverges. n 105) an = 1 + n · 4 4n
A) Diverges
105)
B) Converges
Find the Maclaurin series for the given function. 106) sin 5x x
A) n=1
C) n=0
104)
106)
(-1)2n+1 52n+1 x2n (2n + 1)!
B)
(-1)2n+1 52n+1 x2n (2n + 1)!
D)
n=0
n=1
(-1)n 5 2n+1 x2n (2n + 1)! (-1)n 5 2n+1 x2n (2n + 1)!
Use the Ratio Test to determine if the series converges or diverges. n! e-10n
107)
107)
n=1
A) Diverges
B) Converges
Solve the problem. 108) Find the sum of the infinite series 1 + 2r + 8r2 + 2r3 + r4 + 2r5 + 8r6 + 2a 7 + r8 ...
108)
for those values of r for which it converges. 8r3 + 2r2 + 8r + 1
A)
3
1 - r4
3
2
B) 8r + 2r + 8r + 2 1 - r4
2
3
C) 2r + 8r + 2r + 1
2
D) 2r + 8r + 2r + 8
1 - r4
1 - r4
17
109) If an is a convergent series of nonnegative terms, what can be said about nk a n , where k is a
109)
positive integer? A) May converge or diverge
B) Always converges C) Always diverges Find the values of x for which the geometric series converges. -5 n xn
110)
110)
n=0
B) |x| < 1
A) |x| < 1
C) |x| < 10
5
D) |x| < 5
Find the sum of the series. (-1)n-1
111) n=1
9 4n
111) B) 36
A) 3
C) 9
5
D) 12
5
Use power series operations to find the Taylor series at x = 0 for the given function. 112) x8 ex xn+8
A)
B)
n!
n=0
xn+8
n=0
Provide an appropriate response.
C)
(n + 8)!
3
5
45
945
112)
8nxn
n=0
8 nxn
D)
(n + 8)!
n=0
n!
113) Use the fact that cot x = 1 - x + x + 2x + ... for |x| < to find the first four terms of the series x
for ln(sin x).
2
4
6
180
2
4
15
189
A) - ln|x| + x + x
3
+
x6 + ... 2835
2 4 B) - 1 + 1 + x + 2x + ... x2
3
3
15
4
6
180
Find the Taylor polynomial of order 3 generated by f at a. 114) f(x) = e-8x, a = 0
A) P3(x) = 1 - 8x +
64x2 2
-
B) P3(x) = 1 - 8x +
3
2
2
+
x6 + ... 2835
64x2
512x3
114)
512x3
C) P3(x) = 1 - 64x + 4096x - 262,144x
189
2
D) ln|x| - x + x
C) 1 + 1 + x + 2x + ... x2
113)
3
2
-
18
2
D) P3(x) = 1 - 8x + 64x - 512x
12
2
18
6
3
Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. 9 115) 9 + 9 + 9 + ... + + ... 2·3 3·4 4·5 (n + 1)(n + 2)
A)
9n 9 ; 2(n + 1) 2
B) 9n ; 9
C) 9n ; 9
n+1
D)
n+2
9n 9 ; 2(n + 2) 2
Write the first four elements of the sequence. n 116) 1 + 1 n
A) 0, 2, 9 , 64 4 27
115)
116)
B) 0, 1, 9 , 64
C) 1, 9 , 64 , 625
4 27
4 27
64
D) 2, 9 , 64 , 625 4 27 256
A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. 117) a1 = 6, a n+1 = -a n 117)
A) 6, 0, -6, -12, -18 C) -6, 6, -6, 6, -6
B) 6, -36, 216, -1296, 7776 D) 6, -6, 6, -6, 6
Use the integral test to determine whether the series converges. 2 cos-1 (1/x)
118)
118)
n=1
A) converges
B) diverges
For what values of x does the series converge absolutely?
119)
xn
119)
5 n=2 n(ln n)
A) -1 < x < 1
B) x = 0
C) -1 x 1
D) 0 x <
Determine convergence or divergence of the series.
120)
3
120)
3/2 n=2 n[ln (7n)]
A) Diverges
B) Converges
Find the sum of the series.
121) n=1
1 1 n 2 4n
A) 2
121) B) 2
C) - 2
3
3
19
D) - 2
Solve the problem. 122) If p > 0 and q > 0, what can be said about the convergence of
n=2
122)
1 ? p n (ln n)q
A) Always converges B) Always diverges C) May converge or diverge Find a formula for the nth term of the sequence. 123) 3, 4, 5, 6, 7 (integers beginning with 3) A) an = n - 3 B) an = n + 2
123) C) an = n + 4
D) an = n + 3
Find the first four terms of the binomial series for the given function. 124) (1 + 9x2 )3 A) 1 + 27x2 + 81x4 + 243x6 B) 1 + 27x2 + 243x4 + 729x6
C) 1 + 27x2 + 405x4 + 5103x6
124)
D) 1 + 27x2 + 27x4 + x6
Find the series' radius of convergence. (x - 2)n
125)
125)
n=0
A) 2
B) 0
C) , for all x
D) 1
Find the polynomial of lowest degree that will approximate F(x) throughout the given interval with an error of magnitude less than than 10-3 . x
126) F(x) = 0
tan-1 t dt , [0, 0.5] t 3
5
3
5
A) x - x + x
126) 3
5
3
60
B) x - x + x
3
5
9
25
C) x - x + x
3
5
6
120
D) x - x + x
Determine convergence or divergence of the series.
127) n=1
3n 2 + 2 n 2 + 5n + 5
127)
A) Converges
B) Diverges
Use the Root Test to determine if the series converges or diverges.
128) n=1
n 9n ln n + 5n + 4
128)
A) Converges
B) Diverges
20
Determine if the series converges or diverges. If the series converges, find its sum.
129) n=1
1 1 3/2 n (n + 1)3/2
129)
A) diverges
B) converges; 1
C) converges; 1
D) converges; 1
3 3
6 6
A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. 130) a1 = 1, a 2 = 2, a n+2 = a n+1 - a n 130)
A) 1, 2, 1, 0, -1
B) 1, -2, 3, -4, 5
C) 1, 2, 1, -1, -2
D) 1, -1, 2, -3, 5
Find the first four terms of the binomial series for the given function. -2 131) 1 + x 9
131)
A) 1 - 2 x + 1.407374884e+14 x2 - 5.277655813e+13 x3 9
2.849934139e+15
6.412351813e+15
B) 1 - 2 x + 1.055531163e+14 x2 - 3.518437209e+13 x3 9
2.849934139e+15
6.412351813e+15
C) 1 - 2 x + 1.055531163e+14 x2 - 1.055531163e+14 x3 9
2.849934139e+15
6.412351813e+15
D) 1 - 2 x + 1.407374884e+14 x2 - 7.036874418e+13 x3 9
2.849934139e+15
6.412351813e+15
Find the series' radius of convergence.
132) n=0
(x - 4)n 2 + 9n
A) 2
132) B) 1
C) , for all x
D) 0
Use the Root Test to determine if the series converges or diverges.
133)
(n!)n n9 n=1 n
133)
A) Diverges
B) Converges
Determine if the series converges absolutely, converges, or diverges.
134) n=1
(-9)n 3n 4 + 8n
134)
A) converges conditionally B) Converges absolutely C) Diverges
21
For what values of x does the series converge absolutely? xn n3 + 8
135) n=1
135)
A) -1 < x 1
B) -1 x 1
C) -1 x < 1
D) -1 < x < 1
Find the polynomial of lowest degree that will approximate F(x) throughout the given interval with an error of magnitude less than than 10-3 . x
136) F(x) = 0
e-t dt , [0, 0.5] 2
3
4
2
6
10
x2
x3
x4
A) x - x + x - x C) x +
2
+
6
+
136) 2
3
4
2
6
24
x2
x3
B) x - x + x - x D) x -
10
2
+
12
-
x4 240
Find the sum of the geometric series for those x for which the series converges. sin 6x n
137) n=0
A)
137)
1 1 - sin 6x
B)
1 1 + sin 6x
C)
6 1 - sin 6x
D)
6 1 + sin 6x
Determine if the series converges or diverges. If the series converges, find its sum.
138) n=1
9 n(n + 1)(n + 2)
138)
A) converges; 9
C) converges; 27
B) diverges
4
4
D) converges; 6
Solve the problem. 139) If p > 0 and q > 1, what can be said about the convergence of
n=2
139)
1 ? p n (ln n)q
A) Always converges B) May converge or diverge C) Always diverges Determine if the series converges or diverges. If the series converges, find its sum.
140) n=1
1 n+1
1 n+2
A) converges; 1
3
140) C) converges; 1
B) diverges
2
22
D) converges; 1
6
For what values of x does the series converge absolutely? (x - 10)n
141)
141)
n=0
A) 9 x 11
B) 9 < x < 11
C) 9 x < 11
D) 9 < x 11
Find the Taylor polynomial of order 3 generated by f at a. 142) f(x) = x2 + x + 1, a = 9
142)
A) P3(x) = 1 + 3(x - 9) + 3(x - 9)2 + (x - 9)3
B) P3(x) = 91 + 19(x - 9) + 19(x - 9)2 + 91(x - 9)3 C) P3(x) = 10 + 19(x - 9) + 28(x - 9)2 D) P3(x) = 91 + 19(x - 9) + (x - 9)2 Find a formula for the nth term of the sequence. 143) 1, - 1 , 1 , - 1 , 1 (reciprocals of squares with alternating signs) 4 9 16 25
A) an = (-1)
n+1
n2
B) an = (-1)
2n+1
C) an = (-1)
n2
143) n
n2
Provide an appropriate response. 144) Obtain the first nonzero term of the Maclaurin series for sin x - tan x. 3 3 3 A) - x B) x C) - x 2 3 3
D) an = (-1)
n2
n2
144) D)
x3 2
Find the polynomial of lowest degree that will approximate F(x) throughout the given interval with an error of magnitude less than than 10-3 . x
145) F(x) = A)
0 x3 3
-
tan-1 t2 dt , [0, 0.75] x7 x11 + 21 33
145) 3
7
11
3
21
231
B) x - x + x
3
7
3
7
C) x - x
3
7
3
21
D) x - x
Use the limit comparison test to determine if the series converges or diverges.
146) n=1
1 4 + 9n ln n
146)
A) converges
B) Diverges
23
Determine if the series converges or diverges. If the series converges, find its sum. 1 n+1
147) n=1
1 n+3
147)
A) converges; 1 + 1
B) converges; 1 + 1
C) diverges
D) converges; 1 + 1
3
2
3 2
6 3
Use power series operations to find the Taylor series at x = 0 for the given function. 148) x4 cos x
A) n=0
C) n=0
(-1)n 2nx2n+4 (2n)!
B)
(-1)n 2n+4 x2n+4 (2n)!
D)
n=0
n=0
148)
(-1)n 2n+4 x2n+4 (2n + 8)! (-1)n 2nx2n+4 (2n + 8)!
For what values of x does the series converge conditionally?
149) n=1
xn n3 + 2
A) x = 1
149) B) x = -1
C) x = ±1
D) None
Determine if the sequence is monotonic and if it is bounded. n 150) an = 2 (2n)!
A) nondecreasing; bounded C) decreasing; bounded
150) B) not monotonic; bounded D) increasing; unbounded
Find the Maclaurin series for the given function. 151) 1 x - 10
A) n=0
xn 10n
B) n=0
151)
xn 10n
C) n=0
xn 10n+1
D) n=0
xn 10n+1
Find the series' radius of convergence.
152)
x8n
8 n = 2 (ln n) A) , for all x
152) B) 0
C) 2
Find the quadratic approximation of f at x = 0. 153) f(x) = ln(cos 10x) A) Q(x) = 1 + 50x2
D) 1
153) B) Q(x) = - 50x2 D) Q(x) = 1 - 50x2
C) Q(x) = 50x2 24
Use the Comparison Test to determine if the series converges or diverges.
154) n=1
sin n cos n 5n
154)
A) diverges
B) converges
Find the Taylor polynomial of order 3 generated by f at a. 155) f(x) = ln(x + 1), a = 4 2 3 A) P3(x) = ln 3 - x - 4 + (x - 4) - (x - 4) 3 18 81 2
155) 2
5
3
18
50
375
2
C) P3(x) = ln 3 + x - 4 + (x - 4) + (x - 4) 3
3
B) P3(x) = ln 5 + x - 4 - (x - 4) + (x - 4)
3
D) P3(x) = ln 5 - x - 4 + (x - 4) - (x - 4)
81
5
Provide an appropriate response.
3
5
6
120
50
375
156) For approximately what values of x can sin x be replaced by x - x + x with an error of magnitude no greater than 5 × 10-4 ? A) |x| < 0.86946 B) |x| < 1.16654
C) |x| < 0.88701
156)
D) |x| < 1.14115
Determine convergence or divergence of the series.
157) n=1
4 2/3 n
157)
A) Diverges
B) Converges
Find the first four terms of the binomial series for the given function. 3 158) 1 + x 6
158)
A) 1 - 1 x + 4.222124651e+14 x2 - 1.759218604e+13 x3 2
5.066549581e+15
3.799912186e+15
B) 1 + 1 x + 4.222124651e+14 x2 + 1.759218604e+13 x3 2
5.066549581e+15
3.799912186e+15
C) 1 - 1 x + 1.055531163e+14 x2 - 2.199023256e+12 x3 4
5.066549581e+15
3.799912186e+15
D) 1 + 1 x + 1.055531163e+14 x2 + 2.199023256e+12 x3 4
5.066549581e+15
3.799912186e+15
Solve the problem. 159) If p > 1 and q > 1, what can be said about the convergence of 1
p q n=2 (ln n) (ln ln n)
?
A) May converge or diverge B) Always converges C) Always diverges
25
159)
Find the Taylor polynomial of order 3 generated by f at a. 160) f(x) = 1 , a = 1 10 - x 2
A) P3(x) = 1 - x + 1 + (x + 1) - (x + 1) 11
121
1331
2
121
3
2
B) P3(x) = 1 - x - 1 + (x - 1) - (x - 1)
14,641
C) P3(x) = 1 - x - 1 + (x - 1) - (x - 1) 11
160)
1331
9
3
81
729
6561
2
D) P3(x) = 1 - x + 1 + (x + 1) - (x + 1)
14,641
9
81
3
729
3
6561
Find the interval of convergence of the series. (x - 3)n 3n + 9
161) n=0
161)
A) 2 < x < 4
B) 0 x < 6
C) x < 4
D) 2 x < 4
Determine if the geometric series converges or diverges. If it converges, find its sum. 2 3 4 162) 1 + 2 + 2 + 2 + 2 + . . . 5 5 5 5
B) converges, 7
A) diverges
162)
C) converges, 5
3
D) converges, 2
3
For what values of x does the series converge conditionally? (-1)n+1 (x + 2)n n8 n
163) n=1
163)
A) x = -10, x = 6
B) x = 6
C) x = -10
D) None
Find the first four terms of the binomial series for the given function. -2/3 164) 1 - x 4
164)
A) 1 - 1 x + 5 x2 - 5 x3
B) 1 + 1 x + 5 x2 + 5 x3
C) 1 + 1 x + 5 x2 + 5 x3
D) 1 - 1 x + 5 x2 - 5 x3
6 6
72
144
216
6
648
6
72
144
216
648
Find the sum of the geometric series for those x for which the series converges. (x + 5)n
165) n=0
A)
1 6-x
165) B)
1 -4 + x
C)
26
1 6+x
D)
1 -4 - x
Find the Taylor series generated by f at x = a. 166) f(x) = 1 , a = 2 4-x
166) B)
n=0
(x - 2)n 2n
D)
n=0
(x - 2)n+1 2n
A) C)
n=0
(x - 2)n 2 n+1
n=0
(x - 2)n+1 2 n+1
Answer the question.
167) Estimate the error if sin x3/2 is approximated by x3/2 - x A) 2
B) 1
17·5!
9/2 3!
C) 2
5!
1
in the integral of
15·4!
sin x3/2 dx.
167)
0
D) 6
17·5!
Determine convergence or divergence of the alternating series. (-1)n ln
168) n=1
3n + 5 3n + 4
168)
A) Converges
B) Diverges
Find the sum of the geometric series for those x for which the series converges.
169) n=0
A)
x-3 n 2
169)
3 5+x
B)
2 5-x
C)
2 5+x
D)
3 5-x
Use series to evaluate the limit. -1 170) lim tan 9x - tan 9x x 0 x3
170)
A) - 6.412351813e+15
B) - 2.849934139e+15
C) - 6.412351813e+15
D) - 8.549802418e+15
2.638827907e+13
1.055531163e+14
1.319413953e+13
7.036874418e+13
Find the infinite sum accurate to three decimal places. (-1)n+1
171) n=1
A) -1.036
1 n5
171) C) 1.036
B) -0.972
27
D) 0.972
(-1)n+1
172) n=1
1 (2n)!
A) -0.320
172) B) 0.460
C) 0.859
D) -0.859
Use the Root Test to determine if the series converges or diverges. n 8n 1/n - 1 173) 1/n - 1 n=1 6n A) Converges B) Diverges
173)
Find the limit of the sequence if it converges; otherwise indicate divergence. 4 174) an = 3 + 4n + 9n 5n 4 + 9n 3 + 2
A) 9
B) 9
2
C) 3
5
174) D) Diverges
5
Provide an appropriate response.
175) For an alternating series
(-1)n+1 un where lim un 0, what can be said about the n n=1
175)
convergence or divergence of the series? A) The series always converges.
B) The series may or may not converge. C) The series always diverges. Use the integral test to determine whether the series converges.
176)
3 x-1 e n=1
176)
A) converges
B) diverges
Find the sum of the series as a function of x.
177) n=0
x-5 n 5
A) -
5 x - 10
177) B)
5 x + 10
C) -
Find the Taylor series generated by f at x = a. 178) f(x) = x2 + 5x + 3, a = 3
5 x + 10
D)
5 x - 10
178)
A) (x - 3)2 + 11(x - 3) + 21 C) (x + 3)2 + 11(x + 3) + 21
B) (x + 3)2 + 11(x + 3) + 27 D) (x - 3)2 + 11(x - 3) + 27
28
Use the integral test to determine whether the series converges.
179) n=1
cos 1/n n2
179)
A) diverges
B) converges
Find the sum of the series. (-1)n
180) n=0
5 7n
A) 35 6
180) B) 5
C) 35
6
8
D) 5 8
Use the limit comparison test to determine if the series converges or diverges.
181) n=1
(ln n)3 3 n(7 + 8 n)
181)
A) Converges
B) Diverges
Provide an appropriate response. 182) Obtain the first nonzero term of the Maclaurin series for sin(tan x) - tan(sin x). 7 7 7 7 A) - x B) x C) - x D) x 30 60 60 30
182)
A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. 183) a1 = 1, a n+1 = 5a n 183)
A) 5, 25, 125, 625, 3125 C) 1, 5, 25, 125, 625
B) 5, 6, 7, 8, 9 D) 1, 6, 11, 16, 21
Find the limit of the sequence if it converges; otherwise indicate divergence. n 184) an = 3 n · n
A) 0
B) 1
C) 3
184) D) Diverges
Solve the problem. 185) A ball is dropped from a height of 50 meters. If each bounce brings it to 90% of its previous height, how far will the ball travel before it stops? A) 950 m B) 200 m C) 500 m D) 250 m Find the limit of the sequence if it converges; otherwise indicate divergence. 186) an = 7 + 3n 8 + 7n
A) 25
B) 7
C) 3
8
7
29
185)
186) D) Diverges
Find the Taylor series generated by f at x = a. 187) f(x) = x3 - 10x2 - 2x - 3, a = -3
187)
A) (x + 3)3 + 19(x + 3)2 + 89(x + 3) + 114 C) (x + 3)3 + 19(x + 3)2 + 85(x + 3) + 114
B) (x + 3)3 - 19(x + 3)2 + 85(x + 3) - 114 D) (x + 3)3 - 19(x + 3)2 + 89(x + 3) - 114
Find the smallest value of N that will make the inequality hold for all n > N. 2 188) n < 10-2 2n
A) 15
B) 16
C) 18
188) D) 17
Provide an appropriate response. 189) Let sk denote the kth partial sum of the alternating harmonic series. If e(11) denotes the absolute value of the error in approximating ln 2 by
s11 + s12 , compute floor 2
denotes the integer floor (or greatest integer) function. A) 22 B) 21 C) 23
189)
1 , where floor(x) e(11)
D) 24
Use series to evaluate the limit. 81 2 cos 9x - 1 x 2 190) lim x 0 x2
190) B) 9
A) -81
C) 81
D) -9
Find the Taylor polynomial of order 3 generated by f at a. 191) f(x) = ln x, a = 10 2 3 A) P3(x) = ln 10 - x - 10 + (x - 10) - (x - 10) 10 100 1000 10,000 2
191)
3
B) P3(x) = ln 10 - x - 10 + (x - 10) - (x - 10) 10
200
3000
2
3
C) P3(x) = ln 10 + x - 10 - (x - 10) + (x - 10) 10
200
3000
2
3
D) P3(x) = ln 10 + x - 10 + (x - 10) + (x - 10) 10
100
1000
10,000
Use series to evaluate the limit. 192) lim x2 e3/x2 - 1 x
A) 3
192) B) 1
C) -3
30
D) -1
Find the series' radius of convergence.
193) n=1
(x - 7)n ln (n + 4)
193)
A) 1
B) , for all x
C) 0
D) 2
Find the interval of convergence of the series.
194) n=0
(x - 3)n 2 + 5n
194)
A) - 1 < x < 13 2
B) -2 x 8
2
Solve the problem.
C) 2 x < 4
D) -2 < x < 8
2
195) For what values of x can we replace cos x by 1 - x with an error of magnitude no greater than 2
8 x 10-3 ? A) -0.577 x 0.577
195)
B) -0.423 x 0.423 D) -0.662 x 0.662
C) -0.363 x 0.363
196) A pendulum is released and swings until it stops. If it passes through an arc of 35 inches the first
196)
1 pass, and if on each successive pass it travels the distance of the preceding pass, how far will it 5 travel before stopping? A) 8.75 in.
B) 52.5 in.
C) 78.75 in.
D) 43.75 in.
Determine convergence or divergence of the series. n 6 e-n
197)
197)
n=1
A) Diverges
B) Converges
Use series to evaluate the limit. -3x - 1 + 3x 198) lim e x 0 x2
198) B) 9
A) 0
D) 9
C) 9
4
2
Write the first four elements of the sequence. 1 n
199)
199)
3
A) 1 , - 1 , 1 , - 1
B) 1, 1 , 1 , 1
C) 1 , 1 , 1 , 1
D) 1 , 1 , 1 , 1
3
9 27
81
3 9 27
3 9 27 81
3 3 3 3
31
Use substitution to find the Taylor series at x = 0 of the given function. 3 200) ln 1 + x 1 - x3
A) n=1
(-1)n-1 x3n n
B)
x6n+3 2n + 1
D) 2
C) 2 n=0
n=0
200) x6n+3 6n + 3
n=0
x6n+3 6n + 3
Find the values of x for which the geometric series converges. (x + 5)n
201)
201)
n=0
A) -5 < x < 5
B) -6 < x < -4
C) -6 < x < -5
D) -5 < x < -4
Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. 202) 4 + 4 + 4 + ... + 4 + ... 1·2 2·3 3·4 n(n + 1)
A) 4(n + 1) ; 4 n+2
B) 4n ; 4
C) 4(n + 1) ; 4
n+1
n
202)
D) 4(n + 2) ; 4 n+1
Provide an appropriate response.
203) If an converges, what can be said about a nk , where k is an integer greater than 1? A) The series may converge or diverge. B) The series always diverges. C) The series always converges.
203)
Determine convergence or divergence of the series.
204)
6
204)
7/4 n=1 (2n + 6)
A) Converges
B) Diverges
Solve the problem. 205) A company makes a very durable product. It sells 20,000 in the first year, but will have diminishing sales due to the product's durability, so that each year it can expect to sell only seventy-five percent of the quantity it will have sold the year before. How many products can the company expect to eventually sell? A) 40,000 products B) 35,000 products
C) 26,667 products
D) 80,000 products
32
205)
Find the sum of the series.
206) n=8
A)
1 3n
206)
1 13,122
B) 1,202,315,964,973,057 1,099,511,627,776
C) 1
D) 7,213,895,789,838,341
4374
2,199,023,255,552
Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.
207) n=1
(-1)n+1 (0.6)n n
207)
A) Error < 7.78 × 10-2 C) Error < 1.56 × 10-2
B) Error < 7.78 × 10-3 D) Error < 3.24 × 10-2
Find the Maclaurin series for the given function. 208) cos 6x
A) n=0
208)
(-1)n 6 2n x2n (2n)!
B) n=0
(-1)2n 6 2n x2n
C) n=0
D)
(2n)!
n=0
(-1)2n 6 2n x2n n! (-1)n 6 2n x2n n!
Determine if the sequence is monotonic and if it is bounded. n 209) an = 4 7 n n!
A) increasing; unbounded C) not monotonic; unbounded
209) B) nondecreasing; bounded D) decreasing; bounded
Provide an appropriate response.
210) If the series
(-1)n (x - 10)n is integrated term by term, for what value(s) of x (if any) does the
n=0 new series converge and for which the given series does not converge? A) x = 11 B) x = 9 C) none
33
D) x = 9, x = 11
210)
3
5
360
15120
211) Use the fact that csc x = 1 + x + 7x + 31x + ... (for |x| < ) to find the first four terms of the x
6
series for 2 csc(2x) for |x| <
2
2
4
6
3
90
2835
and thus for ln tan x.
A) ln|x| - x + 7x - 62x + ... 2
4
12
1440
C) ln|x| - x + 7x
-
211)
2
4
6
3
90
2835
B) ln|x| + x + 7x + 62x + ...
31x6 + ... 90720
2
4
12
1440
D) ln|x| + x + 7x
+
31x6 + ... 90720
Use the integral test to determine whether the series converges.
212) n=1
1 6n
212)
A) converges
B) diverges
Find the sum of the series as a function of x. n x2 - 7 213) 2 n=0 A) - 2 B) - 2 2 x +9 x2 - 9
213) C)
2 2 x -9
D)
2 2 x +9
Find the first four nonzero terms in the Maclaurin series for the function. 214) e5x 1 + x
214)
A) 1 - 11 x + 119 x2 - 1273 x3 + . . .
B) 1 + 3 x + 7 x2 + 17 x3 + . . .
C) 2 + 11 x + 99 x2 + 1003 x3 + . . .
D) 1 + 11 x + 119 x2 + 1273 x3 + . . .
2
8
2
48
8
2
48
8
2
Solve the problem. 215) If p > 1 and q > 0, what can be said about the convergence of 1
p q n=2 n (ln n)
48
8
48
215)
?
A) Always diverges B) Always converges C) May converge or diverge Use the Root Test to determine if the series converges or diverges. n
216) n=1
216)
n n 1/n + 6
A) Diverges
B) Converges
34
Find the Taylor series generated by f at x = a. 217) f(x) = 8x + 2, a = 5 A) 8(x - 5) + 38 B) 8(x + 5) + 38
217) C) 8(x + 5) + 42
D) 8(x - 5) + 42
Determine if the series converges or diverges; if the series converges, find its sum.
218)
1
218)
n n=0 ( 5)
A) Converges;
5-5 4
B) Converges; 5 - 5 4
C) Converges; 5 + 5
D) Diverges
4
Use series to evaluate the limit. 219) lim sin 6x - sin 12x x x 0
219)
A) 1
C) 6
B) -1
D) -6
Provide an appropriate response.
3
220) For approximately what values of x can sin x be replaced by x - x with an error of magnitude no 6
greater than 5 × 10-6 ? A) |x| < 0.16438
B) |x| < 0.10466
C) |x| < 0.22679
D) |x| < 0.15651
Write the first four elements of the sequence. 221) n + 1 3n - 1
A) 1, 3 , 1 , 5
A) n=0
(-1)n+1 x2n
C) 1 , 1 , 3 , 2
5 2
Provide an appropriate response.
222) Derive the series for
221)
B) -1, 1, 3 , 1
5 2 11
x
1 + x2
3 2 4 3
for x > 1 by first writing
B) n=0
(-1)n+1 x2n+1
x
1 + x2
=
D) 0, 1 , 1 , 3 3 2 4
1 1 . x 1 + 1/x2
C) n=0
(-1)n x2n
222) D) n=0
(-1)n x2n+1
Find the limit of the sequence if it converges; otherwise indicate divergence. 223) an = 1 + (-1)n + (-1)n(n+1)
A) 3
220)
B) 0
C) 1
35
223) D) Diverges
Find the values of x for which the geometric series converges. (x - 8)n
224)
224)
n=0
A) -8 < x < 8
B) 8 < x < 9
C) 7 < x < 8
Find a formula for the nth term of the sequence. 225) -3, -2, -1, 0, 1 (integers beginning with -3) A) an = n - 4 B) an = n + 3
D) 7 < x < 9
225) C) an = n - 2
Provide an appropriate response.
D) an = n - 3
2
4
2
24
226) For approximately what values of x can cos x be replaced by 1 - x + x with an error of magnitude no greater than 5 × 10-6 ? A) |x| < 0.32453 B) |x| < 0.39149
C) |x| < 0.29042
226)
D) |x| < 0.22679
Find the series' radius of convergence.
227) n=0
(x - 6)2n 9n
227)
A) 3
B) 6
C) , for all x
D) 1
Use the integral test to determine whether the series converges.
228) n=1
3n n2 + 1
228)
A) converges
B) diverges
Find the series' radius of convergence.
229) n=0
n(n + 1)(n + 2) (x - )n 3n
A) 3
229)
B) , for all x
C) 6
D) 1
Provide an appropriate response. 230) Let f(n) = n 2 + 3 - n2 - 3. What is f(n) approximately equal to as n gets large? Hint: Compute various examples on your calculator. A) 3 B) 3 n n
2
4
2
24
C) 3
n2
D) 3
2n
231) If cos x is replaced by 1 - x + x and |x| < 0.7, what estimate can be made of the error? A) |E| < 0.00023343 C) |E| < 0.00098041
B) |E| < 0.00016340 D) |E| < 0.0014006
36
230)
231)
Use power series operations to find the Taylor series at x = 0 for the given function. 232) x2 sin x (-1)n x2n+3
A) C) n=0
B)
(2n + 3)!
n=0
(2n + 1)!
n=0
(-1)n 2 n x2n (2n + 3)!
232)
(-1)n 2 n x2n
D) n=0
(-1)n x2n+3 (2n + 1)!
Solve the problem.
233) Use series to estimate the integral's value to within an error of magnitude less than 10-3. 0.8
233)
2 e-x dx
0
A) 0.3822
B) 1.352
C) 0.6145
D) 0.6577
Find the limit of the sequence if it converges; otherwise indicate divergence. n 234) an = 8n n+1
B) e8
A) 0
C) 8
234) D) Diverges
Solve the problem. 235) A company adopts an advertising campaign to weekly add to its customer base. It assumes that as an average fifty percent of its new customers, those added the previous week, will bring in one friend, but those who have been customers longer will not be very effective as recruiters and can be discounted. A media campaign brings in 10,000 customers initially. What is the expected total number of customers with whom the company can expect to have dealings? A) 15,000 customers B) 30,000 customers
C) 20,000 customers
235)
D) The sum diverges to infinity.
236) If an is a convergent series of nonnegative terms, what can be said about an k , where k is a
236)
positive integer? A) May converge or diverge
B) Always diverges C) Always converges For what values of x does the series converge conditionally? (2x)n
237)
237)
n=0
A) x = 1 2
C) x = - 1
B) x = 0
2
37
D) None
Find the values of x for which the geometric series converges.
238) n=0
(-1)n 1 5 (10 + sin x)n
238)
A) diverges for all x C) x x is not a multiple of 2
B) - < x < D) x x is not a multiple of
For what values of x does the series converge conditionally? (x + 5)n
239)
239)
n=0
A) x = -4
B) x = -5
C) x = -6
D) None
Provide an appropriate response.
240) Derive a series for ln(1 + x2 ) for x > 1 by first finding the series for Hint:
x
1 + x2
=
x
1 + x2
and then integrating.
240)
1 1 x 1 + 1/x2
A) ln x + n=1
(-1)n-1 nx2n
C) 2 ln x + n=1
B) ln x + n=1
(-1)n+1 nx2n
(-1)n+1 nx2n
D) 2 ln x + n=1
(-1)n nx2n
Determine convergence or divergence of the series. n 5 e-3n
241)
4
241)
n=1
A) Converges
B) Diverges
Find the smallest value of N that will make the inequality hold for all n > N. n 242) 2n - 1 < 10-2 A) 733 B) 737 C) 731
242) D) 750
Solve the problem.
243) Use series to estimate the integral's value to within an error of magnitude less than 10-3. 0.8
1 + x3 dx
0
A) 0.8541
B) 0.8480
C) 1.255
38
D) 0.4599
243)
Find the quadratic approximation of f at x = 0. 244) f(x) = sin ln(7x + 1) A) Q(x) = 1 - 7x + 49 x2 2
244) B) Q(x) = 7x + 49 x2 2
C) Q(x) = 1 + 7x + 49 x2
D) Q(x) = 7x - 49 x2
2
2
Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.
245) n=1
(-1)n+1 tn , -1 < t 1 n
245) 4
5
B) Error < t
A) Error < 0.20
C) Error < t
4
5
D) Error < t5
Use power series operations to find the Taylor series at x = 0 for the given function. x10
246)
246)
1 - 8x
(-1)n 8 n xn+11
A) n=0
n=0 (-1)n+1 8 n xn+10
C)
8 n xn+10
B)
8 n xn+11
D)
n=0
n=0
Determine convergence or divergence of the alternating series.
247) n=1
(-1)n 2n 3 + 2
247)
A) Converges
B) Diverges
Provide an appropriate response.
2
248) For approximately what values of x can cos x be replaced by 1 - x with an error of magnitude no 2
greater than 5 × 10-4 ? A) |x| < 0.22894
B) |x| < 0.33098 (-1)n-1
249) Find the sum of the series n=1
C) |x| < 0.14422
D) |x| < 0.23403
n 1 by expressing as a geometric series, n-1 1 +x 2
differentiating both sides of the resulting equation with respect to x, and replacing x by
A) 4 9
B) 1
C) 4
4
39
248)
D) 9 4
249) 1 . 2
By calculating an appropriate number of terms, determine if the series converges or diverges. If it converges, find the limit L and the smallest integer N such that a n - L < 0.01 for n N; otherwise indicate divergence.
250) an = n!
250)
nn
A) L = 0, N = 7
B) L = ln 4, N = 8
C) L = ln 2, N = 6
D) diverges
Find the quadratic approximation of f at x = 0. 251) f(x) = x 36 - x2
A) Q(x) = 36x2
251)
B) Q(x) = 1 + 6x
C) Q(x) = 6x
D) Q(x) = 1 - 6x
Determine if the series converges or diverges. If the series converges, find its sum.
252) n=1
4 (4n - 1)(4n + 3)
A) diverges
252) B) converges; 2
C) converges; 1
3
3
D) converges; 2 9
Find the smallest value of N that will make the inequality hold for all n > N. n 253) 10 < 10-3 n!
A) 21
B) 20
C) 30
253) D) 31
Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. 254) 15 + 25 + 35 + ... + 5(2n + 1) + ... 1 2 ·2 2 2 2 ·3 2 3 2 ·4 2 n 2 (n + 1)2
A) 5n(n + 2) ; 5
B) 5(n + 1)(n + 2) ; 5
(n + 1)2
C)
n2
5n 2 ;5 (n + 1)(n + 2)
D) 5n(n + 1) ; 5 (n + 2)2
Express the number as the ratio of two integers. 255) 0.737373 . . . A) 73 B) 730 99 999
255) C) 73 999
Find a formula for the nth term of the sequence. 256) 0, -1, 0, 1, 0, -1, 0, 1 (0, -1, 0, 1 repeated)
A) an = sin(n )
254)
D) 730 99
256)
B) an = cos n 2
C) an = sin n 2
40
D) an = cos(n )
By calculating an appropriate number of terms, determine if the series converges or diverges. If it converges, find the limit L and the smallest integer N such that a n - L < 0.01 for n N; otherwise indicate divergence. n
257) an = n + 1 A) L = ln 2, N = 394 C) L = 1, N = 394
257) B) L = 1, N = 652 D) diverges
Determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001. (-1)n
258) n=1
1 2n 2 + 1
258)
A) n 21
B) n 31
C) n 22
D) n 23
Solve the problem. 259) Mari drops a ball from a height of 17 meters and notices that on each bounce the ball returns to 7 about of its previous height. About how far will ball travel before it comes to rest? 8
A) 272 m
B) 36.4 m
C) 136 m
259)
D) 255 m
Use the Ratio Test to determine if the series converges or diverges.
260) n=1
(2n)! 8 n n!
260)
A) Diverges
B) Converges
Find the first four terms of the binomial series for the given function. 1/3 261) 1 - x 2
261)
A) 1 - 1 x - 1 x2 - 5 x3
B) 1 - 1 x + 1 x2 - 5 x3
C) 1 - 1 x + 1 x2 - 5 x3
D) 1 - 1 x - 1 x2 - 5 x3
6 6
36
648
36
6
648
6
36
216
12
216
A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. 262) a1 = 5, a n+1 = (-1)na n 262)
A) 5, -5, 5, -5, 5
B) 5, -5, -5, 5, 5
C) -5, 5, -5, 5, -5
D) 5, 5, -5, -5, 5
Determine if the series converges or diverges. If the series converges, find its sum. (tan-1 (n + 1) - tan-1 n)
263)
263)
n=1
A) converges; C) converges; -
B) diverges D) converges;
4
41
4
Use the Ratio Test to determine if the series converges or diverges.
264) n=1
7n! nn
264)
A) Converges
B) Diverges
Provide an appropriate response.
265) Use the fact that tan-1 x = n=1
A) n=1
C)
2
(-1)n-1 x2n-1 for |x| < 1 to find the series for cot-1 x. (2n - 1)
(-1)n-1 x2n-1 (2n - 1)
n=1
B)
(-1)n x2n-1 (2n - 1)
2
n=1
D) n=1
265)
(-1)n-1 x2n-1 (2n - 1)
(-1)n x2n-1 (2n - 1)
Use the Root Test to determine if the series converges or diverges.
266) n=1
1 1 n n3 n5
266)
A) Diverges
B) Converges
Find the infinite sum accurate to three decimal places. (-1)n+1
267) n=1
1 4n
A) -0.571
267) B) 0.200
C) 0.143
Find the quadratic approximation of f at x = 0. 268) f(x) = tan-1 10x
D) 0.800
268)
A) Q(x) 1 + 5x2 C) Q(x) = 1 + 10x2
B) Q(x) = 1 - 10x2 D) Q(x) = 10x
Find the Taylor polynomial of order 3 generated by f at a. 269) f(x) = x2 , a = 5
269)
A) P3(x) = 1 + 10(x - 5) + 15(x - 5)2 + 20(x - 5)3 B) P3(x) = 1 + 50(x - 5) + 375(x - 5)2 + 2500(x - 5)3 C) P3(x) = 25 + 10(x - 5) + (x - 5)2 D) P3(x) = 25 + 10(x - 5) + 15(x - 5)2 + 20(x - 5)3
For what values of x does the series converge conditionally?
270)
xn
270)
6 n=2 n(ln n)
A) x = 1
B) x = -1
C) x = ±1 42
D) None
Determine whether the nonincreasing sequence converges or diverges. 271) a1 = 1, a n+1 = 4a n - 1
A) Diverges
271)
B) Converges
Find the series' radius of convergence.
272) n=0
(x - 4)n 5n + 7
272)
A) 0
B) 1
C) , for all x
D) 2
Determine if the series converges absolutely, converges, or diverges. (cos n )
273) n=1
n! 5n
273)
A) converges conditionally B) converges absolutely C) diverges Find the first four terms of the binomial series for the given function. 274) (1 + 4x)-1/2
274)
A) 1 - 2x + 6x2 - 20x3
B) 1 - 2x + 6x2 - 10x3
C) 1 - 2x - 12x2 - 20x3
D) 1 - 2x + 6x2 - 30x3
Find a formula for the nth term of the sequence. 275) 0, 0, 2, 2, 0, 0, 2, 2 (alternating 0's and 2's in pairs) A) an = 1 + (-1)n(n-1) C) an = 1 + (-1)n(n+1)
275) B) an = 1 - (-1)n(n-1) D) a n = 1 + (-1)n(n + 1)/2
Assume that the sequence converges and find its limit. 276) a1 = 5, a n + 1 = 72 1 + an
A) 4
276)
B) 8
C) 6
D) 5
By calculating an appropriate number of terms, determine if the series converges or diverges. If it converges, find the limit L and the smallest integer N such that a n - L < 0.01 for n N; otherwise indicate divergence.
277) an = cos n A) L = 1, N = 394
277) B) L = 0, N = 394
C) L = 0, N = 237
D) diverges
Find the first four terms of the binomial series for the given function. 278) (1 - 3x)1/2
278)
A) 1 - 3 x - 9 x2 - 7.599824371e+15 x3 2 8 9.007199255e+15
B) 1 - 3 x - 9 x2 - 7.599824371e+15 x3 2 8 4.503599627e+15
C) 1 + 3 x + 9 x2 - 7.599824371e+15 x3
D) 1 - 3 x + 9 x2 - 7.599824371e+15 x3
2
8
9.007199255e+15
2
43
8
4.503599627e+15
Provide an appropriate response.
279) Suppose that a n > 0 and bn > 0 for all n N (N an integer). If lim n
about the convergence of
an bn
= , what can you conclude
279)
an ?
A)
a n converges if
bn converges
B)
a n diverges if
bn diverges
C)
a n diverges if
bn diverges, and
a n converges if
bn converges
D) The convergence of a n cannot be determined. Determine convergence or divergence of the alternating series. (-1)n+1
280) n=1
n+ n n2 + 1
280)
A) Converges
B) Diverges
Determine if the series converges or diverges; if the series converges, find its sum.
281) n=0
cos n 3n
281)
A) Converges; 3 2
B) Converges; 1
C) Converges; 3
2
4
D) Diverges
Determine if the sequence is monotonic and if it is bounded. 282) an = (n + 5)! (n + 1)!
A) not monotonic; unbounded C) nondecreasing; unbounded
282) B) nondecreasing; bounded D) increasing; bounded
Use the limit comparison test to determine if the series converges or diverges.
283) n=1
(ln n)2 n(7n - 10 n)
283)
A) Converges
B) Diverges
Determine if the series converges or diverges; if the series converges, find its sum. 7
284)
284)
n=0
A) Converges; 7 + 1 C) Converges; 7 - 1
B) Converges; 1 - 7 D) Diverges
44
Use the limit comparison test to determine if the series converges or diverges.
285) n=2
1 5 + 3n ln(ln n)
285)
A) Diverges
B) Converges
Determine if the series converges or diverges; if the series converges, find its sum. (n + 1) sin 2
286)
286)
7n
n=0
A) Converges; 7 6
B) Converges; 1
C) Converges; 7
6
D) Diverges
Find the sum of the geometric series for those x for which the series converges.
287) n=0
(-1)n 1 4 (7 + sin x)n
A) 4 + sin x
7(8 + sin x)
287) B) 4 + sin x
C) 7 + sin x
7(8 - sin x)
4(8 - sin x)
D) 7 + sin x
4(8 + sin x)
Find the sum of the series.
288) n=0
1 1 + n 5 3n
A) 19 4
288) C) 11
B) 4
4
D) 7 2
Use the Comparison Test to determine if the series converges or diverges.
289) n=1
4 + 5 cos n n3
289)
A) converges
B) diverges
Find the limit of the sequence if it converges; otherwise indicate divergence. 290) an = ln(n + 5) n 1/n
A) ln 5
B) 0
C) 1
Find the Taylor series generated by f at x = a. 291) f(x) = x4 - 3x2 - 9x + 6, a = -5
290) D) Diverges
291)
A) (x + 5)4 + 20(x + 5)3 + 147(x + 5)2 - 461(x + 5) - 739 B) (x + 5)4 - 20(x + 5)3 + 147(x + 5)2 - 479(x + 5) + 601 C) (x + 5)4 - 20(x + 5)3 + 147(x + 5)2 - 461(x + 5) - 739 D) (x + 5)4 + 20(x + 5)3 + 147(x + 5)2 - 479(x + 5) + 601
45
Find the values of x for which the geometric series converges. 8 n xn
292)
292)
n=0
A) |x| < 1
B) |x| < 16
8
C) |x| < 1
D) |x| < 8
For what values of x does the series converge absolutely? (-1)n (6x + 8)n
293)
293)
n=0
A) - 7 x < - 5 8
B) - 7 < x < - 5
8
8
C) - 3 < x < - 7
8
2
6
D) - 3 < x - 7 2
6
Find the limit of the sequence if it converges; otherwise indicate divergence. n 294) an = 2 + (-1) 2
B) 3
A) 1
C) 0
2
294) D) Diverges
Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. 5 295) 5 + 5 + 5 + ... + + ... 3·1 5·3 7·5 (2n + 1)(2n - 1)
A)
5n 5 ; 2(n + 1) 2
B) 10n ; 5
C) 5n ; 5
2n + 1
n+2
295)
D) 5n ; 5 2n + 1 2
Provide an appropriate response.
296) It can be shown that lim 1 = 0 for c > 0. Find the smallest value of N such that 1 < for all n
nc
nc
n > N if = 0.01 and c = 1.8. A) 13 B) 11
C) 12
296)
D) 10
Determine convergence or divergence of the series. sin
297) n=1
6n 2 + 3 n2 + 3
297)
A) Diverges
B) Converges
For what values of x does the series converge absolutely?
298) n=1
3 n xn n!
A) 0 x <
298) B) - < x <
C) - < x < 0
46
D) 0 < x <
Find the Maclaurin series for the given function. 299) cos (-10x)
A) n=0
299)
(-1)2n 102n x2n (2n)!
B) -
(-1)n 102n x2n (2n)!
D)
C) n=0
n=0
n=0
(-1)2n 102n x2n (2n)!
(-1)n 102n x2n (2n)!
Use the integral test to determine whether the series converges.
300) n=1
1 2n e -1
300)
A) diverges
B) converges
Find the Taylor series generated by f at x = a. 301) f(x) = ex, a = 3
A) n=0
C) n=0
301)
e3 (x - 3)n (n + 1)!
B)
e3 (x - 3)n+1 n!
D)
n=0
n=0
e3 (x - 3)n+1 (n + 1)! e3 (x - 3)n n!
Use power series operations to find the Taylor series at x = 0 for the given function. 302) x2 ln(1 + 3x)
A) n=1
(-1)n-1 3 n xn+2 n
B) n=1
(-1)n 3 n xn+2
C) n=0
D)
n+1
n=0
302)
(-1)n-1 3 n-1 xn+1 n (-1)n-1 3 n xn+2 n+1
Use the nth-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive.
303) n=1
n n+2
303)
A) converges, 1 2
B) converges, 2
C) diverges
D) inconclusive
Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.
304) n=1
(-1)n+1 (0.3)2n+1 2n + 1
304)
A) Error < 1.61 × 10-7 C) Error < 1.77 × 10-6
B) Error < 4.43 × 10-8 D) Error < 2.19 × 10-6
47
Find the polynomial of lowest degree that will approximate F(x) throughout the given interval with an error of magnitude less than than 10-3 . x
305) F(x) =
sin t dt , [0, 0.75] t
0
3
5
9
300
A) x + x + x
305) 3
5
18
600
B) x + x + x
C) x + x
3
D) x + x
18
3
9
Use the Comparison Test to determine if the series converges or diverges.
306) n=1
1 3 n-1 + 1
306)
A) converges
B) diverges
Find the interval of convergence of the series.
307) n=1
(x - 1)n (3n)!
307)
A) x 2
B) - < x <
C) 0 x 2
D) -5 x 7
Determine if the series converges or diverges; if the series converges, find its sum.
308)
n!
308)
n n=0 900
A) Converges; e
C) converges; 1
B) Converges; 1
e
D) Diverges
Find the first four terms of the binomial series for the given function. 309) (1 + 8x)1/2
309)
A) 1 + 4x - 4.503599627e+15 x2 + 9.007199255e+15 x3 5.629499534e+14
2.814749767e+14
B) 1 - 4x + 4.503599627e+15 x2 - 9.007199255e+15 x3 5.629499534e+14
2.814749767e+14
C) 1 - 4x + 4.503599627e+15 x2 - 9.007199255e+15 x3 5.629499534e+14
5.629499534e+14
D) 1 + 4x - 4.503599627e+15 x2 + 9.007199255e+15 x3 5.629499534e+14
5.629499534e+14
Find the polynomial of lowest degree that will approximate F(x) throughout the given interval with an error of magnitude less than than 10-3 . x
310) F(x) =
sin t3 dt , [0, 1]
0
A) x3 - x
7
6
310) 4
8
4
48
B) x - x
C) x3 - x
9
6
48
4
10
4
60
D) x - x
Determine convergence or divergence of the alternating series.
311) n=1
(-1)n n 4/3
311)
A) Diverges
B) Converges
Provide an appropriate response.
3
5
7
3
15
315
312) Use the fact that tan x = x + x + 2x + 17x + ... for |x| < series for ln(cos x). 2 4 6 8 A) x + x + x + 17x + ... 2 12 45 2520
2
4
6
8
2
12
45
2520
2
to find the first four terms of the
4
312)
6
B) 1 + x2 + 2x + 17x + ... 3
45
4
C) - x + x + x + 17x + ...
6
D) - 1 + x2 + 2x + 17x + ... 3
45
By calculating an appropriate number of terms, determine if the series converges or diverges. If it converges, find the limit L and the smallest integer N such that a n - L < 0.01 for n N; otherwise indicate divergence.
313) an = n cos 1
313)
n
A) L = 1, N = 28
B) L = 0, N = 37
C) L = 0, N = 28
D) diverges
Use the Ratio Test to determine if the series converges or diverges.
314) n=1
3(n!)2 (2n)!
314)
A) Diverges
B) Converges
Find a formula for the nth term of the sequence. 315) 4, -4, 4, -4, 4 (4's with alternating signs) A) an = 4(-1)2n-1 B) an = 4(-1)n
315) C) an = 4(-1)n+1
D) an = 4(-1)2n+1
Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. 316) 2 - 14 + 98 - 686 + ... + (-1)n-1 2 · 7 n-1 + ... n A) 2 1 - (-7) ; 1 1+7 4 n
n
C) 2 1 - (-7) ; 1 1-7
316)
n B) 2 1 - (-7) ; series diverges 7-1
D) 2 1 - (-7) ; series diverges
3
1+7
Find the series' radius of convergence.
317) n=1
(x - 6)n (2n)!
A) 2
317) B) , for all x
C) 0
49
D) 1
Solve the problem. 318) Mari drops a ball from a height of 20 meters and notices that on each bounce the ball returns to 7 about of its previous height. About how far will ball travel before it comes to rest? 8
A) 160 m
B) 320 m
C) 300 m
Write the first four elements of the sequence. 319) sin (n ) A) 0, 1, 0, -1 B) 1, 0, -1, 0
318)
D) 42.9 m
319) C) 0, 0, 0, 0
D) 1, 1, 1, 1
Provide an appropriate response. 320) Let sk denote the kth partial sum of the alternating harmonic series. Compute s11, s12, and
320)
s11 + s12 . Which of these is closest to the exact sum (ln 2) of the alternating harmonic series? 2
A) s12
B) s11
C)
s11 + s12 2
Solve the problem.
321) The polynomial 1 + 6x + 15x2 is used to approximate f(x) = (1 + x)6 on the interval -0.01 x 0.01.
321)
Use the Remainder Estimation Theorem to estimate the maximum absolute error. A) 2.061 x 10-4 B) 1.020 x 10-5 C) 2.061 x 10-5
Provide an appropriate response. 322) If an converges conditionally, what can be said about
min {a n , |a n |} ?
322)
A) The series may converge or diverge. B) The series always converges. C) The series always diverges. Determine if the series converges or diverges; if the series converges, find its sum.
323) n=1
4 n+1 9 n-1
323)
A) Converges; 36
B) Converges; 144
C) Converges; 324
D) Diverges
5
5
5
Find the sum of the series. 324) x6 - x8 + x9 - x5 + ...
A)
x5
1+x
324) B)
x6
C)
1+x
50
x6
1-x
D) 6x
1-x
Find the quadratic approximation of f at x = 0. 325) f(x) = tan 3x A) Q(x) = 1 + 3 x2 B) Q(x) = 3x 2
325) D) Q(x) = 1 - 3 x2
C) Q(x) = 1 + 3x2
2
Use the nth-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. ln
326) n=1
10 n
326)
A) inconclusive
B) diverges
C) converges; ln 10
D) converges; ln 1
10
Find the values of x for which the geometric series converges. sin 9x n
327)
327)
n=0
A) x x is not a multiple of C) - < x <
B) x x is not an odd multiple of /18 D) diverges for all x
Determine if the geometric series converges or diverges. If it converges, find its sum. 2 3 4 5 328) 1 + 1 + 1 + 1 + 1 + . . . 5 5 5 5 5
A) diverges
B) converges, 5
C) converges, 1
4
4
328) D) converges, 4 5
Determine convergence or divergence of the series.
329) n=1
5n + 5 6n 2 + 5n + 4
329)
A) Converges
B) Diverges
Express the number as the ratio of two integers. 330) 0.6969696 . . . A) 230 B) 23 33 11
330) C) 23 33
D) 46 33
For what values of x does the series converge conditionally?
331) n=1
4 n xn n!
331)
A) x = - 1 4
B) x = 1
C) x = 0
4
51
D) None
Solve the problem. 332) A child on a swing initially swings through an arc length of 12 meters. The child stops pushing and sits patiently waiting for the swing to stop moving. If friction slows the swing so the length of each arc is 80% of the length of the previous arc, how far will the child have traveled before the swing stops? A) 60 m B) The child will travel an infinite distance.
C) 24 m
332)
D) 25 m
Find the sum of the series. 3 5 7 333) 4 - 4 + 4 - 4 + ... 5 5 3 · 3! 5 5 · 5! 5 7 · 7!
A) ln 4
333)
B) cos 4
5
C) sin 5
5
4
D) sin 4 5
Use the limit comparison test to determine if the series converges or diverges.
334)
2 n=1 7 + 2n (ln n) A) Converges
334)
2
B) Diverges
Solve the problem. 335) Find the value of b for which 1 + eb + e2b + e3b + ... = 10.
A) ln 10 9
335)
B) ln 11
C) ln 10
10
11
D) ln 9
10
Find the sum of the series.
336) n=1
5 4n
A) 5 3
336) B) 20
C) 4
3
D) 1
For what values of x does the series converge absolutely?
337) n=1
(-1)n (x + 2)n n
A) -3 < x -1
337) B) -3 x < -1
C) -3 < x < -1
D) -3 x -1
Determine if the sequence is monotonic and if it is bounded. 338) n! 5n
A) not monotonic; unbounded C) nondecreasing; unbounded
338) B) nonincreasing; unbounded D) nondecreasing; bounded
52
Express the number as the ratio of two integers. 339) 0.88888 . . . A) 8 B) 8 99 9
339) C) 80
D) 80
99
999
Determine if the series converges or diverges; if the series converges, find its sum. e-8n
340)
340)
n=0
A) Converges;
8 B) Converges; e
1 -8 e -1
e8 - 1
-8 C) Converges; e
D) Diverges
e-8 - 1
A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. nan 341) a1 = 1, a n+1 = 341) n+2
A) 1, 1 , 2 , 3 , 4
B) 1, 1 , 2 , 6 , 24
C) 1, 1 , 2 , 6 , 24
D) 1, 1 , 4 , 4 , 24
3 12 60 360 3 4 5
3 12 60 360
6
3 3 15 105
Determine convergence or divergence of the alternating series.
342) n=1
(-5)n 7n 2 + 5n
342)
A) Converges
B) Diverges
Provide an appropriate response.
343) Find the sum of the series
(-1)n-1
n2 1 by expressing as a geometric series, 1+x 10n-1
343)
n=1 differentiating both sides of the resulting equation with respect to x, multiplying both sides by x, 1 differentiating again, and replacing x by . 10
A) 3.562417674e+15
B) 3.298534883e+14
C) 6.333186976e+16
D) 3.298534883e+13
5.853799906e+15
4.878166589e+14
8.514618045e+15
4.878166589e+14
Determine if the series
a n defined by the formula converges or diverges. n=1 344) a1 = 9, a n+1 = 2n + 5 a n 4n - 5
A) Diverges
B) Converges 53
344)
Solve the problem.
345) Use series to estimate the integral's value to within an error of magnitude less than 10-3. 0.6
345)
ln(x2 + 1)dx
0
A) 0.06445
B) 0.1285
C) 0.1845
D) 0.06533
By calculating an appropriate number of terms, determine if the series converges or diverges. If it converges, find the limit L and the smallest integer N such that a n - L < 0.01 for n N; otherwise indicate divergence.
346) an = 2
n
346)
n!
A) L = 0, N = 8 Provide an appropriate response.
B) L = ln 4, N = 8
3
5
6
120
C) L = ln 2, N = 6
D) diverges
347) If sin x is replaced by x - x + x and |x| < 0.3, what estimate can be made of the error? A) |E| < 0.000001012 C) |E| < 0.000000304
B) |E| < 0.000000145 D) |E| < 0.000000043
Find the Maclaurin series for the given function. 348) 1 3+x
348) B)
n=0
(-1)n xn+1 3n
D)
n=0
(-1)n xn 3n
A) C)
n=0
(-1)n xn 3 n+1
n=1
(-1)n xn 3 n+1
Solve the problem. 349) For what value of r does the infinite series 1 + 7r + r2 + 7r3 + r4 + 7r5 + r6 + ...
349)
converge?
A) |r| < 7 2
B) |r| < 7
C) |r| < 1
D) |r| < 1 7
Find the limit of the sequence if it converges; otherwise indicate divergence. n 350) an = 1 + -9 n
A) 1
347)
B) e-9
C) e
350) D) Diverges
Find the infinite sum accurate to three decimal places.
351) n=1
2(-1)n+1 3n
A) 0.500
351) B) 1.000
C) 3.000 54
D) 1.500
Find the series' radius of convergence.
352) n=1
(x - 2)n n54n
352)
A) 4
B) 8
C) 0
D) , for all x
Find the smallest value of N that will make the inequality hold for all n > N. 2 353) n < 10-3 2n
A) 17
B) 19
Provide an appropriate response.
353)
C) 20
D) 18
3
354) If sin x is replaced by x - x and |x| < 0.3, what estimate can be made of the error?
354)
6
A) |E| < 0.000067500 C) |E| < 0.00010125
B) |E| < 0.000020250 D) |E| < 0.00033750
Use the limit comparison test to determine if the series converges or diverges.
355)
9 + 10 ln n
355)
3 n=1 5 + 9n(ln n)
A) Converges
B) Diverges
Provide an appropriate response.
356) It can be shown that lim ln n = 0 for c > 0. Find the smallest value of N such that ln n < for n
nc
nc
all n > N if = 0.01 and c = 1.8. A) 35 B) 25
C) 31
356)
D) 28
A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. 357) a1 = 1, a 2 = 5, a n+2 = a n+1 + a n 357)
A) 1, 1, 2, 3, 5
B) 1, 5, 6, 11, 17
Find the quadratic approximation of f at x = 0. 358) f(x) = 25 - x2
A) Q(x) = 5 +
x2 5
B) Q(x) = 5 -
C) 1, 5, 6, 12, 18
D) 1, 5, 6, 7, 8
358) x2
C) Q(x) = 5 -
10
55
x2 5
D) Q(x) = 5 +
x2 10
Provide an appropriate response.
359) Use the fact that sin-1 x = x + n=1
1 · 3 · 5 · ... · (2n - 1) x2n+1 for |x| < 1 to find the series for 2 · 4 · 6 · ... · (2n)(2n + 1)
359)
cos-1 x.
A)
2
B)
2
C)
2
D)
2
-xn=1 -x+ n=1 -x+ n=1 n=1
1 · 3 · 5 · ... · (2n - 1) x2n+1 2 · 4 · 6 · ... · (2n)(2n + 1) (-1)n · 1 · 3 · 5 · ... · (2n - 1) x2n+1 2 · 4 · 6 · ... · (2n)(2n + 1) (-1)n+1 · 1 · 3 · 5 · ... · (2n - 1) x2n+1 2 · 4 · 6 · ... · (2n)(2n + 1)
1 · 3 · 5 · ... · (2n - 1) x2n+1 2 · 4 · 6 · ... · (2n)(2n + 1)
Find the first four nonzero terms in the Maclaurin series for the function. 360) x sin(2x) A) 2x2 + 4 x4 + 4 x6 + 8 x8 + . . . B) 2x - 4 x4 + 8 x6 - 16 x8 + . . . 3 15 315 3 15 315
C) 2x2 - 4 x4 + 4 x6 - 8 x8 + . . . 3
15
360)
D) 2x - 2x3 + 2 x5 - 16 x7 + . . .
315
3
45
Use the limit comparison test to determine if the series converges or diverges.
361) n=1
3 10n - 7 ln n + 8
361)
A) Diverges
B) Converges
Find the polynomial of lowest degree that will approximate F(x) throughout the given interval with an error of magnitude less than than 10-3 . x
362) F(x) = 0
2 e-t dt , [0, 0.5] 3
5
3
10
A) x - x + x
362) 3
5
3
5
B) x - x + x
3
5
3
5
C) x + x + x
3
5
3
15
D) x - x + x
For what values of x does the series converge conditionally?
363) n=1
xn n2 + 6
A) x = -1
363) B) x = 1
C) x = ±1
56
D) None
Provide an appropriate response.
364) Find the sum of the series n=1
n3 . n!
[Hint: Write the series as 1 + 4 + n=3
A) 3e + 5
364) n3 =5+ n!
n=3
B) 8(e - 1)
n(n - 1)(n - 2) +3 n!
n=3
n(n - 1) + n!
C) 9e - 11
365) If an converges conditionally, what can be said about
n=3
n .] n!
D) 5e
max {a n , |a n |} ?
365)
A) The series may converge or diverge. B) The series always converges. C) The series always diverges. Solve the problem. 366) An object is rolling with a driving force that suddenly ceases. The object then rolls 10 meters in the first second, and in each subsequent interval of time it rolls 80% of the distance it had rolled the second before. This slowing is due to friction. How far will the object eventually roll? A) It will roll an infinite distance. B) 50.0 m
C) 20.0 m
366)
D) 12.2 m
Find the interval of convergence of the series. (x - 7)n ln (n + 4)
367) n=1
A) - < x <
367) B) 6 < x < 8
C) 6 x < 8
D) x < 8
Provide an appropriate response.
368) Obtain the first nonzero term of the Maclaurin series for ln(1 + x2 ) - x2 cos x. A) -
7x6 12
B)
7x6
C) -
12
7x6 24
368) D)
7x6 24
Solve the problem. 369) Use a graphical method to determine the approximate interval for which the second order Taylor polynomial for ln (1 + x) at x = 0 approximates ln (1 + x) with an absolute error of no more than 0.04. A) -0.5525 x 0.2640 B) -0.5525 x 0.5525
C) -0.1928 x 0.7063
369)
D) -0.4310 x 0.5525
Find the sum of the geometric series for those x for which the series converges. (-1)n (2x)2n
370) n=0
A)
1
1 - 4x2
370) B)
4
C)
1 + 4x2
57
1
1 + 4x2
D)
2
1 + 4x2
Provide an appropriate response.
371) Obtain the first nonzero term of the Maclaurin series for sin x - tan-1 x. A)
x3
B)
3
x3
C) -
6
x3
371) D)
3
x3 2
(-1)n (x - 3)n is integrated twice term by term, for what value(s) of x does the
372) If the series
n=0 new series converge? A) 2 < x 4
B) 2 x 4
C) 2 < x < 4
372)
D) 2 x < 4
Find the sum of the series.
373) n=0
1 1 n 6 3n
A) - 9
373) B) - 3
10
C) 9
10
D) 3
10
Find a formula for the nth term of the sequence. 374) 0, 2, 0, 2, 0 (alternating 0's and 2's) A) an = 1 + (-1)n-1 C) an = 1 + (-1)n
10
374) B) an = 1 - (-1)n D) an = 1 + (-1)n+1
Determine whether the nonincreasing sequence converges or diverges. 375) an = 1 + 7n n
A) Diverges
375)
B) Converges
Solve the problem. 376) For what value of r does the infinite series 1 + 3r + 2r2 + 3r3 + r4 + 3r5 + 2r6 + 3r7 + r8 + ... converge?
A) |r| < 5 2
B) |r| < 3
376) C) |r| < 2
2
3
D) |r| < 1
Use the integral test to determine whether the series converges.
377)
1
377)
n n=1 ln 4 A) converges
B) diverges
Find the interval of convergence of the series.
378) n=0
(x - 3)n n3 4n
A) -1 x 7
378) B) 2 x 4
C) -7 < x < 7 58
D) x < 7
Solve the problem.
2
379) Using the Maclaurin series for ln(1 + x), obtain a series for ln(1 + x ) .
379)
x
A) n=0
C) n=0
(-1)n x2n+1 n+1
B)
(-1)n xn+1 2n + 1
D)
n=0
n=0
(-1)n xn+1 n+1 (-1)n x2n+1 2n + 1
Use power series operations to find the Taylor series at x = 0 for the given function. x9
380)
380)
1 + 5x
(-1)n 5 n xn+10
A) n=0
n=0 (-1)n+1 5 n xn+10
C)
(-1)n+1 5 n xn+9
B)
(-1)n 5 n xn+9
D)
n=0
n=0
Use substitution to find the Taylor series at x = 0 of the given function. 381) e-6x
A) n=0
C) n=0
(-1)n (6)n xn n!
B)
(-1)n xn (6)n n!
D)
n=0
n=0
Provide an appropriate response.
382) Derive the series for 1
1-x
A) n=0
1 n+1 x
B) n=0
383) Derive the series for 1
1+x
A) n=0
(-1)n+1 xn+1
for x > 1 by first writing
B) n=0
(-1)n (6)6n x6n n! (6)n xn n!
1 1 1 . = 1 - x x 1 - 1/x
1 xn
C) n=0
for x > 1 by first writing
381)
(-1)n xn
382) D) n=0
(-1)n xn+1
1 1 1 . = 1 + x x 1 + 1/x
(-1)n+1 xn
C) n=0
59
(-1)n xn+1
383) D) n=0
(-1)n xn
384) Let sk denote the kth partial sum of the alternating harmonic series. Compute 2s10 + s11 3
, and
B)
3
s10 + s11 2
C)
s10 + 2s11 3
Find a formula for the nth term of the sequence. 385) 0, 2 , 0, 2 , 0 (alternating 0's and 2 's) 3 3 3
A) an = 1 + (-1)
n
C) an = 1 + (-1)
n+1
385)
2 + (-1)n
B) an = 1 + (-1)
n+1
D) an = 1 - (-1)
n
2 - (-1)n+1
2 + (-1)n+1
2 + (-1)n
Find the limit of the sequence if it converges; otherwise indicate divergence. -1 386) an = tan n 3 n
A)
384)
s10 + 2s11 . Which of these is closest to the exact sum (ln 2) of the alternating 3
harmonic series? 2s10 + s11
A)
s10 + s11 , 2
B) 0
2
C) 1
386) D) Diverges
387) an = (-1)n 1 - 5
387)
n
A) 0
B) 1
C) 5
D) Diverges
Find the values of x for which the geometric series converges. (-1)n
388) n=0
x-3 n 4
A) -5 < x < 11
388) B) -1 < x < 7
C) 5 < x < 11
D) -7 < x < 7
Use the Root Test to determine if the series converges or diverges.
389) n=1
ln n n 6n + 1
389)
A) Converges
B) Diverges
Find the smallest value of N that will make the inequality hold for all n > N. n 390) 2n - 1 < 10-3 A) 9907 B) 9899 C) 9891
60
390) D) 9883
Use the integral test to determine whether the series converges. 1 8n - 1
391) n=1
391)
A) diverges
B) converges
Determine if the series
a n defined by the formula converges or diverges. n=1 392) a1 = 3, a n+1 = 8n + 1 a n 4n - 9
A) Converges
392)
B) Diverges
Find the Maclaurin series for the given function. 393) sin 7x
A) n=0
C) n=0
393)
(-1)n 7 2n+1 x2n+1 n!
B)
(-1)n 7 2n+1 x2n+1 (2n + 1)!
D)
n=0
n=0
(-1)2n+1 72n+1 x2n+1 (2n + 1)! (-1)2n+1 72n+1 x2n+1 n!
Write the first four elements of the sequence. 394) ln (n + 1) n3
394)
A) ln 2 , ln 3 , ln 4 , ln 5
B) 0, ln 2 , ln 3 , ln 4
C) 0, ln 2 , ln 3 , ln 4
D) ln 2, ln 3 , ln 4 , ln 5
8
27
8
64
27
81
27
64
64
8
81
27
64
Find the sum of the geometric series for those x for which the series converges. (9x + 1)n
395)
395)
n=0
A) - 1
9x
B)
1 1 + 9x
C) 1
9x
D)
1 1 - 9x
For what values of x does the series converge absolutely?
396) n=1
(x - 5)n n
A) x = ±5
396) B) x = 5
C) 4 < x < 6
61
D) x = -5
Provide an appropriate response.
397) Find the sum of the series n=1
n2 . n!
[Hint: Write the series as 1 + n=2
A) 4e - 5
n2 =1+ n!
397)
n=2
n(n - 1) + n!
B) 3(e - 1)
n=2
n .] n!
C) 2e
D) e + 3
Find the first three nonzero terms of the Maclaurin series for the given function and the values of x for which the series converges absolutely. 398) f(x) = cos x - 4 398) 1-x
A) -3 - 4x - 9 x2 - ..., -1 < x < 1
B) -3 - x - 9 x2 - ..., -1 < x < 1
C) -3 - 4x + 9 x2 - ..., -1 < x < 1
D) -3 + 4x + 9 x2 + ..., -1 < x < 1
2
2
2
2
A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. 399) a1 = 1, a n+1 = an 6 399)
A) 1, 7, 13, 19, 25 C) 1, 6, 36, 216, 1296
B) 1, 6, 36, 216, 1296, 7776 D) 1, 1, 1, 1, 1
Find the limit of the sequence if it converges; otherwise indicate divergence. 400) an = 7n - 3 3-3 n
A) 7
B) - 7
3
400)
C) 1
3
D) Diverges
Determine if the series converges absolutely, converges, or diverges. (-1)n ln
401) n=1
7n + 4 6n + 3
401)
A) Diverges B) converges absolutely C) Converges conditionally Determine if the series converges or diverges; if the series converges, find its sum. ln
402) n=1
7 n
402)
A) Converges; 1
B) Converges; ln 1
C) Converges; ln 7
D) Diverges
7
62
Find the sum of the series.
403) n=0
1 (-1)n 6n 6n
A) 12
403) B) 6
37
C) 12
37
35
D) 6
35
Solve the problem. 404) If p > 1 and q > 1, what can be said about the convergence of
404)
1 ? p n (ln n)q
n=2 A) May converge or diverge
B) Always diverges C) Always converges Determine if the series converges absolutely, converges, or diverges. (-1)n ( n 2 + 1 - n)
405)
405)
n=1
A) diverges B) converges absolutely C) converges conditionally Use series to evaluate the limit. 406) lim x3 sin 4 - tan 4 x x x
A) - 8
406) C) - 1
B) - 32
2
D) - 2
Find the sum of the series as a function of x.
407) n=1
x-9 n 3
A) - x - 9 x+6
407) B) - x - 9
C) - x - 9
x + 12
x - 21
Find the Taylor polynomial of order 3 generated by f at a. 408) f(x) = x3 , a = 7
D) - x - 9
x - 12
408)
A) P3(x) = 343 + 49(x - 7) + 49(x - 7)2 + (x - 7)3 B) P3(x) = 343 + 147(x - 7) + 21(x - 7)2 + (x - 7)3 C) P3(x) = 1372 + 147(x - 7) + 14(x - 7)2 + (x - 7)3 D) P3(x) = 6 + 3(x - 7) + (x - 7)2 + (x - 7)3
63
Find the quadratic approximation of f at x = 0. 409) f(x) = esin 2x
409)
A) Q(x) = 1 + 2x + 2x2
B) Q(x) = 2x + 2x2
C) Q(x) = 1 - 2x + 2x2
D) Q(x) = 2x - 2x2
Provide an appropriate response. n 410) It can be shown that n!
n for large values of n. Find the smallest value of N such that e
410)
n
n! - 1 < 10-1 for all n > N. n e
A) 33
B) 27
C) 22
D) 29
Find the Taylor polynomial of order 3 generated by f at a. 411) f(x) = 1 , a = 1 x+6 2
A) P3(x) = 1 - x - 1 + (x - 1) - (x - 1) 5
25
125
2
49
3
2
B) P3(x) = 1 - x + 1 + (x + 1) - (x + 1)
625
C) P3(x) = 1 - x + 1 + (x + 1) - (x + 1) 7
411)
343
5
3
25
125
625
2
D) P3(x) = 1 - x - 1 + (x - 1) - (x - 1)
2401
7
49
343
3 3
2401
By calculating an appropriate number of terms, determine if the series converges or diverges. If it converges, find the limit L and the smallest integer N such that a n - L < 0.01 for n N; otherwise indicate divergence.
412) an = tan
-1 n
412)
en
A) L =
e
,N=8
B) L =
2e
,N=6
C) L = 0, N = 5
D) diverges
Find the Taylor series generated by f at x = a. 413) f(x) = 7 x, a = 4
A) n=0
413)
4 7 (ln 7)n (x - 4)n n!
B) n=0
7 4 (ln 7)n (x - 4)n
C) n=0
D)
n!
n=0
Find a formula for the nth term of the sequence. 414) 9, 11, 13, 15, 17 (every other integer starting with 9) A) an = 2n + 7 B) an = n + 17
64
4 7 (ln 7)n (x - 4)n (n + 1)! 7 4 (ln 7)n (x - 4)n (n + 1)!
414) C) an = 2n + 8
D) an = n + 16
Find the polynomial of lowest degree that will approximate F(x) throughout the given interval with an error of magnitude less than than 10-3 . x
415) F(x) = 0
2 e-t dt , [0, 1]
415)
3
5
7
9
3
5
7
9
3
5
7
3
10
42
A) x - x + x - x + x C) x - x + x - x
3
5
7
9
3
10
42
216
3
5
7
3
5
7
B) x - x + x - x + x D) x - x + x - x
Determine if the series
a n defined by the formula converges or diverges. n=1 416) a1 = 1 , a n+1 = n an 10
A) Diverges
416)
B) Converges
Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. 417) 8 + 72 + 648 + ... + 8 · 9 n-1 + ...
A)
8 1 - 9n 9-1
; series diverges
B)
n
8 1 + 9n
;
1+9
4 5
n
C) 8 1 - 9 ; 1
D) 8 1 - 9 ; series diverges
1-9
1-9
Find the limit of the sequence if it converges; otherwise indicate divergence. 418) an = 1 + (0.3)n
A) 2
B) 1
418)
C) 1.3
D) Diverges
Use power series operations to find the Taylor series at x = 0 for the given function. 419) cos2 (8x) (-1)n (16)2nx 2n
A) 1 + 2
C) 1 + 2
n=1
2
(-1)n (16)2nx 2n (2n)!
2(2n)!
n=1
D) 1 + 2
n=0
(-1)n (16)2nx 2n (2n)!
Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. 420) 8 + 8 + 8 + ... + 8 + ... 1·3 2·4 3·5 n(n + 2)
A) 8n(3n + 4) ; 6
B)
C) 8n(3n + 4) ; 6
D) 8n(3n + 5) ; 6
4n(n + 1) 4n(n + 2)
8n(3n + 5) ;6 4(n + 1)(n + 2) 4n(n + 2)
65
419)
(-1)n (16)2nx 2n
B) 1 +
2(2n)!
n=0
417)
420)
Find the sum of the geometric series for those x for which the series converges. -8 n xn
421) n=0
A)
421)
8 1 + 8x
B)
1 1 + 8x
C)
1 1 - 8x
D)
8 1 - 8x
Determine if the sequence is monotonic and if it is bounded. n 422) an = 2 2n
A) decreasing; unbounded C) nondecreasing; unbounded
422) B) not monotonic; unbounded D) nondecreasing; bounded
Find the first three nonzero terms of the Maclaurin series for the given function and the values of x for which the series converges absolutely. 423) f(x) = (cos x) ln(1 + x) 423) 2 3 2 3 A) x - x - x + ..., -1 < x < 1 B) x - x - x + ..., -1 < x < 1 3 5 3 5 2
3
2
6
C) x - x + x + ..., -1 < x < 1
2
3
2
6
D) x - x - x + ..., -1 < x < 1
Use the integral test to determine whether the series converges.
424) n=1
5 n
424)
A) diverges
B) converges
Find the smallest value of N that will make the inequality hold for all n > N. n 425) 0.2 - 1 < 10-2 A) 165 B) 158 C) 163
425) D) 161
Use power series operations to find the Taylor series at x = 0 for the given function. 1 426) (1 - 3x)2 (n + 1)3 n+1 xn+1
A)
n=0 n3 n xn
C)
n3 n+1 xn+1
B)
n=0
426)
(n + 1)3 n xn
D)
n=0
n=0
Find the limit of the sequence if it converges; otherwise indicate divergence. n 427) an = ln 1 + 2 n
A) 0
B) 2
C) ln 2
66
427) D) Diverges
By calculating an appropriate number of terms, determine if the series converges or diverges. If it converges, find the limit L and the smallest integer N such that a n - L < 0.01 for n N; otherwise indicate divergence.
428) an = cos n
428)
n2
A) L = 0, N = 10
B) L = 0, N = 13
C) L = 0, N = 8
D) diverges
Provide an appropriate response. n2 1 by expressing as a geometric series, differentiating n-1 1 -x 10
429) Find the sum of the series
429)
n=1 both sides of the resulting equation with respect to x, multiplying both sides by x, differentiating 1 again, and replacing x by . 10
A) 1.520467508e+15
B) 9.675702324e+14
C) 3.87028093e+16
D) 9.675702324e+15
9.16050259e+14
6.412351813e+15
2.849934139e+15
6.412351813e+15
Determine if the series converges absolutely, converges, or diverges.
430) n=1
(-1)n 7n 5/4 + 7
430)
A) diverges B) Converges absolutely C) Converges conditionally By calculating an appropriate number of terms, determine if the series converges or diverges. If it converges, find the limit L and the smallest integer N such that a n - L < 0.01 for n N; otherwise indicate divergence.
431) an = (1 + 0.153/n)n A) L 1.1653, N = 6 C) L 1.1653, N = 2
431) B) L 1.153, N = 6 D) diverges
Determine if the series converges absolutely, converges, or diverges. (-1)n
432) n=1
7 6 n 4 n
432)
A) Converges absolutely B) converges conditionally C) Diverges Find the sum of the series as a function of x. (x - 2)n
433) n=1
A) x - 2 x-1
433) B) - x - 2
C) - x - 2
x-3
x-1
67
D) x - 2 x-3
Determine if the sequence is monotonic and if it is bounded. 434) an = 8 - 4 n
A) not monotonic; bounded C) increasing; bounded
434) B) nonincreasing; bounded D) decreasing; unbounded
Find the limit of the sequence if it converges; otherwise indicate divergence. n 435) an = 9n
A) 1
B) ln 9
435)
C) 0
D) Diverges
Provide an appropriate response.
436) A sequence of rational numbers {rn } is defined by r1 = 1 , and if rn = a then rn+1 = a + 7b . Find 1
b
a+b
436)
lim rn . Hint: Compute the square of several terms of the sequence on a calculator. n
A) 6
B) 2 2
C) 7
D) 14
Determine convergence or divergence of the series.
437) n=1
ln (6n) n3
437)
A) Diverges
B) Converges
Determine if the series converges or diverges; if the series converges, find its sum. 1+
438) n=0
1 -2n n
A) Converges;
438) 1 1 +1
B) Converges;
C) Converges; e-2
1 1 -1
D) Diverges
Use the Ratio Test to determine if the series converges or diverges.
439) n=1
n7 7n
439)
A) Converges
B) Diverges
By calculating an appropriate number of terms, determine if the series converges or diverges. If it converges, find the limit L and the smallest integer N such that a n - L < 0.01 for n N; otherwise indicate divergence.
440) an = n tan 1
440)
n
A) L = 1, N = 6
B) L = 0, N = 28
C) L = 1, N = 28
68
D) diverges
Use the Ratio Test to determine if the series converges or diverges.
441)
(2n)! n 2 n=1 1 (n!)
441)
A) Converges
B) Diverges
Solve the problem.
442) Use series to estimate the integral's value to within an error of magnitude less than 10-3. 0.8
442)
cos2 x dx
0
A) 0.3748
B) 0.8427
C) 0.6499
D) 0.7758
Find the interval of convergence of the series.
443) n=0
(x - 8)2n 36n
443)
A) 7 < x < 9
B) -14 < x < 14
C) 2 < x < 14
D) x < 14
Find the smallest value of N that will make the inequality hold for all n > N. n 444) 0.6 - 1 < 10-3 A) 511 B) 507 C) 514
D) 509
Express the number as the ratio of two integers. 445) 0.545454 . . . A) 6 B) 60 11 11
D) 20 37
444)
445) C) 2 37
Find the quadratic approximation of f at x = 0. x 446) f(x) = 81 - x2
A) Q(x) = x
446) C) Q(x) = 1 + x
B) Q(x) = 9x
9
D) Q(x) = 1 + 9x
9
Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. 447) 8 + 8 + 8 + ... + 8 + ... 5 25 5 n-1 8 1-
A)
1 15 8 1-
C)
1 5 n-1
1 5n
1 15
;
;
8 1-
20 3
B)
1 5 n-1
18 1-
20 3
D)
69
1 5
1 5n
1 15
; 10
; 10
447)
Determine if the series converges or diverges. If the series converges, find its sum.
448) n=1
2 n(n + 3)
448)
A) converges; 11 9
B) converges; 5
D) converges; 7
C) diverges
3
9
Find the Taylor series generated by f at x = a. 449) f(x) = 1 , a = 10 x
449) B)
n=0
(x - 10)n 10n
D)
n=0
(-1)n (x - 10)n 10n+1
A) C)
Determine if the series
n=0
(-1)n (x - 10)n 10n
n=0
(x - 10)n 10n+1
a n defined by the formula converges or diverges.
n=1 450) a1 = 2, a n+1 = 6n + 5 sin n an 3n - 10 cos n
450)
A) Diverges
B) Converges
Solve the problem.
451) A ball is dropped from a height of 21 m and always rebounds 1 of the height of the previous 3
drop. How far does it travel (up and down) before coming to rest? A) 31.5 m B) 10.5 m C) 42 m
451)
D) 63 m
Provide an appropriate response.
452) Find the sum of the series n=1
n 1 by expressing as a geometric series, differentiating n-1 1 -x 9
both sides of the resulting equation with respect to x, and replacing x by
1 . 9
A) 7.036874418e+15
B) 5.699868278e+15
C) 5.699868278e+15
D) 4.503599627e+15
5.699868278e+15
452)
7.036874418e+15
4.503599627e+15
5.699868278e+15
Use the Root Test to determine if the series converges or diverges.
453) n=1
(n!)3n (3n)! n
453)
A) Converges
B) Diverges
70
Determine convergence or divergence of the alternating series. (-1)n+1
454) n=1
6(n + 1)3/2 n 3/2 + 1
454)
A) Diverges
B) Converges
Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. 455) 4 - 4 + 4 - 4 + ... + (-1)n-1 4 + ... 3 9 27 3 n-1 4 1-
A)
1
(-3)n
1+ 4 1-
C)
1 3
4 1;6
1 (-3)n-1
1+
1 3
B)
1 (-3)n-1 1 3
1+ 4 1;6
D)
;3
1
(-3)n
1+
455)
1 3
;3
Determine if the series converges absolutely, converges, or diverges. (-5)-n
456)
456)
n=1
A) converges conditionally B) converges absolutely C) diverges Use the Root Test to determine if the series converges or diverges.
457)
nn 2 n=1 3 n
457)
A) Diverges
B) Converges
Provide an appropriate response. 458) Obtain the first two terms of the Maclaurin series for sin(tan x). 3 3 3 A) x + x B) x - x C) x - x 3 3 6
458) D) x +
x3 6
Use the Comparison Test to determine if the series converges or diverges.
459) n=1
4 9n + 4 n
459)
A) converges
B) diverges
71
Determine if the series n=1
a n defined by the formula converges or diverges. -1 n
460) a1 = 7, a n+1 = 3 + tan n
an
460)
A) Converges
B) Diverges
Use the Comparison Test to determine if the series converges or diverges. 4
461)
n=1 5 n + 3
3
461) n
A) diverges
B) converges
Solve the problem.
-1 x2
462) Using the Maclaurin series for tan-1 x, obtain a series for tan
x2
A) n=0
(-1)n x4n 2n + 1
B) n=0
(-1)n x2n
C) n=0
D)
2n + 2
n=0
.
462)
(-1)n x4n 2n + 2 (-1)n x2n 2n + 1
463) If an is a convergent series of nonnegative terms, what can be said about an a n+1 ? A) Always diverges B) Always converges C) May converge or diverge
463)
Find the infinite sum accurate to three decimal places. (-1)n
464) n=1
1
464)
(4n + 1)3
A) 0.007
B) -0.007
C) -0.010
D) 0.010
Find the sum of the series.
465) n=0
7 9 + n 9 4n
A) - 33 8
465) B) 27
C) - 9
2
10
D) 159 8
Assume that the sequence converges and find its limit. 466) a1 = -2, an + 1 = 8 + 2a n
A) 8
466)
B) -6
C) -2
72
D) 4
Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. 8 467) 8 + 8 + 8 + ... + + ... 1·2·3 2·3·4 3·4·5 n(n + 1)(n + 2)
A)
8n(n + 2) ;4 2(n + 1)(n + 3)
B) 8(n + 1)(n + 3) ; 4
C)
8n(n + 3) ;2 4(n + 1)(n + 2)
D)
467)
2n(n + 2)
8n(n + 1) ;2 4(n + 2)(n + 3)
Determine if the series converges or diverges; if the series converges, find its sum. (-1)n-1
468) n=1
4 5n
468)
A) Converges; 1
B) Converges; 2
C) Converges; 1
3
D) Diverges
4
Provide an appropriate response.
469) Derive a series for ln(1 + x) for x > 1 by first finding the series for 1
1+x
Hint:
and then integrating.
469)
1 1 1 = 1 + x x 1 + 1/x
B) ln x +
n=1
(-1)n-1 nxn-1
D) ln x +
n=1
(-1)n nxn
A) ln x + C) ln x +
n=1
(-1)n nxn-1
n=1
(-1)n-1 nxn
Find the sum of the geometric series for those x for which the series converges. (x - 10)n
470) n=0
A)
470)
1 -9 - x
B)
1 11 + x
C)
1 -9 + x
D)
1 11 - x
Determine if the series converges or diverges. If the series converges, find its sum.
471) n=1
1 1 ln(n + 5) ln(n + 6)
471) B) converges; 1
A) diverges
ln 6
C) converges; 1
D) converges; ln 6
ln 5
73
Determine if the series converges absolutely, converges, or diverges. (-1)n
472) n=1
5n 6 + 5 4n 8 + 7
472)
A) Converges conditionally B) Converges absolutely C) Diverges For what values of x does the series converge absolutely?
473) n=1
(-1)n+1 (x + 3)n n2 n
A) 0 < x -1
473) B) 0 < x < -1
C) -5 < x < -1
D) -5 x < -1
Find the values of x for which the geometric series converges.
474) n=0
x-5 n 10
A) -15 < x < 15
474) B) -15 < x < 25
C) 15 < x < 25
74
D) -5 < x < 15
Answer Key Testname: CHAPTER 10
1) A 2) A 3) C 4) B 5) C 6) A 7) B 8) C 9) B 10) C 11) C 12) A 13) C 14) D 15) B 16) D 17) A 18) A 19) A 20) C 21) D 22) A 23) A 24) B 25) B 26) B 27) B 28) D 29) A 30) A 31) C 32) A 33) B 34) C 35) D 36) C 37) C 38) D 39) D 40) B 41) C 42) D 75
Answer Key Testname: CHAPTER 10
43) C 44) B 45) B 46) C 47) D 48) A 49) A 50) B 51) C 52) A 53) B 54) C 55) A 56) C 57) A 58) C 59) B 60) A 61) C 62) B 63) C 64) B 65) C 66) D 67) A 68) A 69) A 70) A 71) A 72) B 73) D 74) B 75) D 76) A 77) D 78) B 79) A 80) A 81) A 82) D 83) A 84) A 76
Answer Key Testname: CHAPTER 10
85) C 86) A 87) A 88) D 89) C 90) B 91) D 92) C 93) B 94) D 95) C 96) D 97) B 98) C 99) C 100) A 101) C 102) A 103) D 104) C 105) A 106) B 107) A 108) C 109) A 110) B 111) C 112) A 113) D 114) D 115) D 116) D 117) D 118) B 119) C 120) B 121) B 122) C 123) B 124) B 125) D 126) C 77
Answer Key Testname: CHAPTER 10
127) B 128) B 129) B 130) C 131) B 132) B 133) A 134) C 135) B 136) B 137) A 138) A 139) B 140) C 141) B 142) D 143) A 144) A 145) D 146) B 147) D 148) A 149) D 150) C 151) D 152) D 153) B 154) B 155) B 156) D 157) A 158) B 159) C 160) B 161) D 162) C 163) B 164) C 165) D 166) B 167) A 168) A 78
Answer Key Testname: CHAPTER 10
169) B 170) C 171) D 172) B 173) B 174) B 175) C 176) A 177) A 178) D 179) B 180) C 181) B 182) A 183) C 184) C 185) A 186) C 187) B 188) A 189) D 190) A 191) C 192) A 193) A 194) C 195) D 196) D 197) B 198) D 199) C 200) C 201) B 202) B 203) A 204) A 205) D 206) C 207) C 208) A 209) D 210) A 79
Answer Key Testname: CHAPTER 10
211) B 212) A 213) B 214) D 215) B 216) B 217) D 218) C 219) D 220) C 221) A 222) D 223) D 224) D 225) A 226) B 227) A 228) B 229) A 230) B 231) B 232) D 233) D 234) D 235) C 236) C 237) D 238) B 239) D 240) C 241) A 242) A 243) B 244) D 245) C 246) B 247) A 248) B 249) A 250) A 251) C 252) C 80
Answer Key Testname: CHAPTER 10
253) D 254) A 255) A 256) B 257) B 258) C 259) D 260) A 261) A 262) B 263) D 264) A 265) B 266) B 267) B 268) D 269) C 270) D 271) A 272) B 273) C 274) A 275) D 276) B 277) D 278) B 279) B 280) A 281) C 282) C 283) A 284) D 285) A 286) C 287) D 288) C 289) A 290) D 291) B 292) A 293) C 294) D 81
Answer Key Testname: CHAPTER 10
295) D 296) A 297) A 298) B 299) D 300) B 301) D 302) A 303) C 304) A 305) C 306) A 307) B 308) D 309) A 310) D 311) B 312) C 313) D 314) B 315) C 316) D 317) B 318) C 319) C 320) C 321) C 322) B 323) B 324) B 325) B 326) B 327) B 328) C 329) B 330) C 331) D 332) A 333) D 334) A 335) D 336) A 82
Answer Key Testname: CHAPTER 10
337) C 338) C 339) B 340) B 341) B 342) B 343) B 344) B 345) D 346) A 347) D 348) B 349) C 350) B 351) A 352) A 353) B 354) B 355) A 356) B 357) B 358) B 359) A 360) C 361) A 362) A 363) A 364) D 365) C 366) B 367) C 368) D 369) D 370) C 371) B 372) B 373) B 374) C 375) B 376) D 377) A 378) A 83
Answer Key Testname: CHAPTER 10
379) A 380) D 381) A 382) A 383) C 384) B 385) A 386) B 387) D 388) B 389) A 390) B 391) A 392) B 393) C 394) D 395) A 396) C 397) C 398) A 399) D 400) D 401) A 402) D 403) C 404) C 405) C 406) B 407) D 408) B 409) A 410) B 411) D 412) C 413) C 414) A 415) B 416) A 417) D 418) B 419) A 420) B 84
Answer Key Testname: CHAPTER 10
421) B 422) C 423) D 424) A 425) D 426) D 427) B 428) A 429) D 430) B 431) C 432) C 433) B 434) C 435) A 436) C 437) B 438) D 439) A 440) A 441) B 442) C 443) C 444) A 445) A 446) A 447) D 448) A 449) C 450) A 451) C 452) C 453) A 454) B 455) D 456) B 457) B 458) D 459) B 460) A 461) A 462) A 85
Answer Key Testname: CHAPTER 10
463) B 464) B 465) D 466) D 467) C 468) B 469) D 470) D 471) B 472) B 473) C 474) D
86
Chapter 11
Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Choose the equation that matches the graph.
1)
1)
2
2
25
9
A) x + y = 1
2
2
25
9
B) x - y = 1
2
2
5
3
C) x + y = 1
2
2
9
25
D) x + y = 1
2)
2)
2
2
9
25
A) x + y = 1
2
2
25
9
B) x - y = 1
2
2
9
25
C) x - y = 1
2
2
9
25
D) y - x = 1
3)
3)
A) x2 = -8y
B) x2 = 8y
C) -8x2 = y
1
D) y2 = -8x
4)
4)
A) 9x2 - 16y2 = 144 C) 9x2 + 16y2 = 144
B) 16x2 - 9y2 = 144 D) 16x2 + 9y2 = 144
5)
5)
A) y2 = 4x
B) x2 = 4y
C) y2 = -4x
D) 4y2 = x
6)
6)
2
2
25
9
A) y - x = 1
2
2
9
25
B) y - x = 1
2
2
9
25
C) x - y = 1
2
2
2
9
25
D) y + x = 1
7)
7)
2
2
25
9
A) y - x = 1
2
2
25
9
B) x + y = 1
2
2
5
3
C) y + x = 1
2
2
25
9
D) y + x = 1
8)
8)
A) y2 = 2x
B) y2 = -2x
C) -2y2 = x
9)
D) x2 = -2y 9)
A) 4x2 - 25y2 = 100 C) 25x2 + 4y2 = 100
B) 25x2 - 4y2 = 100 D) 4y2 - 25x2 = 100
3
10)
10)
A) 4x2 = y
B) y2 = 4x
C) x2 = -4y
D) x2 = 4y
11)
11)
A) 6x2 = -y
B) 6x2 = y
C) 6y2 = x
D) x2 = 6y
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 12) Does the spiral r = e- , 0 why not.
< + , have finite length? If yes, give its length. If no, explain
12)
13) How are the graphs of r = 1 + cos( - /6) and r = 1 + cos( - /4) related to the graph of
13)
14) Determine whether or not the point (1, ) lies on the curve r2 cos = 1. Explain.
14)
r = 1 + cos ? In general, how is the graph of r = f( - ) related to the graph of r = f( )?
4
15) (a) Graph the curves r =
1 1 + cos
and r =
1 1 - cos
.
15)
(b) Find polar coordinates of the points of intersection of the curves in part (a). (c) Show that at the points of intersection the tangent lines to the curves are perpendicular to each other.
16) Show that the arc length of one petal of the rose r = cos(n ) is given by 2
/(2n)
16)
1 + (n 2 - 1)sin2 (n ) d
0 and use this formula to help make a conjecture about the limit of such arc lengths as n + .
17) Assuming r = f( ) is continuous for
and < + 2 , what can be said about the relative areas between the origin and the polar curves r = f( ), and r = 2f( ), ? Give reasons for your answer.
17)
Find a parametrization for the curve.
18) The upper half of the parabola x - 3 = y2
18)
Provide an appropriate response. 19) Find the error in the following "proof" that the area inside the lemniscate r2 = a 2 cos 2 is 0 and then find the correct area. 2
A= 0
1 2 r d 2
2
= 0
1 2 a cos 2 d 2
=
1 2 a sin 2 4
2 0
=0
20) A radial line is drawn from the origin to the spiral of Archimedes r = a , 0
, a > 0. Find the area swept out during the second revolution of the radial line that was not swept out during the during the first revolution. In other words, find the area between the first and second turns of the spiral.
5
19)
20)
Find a parametrization for the curve. 21) The line segment with endpoints (-6, -8) and (-8, -11)
21)
Provide an appropriate response.
22) Consider the curves corresponding to polar equations of the form r = 1 - a cos 1 + a cos
where
22)
a > 0. Explain how the graph changes as a changes. Identify any values of a for which the basic shape of the curve changes.
23) Find the left most point on the cardioid r = 1 + sin .
23)
24) The area of the surface formed by revolving the graph of r = f( ) from = to = about
24)
the line
= 0 (the polar axis) is SA =
dr 2 r2 + d . The area of the d
2 r sin
surface formed by revolving the graph of r = f( ) from is SA =
2 r cos
=
to
=
about the line
=
2
dr 2 r2 + d . Use the idea underlying these two integral d
formulas to find the surface area of the torus formed by revolving the circle r = a about the line r = b sec where 0 < a < b.
25) Find any horizontal or vertical asymptotes for the spiral r = 1 , > 0. (Suggestion: Express
25)
the curve in parametric form (x = r cos , y = r sin ) and investigate what happens to x and y as varies.)
Find a parametrization for the curve. 26) The ray (half line) with initial point (9, -8) that passes through the point (3, -2) Provide an appropriate response. 27) Find any horizontal or vertical asymptotes for the curve r = 4 tan , 0 2 . (Suggestion: Express the curve in parametric form (x = r cos , y = r sin ) and investigate what happens to x and y as varies.)
26)
27)
28) Show that if a and b are not both 0, the graph of the equation r = a cos + b sin is a
28)
29) Graph the equation r = 1 + a cos for several values of a in order to get an idea of what
29)
30) Find the area of the region enclosed by the rose r = a cos n for n = 1, 2, 3, . . . .
30)
circle. Find the radius and the rectangular coordinates of the center of this circle.
these curves look like. (Try both a < 1 and a > 1.) Show that the graph will have an inner loop for every a such that a > 1 and find the values of that correspond to the inner loop.
6
31) Consider the inequality x2 + y2 < 3x + y.
31)
(a) Find an equivalent inequality in polar coordinates. (b) Use an integral with respect to to find the area of the region defined by the inequality. (c) Find the length of the curve that forms the boundary of the region defined by the inequality. (d) Find the slope of this curve at the point (2, 2).
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the directrices of the hyperbola. 32) 81y2 - 16x2 = 1296
A) y = ± 16 65
32) B) x = ± 16 97
C) y = ± 16 97
D) y = ± 81 97
The polar equation of a circle is given. Give polar coordinates for the center of the circle and identify its radius. 33) r = -6 sin 33)
A) C 6, C) C -6,
2 2
, radius = 6
B) C 3,
, radius = 6
D) C -3,
, radius = 3
2 2
, radius = 3
Graph the set of points whose polar coordinates satisfy the given equation or inequality. 34) 0 ,r 3
A)
B)
7
34)
C)
D)
Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. 35) Foci at (-5, 0), (5, 0); asymptotes: y = 4 x, y = - 4 x 35) 3 3 2
2
16
9
A) x - y = 1 Graph.
2
2
9
16
B) x - y = 1
2
2
9
16
C) y - x = 1
36) 25y2 - 16x2 = 400
2
2
16
9
D) y - x = 1
36)
A)
B)
8
C)
D)
Graph the pair of parametric equations with the aid of a graphing calculator. 37) x = 3 cos t + cos 3t, y = 3 sin t - sin 3t, 0 t 2
A)
B)
9
37)
C)
D)
Find the slope of the polar curve at the indicated point. 38) r = 9 , = 3
A) -3
38)
B) 3
C) Undefined
Graph the polar equation. 39) r2 = 31 sin 4
D) 0
39)
A)
B)
10
C)
D)
Find the standard-form equation of the ellipse centered at the origin and satisfying the given conditions. 40) An ellipse with length of major axis 16 and y-intercepts (0, ±7) 2 2 2 2 2 2 2 2 A) x + y = 1 B) x + y = 1 C) x + y = 1 D) x + y = 1 49 64 8 7 64 49 7 8 Find the area of the specified region. 41) Inside the three-leaved rose r = 4 cos 3 A) 4 B) 4 3
40)
41) C) 8
Graph the polar equation. 42) r = 7 cos 7
D) 2
42)
11
A)
B)
C)
D)
Find the length of the curve. 43) The cardioid r = 5(1 + sin )
A) 40
43) B) 75
C) 75
2
D) 40
2
Describe the graph of the polar equation. 44) r = -8 csc A) Vertical line through (-8, 0)
44) B) Line through origin with slope -8 D) Horizontal line through (0, -8)
C) Circle of radius 8 centered at origin
Solve the problem. 45) Find the volume generated by revolving about the y-axis the area bounded by the curves: x2 - 16y2 = 16, x = 7
A) 65 3
11
B) 11 2
33
C) 11
12
33
D) 1
11
45)
Find a Cartesian equation for the line whose polar equation is given. 46) r cos + 4 = 4 3 3
A) y = 3x + 8 C) y =
46)
B) y = 3x + 4
3 x-4 3 3
D) y =
3 x+8 3
Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. 8 47) r = 2 - cos
A)
B)
C)
D)
13
47)
Find the focus and directrix of the parabola. 48) x2 = -16y
A) (0, -4); y = 4
48)
B) (0, -4); x = 4
C) (-4, 0); y = 4
D) (0, -4); y = -4
t
3 , 0 . (Hint: 4
Solve the problem.
49) Find the point on the curve x = 2 sin t, y = cos t, -
2
2
, closest to the point
Minimize the square of the distance as a function of t.) A) 3 , 1 B) 1, 3 C) (-2, 0) 2 2
49)
D) (2, 0)
The polar equation of a circle is given. Give polar coordinates for the center of the circle and identify its radius. 50) r = -8 cos 50)
A) C 4,
2
, radius = 4
B) C 8,
C) C(-8, 0), radius = 8 Find the directrices of the hyperbola. 51) x2 - y2 = 49
A) y = ± 7 2
2
, radius = 8
D) C(-4, 0), radius = 4
51) B) x = ± 7
C) y = ± 7
2
2
D) x = ± 49
2
Replace the Cartesian equation with an equivalent polar equation. 52) x2 + y2 = 49
A) r = 7 sin
B) r = 49
C) r = 7 or r = -7
Solve the problem. 53) Find the foci and asymptotes of the following hyperbola: x2 - y2 = 18
A) Foci: (6, 0), (-6, 0); Asymptotes: y =2x, y = - 2x B) Foci: (3, 0), (-3, 0); Asymptotes: y = x, y = - x C) Foci: (6, 0), (-6, 0); Asymptotes: y = x, y = - x D) Foci: (0, 6), (0, -6); Asymptotes: y = x, y = - x Graph the pair of parametric equations with the aid of a graphing calculator.
14
52) D) r = 7 cos
53)
54) x = 4 sin t - sin 4t, y = 4 cos t - cos 4t, 0 t 2
54)
A)
B)
C)
D)
Find the standard-form equation for a hyperbola which satisfies the given conditions.
55) A hyperbola centered at the origin having focus at (6, 0) and eccentricity equal to 3
55)
2
2
2
16
20
A) y + x = 1
2
2
16
20
B) x + y = 1
2
2
16
20
C) x - y = 1
2
2
16
20
D) y - x = 1
Replace the polar equation with an equivalent Cartesian equation. 56) r2 + 2r2 sin cos = 323
A) x2 + y2 = 323
B) x2 + 3y2 = 323
C) x + y = ±18
15
56) D) x + y = 18
Find the length of the curve. 57) x = 3 cos t, y = 3 sin t, 0 t A) 3
57) B) 9
C)
Find the Cartesian coordinates of the given point. 58) (1, 0) A) (0, -1) B) (-1, 0)
D) 6
58) C) (0, 1)
Find all the polar coordinates of the point. 59) (-6, /3) A) (6, 4 /3 + 2n ), (-6, 2 /3 + 2n )
D) (1, 0)
59) B) (6, 4 /3 + 2n ), (-6, /3 + 2n ) D) (6, /3 + 2n ), (6, 4 /3 + 2n )
C) (6, 4 /3 + 2n ), (-6, - /3 + 2n ) 60) (10, /4) A) (10, /4 + 2n ), (10, -3 /4 + 2n ) C) (10, /4 + 2n ), (-10, -3 /4 + 2n )
60) B) (10, /4 + 2n ), (-10, - /4 + 2n ) D) (10, /4 + 2n ), (-10, 3 /4 + 2n )
Find the focus and directrix of the parabola. 61) x2 = 16y
A) (0, 4); y = -4
61)
B) (0, -4); x = - 4
C) (4, 0); y = 4
Graph the pair of parametric equations with the aid of a graphing calculator. 62) x = t - sin t, y = 1 - cos t, -4 t 4
16
D) (4, 0); x = 4
62)
A)
B)
C)
17
D)
Graph.
2
2
4
9
63) x - y = 1
63)
A)
B)
18
C)
D)
Find the slope of the polar curve at the indicated point.
64) r = 1 - sin , =
64)
4
A) 2 Find the directrices of the ellipse. 65) 4x2 + y2 = 4
A) x = ± 4 5
B) 2 - 1
C) 3
D) 1
B) y = ± 2 3
C) x = ± 4 3
D) y = ± 4 3
2
65)
Determine the symmetries of the curve. 66) r2 = 7 sin 3
66)
A) x-axis only C) Origin only
B) y-axis only D) x-axis, y-axis, origin
Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. 67) Vertices at (9, 0) and (-9, 0); foci at (11, 0) and (-11, 0) 67) 2 2 2 2 2 2 2 2 A) x - y = 1 B) x - y = 1 C) x - y = 1 D) x - y = 1 121 81 81 121 81 40 40 81 If the equation represents a hyperbola, find the center, foci, and asymptotes. If the equation represents an ellipse, find the center, vertices, and foci. If the equation represents a circle, find the center and radius. If the equation represents a parabola, find the focus and directrix. 68) 9x2 + 90x + 25y2 + 150y + 225 = 0 68)
A) C: (-5, -3); V: (-10, -3), (0, -3); F: (-9, -3), (-1, -3) B) C: (-5, -3); V: (-10, -3), (0, -3); F: (-8, -3), (-2, -3) C) C: (-5, -3); V: (-3, -10), (-3, 0); F: (-3, -9), (-3, -1) D) C: (-5, -3); V: (-3, -10), (-3, 0); F: (-3, -8), (-3, -2)
19
Find the area of the specified region. 69) Inside the outer loop and outside the inner loop of the limacon r = 8 sin A) 8(4 + 3 3) B) 8(2 - 3 3) C) 8 (4 - 3 3) 3
-4
69) D) 16( + 3 3)
Solve the problem. 70) Find the volume generated by revolving about the x-axis the region bounded by the following graph: y = 64 - x2 , x = 0, x = 8
A) 3.002399752e+15
B) 16
C) 256
D) 3.002399752e+15
4.398046511e+12
8.796093022e+12
Find a polar equation for the circle. 71) (x + 6)2 + y2 = 36
A) r = -12 sin Find the directrices of the ellipse. 72) 25x2 + 36y2 = 900
A) y = ± 6 11
71) B) r = 12 cos
C) r = -12 cos
D) r cos = -12
B) x = ± 36 61
C) x = ± 36 11
D) y = ± 36 11
72)
Find a polar equation for the circle. 73) x2 + (y - 7)2 = 49
A) r = 14 sin
73) B) r cos = -14
Find the area of the specified region. 74) Inside the cardioid r = (1 + sin ),
A) 2
70)
B)
3 2 2
C) r = -14 sin
D) r = 14 cos
3 2
3 2
>0
74)
-2 2
C)
2
D)
2
+2 2
Assuming that the equations define x and y implicitly as differentiable functions x = f(t), y = g(t), find the slope of the curve x = f(t), y = g(t) at the given value of t. 75) 2x + 4x3/2 = t3 + t, y(t + 1) - 4t y = 4, t = 0 75)
A) -8
B) 2
C) 8
Sketch the region which is defined by the given conditions.
20
D) 4
76) 0
4
,0 r
4
76)
A)
B)
C)
D)
Replace the polar equation with an equivalent Cartesian equation. 77) r2 = 42r cos - 6r sin - 9
A) (x - 21)2 + (y + 3)2 = 9 C) (x + 42)2 + (y - 6)2 = 9
77)
B) (x - 21)2 + (y + 3)2 = 441
D) 42x - 6y = 9
Find the foci of the ellipse. 78) 64x2 + 36y2 = 2304
A) (0, ±6)
78) C) (±2 7, 0)
B) (0, ±8)
21
D) (0, ±2 7)
Determine the symmetries of the curve. 79) r2 = -4 cos 5
79)
A) x-axis, y-axis, origin C) x-axis only
B) Origin only D) No symmetry
Describe the graph of the polar equation. 80) r2 = 32r cos
80)
A) Horizontal line passing through (0, 32) C) Vertical line passing through (32, 0) Find the polar coordinates, 0 81) (- 3, - 3) A) 3, 5 4
B) Circle of radius 16 and center (16, 0) D) Circle of radius 16 and center (0, 16)
< 2 and r 0, of the point given in Cartesian coordinates.
81) 3,
B)
C) 3,
4
D) 6, 5 4
4
Find the eccentricity of the ellipse. 82) 4x2 + y2 = 36
A) 5
82) B) 3
C)
3 2
D)
5 2
The eccentricity is given of a conic section with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for the conic section. 83) e = 3, y = 7 83) 21 21 21 21 A) r = B) r = C) r = D) r = 1 + 7 sin 1 + 3 sin 1 + 3 cos 1 - 3 sin
Describe the graph of the polar equation. 84) r = 48 sin A) Circle with radius 24 and center (0, 24)
84)
B) Parabola opening upward with vertex at (0, 24) C) Ellipse centered at the origin with intercepts (±1, 0) and (0, ±24) D) Horizontal line passing through (0, 48) Find the standard-form equation for an ellipse which satisfies the given conditions. 85) An ellipse centered at the origin having focus ( 13, 0) and directrix x = 49 13 2
2
36
49
A) x + y = 1
2
2
49
36
B) x + y = 1
2
2
7
6
C) x + y = 1
85) 2
D) x + y2 = 1 49
Assuming that the equations define x and y implicitly as differentiable functions x = f(t), y = g(t), find the slope of the curve x = f(t), y = g(t) at the given value of t. 86) 2tx + 3t2 = 3, 2y - t2 - t = 0, t = 1 86)
A) 1 2
B) 3
C) - 1
2
2
22
D) - 9 2
Describe the graph of the polar equation. 87) r2 = 34r cos - 6r sin - 9
87)
A) Circle with radius 3 and center (34, -6) B) Parabola opening to the right with vertex (34, -6) C) Circle with radius 17 and center (17, -3) D) Parabola opening upward with vertex (34, -6) Replace the Cartesian equation with an equivalent polar equation. 88) x2 + y2 - 4x = 0
Graph.
A) r sin2 = 4 cos
B) r cos2 = 4 sin
C) r = 4 cos
D) r = 4 sin
89) 16x2 - 4y2 = 64
88)
89)
A)
B)
23
C)
D)
Graph the polar equation. 90) r2 = 70 cos 3
90)
A)
B)
24
C)
Find the directrices of the hyperbola. 91) 36x2 - 25y2 = 900
A) y = ± 36 61
D)
91) B) y = ± 25 61
C) x = ± 36 61
Graph the set of points whose polar coordinates satisfy the given equation or inequality. 92) r 4
A)
B)
25
D) x = ± 25 61
92)
C)
D)
Find the area of the specified region. 93) Inside the graph of r = 1 + cos 3
A) 3
93)
B) 3 2
C) 3 + 3 2 8
D) 3 + 3 4
Find an equation for the line tangent to the curve at the point defined by the given value of t. 94) x = 5 sin t, y = 5 cos t, t = 3 4
A) y = 5 2x + 1 Graph.
B) y = 5x + 5 2
C) y = x - 5 2
95) x2 + 25y2 = 25
94)
D) y = - x + 5 2
95)
A)
B)
26
C)
D)
Find the polar coordinates, 0 96) 1 , 3 6 6
< 2 and r 0, of the point given in Cartesian coordinates.
96)
A) 1 , 2 3
B) 1 ,
3
Solve the problem.
C) 1 ,
3 6
2
2
16
9
D) 1 ,
3 3
6 6
97) The ellipse x + y = 1 is shifted up 5 units and left 5 units. Find an equation for the new ellipse and find the new center. 2 2 A) (x + 5) + (y - 5) = 1; center: (-5, 5) 16 9
2
2
2
B) (x + 5) + (y - 5) = 1; center: (5, -5) 7
2
3
2
C) (x + 5) + (y - 5) = 1; center: (5, 5) 7
97)
2
D) (x - 5) + (y + 5) = 1; center: (-5, 5)
3
16
9
98) Find the foci and asymptotes of the following hyperbola:
98)
y2 x2 =1 9 16
A) Foci: (0, -4), (0, 4); Asymptotes: y = 3 x, y = - 3 x 4
4
B) Foci: (-4, 0), (4, 0); Asymptotes: y = 4 x, y = - 4 x 3
3
C) Foci: (0, -5), (0, 5); Asymptotes: y = 3 x, y = - 3 x 4
4
D) Foci: (-5, 0), (5, 0); Asymptotes: y = 4 x, y = - 4 x 3
3
Find the eccentricity of the ellipse. 99) x2 + 2y2 = 6
A)
2 2
99) B) 2
C) 6
27
D)
6 2
Replace the polar equation with an equivalent Cartesian equation. 100) 4r cos + 9r sin = 1
100)
A) 4x + 9y = 1
B) 4y + 9x = 1
C) 4x + 9y = x2 + y2
D) x + y = 1 4
Find the value of d2 y/dx2 at the point defined by the given value of t. 101) x = t + 3, y = - t, t = 22 A) - 1 B) 1 C) 10 2 5
9
101) D) -2
Graph.
102) x2 + 9y2 = 9
102)
A)
28
B)
C)
D)
Find the area of the specified region. 103) Inside the circle r = -6 sin and outside the circle r = 3 A) 9 B) 3 (4 - 3 3) C) 3 (2 + 3 3) 2 2 2 Graph the polar equation.
29
103) D) 3
104) r = 3
104)
A)
B)
C)
D)
Plot the point whose polar coordinates are given.
30
105) (2, 2 /3)
105)
A)
B)
31
C)
D)
Find the length of the curve. 106) The spiral r = 4 2 , 0
A) 256 5 5
2 3
106) B) 2304 5
C) 224
3
D) 32 (( 2 + 1)3/2 - 1)
3
3
Solve the problem.
107) The parabola y2 = -28x is shifted 7 units down and 2 units to the right. Find the focus and directrix of the new parabola. A) Focus: (-7, -5); Directrix: y = 9
B) Focus: (9, -7); Directrix: x = 9 D) Focus: (-5, -7); Directrix: x = 9
C) Focus: (-5, -7); Directrix: x = -5
Graph the pair of parametric equations with the aid of a graphing calculator.
32
107)
108) x = cos t + 4 cos 3t, y = 5 cos t - 4 sin 3t, 0 t 2
108)
A)
B)
C)
D)
Find the area of the specified region. 109) Inside the circle r = 4 cos + 3 sin A) 25 B) 25 2 4
109) C) 6
33
D) 25 8
+6
Find the value of d2 y/dx2 at the point defined by the given value of t.
110) x = csc t, y = 6 cot t, t =
110)
3
A) -12 3
B) -18 3
Solve the problem.
2
2
400
225
C) 6 3
D) 12 3
111) The hyperbola y - x = 1 is shifted horizontally and vertically to obtain the hyperbola
111)
(y - 5)2 (x + 3)2 =1 400 225 Find the asymptotes of the new hyperbola. A) y = ± 16 x 9
B) y - 5 = ± 16 (x + 3) 9
C) y - 5 = ± 4 (x + 3) 3
D) y = ± 4 x 3
Find the area of the specified region.
112) Inside one loop of the lemniscate r2 = 5 cos 2
112)
B) 5 2
A) 10
D) 5 4
C) 5
Find a polar equation in the form r cos( - 0 ) = r0 for the given line.
113) x = -6
113)
A) r cos = 6 C) r cos
+
=6
Find the Cartesian coordinates of the given point. 114) 5, 4 /3 A) -5 3 , -5 B) -5 , -5 3 2 2 2 2
B) r cos
-
D) r cos
+
2 2
=6 =6
114) C)
-5 -5 , 2 2
D)
5 5 , 2 2
Determine the symmetries of the curve. 115) r2 = 8 sin 6
115)
A) y-axis only C) Origin only
B) x-axis, y-axis, origin D) x-axis only
Assuming that the equations define x and y implicitly as differentiable functions x = f(t), y = g(t), find the slope of the curve x = f(t), y = g(t) at the given value of t. 116) 2x - t2 - t = 0, 2ty + 4t2 = 4, t = 1 116)
A) - 6
B) - 8
C) 6
3
34
D) - 4
Find the area of the specified region. 117) Inside the limacon r = 3 + 2 sin A) 22 B) 11 + 12
117) C) 11
D) 22 - 12
Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. 2 118) r = 1 - cos
A)
B)
C)
D)
35
118)
Find the vertices and foci of the ellipse. 119) 169x2 + 81y2 = 13,689
119)
A) Vertices: (0, ±9); Foci: (0, ±2 22) C) Vertices: (±13, 0); Foci: (±2 22, 0)
B) Vertices: (±9, 0); Foci: (±2 22, 0) D) Vertices: (0, ±13); Foci: (0, ±2 22)
Solve the problem.
120) The parabola x2 = -32y is shifted 5 units up and 7 units to the left. Find the focus and directrix of the new parabola. A) Focus: (7, -13); Directrix: y = -13
B) Focus: (-7, -3); Directrix: y = 13 D) Focus: (-7, 13); Directrix: y = 13
C) Focus: (-7, -3); Directrix: y = -3 2
2
11
5
121) The hyperbola y - x = 1 is shifted down 8 units and right 3 units. Find an equation for the new hyperbola and find the new vertices. 2 2 A) (y - 8) - (x + 3) = 1; vertices: (3, -8 ± 11 5
2
120)
121)
11 )
2
B) (y + 8) - (x - 3) = 1; vertices: (3 ± 5 , -8 ) 11
5
2
2
C) (y + 3) - (x - 8) = 1; vertices: (3 ± 5 , -8 ) 11
5
2
2
D) (y + 8) - (x - 3) = 1; vertices: (3, -8 ± 11 ) 11
5
122) Find the foci and asymptotes of the following hyperbola:
122)
4y2 - 25x2 = 100
A) Foci: ( 21, 0), (- 21, 0); Asymptotes: y = 2 x, y = - 2 x 5
5
B) Foci: (0, 21), (0, - 21); Asymptotes: y = 5 x, y = - 5 x 2
2
C) Foci: (0, 29), (0, - 29); Asymptotes: y = 5 x, y = - 5 x 2
2
D) Foci: ( 29, 0), (- 29, 0); Asymptotes: y = 25 x, y = - 25 x 4
4
Find the length of the curve.
123) The curve r = 6 sin3 A) 12(3 3 - )
3
,0
123)
2
B) 3 (2 - 3 3)
C) 9 (5 - 9 3)
4
16
Graph the polar equation.
36
D) 9 (5 - 8) 2
124) r = 5 sin 2
124)
A)
B)
C)
D)
Sketch the region which is defined by the given conditions.
37
125) 0
,r 3
125)
A)
B)
C)
D)
Find the area of the specified region. 126) Inside the circle r = a sin and outside the cardioid r = a(1 - sin ), a > 0 2 2 2 A) a (2 - 3 3) B) a (3 3 - ) C) a (6 - ) 12 3 6
126) D)
a2 6
(4 - 3 3)
Find the standard-form equation for an ellipse which satisfies the given conditions.
127) An ellipse centered at the origin having vertex at (0, -6) and eccentricity equal to 1
127)
3
2
2
36
32
A) y + x = 1
2
2
36
32
B) y - x = 1
2
2
36
32
C) x + y = 1
38
2
2
36
32
D) x - y = 1
Find the polar coordinates, 0 128) (-4, 0)
< 2 and r 0, of the point given in Cartesian coordinates.
128) B) 4,
A) (4, ) Solve the problem.
2
2
25
36
C) 4,
2
2
D) (-4, )
129) The hyperbola x - y = 1 is shifted horizontally and vertically to obtain the hyperbola (x + 3)2 (y - 4)2 = 1. Graph the new hyperbola. 25 36
A)
B)
C)
D)
39
129)
Provide an appropriate response. 130) True or false? If n > 0 is an even integer, then the area of the region enclosed by r = sin n is twice the area of the region enclosed by r = sin[(n + 1) ]. A) False B) True Find the directrices of the ellipse. 131) 49x2 + 36y2 = 1764
131)
A) x = ± 49 13
Solve the problem.
130)
B) y = ± 49 13
2
2
625
225
C) y = ± 36 13
D) x = ± 49 85
132) The hyperbola x - y = 1 is shifted horizontally and vertically to obtain the hyperbola
132)
(x - 2)2 (y - 3)2 = 1. 625 225 Find the center and vertices of the new hyperbola. A) Center: (3, 2); Vertices: (-23, 3), (27, 3)
C) Center: (2, 3); Vertices: (3, -23), (3, 27) Find the polar coordinates, 0 133) ( 5, - 5) A) 5, 7 4
B) Center: (3, 2); Vertices: (3, -23), (3, 27) D) Center: (2, 3); Vertices: (-23, 3), (27, 3)
< 2 and r 0, of the point given in Cartesian coordinates.
133) B) ( 10, 5 4
C) 5, 5 4
D) 5, 7 4
Find the focus and directrix of the parabola. 134) y2 = -4x
A) (0, -1); y = 1
134)
B) (1, 0); x = -1
C) (-1, 0); x = 1
D) (-1, 0); y = 1
Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. 6 135) r = 2 + 2 cos
40
135)
A)
B)
C)
D)
Determine the symmetries of the curve. 136) r = 2 sin - 5 cos A) x-axis only
136) B) y-axis only D) No symmetry
C) x-axis, y-axis, origin
Assuming that the equations define x and y implicitly as differentiable functions x = f(t), y = g(t), find the slope of the curve x = f(t), y = g(t) at the given value of t. 137) x(t + 1) - 4t x = 25, 2y + 4y3/2 = t3 + t, t = 0 137)
A) -10
C) - 1
B) -5
5
D) - 1
10
Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. 138) Vertices at (0, 7) and (0, -7); foci at (0, 11) and (0, -11) 138) 2 2 2 2 2 2 2 2 A) y - x = 1 B) y - x = 1 C) y - x = 1 D) y - x = 1 72 49 49 121 49 72 121 49
41
Find the foci of the hyperbola. 139) 4y2 - 25x2 = 100
139)
A) (0, 21), (0, - 21) C) (0, 29), (0, - 29)
B) ( 29, 0), (- 29, 0) D) ( 21, 0), (- 21, 0)
Find the length of the curve. 140) x = 3 sin3 t, y = 3 cos3 t, 0 t A) 6 B) 3
140) C) 9
D) 18
Find the eccentricity of the hyperbola. 141) y2 - 36x2 = 36
A)
37 6
141) B)
Find the length of the curve.
142) The parabolic segment r =
35 6
5 1 + cos
,0
C) 37
D) 35
142)
2
A) 50
B) 5 ( 2 - ln( 2 - 1))
C) 25
D) 5 ( 2 + ln( 2 + 1))
3
2
3
2
Find the vertices and foci of the ellipse. 143) 9x2 + 25y2 = 225
143)
A) Vertices: (±5, 0); Foci: (±4, 0) C) Vertices: (0, ±5); Foci: (0, ±4)
B) Vertices: (±9, 0); Foci: (±4, 0) D) Vertices: (0, ±9); Foci: (0, ±5)
Find the foci of the hyperbola. 144) x2 - y2 = 8
144)
A) (0, 2), (0, -2)
B) (4, 0), (-4, 0)
C) (2, 0), (-2, 0)
D) (0, 4), (0, -4)
Find an equation for the line tangent to the curve at the point defined by the given value of t.
145) x = sin t, y = 10 sin t, t =
145)
3
B) y = -10x + 10 3
A) y = 10x C) y = 10x +
3 2
D) y = 10x - 10 3
Find the area of the specified region. 146) Shared by the cardioids r = 9(1 + sin ) and r = 9(1 - sin ) A) 81 (5 + 8) B) 1681 C) 81 (3 + 8) 4 2
42
146) D) 81 (3 - 8) 2
Find the foci of the ellipse. 147) 25x2 + 169y2 = 4225
A) (±12, 0)
147) B) (±25, 0)
C) (±13, 0)
D) (0, ±12)
Find the standard-form equation for a hyperbola which satisfies the given conditions. 148) A hyperbola centered at the origin having vertex at (0, -3) and eccentricity equal to 2 2 2 2 2 2 2 2 2 A) y - x = 1 B) y - x = 1 C) x - y = 1 D) y + x = 1 9 9 9 27 9 27 9 27 Solve the problem.
149) Find the point on the parabola x = t2 , y = t,
< t < , closest to the point
1 , 54 . (Hint: Minimize 2
C) (36, 6)
D) 9 , 3
the square of the distance as a function of t.)
A) (9, 3)
B) (0, 0)
148)
149)
4 2
Find the Cartesian coordinates of the given point.
150) 14, -
150)
4
A) (7 2, -7 2) C) (-14 2, -14 2)
B) (7 3, -7) D) (-7 2, -7 2)
Determine if the given polar coordinates represent the same point. 151) (3, /4), (-3, - /4) A) Yes B) No Graph the set of points whose polar coordinates satisfy the given equation or inequality. 152) r = 3
43
151)
152)
A)
B)
C)
D)
Find the length of the curve. 153) x = 2 (t2 + 6) 3/2, y = 6t, 0 t 1 3
A) 20
153)
B) 8
C) 19
3
3
D) 20 3
Replace the polar equation with an equivalent Cartesian equation. 154) r = 32 sin A) (x - 16)2 + y2 = 256 B) x2 + (y - 32)2 = 256
C) x2 + (y - 16)2 = 256
154)
D) y = 32
Find the length of the curve. 155) x = 1 (t3 - 3t), y = t2 + 2, 0 t 4 3
A) 17
155)
B) 64
C) 68
3
3
44
D) 76 3
Solve the problem. 156) Find the foci and asymptotes of the following hyperbola: x2 y2 =1 256 144
156)
A) Foci: (-20, 0), (20, 0); Asymptotes: y = 4 x, y = - 4 x 3
3
B) Foci: (0, -20), (0, 20); Asymptotes: y = 3 x, y = - 3 x 4
4
C) Foci: (-20, 0), (20, 0); Asymptotes: y = 3 x, y = - 3 x 4
4
D) Foci: (0, -12), (0, 12); Asymptotes: y = 4 x, y = - 4 x 3
3
Replace the Cartesian equation with an equivalent polar equation. 157) x = 12 A) r = 12 sin B) r = 12 csc C) r = 12 sec
D) r = 12 cos
Find the area of the specified region. 158) Shared by the circles r = 4 and r = 8 cos A) 16 B) 8 (2 + 3 3) 3 3
D) 8 (4 - 3 3) 3
157)
158) C) 8
159) Inside the circle r = 6 and to the right of the line r = 3 sec A) 6 B) 6(2 - 3 3) C) 3(4 - 3 3)
159) D) 3(2 + 3 3)
Find a polar equation in the form r cos( - 0 ) = r0 for the given line.
160) y = -6 A) r cos
160) -
2
=6
C) r cos = 6
B) r cos
+
D) r cos
+
=6 2
=6
Find the value of d2 y/dx2 at the point defined by the given value of t.
161) x = t + cos t, y = 2 - sin t, t = A) -2
161)
6
C) 1
B) -4
4
Graph the pair of parametric equations with the aid of a graphing calculator.
45
D) - 3
162) x = cos 5t, y = sin 2t, 0 t 2
162)
A)
B)
C)
D)
Find the length of the curve.
163) The curve r = a cos2 A) 2a
2
,0
,a>0
163) C) 3 a
B) 2 a
16
46
2
D) 3a
2
16
Replace the polar equation with an equivalent Cartesian equation. 164) r2 = 34r cos
164)
B) (x - 17)2 + y2 = 288 D) (x - 17)2 + y2 = 0
A) x = 34 C) x2 + (y - 17)2 = 288
Find a polar equation in the form r cos( - 0 ) = r0 for the given line.
165) 2x - 2y = 8 A) r cos
+
C) r cos
-
165) 3 4
=8
B) r cos
+
=4
D) r cos
+
4 3 4
=4 =8
Find an equation for the line tangent to the curve at the point defined by the given value of t. 166) x = 8t2 - 5, y = t3 , t = 1
A) y = 3 x - 7 16 16
B) y = 3 x + 1 8
C) y = 16 x - 7 3 16
Replace the polar equation with an equivalent Cartesian equation. 1 167) r = 5 cos - 2 sin
A) x + y = 1 5
2
B) 5y - 2x = 1
C)
166)
D) y = 3 x + 7 16 16
167) 1 =1 5x - 2y
D) 5x - 2y = 1
Solve the problem.
168) The parabola x2 = -28y is shifted 4 units up and 6 units to the left. Graph the new parabola.
47
168)
A)
B)
C)
48
D)
49
Graph the polar equation. 169) r = 3 + cos 2
169)
A)
B)
C)
D)
Find the area of the surface generated by revolving the curves about the indicated axis. 170) x = sin t, y = 9 + cos t, 0 t 2 ; x-axis A) 36 2 B) 9 2 C) 27 2 Replace the Cartesian equation with an equivalent polar equation. 171) 2x + 3y = 6 A) 2 sin + 3 cos = 6r B) r(2 cos
C) r(2 sin + 3 cos ) = 6
171) + 3 sin ) = 6
D) 2 cos + 3 sin = 6r
50
170) D) 54 2
Solve the problem. 172) Find the foci and asymptotes of the following hyperbola: y2 x2 =1 36
172)
A) Foci: ( 37, 0), (- 37, 0); Asymptotes: y = 6x, y = -6x B) Foci: ( 37, 0), (- 37, 0); Asymptotes: y = 1 x, y = - 1 x 6
6
C) Foci: (6, 0), (-6, 0); Asymptotes: y = 1 x, y = - 1 x 6
6
D) Foci: (0, 37), (0, - 37); Asymptotes: y = 6x, y = -6x Find the length of the curve.
173) The curve r = 2 1 + sin 2 , 0 A) 1 2
173)
4
B) 1 ( + 2)
2
2
D) 2( + ln( + 2))
C) 2
51
Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. 20 174) r = 8 - 4 sin
A)
B)
C)
D)
174)
Find the Cartesian coordinates of the given point.
175) 4 2 ,
175)
4
A) (2 2, 2 2)
C) (4 2, 4 2)
B) (4, 4)
Graph the parabola.
52
D) (2 3, 2)
176) y2 = 2x
176)
A)
B)
C)
D)
Find the area of the specified region. 177) Shared by the circles r = 5 cos and r = 5 sin A) 25 ( - 2) B) 25 (4 - 3 3) 8 24
177) C) 25 4
D) 5 (2 - 2) 2
Find the eccentricity of the hyperbola. 178) 8x2 - 9y2 = 72
A) 3 2
178) B)
13 3
C) 1 2
53
D)
17 3
Find the eccentricity of the ellipse. 179) 4x2 + y2 = 4
A)
3 2
179) B) 5
C) 3
Describe the graph of the polar equation. 180) r = 3cot csc A) Parabola with vertex at origin opening upward
D)
5 2
180)
B) Parabola with vertex at origin opening to the right C) Line with slope 3 passing through origin D) Circle with radius 3 centered at the origin Determine the symmetries of the curve. 181) r = 2 cos 7 A) y-axis only
181) B) x-axis, y-axis, origin D) Origin only
C) x-axis only Graph the polar equation. 182) r = 5 - cos
182)
A)
B)
54
C)
D)
Solve the problem.
183) Find the coordinates of the centroid of the area bounded by y = x2 and y = 9 A) 0, 54 5
C) 0, 243 5
B) 0, 15
183) D) 0, 27 5
Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. 184) Asymptotes y = 2 x, y = - 2 x; one vertex is (11, 0) 184) 11 11 2
2
121
4
2
2
121
4
A) y - x = 1
2
2
121
4
2
2
121
4
B) x - y = 1
C) x - y = 11
D) x - y = 484
Find the length of the curve.
185) The line segment r = 5 sec , 0
185)
4
A) 5 ln tan 3
B)
C) 5 ( 2 + ln( 2 + 1))
D) 5
8
2
Graph the polar equation.
55
4
186) r = 2(1 -sin )
186)
A)
B)
C)
D)
Find the length of the curve. 187) The spiral r = e3 , 0
A) 1 (e6 - 1) 12
187) B) 8( 2 - 1)
C) e6 - 3
Graph the set of points whose polar coordinates satisfy the given equation or inequality.
56
D)
10 3 (e - 1) 3
188) 2 r 5
188)
A)
B)
C)
D)
Find the length of the curve. 189) x = t3 , y = 2t2 , 0 t 1 A) 125 27
189) B) 1 27
C) 122 3
D) 61 27
Find the eccentricity of the hyperbola. 190) x2 - y2 = 5
A) 2 5
190) B) 0
C)
57
5 2
D) 2
Find the area. 191) Find the area enclosed by the ellipse x = 7 cos t, y = 3 sin t, 0 t 2 . A) 42 B) 21 2 C) 49 2 + 9
191) D) 21
Determine if the given polar coordinates represent the same point. 192) (3, /2), (-3, 3 /2) A) Yes B) No Find the Cartesian coordinates of the given point. 193) (5, ) A) (-5, 0) B) (0, 5)
192)
193) C) (0, -5)
D) (5, 0)
Graph.
194) 48x2 + 27y2 = 432
194)
A)
58
B)
C)
D)
Find the eccentricity of the ellipse. 195) Find the eccentricity of an ellipse centered at the origin having a focus of (0, 4 corresponding directrix y = 3
A) 3
B)
5 2
C)
59
3 2
3) and
D)
195)
3 4
Plot the point whose polar coordinates are given. 196) (2, /2)
196)
A)
B)
60
C)
D)
Find the foci of the ellipse. 197) 9x2 + 64y2 = 576
A) (±8, 0)
197) B) (± 55, 0)
C) (±3, 0)
Find the Cartesian coordinates of the given point. 198) ( 3, /6) A) 3 , 3 B) 3 , 1 2 2 2 2
D) (0, ± 55)
198) C) 1 , 2
Determine the symmetries of the curve. 199) r = 9 A) x-axis only
3 2
D) 3 , 2
3 2
199) B) x-axis, y-axis, origin D) No symmetry
C) y-axis only
Replace the Cartesian equation with an equivalent polar equation. 200) x2 + (y - 20)2 = 399
A) r = 20 sin
C) r2 = 40 cos
B) r = 40 cos
Graph. 61
200) D) r = 40 sin
201) 7x2 + 4y2 = 56
201)
A)
B)
62
C)
D)
If the equation represents a hyperbola, find the center, foci, and asymptotes. If the equation represents an ellipse, find the center, vertices, and foci. If the equation represents a circle, find the center and radius. If the equation represents a parabola, find the focus and directrix. 202) x2 + y2 - 12x - 14y = -76 202)
A) (6, 7); r = 3
B) (7, 6); r = 3
C) (-7, -6); r = 9
D) (-6, -7); r = 9
Find the value of d2 y/dx2 at the point defined by the given value of t. 203) x = 5 sin t, y = 5 cos t, t = 3 4
A)
2 5
203)
C) - 2 2
B) -2
5
D) 2 2
Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin.
63
204) r =
6 3 - 3 sin
204)
A)
B)
C)
D)
Solve the problem.
205) The parabola y2 = -8x is shifted down 4 units and left 2 units. Find an equation for the new parabola and find the new vertex. A) (y - 4) 2 = -8(x - 2); vertex; (-2, -4)
B) (y - 2) 2 = -8(x - 4); vertex; (-4, -2) D) (y + 2) 2 = -8(x + 4); vertex; (-4, -2)
C) (y + 4) 2 = -8(x + 2); vertex; (-2, -4) 64
205)
Find the value of d2 y/dx2 at the point defined by the given value of t. 206) x = 7t2 - 4, y = t5 , t = 1
A) - 15 14
B) - 15
C) 15
196
14
Plot the point whose polar coordinates are given. 207) (4, 9 /4)
206) D) 15
196
207)
A)
B)
65
C)
D)
Find the Cartesian coordinates of the given point. 208) (-4, - /3) A) (-2, 2 3) B) (-2 3, -2)
208) C) (2, 2 3)
Replace the polar equation with an equivalent Cartesian equation. 209) r cos = 6 A) 6x = 1 B) x = 6 C) y = 6
D) (-2 3, 2)
209) D) 6y = 1
Find the standard-form equation for an ellipse which satisfies the given conditions. 210) An ellipse centered at the origin having focus ( 15, 0) and directrix x = 16 15 2
2
16
4
A) x + y = 1
2
2
B) x + y2 = 1
C) x + y2 = 1
4
16
66
210) 2
D) x2 + y = 1 16
The polar equation of a circle is given. Give polar coordinates for the center of the circle and identify its radius. 211) r = 6 sin 211)
A) C 3,
2
, radius = 3
B) C(3, 0), radius = 3 D) C 3,
C) C(3, ), radius = 3 Find the slope of the polar curve at the indicated point. 212) r = 4 - 5 sin , = 0 A) 5 B) 4 4 5
2
, radius = 3
212) C) - 4 5
D) - 5 4
Describe the graph of the polar equation. 213) r2 + 2r2 sin cos = 575
213)
A) Two straight lines of slope -1, y-intercepts (0, ±24) B) Parabola opening to the right with vertex at (24, 0) C) Two horizontal lines passing through (0, ±24) D) Parabola opening upward with vertex at (0, 24) Find the standard-form equation of the ellipse centered at the origin and satisfying the given conditions. 214) An ellipse with vertices (0, ±9) and foci at (0, ±2 3). 2 2 2 2 2 2 2 2 A) x + y = 1 B) x + y = 1 C) x + y = 1 D) x + y = 1 69 81 69 93 12 81 81 69 Find the area of the specified region. 215) Inside the circle r = 3 sin A) 3 3 8 Find the polar coordinates, 0 216) 0, 1 4
A) - 1 , 0 4
and outside the cardioid r = 1 + cos B) 3 3 C) 3 4 4
215) D) 3 3 2
< 2 and r 0, of the point given in Cartesian coordinates.
216) B) 1 ,
C) 1 ,
4
4 2
D) 1 , 0 4
Find the value of d2 y/dx2 at the point defined by the given value of t. 217) x = tan t, y = 7 sec t, t = 3 4
A) - 7 2
214)
B)
2 4
217)
C) - 7 2 4
Replace the polar equation with an equivalent Cartesian equation. 218) r = -13 csc A) -13y = 1 B) y = -13 C) x = -13
67
D) 7 2
218) D) -13x = 1
Determine if the given polar coordinates represent the same point. 219) (10, /4), (10, 5 /4) A) Yes B) No Find the directrices of the hyperbola. 220) 25x2 - 144y2 = 3600
A) y = ± 144 13
219)
220) B) x = ± 144 13
C) x = ± 25 13
D) y = ± 25 13
Replace the Cartesian equation with an equivalent polar equation. 221) y = 5 A) r = 5 sin B) r = 5 sec C) r = 5 cos
222) xy = 1 A) 2r sin cos = 1 C) r2 sin 2 = 2 Solve the problem.
221) D) r = 5 csc 222)
B) 2r2 sin
cos
=1
D) r sin 2 = 2
2
2
144
81
223) The hyperbola x - y = 1 is shifted horizontally and vertically to obtain the hyperbola
223)
(x - 1)2 (y - 2)2 =1 144 81 Find the asymptotes of the new hyperbola. A) y - 2 = ± 3 (x - 1) 4
B) y - 2 = ± 9 (x - 1) 16
C) y - 2 = ± 4 (x - 1)
D) y = ± 3 x
3
4
Parametric equations and and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion. 224) x = 2t + 2, y = 6t + 15, 224) t
68
A) y = 3x - 9; Entire line, from right to left
B) y = 3x + 9; Entire line, from right to left
C) y = 3x + 9; Entire line, from left to right
69
D) y = 3x - 9; Entire line, from left to right
Find the area.
225) Find the area of the region between the curve x = e13t , y = 1 e-12t and the x-axis, 0 t ln 9. 13
A) 12
B) 9
C) 8
225)
D) 13
Describe the graph of the polar equation. 226) 9r cos + r sin = 3 A) Parabola with vertex (9, 3) opening upward
226)
B) Line with slope 3 and y-intercept (0, 9) C) Line with slope -9 and y-intercept (0, 3) D) Vertical line passing through (9, 0) Find the vertices and foci of the ellipse. 227) 81x2 + 25y2 = 2025
227)
A) Vertices: (±5, 0); Foci: (±2 14, 0) C) Vertices: (0, ±5); Foci: (0, ±2 14)
B) Vertices: (0, ±9); Foci: (0, ±2 14) D) Vertices: (±9, 0); Foci: (±2 14, 0)
Find the length of the curve. 228) x = 3t, y = 2 t 3/2, 0 t 16 3
A) 49
228) B) 98
C) 244
3
3
D) 196 3
Parametric equations and and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion.
70
229) x = 4 sin t, y = 3 cos t, 0 t 2
2
2
9
16
229)
A) x + y = 1; Counterclockwise from (3, 0) to (3, 0), one rotation
2
2
9
16
B) x + y = 1; Counterclockwise from (0, 4) to (0, 4), one rotation
71
2
2
16
9
C) x + y = 1; Counterclockwise from (0, 3) to (0, 3), one rotation
2
2
16
9
D) x + y = 1; Counterclockwise from (4, 0) to (4, 0), one rotation
Graph.
230) 25x2 - 4y2 = 100
230)
72
A)
B)
C)
D)
If the equation represents a hyperbola, find the center, foci, and asymptotes. If the equation represents an ellipse, find the center, vertices, and foci. If the equation represents a circle, find the center and radius. If the equation represents a parabola, find the focus and directrix. 231) y2 - 4x - 10y + 17 = 0 231)
A) F: (-1, 5); D: x = -3
B) F: (2, -2); D: y = 6
C) F: (-3, -2); D: y = -1
D) F: - 7 , 5 ; D: y = - 9 4
4
Find the area of the specified region.
232) Inside the six-leaved rose r2 = 5 cos 3 A) 5
232)
B) 5
C) 10
3
233) Inside the lemniscate r2 = a 2 sin 2 , a > 0 A) 0
D) 5 2
233)
B) a2
C)
73
a2 2
D) a 2
Find the eccentricity of the hyperbola. 234) 25x2 - 144y2 = 3600
A)
119 12
234) B)
119 5
C) 13 5
D) 13 12
Find the eccentricity of the ellipse. 235) 81x2 + 64y2 = 5184
A)
145 9
235) B) 17 9
C)
Graph the parabola. 236) y = -5x2
17 9
D)
17 8
236)
A)
B)
C)
D)
74
Replace the Cartesian equation with an equivalent polar equation. 237) (x - 23)2 + (y + 3)2 = 529
A) r2 = -23r cos + 3r sin + 529 C) r2 = 46r cos - 6r sin - 9
237)
B) r = 46 cos - 6 sin - 9 D) r2 = 46r sin - 6r cos - 9
Graph the polar equation. 238) r = 3(2 + 2 sin )
238)
A)
B)
C)
D)
75
Find the polar coordinates, 0 239) (0, 6)
A)
6,
< 2 and r 0, of the point given in Cartesian coordinates.
2
239) B) 6,
C) 6,
2
D)
2
6,
2
Find the foci of the hyperbola. 240) 25x2 - y2 = 25
240)
A) (0, 26), (0, - 26) C) ( 26, 0), (- 26, 0)
B) (5, 0), (-5, 0) D) (0, 5), (0, -5)
If the equation represents a hyperbola, find the center, foci, and asymptotes. If the equation represents an ellipse, find the center, vertices, and foci. If the equation represents a circle, find the center and radius. If the equation represents a parabola, find the focus and directrix. 241) x2 - 4x + 4y - 16 = 0 241)
A) F: (2, 1); D: y = -9
B) F: 2, 19 ; D: y = 21
C) F: (2, 4); D: y = 6
D) F: (2, 6); D: y = 4
4
4
Find a Cartesian equation for the line whose polar equation is given. 242) r cos - 3 = 2 2 4
A) y = x + 4
B) y = 2x - 4
C) y = x + 2 2
Graph the pair of parametric equations with the aid of a graphing calculator. 243) x = 5 cos 2t + 4 cos 6t, y = 5 sin 2t - 4 sin 6t, 0 t
76
242) D) y = x - 2
243)
A)
B)
C)
D)
Graph the set of points whose polar coordinates satisfy the given equation or inequality.
244) 0
4
,0 r
4
244)
77
A)
B)
C)
D)
Find the area of the specified region. 245) Inside the lemniscate r2 = 4 sin 2 and outside the circle r = 2 A) B) 2 (3 3 - + 6) C) 2 (3 3 - ) 3 3 3
245) D) 1 (2 + 3 3) 2
Find the vertices and foci of the ellipse. 246) 9x2 + 64y2 = 576
246)
A) Vertices: (±3, 0); Foci: (± 55, 0) C) Vertices: (0, ±8); Foci (0, ± 55)
B) Vertices: (±8, 0); Foci: (± 55, 0) D) Vertices: (0, ±3); Foci (0, ± 55)
Find the length of the curve. 247) x = 1 t2, y = 1 (2t + 1) 3/2, 0 t 1 2 3
A) 2
247)
B) 3
C) 1
2
Determine the symmetries of the curve. 248) r2 = 3 cos 5
D) 3
248)
A) y-axis only C) Origin only
B) x-axis only D) x-axis, y-axis, origin
78
Find the vertices and foci of the ellipse. 2 2 249) x + y = 1 144 225
249)
A) Vertices: (±15, 0); Foci: ( ±12, 0) C) Vertices: (0, ±15); Foci: (0, ±12) Solve the problem.
2
2
36
4
B) Vertices: (±12, 0); Foci: ( ±9, 0) D) Vertices: (0, ±15); Foci: (0, ±9)
250) The ellipse x + y = 1 is shifted horizontally and vertically to obtain the ellipse (x + 3)2 (y + 3)2 + = 1. Graph the new ellipse. 36 4
A)
B)
C)
D)
79
250)
Determine the symmetries of the curve. 251) r = 1 - 5 cos A) No symmetry B) Origin only
251) C) x-axis only
D) y-axis only
Find the value of d2 y/dx2 at the point defined by the given value of t.
252) x = sin t, y = 8 sin t, t =
252)
3
A) 0
B) 16
Solve the problem.
2
2
5
16
C) -8
D) -16
253) The hyperbola x - y = 1 is shifted down 3 units and right 4 units. Find an equation for the new hyperbola and find the new center. 2 2 A) (x + 3) - (y - 4) = 1; center: (3, -4) 5 11
2
2
2
B) (x + 4) - (y - 3) = 1; center: (4, -3) 5
2
11
2
C) (x - 4) - (y + 3) = 1; center: (4, -3) 5
253)
2
D) (x - 3) + (y + 4) = 1; center: (-3, 4)
16
5
11
Find the standard-form equation for an ellipse which satisfies the given conditions.
254) An ellipse centered at the origin having focus at (6, 0) and eccentricity equal to 1
254)
2
2
2
144
108
A) y + x = 1
2
2
36
27
B) x + y = 1
2
2
144
108
C) x + y = 1
2
D) x + y2 = 1 36
Graph.
255) y2 - x2 = 1
255)
16
80
A)
B)
C)
D)
Assuming that the equations define x and y implicitly as differentiable functions x = f(t), y = g(t), find the slope of the curve x = f(t), y = g(t) at the given value of t.
256) x cos t + x = 2t, y = t sin t + t, t = A) 2
B)
256)
2 1 1+
C) 2 +
D)
2 2+
Find a polar equation in the form r cos( - 0 ) = r0 for the given line.
257) 3x + y = 6
257)
A) r cos
-
C) r cos
-
6 3
=3
B) r cos
+
=3
D) r cos
+
81
6 2 3
=3 =3
Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. 5 258) r = 2 + sin
A)
B)
C)
D)
258)
Solve the problem.
259) Find the volume of the solid generated by revolving the region enclosed by the ellipse 4x2 + 25y2 = 100 about the x-axis.
A) 4 2
B) 40
C) 160
3
3
82
D) 80
3
259)
If the equation represents a hyperbola, find the center, foci, and asymptotes. If the equation represents an ellipse, find the center, vertices, and foci. If the equation represents a circle, find the center and radius. If the equation represents a parabola, find the focus and directrix. 260) 9x2 + 4y2 + 108x - 64y + 544 = 0 260)
A) C: (-6, 8); V: (-3, 8), (-9, 8); F: (-3.8, 8), (-8.2, 8) B) C: (-6, 8); V: (-4, 8), (-8, 8); F: (-2.4, 8), (-9.6, 8) C) C: (-6, 8); V: (-6, 11), (-6, 5); F: (-6, 10.2), (-6, 5.8) D) C: (-6,8); V: (-6, 10), (-6, 6); F: (-6, 11.6), (-6, 4.4) Graph the set of points whose polar coordinates satisfy the given equation or inequality. 3 261) , r = -4 2
A)
B)
C)
D)
83
261)
Find the eccentricity of the hyperbola. 262) 81x2 - 64y2 = 5184
A)
17 8
262) B)
17 9
C)
145 9
D)
145 8
Provide an appropriate response. 263) Which of the statements below is true? A: If (r1 , 1 ) and (r2 , 2 ) represent the same point in polar coordinates, then r1 = r2 .
263)
B: If (r1 , 1) and (r2 , 2 ) represent the same point in polar coordinates, then 1 = 2 + 2 n for some integer n. A) A only B) Neither A nor B
C) Both A and B
D) B only
Find the eccentricity of the hyperbola. 264) 81y2 - 64x2 = 5184
A)
145 9
264) B)
145 8
Find the Cartesian coordinates of the given point. 265) 3, 5 /6 A) -3 3 , 3 B) -3 3 , -3 2 2 2 2
C)
17 9
C)
-3 3 , 2 2
D)
17 8
D)
3 -3 3 , 2 2
265)
Graph.
266) x2 - y2 = 50
266)
84
A)
B)
C)
D)
Find the slope of the polar curve at the indicated point.
267) r = 6(1 + cos ), = A)
267)
3
3 3
C) 3 2
B) 0
D) -2
Find an equation for the line tangent to the curve at the point defined by the given value of t. 268) x = t, y = 2t, t = 2 A) y = - 1 x - 1 B) y = 1 x C) y = - 1 x D) y = 1 x + 1 2 2 2 2 Solve the problem.
2
2
144
81
269) The hyperbola x - y = 1 is shifted horizontally and vertically to obtain the hyperbola (x + 2)2 (y + 2)2 =1 144 81 Find the foci of the new hyperbola. A) (-2, -17), (-2, 13)
B) (-11, -2), (7, -2) D) (-2, -11), (-2, 7)
C) (-17, -2), (13, -2) Graph.
85
268)
269)
270) 16x2 + 36y2 = 576
270)
A)
B)
86
C)
D)
Find the eccentricity of the hyperbola. 271) x2 - 2y2 = 6
A)
2 2
Solve the problem.
271) B) 2
2
2
400
625
C) 6
D)
272) The ellipse x + y = 1 is shifted horizontally and vertically to obtain the ellipse (x + 2)2 (y + 2)2 + = 1. 400 625 Find the foci of the new ellipse. A) (-17, -2), (13, -2)
B) (-2, -17), (-2, 13) D) (-22, -2), (18, -2)
C) (-2, -22), (-2, 18) Graph the parabola.
87
6 2
272)
273) x2 = 4y
273)
A)
B)
C)
D)
Find the area of the specified region. 274) Inside one leaf of the four-leaved rose r = 3 sin 2 A) 9 B) 9 - 9 8 8 16
274) C) 9
2
D) 3
2
Find a Cartesian equation for the line whose polar equation is given. 275) r cos - 5 = 5 6
A) y = 3x - 5
B) y = 2x - 5
C) y = x - 10
88
275) D) y = 3x + 10
Find the polar coordinates, 0 276) (0, -3)
A) 3,
2
< 2 and r 0, of the point given in Cartesian coordinates.
276) C) 3,
B) (3, 0)
2
Graph the pair of parametric equations with the aid of a graphing calculator. 277) x = 5 tan t, y = 4 sec t; - /2 t /2
A)
B)
C)
D)
89
D) (3, )
277)
Solve the problem.
278) The parabola x2 = -32y is shifted 4 units up and 6 units to the right. Find the focus and directrix of the new parabola. A) Focus: (6, 12); Directrix: y = 12
B) Focus: (-4, 6); Directrix: x = 12 D) Focus: (6, -4); Directrix: y = -12
C) Focus: (6, -4); Directrix: y = 12 Find the focus and directrix of the parabola. 279) - 1 x2 = y 36
A) (0, -9); y = -9
279)
B) (-18, 0); x = 9
C) (0, 9); y = -9
280) x = 10y2
D) (0, -9); y = 9 280)
A) 0, 1 , y = - 1
B) 1 , 0 , x = - 1
C) 1 , 0 , x = - 1
D) 0, 1 , x = 1
40
40
40
10
40
10
40
40
Solve the problem. 281) Find the volume generated by revolving about the y-axis the area bounded by the curves 2x2 - y2 = 8, x = 5
A) 28
278)
21
B) 28
42
C) 28
Sketch the region which is defined by the given conditions. 3 282) , r = -4 2
14
D) 14
281)
42
282)
90
A)
B)
C)
D)
Replace the Cartesian equation with an equivalent polar equation. 283) x2 - y2 = 4
A) r2 cos 2 = 4 C) cos2 - sin2 = 4r
283)
B) r cos 2 = 4 D) cos2 - sin2 = 4
Graph.
284) 25x2 + 9y2 = 225
284)
91
A)
B)
C)
92
D)
Find the eccentricity of the ellipse. 285) x2 + 144y2 = 144
A)
143 12
285) B)
Find the directrices of the ellipse. 286) x2 + 16y2 = 16
145 12
C) 145
D) 143
C) x = ± 16 15
D) x = ± 16 17
286)
A) y = ± 4 15
B) y = ± 16 15
Find the coordinates of the centroid of the curve.
287) Find the coordinates of the centroid of the curve x = e2t cos 2t, y = e2t sin 2t, 0 t A) (x, y) =
e -2
,
2e + 1
5(e - 1) 5(e - 1)
C) (x, y) = 2e - 1 , e + 1
e /2 - 1 e /2 - 1
Find the focus and directrix of the parabola. 288) y = -6x2
e +2 2e - 1 , 5(e /2 - 1) 5(e /2 - 1)
D) (x, y) =
e -2 2e + 1 , /2 5(e - 1) 5(e /2 - 1)
287)
288) B) 0, - 1 , y = - 1
C) - 1 , 0 , x = 1
D) 0, - 1 , x = 1
24
.
B) (x, y) =
A) 0, - 1 , y = 1 24
4
24
24
24
6
24
6
Find the foci of the hyperbola. 289) 25y2 - x2 = 25
289)
A) (5, 0), (-5, 0) C) (0, 26), (0, - 26)
B) (0, 5), (0, -5) D) ( 26, 0), (- 26, 0)
93
Determine the symmetries of the curve. 290) r = -2 + 4 sin A) No symmetry B) Origin only
290) C) x-axis only
D) y-axis only
If the equation represents a hyperbola, find the center, foci, and asymptotes. If the equation represents an ellipse, find the center, vertices, and foci. If the equation represents a circle, find the center and radius. If the equation represents a parabola, find the focus and directrix. 291) y2 - x2 - 12y + 8x - 29 = 0 291)
A) C: (-3, 6); F: (7 2, 0), (- 7 2, 0); A: y = x, y = -x B) C: (4, 6); F: (4, 6 + 7 2), (4, 6 - 7 2); A: y - 6 = x - 4, y - 6 = -(x - 4) C) C: (4, -1); F: (7 2, 0), (- 7 2, 0); A: y = 1 x, y = - 1 x 7
7
D) C: (4, 6); F: (0, 6 + 7 2), (0, 6 - 7 2); A: y - 6 = x - 4, y - 6 = -(x - 4) Find a polar equation for the circle. 292) x2 - 6x + y2 = 0
A) r = 6 sin
292) B) r = -6 sin
C) r = 6 cos
D) r = -3 cos
Graph.
293) 9y2 - 36x2 = 324
293)
A)
B)
94
C)
D)
Find the eccentricity of the ellipse. 294) 49x2 + 100y2 = 4900
A)
51 7
294) B) 51
C)
10
149 10
D)
51 10
The eccentricity is given of a conic section with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for the conic section. 295) e = 5, y = -8 295) 40 40 40 40 A) r = B) r = C) r = D) r = 1 + 8 cos 1 - 5 cos 1 - 5 sin 1 + 5 sin
Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. 296) Foci at ( 17, 0), (- 17, 0); asymptotes y = 4x, y = -4x 296) 2 2 2 2 A) x - y2 = 1 B) x2 - y = 1 C) y - x2 = 1 D) y2 - x = 1 16 16 16 16 Solve the problem.
2
2
9
36
297) The hyperbola y - x = 1 is shifted horizontally and vertically to obtain the hyperbola (y - 1)2 (x + 4)2 = 1. Graph the new hyperbola. 9 36
95
297)
A)
B)
C)
D)
If the equation represents a hyperbola, find the center, foci, and asymptotes. If the equation represents an ellipse, find the center, vertices, and foci. If the equation represents a circle, find the center and radius. If the equation represents a parabola, find the focus and directrix. 298) 6x2 - y2 - 36x + 6y - 249 = 0 298)
A) C: (3, 3); F: (3 + 7 7, 3), (3 - 7 7, 3); A: y - 3 = 6(x - 3), y - 3 = - 6(x - 3) B) C: (3, 3); F: (7 7, 0), (- 7 7, 0); A: y - 3 = 6(x - 3), y - 3 = - 6(x - 3) C) C: (-4, 3); F: (7 7, 0), (-7 7, 0); A: y = 6x, y = -6x D) C: (3, 3); F: (3 + 7 7, 3), (3 - 7 7, 3); A: y - 3 = 6(x - 3), y - 3 = -6(x - 3) Determine if the given polar coordinates represent the same point. 299) (3, -4 /3), (-3, 5 /3) A) Yes B) No Find the directrices of the ellipse. 300) 64x2 + 4y2 = 256
A) y = ± 32 17
299)
300) B) x = ± 4 15
C) y = ± 32 15
96
D) x = ± 32 15
Find the slope of the polar curve at the indicated point.
301) r = 7 csc , =
301)
2
B) Undefined
A) -1 302) r = 4 cos 3 , = A) -
C) 0
D) 1 302)
6
3 3
B) - 3
Find the directrices of the hyperbola. 303) y2 - 144x2 = 144
A) y = ± 12
143
C)
3 3
D) 3
303) B) y = ± 144
C) x = ± 144
145
145
D) y = ± 144
143
If the equation represents a hyperbola, find the center, foci, and asymptotes. If the equation represents an ellipse, find the center, vertices, and foci. If the equation represents a circle, find the center and radius. If the equation represents a parabola, find the focus and directrix. 304) x2 + y2 + 18x - 12y + 117 = 25 304)
A) (9, -6); r = 25
B) (-6, 9); r = 25
C) (-9, 6); r = 5
D) (6, -9); r = 5
Determine if the given polar coordinates represent the same point. 305) (9, /4), (-9, /4) A) No B) Yes Find the polar coordinates, 0 306) (4, 4) A) 4 2, 3 4 Solve the problem.
305)
< 2 and r 0, of the point given in Cartesian coordinates.
306) B) 4 2,
2
2
7
3
C) 4,
4
2
D) 4,
4
307) The ellipse x + y = 1 is shifted up 5 units and left 2 units. Find an equation for the new ellipse and find the new vertices. 2 2 A) (x - 2) + (y + 5) = 1; vertices: (-2 ± 7 3
2
7, 5)
2
B) (x + 2) + (y - 5) = 1; vertices: (-2 , 5 ± 3) 7
3
2
2
C) (x + 2) + (y - 5) = 1; vertices: (-2 ± 7, 5) 7
3
2
2
D) (x + 5) + (y - 2) = 1; vertices: (-2 , 5 ± 3) 7
3
97
307)
Find the slope of the polar curve at the indicated point.
308) r = 3 + 2 cos , = A) - 3 2
308)
2
B) 3
C) 2
2
3
D) - 2 3
The eccentricity is given of a conic section with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for the conic section. 309) e = 2, x = 5 309) 10 10 10 10 A) r = B) r = C) r = D) r = 1 + 5 cos 1 + 2 cos 1 + 2 sin 1 - 2 cos
Graph.
2
2
9
25
310) y - x = 1
310)
A)
B)
98
C)
D)
Replace the Cartesian equation with an equivalent polar equation. 311) x2 + 4y2 = 4
A) r2 (cos2 + 4 sin2 ) = 4 C) 4 cos2 + sin2 = 4r
311)
B) r2 (4 cos2 + sin2 ) = 4 D) cos2 + 4 sin2 = 4r
Find the coordinates of the centroid of the curve. 312) Find the coordinates of the centroid of the curve x = 4 cos t + 4t sin t, y = 4 sin t - 4t cos t, 0 t A) (x, y) = 48 - 96 , 96 - 8 B) (x, y) = 8 - 48 , 24 2 2 2
C) (x, y) = 8 - 96 , 12
2
2
100
36
312)
D) (x, y) = 8 - 48 , 24
2
Solve the problem.
.
2
313) The ellipse x + y = 1 is shifted horizontally and vertically to obtain the ellipse
313)
(x + 2)2 (y - 1)2 + = 1. 100 36 Find the foci of the new ellipse. A) (-10, 1), (6, 1) B) (1, -8), (1, 4)
C) (-8, 1), (4, 1)
D) (1, -10), (1, 6)
314) The parabola y2 = 6x is shifted up 3 units and left 3 units. Find an equation for the new parabola and find the new vertex. A) (y + 3) 2 = 6(x - 3); vertex; (-3, 3)
314)
B) (y - 3) 2 = 6(x + 3); vertex; (3, -3) D) (y + 3) 2 = 6(x - 3); vertex; (3, -3)
C) (y - 3) 2 = 6(x + 3); vertex; (-3, 3)
Parametric equations and and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion.
99
315) x = 4 cos t, y = 4 sin t,
t 2
315)
A) x2 + y2 = 16; Counterclockwise from (4, 0) to (4, 0), one rotation
B) x2 + y2 = 1; Clockwise from (1, 0) to (1, 0), one rotation
100
C) x2 + y2 = 16; Counterclockwise from (-4, 0) to (4, 0)
D) x2 + y2 = 16; Clockwise from (4, 0) to (-4, 0)
Find the focus and directrix of the parabola. 316) y2 = 16x
A) (4, 0); x = 4
316)
B) (4, 4); x = 4
C) (0, 4); y = -4
Determine if the given polar coordinates represent the same point. 317) (r, ), (-r, + ) A) No B) Yes
101
D) (4, 0); x = -4
317)
Solve the problem.
2
2
625
400
318) The ellipse x + y = 1 is shifted horizontally and vertically to obtain the ellipse
318)
(x + 3)2 (y - 3)2 + = 1. 625 400 Find the center and vertices of the new ellipse. A) Center: (-3, 3); Vertices: (3, -28), (3, 22)
B) Center: (-3, 3); Vertices: (-28, 3), (22, 3) D) Center: (3, -3); Vertices: (-28, 3), (22, 3)
C) Center: (3, -3); Vertices: (3, -28), (3, 22)
If the equation represents a hyperbola, find the center, foci, and asymptotes. If the equation represents an ellipse, find the center, vertices, and foci. If the equation represents a circle, find the center and radius. If the equation represents a parabola, find the focus and directrix. 319) 16x2 + 25y2 - 256x + 100y + 724 = 0 319)
A) C: (8,-2); V: (8, 2), (8, -6); F: (8, 4.4), (8, -8.4) B) C: (8, -2); V: (13, -2), (3, -2); F: (11.0, -2), (5.0, -2) C) C: (8, -2); V: (12, -2), (4, -2); F: (14.4, -2), (1.6, -2) D) C: (8, -2); V: (8, 3), (8, -7); F: (8, 1.0), (8, -5.0) Find the focus and directrix of the parabola. 320) y = 2x2
A) 0, 1 , y = - 1 8 8
B)
320)
1 1 ,0 ,x=2 2
C) 0, 1 , y = 1 8 8
102
D)
1 1 ,0 ,x=8 8
Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. 16 321) r = 8 + 4 cos
A)
B)
C)
D)
Find the area of the surface generated by revolving the curves about the indicated axis. 2 322) x = t + 6, y = t + 6t, - 6 t 6; y-axis 2
A) 248 3
B) 124
C) 248
3
5
103
321)
322) D) 124
Find the standard-form equation for an ellipse which satisfies the given conditions.
323) An ellipse centered at the origin having focus (0, -2 3) and directrix y = - 8
323)
3
2
2
8
16
A) x + y = 1
2
2
4
16
B) x + y = 1
2
2
2
4
C) x + y = 1
2
2
16
4
D) x + y = 1
Graph.
324) x2 - y2 = 32
324)
A)
B)
C)
D)
104
Find the length of the curve. 325) The spiral r = 9 , 0
325) B) 729
A) 24(( 2 + 1)3/2 - 4) 2 + 1 + ln 1 +
C) 9 Solve the problem.
2
2
7
3
4
2+1
D) 9 ( 2
2 + 1 + ln ( +
2 + 1))
326) The hyperbola x - y = 1 is shifted up 3 units and left 8 units. Find an equation for the new hyperbola and find the new vertices. 2 2 A) (x + 8) - (y - 3) = 1; vertices: (-8 ± 7 3
2
326)
7, 3)
2
B) (x - 8) - (y + 3) = 1; vertices: (-8 ± 7, 3) 7
3
2
2
C) (x + 3) - (y - 8) = 1; vertices: (-8 , 3 ± 3) 7
3
2
2
D) (x + 8) - (y - 3) = 1; vertices: (-8 , 3 ± 3) 7
3
Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. 327) Asymptotes y = 9 x, y = - 9 x; one vertex is (0, 18) 327) 4 4 2
2
324
16
A) y - x = 1
2
2
324
64
B) y - x = 1
2
2
81
64
C) y - x = 1
2
2
324
64
D) x - y = 1
Determine if the given polar coordinates represent the same point. 328) (5, /6), (5, 7 /6) A) No B) Yes
328)
Find the eccentricity of the ellipse. 329) Find the eccentricity of an ellipse centered at the origin having a focus of (- 39, 0) and 64 corresponding directrix x = 39
A)
39 5
B)
39 64
C)
39 8
D)
89 8
Find the foci of the ellipse. 330) 225x2 + 64y2 = 14,400
A) (± 161, 0)
329)
330) B) (0, ±15)
C) (0, ±8)
105
D) (0, ± 161)
Find the polar coordinates, 0 331) 1 , - 3 5 5
< 2 and r 0, of the point given in Cartesian coordinates.
331)
A) 2 , 4 5
B) 2 , 5
3
5
C) 2 , 7
3
5
D) 2 , 11
6
5
6
Find the slope of the polar curve at the indicated point.
332) r = 5 cos - 6 sin , =
332)
2
A) - 6
B) 5
5
C) - 5
6
D) 6
6
5
Find the Cartesian coordinates of the given point. 333) 6, 1 2
A) (0, 6)
333)
B) (0, -6)
C) (6, 0)
D) (-6, 0)
334) 20, 11
334)
3
A) (-10 3, -10)
B) (10 3, -10)
C) (-10, -10 3)
D) (10, -10 3)
Solve the problem. 335) Find the vertices and asymptotes of the following hyperbola: y2 x2 =1 64 324
335)
A) Vertices: (0, 8), (0, -8); Asymptotes: y = ± 4 x 9
B) Vertices: (8, 0), (-8, 0); Asymptotes: y = ± 4 x 9
C) Vertices: (0, 8), (0, -8); Asymptotes: y = ± 8x D) Vertices: (0, 64), (0, -64); Asymptotes: y = ± 1 x 8
2
2
5
16
336) The ellipse x + y = 1 is shifted down 4 units and right 8 units. Find an equation for the new ellipse and find the new center. 2 2 A) (x - 4) + (y + 8) = 1; center: (-4, 8) 5 11
2
2
5
2
11
2
C) (x + 8) + (y - 4) = 1; center: (8, -4) 5
2
B) (x + 4) + (y - 8) = 1; center: (4, -8) 2
D) (x - 8) + (y + 4) = 1; center: (8, -4)
11
5
Graph the pair of parametric equations with the aid of a graphing calculator.
106
16
336)
337) x = 7t - 7 sin t, y = 7 - 7 cos t, -2
t 2
337)
A)
B)
C)
D)
The eccentricity is given of a conic section with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for the conic section. 338) e = 1 , y = -2 338) 4
A) r =
2 4 - sin
B) r =
2 1 + 4 cos
C) r =
107
2 4 + sin
D) r =
2 4 - cos
Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. 339) Foci at (0, -25), (0, 25); asymptotes: y = 3 x, y = - 3 x 339) 4 4 2
2
225
400
A) y - x = 1
2
2
400
225
B) y - x = 1
2
2
400
225
C) x - y = 1
2
2
225
400
D) x - y = 1
Find the slope of the polar curve at the indicated point.
340) r = 9, =
340)
4
A) -1
B) -
3 3
C) - 3
D) 1
Solve the problem. 341) Find the vertices and foci of the following hyperbola: x2 y2 =1 16 65
341)
A) Vertices: (0, 9), (0, -9); Foci: (0, 4), (0, -4) B) Vertices: (16, 0), (-16, 0); Foci: (9, 0), (-9, 0) C) Vertices: (4, 0), (-4, 0); Foci: (9, 0), (-9, 0) D) Vertices: (0, 4), (0, -4); Foci: (0, 9), (0, -9) Determine the symmetries of the curve. 342) r = 6 , > 0
A) No symmetry
342)
B) y-axis only
C) x-axis only
D) Origin only
Graph.
343) 36x2 + 12y2 = 192
343)
108
A)
B)
C)
D)
Solve the problem. 344) Find the foci and asymptotes of the following hyperbola: 9x2 - y2 = 9
344)
A) Foci: ( 10, 0), (- 10, 0); Asymptotes: y = 1 x, y = - 1 x 3
3
B) Foci: (3, 0), (-3, 0); Asymptotes: y = 1 x, y = - 1 x 3
3
C) Foci: (0, 10), (0, - 10); Asymptotes: y = 3x, y = -3x D) Foci: ( 10, 0), (- 10, 0); Asymptotes: y = 3x, y = -3x Replace the polar equation with an equivalent Cartesian equation. 345) r2 sin 2 = 20
A) y = 10x
B) y = 10 x
C) y = 20x2
Find the area of the specified region. 346) Shared by the circle r = 8 and the cardioid r = 8(1 + sin ) A) 16(5 + 8) B) 64 C) 0 Graph the parabola.
109
345) D) y = 20 x
346) D) 16(5 - 8)
347) x = 3y2
347)
A)
B)
C)
D)
Find a polar equation for the circle. 348) x2 + (y + 8)2 = 64
A) r = 8 cos
348) B) r = -16 sin
C) r = 8 sin
Plot the point whose polar coordinates are given.
110
D) r = -16 cos
349) (-2, 0)
349)
A)
B)
111
C)
D)
Find the vertices and foci of the ellipse. 2 2 350) x + y = 1 625 225
350)
A) Vertices: (±25, 0); Foci: (±15, 0) C) Vertices: (±25, 0); Foci: (±20, 0)
B) Vertices: (0, ±25); Foci: (0, ±15) D) Vertices: (0, ±25); Foci: (0, ±20)
Find the standard-form equation of the ellipse centered at the origin and satisfying the given conditions. 351) An ellipse with vertices (0, ±9) and foci (0, ±5) 2 2 2 2 2 2 2 2 A) y + x = 1 B) x + y = 1 C) y + x = 1 D) x + y = 1 56 81 25 81 25 81 56 81 Find the length of the curve. 352) x = 2 sin t + 2t, y = 2cos t, 0 t A) 2 B) 12
352) C) 8
D) 4
Find a polar equation for the circle. 353) x2 + y2 - 18y = 0
A) r = -9 sin
351)
353) B) r = -18 sin
C) r = 18 sin 112
D) r = -18 cos
Find the area. 354) Find the area under one arch of the cycloid x = 6(t - sin t), y = 6(1 - cos t). A) 108 2 B) 72 C) 108
354) D) 18
Find an equation for the line tangent to the curve at the point defined by the given value of t.
355) x = t + cos t, y = 2 - sin t, t = A) y = 3x -
3 6
C) y = - 3x +
3 6
355)
6
B) y = - 2x -
2 4
+2
D) y = - 3x + 3
+3
2
Find the eccentricity of the ellipse. 356) 64x2 + 4y2 = 256
A) Graph.
15 8
2
2
25
49
356) B)
15 4
C) 15
357) x + y = 1
D)
17 4
357)
A)
B)
113
C)
D)
Find the focus and directrix of the parabola. 358) y2 - 32x = 0
A) (8, 0); x = -8
358)
B) (0, 8); y = -8
C) (8, 0); y = -8
D) (8, 0); x = 8
The eccentricity is given of a conic section with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for the conic section. 359) e = 1 , x = 8 359) 5
A) r =
8 5 - cos
B) r =
8 5 + cos
C) r =
8 1 + 5 cos
D) r =
8 5 + sin
Graph.
360) 7x2 + 4y2 = 56
360)
114
A)
B)
C)
D)
Graph the pair of parametric equations with the aid of a graphing calculator. 361) x = 4(t - sin t), y= 4(1 - cos t), 0 t 4
115
361)
A)
B)
C)
D)
Find the area of the specified region. 362) Inside the smaller loop of the limacon r = 3 + 6 sin 2 A) B) 3(3 2 - ) 3
362) C) 9
Determine if the given polar coordinates represent the same point. 363) (4, /3), (-4, -2 /3) A) Yes B) No
D) 9 (2 - 3 3) 2
363)
Parametric equations and and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion.
116
364) x = 4t2, y = 2t,
t
364)
A) x = y2; Entire parabola, bottom to top (from 4th quadrant to origin to 1st quadrant)
B) x = y2; Entire parabola, top to bottom (from 1st quadrant to origin to 4th quadrant)
117
C) y = x2; Entire parabola, right to left (from 1st quadrant to origin to 2nd quadrant)
D) y = x2; Entire parabola, left to right (from 2nd quadrant to origin to 1st quadrant)
The eccentricity is given of a conic section with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for the conic section. 365) e = 5, x = -2 365) 10 10 10 10 A) r = B) r = C) r = D) r = 1 - 5 cos 1 + 5 cos 1 + 2 cos 1 - 5 sin Assuming that the equations define x and y implicitly as differentiable functions x = f(t), y = g(t), find the slope of the curve x = f(t), y = g(t) at the given value of t.
366) x = t sin t + t, y cos t + y = 2t, t = A)
2 2+
366)
2
B) 2 +
C) -
118
2 2+
D) 2 +
2
Solve the problem. 367) Find the foci and asymptotes of the following hyperbola: 4y2 - x2 = 4
367)
A) Foci: (0, 5), (0, - 5); Asymptotes: y = 2x, y = -2x B) Foci: ( 5, 0), (- 5, 0); Asymptotes: y = 1 x, y = - 1 x 2
2
C) Foci: (0, 5), (0, - 5); Asymptotes: y = 1 x, y = - 1 x 2
2
D) Foci: (0, 5), (0, -5); Asymptotes: y = 2x, y = -2x Graph.
2
2
64
36
368) x - y = 1
368)
A)
B)
C)
D)
119
The eccentricity is given of a conic section with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for the conic section. 369) e = 1 , x = -5 369) 4
A) r =
5 1 + 4 cos
Find the polar coordinates, 0 370) (3, -3) A) 3 2, 5 4
B) r =
5 4 + cos
C) r =
5 4 - sin
D) r =
5 4 - cos
< 2 and r 0, of the point given in Cartesian coordinates.
370) B) 3 2,
C) 3 2, 7
4
D) 3 2,
4
4
Find a polar equation for the circle. 371) (x - 5)2 + y2 = 25
A) r = 10 sin
371) B) r = - 10 cos
C) r = 5 cos
D) r = 10 cos
Find the standard-form equation of the ellipse centered at the origin and satisfying the given conditions. 372) An ellipse with intercepts (±3, 0) and (0, ±8) 2 2 2 2 2 2 2 2 A) x + y = 1 B) x + y = 1 C) x + y = 1 D) x + y = 1 3 8 8 3 9 64 64 9 Find the value of d2 y/dx2 at the point defined by the given value of t. 373) x = t, y = 2t, t = 3 A) - 1 B) - 1 C) 1 6 6 6 6
372)
373) D) - 1 6
Find a polar equation in the form r cos( - 0 ) = r0 for the given line.
374) x - 3y = 6
374)
A) r cos
+
C) r cos
+
3 2 3
=3
B) r cos
-
=6
D) r cos
+
Graph the pair of parametric equations with the aid of a graphing calculator.
120
6 6
=6 =3
375) x = 6 cos t + 2 cos 3t, y = 6 sin t - 2 sin3t, 0 t 2
375)
A)
B)
C)
D)
Find a Cartesian equation for the line whose polar equation is given.
376) r cos -
6
=2
A) y = 4 -
3 x 3
376) B) y = 2 - 2x
C) y = 3x + 2
121
D) y = 4 - 3x
Describe the graph of the polar equation. 377) r cos = 7 A) Line with slope 7 passing through the origin
377)
B) Circle centered at origin with radius 7 C) Vertical line through (7, 0) D) Horizontal line through (0, 7) Determine if the given polar coordinates represent the same point. 378) (8, /6), (-8, 7 /6) A) Yes B) No
378)
Solve the problem.
379) The parabola y2 = -24x is shifted 7 units down and 4 units to the right. Graph the new parabola.
A)
122
379)
B)
C)
D)
123
Find the standard-form equation of the ellipse centered at the origin and satisfying the given conditions. 380) An ellipse with vertices (±6, 0) and foci at (±4, 0) 2 2 2 2 2 2 2 2 A) x + y = 1 B) x + y = 1 C) x + y = 1 D) x + y = 1 20 20 36 20 20 36 20 52
380)
Find an equation for the line tangent to the curve at the point defined by the given value of t.
381) x = csc t, y = 6 cot t, t = A) y = 12x - 6 3 Solve the problem.
2
2
5
11
381)
3
B) y = 12x + 2 3
C) y = 2 3x - 12
D) y = -12x + 6 3
382) The ellipse x + y = 1 is shifted down 7 units and right 6 units. Find an equation for the new ellipse and find the new vertices. 2 2 A) (x - 6) + (y + 7) = 1; vertices: (6, -7 ± 5 11
2
382)
11 )
2
B) (x - 6) + (y + 7) = 1; vertices: (6 ± 5 , -7 ) 5
11
2
2
C) (x - 7) + (y + 6) = 1; vertices: (6 ± 5 , -7 ) 5
11
2
2
D) (x + 6) + (y - 7) = 1; vertices: (6, -7 ± 11 ) 5
11
Describe the graph of the polar equation. 6 383) r = 2 cos - sin
383)
A) Ellipse centered at origin with intercepts (±2, 0) and (0, ±1) B) Line with slope 2 and y-intercept (0, -6) C) Line with slope 6 and y-intercept (0, -2) D) Hyperbola passing through (6, 2) 384) r2sin 2 = 26 A) Parabola centered at the origin opening to the right B) Parabola centered at the origin opening upward C) Hyperbola with the x- and y-axes as asymptotes D) Hyperbola with asymptotes y = ±13x
384)
Find the slope of the polar curve at the indicated point.
385) r = 4 sin 4 , = A) 1
385)
4
C) 1 + 2
B) -1
124
D) 3
The eccentricity is given of a conic section with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for the conic section. 386) e = 1 , y = 8 386) 2
A) r =
8 1 + 2 cos
B) r =
8 2 + cos
C) r =
8 2 - sin
D) r =
8 2 + sin
Replace the polar equation with an equivalent Cartesian equation. 387) r = 3 cot csc
A) y = 3x2
387)
C) y = 3
B) y2 = 3x
x
Graph the polar equation. 388) r = - 1 - sin
D) y = 3x
388)
A)
B)
C)
D)
125
Solve the problem.
389) The parabola x2 = -6y is shifted down 3 units and right 4 units. Find an equation for the new parabola and find the new vertex. A) (x + 3) 2 = -6(y - 4); vertex; (-3, 4)
389)
B) (x + 4) 2 = -6(y - 3); vertex; (4, -3) D) (y + 3) 2 = -6(x - 4); vertex; (-3, 4)
C) (x - 4) 2 = -6(y + 3); vertex; (4, -3)
Find a Cartesian equation for the line whose polar equation is given.
390) r cos +
=5 2
4
390) B) y = 2x - 10
A) y = x + 5 Solve the problem.
2
2
9
16
C) y = x + 5 2
D) y = x - 10
391) The hyperbola y - x = 1 is shifted horizontally and vertically to obtain the hyperbola
391)
(y + 5)2 (x - 4)2 =1 9 16 Find the foci of the new hyperbola. A) (-9, -5), (-1, -5)
B) (4, 0), (4, -10) D) (-5, -10), (-5, 0)
C) (-5, -9), (-5, -1) Find a polar equation for the circle. 392) x2 + 8 x + y2 = 0 5
A) r = - 8 sin 5
392) B) r = - 8 cos
C) r = 8 sin
5
5
D) r = 8 cos 5
Find the length of the curve.
393) x = 3 sin t - 3t cos t, y = 3cos t + 3t sin t, 0 t A) 3 2
393)
4
B) 3
8
C) 9 2
8
D) 3 2
32
32
The polar equation of a circle is given. Give polar coordinates for the center of the circle and identify its radius. 394) r = 8 cos 394)
A) C 4,
2
, radius = 4
B) C 8,
C) C(4, 0), radius = 4
2
, radius = 8
D) C(8, 0), radius = 8
Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. 395) Vertices at (0, 10) and (0, -10); asymptotes y = 5 x and y = - 5 x 395) 4 4 2
2
16
25
A) y - x = 1
2
2
64
100
B) y - x = 1
2
2
100
16
C) y - x = 1
2
2
100
64
D) y - x = 1
Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. 126
396) r =
6 3 + 3 sin
396)
A)
B)
C)
D)
Find the length of the curve. 397) The circle r = 9 cos
A) 18
397) B) 9 2
C) 81
127
D) 9
398) x = cos 2t, y = sin 2t, 0 t 2 A) 6 B) 4
398) C)
128
D) 2
Answer Key Testname: CHAPTER 11
1) A 2) C 3) A 4) D 5) A 6) B 7) D 8) B 9) B 10) D 11) B +
12) Yes; L =
2e- d
=
2
0
13) They are rotated counterclockwise about the origin through angles of
, , and , respectively. 6 4
14) Yes; (1, ) = (-1, 0) and the pair r = -1, = 0 satisfies the equation. 15) (a)
(b) The curves intersect in two points. In polar coordinates, these points may be represented by 1, (c) For r =
1 1 + cos
,
dy 1 = -cot . For r = dx 2 1 - cos
,
2
and 1,
3 . 2
dy = tan . Thus the slopes of the tangent lines to the curves dx 2
at both of the points of intersection are negative reciprocals of each other so the tangent lines are perpendicular. /(2n) /(2n) 16) L = cos2 (n ) + n 2 sin2 (n ) d = 2 1 + (n 2 - 1)sin2 (n ) d ; 0 - /(2n) As n + , L 2. 17) The area between the origin and the curve r = 2f( ) is four times greater than the area between the origin and the
curve r = f( ) since
1 [2f( )]2 d 2
=4
1 [f( )]2 d . 2
18) Answers will vary. Possible answer: x = t2 - 3, y = t, t 0
129
Answer Key Testname: CHAPTER 11
19) The error is in the limits of integration. There are no points on the curve corresponding to values of such that 4
<
<
3 5 or < 4 4
<
7 since cos 2 is negative on these intervals. The correct area is a 2 . 4
20) 8 3a 2 21) Answers will vary. Possible answer: x = -6 - 2t, y = -8 - 3t, 0 t 1 22) For 0 < a < 1, the curve is an oval. As a 1- , the oval develops a dimple. When a = 1, the graph resembles a parabola with a dimple at the vertex. When a > 1, the curve has two parts that resemble the branches of a hyperbola, although the left-hand "branch" has a loop and, as a increases, the right-hand "branch" develops a bulge. 23) Express curve in parametric form, differentiate x( ), find critical values, test to see which gives minimum. The left 3 5 -3 3 3 most point on the cardioid is , in polar coordinates or , in rectangular coordinates. 2 6 4 4
24) 4 2ab 25) As , x 0 and y 0. As
0 +, x
+ and y 1 (by L'Hospital's Rule). Thus, the line y = 1 is a horizontal asymptote. 26) Answers will vary. Possible answer: x = 9 - 6t, y = -8 + 6t, t 0 + 3 27) As 0 + or 2 - , x 0 and y 0. As , x 4 and y + . As , x 4 and y - . As , x -4 and 2 2 2 y - . As
3 + , x -4 and y + . Thus the lines x = -4 and x = 4 are vertical asymptotes. 2
28) Radius: 1 a 2+b2 ; center: 1 a, 1 b 2
2
2
29) For any a, the graph will be traced out as varies from 0 to 2 . The loop will occur when r is negative. For a r 0 for all 0
-1 so there is no loop. For a > 1, r < 0 when cos-1 < a
-1 -1 < cos-1 < and 2 - cos-1 a a
2 .
2 2 30) For n odd, A = a . For n even, A = a . 4
2
31) (a) r < 3 cos + sin (b)
5 2
(c)
10 1 (d) 3
32) C 33) D 34) B 35) B 36) B 37) A 38) A 39) C 40) C 130
-1 . For a < -1, r < 0 when < 2 - cos-1 a
1,
Answer Key Testname: CHAPTER 11
41) B 42) D 43) D 44) D 45) C 46) D 47) C 48) A 49) B 50) D 51) B 52) C 53) C 54) B 55) C 56) C 57) A 58) D 59) B 60) C 61) A 62) D 63) B 64) B 65) D 66) D 67) C 68) A 69) D 70) D 71) C 72) C 73) A 74) C 75) C 76) B 77) B 78) D 79) A 80) B 81) A 82) C 131
Answer Key Testname: CHAPTER 11
83) B 84) A 85) B 86) C 87) C 88) C 89) C 90) C 91) D 92) C 93) B 94) C 95) C 96) C 97) A 98) C 99) A 100) A 101) D 102) C 103) C 104) A 105) A 106) C 107) D 108) A 109) B 110) B 111) C 112) B 113) C 114) B 115) C 116) B 117) C 118) A 119) D 120) B 121) D 122) C 123) B 124) D 132
Answer Key Testname: CHAPTER 11
125) C 126) B 127) A 128) A 129) C 130) B 131) B 132) D 133) D 134) C 135) D 136) D 137) D 138) C 139) C 140) C 141) A 142) D 143) A 144) B 145) A 146) D 147) A 148) B 149) A 150) A 151) B 152) D 153) D 154) C 155) D 156) C 157) C 158) D 159) C 160) D 161) B 162) A 163) A 164) B 165) B 166) D 133
Answer Key Testname: CHAPTER 11
167) D 168) C 169) A 170) A 171) B 172) A 173) A 174) A 175) B 176) A 177) A 178) D 179) A 180) B 181) C 182) D 183) D 184) B 185) D 186) A 187) D 188) B 189) D 190) D 191) D 192) A 193) A 194) D 195) C 196) C 197) B 198) D 199) D 200) D 201) B 202) A 203) C 204) C 205) C 206) D 207) B 208) A 134
Answer Key Testname: CHAPTER 11
209) B 210) C 211) D 212) C 213) A 214) A 215) B 216) C 217) C 218) B 219) B 220) B 221) D 222) C 223) A 224) C 225) C 226) C 227) B 228) D 229) C 230) B 231) A 232) C 233) B 234) D 235) C 236) B 237) C 238) D 239) D 240) C 241) C 242) A 243) C 244) B 245) C 246) B 247) B 248) D 249) D 250) A 135
Answer Key Testname: CHAPTER 11
251) C 252) A 253) C 254) C 255) D 256) D 257) A 258) C 259) D 260) C 261) C 262) D 263) A 264) B 265) A 266) B 267) B 268) D 269) C 270) A 271) D 272) B 273) A 274) A 275) D 276) C 277) C 278) C 279) D 280) C 281) B 282) C 283) A 284) D 285) A 286) C 287) D 288) A 289) C 290) D 291) B 292) C 136
Answer Key Testname: CHAPTER 11
293) D 294) D 295) C 296) B 297) A 298) A 299) A 300) C 301) C 302) A 303) B 304) C 305) A 306) B 307) C 308) C 309) B 310) D 311) A 312) D 313) A 314) C 315) C 316) D 317) B 318) B 319) B 320) A 321) C 322) A 323) B 324) C 325) D 326) A 327) B 328) A 329) C 330) D 331) B 332) C 333) A 334) D 137
Answer Key Testname: CHAPTER 11
335) A 336) D 337) C 338) A 339) A 340) A 341) C 342) A 343) A 344) D 345) B 346) D 347) B 348) B 349) D 350) C 351) D 352) C 353) C 354) C 355) C 356) B 357) A 358) A 359) B 360) C 361) A 362) D 363) A 364) A 365) A 366) D 367) C 368) D 369) D 370) C 371) D 372) C 373) A 374) A 375) D 376) D 138
Answer Key Testname: CHAPTER 11
377) C 378) A 379) B 380) B 381) A 382) A 383) B 384) C 385) A 386) D 387) B 388) A 389) C 390) D 391) B 392) B 393) D 394) C 395) D 396) C 397) D 398) B
139
Chapter 12
Exam
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Sketch the coordinate axes and then include the vectors A, B, and A × B as vectors starting at the origin. 1) u = i, v = k 1) Determine whether the following is always true or not always true. Given reasons for your answers. 2) (u × v) · v = u · (u × v)
2)
3) u × v = -(v × u)
3)
Sketch the given surface. 4) y = x2
4)
Determine whether the following is always true or not always true. Given reasons for your answers. 5) c(u · v) = cu · cv (any number c)
5)
6) u × (v + w) = u × v + u × w
6)
1
Sketch the given surface. 7) x2 + y2 = z 2
7)
Determine whether the following is always true or not always true. Given reasons for your answers. 8) |u| = u ·u
8)
Sketch the given surface. 9) x2 + y2 = 4
9)
Sketch the coordinate axes and then include the vectors A, B, and A × B as vectors starting at the origin. 10) u = i + k, v = i - k 10)
2
Sketch the given surface. 11) z = x2 + 4y2
11)
Provide an appropriate response. 12) The unit vectors u and v are combined to produce two new vectors a = u + v and b = u - v. Show that a and b are orthogonal. Assume u v.
13) Show that the vectors a b + b a and a b - b a are orthogonal. Determine whether the following is always true or not always true. Given reasons for your answers. 14) u × 0 = 0 Provide an appropriate response. 15) Show that the midpoint of the hypotenuse of a right triangle is equidistant from all three a+b a-b vertices. [Hint: See the figure below. Show that .] = 2 2
3
12)
13)
14)
15)
Sketch the given surface. 16) 4x2 + y2 + z 2 = 4
16)
Determine whether the following is always true or not always true. Given reasons for your answers. 17) (u × v) · v = 0
17)
18) (u × v) · w = u · (w × v)
18)
Sketch the coordinate axes and then include the vectors A, B, and A × B as vectors starting at the origin. 19) u = i - j, v = k 19) Determine whether the following is always true or not always true. Given reasons for your answers. 20) c(u × v) = cu × cv (any number c)
20)
Sketch the given surface. 21) x = 1 - y2 - z2
21)
Provide an appropriate response. 22) Show that A = au + bv is orthogonal to B = bu - av, where u and v are orthogonal unit vectors. 4
22)
Sketch the coordinate axes and then include the vectors A, B, and A × B as vectors starting at the origin. 23) u = 2i + j, v = i - 2j 23)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the center and radius of the sphere. 24) (x + 4)2 + (y + 2)2 + (z + 2)2 = 81 49
24)
A) C(-4, -2, -2), a = 9
B) C(4, 2, 2), a = 81
C) C(4, 2, 2), a = 9
D) C(-4, -2, -2), a = 81
7
49
7
49
Write the equation for the plane. 25) The plane through the point A(6, 3, 2) perpendicular to the vector from the origin to A. A) 6x + 3y + 2z = -49 B) 6x + 3y + 2z = 49
25)
D) 6x + 3y + 2z = 49
C) 6x + 3y + 2z = 11
Solve the problem. 26) Show that the point P(-3, -1, -6) is equidistant from the points A(-4, -3, -5) and B(-2, 1, -7). A) The distance between P and A is 2; the distance between P and B is 2
26)
B) The distance between P and A is 3; the distance between P and B is 3 C) The distance between P and A is 5; the distance between P and B is 5 D) The distance between P and A is 6; the distance between P and B is 6 Find the distance between points P1 and P2 .
27) P1 (4, 2, 4) and P2(7, 6, 9) A) 5 2;
27) B) 10
C) 50
D) 25
Describe the given set of points with a single equation or with a pair of equations. 28) The circle in which the plane through the point (-4, 4, 4) perpendicular to the y-axis meets the sphere of radius 5 centered at the origin. A) x 2 + y2 + z 2 = 9 and y = 4 B) x 2 + y2 + z 2 = 16 and y = 4
C) x 2 + y2 + z 2 = 7 and y = 4
28)
D) x 2 + y2 + z 2 = 25 and y = 4
Find the center and radius of the sphere. 29) x2 + (y + 1)2 + (z - 4)2 = 25
29)
A) C(0, -1, 4), a = 25 C) C(0, -1, 4), a = 5
B) C(0, 1, -4), a = 25 D) C(0, 1, -4), a = 5
5
Identify the type of surface represented by the given equation. 30) y2 + z 2 = 9
30)
A) Paraboloid B) Ellipsoid C) Parabolic cylinder D) Cylinder 2
2
2
10
6
9
31) x - y - z = 1
31)
A) Hyperboloid of two sheets C) Hyperboloid of one sheet
B) Elliptical cone D) Ellipsoid
Find the vector projv u.
32) v = 3j, u = 3i + 4k
A) 1 i + 1 j + 4 k 3 3 9
32) B) 3 j 10
C) 1 i + 1 j 3 3
D) 0
Write one or more inequalities that describe the set of points. 33) The half-space consisting of the points behind the yz-plane A) y < 0, z < 0 B) x 0 C) x < 0 Find the length and direction (when defined) of u × v. 34) u = 4i + 2j + 8k, v = -i - 2j - 2k A) 180; 2 5 i + 15 j + 5 k 15 15 15
D) y 0, z 0
34) B) 6 5; 2 5 i +
5 k 5
D) 6 5; 2 5 i -
5 k 5
5
C) 180; 1 i + 1 k 15
33)
30
5
Find the center and radius of the sphere. 35) x2 + y2 + z 2 - 12x - 16y - 2z = -65
35)
A) C(6, 8, 1), a = 6 C) C(-6, -8, -1), a = 6
B) C(-6, -8, -1), a = 36 D) C(6, 8, 1), a = 36
Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian. 36) 7x - 3y - 6z = -6 and- 4x + 7y - 10z = 9 A) 0.088 rad B) 1.555 rad C) 1.482 rad D) 1.003 rad Solve the problem. 37) Find a unit vector perpendicular to plane PQR determined by the points P(2, 1, 1), Q(1, 0, 0) and R(2, 2, 2). A) 1 (j - k) B) 1 (i - j) C) 1 (j + k) D) 1 (i + j) 2 2 2 2
6
36)
37)
38) Find the work done by a force of 16i (newtons) in moving as object along a line from the origin to the point (3, 3) (distance in meters). A) 48 2 joules B) 48 joules
Find the indicated vector. 39) Let u = -2, -9 . Find 5u. A) 10, 45
C) 16 2 joules
D) 33.94 joules
39) C) -10, 45
B) -10, -45
D) 10, -45
Describe the given set of points with a single equation or with a pair of equations. 40) The plane through the point (6, -2, 6) and parallel to the yz-plane A) -2y + 6z = 0 and x = 6 B) x = 6
C) y = -2 and z = 6
40)
D) 6 + y + z = 0
Find an equation for the sphere with the given center and radius. 41) Center (0, -5, -4), radius = 9 A) x2 + y2 + 9z2 + 10y - 8z = 40 B) x2 + y2 + 9z2 + 10y + 8z = 40
C) x2 + y2 + z 2 + 10y + 8z = 40
41)
D) x2 + y2 + z 2 + 10y - 8z = 40
Describe the given set of points with a single equation or with a pair of equations. 42) The circle of radius 8 centered at the point (3, -4, 64) and lying in a plane parallel to the xy-plane
A) (x - 3)2 + (y - 4)2 = 64 and x + y = -1
2 2 B) x + y + 1 = 64
2 2 C) x + y = 64 and z = 64
D) (x - 3)2 + (y - 4)2 = 64 and z = 64
3
38)
3
-4
42)
-4
Find the component form of the specified vector.
43) The vector PQ , where P = (10, -5) and Q = (2, 5) A) 5, 5 B) 8, -10
43) C) -8, 10
D) 12, 0
Describe the given set of points with a single equation or with a pair of equations. 44) The plane through the point (-7, -2, 1) and perpendicular to the x-axis A) y = -2 and z = 1 B) -7 + y + z = 0
C) x = -7
44)
D) y + z = -1
Write one or more inequalities that describe the set of points. 45) The slab bounded by the planes x = -9 and x = 7 (planes included) A) x = -9 and x = 7 B) x -9 and x 7
C) - < y < and - < z <
45)
D) -9 x 7
Solve the problem. 46) A bullet is fired with a muzzle velocity of 1452 ft/sec from a gun aimed at an angle of 33° above the horizontal. Find the horizontal component of the velocity. A) -19.28 ft/sec B) 942.9 ft/sec C) 790.8 ft/sec D) 1218 ft/sec
7
46)
Give a geometric description of the set of points whose coordinates satisfy the given conditions. 47) (x - 8)2 + (y - 10)2 + (z - 7)2 < 16
47)
A) All points outside the lower hemisphere centered at (8, 10, 7) B) All points within the lower hemisphere centered at (8, 10, 7) C) All points on the lower hemisphere centered at (8, 10, 7) D) No set of points satisfy the given relations. Find the angle between u and v in radians. 48) u = 9i + 9j + 4k, v = 5i + 7j + 9k A) 0.52 B) 1.50
48) C) 1.05
D) 1.12
Give a geometric description of the set of points whose coordinates satisfy the given conditions. 49) y2 + z 2 = 2, x = -2
49)
A) The line tangent to the circle y2 + z2 = 2 at the point x = -2 B) The cylinder with the radius 2 along the x-axis C) All points more than 2 units from the origin D) The circle y2 + z 2 = 2 in the plane x = -2
Find parametric equations for the line described below. 50) The line through the point P(-1, 0, 6) and parallel to the line x = 4t + 6, y = 3t + 4, z = 5t - 5 A) x = 4 + 1, y = 3t, z = 5t - 6 B) x = -1, y = 5t, z = -3t + 6
C) x = 4t - 1, y = 3t, z = 5t + 6
50)
D) x = 3t - 1, y = -4t, z = 6
Solve the problem. 51) How much work does it take to slide a box 25 meters along the ground by pulling it with a 106 N force at an angle of 30° from the horizontal? A) 2650 joules B) 2650 joules C) 1325 3 joules D) 2650 3 joules 3
51)
Find v · u.
52) v = 8i + 8j and u = 3i + 2j A) 8
52) B) 24i + 16j
C) 11i + 10j
D) 40
Find the distance between points P1 and P2 .
53) P1(-4, 4, -1) and P2 (-3, 3, -2) A) 3 B) 3
53) C) 2
Identify the type of surface represented by the given equation. 2 2 54) x + y = 7 4 7
A) Ellipsoid C) Parabolic cylinder
54) B) Paraboloid D) Elliptical cylinder
8
D) 9
Find the magnitude. 55) Let u = 2, -3 and v = 4, 1 . Find the magnitude (length) of the vector: -4u - v. A) 187 B) 265 C) 187 D)
55) 265
Solve the problem. 56) A garden hose is spraying a stream of water at a box on the ground with a force of 4.9 pounds. The water stream makes an angle of 60 degrees with the ground. What is the horizontal component of the force? A) -4.667 B) 2.450 C) 4.244 D) 0.5000
56)
57) Find the vector from the origin to the center of mass of a thin triangular plate (uniform density) whose vertices are A(2, 6, 7), B(3, 10, 10), and C(3, 9, 4). A) - 1 i - 1j + 1k B) - 1 i - 2 j + 2 k C) 8 i + 25 j + 7k 2 3 3 3 3 3
Find the intersection. 58) x = -6 + 6t, y = 6 + 5t, z = 2 + 6t ; 9x + 6y + 2z = -4 A) - 53 , 79 , 3 8 24 8 48
D) - 1 i - 1 j + 1 k 6
3
3
58) B) (0, 11, 8)
C) - 43 , 313 , 21 8
57)
D) (-12, 1, -4)
8
Find the vector projv u.
59) v = k, u = 2i + 6j + 9k
59)
A) 18 i + 54 j + 81 k 121 121 121
B) 9k
C) 18 i + 54 j + 81 k
D) 9 k
11
11
11
121
Find the acute angle, in radians, between the lines. 60) 6x + 8y = -8 and 6x + 6y = 8 A) 0.9899 radians B) 0.1422 radians
60) C) 1.429 radians
D) 1.713 radians
Calculate the direction of P1 P2 and the midpoint of line segment P1 P2 .
61) P1 (8, -8, -8) and P2 (11, -12, -13)
61)
A) 3 i + -4 j + -1 k; 11 , - 6, - 13
B) 3 i + -4 j + -1 k; 19 , - 10, - 21
C) 3 i - 2 j - 1 k; 4, - 4, - 4
D) 3 i - 2 j - 1 k; 19 , - 10, - 21
5 2
50
5 2
25
2
2
2
5 2
10
50
Find the intersection. 62) x + y + z = -9, x + y = -5 A) x = -1, y = 1 - 5t, z = -4t
5 2
25
2
10
2
2
2
2
62) B) x = -t, y = -5 + t, z = -4 D) x = -t, y = -5 + t, z = 4
C) x = t, y = -5 - t, z = -4
9
Express the vector as a product of its length and direction. 63) - 5 i - 6j 2
63)
A) 13 ( -i - j)
B) 13 - 5 i - 12 j
C) 13 - 5 i - 6j
D) 169 - 5 i - 6j
2 2
2
2
13
4
Find the indicated vector. 64) Let u = -4, 6 , v = 4, 3 . Find u + 2v. A) 2, 14 B) -12, 0
13
2
64) C) 0, 18
Find the angle between u and v in radians. 65) u = -6i + 9j - 6k, v = 7i + 4j - 4k A) 0.16 B) 1.41
D) 4, 12
65) C) 1.57
D) 1.49
Find parametric equations for the line described below. 66) The line through the points P(-1, -1, 2) and Q(1, 2, -3) A) x = t + 2, y = t + 3, z = 2t - 5 B) x = 2t + 1, y = 3t + 1, z = -5t - 2
C) x = t - 2, y = t - 3, z = 2t + 5
D) x = 2t - 1, y = 3t - 1, z = -5t + 2
Identify the type of surface represented by the given equation. 2 2 2 67) x + y - z = 1 8 10 9
A) Elliptical cone C) Ellipsoid
67) B) Hyperboloid of two sheets D) Hyperboloid of one sheet
Find the intersection. 68) -6x - 8y - 7z = -8, -3x - 8y + 8z = -8 A) x = -120t - 0, y = 69t + 24, z = -24t
68) B) x = -120t + 0, y = 69t + 1, z = 24t D) x = 120t + 0, y = -69t + 24, z = -24t
C) x = -2880t + 0, y = 1656t - 24, z = -24t
Solve the problem. 69) Find the perimeter of the triangle with vertices A(4, 6, 2), B(2, -2, 6), and C(3, 5, 7). A) 132+ 51 + 83 B) 84 + 51 + 27
C) 3 4 + 51 + 7 3
70)
A) 1 6 i + 6 j + 6 k
B) 2
C) 1 1 i + 2 j + 2 k
D) 1 1 i + 1 j + 1 k
2 3
6
3
3
69)
D) 116 + 51 + 75
Express the vector as a product of its length and direction. 70) 1 i + 1 j + 1 k 6 3 2 6
66)
3
3 2 6
2
2
6
10
i+
3 2 3 3
j+
2
3 2 2
k
71) -10i + 10j + 10k
71) B) 10 3 -
A) 10( -i + j + k) C) 10 3 - 1 i + 1 j + 1 k 300
300
D)
300
3 3 3 i+ j+ k 3 3 3
3 3 3 3 i+ j+ k 30 3 3 3
Find the distance between points P1 and P2 .
72) P1(-10, 9, -2) and P2 (-6, 17, 6) A) 12 B) 14
72) C) 18
D) 10
Solve the problem. 73) A force of magnitude 9 pounds pulling on a suitcase makes an angle of 60° with the ground. Express the force in terms of its i and j components. A) 0.5000i + + 0.8660j B) 7.794i + 4.500j
C) 4.500i + 7.794 j
73)
D) -8.572i - 2.743j
Give a geometric description of the set of points whose coordinates satisfy the given conditions. 74) x2 + y2 + z 2 > 36
74)
A) All points in space B) All points outside the cylinder with radius 6 C) All points outside the sphere of radius 6 D) All points on the surface of the cylinder with radius 6 Find the indicated vector. 75) Let u = 5, -4 . Find -9u. A) -45, -36
75) B) 45, -36
C) -45, 36
D) 45, 36
Solve the problem. 76) A bullet is fired with a muzzle velocity of 1249 ft/sec from a gun aimed at an angle of 16° above the horizontal. Find the horizontal component of the velocity. A) -1196 ft/sec B) 344.3 ft/sec C) 358.1 ft/sec D) 1201 ft/sec Find the intersection. 77) x = -1 - 6t, y = -6 + 9t, z = -1 + 9t ; -6x - 3y - 5z = -7 A) (5, -15, -10) B) - 7, 3, 8 C) (-7, 3, 8)
76)
77) D) 5, - 15, - 10
Find the vector projv u.
78) v = i + j + k, u = 3i + 4j + 12k
78)
A) 20 i + 20 j + 20 k 3 3 3
B) 19 i + 19 j + 19 k 169 169 169
C) 19 i + 19 j + 19 k
D) 19 i + 19 j + 19 k
13
13
13
3
11
3
3
Write one or more inequalities that describe the set of points. 79) The closed region bounded by the spheres of radius 6 and 10, both centered at the origin, and the planes x = 4 and x = 8 A) 6 x2 + y2 + z 2 10 and x = 4 and x = 8 B) 6 < x2 + y2 + z 2 < 10 and x = 4 and x = 8
C) 36 x2 + y2 + z 2 100 and 4 x 8
79)
D) 36 < x2 + y2 + z 2 < 100 and 4 < x < 8
Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian. 80) 3x - 9y + 6z = 7 and -3x - 9y + 4z = -3 A) 0.980 rad B) 1.571 rad C) 0.165 rad D) 0.590 rad
80)
Solve the problem. 81) Let u = 5i + j, v = i + j, and w = i - j. Find scalars a and b such that u = av + bw. A) 0.3333 v + 0.5000 w B) 3.000 v + 2.000w
81)
C) 5v - w
D) 5v + w
Find the intersection. 82) -5x - 5y = 8, 5y + 9z = -9 A) x = -45t + 1 , y = 45t - 9 , z = -25t 5 5
B) x = -45t + 1 , y = 45t - 9 , z = -25 5 5
C) x = -45t - 5, y = 45t - 9, z = -25t
D) x = -45t - 1 , y = 45t + 9 , z = 25t
82)
5
5
Express the vector in the form v = v1 i + v2 j + v3 k.
83) 3u - 4v if u = 1, 1, 0 and v = 3, 0, 1 A) v = 15i + 3j - 4k C) v = -9i + 3j - 4k
83) B) v = -9i + 7j - 4k D) v = 3i + 3j - 4k
Find the magnitude. 84) Let u = -1, 2 . Find the magnitude (length) of the vector: 6u. A) -6 5 B) 6 10 C) 10
84) D) 6 5
Solve the problem. 85) Let u = 2i + 6j, v = 2i + 3j, and w = i - j. Write u = u1 + u2 where u1 is parallel to v and u2 is parallel to w. A) u1 = 0.6250 v u2 = -0.8333w
B) u1 = 4v u2 = -18w
12
C) u1 = 1.600 v
D) u1 = 3v
u2 = -1.200w
u2 = 2w
85)
Match the equation with the surface it defines. 2 2 2 86) z - x - y = 1 4 49 49
A) Figure 2
86)
B) Figure 1
C) Figure 4
D) Figure 3
Solve the problem. 87) Find the magnitude of the torque in foot-pounds at point P for the following lever:
PQ = 9 in. and F = 25 lb A) 17 2 ft-lb 12
B) 75 2 ft-lb
C) - 75 2 ft-lb
D) 1.271035442e+15 2 ft-lb
8
8
2.638827907e+13
Find the center and radius of the sphere. 88) 3x2 + 3y2 + 3z 2 - 2x + 2y = 9
88)
A) C - 1 , 1 , 0 , a = 29 3 3 9
B) C 1 , - 1 , 0 , a = 3 3
29 3
C) C - 1 , 1 , 0 , a =
D) C - 1 , 1 , 0 , a =
29 3
9 9
87)
29 9
3 3
13
Find v · u.
89) v =
1 1 , and u = 23 23
A) 2
23
1 -1 , 23 23
89) C) 1 i - 1 j
B) 0
23
23
D) 2 i - 2 j 23
23
Solve the problem. 90) Find the area of the parallelogram determined by the points P(7, -5, 5), Q(-5, -2, 4), R(-6, -10, 3) and S(-18, -7, 2). A) 10043 B) 1811 C) 10043 D) 1811 2 2
14
90)
Use the vectors u, v, w, and z head to tail as needed to sketch the indicated vector.
91) 3w
91)
A)
B)
C)
D)
15
Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian. 92) -7x - 3y - 10z = -8 and 7x + 7y - 10z = -8 A) 1.458 rad B) 1.4 rad C) 0.762 rad D) 0.170 rad Solve the problem.
2
2
49
4
93) Find the volume of the solid bounded by the paraboloid x + y = z and the planes z = 0 and z = 3. (The area of an ellipse with semiaxes a and b is ab.) A) 63 units3 B) 63 2 units3 C) 63 4 4 2
4
4
2
2
2
2
9
4
9
96) Find the volume of the solid bounded by the ellipsoid x + y + z = 1 and the planes z = -2 and z = 2. (The area of an ellipse with semiaxes a and b is ab.) A) 27.56 units3 B) 13.33 units3 C) 20.44 units3
96)
D) 122.67 units3
Find the length and direction (when defined) of u × v. 97) u = - 1 i + 3 j + k, v = i + j + 2k 2 2
97)
A) 8; 1 i - 1 j + 1 k C) 2 2; -
95)
D) 18 13.9567901
C) 18 13.9567901
4
94)
D) 8x + 2y + 7z = -12
Solve the problem. 95) Find the area of the triangle determined by the points P(1, 1, 1), Q(8, 2, 4), and R(3, 10, 4). A) 5854 B) 5854 2
4
93)
D) 63 units3
units3
Write the equation for the plane. 94) The plane through the point P(-8, -6, 2) and perpendicular to the line x = 1 + 8t, y = 9 + 2t, z = 9 + 7t A) 8x + 2y + 7z = 17 B) 8x + 2y + 7z = 62
C) 8x + 2y + 7z = -62
92)
B) 8; 1 i - 1 j - 1 k
2
2
2 2 2 i+ jk 2 2 2
D) 2 3;
16
4
4
3 3 3 i+ jk 3 3 3
Match the equation with the surface it defines. 98) x2 + y2 = 49
A) Figure 1
98)
B) Figure 4
C) Figure 3
D) Figure 2
C) -75
D) -51
Find v · u.
99) v = 7i - 6j and u = -9i - 2j A) -63i + 12j B) -2i - 8j
99)
Find the length and direction (when defined) of u × v. 100) u = 2i + 2j - k, v = -i + k A) 9; 2 i + 1 j - 2 k 9 9 9
100) B) 9; 2 i - 1 j + 2 k 9 9 9
C) 3; 2 i - 1 j + 2 k 3
3
D) 3; 2 i + 1 j - 2 k
3
3
17
3
3
Use the vectors u, v, w, and z head to tail as needed to sketch the indicated vector.
101) v - w
101)
A)
B)
C)
D)
18
Give a geometric description of the set of points whose coordinates satisfy the given conditions. 102) 7 y 8, 7 z 8 A) The cube located in the first quadrant and with sides 7 units in length
B) The square with corners at (0, 7, 7), (0, 7, 8), (0, 8, 7), and (0, 8, 8) C) The line between the points (0, 7, 7) and (0, 8, 8) D) The infinitely long square prism parallel to the x-axis
19
102)
Use the vectors u, v, w, and z head to tail as needed to sketch the indicated vector.
103) 2u - z - w
103)
A)
B)
C)
D)
20
Find the component form of the specified vector. 104) The unit vector that makes an angle 4 /3 with the positive x-axis A) -1 , -1 B) 1 , 3 C) -1 , - 3 2 2 2 2 2 2
104) D)
3 1 , 2 2
Find an equation for the line that passes through the given point and satisfies the given conditions. 105) P = (8, 4); parallel to v = 3i + 7j
A) 3x + 7y = 52
B) 3x + 7y = 58
C) 7x - 3y = 44
D) y - 4 = - 3 (x - 3)
105)
5
Find a parametrization for the line segment joining the points. 106) (3, 3, -4), (0, 3, 1) A) x = -3t + 3, y = 3t, z = 5t - 4, 0 t 1 B) x = 3t, y = 3t, z = -5t + 1, 0 t 1
C) x = 3t, y = 3, z = -5t + 1, 0 t 1
106)
D) x = -3t + 3, y = 3, z = 5t - 4, 0 t 1
Express the vector as a product of its length and direction. 107) -2i - 8j + 8 k 3
107)
A) 26 (-i - j + k)
B) 26 - 3 i - 12 j + 4 k
C) 676 - 3 i - 12 j + 4 k 9 13 13 13
D) 3 26
3
3
13
13
13
Solve the problem. 108) Find a formula for the distance from the point P(x, y, z) to the y-axis. A) x2 + z 2 B) x + z C) y2 + z 2
108) D) y + z
Give a geometric description of the set of points whose coordinates satisfy the given conditions. 109) x2 + y2 + z 2 = 49, z = -4
109)
A) All points on the sphere x2 + y2 + z 2 = 49 and above the plane z = -4
B) All points within the sphere x2 + y2 + z 2 = 49 and above the plane z = -4 C) The sphere x2 + y2 + z 2 = 16 D) The circle x2 + y2 = 33 in the plane z = -4 Find the indicated vector. 110) Let u = 6, 2 . Find -2u. A) -12, -4
110) B) -12, 4
C) 12, 4
21
D) 12, -4
Solve the problem. 111) An airplane is flying in the direction 30° east of south at 624 km/hr. Find the component form of the velocity of the airplane, assuming that the positive x-axis represents due east and the positive y-axis represents due north. A) -616.5, -96.25 B) 0.5000, -0.8660
C) 312.0, -540.4
111)
D) 540.4, -312.0
Find an equation for the sphere with the given center and radius. 112) Center (-1, 7, 0), radius = 9 A) x2 + y2 + z 2 + 2x - 14y = 31 B) x2 + y2 + z 2 - 2x + 14y = 31
C) x2 + y2 + z 2 - 2x - 14y = 31
112)
D) x2 + y2 + z 2 + 2x + 14y = 31
Solve the problem. 113) Find a unit vector perpendicular to plane PQR determined by the points P(2, 1, 3), Q(1, 1, 2) and R(2, 2, 1). A) 1 (-i - 2j - k) B) 1 (i - 2j - 2k) C) 1 (i - 2j - k) D) 1 (i - 2j + k) 6 6 6 6
113)
Find v · u.
114) v = 5i - 6j and u = -7 7i + 2j A) (5 - 7 7)i - 4j B) -12 7 + 35
114) C) -12 7i - 35j
D) -12 - 35 7
Match the equation with the surface it defines. 2 2 115) y + z = 1 64 16
A) Figure 3
115)
B) Figure 2
C) Figure 4
22
D) Figure 1
Write the equation for the plane. 116) The plane through the point P(-2, 7, 6) and normal to n = -5i - 5j + 8k. A) -2x + 7y - 6z = -3 B) 5x + 5y - 8z = -3
C) 2x - 7y + 6z = -3
116)
D) -5x - 5y + 8z = 23
Solve the problem. 117) A bullet is fired with a muzzle velocity of 1265 ft/sec from a gun aimed at an angle of 16° above the horizontal. Find the vertical component of the velocity. A) 348.7 ft/sec B) 1216 ft/sec C) -1211 ft/sec D) 362.7 ft/sec Write the equation for the plane. 118) The plane through the point P(3, 4, 5) and parallel to the plane -2x - 5y + 6z = 1. A) -2x - 5y + 6z = -4 B) -2x - 5y + 6z = 4
C) 4y = 4
118)
D) -2x - 5y + 6z = 56
Match the equation with the surface it defines. 2 2 2 119) x + y + z = 1 25 100 25
A) Figure 3
119)
B) Figure 1
C) Figure 2
D) Figure 4
Find the length and direction (when defined) of u × v. 120) u = 3i - 5j - 3k, v = -6i + 10j + 6k
120)
A) 0; -3i+ 5j + 3k C) 129;
117)
B) 0; no direction
1 (-3i + 5j + 3k) 129
D) 215;
23
1 (-3i + 5j + 3k) 215
Find the distance between points P1 and P2 .
121) P1(6, -8, -2) and P2 (9, -6, 4) A) 7 B) 5
121) D) 7
C) 6
Find the indicated vector.
122) Let u = -1, 4 , v = 6, -3 . Find 4 u + 3 v. 5
A) 4, 3
122)
5
B) 8 , 3
5
C) 7 , 14
5
5
5
D) 14 , 7 5
5
Solve the problem. 123) A bird flies from its nest 3 km in the direction 10° north of east, where it stops to rest on a tree. It then flies 6 km in the direction 4° south of west and lands atop a telephone pole. With an xy-coordinate system where the origin is the bird's nest, the x-axis points east, and the y-axis points north, at what point is the tree located? A) (0.9848, 0.1736) B) (2.954, 0.5209) C) (-2.517, -1.632) D) (0.5209, 2.954) Find the acute angle, in degrees, between the lines. 124) (1 + 3) x + (1 - 3) y = -3 and 3 x + y = -14 A) 30° B) 45°
123)
124) C) 60°
D) 75°
Solve the problem. 125) Find the area of the triangle determined by the points P(-3, 6, -4), Q(2, -2, 2), and R(10, 4, -7). A) 18781 B) 3 1901 C) 3 1901 D) 18781 2 2
125)
Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian. 126) 8x + 8y + 10z = -10 and 5x + 3y + 6z = -10 A) 0.192 rad B) 1.215 rad C) 1.364 rad D) 1.378 rad
126)
Give a geometric description of the set of points whose coordinates satisfy the given conditions. 127) x = 6, z = 1 A) The line through the point (6, 1, 0) and parallel to the z-axis
127)
B) The line through the point (6, 0, 1) and parallel to the y-axis C) All points in the x-z plane D) The point (6, 1) Find an equation for the line that passes through the given point and satisfies the given conditions. 128) P = (-8, -7); parallel to v = -2i + 6j
A) -2x + 6y = 40
B) -2x + 6y = -26
C) y + 7 = 13 (x + 2)
D) 6x + 2y = -62
6
24
128)
Solve the problem. 129) How much work does it take to slide a box 19 meters along the ground by pulling it with a 223 N force at an angle of 26° from the horizontal? A) 2741 joules B) 200.4 joules C) 4237 joules D) 3808 joules Find parametric equations for the line described below. 130) The line through the point P(-3, -6, -5) and perpendicular to the plane 2x + 2y + 2z = 3 A) x = -2t + 3, y = -2t + 6, z = -2t + 5 B) x = 2t - 3, y = 2t - 6, z = 2t - 5
C) x = -2t - 3, y = 2t - 6, z = -5
129)
130)
D) x = 2t + 3, y = 2t + 6, z = 2t + 5
Find the vector projv u.
131) v = 7i - 3j + k, u = -3j + 4k
131)
A) 91 i - 39 j + 13 k 59 59 59
B) - 39 j + 52 k 25 25
C) 91 i - 39 j + 13 k
D) - 39 j + 52 k
25
25
25
59
59
Identify the type of surface represented by the given equation. 2 2 132) x + z = y 2 3 6
A) Elliptical paraboloid C) Hyperbolic paraboloid
132) B) Ellipsoid D) Elliptical cone
Solve the problem. 133) Find a vector of magnitude 11 in the direction of v = 12i - 5k. A) 11 (12i - 5k) B) 1 (12i - 5k) C) 11(12i - 5k) 13 13
25
133) D) 11(i - k)
Match the equation with the surface it defines. 2 2 134) y - x = z 64 64 10
A) Figure 4
134)
B) Figure 1
C) Figure 2
D) Figure 3
Express the vector in the form v = v1 i + v2 j + v3 k.
135) P1 P2 if P1 is the point (-1, -4, 2) and P2 is the point (1, -7, -2) A) v = -2i - 3j + 4k B) v = -2i + 3j - 4k C) v = 2i - 3j - 4k D) v = -2i + 3j + 4k
135)
Write the equation for the plane. 136) The plane through the point P(-5, 5, -5) and parallel to the plane 5x + 6y + 6z = -21. A) -5x + 5y - 5z = -25 B) 5x + 6y + 6z = 25
C) 5x + 6y + 6z = -25
136)
D) 6x + 6y + 5z = -25
Calculate the requested distance. 137) The distance from the point S(-7, -2, -10) to the plane 2x + 2y + z = -5 A) 13 B) 13 C) 23 9 3 9
26
137) D) 23 3
Use the vectors u, v, w, and z head to tail as needed to sketch the indicated vector.
138) u + z
138)
A)
B)
C)
D)
27
Find parametric equations for the line described below. 139) The line through the point P(1, 2, 3) parallel to the vector -8i + 5j - 6k A) x = -8t + 1, y = 5t + 2, z = -6t + 3 B) x = 8t + 1, y = 5t + 2, z = -6t + 3
C) x = 8t - 1, y = 5t - 2, z = -6t - 3
139)
D) x = -8t - 1, y =5t - 2, z = -6t - 3
Find v · u.
140) v = 9i - 2j and u = -7i + 8j A) -63i - 16j B) -79
140) C) 2i + 6j
D) -47
Solve the problem. 141) Find a formula for the distance from the point P(x, y, z) to the xy plane. A) x2 + y2 B) y C) x
141) D) z
Match the equation with the surface it defines. 2 2 2 142) - x + y + z = 1 49 36 36
A) Figure 2
142)
B) Figure 3
C) Figure 4
D) Figure 1
Find the center and radius of the sphere. 143) 2x2 + 2y2 + 2z 2 - x + y - z = 9
143)
A) C 1 , - 1 , 1 , a = 5 6
B) C - 1 , 1 , - 1 , a = 75
C) C 1 , - 1 , 1 , a = 5 6
D) C 1 , - 1 , 1 , a = 5 3
4 4
4 4 4 4
4 4
2
4
4
Find the acute angle, in degrees, between the lines. 144) 3x - y = 19 and 2x + y = 10 A) 75° B) 60°
4 4
4
8
4
144) C) 30°
28
D) 45°
Find the angle between u and v in radians. 145) u = -6i - 2j, v = 7i + 10j + 10k A) -0.67 B) 1.58
145) C) 1.79
D) 2.24
Calculate the requested distance. 146) The distance from the point S(-6, -2, 8) to the line x = 6 + 6t, y = -3 + 9t, z = -8 + 2t A) 6.536596627e+15 B) 6.536596627e+15 1.663011337e+13 1.511828488e+12 47,560 121
C)
146)
47,560 11
D)
Give a geometric description of the set of points whose coordinates satisfy the given conditions. 147) x2 + y2 25, z = -9
147)
A) All points on or within the circle x2 + y2 = 25 and in the plane z = -9
B) All points on the cylinder with radius 5 along the z-axis C) All points on or outside of the circle x2 + y2 = 25 and in the plane z = -9 D) All points within the parabola x2 + y2 = 25 in the plane z = -9 Identify the type of surface represented by the given equation. 2 2 148) z - x = y 8 5 6
A) Paraboloid C) Hyperbolic paraboloid
148) B) Parabolic cylinder D) Ellipsoid
Describe the given set of points with a single equation or with a pair of equations. 149) The plane perpendicular to the y-axis and passing through the point (4, -1, 3) A) x = 4 and z = 3 B) y = -1
C) 4x + 3z = 0 and y = -1
149)
D) x - 1 + z = 0
Find the vector projv u.
150) v = 2i - 2j - 4k, u = 12i - 5k
150)
A) 11 i - 11 j - 22 k
B) 11 i - 11 j - 11 k
C) 528 i - 220 k
D) 528 i - 220 k
3
169
3
3
6
169
13
6
6
169
Write the equation for the plane. 151) The plane through the point P(5, 5, -7) and normal to n = 6i + 6j + 4k. A) 5x + 5y - 7z = 32 B) -6x - 6y - 4z = 32
C) -5x - 5y + 7z = 32
D) 6x + 6y + 4z = 32
29
151)
Calculate the requested distance. 152) The distance from the point S(-4, 9, 4) to the line x = -5 + 2t, y = -2 + 2t, z = 10 + 2t A) 3.852688744e+15 B) 2 438 1.97912093e+13 9
C) 2 438
152)
D) 3.852688744e+15 6.597069767e+12
3
Find the indicated vector.
153) Let u = 7, 1 , v = -1, 4 . Find 5 u - 12 v. 13
A) - 43 , 47
153)
13
B) 40 , - 36
13 13
13
C) 23 , - 53
31
13
13
D) 47 , - 43 13
13
Find parametric equations for the line described below. 154) The line through the point P(-7, -5, 5) parallel to the vector 2i - 2j - 2k A) x = -2t + 7, y = 2t + 5, z = -2t - 5 B) x = 2t - 7, y = -2t - 5, z = -2t + 5
C) x = 2t + 7, y = -2t + 5, z = -2t - 5
154)
D) x = -2t- 7, y = -2t - 5, z = 2t + 5
Express the vector as a product of its length and direction. 155) 5j + 8 k 3
A) 17 15 j + 8 k 3 17
17
155)
B) 17 (j + k)
C) 17 5j + 8 k
3
3
Find the acute angle, in degrees, between the lines. 156) y = 3 x - 1 and y = - 3 x - 2 A) 75° B) 45°
3
D) 289 5j + 8 k 9
3
156) C) 60°
D) 30°
Calculate the requested distance. 157) The distance from the point S(1, 9, -8) to the line x = -7 + 11t, y = -4 + 2t, z = -4 + 10t A) 50,549 B) 6.947401659e+15 3.092376453e+13 225
C) 6.947401659e+15
D)
2.061584302e+12
50,549 15
Find a parametrization for the line segment joining the points. 158) (3, 0, -5), (0, -4, 0) A) x = -3t + 3, y = -4t, z = 5t - 5, 0 t 1 B) x = 3t, y = 4t - 4, z = -5t, 0 t 1
C) x = -2t, y = 5t - 4, z = 6t, 0 t 1
D) x = -2t + 3, y = 5t, z = 6t - 5, 0 t 1
30
157)
158)
Express the vector as a product of its length and direction. 159) 1 i - 1 j - 1 k 6 6 6 1 2
A)
159)
1 1 1 ijk 3 3 3
B) 1 6 i - 6 j - 6 k 6
C) 1 3 i - 3 j - 3 k 3
6
6
D)
6
Find the indicated vector. 160) Let u = 5, 8 , v = 6, 5 . Find v - u. A) 3, -1 B) 1, -3
6
1 2
6
6
1 1 1 ijk 6 6 6
160) C) 0, -2
D) 11, 13
Solve the problem.
161) Find a vector of magnitude 9 in the direction opposite to the direction of v = 1 i + 1 j - 1 k. 2
A) 6 - 1 i - 1 j + 1 k
B) 6 3 - 1 i - 1 j + 1 k
C) 6 3 1 i + 1 j - 1 k
D) 9 1 i + 1 j - 1 k
2
2
2
2
2
2
2
2
Find the angle between u and v in radians. 162) u = -9j and v = 7i - 2k A) 1.57 B) 0.14
2
2
2
2
161)
2
2
162) C) 1.71
Solve the problem.
D) 0.00
2
2
2
16
64
9
163) Find the volume of the solid bounded by the elliptical cone x + y = z and the planes z = 0 and z = 5. (The area of an ellipse with semiaxes a and b is ab.) A) 4000 units3 B) 4000 units3 C) 800 units3 27 9 27
31
D) 4000 units3 27
163)
Match the equation with the surface it defines. 2 2 164) x + z = y 36 36 3
A) Figure 3
164)
B) Figure 2
C) Figure 4
Find the triple scalar product (u x v) · w of the given vectors. 165) u = -5i - 3j + 4k; v = 4i - 10j + 10k; w = 8i - 7j - 3k A) -568 B) -16
D) Figure 1
165) C) -76
D) -1136
Write the equation for the plane. 166) The plane through the point P(4, 8, 7) and perpendicular to the line x = 1 + 8t, y = -3 + 7t, z = 4 - t. A) 8x + 7y - z = 81 B) 8x + 7y - z = 22
C) 8x + 7y - z = -81
166)
D) 8x + 7y + z = 81
Find the acute angle, in radians, between the lines. 167) 2x - 2y = 1 and 7x + 5y = 8 A) 0.1644 radians B) 1.406 radians
167) C) 0.1651 radians
D) 2.976 radians
Describe the given set of points with a single equation or with a pair of equations. 168) The circle in which the plane through the point (5, -8, 10) perpendicular to the x-axis meets the sphere of radius 13 centered at the origin. A) x2 + y 2 + z 2 = 119 and y = 5 B) x2 + y2 + z 2 = 144 and y = 5
C) x2 + y2 + z 2 = 169 and z = 5
D) x2 + y2 + z 2 = 169 and x = 5
32
168)
Solve the problem.
2
2
2
16
100
25
169) Find the volume of the solid bounded by the hyperboloid of one sheet x + y - z = 1 and the planes z = -2 and z = 2. (The area of an ellipse with semiaxes a and b is ab.) A) 151.47 units3 B) 168.53 units3
C) 164.27 units3
D) 6741.33 units3
33
169)
Use the vectors u, v, w, and z head to tail as needed to sketch the indicated vector.
170) z - v
170)
A)
B)
C)
D)
34
Find an equation for the line that passes through the given point and satisfies the given conditions. 171) P = (11, 8); perpendicular to v = 8i - 5j
A) 8x - 5y = 48
B) 8x - 5y = 89
C) y - 8 = 13 (x - 8)
D) -5x - 8y = -119
3
Write one or more inequalities that describe the set of points. 172) The rectangular solid in the first octant bounded by the planes x = 3, x = 5, y = 1, y = 5, z = 6 and z = 10 (planes excluded) A) The given planes do not form a rectangular solid
171)
172)
B) x < 3, x > 5; y < 1, y > 5; z < 6, z > 10 C) x < y < z D) 3 < x < 5; 1 < y < 5; 6 < z < 10 Solve the problem. 173) A bird flies from its nest 4 km in the direction 3° north of east, where it stops to rest on a tree. It then flies 7 km in the direction 4° south of west and lands atop a telephone pole. With an xy-coordinate system where the origin is the bird's nest, the x-axis points east, and the y-axis points north, at what point is the telephone pole located? A) (0.6155, 7.494) B) (3.995, 0.2093)
C) (-6.983, -0.4883)
173)
D) (-2.988, -0.2790)
Write the equation for the plane. 174) The plane through the point A(-5, -4, 10) perpendicular to the vector from the origin to A. A) -5x - 4y + 10z = -141 B) 5x - 4y - 10z = -1
174)
D) 5x + 4y - 10z = 141
C) 5x + 4y - 10z = -141 Find the acute angle, in degrees, between the lines. 175) x - 3 y = 18 and 3 x - y = -6 A) 30° B) 75°
175) C) 45°
D) 60°
Find an equation for the line that passes through the given point and satisfies the given conditions. 176) P = (-7, 8); parallel to v = -2i - 7j A) -2x - 7y = 53 B) -7x + 2y = 65
C) y - 8 = - 3(x + 2)
176)
D) -2x - 7y = -42
Express the vector as a product of its length and direction. 177) 3j A) 3 1 j B) 1 j 3 3
177) C) 3j
35
D) 3(3j)
178) 18 i - 12 j - 4 k 5
5
178)
5
A) 22 (i - j - k)
B) 22 18 i - 12 j - 4 k
5
5
C) 22 9 i - 6 j - 2 k 5 11
11
5
5
5
D) 22 9 i - 6 j - 2 k
11
5 121
Find the triple scalar product (u x v) · w of the given vectors. 179) u = i + j + k; v = 9i + 8j + 7k; w = 2i + 9j + 10k A) 6 B) -124
121
179) C) 158
Calculate the requested distance. 180) The distance from the point S(4, 2, 10) to the plane 4x + 3y = -2 A) 8 B) 24 C) 24 25 25 5
181) The distance from the point S(-7, -2, -7) to the plane 11x + 2y + 10z = 1 A) 152 15
B) 2
C) 2
3
45
182) The distance from the point S(-7, -5, 10) to the plane -9x + 6y + 2z = 5 A) 18 11
B) 48 121
C) 48 11
Find the magnitude. 183) Let u = 3, 4 . Find the magnitude (length) of the vector: -3u. A) -15 5 B) -15 C) 15 Solve the problem. 184) For the vectors u and v with magnitudes u = 4 and v = 8, find the angle which makes projv u = 1
A) 14.48°
121
B) 75.52°
C) 60.00°
D) 136
180) D) 12 5
181) D) 152 225
182) D) 18 121
183) D) 75
between u and v
D) 82.82°
Find an equation for the line that passes through the given point and satisfies the given conditions. 185) P = (-6, -9); perpendicular to v = -5i + 4j A) y + 9 = 13(x + 5) B) 4x + 5y = -69
C) -5x + 4y = -6
184)
185)
D) -5x + 4y = 41
Describe the given set of points with a single equation or with a pair of equations. 186) The set of points equidistant from the points (-10, 0, 0) and (2, 0, 0) A) x = -4 B) y + z = 0 and -10 < x < 2
C) x > -10 and x < 2
D) y + z = -4
36
186)
Write the equation for the plane. 187) The plane through the points P(-1, 7, -60) , Q(2, 1, 6) and R(1, -1, 16). A) 6x + 8y + z = -2 B) 6x - 8y - z = -2
C) 6x + 8y +z = 2
187)
D) 6x - 8y - z = 2
Describe the given set of points with a single equation or with a pair of equations. 188) The plane through the point (7, 3, 8) and parallel to the xy-plane A) x + y + 8 = 0 B) z = 8 C) x + y = 10 Find the acute angle, in radians, between the lines. 189) -7x - 6y = 1 and 4x - 2y = 9 A) 0.3986 radians B) 1.172 radians
188) D) x = 7 and y = 3
189) C) 2.743 radians
D) 0.3881 radians
Solve the problem. 190) For the triangle with vertices located at A(3, 5, 4), B(2, 5, 3), and C(1, 1, 1) , find a vector from vertex C to the midpoint of side AB. A) 1 i + 3 j + 1k B) 3 i + 4j + 5 k C) 7 i + 6j + 9 k D) 5 i + 5j + 7 k 2 2 2 2 2 2 2 2 Find the length and direction (when defined) of u × v. 191) u = 8i, v = 3j A) 24; -k B) 24; k
190)
191) C) 24; 24k
D) 8; 3k
Identify the type of surface represented by the given equation. 2 2 192) x + z = y 5 5 9
A) Elliptical cone C) Elliptical paraboloid
192) B) Ellipsoid D) Hyperbolic paraboloid
Find v · u.
193) v = -9i + 5j and u = 3i + 2j A) -17 B) -37
193) C) -27i + 10j
D) -6i + 7j
Solve the problem. 194) Find the magnitude of the torque in foot-pounds at point P for the following lever:
PQ = 4 in. and F = 30 lb A) 120 ft-lb
B) 14.37 ft-lb
C) 9.06 ft-lb
37
D) 4.23 ft-lb
194)
Find the indicated vector. 195) Let u = -8, -6 , v = -9, 6 . Find u - v. A) -14, 3 B) 1, -12
195) C) -17, 0
Find the acute angle, in radians, between the lines. 196) 9x - 7y = -4 and 4x - 5y = 6 A) 0.2351 radians B) 1.336 radians
D) -2, -15
196) C) 1.806 radians
Find the angle between u and v in radians. 197) u = 8i, v = 5i - 9j A) 1.56 B) 0.51
D) 0.9725 radians
197) C) 1.33
D) 1.06
Express the vector as a product of its length and direction. 198) 2j - 3 k 2
A) 5 (j - k)
198)
B) 5 2j - 3 k
2
2
C) 25 4 j - 3 k
2
4 5
Find the angle between u and v in radians. 199) u = 8i - 8j - 3k, v = 4i + 10j - 5k A) 1.69 B) 1.57
D) 5 4 j - 3 k
5
2 5
5
199) D) 1.81
C) -0.24
Calculate the direction of P1 P2 and the midpoint of line segment P1 P2 .
200) P1(-9, -10, -1) and P2(1, 0, 4)
200)
A) 3i + 10 j + 1 k; - 9 , - 5, - 1
B) 2 i + 2 j + 1 k; 5, 5, 5
C) 2 i + 2 j + 1 k; - 4, - 5, 3
D) 2 i + 2 j + 1 k; 1 , 0, 2
3
3
3
3
2
3
2
3
2
3
3 3
3 3
2
2
Find parametric equations for the line described below. 201) The line through the point P(-4, 1, -3) and parallel to the line x = 3t - 3, y = 2t + 6, z = 2t + 4 A) x = -4, y = 2t + 1, z = -2t - 3 B) x = 3t + 4, y =2t - 1, z = 2t + 3
C) x = 3t - 4, y = 2t + 1, z = 2t - 3
201)
D) x = 2t - 4, y = -3t, z = -3
202) The line through the point P(-6, -2, 0) and perpendicular to the plane 2x + 4y + 4z = 2 A) x = -2t + 6, y = -4t + 2, z = -4t B) x = 2t + 6, y = 4t + 2, z = 4t C) x = 2t - 6, y = 4t - 2, z = 4t D) x = -4t - 6, y = -4t - 6, z = 0 Find an equation for the sphere with the given center and radius. 203) Center (-3, 0, 0), radius = 9 A) x2 + y2 + z 2 + 6x = 9 B) x2 + y2 + z 2 - 6x = 9
C) x2 + y2 + z 2 + 6x = 72
D) x2 + y2 + z 2 - 6x = 72
38
202)
203)
Find the component form of the specified vector. 204) The vector from the point A(5, 9) to the origin A) 5, -9 B) -5, 9
204) C) 5, 9
D) -5, -9
Find parametric equations for the line described below. 205) The line through the point P(4, -2, 3) and perpendicular to the vectors u = -7i + 4j + 2k and v = 8i + 8j - 7k A) x = -44t + 4, y = -33t - 2, z = -88t + 3 B) x = -44t + 4, y = -33t - 2, z = -3t + 3
C) x = -44t - 4, y = -33t + 2, z = -3t - 3
205)
D) x = -44t + 4, y = 33t - 2, z = -88t + 3
Find the length and direction (when defined) of u × v. 206) u = -5i + 3i - 4k, v = 0
206)
A) 0; -5i + 3i - 4k
B) 0; 1 (-5i + 3i - 4k)
C) 0; no direction
D) 5 2; 1 (-5i + 3i - 4k)
5 2
5 2
Find a parametrization for the line segment joining the points. 207) (0, 0, 0), (3, 3, 4) A) x = 3t + 3, y = 3t + 3, z = 4t + 4 B) x = 3t - 3, y = 3t - 3, z = 4t - 4
C) x = -3t, y = -3t, z = -4t
207)
D) x = 3t, y = 3t, z = 4t
Find the vector projv u.
208) v = 3i - j + 3k, u = 10i + 11j + 2k
208)
A) 141 i - 47 j + 141 k
B) 50 i + 55 j + 10 k
C) 75 i - 25 j + 75 k
D) 10 i + 11 j + 2 k
19
19
19
19
19
3
19
9
3 9
3
9
Solve the problem. 209) An airplane is flying in the direction 80° west of north at 869 km/hr. Find the component form of the velocity of the airplane, assuming that the positive x-axis represents due east and the positive y-axis represents due north. A) -0.9848, 0.1736 B) -150.9, 855.8
C) 863.7, -95.93
D) -855.8, 150.9
39
209)
Match the equation with the surface it defines. 2 2 2 210) x + y = z 4 4 16
A) Figure 1
210)
B) Figure 3
C) Figure 4
D) Figure 2
Find the center and radius of the sphere. 211) x2 + y2 + z 2 - 8x - 12y + 14z = -1
211)
A) C(4, 6, -7), a = 100 C) C(4, 6, 7), a = 10
B) C(4, 6, -7), a = 10 D) C(-4, -6, 7), a = 10
Write the equation for the plane. 212) The plane through the points P(5, -1, -23) , Q(-3, -6, 53) and R(-1, -2, 23). A) 4x + y + 7z = 8 B) 7x + 4y + z = -8
C) 4x + y + 7z = -8
D) 7x + 4y + z = 8
Calculate the requested distance. 213) The distance from the point S(5, -7, 2) to the line x = 4 + 3t, y = -2 + 12t, z = -3 + 4t A) 3.985729651e+15 B) 3.985729651e+15 7.146825581e+12 9.290873255e+13
C)
212)
7250 13
D)
7250 169
Solve the problem. 214) Find the work done by a force of 7i (newtons) in moving an object along a line from the origin to the point (8, 5) (distance in meters). A) 53.55 joules B) 8.255 joules C) 66.04 joules D) 56.00 joules
40
213)
214)
Express the vector in the form v = v1 i + v2 j + v3 k.
215) AB if A is the point (-3, -9, -4) and B is the point (2, -16, -1) A) v = 5i + 7j + 3k B) v = 5i - 7j - 3k C) v = 5i - 7j + 3k
215) D) v = 5i + 7j - 3k
Describe the given set of points with a single equation or with a pair of equations. 216) The circle of radius 9 centered at the point (8, -7, 6) and lying in a plane perpendicular to the x-axis
A) (y + 7)2 + (z - 6)2 = 81 and y + z = -1 C)
B) (y + 7)2 + (z - 6)2 = 81 and x = 8
y 2 z 2 + = 81 and x = 8 -7 6
D)
y 2 z 2 + + 1 = 81 -7 6
Solve the problem. 217) A ramp leading to the entrance of a building is inclined upward at an angle of 4°. A suitcase is to be pulled up the ramp by a handle that makes an angle of 31° with the horizontal. How much force must be applied in the direction of the handle so that the component of the force parallel to the ramp is 50 lbs.? A) 54.23 lbs B) 58.19 lbs C) 44.55 lbs D) -35.73 lbs Find the indicated vector. 218) Let u = -1, 3 . Find 2u. A) 2, 6
D) -2, 6
219) B) 9 1 i + 2 j + 2 k 27 27 27
A) 9(i + j + k) C) 9 1 i + 2 j + 2 k 3
217)
218) C) 2, -6
B) -2, -6
Express the vector as a product of its length and direction. 219) 3i + 6j + 6k
3
216)
D) 9(3i + 6j + 6k)
3
Find an equation for the line that passes through the given point and satisfies the given conditions. 220) P = (8, 9); parallel to v = 2i - 2j
A) 2x - 2y = -2
B) -2x - 2y = -34
C) 2x - 2y = 8
D) y - 9 = 11 (x - 2)
220)
6
Calculate the direction of P1 P2 and the midpoint of line segment P1 P2 .
221) P1(-8, -4, 7) and P2 (-7, -3, 8)
221)
A) 1 i + 1 j + 1 k; - 15 , - 7 , 15
B) 1 i + 1 j + 1 k; - 15 , - 7 , 15
C) 1 i + 1 j + 1 k; - 7 , - 3 , 4
D) 8 i + 4 j + 7 k; - 4, - 2, 7
3
3
3
3
3
3
2 2
2
2
3
2
3
3
41
3
3
2
3
2
2
2
Find the triple scalar product (u x v) · w of the given vectors. 222) u = 4i + 2j - k; v = 10i + 5j - 10k; w = 3i + 5j - 4k A) 105 B) 65
222) C) 75
D) -205
Identify the type of surface represented by the given equation. 2 2 2 223) x + y = z 8 5 2
A) Ellipsoid C) Elliptical cone
223) B) Hyperbolic paraboloid D) Paraboloid
Find the intersection. 224) x = -10 + 3t, y = -4 - 4t, z = -6 + 3t ; 9x + 6y - 6z = -5 A) - 123 , 232 , - 103 B) (-7, -8, -3) 5 15 5
C) 23 , - 352 , 151 5
15
224)
D) (-13, 0, -9)
5
Find the angle between u and v in radians. 225) u = 10j - 6k, v = 8i - 6j - 8k A) 1.57 B) 1.65
225) C) -0.08
D) 1.61
Find an equation for the line that passes through the given point and satisfies the given conditions. 226) P = (9, 8); perpendicular to v = 5i + 6j A) y - 8 = 1 (x - 5) B) 5x + 6y = 61 2
C) 6x - 5y = 14
226)
D) 5x + 6y = 93
Describe the given set of points with a single equation or with a pair of equations. 227) The set of points equidistant from the points (0, 0, -3) and (0, 0, -1) A) x + y = 0 and -3 < z < -1 B) x + y = -2
C) z > -3 and z < -1
227)
D) z = -2
Identify the type of surface represented by the given equation. 228) x = -7z 2 , no limit on y
A) Parabolic cylinder C) Hyperboloid of two sheets
228) B) Cylinder D) Sphere
Find an equation for the line that passes through the given point and satisfies the given conditions. 229) P = (-9, 2); perpendicular to v = -5i - 5j A) -5x - 5y = 35 B) y - 2 = - 7 (x + 5) 4
C) -5x - 5y = 50
D) -5x + 5y = 55
42
229)
Find the triple scalar product (u x v) · w of the given vectors. 230) u = 2i - 4j + 3k; v = -9i - 5j + 5k; w = 3i - 10j + 10k A) -105 B) 210
230) D) 735
C) -275
Identify the type of surface represented by the given equation. 2 2 2 231) x + y + z = 1 7 6 9
A) Ellipsoid
B) Paraboloid
231) C) Sphere
D) Elliptical cone
Write one or more inequalities that describe the set of points. 232) The exterior of the sphere of radius 4 centered at the point (-2, -2, -5) A) (x - 2)2 + (x - 2)2 + (x - 5)2 < 16 B) (x + 2)2 + (x + 2)2 + (x + 5)2 < 16
C) (x + 2)2 + (x + 2)2 + (x + 5)2 > 16
232)
D) (x - 2)2 + (x - 2)2 + (x - 5)2 16
Give a geometric description of the set of points whose coordinates satisfy the given conditions. 233) 5 z 6 A) The line from z = 5 to z = 6
233)
B) All points between z = 5 and z = 6 in the x-y plane C) No set of points satisfies the given relation D) The slab between the planes z = 5 and z = 6 (including planes) Find a parametrization for the line segment joining the points. 234) (-2, 2, 2), 0, 2, 7 6
234)
A) x = 2t - 2, y = 2t, z = - 5 t + 2, 0 t 1
B) x = -2t, y = 2t, z = 5 t + 7, 0 t 1
C) x = -2t, y = 2, z = 5 t + 7, 0 t 1
D) x = 2t - 2, y = 2, z = - 5 t + 2, 0 t 1
6
6
6
6
Write one or more inequalities that describe the set of points. 235) The interior of the sphere x2 + y2 + x2 = 36
A) x2 + y2 + x2 36 C) x2 + y2 + x2 < 36
235) B) x2 + y2 + x2 6 D) x2 + y2 + x2 < 6
Find an equation for the sphere with the given center and radius. 236) Center (0, 0, 7), radius = 2 A) x2 + y2 + 2 z2 - 14z = -45 B) x2 + y2 + z 2 - 14z = -45
C) x2 + y2 + z 2 + 14z = -45
236)
D) x2 + y2 + 2 z2 + 14z = -45
Find the length and direction (when defined) of u × v. 237) u = 2i + 5j , v = i - j A) 7; -k B) 7; -7k
237) C) 3; 3k
43
D) 3; -3k
Find the indicated vector. 238) Let u = -7, 7 , v = 6, 8 . Find u + v. A) -13, -1 B) 0, 14
238) C) -1, 15
D) 1, 13
Calculate the direction of P1 P2 and the midpoint of line segment P1 P2 .
239) P1(-6, -2, 1) and P2 (-3, 4, -1)
239)
A) 3 i + 6 j - 2 k; - 9 , 1, 0
B) 6 i + 2 j + 1 k; - 3, - 1, 1
C) 3 i + 6 j - 2 k; - 9 , 1, 0
D) 3 i + 6 j - 2 k; - 3 , 2, - 1
7
7
7
7
7
2
7
2
7
7
7
7
7
2
7
2
2
Find v · u.
240) v = 1 , 1 and u = 1 , -1 3
7
A) 2 i 3
3
240)
7
C) 4
B) 0
21
44
D) 1 i - 1 j 3
7
Answer Key Testname: CHAPTER 12
1)
2) Not always true; The statement is false if u v. 3) Always true by definition of the cross product 4)
5) Not always true; The statement if false if c 0,1. 6) Always true by distributive property
45
Answer Key Testname: CHAPTER 12
7)
8) Always true by definition 9)
46
Answer Key Testname: CHAPTER 12
10)
11)
12) u = ux i + uyj and v = vx i + vyj , so a = u + v = (ux + vx ) i + (uy + vy)j and b = u - v = (ux - vx) i + (uy - vy)j Take the dot product a · b: a · b = (u + v) · (u - v) = (ux + vx)(ux - vx) + (uy + vy)(uy - vy) 2 2 2 2 2 2 2 2 = u x - v x + u y - v y = (u x + u y) - (v x + v y) = u - v =1-1=0 Since the dot product of the two non-zero vectors is zero they are orthogonol. 13) Take the dot product: ( a b + b a) · ( a b - b a) = ( a b + b a) · a b - ( a b + b a) · b a = ( a b) · ( a b) + ( b a) · ( a b) - ( a b) · ( b a) - ( b a) · ( b a) = a 2 b·b + a b a · b - a b a · b - b 2 a·a
= a2 b2- b2 a2=0 Since the dot product is zero and since neither vector is identically zero, then the vectors are orthogonal. 14) Always true by definition of 0
47
Answer Key Testname: CHAPTER 12
15) Verify that a + b = a - b . 2
2
Cancel the 2's and square both sides: a + b 2 = a - b 2 or
(a + b)·(a + b) = (a - b)·(a - b) or a·a + 2a·b + b·b = a·a - 2a·b + b·b [2a·b = 0 since a and b are orthogonal] a·a + b·b = a·a + b·b Verified. Thus, the midpoint is equidistant from all three vertices.
16)
17) Always true because u × v and v are orthogonal 18) Not always true; (u × v) · w = u · (v × w), but v × w = -(w × v) from which it follows that the original equation false if w × v 0.
19)
20) Not always true; The statement if false if c 0,1.
48
Answer Key Testname: CHAPTER 12
21)
22) A·B = (au + bv)·(bu - av) = abu·u - a 2 u·v + b2 u·v - abv·v = ab - ab = 0
Since the dot product is zero and since neither A nor B is identically zero, A and B are orthogonal.
23)
24) A 25) B 26) D 27) A 28) D 29) C 30) D 31) A 32) D 33) C 34) D 35) A 36) C 37) A 38) B 49
Answer Key Testname: CHAPTER 12
39) B 40) B 41) C 42) D 43) C 44) C 45) D 46) D 47) B 48) A 49) D 50) C 51) C 52) D 53) A 54) D 55) D 56) B 57) C 58) C 59) B 60) B 61) B 62) B 63) B 64) D 65) B 66) D 67) D 68) B 69) B 70) D 71) B 72) A 73) C 74) C 75) C 76) D 77) B 78) D 79) C 80) D 50
Answer Key Testname: CHAPTER 12
81) B 82) A 83) C 84) D 85) C 86) C 87) B 88) B 89) B 90) A 91) D 92) B 93) A 94) C 95) D 96) C 97) D 98) D 99) D 100) C 101) A 102) D 103) B 104) C 105) C 106) D 107) B 108) A 109) D 110) A 111) C 112) A 113) C 114) D 115) A 116) D 117) A 118) B 119) B 120) B 121) A 122) D 51
Answer Key Testname: CHAPTER 12
123) B 124) B 125) A 126) A 127) B 128) D 129) D 130) B 131) A 132) A 133) A 134) D 135) C 136) C 137) D 138) A 139) A 140) B 141) D 142) B 143) D 144) D 145) D 146) D 147) A 148) C 149) B 150) A 151) D 152) C 153) D 154) B 155) A 156) C 157) D 158) A 159) A 160) B 161) B 162) D 163) D 164) C 52
Answer Key Testname: CHAPTER 12
165) A 166) A 167) B 168) D 169) B 170) D 171) A 172) D 173) D 174) C 175) A 176) B 177) C 178) C 179) A 180) C 181) A 182) C 183) C 184) B 185) C 186) A 187) B 188) B 189) B 190) B 191) B 192) C 193) A 194) D 195) B 196) A 197) D 198) D 199) D 200) C 201) C 202) C 203) C 204) D 205) A 206) C 53
Answer Key Testname: CHAPTER 12
207) D 208) C 209) D 210) A 211) B 212) D 213) C 214) D 215) C 216) B 217) B 218) D 219) C 220) B 221) A 222) A 223) C 224) A 225) B 226) D 227) D 228) A 229) A 230) A 231) A 232) C 233) D 234) D 235) C 236) B 237) A 238) C 239) A 240) C
54
Chapter 13
Exam
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response.
1) Prove that if v and u are differentiable functions of t, then d u + v = du + dv .
1)
2) A golf ball leaves the ground at a 26° angle at a speed of 90 ft/sec. Will it clear the top of a
2)
3) A baseball is hit when it is 3.2 feet above the ground. It leaves the bat with an initial
3)
4) Show that if a particle's velocity vector is always orthogonal to its acceleration vector then
4)
dt
dt
dt
29 ft tree that is in the way, 130 ft down the fairway? Explain.
velocity of 143 ft/sec at a launch angle of 25°. At the instant the ball is hit, an instantaneous gust of wind blows against the ball, adding a component of 15i ft/sec to the ball's initial velocity. Find the range and flight time of the ball, assuming that the ball is not caught.
the particle's speed is constant. b
5) Prove that a
(r1 (t) + r2 (t)) dt =
b a
r1 (t) dt +
b a
r2 (t) dt
6) A human cannonball is to be fired with an initial speed of 90 ft/sec. The circus performer
5)
6)
hopes to land on a cushion located 200 feet downrange at the same height as the muzzle of the cannon. The circus is being held in a large room with a flat ceiling 35 feet higher than the muzzle. Can the performer be fired to the cushion without striking the ceiling? If so, what is the proper firing angle? (g = 32 ft/sec2 )
Solve the problem. 7) Show that a planet in a circular orbit moves with constant speed. Provide an appropriate response. 8) Prove that if u is a differentiable function of t and f is a differentiable scalar function of t, d df du then (fu) = u + f . dt dt dt
1
7)
8)
9) Derive the equations
9)
x = x0 + (v0 cos )t, y = y0 + (v0 sin )t -
1 2 gt 2
by solving the following initial value problem for a vector r in the plane. d2 r Differential equation: = -gj dt2 Initial conditions:
r(0) = x0 i + y0j dr (0) = v0 cos )i + (v0 sin )j dt
b
10) Prove that
b
kr(t) dt = k
a
r(t) dt for any scalar constant k.
10)
a
11) Derive the equations
11)
x = x0 +
v0 (1 - e-kt)cos , k
y = y0 +
v0 (1 - e-kt)sin k
+
g (1 - kt - e-kt) k2
by solving the following initial value problem for a vector r in the plane. d2 r dr = -gj - kv = -gj - k Differential equation: 2 dt dt Initial conditions:
r(0) = x0 i + y0j dr (0) = v0 = (v0 cos )i + (v0 sin )j dt
The drag coefficient k is a positive constant representing resistance due to air density, v0 and are the projectile's initial speed and launch angle, and g is the acceleration of gravity.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Calculate the arc length of the indicated portion of the curve r(t). 12) r(t) = (4 - 2t)i + (2 + 9t)j + (6t - 6)k, -1 t 8 A) 847 B) 1089 C) 99 Find the unit tangent vector of the given curve. 13) r(t) = (6 - 2t)i + (2t - 1)j + (4 + t)k A) T = 2 i - 2 j - 1 k 3 3 3 3
D) 77
13) B) T = - 2 i + 2 j + 1 k 9
C) T = - 2 i + 2 j + 1 k 3
12)
9
9
D) T = 2 i - 2 j - 1 k
3
9
2
9
9
Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured from the horizontal, and that the projectile is launched from the origin over a horizontal surface 14) An ideal projectile is launched from level ground at a launch angle of 26° and an initial speed of 14) 48 m/sec. How far away from the launch point does the projectile hit the ground? A) 60 m B) 185 m C) 230 m D) 290 m
The position vector of a particle is r(t). Find the requested vector. 15) The velocity at t = 0 for r(t) = ln(t3 - 9t2 + 4)i - 2t + 16j - 4cos(t)k
15)
A) v(0) = 1 i - 1 j + 4k 4 8
B) v(0) = 1 j 4
C) v(0) = - 1 j
D) v(0) = 1 i - 1 j
4
4
4
The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors at time t = 0. 16) r(t) = sin-1(5t)i + ln(5t2 + 1)j + 7t2 + 1k 16)
A)
B) 0
3
C)
D)
4
2
Find T, N, and B for the given space curve. 17) r(t) = 2 + 5 sin 4 t i + 4 + 5 cos 4 t j + 3tk 5 5
17)
A) T = 4 (sin 0.8t)i - 4 (cos 0.8t)j ; N = (-sin 0.8t)i - (cos 0.8t)j ; B = 3 (cos 0.8t)i - 3 (sin 0.8t)j 5
5
5
5
4 k 5
B) T = 4 (cos 0.8t)i - 4 (sin 0.8t)j + 3 k ; N = (-sin 0.8t)i - (cos 0.8t)j ; B = - 4 k 5
5
5
5
C) T = 4 (cos 0.8t)i - 4 (sin 0.8t)j ; N = (-sin 0.8t)i - (cos 0.8t)j ; B = 3 (cos 0.8t)i - 3 (sin 0.8t)j 5
5
5
5
4 k 5
D) T = 4 (cos 0.8t)i - 4 (sin 0.8t)j + 3 k ; N = (-sin 0.8t)i - (cos 0.8t)j ; B = 3 (cos 0.8t)i - 3 (sin 5
0.8t)j -
5
5
5
5
4 k 5
t
Find the arc length parameter along the curve from the point where t = 0 by evaluating s =
18) r(t) = (etcos t)i + (etsin t)j + 6etk A) 34et - 38 B)
|v( )| d .
0
18) 38et -
38
C)
3
38et -
34
D)
34et -
34
Find the acceleration vector in terms of ur and u .
19) r = 2 cos 4t and = 3t A) a = (-50 cos 4t)ur + 48 sin 4tu C) a = (-50 cos 3t)ur - 48 sin 3tu
19) B) a = (-50 sin 4t)ur - 48 cos 4tu D) a = (-50 cos 4t)ur - 48 sin 4tu
Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured from the horizontal, and that the projectile is launched from the origin over a horizontal surface 20) A spring gun at ground level fires a tennis ball at an angle of 50°. The ball lands 18 m away. What 20) was the ball's initial speed? Round your answer to the nearest tenth. A) 179.1 m/sec B) 13.4 m/sec C) 4.3 m/sec D) 15.2 m/sec
Provide an appropriate response. 21) A baseball is hit when it is 2.7 feet above the ground. It leaves the bat with an initial speed of 153 ft/sec, making an angle of 22° with the horizontal. Assuming a drag coefficient k = 0.10, how high does the ball go, and when does it reach maximum height?
21)
For projectiles with linear drag: v0 x = x0 + (1 - e-kt) cos , k y = y0 +
v0 (1 - e-kt) sin k
+
g (1 - kt - e-kt) k2
where k is the drag coefficient, v0 and
are the projectile's initial speed and launch angle, and g is
the acceleration of gravity (32 ft/sec2 ). A) Maximum height = 3.05 feet; time to maximum height = 0.05 sec C) Maximum height = 8.27 feet; time to maximum height = 0.74 sec
B) Maximum height = 3.07 feet;
time to maximum height = 0.02 sec
D) Maximum height = -0.23 feet;
time to maximum height = -0.21 sec
Find the length of the indicated portion of the curve. 22) r(t) = (1 + 3t)i + (1 + 6t)j + (2 - 2t)k, -1 t 0 A) 2 10 B) 3 5
22) C) 7
D) 13
Find the velocity vector in terms of ur and u .
23) r = ea and d = 3
23)
dt
A) v = 3ea ur + 3aea u
B) v = 3aea ur + 3aea u
C) v = 3aea ur + 3ea u
D) v = 3ea ur + 3ea u
4
Provide an appropriate response. 24) The following equations each describe the motion of a particle. For which path is the particle's velocity vector always orthogonal to its acceleration vector? (1) r(t) = t6 i + t3 j
24)
(2) r(t) = cos (2t)i + sin (6t)j (3) r(t) = ti + t3 j
(4) r(t) = cos (-8t)i + sin (-8t)j A) Path (2) and Path (4)
B) Path (4) D) Path (3)
C) Path (1)
25) Increasing the initial speed of a projectile by a factor of 7 increases its range by what factor?
25)
Assume the elevation is the same. A) The elevation must be specified before an answer can be found.
B) Factor of 2.65 C) Factor of 7 D) Factor of 49 For the curve r(t), find an equation for the indicated plane at the given value of t. 26) r(t) = (10 sin 6 t + 9)i + (10 cos 13t) - 3)j + 5tk; osculating plane at t = 2.5 . 5
26)
A) 5 (y + 3) + 10 z + 25 = 0
B) 5 (x - 9) + 10 z - 25
C) 5 (y + 3) + 10 z - 25
D) 5 (y - 3) + 10 z - 25
13
2
13
2
13
=0
169
=0
2
13
27) r(t) = (7t sin t + 7 cos t)i + (7t cos t - 7 sin t)j + 9k; normal plane at t = 1.5 . A) y = 7 B) x + y + z = -7 C) y = -7
2
=0
27) D) x - y + z = -7
Find the acceleration vector in terms of ur and u .
28) r = a(9 - cos ) and d = 8
28)
dt
A) a = 64a(2 cos - 9)ur + (128a sin )u B) a = 64a( cos + 9a - a cos )ur - (128a sin )u C) a = 64a( cos - a + a cos )ur + (128a sin )u D) a = 81a( cos - 8a + a cos )ur + (162a sin )u Evaluate the integral. 1 29) (30t2 - 7)i + 0
4t j2 t +1
t k dt 2 t +1
29)
A) 3i + 2 ln 2j + (1 - 2)k
B) 17i + 2 ln 2j + (1 + 2)k
C) 17i + 2 ln 2j + 1 - 2 k
D) 17i + 2 ln 2j + (1 - 2)k
2
5
Solve the problem. 30) The orbit of a satellite has a period of T = 1436.8 minutes. Calculate the semimajor axis of the satellite. (Earth's mass = 5.975 × 1024 kg and G = 6.6720 × 10-11 Nm 2 kg-2).
A) 2.751 × 106 km
B) 2751 km
C) 42,170 km
30)
D) 4.217 × 107 km t
Find the arc length parameter along the curve from the point where t = 0 by evaluating s =
|v( )| d .
0
31) r(t) = (3 + 2t)i + (3 + 3t)j + (4 - 6t)k A) 5t B) 9t
31) C) 8t
D) 7t
Find the principal unit normal vector N for the curve r(t). 32) r(t) = (9t sin t + 9 cos t)i + (9t cos t - 9 sin t))k
32)
A) N = (sin t)i + (cos t)k C) N = -
B) N = (-sin t)i - (cos t)k
2 2 (cos t)i (sin t)k 2 2
D) N = (cos t)i - (sin t)k
33) r(t) = (6 + t)i + (5 + ln(cos t))k, - /2 < t < /2 A) N = (-cos t)i - (tan t)k C) N = (-cos t)i - (ln(cos t))k
33) B) N = (-sin t)i - (cos t)k D) N = (cos t)i + (sin t)k
The position vector of a particle is r(t). Find the requested vector.
34) The acceleration at t = A) a C) a
8 8
8
for r(t) = (7 sin 4t)i - (8 cos 4t)j + (2 csc 4t)k
= -112i - 32k
B) a
= 112i + 32k
D) a
Solve the initial value problem.
35) Differential Equation: dr = (3 sec 3t tan 3t) i + dt
Initial Condition: r(0) = 3i +
2 (t2 + 4)
55 j - 3k 8
A) r(t) = (csc 3t + 2)i + 7 -
1 t3 -3k j+ 2 3 2t + 8
B) r(t) = (sec 3t + 2)i + 7 -
1 t3 -3k j+ 3 2t2 + 8
C) r(t) = (csc 3t + 2)i + 7 -
1 j + (t3 - 3)k 2 2t + 8
D) r(t) = (sec 3t + 2)i -
t
1 t3 -3 k j+ 3 2t2 + 8
6
8 8
j + t2 k
34)
= -112i + 32k = 128j + 32k
35)
Find the unit tangent vector of the given curve. 36) r(t) = 3t4 i - 12t4j + 4t4 k
36)
A) T = 3 i + 12 j + 4 k 13 13 13
B) T = 3 i - 12 j - 4 k 13 13 13
C) T = 3 i - 12 j + 4 k
D) T = 3 i - 12 j + 4 k
169
169
169
13
Find the torsion of the space curve. 37) r(t) = (cosh t)i + tj + (sinh t)k 2 A) = tanh t B) = 0 2 Find T, N, and B for the given space curve. 38) r(t) = (t2 - 2)i + (2t - 2)j + 10k
A) T = B) T =
t
t2 + 1
t2 + 1
j; N = -
37) C) = -
tanh2 t 2
D)
sech2 t =2
38) 1
t2 + 1
i -
t i + 2 t +1
1 j; N = 2 t +1
t
t2 + 1
1 i 2 t +1
j; B = -k
t j; B = -k 2 t +1
t 1 1 t i + j; N = i j; B = -k 2 2 2 2 t +1 t +1 t +1 t +1
Solve the problem.
39) Find the values of v0 in the equation e = r=
13
t 1 1 t i + j; N = i j; B = k 2 2 2 2 t +1 t +1 t +1 t +1
C) T = D) T =
i -
1
13
(1 + e)r0
r0 v0 2 - 1 that make the orbit described by GM
39)
parabolic.
1 + e cos
A) v0 = 2GM r0
B) v0 = 2GMr0
C) v0 = 2GM
D) v0 =
2GM r0
For the curve r(t), write the acceleration in the form a T T + a NN.
40) r(t) = (t + 3)i + (ln(cos t) - 7)j + 6k A) a = (sec t tan t)T - (sec t)N C) a = (-sec t tan t)T + (sec t)N
40) B) a = (-sec t tan t)T - (sec t)N D) a = (sec t tan t)T + (sec t)N
7
Solve the initial value problem.
41) Differential Equation: dr = i + (5t3 - 3t)j + 1 k dt
41)
t+9
Initial Condition: r(0) = 2j + (ln 9)k 2 A) r(t) = ti + t (5t2 - 6) + 2 j + ln(t + 9)k 4
2
B) r(t) = ti + t (5t2 - 6) + 1 j + (ln 9)k 4
2
2
C) r(t) = i + t (5t2 - 6) + 2 j + (ln 9)k
D) r(t) = ti + t (5t2 - 6) + 2 j + (ln t)k
4
2
Find the length of the indicated portion of the curve. 42) r(t) = (4 + 2t)i + (2 + 3t)j + (5 - 6t)k, -1 t 0 A) 5 B) 7
42) C) 8
D) 9
Solve the initial value problem.
43) Differential Equation: dr = 3 (t + 3)1/2i + etj dt
43)
2
Initial Condition: r(0) = 0 A) r(t) = [(t + 3)3/2 - 3 3/2]i + (et - 1)j
B) r(t) = [(t + 3)5/2 - 3 5/2]i + (et - 1)j D) r(t) = (t + 3)3/2i + etj
C) r(t) = [(t + 3)3/2 + 3 3/2]i + (et + 1)j
Solve the problem. 44) A space shuttle is in a circular orbit 250 km above the Earth's surface. Use Kepler's third law( with a = Earth's radius + 250 km) to find the orbital period of the satellite.(G = 6.67 × 10-11 Nm 2 kg-2 ,
44)
M = 5.97 × 1024 kg and Earth's radius is 6.37 × 106 m). A) T = 2 hr B) T = 9 min
C) T = 1 hr
D) T = 90 min
Calculate the arc length of the indicated portion of the curve r(t). 45) r(t) = 2t2 i + 11t2j + 10t2 k, 1 t 2
A) 22.5
B) 337.5
45)
C) 675
D) 45
For the curve r(t), find an equation for the indicated plane at the given value of t. 46) r(t) = (cosh t)i + (sinh t)j + tk; rectifying plane at t = 0. A) x - y = -1 B) x = 1 C) x - y = 1 Evaluate the integral. 4 2t 47) j + 6 tk dt -i 2 )2 + (9 t 0 A) -4i - 16 j - 32k 225
46) D) x = -1
47) B) -4i - 16 j + 32k 225
C) -4i + 4 j + 32k
D) +4i + 16 j + 32k
117
225
8
Solve the initial value problem.
48) Differential Equation: dr = (12t2 - 6)i - j + dt
1 k 1+t
48)
Initial Condition: r(0) = -4i + 8j + 8k A) r(t) = (4t3 + 6t + 4)i + (8 + t)j + ( 1 + t + 6)k B) r(t) = (4t3 - 6t - 4)i + (8 - t)j + ( 1 + t - 6)k
C) r(t) = (12t3 - 6t - 4)i + (8 - t)j + ( 1 + t + 6)k D) r(t) = (4t3 - 6t - 4)i + (8 - t)j + (2 1 + t + 6)k The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors at time t = 0. 49) r(t) = (5t2 + 10)i + (2t3 - 7t)k 49)
A)
4
C) 0
B)
D)
2
If r(t) is the position vector of a particle in the plane at time t, find the indicated vector. 50) Find the velocity vector. r(t) = (cot t)i + (csc t)j A) v = (csc2 t)i + (cot t csc t)j B) v = (-sec2 t)i - (tan t sec t)j
C) v = (-csc2 t)i - (cot t csc t)j
D) v = (sec2 t)i + (tan t sec t)j
Find the torsion of the space curve. 51) r(t) = (t - 1)i + (ln(sec t) + 1)j - 4k, - /2 < t < /2 A) Undefined B) = -1
51) C) = 0
D) = 1
Find T, N, and B for the given space curve. 52) r(t) = (cosh t)i + (sinh t)j + tk A) T = - 2 (tanh t)i + 2 j + 2 (sech t)k; N = (-sech t)i - (sinh t)k ; 2 2 2 B=-
B) T = B=
C) T = B=-
D) T = B=
50)
2 2 2 ( sinh t)i + j(sech t)k 2 2 2 2 2 2 (tanh t sech t)i + j + (sech t)k; N = (-sech t)i - (tanh t)k; 2 2 2 2 2 2 (sinh t)i j(sech t)k 2 2 2 2 2 2 (tanh t)i + j + (sech t)k; N = (sech t)i - (tanh t)k; 2 2 2 2 2 2 (tanh t)i + j(sech t)k 2 2 2 2 2 2 (tanh t sech t)i + j + (sech t)k; N = (sech2 t)i - (sinh t)k; 2 2 2 2 2 2 (sinh t)i + j+ (sech t)k 2 2 2
9
52)
If r(t) is the position vector of a particle in the plane at time t, find the indicated vector. 53) Find the acceleration vector. r(t) = (cos 5t)i + (2 sin t)j A) a = (25 cos 5t)i + (-2 sin t)j B) a = (-25 cos 5t)i + (-2 sin t)j
C) a = (-5 cos 5t)i + (2 sin t)j
53)
D) a = (-25 cos 5t)i + (-4 sin t)j
Provide an appropriate response. 54) The following equations each describe the motion of a particle. For which path is the particle's speed constant? (1) r(t) = t3 i + t5 j
54)
(2) r(t) = cos (9t)i + sin (3t)j (3) r(t) = ti + tj (4) r(t) = cos (-7t2 )i + sin (-7t2 )j
A) Path (2) and Path (3) C) Path (3)
B) Path (4) and Path (2) D) Path (1)
Calculate the arc length of the indicated portion of the curve r(t). 55) r(t) = (8t sin t + 8 cos t)i + (8 t cos t - 8 sin t)j ; -3 t 5 A) 64 B) 128 C) 272
55) D) 136
For the curve r(t), write the acceleration in the form a T T + a NN.
56) r(t) = (t2 - 5)i + ( 2t - 4)j + 8k
56)
A) a = 2t T + 2 N t2 + 1 t2 + 1 C) a =
t
t2 + 1
T+
1
t2 + 1
N
Find the length of the indicated portion of the curve. 57) r(t) = (2cos t)i + (2sin t)j + 5tk, 0 t /2 A) 29t B) 29t 2 4
B) a =
2t T+ t2 + 1
2 N t2 + 1
D) a =
t T+ t2 + 1
1 N t2 + 1
57) C)
33t 2
D)
7t 2
Calculate the arc length of the indicated portion of the curve r(t). 58) r(t) = (10 + 2t2 )i + (2t2 - 7)j + (8 - t2 )k, -1 t 2
A) 9
B) 27
58)
C) 45
D) 15
For the curve r(t), write the acceleration in the form a T T + a NN.
59) r(t) = 8 (1 + t)3/2i + 8 (1 - t)3/2j + 2 tk 9
9
A) a = 2 3
C) a = 1 3
2
1 - t2 1
1 - t2
59)
3
T
B) a = 2
N
D) a = T + 2
3
2
1 - t2 3
10
N 2
1 - t2
N
Solve the problem. 60) The orbit of a satellite had a period of T = 93.11 minutes. Calculate the semimajor axis of the orbit. (Earth's mass = 5.975 × 1024 kg and G = 6.6720 × 10-11 Nm 2 kg-2 ).
A) 4.440 × 104 km
C) 6.805 × 106 km
B) 444 km
D) 6805 km
Provide an appropriate response. 61) What two angles of elevation will enable a projectile to reach a target 18 km downrange on the same level as the gun if the projectile's initial speed is 440 m/sec? Assume there is no wind resistance. A) 0.03° and 89.97° B) 32.83° and 57.17°
C) 65.66° and 24.34°
60)
61)
D) 32.83°
Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured from the horizontal, and that the projectile is launched from the origin over a horizontal surface 62) A projectile is fired with an initial speed of 500 m/sec at an angle of 45°. What is the greatest 62) height reached by the projectile? Round your answer to the nearest tenth. A) 6377.6 m B) 25,510.2 m C) 62,500.0 m D) 72.2 m
Solve the initial value problem.
63) Differential Equation: dr = (t4 + 8t2 )i + 4tj
63)
dt
Initial Condition: r(0) = -2i + 5j 5 3 A) r(t) = t + 8t + 5 i + (2t2 - 2)j 5 3
5
3
5
3
5
3
5
3
5
3
5
3
B) r(t) = t + 8t
C) r(t) = t + 8t - 2 i + (t2 + 5)j
i + 2t2 j
D) r(t) = t + 8t - 2 i + (2t2 + 5)j
Calculate the arc length of the indicated portion of the curve r(t). 64) Following the curve r(t) = 12ti + (5 cos t)j + (5 sin t)k in the direction of increasing of arc length, find the point that lies 91 units away from the point where t = 0. A) (84 , -5, 0) B) (84 , 0, -5) C) (84, -5, 0) D) (84 , 0, 0) The position vector of a particle is r(t). Find the requested vector. 65) The velocity at t = 4 for r(t) = (8 - 1t2)i + (5t + 7)j - e-4tk
A) v(4) = -4i +5j + 4e-16k C) v(4) = -8i + 5j + 4e-16k
64)
65)
B) v(4) = 8i + 5j + 4e-16k D) v(4) = -8i + 5j - 4e-16k
Find T, N, and B for the given space curve. 66) r(t) = (ln(cos t) + 9)i + 3j + (2 + t )k, - /2 < t < /2 A) T = (sin t)i + (cos t)k; N = (cos t)i - (sin t)k; B = j
B) T = (-sin t)i + (cos t)k; N = (-cos t)i - (sin t)k; B = -j C) T = (sin t)i + (cos t)k; N = (cos t)i - (sin t)k; B = -j D) T = (-sin t)i + (cos t)k; N = (-cos t)i - (sin t)k; B = j
11
66)
The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors at time t = 0.
67) r(t) = ti + ( 3t + A)
6
t2 )k
67) B) 0
6
C)
Evaluate the integral. 2 68) ((5 - 7t)i + 5 t j) dt 1 A) 5 2 - 2 j 4
1
69)
3
1
i + te10tj -
2t - t2
0
B) - 11 i + 10 (2 2 - 1) j 2
3
D) - 11 + 10 (2 2 - 1) 2
9t2
2 (10 + 3t3 )
3
k dt
69)
10 A) 9e + 1 j + 3 k
B)
10 C) - 3 i + 9e + 1 j + 3 k
D) - 3 i + 9e j + 3 k
100
2
2
68)
C) - 5 i + 10 (2 2 - 1) j 2
D)
40
100
2
i+
9e10 + 1 3 jk 100 130 10
10
2
Find the curvature of the curve r(t). 70) r(t) = (3 + ln(sec t))i + (8 + t)k, - /2 < t < /2 A) = -cos t B) = sin t
100
40
70) C) = 1 - cos t
D) = cos t
71) r(t) = (4 + cos 4t - sin 4t)i + (3 + sin 4t + cos 4t)j + 7k A)
2 = 2
B) = 2
71) C) = 2
D) = 2 2
The position vector of a particle is r(t). Find the requested vector. 72) The velocity at t = 2 for r(t) = (5t2 + 5t + 2)i - 7t3 j + (8 - t2)k
A) v(2) = 15i - 84j - 4k C) v(2) = 25i - 84j - 4k
72)
B) v(2) = 25i + 84j + 4k D) v(2) = 15i - 28j - 2k
For the curve r(t), write the acceleration in the form a T T + a NN.
73) r(t) = (t + 1)i + (ln(sec t) - 3)j + 4k, - /2 < t < /2 A) a = (csc t)T + (sec t)N C) a = (cos t)T + (cos t)N
73) B) a = (sec2 t)T + (cos t)N D) a = (sec t tan t)T + (sec t)N
12
For the curve r(t), find an equation for the indicated plane at the given value of t. 74) r(t) = 4(1 + t)3/2i + 4(1 - t)3/2j + 3tk; osculating plane at t = 0.
A) x - y = - 4
B) x - y = 0
Find the curvature of the space curve. 75) r(t) = ti + (sinh t)j + (cosh t)k
A) = sinh2 t
C) x - y = 4
74) D) x - y = 8
75) B) =
sech2 t
C) = cosh2 t
2
Find the principal unit normal vector N for the curve r(t). 76) r(t) = (10 sin t)i + (10 cos t)j + 8k A) N = (-cos t)i - (sin t)j
D) =
tanh2 t 2
76) B) N = (-sin t)i - (cos t)j D) N = (-sin t)i + (cos t)j
C) N = (sin t)i + (cos t)j
Provide an appropriate response. 77) A baseball is hit when it is 2.7 feet above the ground. It leaves the bat with an initial speed of 150 ft/sec, making an angle of 18° with the horizontal. Assuming a drag coefficient k = 0.12, find the range and the flight time of the ball.
77)
For projectiles with linear drag: v0 x = x0 + (1 - e-kt)cos , k y = y0 +
v0 (1 - e-kt)sin k
+
g (1 - kt - e-kt) k2
where k is the drag coefficient, v0 and
are the projectile's initial speed and launch angle, and g is
the acceleration of gravity (32 ft/sec2 ). A) Flight time = 2.93 sec; range = 351.6 feet
B) Flight time = 2.81 sec; range = 340.3 feet D) Flight time = 3.16 sec; range = 362.5 feet
C) Flight time = 2.01 sec; range = 331.6 feet
Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured from the horizontal, and that the projectile is launched from the origin over a horizontal surface 78) A baseball is hit when it is 2.4 ft above the ground. It leaves the bat with an initial velocity of 78) 137 ft/sec at a launch angle of 29°. At the instant the ball is hit, an instantaneous gust of wind blows against the ball, adding a component of -14.2i (ft/sec) to the ball's initial velocity. Find a vector equation for the path of the baseball. A) r = (137 sin 29° - 14.2)ti + (2.4 + (137 cos 29°)t - 16t2 )j
B) r = (137 cos 29° - 14.2)ti + (2.4 + (137 sin 29°)t + 16t2 j C) r = ((137 cos 29°)t - 14.2)i + (2.4 + (137 sin 29°)t - 16t2)j D) r = (137 cos 29° - 14.2)ti + (2.4 + (137 sin 29°)t - 16t2 )j Calculate the arc length of the indicated portion of the curve r(t). 79) r(t) = 4 2t3/2i + (6t sin t)j + (6t cos t)k ; -4 t 6
A) 120
B) 180
C) 324
13
79) D) 168
Find the curvature of the curve r(t).
80) r(t) = (8t + 5)i - 9j + (2 - 4t2 )k
80)
A) = 8 1 + t2 C) =
B)
1
3/2 = 8 1 + t2
D) =
8 1 + t2
1
3/2 8 1 + t2
Find T, N, and B for the given space curve. 81) r(t) = (8 + t)i + (4 + ln(sec t))j - 8k, - /2 < t < /2 A) T = (cos t)i + (sin t)j; N = (- sin t)i + (cos t)j ; B = k
81)
B) T = (cos t)i - (sin t)j; N = (-sin t)i - (cos t)j ; B = -k C) T = (-cos t)i - (sin t)j; N = (sin t)i - (cos t)j ; B = -k D) T = (-cos t)i - (sin t)j; N = (-cos t)i + (sin t)j ; B = k Evaluate the integral. /4 82) (9 cos t i + 3 sin t j) dt - /4 A) 3 2 i B) 0
82) C) 9 2 i + 3 2 j
D) 9 2 i
Solve the problem. 83) The orbit of a satellite had a semimajor axis of a = 6955 km. Calculate the period of the satellite. (Earth's mass = 5.975 × 1024 kg and G = 6.6720 × 10-11 Nm 2 kg-2 ).
A) 124 min
B) 96.2 min
C) 0.18 sec
83)
D) 9250 hr
Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured from the horizontal, and that the projectile is launched from the origin over a horizontal surface 84) A baseball is hit when it is 3.4 ft above the ground. It leaves the bat with an initial velocity of 84) 152 ft/sec at a launch angle of 26°. At the instant the ball is hit, an instantaneous gust of wind blows against the ball, adding a component of -9.9i (ft/sec) to the ball's initial velocity. How high does the baseball go? Round your answer to the nearest tenth. A) 72.8 ft B) 69.4 ft C) 74.8 ft D) 66.6 ft
For the curve r(t), write the acceleration in the form a T T + a NN.
85) r(t) = 6 sin 2t + 9 i + 6 cos 2t - 3 j + 5tk A) a = T + 144N C) a = 24T
85) B) a = 144T + 144N D) a = 24N
Find the curvature of the curve r(t). 86) r(t) = (8 + 7 cos 8t) i - (3 + 7 sin 8t)j + 10k A) = 1 B) = 1 7 49
86) C) = 7
14
D)
8 = 7
Evaluate the integral. 1 6 87) 4ti + 12t2 j k dt (1 + t)4 0 A) 2i + 4j - 7 k B) 4i + 8j - 2k 4
87) C) 2i - 4j - 7 k
D) 4i - 8j + 2k
4
The position vector of a particle is r(t). Find the requested vector.
88) The acceleration at t = A) a C) a
8 8
=
8
for r(t) = (10t - 2t3 )i + 3tan(2t)j + e4tk
3 i + 48j + 16e1/2 k 2
=-
B) a
3 i + 48j + 16e1/2 k 2
D) a
8 8
88) =-
3 i - 48j + 16k 2
=-
3 i - 48j + 16e1/2 k 2
Find the length of the indicated portion of the curve. 89) r(t) = (etcos t)i + (etsin t)j + 5etk, -ln 2 t 0
A) 3 3 4
B)
89)
23 2
C)
23 4
D) 3 3 2
Find the torsion of the space curve. 90) r(t) = (2t - 3)i + (t2 - 9)j + 8k
A) = -1
90) B) Undefined
C) = 1
D) = 0
Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured from the horizontal, and that the projectile is launched from the origin over a horizontal surface 91) A fan in the bleachers at Wrigley Field throws an opposing player's home run baseball back onto 91) the playing field. Assume that the fan is 30 feet above the field and that the ball is launched at an angle of 35°. When will the ball hit the ground if its initial speed is 27 ft/sec? Round your answer to the nearest tenth. A) 1.9 sec B) 4.5 sec C) 2.1 sec D) 1.0 sec
Find the unit tangent vector of the given curve. 92) r(t) = (3 + 11t4 )i + (9 + 2t4 )j + (2 + 10t4)k
92)
A) T = 11 i + 2 j + 2 k 15 15 3
B) T = 44 i + 8 j + 8 k 15 15 3
C) T = 11i + 2j + 10k
D) T = 11 i + 2 j + 2 k 225
15
225
45
Solve the initial value problem.
2
93) Differential Equation: d r = (4t - 9)i
93)
dt2
Initial Conditions:
dr = -k, r(0) = 8i + 4j + 8k dt r = 0
A) r(t) = 4 t3 - 9t2 + 8 i + 4j + (8 - t)k
B) r(t) = 2 t3 + 9 t2 + 8 i - 4j + (8 - t)k
C) r(t) = 4 t3 + 9 t2 + 8 i - 4j + (t - 8)k
D) r(t) = 2 t3 - 9 t2 + 8 i + 4j + (8 - t)k
3 3
3
2
3
2 2
Find the unit tangent vector of the given curve. 94) r(t) = (2t cos t - 2 sin t)j + (2t sin t + 2 cos t)k
94)
A) T = (2 cos t)j - (2 sin t)k
B) T = (-sin t)j + (cos t)k
C) T = - 1 (sin t)j + 1 (cos t)k
D) T = (-2 sin t)j + (2 cos t)k
2
2
t
Find the arc length parameter along the curve from the point where t = 0 by evaluating s =
|v( )| d .
0
95) r(t) = (1 + 2t)i + (1 + 8t)j + (5 - 5t)k A) 93t B) 89t
95) C) 29t
D) 2 17t
For the smooth curve r(t), find the parametric equations for the line that is tangent to r at the given parameter value t = t0 .
96) r(t) = (2 tan t)i - (3 sin t)j + (7 cos2 t)k ; t0 =
96)
4
A) x = 2 + 4t, y = - 3 2 - 3 2 t, z = 7 - 7t
B) x = 2 + 4t, y = - 3 - 3 t, z = 7 - 7t
C) x = 2 + 4t, y = - 3 2 + 3 2 t, z = 7 + 7t
D) x = 4 + 2t, y = - 3 2 - 3 2 t, z= -7 + 7 t
2
2
2
2
2
2
2
2
Find the unit tangent vector of the given curve. 97) r(t) = 4 cos 3t i + 4 sin 3t j - 5tk A) T = - 4 sin 3t i + 4 cos 3t j - 5 k 13 13 13 169
169
C) T = - 12 cos 3t i + 12 sin 3t j - 5 k 13
13
2
2
97)
B) T = - 12 cos 3t i + 12 sin 3t j - 5 k 169
2
13
D) T = - 12 sin 3t i + 12 cos 3t j - 5 k 13 13 13
16
Solve the initial value problem.
2
98) Differential Equation: d r = 2t2 i - tj
98)
dt2
Initial Conditions:
dr = -10i, r(0) = 6i - 3k dt r = 0
3
3
A) r(t) = 1 t4 + 10t + 6 i + t j - 3k 6
B) r(t) = 1 t4 - 10t + 6 i - t j - 3k
6
6
3
3
C) r(t) = 1 t4 - 10t + 6 i - t j - 3k 3
6
D) r(t) = 1 t4 - 10t + 6 i - t j - 3k
6
6
2
Find T, N, and B for the given space curve. 99) r(t) = (6t sin t + 6cos t)i + (6t cos t - 6 sin t)j - 5k A) T = (cos t)i - (sin t)j; N = (-sin t)i - (cos t)j; B = -k
99)
B) T = (-cos t)i - (sin t)j; N = (sin t)i - (cos t)j; B = -k C) T = (-cos t)i - (sin t)j; N = (sin t)i - (cos t)j; B = -5k D) T = (cos t)i + (sin t)j; N = (-sin t)i + (cos t)j; B = k The position vector of a particle is r(t). Find the requested vector. 100) The acceleration at t = 1 for r(t) = t5 i + 6ln 1 j + 2 k 4+t t
100)
A) a(1) = 20i + 6j + 4k
B) a(1) = 20i - 6j - 4k
C) a(1) = 20i + 8.444249301e+14 j - 4k
D) a(1) = 20i + 8.444249301e+14 j + 4k
3.518437209e+15
3.518437209e+15
Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured from the horizontal, and that the projectile is launched from the origin over a horizontal surface 101) Find the muzzle speed of a gun whose maximum range is 18.7 km. Round your answer to the 101) nearest tenth. A) 183.8 km/sec B) 13.5 km/sec C) 183.3 km/sec D) 14.0 km/sec
Solve the problem. 102) At time t = 0 a particle is located at the point (1, -2, 4). It travels in a straight line to the point (3, 1, 2), has speed 3 at (1, -2, 4) and constant acceleration 4i - j - k. Find an equation for the position vector r(t) of the particle at time t. A) r(t) = 2t2 + 2 t + 1 i - 1 t2 - 3 t + 2 j - 1 t2 + 2 t - 4 k 2 2 17 17 17
B) r(t) = 4t2 + 6 t + 1 i - t2 - 9 t + 2 j - t2 + 17
17
C) r(t) = 2t2 + 6 t + 1 i - 1 t2 17
2
6 t-4 k 17
9 1 t + 2 j - t2 + 2 17
6 t-4 k 17
D) r(t) = 2t2 + 3 t + 1 i - 1 t2 + 6 t + 2 j - 1 t2 - 12 t - 4 k 21
2
21
2
17
21
102)
Evaluate the integral. /4 103) [(4sec2 t)i - (9 + sin t)j - (8sec t tan t)k)]dt 0 A) 4i + 2 2 - 9 - 4 j + 8(1 + 2)k 2
103) B) 4i + 2 2 + 9 + 4 j + 8(1 + 2)k 2
C) 4i + 2 2 - 9 - 4 j + 8(1 - 2)k
D) 4i + 2 2 - 9 - 4 j + 8(1 - 2)k
4
2
If r(t) is the position vector of a particle in the plane at time t, find the indicated vector. 104) Find the velocity vector. 1 3 r(t) = (-7t2 + 15)i + t j 15
A) v = (-14)i + 2 t j
B) v = (-14t)i + 1 t2 j
C) v = 1 t2 i + (-14t)j
D) v = (-14t)i - 1 t2 j
5
104)
5
5
5
For the curve r(t), write the acceleration in the form a T T + a NN.
105) r(t) = (5t sin t + 5 cos t)i + (5t cos t - 5 sin t)j + 3k A) a = 5tN
105)
B) a = 5T + 1 N 5t
C) a = 1 N 5t
D) a = 5T + 5tN
Find the velocity vector in terms of ur and u .
106) r = 2 cos 2t and = 3t A) v = (-2 sin 2t)ur + 6cos 2tu C) v = (-4 sin 2t)ur + 6cos 2tu
106) B) v = (-6 sin 3t)ur + 4cos 3tu D) v = (4 sin 2t)ur + 6cos 2tu
The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors at time t = 0. 107) r(t) = e6ti + (4 + e-6t)j + (8 cos 6t)k 107)
A)
B) 0
4
C)
D)
2
3
Calculate the arc length of the indicated portion of the curve r(t). 1 108) r(t) = (3cos3 6t)j + (3sin3 6t)k; 1 t 6 4
A) 0
108)
C) 3
B) 9
D) 9
2
2
Find the curvature of the space curve. 109) r(t) = 5ti + 6 + 10 cos 6 t j + 9 + 10 sin 6 k 5 5
A) = 72 65
109)
B) = 72
C) = 12
845
13
18
D) = 12
169
The position vector of a particle is r(t). Find the requested vector.
110) The velocity at t = A) v C) v
4 4
4
for r(t) = 5sec2 (t)i - 5tan(t)j + 10t2 k
= 20i + 10j - 5 k
B) v
= 20i - 10j + 5 k
D) v
110) = -10j - 5 k
4
= -10j + 5 k
4
Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured from the horizontal, and that the projectile is launched from the origin over a horizontal surface 111) A projectile is fired at a speed of 920 m/sec at an angle of 40°. How long will it take to get 23 km 111) downrange? Round your answer to the nearest whole number. A) 33 sec B) It will never get that far downrange.
C) 31 sec
D) 35 sec
Find the curvature of the space curve. 112) r(t) = 8 (1 + t)3/2i + 8 (1 - t)3/2j + 2tk 3 3
A) = 1
18
2
112)
B) = 1
1 + t2
3
2
C) = 1
1 + t2
18
2
1 - t2
D) = 1 3
1 - t2
Provide an appropriate response. 113) The position of a particle is given by r(t) = sin 8t i + cos 6t j. Find the velocity vector for the particle. A) 8 i + 6 j B) cos 8t i - sin 6t j
C) 8 sin 8t i + 6 cos 6t j
2
113)
D) 8 cos 8t i - 6 sin 6t j
Find the velocity vector in terms of ur and u .
114) r = a(8 - cos ) and d = 6
114)
dt
A) v = (6a cos )ur + 6a(8 - sin )u C) v = (6a sin )ur + 6a(8 - cos )u
B) v = 6a(8 - cos )ur + (6a sin )u D) v = (8a sin )ur + 8a(6 - cos )u
Find the curvature of the space curve. 115) r(t) = -9i + (t + 4)j +(ln(cos t) + 6)k A) = csc t B) = sec t
C) = cos t
D) = sin t
116) r(t) = (t + 10)i + 6j + (ln(sec t) + 6)k A) = sin t B) = csc t
C) = sec t
D) = cos t
115)
116)
Find the torsion of the space curve. 117) r(t) = (t - 10)i + (ln(cos t) - 5)j + 5k, - /2 < t < /2 A) Undefined B) = 1
117) C) = 0
19
D) = -1
118) r(t) = 9 + 2 sin 3 t i + 4tj + 9 + 2 cos 3 t k 2
118)
2
A) = 12
B) = 6
5
C) = 6
5
D) = - 6
25
5
For the curve r(t), find an equation for the indicated plane at the given value of t. 119) r(t) = (t - 6)i + (ln(sec t) - 4)j + 6k, - /2 < t < /2; normal plane at t = 6 . A) x = 6 - 6 B) x = 6 + 6 C) x = -6 - 6
119) D) x = 6 - 6
If r(t) is the position vector of a particle in the plane at time t, find the indicated vector. 120) Find the acceleration vector. r(t) = (3 cos t)i + (9 sin t)j A) a = (3 sin t)i + (9 cos t)j B) a = (-3 cos t)i + (-9 sin t)j
C) a = (-3 sin t)i + (-9 cos t)j
120)
D) a = (3 cos t)i + (9 sin t)j
Evaluate the integral. 3 5 8t 121) i - 9t2 j + k dt 2 1+t (1 + t2 ) 0 A) 5i - 81j + 18 k B) 10i + 81j + 9 k 5 10
121) C) 10i - 81j + 18 k
D) 10i - 81j + 9 k
C) Undefined
D) = 1
5
Find the torsion of the space curve. 122) r(t) = (3t sin t + 3 cos t)i + (3t cos t) - 3 sin t)j - 8k A) = -1 B) = 0
10
122)
For the smooth curve r(t), find the parametric equations for the line that is tangent to r at the given parameter value t = t0 .
123) r(t) = (10 sin t)i - (9 cos 5t)j +e-4tk ; to = 0 A) x = 10t, y = 9, z = 1 + t C) x = 10, y = -9t, z = -4 + t
123) B) x = 10t, y = -9, z = 1 - t D) x = 10t, y = -9, z = 1 - 4t
Find the principal unit normal vector N for the curve r(t). 124) r(t) = (t2 + 7)j + (2t - 4)k
A) N = C) N =
1 j+ t2 + 1
1 3 (t2 + 1)
j-
124)
t k t2 + 1
t 3 (t2 + 1)
B) N = k
D) N = -
20
1 jt2 + 1
1 3 (t2 + 1)
t k t2 + 1
j+
t 3 (t2 + 1)
k
If r(t) is the position vector of a particle in the plane at time t, find the indicated vector. 125) Find the acceleration vector. r(t) = (3 ln(5t))i + (9t3 )j
A) a = 3 t-2 i + 54tj
B) a = 3 i + 27tj
C) a = - 3 i + 54tj
D) a = - 3 t-2 i + 54tj
t2
125)
t
t2
5
Find T, N, and B for the given space curve. 126) r(t) = 8 (1 + t)3/2i + 8 (1 - t)3/2j + 2tk 3 3
126)
A) T = 2 1 + ti - 2 1 - tj + 1 k; N = 1 2 - 2ti + 1 2 + 2tj ; B = - 1 2 + 2ti + 1 2 - 2tj + 3
3
3
2
2
6
6
2 2 k 3
B) T = 2 1 + ti + 2 1 - tj + 1 k; N = 1 2 - 2ti + 1 2 + 2tj; B = 1 2 + 2ti + 1 2 - 2tj + 3
3
3
2
2
6
6
2 2 k 3
C) T = 4 1 + ti - 4 1 - tj + 2k; N = 3 2 - 2ti + 3 2 + 2tj; B = - 1 2 + 2ti + 1 2 - 2tj + 4 2k D) T = 1 1 + ti - 1 1 - tj + 1 k; N = 1 2 - 2ti + 1 2 + 2tj; B = - 1 2 + 2ti + 1 2 - 2tj + 3
3
3
2
2
6
6
2 k 3
Solve the problem. 127) The orbit of a satellite had a semimajor axis of a = 8872 km. Calculate the period of the satellite. (Earth's mass = 5.975 × 1024 kg and G = 6.6720 × 10-11 Nm 2 kg-2 ).
A) 78.2 min
B) 138.6 min
C) 1.98 hr
127)
D) 12 hr
Solve the initial value problem.
128) Differential Equation: dr = (-cos t)i + (4t3- 3)j
128)
dt
Initial Condition: r(0) = -3j A) r(t) = (sin t)i + (12t2 - 3)j
B) r(t) = (-sin t)i + (t4 - 3)j D) r(t) = (-sin t)i + (t4 )j
C) r(t) = (-sin t)i + (t4 - 3t - 3)j 129) Differential Equation: dr = -9ti + 3tj + 9tk
129)
dt
Initial Condition: r(0) = -4i + 6k 2 2 A) r(t) = -9t - 8 i + 3 t2 j + 9t + 12 k 2 2 2
B) r(t) = -9t - 4 i + 3 t2 j - 9t + 6 k
C) r(t) = (-9t2 - 8)i + 3t2 j + (9t2 + 12)k
D) r(t) = -9t - 4 i + 3 t2 j + 9t + 6 k
2
2
2
2
2
2
21
2
2
2
2
The position vector of a particle is r(t). Find the requested vector. 3 130) The velocity at t = 0 for r(t) = cos(2t)i + 5ln(t - 10)j - t k 5
A) v(0) = 2i - 1 j 2
B) v(0) = - 1 j
130)
C) v(0) = 1 j
2
2
D) v(0) = -2i - 1 j 2
For the curve r(t), find an equation for the indicated plane at the given value of t. 131) r(t) = (t2 - 7)i + (2t - 9)j + 10k; osculating plane at t = 3.
A) z = -10 C) x + y + (z + 10) = 0
131)
B) x + y + (z - 10) = 0 D) z = 10
Find the curvature of the space curve. 132) r(t) = (2 t sin t + 2 cos t)i + 2j + (2t cos t - 2 sin t)k A) = 1 B) = 2t 2t
132) C)
1 =2t
D)
1 = 4t2
Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured from the horizontal, and that the projectile is launched from the origin over a horizontal surface 133) An athlete puts a 16-lb shot at an angle of 43° to the horizontal from 5.5 ft above the ground at an 133) initial speed of 43 ft/sec. How far forward does the shot travel before it hits the ground? Round your answer to the nearest tenth. A) 2 ft B) 193.9 ft C) 63 ft D) 5.4 ft
Calculate the arc length of the indicated portion of the curve r(t). 134) r(t) = 4ti + 7 cos 3 t j + 7 sin 3 t k; -5 t 9 7 7
A) 70
B) 20
C) 100
134) D) 350
Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured from the horizontal, and that the projectile is launched from the origin over a horizontal surface 135) An ideal projectile is launched from the origin at an angle of radians to the horizontal and an 135) initial speed of 50 ft/sec. Find the position function r(t) for this projectile. A) r(t) = (50t sin )i + (50t cos - 16t2 )j B) r(t) = (50t cos - 32t2 )i + (50t sin )j
C) r(t) = (50t sin - 16t2 )i + (50t cos )j
D) r(t) = (50t cos )i + (50t sin - 16t2 )j
22
For the curve r(t), find an equation for the indicated plane at the given value of t.
136) r(t) = (t + 6)i + (ln(cos t) - 10)j + 6k, - /2 < t < .2; rectifying plane at t = A) -
4
.
2 2 2 xy - ln +6 + - 10 = 0 2 4 2 2
B)
2 2 2 xy - ln +6 -10 = 0 2 4 2 2
C)
2 2 2 xy - ln +6 + - 10 = 0 2 4 2 2
D)
2 (x - (1 + 6)) = 0 2
Find the unit tangent vector of the given curve. 137) r(t) = (3 sin3 3t)i + (3 cos3 3t)j
137)
A) T = (sin 3t)i - (cos 3t)j C) T = (27 sin 3t)i -(27 cos 3t)j
B) T = (3 cos 3t)i - (3 sin 3t)j D) T = (3 sin 3t)i - (3 cos 3t)j
The position vector of a particle is r(t). Find the requested vector. 138) The acceleration at t = 2 for r(t) = (5t - 2t4 )i + (8 - t)j + (4t2 - 6t)k
A) a(2) = -24i + 8k C) a(2) = 96i + 8k
139)
1 = 2 t2 + 1
C) = -
138)
B) a(2) = -96i - j + 8k D) a(2) = -96i + 8k
Find the curvature of the space curve. 139) r(t) = -4i + (5 + 2t)j + (t2 + 6)k
A)
136)
B) =
1
D) =
3/2 2(t2 + 1)
1
3/2 2(t2 + 1) 1
3/2 (t2 + 1)
The position vector of a particle is r(t). Find the requested vector. 140) The acceleration at t = 0 for r(t) = t2 i + (6t3 - 10)j + 16 - 2tk
140)
A) a(0) = 2i + 1 k 64
B) a(0) = 2i - 1 k 128
C) a(0) = 2i - 1 k
D) a(0) = 2i - 1 k
64
4
For the curve r(t), write the acceleration in the form a T T + a NN.
141) r(t) = (cosh t)i + (sinh t)j + tk A) a = (-sinh t)T + N C) a = (- 2 sinh t)T + N
141) B) a = ( 2 sinh t)T + N D) a = (sinh t)T + N
23
For the smooth curve r(t), find the parametric equations for the line that is tangent to r at the given parameter value t = t0 .
142) r(t) = (7t2 - 7t)i + (t + 7)j + k ; t0 = 2 A) x = 14 + 21t, y = 9 + t, z = 0 C) x = 14 + 21t, y = 9 + t, z = 1
142) B) x = 14 + t, y = t, z = t D) x = 21t, y = t, z = t t
Find the arc length parameter along the curve from the point where t = 0 by evaluating s =
|v( )| d .
0
143) r(t) = (5cos t)i + (5sin t)j + 2tk A) 33t
B)
143) 29 t 2
C) 29t
24
D) 3 6t
Answer Key Testname: CHAPTER 13
1) d u + v = d uxi + uyj + vxi + vyj = d (ux + vx)i + (uy + vy)j dt =
dt
dt
d d d d du dv + (u + vx)i + (u + vy)j = (u i + uyj) + (v i + vyj) = dt x dt y dt x dt x dt dt
2) Flight time to tree: t = distance = 1.607 seconds; v0 cos
height at 1.607 seconds: y = (v0 sin )t -
1 2 gt = 22.09 ft. Tree is 29 ft high 2
3) Immediately after the wind gust, the velocity is v = 144.6i + 60.43j. Flight time 3.78 sec Range 489.53 ft
4) Given: v and a are orthogonal. Thus v·a = v·(dv/dt) = 1 d(v·v)/dt = 0. 2
So, v·v = v 2 = constant and thus v = constant. b b 5) (r1 (t) + r2 (t)) dt = (x1 (t)i + y1 (t)j) + (x2 (t)i + y2 (t)j) dt a a b b x 1(t)i + y1 (t)j dt + x2 (t)i + y2 (t)j dt = a a b b r1 (t) dt + r2 (t) dt = a a 6) Yes: firing angle = 26.1° under these conditions.
7) Kepler's first law says that the eccentricity of a planet's elliptical orbit e = and solving for the speed v0 with e = 0 gives v0 =
r0 v0 2 -1. The eccentricity of a circle is 0, GM
GM . Since r0 is a fixed radius for a circular orbit, all three terms r0
for computing v0 are constant, so v0 is also constant.
8) d (f u) = dt
d(f(uxi + uyj)) dt
=
d(fuxi + fuyj) dt
=
d(fuxi) dt
+
d(fuyj) dt
=
dux duy dux duy df df df df ux + f i+ uy + f j= ux i + uy j + f i+ f j dt dt dt dt dt dt dt dt
=
dux duy df df du (uxi + uyj) + f i+ j = u + f dt dt dt dt dt
25
Answer Key Testname: CHAPTER 13
9) x-coordinate:
d2 x dx = 0, so = constant = v0,x = v0 c os 2 dt dt dx dt = dt
Then, x =
y-coordinate: d2 y dy = -g , so = 2 dt dt dy dt = dt
y= b
10)
b
kr(t) dt =
a
b
=k a =k
a
dt = v0 cos
d2 y dt = dt2
v0 sin
b
x(t)i dt + k
- gt dt = v0 sin b a
y(t)j dt = k
a
b
(x(t)i + y(t)j) dt = k
t + x0
-g dt = -gt + vy,0 = v0 sin
k(x(t)i + y(t)j) dt =
a
b
v0 cos
b
t-
1 2 gt + y0 2
kx(t)i + ky(t)j dt b
x(t)i dt +
a
a
r(t) dt
a
26
y(t)j dt
- gt
Answer Key Testname: CHAPTER 13
11) Problem is separable.
x-coordinate: dvx vx d2 x dvx = = -kvx or = -k dt. Integrate to get ln = -kt or 2 dt v v x x,0 dt vx = vx,0e-kt Next, x =
vx dt =
At t = 0, x(0) = x0 = Thus, x = x0 + x = x0 +
vx,0e-kt dt = -
vx,0 e-kt + C k
vx,0 vx,0 + C or C = x0 + k k
vx,0 (1 - e-kt). Also, vx,0 = v0 cos , so k
v0 (1 - e-kt)cos . k
y-coordinate: g + kvy d2 y dvy dw dvy = = - g - kvy. Let w = = , then . Make substitutions to get: dt k dt dt dt2 dw dw w = - kw or = -k dt. Integrate to get: ln = -kt or w = w0 e-kt. dt w w0 Replace w: g + kvy k Finally, y =
=
g + kvy,0 k vy dt =
At t = 0, y = y0 = y = y0 -
1 g e-kt. Solve for vy to get vy = (g + kvy,0)e-kt - . k k 1 g 1 g (g + kvy,0)e-kt dt = (g + kvy,0)e-kt - t + C. k k 2 k k
1 1 (g + kvy,0) + C or C = y0 + (g + kvy,0). Thus, k2 k2
1 g 1 (g + kvy,0)e-kt - t + (g + kvy,0) k k2 k2
= y0 +
1 g (g + kvy,0)(1 - e-kt) - t k k2
Also, vy,0 = v0 sin . Substituting and rearranging: y = y0 +
v0 (1 - e-kt)sin k
+
g (1 - kt - e-kt) k2
12) C 13) C 14) B 15) C 16) D 17) D 18) B 27
Answer Key Testname: CHAPTER 13
19) D 20) B 21) B 22) C 23) C 24) B 25) D 26) C 27) C 28) A 29) A 30) C 31) D 32) B 33) B 34) B 35) B 36) D 37) D 38) C 39) D 40) D 41) A 42) B 43) A 44) D 45) D 46) B 47) B 48) D 49) D 50) C 51) C 52) C 53) B 54) C 55) A 56) B 57) A 58) A 59) B 60) D 28
Answer Key Testname: CHAPTER 13
61) B 62) A 63) D 64) A 65) C 66) B 67) A 68) B 69) B 70) D 71) A 72) C 73) D 74) B 75) B 76) B 77) B 78) D 79) A 80) D 81) A 82) D 83) B 84) A 85) D 86) A 87) A 88) C 89) D 90) D 91) A 92) A 93) D 94) B 95) A 96) A 97) D 98) B 99) A 100) A 101) B 102) C 29
Answer Key Testname: CHAPTER 13
103) C 104) B 105) D 106) C 107) C 108) D 109) B 110) C 111) A 112) C 113) D 114) C 115) C 116) D 117) C 118) C 119) A 120) B 121) C 122) B 123) D 124) B 125) C 126) A 127) B 128) C 129) A 130) B 131) D 132) A 133) C 134) A 135) D 136) C 137) A 138) D 139) B 140) C 141) B 142) C 143) C 30
Chapter 14
Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Match the surface show below to the graph of its level curves.
1)
1)
A)
B)
C)
D)
1
2)
2)
A)
B)
C)
D)
2
3)
3)
A)
B)
C)
D)
3
4)
4)
A)
B)
C)
D)
4
5)
5)
A)
B)
C)
D)
5
6)
6)
A)
B)
C)
D)
6
7)
7)
A)
B)
C)
D)
7
8)
8)
A)
B)
C)
D)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0). 9) f(x, y) = xy 9) xy
8
Give an appropriate answer. 10) Given the function f(x, y) and the positive number as in the formal definition of a limit, find a positive number as in the definition that insures f(x, y) - f(0, 0) < .
10)
f(x,y) = x + y ; = 0.08
Use the limit definition of the partial derivative to compute the indicated partial derivative of the function at the specified point. 11) Find f at the point (2, -2, -10): f(x, y, z) = 10x2 y + 8y2 + 8z 11) x
Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0). 2 12) f(x, y) = x - y 12) x2 + y 2 13) f(x, y) = y
13)
y2 - x 5
10
14) f(x, y) = x + y
14)
x2 x4 + y2
15)
x5
15) f(x, y) =
Provide an appropriate response. -1 16) Does knowing that 5 - x2 y3 < 5 tan xy < 5 tell you anything about xy
16)
lim 5 tan-1 xy ? Give reasons for your answer. (x, y) (0, 0) xy
Use the limit definition of the partial derivative to compute the indicated partial derivative of the function at the specified point. 17) Find f at the point (10, 2): f(x, y) = 6 - 8xy + 3xy2 17) x
18) Find f at the point (7, 7): f(x, y) = 4 - 10xy + 2xy2
18)
y
9
Give an appropriate answer. 19) Given the function f(x, y) and the positive number as in the formal definition of a limit, find a positive number as in the definition that insures f(x, y) - f(0, 0) < . f(x, y) =
19)
x+y ; = 0.04 2 x + y2 + 1
20) Given the function f(x, y) and the positive number as in the formal definition of a limit, find a positive number
20)
as in the definition that insures f(x, y) - f(0, 0) < .
f(x, y) = (1 + cos x)(x + y); = 0.04
21) Given the function f(x, y, z) and the positive number as in the formal definition of a limit, find a positive number
21)
as in the definition that insures f(x, y, z) - f(0, 0, 0) < .
f(x, y, z) = x + y + z ; = 0.06
Provide an appropriate response.
22) Show that f(x, y, z) = x3y5 z 3 is continuous at every point (x0, y0, z 0).
22)
23) Show that f(x, y, z) = ex2 + y2 + z 2 is continuous at the origin.
23)
Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0). 24) f(x, y) = xy 24) x2 + y2 Provide an appropriate response. 25) If f(x, y, z) is differentiable, x = r - s, y = s - t, and z = t - r, show that f f f + + = 0. r s t
25)
Use the limit definition of the partial derivative to compute the indicated partial derivative of the function at the specified point. 26) Find f at the point (-3, -5, 3): f(x, y, z) = xyz - 6y2 - 5z 26) y
Provide an appropriate response.
27) Does knowing that cos 1
y
1 tell you anything about
lim 1 sin(x) cos ? (x, y) (0, 0) y
27)
Give reasons for your answer.
Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0). 3y 28) f(x, y) = 28) 4x2 + 9y2
10
Give an appropriate answer. 29) Given the function f(x, y) and the positive number as in the formal definition of a limit, find a positive number as in the definition that insures f(x, y) - f(0, 0) < . f(x, y) =
29)
2x + y ; = 0.09 x2 y2 + 1
30) Given the function f(x, y, z) and the positive number as in the formal definition of a limit, find a positive number f(x, y, z) =
30)
as in the definition that insures f(x, y, z) - f(0, 0, 0) < .
x2 + y2 + z 2 ; = 0.02 x+1
Use the limit definition of the partial derivative to compute the indicated partial derivative of the function at the specified point. 31) Find f at the point (4, 8): f(x, y) = 2x2 + 5xy + 3y2 31) x
Give an appropriate answer. 32) Given the function f(x, y, z) and the positive number as in the formal definition of a limit, find a positive number as in the definition that insures f(x, y, z) - f(0, 0, 0) < .
32)
f(x, y, z) = x + y - z ; = 0.09
Use the limit definition of the partial derivative to compute the indicated partial derivative of the function at the specified point. 33) Find f at the point (-7, -5): f(x ,y) = 10x2 + 2xy + 10y2 33) y
Provide an appropriate response.
34) Does knowing that sin 1
y
1 tell you anything about
lim 1 sin(x) sin ? Give (x, y) (0, 0) y
34)
reasons for your answer.
Use the limit definition of the partial derivative to compute the indicated partial derivative of the function at the specified point. 35) Find f at the point (-5, -9, 6): f(x, y, z) = xyz - 5y2 - 3z 35) z
36) Find f at the point (6, 10, 2): f(x, y, z) = 10x2 y + 7y2 + 3z y
11
36)
Give an appropriate answer. 37) Given the function f(x, y, z) and the positive number as in the formal definition of a limit, find a positive number as in the definition that insures f(x, y, z) - f(0, 0, 0) < .
37)
f(x, y, z) = sin2 x + sin2 y + sin2 z ; = 0.12
Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0). 2 38) f(x, y) = x y 38) x4 + y2 Provide an appropriate response. 39) We say that a function f(x, y, z) approaches the limit L as (x, y, z) approaches (x0 , y0 , z 0 )
39)
and write lim f(x, y, z) = L (x, y, z) (x0 , y0 , z 0 ) if for every number > 0, there exists a corresponding number > 0 such that for all (x, y, z) in the domain of f, 0 < (x - x0 ) 2 + (y - y0 )2 + (z - z 0 )2 < f(x,y,z) - L < . Show that the - requirement in this definition is equivalent to 0 < x - x0 < , 0 < y - y0 < , and 0 < z - z 0 < f(x,y,z) - L < .
40) If f(x0 , y0 ) = -2 and the limit of f(x, y) exists as (x,y) approaches (x0 , y0 ), what can you say
40)
about the continuity of f(x, y) at the point (x0 , y0 )? Give reasons for your answer.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the domain and range and describe the level curves for the function f(x,y). 41) f(x, y) = 100 - x2 - y2
41)
A) Domain: all points in the xy-plane satisfying x2 + y2 = 100; range: real numbers 0 z 10; level curves: circles with centers at (0, 0) and radii r, 0 < r 10 B) Domain: all points in the xy-plane satisfying x2 + y2 100; range: real numbers 0 z 10; level curves: circles with centers at (0, 0) and radii r, 0 < r 10 C) Domain: all points in the xy-plane; range: real numbers 0 z 10; level curves: circles with centers at (0, 0) and radii r, 0 < r 10 D) Domain: all points in the xy-plane; range: all real numbers; level curves: circles with centers at (0, 0)
At what points is the given function continuous? 42) f(x, y) = x - y 2x2 + x - 6
42)
A) All (x, y) such that x 3 and x -2
B) All (x, y) satisfying x - y 0
C) All (x, y)
D) All (x, y) such that x 0
2
12
Provide an appropriate response. 43) Find any local extrema (maxima, minima, or saddle points) of f(x, y) given that fx = -9x - 2y and fy = -4x - 4y.
A) Local minimum at - 2 , - 1
B) Local maximum at (0, 0)
C) Saddle point at (0, 0)
D) Local maximum at - 2 , - 1
9
At what points is the given function continuous? 44) f(x, y) = tan (x + y) A) All (x, y) (0, 0) B) All (x, y) such that x + y (2n + 1) 2
C) All (x, y)
2
,
43)
9
44) , where n is an integer
2
D) All (x, y) Sketch the surface z = f(x,y). 45) f(x, y) = -x2 - y2
45)
A)
B)
C)
D)
13
Solve the problem.
46) Evaluate w at (u, v) = (5, 2) for the function w = xz + yz - z2 ; x = uv, y = uv, z = u.
46)
u
A) 0
B) 30
C) -10
D) 40
Show that the function is a solution of the wave equation. 47) w(x, t) = e-2ct cos -2x
47)
A) No
B) Yes
At what points is the given function continuous? x 48) f(x, y, z) = e ey+z
48)
A) All (x, y, z) such that x 0 and y + z 0 C) All (x, y, z) such that x - y - z 0
B) All (x, y, z) D) All (x, y, z) in the first octant
Find the requested partial derivative. 49) z at (x, y, z) = (1, 1, 1) if z 3 = z + xy - 1 and y3 = x + y - 1 y
A) 1 2
B) 1
C) 3
4
49) D) 2
2
3
Find an upper bound for the magnitude |E| of the error in the approximation f(x, y) L(x, y) at the given point over the given region R. 50) f(x, y) = -8x2y3 at (2, 1); R: |x - 2| 0.2, |y - 1| 0.2 50)
A) |E| 17.23392
B) |E| 12.16512
C) |E| 22.30272
D) |E| 17.47104
Find the specific function value.
51) Find f(7, 2) when f(x, y) = 8y2 - 3xy. A) 352 B) 350
51) C) -17
D) -10
Solve the problem.
52) Find the derivative of the function f(x, y, z) = x + y + z at the point (6, -6, 6) in the direction in y
z
which the function decreases most rapidly. A) - 1 2 B) - 1 3 3 3
x
C) - 1 2 2
D) - 1 3 2
Provide an appropriate response. 53) Find dw at t = 0 for the function w = sin(x) cos(y) ln(z) where x = 6t2 + 6t + , y = t, z = e-5t. dt 2
A) 1
B) -10
C) -5
14
52)
D) -5e-5
53)
Show that the function is a solution of the wave equation. 54) w(x, t) = sin (-5x - 5ct) A) Yes
54) B) No
Answer the question. 55) The graph below shows the level curves of a differentiable function f(x, y) (thin curves) as well as 3 the constraint g(x, y) = x2 + y2 - = 0 (thick circle). Using the concepts of the orthogonal 2 gradient theorem and the method of Lagrange multipliers, estimate the coordinates corresponding to the constrained extrema of f(x,y).
A) (1.1, 1.1), (-1.1, 1.1), (-1.1,-1.1), (1.1,-1.1) B) (1.5, 0.2), (0.7, 1.3), (-1.5, 0.2), (-0.7, 1.3), (-1.5, -0.2), (-0.7, -1.3), (1.5, -0.2), (0.7, -1.3) C) (1.3, 0.7), (-1.3, 0.7), (-1.3,-0.7), (1.3,-0.7) D) (1.5, 0), (0, 1.5), (-1.5, 0), (0, -1.5)
15
55)
Find the equation for the level surface of the function through the given point. zy + x n , (1, 4, 1) 2 nxn
56) f(x, y, z) = n=0
A) n=0
B) zy + x
56)
zy + x n 5 = n n 2 2 x
2nxn
n
=
5 2
C) zy + x = 5 2x
2
D) The level surfaces cannot be determined. Find the absolute maxima and minima of the function on the given domain. 57) f(x, y) = 8x + 7y on the trapezoidal region with vertices (0, 0), (1, 0), (0, 2), and (1, 1) A) Absolute maximum: 15 at (1, 1); absolute minimum: 0 at (0, 0)
57)
B) Absolute maximum: 15 at (1, 1); absolute minimum: 8 at (1, 0) C) Absolute maximum: 16 at (2, 0); absolute minimum: 8 at (1, 0) D) Absolute maximum: 14 at (0, 2); absolute minimum: 0 at (0, 0) Show that the function is a solution of the wave equation. 58) w(x, t) = ex - ct
58)
A) No
B) Yes
At what points is the given function continuous? 1 59) f(x, y, z) = x + 10 + y - 3 + z - 6
A) All (x, y, z) C) All (x, y, z)
59)
(-10, 3, 6)
B) All (x, y, z) (0, 0, 0) D) All (x, y, z) ± (-10, 3, 6)
Find the extreme values of the function subject to the given constraint. 60) f(x, y) = x2 + 4y3, x2 + 2y2 = 2
60)
A) Maximum: 8 at (2, 1); minimum: -31 at (1, -2) B) Maximum: 4 at (0, 1); minimum: -4 at (0, -1) C) Maximum: 4 at (0, 1); minimum: -31 at (1, -2) D) Maximum: 8 at (2, 1); minimum: -4 at (0, -1) 61) f(x, y, z) = x + 2y - 2z, x2 + y2 + z2 = 9 A) Maximum: 8 at (2, 1, -2); minimum: -8 at (-2, -1, 2) B) Maximum: 1 at (-1, -2, -3); minimum: -1 at (1, 2, 3) C) Maximum: 1 at (1, -2, -2); minimum: -1 at (-1, 2, 2) D) Maximum: 9 at (1, 2, -2); minimum: -9 at (-1, -2, 2)
16
61)
Find an upper bound for the magnitude |E| of the error in the approximation f(x, y) L(x, y) at the given point over the given region R. 62) f(x, y) = -10x2 + 8y2 + 5 at (4, 5); R: |x - 4| 0.1, |y - 5| 0.1 62)
A) |E| 0.08
B) |E| 0.32
C) |E| 0.16
D) |E| 0.24
Provide an appropriate response. 63) Which order of differentiation will calculate fxy faster, x first or y first? f(x, y) = y ln(x) - 6 sin(x) A) y first
63)
B) x first
Compute the gradient of the function at the given point. 64) f(x, y, z) = 7x - 9y - 10z, (-9, 7, 4) A) f = -63i - 63j - 40k
64) B) f = -63i + 63j - 40k D) f = -9i + 7j + 4k
C) f = 7i - 9j - 10k
Solve the problem. 65) Find the point on the line x - 3y = 6 that is closest to the origin. A) - 3 , - 11 B) 3 , - 9 C) 9 , - 7 5 5 5 5 5 5
65) D) - 9 , - 13 5 5
66) Find an equation for the level surface of the function f(x, y, z) = ln xy that passes through the z
point e4, e12, e3 . A) ln xy = 1 z 16
B) xy = e13
C) xy = e16
z
D) ln xy = 16
z
z
y
67) Find an equation for the level surface of the function f(x, y, z) =
e d
z through the point (0, ln 6, ln 9) . 2 A) ey - ez + x = -3 2
66)
2
-
t dt that passes
67)
x
2
B) ey + ez - x = -3 2
2
2
C) ey + ez - x = 3
D) ey - ez + x = 3
2
2
Find the linearization of the function at the given point. 68) f(x, y) = 4x2 - 7y2 + 8 at (7, -10)
68)
A) L(x, y) = 56x + 140y + 512 C) L(x, y) = 8x - 14y + 512
B) L(x, y) = 56x + 140y - 496 D) L(x, y) = 8x - 14y - 496
17
69) f(x, y, z) = e10x + 4y + 9z at (0, 0, 0)
69)
A) L(x, y, z) = 5x + 2y + 9 z
B) L(x, y, z) = 10x + 4y + 9z + 1
C) L(x, y, z) = 10x + 4y + 9z
D) L(x, y, z) = 5x + 2y + 9 z + 1
2
2
Provide an appropriate response.
70) For the space curve x = t, y = t2 , z = t , find the points at which the function f(x, y) takes on extreme values if fx = 16, fy =
70)
t , and fz = 48t. 2
A) t = - 2 and t = - 3 C) t = 0
B) t = -8 and t = -12 D) t = -4 and t = -6
Solve the problem.
71) Find the point on the sphere x2 + y2 + z 2 = 4 that is closest to the point (3, 1, -1). A)
6 2 2 , , 11 11 11
B) -
C) - 6 , 2 , - 2 11
11
D)
11
71)
6 2 2 , , 11 11 11
6 2 , ,11 11
2 11
Provide an appropriate response. 72) Determine the point on the plane 10x + 10y + 3z = 19 that is closest to the point (22, 19, 9). A) (2, -1, 3) B) (2, 1, -3) C) (-2, -1, -3) D) (-2, 1, -3)
72)
Find fx , fy, and fz . z
73) f(x, y, z) = A) fx = B) fx = C) fx = D) fx =
73)
x + y2
z
yz ; fy = ;f =3/2 3/2 z 2(x + y2 ) (x + y2 ) z
3/2 2(x + y2 )
; fy = -
yz
3/2 (x + y2 )
x + y2 1
; fz =
z
yz ; fy = ;f = 3/2 3/2 z 2(x + y2 ) (x + y2 ) z
1
yz ; fy = ;f = 3/2 3/2 z 2(x + y2 ) (x + y2 )
x + y2 1
x + y2 1 x + y2
Find the absolute maxima and minima of the function on the given domain. 74) f(x, y) = 3x + 6y on the closed triangular region with vertices (0, 0), (1, 0), and (0, 1) A) Absolute maximum: 6 at (0, 1); absolute minimum: 3 at (1, 0)
B) Absolute maximum: 9 at (1, 1); absolute minimum: 3 at (1, 0) C) Absolute maximum: 3 at (1, 0); absolute minimum: 0 at (0, 0) D) Absolute maximum: 6 at (0, 1); absolute minimum: 0 at (0, 0)
18
74)
Find the domain and range and describe the level curves for the function f(x,y). 75) f(x, y) = y + 5 x2
75)
A) Domain: all points in the xy-plane excluding x = 0; range: all real numbers; level curves:
parabolas y = cx2 - 5 B) Domain: all points in the xy-plane excluding x = 0; range: real numbers z 0; level curves: parabolas y = cx2 - 5
C) Domain: all points in the xy-plane; range: all real numbers; level curves: parabolas
y = cx2 - 5 D) Domain: all points in the xy-plane; range: real numbers z 0; level curves: parabolas y = cx2 - 5 Find
f f and . x y
76) f(x, y) = (4x5 y3 - 8)2
76)
A)
f f = 2(4x5y3 - 8); = 2(4x5 y3 - 8) x y
B)
f f = 24x5 y2 (4x5 y3 - 8); = 40x4 y3 (4x5 y3 - 8) x y
C) f = 20x4 y3 ; f = 12x5 y2 x
y
D) f = 40x4 y3 (4x5 y3 - 8); f = 24x5 y2 (4x5y3 - 8) x
y
Solve the problem. 77) Write parametric equations for the tangent line to the curve of intersection of the surfaces z = 7x2 + 3y2 and z = x + y + 8 at the point (1, 1, 10).
A) x = -13t + 1, y = 15t + 1, z = 8t + 10 C) x = -13t + 1, y = 13t + 1, z = 8t + 10
77)
B) x = -5t + 1, y = 15t + 1, z = 8t + 10 D) x = -5t + 1, y = 13t + 1, z = 8t + 10
78) Write an equation for the tangent line to the curve xy = 45 at the point (9, 5). A) 9x + 5y = 90 B) 5x + 9y = 90 C) 5x + 9y = 45 D) 9x + 5y = 45
19
78)
Find the linearization of the function at the given point. 79) f(x, y, z) = tan-1 xyz at (10, 10, 10)
79)
A) L(x, y, z) = 13,743,895,347,200 x + 13,743,895,347,200 y + 13,743,895,347,200 z + 1,374,526,973,673,473 1,374,526,973,673,473 1,374,526,973,673,473 tan-1 1000 -
B) L(x, y, z) =
858,993,459,200 858,993,459,200 858,993,459,200 x+ y+ z+ 8,589,943,181,934,612 8,589,943,181,934,612 8,589,943,181,934,612
tan-1 1000 -
C) L(x, y, z) =
41,231,686,041,600 1,374,526,973,673,473
25,769,803,776,000 8,589,943,181,934,612
858,993,459,200 858,993,459,200 858,993,459,200 x+ y+ z+ 8,589,943,181,934,612 8,589,943,181,934,612 8,589,943,181,934,612 25,769,803,776,000 8,589,943,181,934,612
tan-1 100 -
D) L(x, y, z) = 13,743,895,347,200 x + 13,743,895,347,200 y + 13,743,895,347,200 z + 1,374,526,973,673,473
tan-1 100 -
1,374,526,973,673,473
1,374,526,973,673,473
41,231,686,041,600 1,374,526,973,673,473
Compute the gradient of the function at the given point. 80) f(x, y, z) = -3xy3z 2 , (-3, -27, 9)
80)
A) f = -1,594,323i + 1,062,882j - 531,441k C) f = 4,782,969i + 1,594,323j - 3,188,646k
B) f = 4,782,969i + 1,062,882j - 4,782,969k D) f = -1,594,323i + 1,594,323j - 354,294k
Find the equation for the level surface of the function through the given point. 2 81) f(x, y, z) = x y , (6, 1, 3) xz + y2
A) 12 = 5
x2y
B) 6 =
xz + y2
C) 19 xz + y2 = 36x2 y
x2y
19
xz + y2
D) 36 =
x2y
19
81)
xz + y2
Solve the problem.
82) A rectangular box is to be inscribed inside the ellipsoid 2x2 + y2 + 4z 2 = 12. Find the largest possible volume for the box. A) 15 2 B) 16 2
C) 18 2
83) Find the least squares line for the points (1, 1), (2, 4), (3, 9), (4, 16). A) y = -5 + 5x
B) y = -3 + 4x
C) y = -2 + 3x
84) Maximize f(x, y) = 2x2 + 7xy + 9y2 subject to x + y = 1 and x + 4y = 9. A) 54
B) 22 3
C) 206 9
20
82)
D) 12 2 83) D) y = - 10 + 4x 3
84) D) 346 9
Find
f f and . x y
85) f(x, y) =
1 2 x + y2
A) f = x
B) f = x
C) f = x
x
3/2 2(x2 + y2 )
;
f = y
y
3/2 2(x2 + y2 )
1 f 1 ; = 3/2 y 3/2 2 2 2 2(x + y ) 2(x + y2 ) x
3/2 2(x2 + y2 )
D) f = x
85)
;
f = y
y
3/2 2(x2 + y2 )
x f y ; = 3/2 y 3/2 2 2 2 (x + y ) (x + y2 )
Solve the problem.
86) Find an equation for the level curve of the function f(x, y) = y2 - 100 that passes through the point 0, 10 . A) y = -100
B) y = 10
87) Evaluate dw at t = 2 dt
C) y = ± 10
D) y = 100
for the function w = x2 - y2 - 9x; x = cost, y = sin t.
B) 0
A) -11
87)
C) -13
D) -7
88) Find the equation for the tangent plane to the surface z = e7x2 + 6y2 at the point (0, 0, 1). A) z = -1 B) z = 0 C) z = 1 D) z = 2 Find the linearization of the function at the given point. 89) f(x, y, z) = ln(-4x + 5y + 6z) at - 1 , 1 , 1 4 5 6
88)
89)
A) L(x, y, z) = - 2x + 5 y + 3z + ln 3 - 1
B) L(x, y, z) = - 2x + 5 y + 3z + ln 2 - 1
C) L(x, y, z) = - 4 x + 5 y + 2z + ln 2 - 1
D) L(x, y, z) = - 4 x + 5 y + 2z + ln 3 - 1
2
3
86)
2
3
3
21
3
Sketch the surface z = f(x,y). 90) f(x, y) = 2 -x2 - y2
90)
A)
B)
C)
D)
Solve the problem.
91) Find the maximum value of f(x, y, z, w) = x + y + z + w subject to x2 + y2 + z2 + w2 = 1. A) 4 B) 2 C) 10 D) 1
Provide an appropriate response. 92) Find any local extrema (maxima, minima, or saddle points) of f(x, y) given that fx = 6x - 8y and fy = -2x + 5y.
A) Saddle point at (0, 0)
B) Local maximum at 4 , 5
C) Local minimum at 4 , 5
D) Local minimum at (0, 0)
3 2
3 2
22
91)
92)
Find the linearization of the function at the given point. 93) f(x, y, z) = -4x2 + 3y2 + 8z 2 at (1, -2, 3)
93)
A) L(x, y, z) = -8x + 12y + 48z + 80 C) L(x, y, z) = -8x - 12y + 48z - 80
B) L(x, y, z) = -8x + 12y + 48z - 80 D) L(x, y, z) = -8x - 12y + 48z + 80
Solve the problem. 94) Find the distance between the skew lines x = t - 6, y = t, z = 2t and x = t, y = t, z = -t. A) 3 3 B) 3 2
94)
C) 2 2
D) 2 3
2
2
25
16
5 4 , . 2 2
95) Write an equation for the tangent line to the curve x + y = 1 at the point A) x + y = 1 4
B) x + y = 2
5
5
C) x + y = 2
4
4
95)
D) x + y = 1
5
5
4
Find the absolute maxima and minima of the function on the given domain. 96) f(x, y) = x2 + xy + y2 on the square -5 x, y 5
96)
A) Absolute maximum: 75 at (5, 5) and (-5, -5); absolute minimum: 0 at (0, 0) B) Absolute maximum: 25 at (5, -5) and (-5, 5); absolute minimum: 0 at (0, 0) C) Absolute maximum: 25 at (5, -5) and (-5, 5); absolute minimum: 2.638827907e+15 at 1.407374884e+14
-
5 5 5 5 ,5 , , -5 , 5, - , and -5, 2 2 2 2
D) Absolute maximum: 75 at (5, 5) and (-5, -5); absolute minimum: 25 at (5, -5) and (-5, 5) Write a chain rule formula for the following derivative. 97) u for u = f(p, q); p = g(x, y, z), q = h(x, y, z) x
97)
A) u = u p + u q
B) u = u x + u x
C) u = u q
D) u = u p + u q + u q
x x
p q
x
q
x
x
x
x
p p
p x
q q
q y
q
z
Solve the problem.
98) Find parametric equations for the normal line to the surface z = e6x2 + 9y2 at the point (0, 0, 1). A) x = 0, y = 0, z = t - 1 B) x = t, y = t, z = t - 1 C) x = 0, y = 0, z = t + 1 D) x = t, y = t, z = -t - 1
23
98)
Use Taylor's formula to find the requested approximation of f(x, y) near the origin. 99) Cubic approximation to f(x, y) = ex + 8y
99)
A) 1 + x + 8y + 1 x2 + 4xy + 32y2 + 1 x3 + 4x2 y + 32xy2 + 256 y3 2 6 3
B) 1 + x + 8y + 1 x2 + 4xy + 32y2 + 1 x3 + 8 x2 y + 64 xy2 + 256 y3 2
6
3
3
3
C) 1 + x + 8y + 1 x2 + 8xy + 32y2 + 1 x3 + 8 x2 y + 64 xy2 + 256 y3 2
6
3
3
3
D) 1 + x + 8y + 1 x2 + 8xy + 32y2 + 1 x3 + 4x2 y + 32xy2 + 256 y3 2
6
3
Provide an appropriate answer. 100) Find w when r = 4 and s = -5 if w(x, y, z) = xz + y^2, x = 2r + 8, y = r + s, and z = r - s. r
A) w = 18
B) w = 19
r
C) w = 28
r
100)
D) w = 32
r
r
Find the requested partial derivative.
101) ( w/ z)x at (x, y, z, w) = (1, 2, 9, 158) if w = x2 + y2 + z 2 + 4xyz and z = x3 + y3 A) 44
C) 88
B) 88
A)
B)
2f
x2 y2 )] 2f x2
= y2 sin xy2 ;
= - y2 sin xy2 ;
2f
y2
= 2[2y2 cos (xy2 ) - sin (xy2 )] ;
2f
y2
D) 176
3
Find all the second order partial derivatives of the given function. 102) f(x, y) = cos (xy2 )
= 2[ sin (xy2 )- 2y2 cos (xy2 )] ;
101) 3
102) 2f
y x 2f
y2 cos xy2 - sin xy2 2 2 D) f = -y4 cos xy2 ; f = - 2x[2xy2 cos (xy2 ) + sin(xy2 )]; x2 y2
x y
=
y x
(xy2 )] 2 2 C) f = - y2 sin xy2 ; f = 2y 2y2 cos (xy2) - sin (xy2 ) ; x2 y2
2f
=
2f
y x
2f
x y 2f
=
2f
y x
= 2y[y2 cos (xy2 ) - sin (x
= 2y [sin (xy2 )-y2 cos
x y
=
=2
2f
x y
=
- 2y[xy2 cos (xy2 ) + sin(xy2 )];
Show that the function is a solution of the wave equation. 103) w(x, t) = ln -7cxt A) Yes
103) B) No
24
Solve the problem. 104) The resistance R produced by wiring resistors of R 1 and R2 ohms in parallel can be calculated
104)
from the formula 1 1 1 = + . R R1 R2 If R 1 and R2 are measured to be 9 ohms and 3 ohms respectively and if these measurements are accurate to within 0.05 ohms, estimate the maximum percentage error in computing R. A) 2.78% B) 1.67% C) 2.22% D) 1.39%
Find the limit. lim 105) (x, y) (0, 0) 2 y - 2 x + xy - x y-
x y
105)
x
A) 0
B) 2
C) 4
D) No limit
Write a chain rule formula for the following derivative. 106) dz for z = f(r, s); r = g(t), s = h(t) dt
106)
A) dz = dr + ds
B) dz = z dr + z ds
C) dz = z dt + z dt
D) dz = z dr
dt dt
dt
dt
r dr
dt
s ds
dt
r dt
s dt
r ds
Find the domain and range and describe the level curves for the function f(x,y). 1 107) f(x, y) = 2 9x + 5y2
107)
A) Domain: all points in the xy-plane; range: all real numbers; level curves: ellipses
9x2 + 5y2 = c B) Domain: all points in the xy-plane except (0, 0); range: all real numbers; level curves: ellipses 9x2 + 5y2 = c
C) Domain: all points in the xy-plane except (0, 0); range: real numbers > 0; level curves:
ellipses 9x2 + 5y2 = c D) Domain: all points in the xy-plane; range: real numbers > 0; level curves: ellipses 9x2 + 5y2 = c
Write a chain rule formula for the following derivative. 108) u for u = f(r, s, t); r = g(y), s = h(z), t = k(x, z) x
A) u = u t x
t
x
108)
B) u = u x
C) u = 0
t
x
25
D) u = t x
x
Sketch a typical level surface for the function. 109) f(x, y, z) = x - y2 - z 2
109)
A)
B)
C)
D)
Compute the gradient of the function at the given point. 110) f(x, y) = -3x - 5y, (-3, -5) A) f = -15i + 25j B) f = 5i - 5j
110) C) f = -15i - 25j
Find the extreme values of the function subject to the given constraint. 111) f(x, y, z) = x2 + y2 + z 2 , x + 2y + 3z = 6
A) Maximum: none; minimum: 2 at 1 , 4 , 9 7 7 7
B) Maximum: none; minimum: 2 at 1 , 2 , 3 7
7 7 7
C) Maximum: none; minimum: 18 at 3 , 6 , 9 7
7 7 7
D) Maximum: none; minimum: 72 at 6 , 12 , 18 7
7
7
7
26
D) f = -3i - 5j
111)
Solve the problem. 112) The resistance R produced by wiring resistors of R 1, R2 , and R 3 ohms in parallel can be
112)
calculated from the formula 1 1 1 1 = + + . R R1 R2 R 3 If R 1 , R2 , and R 3 are measured to be 10 ohms, 9 ohms, and 4 ohms respectively, and if these measurements are accurate to within 0.05 ohms, estimate the maximum possible error in computing R. A) 0.020 B) 0.024 C) 0.016 D) 0.012
Provide an appropriate response. 113) Find the direction in which the function is increasing most rapidly at the point P0 .
113)
f(x, y) = xey - ln(x), P0 (-1, 0)
A) - 2 i
B) - 1 i + 2 j
5
5
C) 2 i - 1 j 5
5
D) - 2 i - 1 j
5
5
5
Determine whether the given function satisfies a Laplace equation. 114) f(x, y) = e-3ysin -3x
A) Yes
114)
B) No
Find all the local maxima, local minima, and saddle points of the function. 115) f(x, y) = x3 + y3 - 27x - 300y - 5
115)
A) f(-3, -10) = 2049, local maximum; f(3, 10) = -2059, local minimum B) f(-3, -10) = 2049, local maximum C) f(3, -10) = 1941, saddle point; f(-3, 10) = -1951, saddle point D) f(3, 10) = -2059, local minimum; f(3, -10) = 1941, saddle point; f(-3, 10) = -1951, saddle point; f(-3, -10) = 2049, local maximum
Find the limit. lim
2
2
116) (x, y) (1, -1) 3x + 6xy + 3y x -y
A) 1 2
116)
x+y
B) 0
C) 1
D) No limit
Solve the problem.
y
117) Find an equation for the level curve of the function f(x, y) = -8, -2 . A) y2 + x2 = 68
t dt that passes through the point
x
B) y2 - x2 = -60
C) y2 - x2 = 68
27
D) y2 - x2 = 60
117)
Find the extreme values of the function subject to the given constraint. 118) f(x, y) = 12x + 3y, xy = 4, x > 0, y > 0 A) Maximum: 30 at (2, 2); minimum: 24 at (1, 4)
118)
B) Maximum: none; minimum: 30 at (2, 2) C) Maximum: none; minimum: 24 at (1, 4) D) Maximum: 51 at (4, 1); minimum: 30 at (2, 2) Find all the local maxima, local minima, and saddle points of the function. 119) f(x, y) = 10 - x4y4
119)
A) f(0, 0) = 10, local maximum B) f(10, 10) = -99,999,990, local minimum C) f(10, 0) = 10, saddle point; f(0, 10) = 10, saddle point D) f(0, 0) = 10, local maximum; f(10, 10) = -99,999,990, local minimum Sketch the surface z = f(x,y). 120) f(x, y) = x + y
120)
A)
B)
C)
D)
28
Solve the problem.
121) Find the derivative of the function f(x, y, z) = x + y + z at the point (9, -9, 9) in the direction in y
z
which the function increases most rapidly. A) 2 3 B) 1 3 9 3
x
C) 2 2 9
Find the absolute maxima and minima of the function on the given domain. 122) f(x, y) = x2 + 6x + y2 + 12y + 9 on the rectangular region -1 x 1, -2 y 2 A) Absolute maximum: 44 at (1, 2); absolute minimum: -16 at (-1, -2)
121)
D) 1 2 3
122)
B) Absolute maximum: 53 at (2, 2); absolute minimum: 9 at (0, 0) C) Absolute maximum: 44 at (1, 2); absolute minimum: 9 at (0, 0) D) Absolute maximum: 53 at (2, 2); absolute minimum: -16 at (-1, -2) Sketch a typical level surface for the function. 123) f(x, y, z) = e(x2 + y2 + z2 )
123)
A)
B)
C)
D)
29
Solve the problem. 124) Find the least squares line through the points (1, -1) and (2, -3). A) y = -2x + 1 B) y = -4x - 5 C) y = -4x + 1 Find the requested partial derivative. 125) ( w/ x)y,z if w = x3 + y3 + z 3 + 9xyz A) 3(x2 + 6xyz) B) 3(x2 + 6yz)
124) D) y = -2x - 5
125) C) 3(x2 + 3xyz)
D) 3(x2 + 3yz)
Write a chain rule formula for the following derivative. 126) w for w = f(p, q); p = g(x, y), q = h(x, y) x
126)
A) w = w + w x
p
B) w = w p
q
x
C) w = w p + w q x
p
x
q
p
x
D) w = w x + w x
x
x
x
p
x
q
Find the absolute maximum and minimum values of the function on the given curve. 127) Function: f(x, y) = xy; curve: x2 + y2 = 16, x 0, y 0. (Use the parametric equations x = 4 cos t,
127)
y = 4 sin t.)
A) Absolute maximum: 4 at t = B) Absolute maximum: 8 at t = C) Absolute maximum: 8 at t = D) Absolute maximum: 4 at t =
4 4 4 4
; absolute minimum: 0 at t = 0 and t = ; absolute minimum: 0 at t = 0 and t = ; absolute minimum: 0 at t = 0 and t = ; absolute minimum: 0 at t = 0 and t =
4 4 2 2
Write a chain rule formula for the following derivative. 128) w for w = f(x, y, z); x = g(r, s, t), y = h(r, s, t), z = k(r, s, t) t
128)
A) w = w x + w y
B) w = w x + w y + w z
C) w = w + w + w
D) w = w x + w y + w z
t t
x x
t
y
y
t
t
z
t
x x
r t
y y
s t
z z
t t
Solve the problem.
129) Find the points on the curve xy2 = 1024 that are closest to the origin. A) (8, 8 2) B) (512, 2) C) (512, ± 2)
Find the linearization of the function at the given point. 130) f(x, y, z) = -9xy + 3yz + 4zx at (1, 1, 1) A) L(x, y, z) = -5x - 6y + 7z + 2
129) D) (8, ±8 2)
130) B) L(x, y, z) = -9x + 3y + 4z + 4 D) L(x, y, z) = -5x - 6y + 7z + 4
C) L(x, y, z) = -9x + 3y + 4z + 2
30
Provide an appropriate response. 131) Find any local extrema (maxima, minima, or saddle points) of f(x, y) given that fx = 9x - 27 and fy = 10y - 20.
A) Local maximum at (2, 3) C) Local maximum at (3, 2)
131)
B) Local minimum at (3, 2) D) Saddle point at (2, 3)
Find fx , fy, and fz .
132) f(x, y, z) = x2 y + y2 z + xz 2 A) fx = 2y + z2 ; fy = x2 + 2z; fz = y2 + 2x
132) B) fx = 2xy; fy = x2 + 2yz; fz = y2 + 2xz D) fx = 2xy + z 2 ; fy = x2 + 2yz; fz = y2 + 2xz
C) fx = 2xy + z 2 ; fy = x2 + yz; fz = y2 + xz Find the limit. lim
3
2
133) (x, y) (4, -10) 9x - 6xy
133)
9x2 - 6y2
9x2 - 6y2 0
A) 0
B) 4
D) -10
C)
Provide an appropriate response. 134) Which order of differentiation will calculate fxy faster, x first or y first? f(x, y) = x2 y + A) y first
134)
y2 + 1
B) x first
135) Find the direction in which the function is increasing most rapidly at the point P0 .
135)
f(x, y) = xy2 - yx2 , P0 (-2, 1)
A) - 8
i+
5 j 89
B)
5 i89
8 j 89
D) 5 89 i - 8 89 j
89
C)
5 i+ 89
8 j 89
Find the extreme values of the function subject to the given constraint. 136) f(x, y, z) = (x - 1)2 + (y - 2)2 + (z - 2)2 , x2 + y2 + z2 = 36
136)
A) Maximum: 77 at (-4, -4, -2); minimum: 13 at (4, 4, 2) B) Maximum: 81 at (-2, -4, -4); minimum: 9 at (2, 4, 4) C) Maximum: 49 at (-2, 4, -4); minimum: 41 at (2, -4, 4) D) Maximum: 77 at (-4, -2, -4); minimum: 13 at (4, 2, 4) Estimate the error in the quadratic approximation of the given function at the origin over the given region. 137) f(x, y) = sin 4x sin 2y, x 0.1, y 0.1 A) |E(x, y)| 0.0360 B) |E(x, y)| 0.0540
C) |E(x, y)| 0.0120
D) |E(x, y)| 0.0270
31
137)
Find the domain and range and describe the level curves for the function f(x,y). 138) f(x, y) = ex+y
138)
A) Domain: all points in the xy-plane; range: all real numbers; level curves: lines x + y = c B) Domain: all points in the first quadrant of the xy-plane; range: real numbers z > 0; level
curves: lines x + y = c C) Domain: all points in the first quadrant of the xy-plane; range: all real numbers; level curves: lines x + y = c D) Domain: all points in the xy-plane; range: real numbers z > 0; level curves: lines x + y = c
Find the limit.
139)
P
lim 2 z ey sin-1 (6, 0, -6) x
A)
139) C) 0
B)
2
D) 1
Solve the problem.
140) Find the derivative of the function f(x, y) = x2 + xy + y2 at the point (5, 6) in the direction in which the function decreases most rapidly. A) - 545 B) -3 62
Find
C) - 461
140)
D) - 455
f f and . x y
141) f(x, y) = x
141)
x+y
A) f = x
y
(x + y)2
f x = y (x + y)2
;
C) f = 2x + y ; f = x
(x + y)2
y
B) f = 2x + y ; f = -
x
(x + y)2
x
(x + y)2
D) f =
y
x
(x + y)2
y
;
x
(x + y)2
f x = y (x + y)2
Determine whether the given function satisfies a Laplace equation. 142) f(x, y, z) = cos (-4x) sin (-4y) e( 32z)
A) No
142)
B) Yes
At what points is the given function continuous? z 143) f(x, y, z) = 2 x + y2 - 4
143)
A) All (x, y, z) such that x2 + y2 4 C) All (x, y, z) such that x2 + y2 0
B) All (x, y, z) D) All (x, y, z) such that x2 + y2 16
32
Provide an appropriate response.
144) Find F (x) if F(x) =
x2
0 x2
A) x x2 + 1 + 0
0
144)
x dt t2 + x2 x2
C) 2x2 x2 + 1 +
t2 + x2 dt. x2
B) 2x2 x2 + 1 + 0
x2 dt t2 + x2
x2
D) 2x x2 + 1 + 0
x dt t2 + x2 x dt t2 + x2
Solve the problem.
145) Find an equation for the level curve of the function f(x, y) = 100 - x2 - y2 that passes through the point
3, 3 . 2 A) x + y2 = - 6
B) x2 + y2 = 6
C) x2 + y2 = 106
145)
D) x2 - y2 = 6
146) If the length, width, and height of a rectangular solid are measured to be 6, 2, and 6 inches
146)
respectively and each measurement is accurate to within 0.1 inch, estimate the maximum percentage error in computing the volume of the solid. A) 10.00% B) 7.50% C) 6.67% D) 8.33%
Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0). 147) f(x, y) = x + y x2 + y + y2
A) 1
B) 0
C) 2
33
147) D) No limit
Sketch a typical level surface for the function. 148) f(x, y, z) = y - x2 - z 2
148)
A)
B)
C)
D)
Solve the problem. 149) Find parametric equations for the normal line to the surface -3x + 10y + 8z = 3 at the point (1, -1, 2). A) x = -3t + 1, y = 10t - 1, z = 8t + 2 B) x = t - 3, y = -t + 10, z = 2t + 8
C) x = -t - 3, y = t + 10, z = -2t + 8
D) x = -3t - 1, y = 10t + 1, z = 8t - 2
Compute the gradient of the function at the given point. 150) f(x, y) = ln(-2x - 6y), (-3, -7) A) f = - 1 i - 7 j 16 48
150) B)
C) f = - 1 i - 1 j 24
149)
1 1 f= i+ j 48 48
D) f = - 7 i - 1 j
8
48
34
16
Find all the second order partial derivatives of the given function. 151) f(x, y) = ln (x2 y - x)
A) B) C) D)
2f
x2 2f
x2 2f
x2 2f
x2
=
xy - x2 y2 - 1 2 (x2 y - x)
;
2f
y2
=-
x4
(x2 y - x)
2
;
=
2xy - 2x2 y2 - 1 2 f x4 = ; ; 2 y2 (x2 y - x)2 (x2 y - x)
=
2xy - 2x2 y2 - 1 2 f x2 =; ; 2 2 y2 (x2 y - x) (x2 y - x)
=
2xy - 2x2 y2 - 1 2 f x4 =; ; 2 2 2 y (x2 y - x) (x2 y - x)
2f
y x 2f
y x
=
151) 2f
x y 2f
=
2f
y x
x y =
= -
=
2f
x y
x2 (x2 y - x)
2
x2
2 (x2 y - x) = -
2f 2f = = y x x y
x2 (x2 y - x) x2 (x2 y - x)
Solve the problem. 152) Find the least squares line through the points (1, -60), (2, -12), and (3, 24). A) y = 42x - 100 B) y = -18x - 68 C) y = -18x - 100
2 2
152) D) y = 42x - 68
Find the derivative of the function at P0 in the direction of u.
153) f(x, y) = tan-1 5x , P0 (3, -4), u = 12i - 5j
153)
y
A) - 15
B) - 105
241
C) - 120
3133
3133
D) - 165
3133
Find fx , fy, and fz .
154) f(x, y, z) = cos y
154)
xz 2
A) fx = - cos y ; fy = - sin y ; fz = - 2 cos y x2z 2
xz 2
xz 3
B) fx = cos y ; fy = sin y ; fz = 2 cos y z2
xz 2
xz
C) fx = - cos y ; fy = - sin y ; fz = - 2 cos y z2
xz 2
xz
D) fx = cos y ; fy = sin y ; fz = 2 cos y x2 z 2
xz 2
xz 3
Provide an appropriate response. 155) Find the direction in which the function is increasing most rapidly at the point P0 .
155)
f(x, y, z) = xy - ln(z), P0 (1, 1, 1)
A) 1 ( i - j + k) 3
B) 3 ( i + j - k)
C) 1 ( i + j - k)
3
3
35
D) 1 ( i + j - k) 3
Find the extreme values of the function subject to the given constraint. 156) f(x, y) = 3x - y + 1, 3x2 + y2 = 9
156)
A) Maximum: 7 at 3 , - 3 ; minimum: -5 at - 3 , 3 2 2 2 2
B) Maximum: 7 at 3 , - 3 ; minimum: -2 at - 3 , - 3 2
2
2
2
C) Maximum: 4 at 3 , 3 ; minimum: -2 at - 3 , - 3 2 2
2
2
D) Maximum: 4 at 3 , 3 ; minimum: -5 at - 3 , 3 2 2
2 2
Write a chain rule formula for the following derivative. 157) u for u = f(x); x = g(p, q, r) r
157)
A) u = du x + du x + du x
B) u = du
C) u = x
D) u = du x
r
dx
r
p
dx
q
dx
r
r
r
r
dx dx
r
Provide an appropriate response. 158) Find any local extrema (maxima, minima, or saddle points) of f(x, y) given that fx = -3x + 8y and fy = 3x + 5y.
158)
A) Local maximum at 8 , - 5
B) Saddle point at (0, 0)
C) Saddle point at (-15, 24)
D) Local minimum at 8 , - 5
3
3
3
3
Find all the local maxima, local minima, and saddle points of the function. 159) f(x, y) = 6x2 y + 4xy2
159)
A) f 1 , 1 = 1 , local minimum 6 4 12 B) f(0, 0) = 0, saddle point
C) f 1 , 1 = 3.501602581e+13 , local minimum 4 6
3.878698244e+14
D) f(24, 24) = 138,240, local maximum Solve the problem.
160) Evaluate z at (u, v) = (1, 1) for the function z = xy - y2; x = u - v, y = uv.
160)
u
A) -1
B) 2
C) 1
36
D) 0
Find
f f and . x y
161) f(x, y) = x3 - 4x2y - 6xy3
161)
f f = 3x2 + 2xy - 6y3 ; = -4x2 + 3xy2 x y
A)
B)
C) f = 3x2 ; f = -4x2 - 18xy2 x
f f = x2 - 4xy - 6y3 ; = -4x2 - 6xy2 x y
D) f = 3x2 - 8xy - 6y3 ; f = -4x2 - 18xy2
y
x
y
Find the limit.
162)
2 2 lim e-x - y (x, y) (3, 3)
A) e18
162) B) e-18
C) 0
D) 1
Estimate the error in the quadratic approximation of the given function at the origin over the given region. 163) f(x, y) = e4x cos 2y, x 0.1, y 0.1
A) |E(x, y)| 0.0806 C) |E(x, y)| 0.0179
163)
B) |E(x, y)| 0.0537 D) |E(x, y)| 0.0403
Find the limit.
164)
x5 + y6 lim sin x-y+9 (x, y) (0, 0)
B) 1
A) 1
Find
164) C) 0
9
D) No limit
f f and . x y
165) f(x, y) = xye-y A)
165)
f f = ye-y; = xe-y x y
B)
C) f = ye-y; f = - xye-y x
f f = ye-y; = xe-y(y - 1) x y
D) f = ye-y; f = xe-y(1 - y)
y
x
y
Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0). 2 166) f(x, y) = cos x x2 + y2
A)
2
B) 0
C) 1
37
166) D) No limit
Sketch the surface z = f(x,y). 167) f(x, y) = 4x2 + 4y2 + 2
167)
A)
B)
C)
D)
Find an upper bound for the magnitude |E| of the error in the approximation f(x, y) L(x, y) at the given point over the given region R. 168) f(x, y) = ln(8x + 5y) at (1, 1); R: |x - 1| 0.1, |y - 1| 0.1 168) A) |E| 0.0118 B) |E| 0.0063 C) |E| 0.0076 D) |E| 0.0094
Provide an appropriate response.
2 2 169) Define f(0,0) in a way that extends f(x, y) = x y to be continuous at the origin. x2 + y2
169)
A) f(0, 0) = 1 B) f(0, 0) = 2 C) f(0, 0) = 0 D) No definition makes f(x, y) continuous at the origin. Solve the problem.
170) Evaluate u at (x, y, z) = (4, 3, 5) for the function u = p2 q2 - r; p = y - z, q = x + z, r = x + y. z
A) 198
B) 396
C) -252
38
D) 792
170)
Find the limit.
171)
lim (x, y) (0, 1)
y4 sin x x
B) 0
A) 172)
171)
lim ln (x, y) (1, 1)
C) 1
D) No limit
x+y xy
A) 0
172) B) ln 2
C) -ln 2
D) No limit
Find all the local maxima, local minima, and saddle points of the function. 173) f(x, y) = x2 + 10x + y2 + 20y - 8
A) f(-5, 10) = 267, saddle point C) f(5, 10) = 367, local maximum
173)
B) f(-5, -10) = -133, local minimum D) f(5, -10) = -33, saddle point
Use Taylor's formula to find the requested approximation of f(x, y) near the origin. 174) Cubic approximation to f(x, y) = sin(6x + y) A) 6x + y - 36x3 - 1 y3 B) 6x + y - 72x3 - 18x2y - 3xy2 - 1 y3 6 3
C) 6x + y - 36x3 - 18x2y - 3xy2 - 1 y3
174)
D) 6x + y - 72x3 - 1 y3
6
3
Provide an appropriate response. 175) Determine whether the function f(x, y) = 8x2 y2 + 3x4 y4
175)
has a maximum, a minimum, or neither at the origin. A) Maximum B) Minimum
C) Neither
Solve the problem.
176) Find the equation for the tangent plane to the surface z = -6x2 - 7y2 at the point (2, 1, -31). A) 2x + y - 31z = 1 B) 2x + y - 31z = -28 C) -24x - 14y - z = -31 D) -24x - 14y - z = -40
176)
Provide an appropriate answer.
2 177) Find w when u = -5 and v = 2 if w(x, y, z) = xy , x = u , y = u + v, and z = u · v. u
z
A) w = - 15 u
2
B) w = - 3 u
177)
v
C) w = - 3
2
u
4
D) w = - 12 u
125
Solve the problem. 178) Find the derivative of the function f(x, y, z) = ln(xy + yz + zx) at the point (-6, -12, -18) in the direction in which the function decreases most rapidly. A) - 5 2 B) - 5 2 C) - 5 2 D) - 5 2 66 102 78 42
39
178)
179) A simple electrical circuit consists of a resistor connected between the terminals of a battery. The
179)
voltage V (in volts) is dropping as the battery wears out. At the same time, the resistance R (in ohms) is increasing as the resistor heats up. The power P (in watts) dissipated by the circuit is V2 given by P = . Use the equation R dP P dV P dR = + dt V dt R dt to find how much the power is changing at the instant when R = 3 ohms, V = 2 volts, dR/dt = 2 ohms/sec and dV/dt = -0.02 volts/sec. A) -0.92 watts B) 0.92 watts C) -0.86 watts D) 0.86 watts
Find the derivative of the function at P0 in the direction of u.
180) f(x, y, z) = -10x + 7y - 9z, P0 (4, -10, 4), u = 3i - 6j - 2k A) - 33 7
B) - 54
180)
C) - 26
7
D) - 75
7
7
Find the absolute maximum and minimum values of the function on the given curve. 2 2 181) Function: f(x, y) = xy; curve: x + y = 1, y 0. (Use the parametric equations x = 6 cos t, 36 64 y = 8 sin t.)
A) Absolute maximum: 12 at t = B) Absolute maximum: 12 at t = C) Absolute maximum: 24 at t = D) Absolute maximum: 24 at t =
4 4 4 4
; absolute minimum: - 12 at t =
3 4
; absolute minimum: - 24 at t =
3 4
; absolute minimum: - 12 at t =
3 4
; absolute minimum: - 24 at t =
3 4
181)
Find fx , fy, and fz .
182) f(x, y, z) = z(ex)y A) fx = zyexy; fy = zxexy; fz = exy
182) B) fx = zexy; fy = zexy; fz = exy
C) fx = zyexy; fy = zxexy; fz = zexy
D) fx = zxexy; fy = zyexy; fz = exy
Solve the problem.
183) Evaluate dw at t = 5 for the function w = exyz 2 ; x = t, y = t, z = 1 . dt
A) 8 5
183)
t
C) - 8 e
B) 0
5
40
D) 8 e 5
Provide an appropriate response. 184) Which order of differentiation will calculate fxy faster, x first or y first? f(x, y) =
1 , g(y) 0 g(y)
A) y first Find the limit.
185)
P
184)
B) x first
lim sec2 x - tan2 y + z (8, 8, 4)
A) 9
185)
B) 4
C) 5
D) 8
Solve the problem.
186) Find parametric equations for the normal line to the surface z = ln(8x2 + 6y2 + 1) at the point (0, 0, 0). A) x = 1, y = 1, z = -t
B) x = t , y = t, z = 0 D) x = 0, y = 0, z = t
C) x = t, y = t, z = -1 Compute the gradient of the function at the given point. 187) f(x, y) = tan-1 -10x , (6, 7) y
B) f = - 35 i + 30 j
C) f = - 35 i - 30 j
D) f = - 70 i - 60 j
1849
Find
187)
A) f = - 70 i + 60 j 3649
186)
3649
1849
1849
3649
1849 3649
f f and . x y
188) f(x, y) = 4x - 5y2 - 2 A)
188)
f f = -10y; =4 x y
B)
C) f = 4; f = -10y x
f f = 4x; = -10y x y
D) f = 2; f = -10y - 2
y
x
y
Solve the problem. 189) Find the point on the line 2x + 3y = 5 that is closest to the point (1, 2). A) 5 , 15 B) 7 , 13 C) 7 , 17 13 13 17 17 13 13 Find the limit.
190)
P
189) D) 5 , 15 17 17
lim ln (z x2 + y2 ) (6, 6, 1)
A) ln 36
190) B) ln 6 2
C) ln 2
41
D) 0
Sketch the surface z = f(x,y). 191) f(x, y) = x2
191)
A)
B)
C)
D)
Find an upper bound for the magnitude |E| of the error in the approximation f(x, y) L(x, y) at the given point over the given region R. 192) f(x, y, z) = ln(7x + 9y + 3z) at (1, 1, 1); R: |x - 1| 0.1, |y - 1| 0.1, |z - 1| 0.1 192) A) |E| 0.014 B) |E| 0.015 C) |E| 0.0187 D) |E| 0.0125
At what points is the given function continuous? 193) f(x, y, z) = ln(x + y + z - 5) A) All (x, y, z) in the first octant
193) B) All (x, y, z) D) All (x, y, z) such that x + y + z > 5
C) All (x, y, z) such that x + y + z 5
42
Answer the question. 194) Consider a function f(x, y, z), where the independent variables are constrained to lie on the curve
194)
r (t) = x(t)i + y(t)j + z(t)k. What mathematical fact forms the basis for the method of Lagrange multipliers?
A) f g = 0
B) f·v = 0
C) g = 0
D) f approaches a local extremum as
0.
Solve the problem.
195) Write an equation for the tangent line to the curve x2 - 9xy + y2 = 11 at the point (-1, 1). A) y = x + 1 B) x + y = 1 C) x - y + 2 = 0 D) y = x - 2
Find the limit.
sec x + 9 2x - tan y
lim
196) (x, y)
0, -
195)
196)
4
A) - 2 - 9
B) 10
C) -10
D) 2 + 9
Solve the problem.
197) Find the extreme values of f(x, y, z) = x2 + y2 + z2 subject to 3x - y + z = 6 and x + 2y + 2z = 2.
197)
A) Maximum: none; minimum: 148 at 74 , 20 , - 28 45 45 45 45 B) Maximum: none; minimum: 148 at - 74 , 20 , 28 45
45 45 45
C) Maximum: none; minimum: 148 at 74 , - 20 , 28 45
45
45 45
D) Maximum: none; minimum: 148 at 74 , - 20 , - 28 45
45
45
45
198) Find parametric equations for the normal line to the surface x2 + 5xyz + y2 = 7z 2 at the point (1, 1, 1). A) x = 7t + 1, y = -7t + 1, z = -9t + 1
198)
B) x = 7t + 1, y = 7t + 1, z = -9t + 1 D) x = t - 7, y = t - 7, z = t + 9
C) x = t + 7, y = t + 7, z = t - 9 Find the requested partial derivative.
199) ( z/ x)y at (x, y, z) = (1, 1, 1) if z 3 + 15xyz = 16 A) - 5
11
199)
B) - 5
C) - 5
3
6
D) - 5 7
Use implicit differentiation to find the specified derivative at the given point. 200) Find dy at the point (1, 1) for 7x2 + 3y3 + 5xy = 0. dx
A) - 3 2
B) 19
C) - 19
14
14
43
200) D) - 19 8
Sketch the surface z = f(x,y). 201) f(x, y) = 1 - x
201)
A)
B)
C)
D)
Find the derivative of the function at P0 in the direction of u.
202) f(x, y) = ln(6x + 7y), P0 (3, 7), u = 6i + 8j A) 52
202)
B) 8
335
C) 34
67
335
D) 46
335
Find the limit. lim
203) (x, y) (5, 4) xy + 10y - 4x - 40 y 4
A) -5
203)
y- 4
B) 0
C) 1
D) 15
Solve the problem.
204) Write parametric equations for the tangent line to the curve of intersection of the surfaces x = y2 and y = 7z 2 at the point (49, 7, 1).
A) x = 98t + 49, y = 7t + 7, z = t + 1 C) x = 98t + 49, y = 14t + 7, z = t + 1
B) x = 196t + 49, y = 7t + 7, z = t + 1 D) x = 196t + 49, y = 14t + 7, z = t + 1 44
204)
Use Taylor's formula to find the requested approximation of f(x, y) near the origin. 205) Quadratic approximation to f(x, y) = sin(6x + y) A) 6x + y + 3x2 + 1 xy + 1 y2 B) 6x + y + 3x2 + 1 y2 2 2 2
C) 6x + y + 3x2 + y2
205)
D) 6x + y
Solve the problem.
206) Find the point on the paraboloid z = 2 - x2 - y2 that is closest to the point (1, 1, 2). A)
1 1 3 ,- , 2 2 2
B)
1 1 3 , , 2 2 2
C) - 1 , - 1 , 3 2 2 2
Sketch a typical level surface for the function. 207) f(x, y, z) = 7e9(y2 + z 2 )
206)
D) - 1 , 1 , 3 2 2 2
207)
A)
B)
C)
D)
45
Sketch the surface z = f(x,y). 208) f(x, y) = 3 - x2
208)
A)
B)
C)
D)
Find the linearization of the function at the given point. 209) f(x, y) = -5x2y3 at (-10, 7)
209)
A) L(x, y) = 34,300x + 7350y + 686,000 C) L(x, y) = -3430x + 7350y + 686,000
B) L(x, y) = 34,300x - 73,500y + 686,000 D) L(x, y) = -3430x - 73,500y + 686,000
Solve the problem.
210) About how much will f(x, y) = tan-1 xy change if the point (x, y, z) moves from 5 2, 7
2
distance of ds =
A)
a
210)
1 unit in the direction of i + j? 10
27 24520
B)
11 12260
C) 3
D)
6130
17 24520
Find fx , fy, and fz .
211) f(x, y, z) = ln (xy)z
211)
A) fx = - z ; fy = - z ; fz = ln (xy) x y
B) fx = z ln z ; fy = z ln z ; fz = ln (xy) x y
C) fx = z ; fy = z ; fz = ln (xy)
D) fx = z ; fy = z ; fz = z ln (xy)z - 1
x
y
x
46
y
Find all the local maxima, local minima, and saddle points of the function. 212) f(x, y) = 2xy + 2x + 8y A) f - 4, 1 = - 8, saddle point; f 4, - 1 = - 8, saddle point
212)
B) f - 4, - 1 = - 8, saddle point C) f - 4, 1 = - 8, local minimum; f 4, - 1 = - 8, local minimum D) f 4, 1 = 24, local maximum Solve the problem.
213) Find the extreme values of f(x, y, z) = x + 2y subject to x + y + z = 1 and y2 + z 2 = 4. A) Maximum: 1 + 2 2 at 1, 2, 2 ; minimum: 1 - 2 2 at 1, - 2, 2 B) Maximum: 1 + 2 2 at 1, 2, - 2 ; minimum: 1 - 2 2 at 1, - 2, - 2 C) Maximum: 1 + 2 2 at 1, 2, 2 ; minimum: 1 - 2 2 at 1, - 2, - 2 D) Maximum: 1 + 2 2 at 1, 2, - 2 ; minimum: 1 - 2 2 at 1, - 2, 2
213)
214) Find the extreme values of f(x, y, z) = xyz subject to x2 + y2 + z2 = 4 and x + y = 2. A) Maximum: 2 at 1, -1, - 2 ; minimum: - 2 at 1, -1, 2 B) Maximum: 2 at 1, 1, 2 ; minimum: - 2 at 1, 1, - 2 C) Maximum: 3 at 1, -1, - 3 ; minimum: - 3 at 1, -1, 3 D) Maximum: 3 at 1, 1, 3 ; minimum: - 3 at 1, 1, - 3
214)
Find the derivative of the function at P0 in the direction of u.
215) f(x, y, z) = ln(x2 + 10y2 + 6z 2 ), P0 (10, 10, 10), u = 3i + 4j A) 129
215)
B) 129
C) 86
D) 43
216) Find f(3, 6) when f(x, y) = 3x + y2. A) 10 B) 9
C) 3 5
D) 5 3
850
425
425
425
Find the specific function value.
216)
Find the domain and range and describe the level curves for the function f(x,y). 217) f(x, y) = ln (5x + 7y) A) Domain: all points in the xy-plane; range: all real numbers; level curves: lines 5x + 7y = c
B) Domain: all points in the xy-plane satisfying 5x + 7y > 0; range: all real numbers; level curves: lines 5x + 7y = c
C) Domain: all points in the xy-plane satisfying 5x + 7y > 0; range: real numbers z 0; level
curves: lines 5x + 7y = c D) Domain: all points in the xy-plane satisfying 5x + 7y 0; range: all real numbers; level curves: lines 5x + 7y = c
47
217)
Find fx , fy, and fz .
218) f(x,y,z) = xe(x2 + y2 + z 2)
218)
A) fx = (1 + 2x2 ) e(x2 + y2 + z 2 ); fy = xy2 e(x2 + y2 + z2 ); fz = xz2 e(x2 + y2 + z 2 ) B) fx = 2x2 e(x2 + y2 + z 2 ); fy = xye(x2 + y2 + z 2); fz = 2xze(x2 + y2 + z 2 ) C) fx = (1 + 2x2 ) e(x2 + y2 + z 2 ); fy = 2xye(x2 + y2 + z2 ); fz = 2xze(x2 + y2 + z 2) D) fx = (1 + 2x2 ) e(x2 + y2 + z 2 ); fy = xe(x2 + y2 + z 2 ); fz = xe(x2 + y2 + z 2 )
Find the extreme values of the function subject to the given constraint. 219) f(x, y) = x2 y, x2 + 2y2 = 6
219)
A) Maximum: 18 at (3, 2); minimum: -18 at (-3, -2) B) Maximum: 4 at (2, 1); minimum: -4 at (-2, -1) C) Maximum: 4 at (±2, 1); minimum: -4 at (±2, -1) D) Maximum: 18 at (±3, 2); minimum: -18 at (±3, -2) Use implicit differentiation to find the specified derivative at the given point. 220) Find dy at the point (1, -1) for 3xy2 - 5x2 y + 5x = 0. dx
A) 13 8
B) - 9
C) 18
4
11
220) D) - 18
Find the extreme values of the function subject to the given constraint. 221) f(x, y, z) = x2 + 2y2 + 3z2 , x - y - z = 1
11
221)
A) Maximum: none; minimum: 6 at - 6 , - 3 , - 2 11
11
11
11
B) Maximum: none; minimum: 6 at - 6 , - 3 , 2 11
11
11 11
C) Maximum: none; minimum: 6 at - 6 , 3 , - 2 11
11 11
11
D) Maximum: none; minimum: 6 at 6 , - 3 , - 2 11
11
11
11
Provide an appropriate response. 222) Find any local extrema (maxima, minima, or saddle points) of f(x, y) given that fx = 64x2 - 64 and fy = 2y + 12.
A) Local minimum at (-1, -6); saddle point at (1, -6) B) Local maximum at (1, -6); saddle point at (-1, -6) C) Local maximum at (-1, -6); saddle point at (1, -6) D) Local minimum at (1, -6); saddle point at (-1, -6)
48
222)
Find an upper bound for the magnitude |E| of the error in the approximation f(x, y) L(x, y) at the given point over the given region R. 223) f(x, y) = 7x - 9y - 4 at (5, 2); R: |x - 5| 0.1, |y - 2| 0.1 223) A) |E| 0.04 B) |E| 0.02 C) |E| 0.01 D) |E| 0
Find the derivative of the function at P0 in the direction of u.
224) f(x, y) = 2x2 - 8y, P0(-5, 7), u = 3i - 4j A) - 68
224)
B) - 88
5
C) - 28
5
5
D) - 48 5
Use implicit differentiation to find the specified derivative at the given point. 225) Find y at the point (6, 1, 4) for -5x2 - 5 ln xz + 5yz 2 + 2ez = 0. x
A) 71
B) 73
96
C) - 73
24
96
225) D) 73 96
Find all the second order partial derivatives of the given function. 226) f(x, y) = x ln (y - x) 2 2 2f 2f y A) f = x - 2y ; f = - x ; = = 2 2 2 2 y x x y x (y - x) y (y - x) (y - x)2
B) C) D)
2f
x2 2f
x2 2f
x2
=
2f x - 2y x = ; ; (y - x)2 y2 (y - x)2
=
2f 2y - x x = ; ; 2 2 (y - x) y (y - x)2
=
2f x - 2y x = ; ; 2 2 (y - x) y (y - x)2
2f
y x 2f
y x 2f
y x
=
= =
2f
=
x y 2f
=
x y 2f
x y
=
y
(y - x)2 y
(y - x)2 y
(y - x)2
Estimate the error in the quadratic approximation of the given function at the origin over the given region. 227) f(x, y) = e4x sin y, x 0.1, y 0.1
A) |E(x, y)| 0.0311 C) |E(x, y)| 0.0622
226)
227)
B) |E(x, y)| 0.0162 D) |E(x, y)| 0.0466
Solve the problem. 228) The surface area of a hollow cylinder (tube) is given by S = 2 (R1 + R2 )(h + R 1 - R2 ), where h is the length of the cylinder and R1 and R 2 are the outer and inner radii. If h, R1 , and R2 are measured to be 9 inches, 7 inches, and 10 inches respectively, and if these measurements are accurate to within 0.1 inches, estimate the maximum percentage error in computing S. A) 0.060% B) 0.046% C) 0.039% D) 0.032%
49
228)
Determine whether the given function satisfies a Laplace equation. 229) f(x, y) = cos(x) sin(-y) A) No B) Yes
229)
Provide an appropriate response. 230) Let T(x, y) = 16x2 - 2xy + 9y2 be the temperature at the point (x, y) on the ellipse x = 3 cos t,
230)
y = 4 sin t, 0 t 2 . Find the minimum and maximum temperatures, Tmin and Tmax, respectively, on the ellipse.
B) Tmin =
C) Tmin = 0; Tmax = 24
D) Tmin = 276; Tmax = 300
4
; Tmax =
3 4
A) Tmin = 132; Tmax = 156
Find the derivative of the function at P0 in the direction of u.
231) f(x, y) = 7x + 6y, P0 (-8, -9), u = 4i - 3j A) - 2
231) C) 46
B) 2
5
D) 58
5
5
Use Taylor's formula to find the requested approximation of f(x, y) near the origin. 232) Quadratic approximation to f(x, y) = ex + 6y
B) x + 6y + 1 x2 + 3xy + 18y2 2
C) 1 + x + 6y + 1 x2 + 6xy + 18y2
D) 1 + x + 6y + 1 x2 + 18y2
2
Find
232)
A) x + 6y + 1 x2 + 18y2 2
2
f f and . x y 4
233) f(x, y) = ln y
233)
x7
A) f = 4 ; f = 7
B) f = - 7 ; f = 4
C) f = -ln 7 ; f = ln 4
4 3 D) f = -ln 7y ; f = ln 4y
x x
y
y
x
x y
x
y
x
50
x
y
x8
y y
x7
Find the absolute maxima and minima of the function on the given domain. 234) f(x, y) = x2 + y2 on the diamond-shaped region |x| + |y| 3
A) Absolute maximum: 9 at 3 , 3 , 2 2 2
3 3 ,- , 2 2
3 3 - , , and 2 2
234) 3 3 - , - ; absolute minimum: 2 2
0 at (0, 0)
B) Absolute maximum: 9 at (3, 0), (-3, 0), (0, 3), and (0, -3); absolute minimum: 9 at 2
3 3 , , 2 2
3 3 ,- , 2 2
-
3 3 , , and 2 2
-
3 3 ,- , 2 2
C) Absolute maximum: 9 at (3, 0) and (0, 3); absolute minimum: 0 at (-3, 0) and (0, -3) D) Absolute maximum: 9 at (3, 0), (-3, 0), (0, 3), and (0, -3); absolute minimum: 0 at (0, 0) Write a chain rule formula for the following derivative. 235) dw for w = f(p, q, r); p = g(t), q = h(t), r = k(t) dt
235)
A) dw = dp + dq + dr
B) dw = w dt + w dt + w dt
C) dw = w + w + w
D) dw = w dp + w dq + w dr
dt dt
dt
p
dt
q
dt
dt
r
dt
p dp p dt
q dq q dt
r dr r dt
Find fx , fy, and fz .
236) f(x, y, z) = cos x sin2 (yz) A) fx = sin x sin2 (yz); fy = - cos x sin (yz) cos (yz); fz = -cos x sin (yz) cos (yz) B) fx = sin x sin2 (yz); fy = -2z cos x sin (yz) cos (yz); fz = - 2y cos x sin (yz) cos (yz) C) fx = -sin x sin2 (yz); fy = 2z cos x sin (yz) cos (yz); fz = 2y cos x sin (yz) cos (yz) D) fx = -sin x sin2 (yz); fy = z cos x sin (yz) cos (yz); fz = y cos x sin (yz) cos (yz)
51
236)
Sketch the surface z = f(x,y). 237) f(x, y) = 4 - x2 - y2
237)
A)
B)
C)
D)
Show that the function is a solution of the wave equation. 238) w(x, t) = cos (4ct) sin (4x) A) Yes
238) B) No
Find the limit.
239)
lim (x, y) (5, 7)
A)
35 35
1 xy
239) B) 35
C) 35
D) No limit
Solve the problem.
240) Find the derivative of the function f(x, y) = tan-1 y at the point (4, -4) in the direction in which x
the function decreases most rapidly. A) - 2 B) - 2 12 8
C) -
52
3 12
D) -
3 8
240)
Find the specific function value.
241) Find f(0, 1, -1) when f(x, y, z) = 5 x - 2yz + 9x. A) 2 B) 3
241) C) -1
D) -2
Find the linearization of the function at the given point.
242) f(x, y) = 2 sin x - 9 cos y at (0,
2
)
242)
A) L(x, y) = 2x + 9y - 9
B) L(x, y) = -2x - 9y - 9
2
2
C) L(x, y) = -2x + 9y - 9
D) L(x, y) = 2x - 9y - 9
2
2
Determine whether the given function satisfies a Laplace equation. 2 243) f(x, y) = x y
A) No
243)
B) Yes
Find the extreme values of the function subject to the given constraint. 244) f(x, y, z) = 4x - 3y + 2z, x2 + y2 = 6z
244)
A) Maximum: none; minimum: - 33 at 6, 9 , - 225 4 2 24 B) Maximum: none; minimum: 225 at 6, - 9 , 225 4
2
24
C) Maximum: none; minimum: - 75 at -6, 9 , 225 4
2
24
D) Maximum: none; minimum: 117 at 6, 9 , 225 4
2
24
Find the limit. lim
245) (x, y) (3, -9) y -9
A) 0
y+ 9 2 x y + 8y + 9x2 + 72
245)
B) 1
C) 17
Find all the local maxima, local minima, and saddle points of the function. 246) f(x, y) = 5xy(x + y) + 3 A) f(5, 5) = 1253, local maximum
B) f(0, 0) = 3, saddle point; f(5, 5) = 1253, local maximum C) f(0, 0) = 3, test is inconclusive D) f(5, 0) = 3, local minimum; f(0, 5) = 3, local minimum
53
D) 1
17
246)
247) f(x, y) = (x2 - 4)2 + (y2 - 49)2 A) f(0, 0) = 2417, local maximum; f(-2, -7) = 0, local minimum B) f(0, 0) = 2417, local maximum; f(0, 7) = 16, saddle point; f(2, 0) = 2401, saddle point;
247)
f(2, 7) = 0, local minimum; f(-2, -7) = 0, local minimum
C) f(0, 0) = 2417, local maximum; f(2, 7) = 0, local minimum; f(2, -7) = 0, local minimum;
f(-2, 7) = 0, local minimum; f(-2, -7) = 0, local minimum D) f(0, 0) = 2417, local maximum; f(0, 7) = 16, saddle point; f(0, -7) = 16, saddle point; f(2, 0) = 2417, saddle point; f(2, 7) = 0, local minimum; f(2, -7) = 0, local minimum; f(-2, 0) = 2401, saddle point; f(-2, 7) = 0, local minimum; f(-2, -7) = 0, local minimum
Use implicit differentiation to find the specified derivative at the given point. 248) Find dy at the point (-1, 1) for 6x - 3 + 7x2y2 = 0. dx y
A) - 8
17
B) 4 9
C) 8
D) - 20
C) 60
D) 30
17
Find the specific function value. 249) Find f(100, 3) when f(x, y) = y log x. A) 3 B) 6
248) 17
249)
Use implicit differentiation to find the specified derivative at the given point. 250) Find dy at the point (1, 0) for cos xy + yex = 0. dx
A) 0
B) 1
C) e
250) D) 1 e
Solve the problem.
251) A rectangular box with square base and no top is to have a volume of 32 ft3 . What is the least amount of material required? A) 42 ft2 B) 48 ft2
C) 36 ft2
D) 40 ft2
Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0). 3 2 252) f(x, y) = cos-1 x - xy x2 + y2
A)
B)
C) 1
2
251)
252) D) No limit
Solve the problem.
253) Find an equation for the level surface of the function f(x, y, z) = x2 + y2 + z 2 that passes through the point 4, 12, 3 . A) x2 + y2 + z 2 = 169
B) x + y + z = 13 D) x + y + z = ± 13
C) x2 + y2 + z 2 = 13
54
253)
Find the limit. lim
254) (x, y)
2, 2 x+y 4
x+y-4 x+y-2
A) 2
254) B) 4
C) 0
D) No limit
Find fx , fy, and fz .
255) f(x ,y, z) = sin (xy) cos (yz 2 ) A) fx = y cos (xy) cos (yz2 ); fy = x cos (xy) cos (yz2 ) - z2 sin (xy) sin (yz 2 ); fz = -2yz sin (xy) sin
255)
(yz 2 )
B) fx = y cos (xy) cos (yz2 ); fy = z 2 sin (xy) sin (yz2 )- x cos (xy) cos (yz 2 ); fz = 2yz sin (xy) sin (y z2 )
C) fx = y cos (xy) cos (yz2 ); fy = x cos (xy) cos (yz2 ) - z2 sin (xy) sin (yz 2 ); fz = 2yz sin (xy) sin (yz 2 )
D) fx = y cos (xy) cos (yz2 ); fy = x cos (xy) cos (yz2 ); fz = -2yz sin (xy) sin (yz 2 ) Find the specific function value.
256) Find f(3, 0, 6) when f(x, y, z) = 3x2 + 3y2 - z 2 . A) -6 B) -9
256) C) -27
D) 63
Use implicit differentiation to find the specified derivative at the given point. 257) Find x at the point 1, , 1 for ex2 cos yz = 0. y 4
A) 1 2
B) - 1
C) - 2
2
257) D) 2
Solve the problem.
258) Find the equation for the tangent plane to the surface x2 - 4xyz + y2 = 6z 2 at the point (-1, -1, -1). A) -6x - 6y + 8z = 1
258)
B) -6x - 6y + 8z = 4 D) x + y + z = 1
C) x + y + z = 4
Use Taylor's formula to find the requested approximation of f(x, y) near the origin. 1 259) Cubic approximation to f(x, y) = (1 + x + 5y)2
A) 1 + 2x + 10y - 3x2 - 30xy - 75y2 + 4x3 + 60x2y + 300xy2 + 500y3 B) 1 - 2x - 10y + 3x2 + 30xy + 75y2 - 4x3 - 60x2y - 300xy2 - 500y3 C) 1 - 2x - 10y + 3x2 + 40xy + 75y2 - 4x3 - 60x2y - 300xy2 - 500y3 D) 1 + 2x + 10y - 3x2 - 40xy - 75y2 + 4x3 + 60x2y + 300xy2 + 500y3
55
259)
Solve the problem. 260) The table below summarizes the construction cost of a set of homes (excluding the lot cost) along with the square footage of the home's floor space. Find a linear equation that relates the construction cost in thousands of dollars to the floor space in hundreds of square feet by finding the least squares line for the data. Floor Space Construction Cost (100's of sq. ft.) (1000's of dollars) 11 76.6 12 86.5 14 101.2 15 96.3 18 127.9 22 159.0 26 198.5 A) y = 8.13x - 12.79
B) y = 7.97x - 13.54 D) y = 7.87x - 13.08
C) y = 8.13x
Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0). 261) f(x, y) = 3xy x2 + y2
A) 1
260)
C) 0
B)
261) D) -1
Find an upper bound for the magnitude |E| of the error in the approximation f(x, y) L(x, y) at the given point over the given region R. 262) f(x, y, z) = 7xy + 6yz + 8zx at (1, 1, 1); R: |x - 1| 0.1, |y - 1| 0.1, |z - 1| 0.1 262) A) |E| 0.3 B) |E| 0.336 C) |E| 0.36 D) |E| 0.28
Find the limit.
263)
2 5 lim (x, y) (5, 4) x y
A) - 17 20
263) C) 17
B) -17
20
D) No limit
Find the absolute maxima and minima of the function on the given domain. 264) f(x, y) = 8x2 + 10y2 on the closed triangular region bounded by the lines y = x, y = 2x, and x+y=6 A) Absolute maximum: 162 at (3, 3); absolute minimum: 72 at (2, 2)
B) Absolute maximum: 192 at (2, 4); absolute minimum: 162 at (3, 3) C) Absolute maximum: 162 at (3, 3); absolute minimum: 0 at (0, 0) D) Absolute maximum: 192 at (2, 4); absolute minimum: 0 at (0, 0)
56
264)
Find the domain and range and describe the level curves for the function f(x,y). 265) f(x, y) = sin-1 (x2 + y2)
A) Domain: all points in the xy-plane; range: real numbers -
2
z
265) 2
; level curves: circles
with centers at (0, 0)
B) Domain: all points in the xy-plane satisfying x2 + y2 1; range: real numbers -
2
z
2
;
level curves: circles with centers at (0,0) and radii r, 0 < r 1 C) Domain: all points in the xy-plane; range: all real numbers; level curves: circles with centers at (0, 0)
D) Domain: all points in the xy-plane satisfying x2 + y2 1; range: real numbers -
2
z
2
;
level curves: circles with centers at (0, 0)
Find fx , fy, and fz .
266) f(x, y, z) = xz x + y
266) x ; fy = x+y
A) fx = z x + y +
xz ; fz = x x + y x+y
B) fx = z x + y -
x xy ; fy = ; fz = x x + y 2 x+y 2 x+y
C) fx = z x + y +
x xz ; fy = ; fz = x x + y 2 x+y 2 x+y
D) fx = z x + y -
x ; fy = x+y
xy ; fz = x x + y x+y
Solve the problem. 267) Write parametric equations for the tangent line to the curve of intersection of the surfaces x + y2 + 4z = 6 and x = 1 at the point (1, 1, 1).
A) x = 1, y = 8t + 1, z = -t + 1 C) x = 1, y = 4t + 1, z = -t + 1
267)
B) x = 1, y = 8t + 1, z = -2t + 1 D) x = 1, y = 4t + 1, z = -2t + 1
Find the extreme values of the function subject to the given constraint. 268) f(x, y) = xy, x2 + y2 = 648
268)
A) Maximum: 324 at (18, -18) and (-18, 18); minimum: -324 at (18, 18) and (-18, -18) B) Maximum: 324 at (18, 18); minimum: 0 at (0, 0) C) Maximum: 324 at (18, 18); minimum: -324 at (-18, -18) D) Maximum: 324 at (18, 18) and (-18, -18); minimum: -324 at (18, -18) and (-18, 18) Provide an appropriate response.
269) Determine the point on the paraboloid z = 5x2 + 4y2 that is closest to the point (33, -18, 60). A) (3, 2, 61) B) (3, -2, 61) C) (-2, 3, 56) D) (2, 3, 56)
57
269)
Solve the problem.
270) Find the derivative of the function f(x, y) = x2 + xy + y2 at the point (-2, -1) in the direction in which the function increases most rapidly. A) 83 B) 3 6
C) 41
270)
D) 77
Answer the question. 271) Describe the results of applying the method of Lagrange multipliers to a function f(x, y) if the points (x, y) are constrained to follow a curve g(x, y) = c that is everywhere perpendicular to the level curves of f. Assume that both f(x, y) and g(x, y) satisfy all the requirements and conditions for the method to be applicable. A) Since f is everywhere parallel to g, every point on g(x, y) = c is a local extremum. Applying the method of Lagrange multipliers should yield the equation g(x, y) = c. B) The results cannot be generally predicted. Specific expressions for f(x, y) and g(x, y) = c are required. C) Since f is everywhere parallel to g, there will be a single local minimum and a single local maximum along g(x, y) = c. Applying the method of Lagrange multipliers should identify the locations of these two local extrema. D) Generally, local extrema of f(x, y) occur at points on the curve g(x, y) = c where the curve becomes tangent to a level curve of f(x, y). Since the curve defined by g(x, y) = c is everywhere perpendicular to the level curves of f(x, y) for this particular case, it is never tangent to a level curve, and there are no local extrema along g(x,y) = c. The method of Lagrange multipliers will fail to find any local extrema since there are none.
271)
Find the derivative of the function at P0 in the direction of u.
272) f(x, y, z) = tan-1 A) - 37
273
3x , P0 (-7, 0, 0), u = 12i - 3j + 4k 9y + 2z
B) - 46
272)
C) - 4
273
39
D) - 19
273
Find all the local maxima, local minima, and saddle points of the function. 273) f(x, y) = (x2 - 25)2 - (y2 - 49)2 A) f(0, 0) = -1776, saddle point; f(0, 7) = 625, local maximum; f(0, -7) = 625, local maximum; f(5, 0) = -2401, local minimum; f(5, 7) = 0, saddle point; f(5, -7) = 0, saddle point; f(-5, 0) = -2401, local minimum; f(-5, 7) = 0, saddle point; f(-5, -7) = 0, saddle point B) f(0, 7) = 625, local maximum; f(0, -7) = 625, local maximum; f(5, 0) = -2401, local minimum; f(-5, 0) = -2401, local minimum C) f(0, 0) = -1776, saddle point; f(0, 7) = 625, local maximum; f(5, 0) = -2401, local minimum
D) f(0, 0) = -1776, saddle point; f(5, 7) = 0, saddle point; f(5, -7) = 0, saddle point; f(-5, 7) = 0, saddle point; f(-5, -7) = 0, saddle point
58
273)
Find the extreme values of the function subject to the given constraint. 274) f(x, y, z) = x + y + z, 1 + 1 + 1 = 1 x y z
274)
A) Maximum: 11 at (6, 3, 2), (6, 2, 3), (3, 2, 6), (3, 6, 2), (2, 3, 6), (2, 6, 3); minimum: 1 at (1, 1, -1), (1, -1, 1), (-1, 1, 1)
B) Maximum: 9 at (3, 3, 3); minimum: 3 at (1, 1, 1) C) Maximum: 6 at (2, 2, 2); minimum: -3 at (-1, -1, -1) D) Maximum: 6 at (2, 2, 2); minimum: 3 at (1, 1, 1) Solve the problem. 275) The resistance R produced by wiring resistors of R 1, R2 , and R 3 ohms in parallel can be
275)
calculated from the formula 1 1 1 1 = + + . R R1 R2 R 3 If R 1 , R2 , and R 3 are measured to be 8 ohms, 5 ohms, and 3 ohms respectively, and if these measurements are accurate to within 0.05 ohms, estimate the maximum percentage error in computing R. A) 1.27% B) 1.52% C) 0.76% D) 1.01%
At what points is the given function continuous? 276) f(x, y) = xy x+y
276)
A) All (x, y) (0, 0) C) All (x, y)
B) All (x, y) such that x y D) All (x, y) such that x - y
Solve the problem.
277) Evaluate u at (x, y, z) = (2, 3, 0) for the function u = epq cos(r); p = 1 , q = x2 ln y, r = z. y
A) 18
Find
x
B) 2
C) 6
3
277)
D) 0
f f and . x y
278) f(x, y) = sin2 (3xy2 - y) A)
278)
f f = 6y2 sin (3xy2 - y) cos (3xy2 - y); = 2sin (3xy2 - y) cos (3xy2 - y) x y
B) f = 6y2 sin (3xy2 - y) cos (3xy2 - y); f = (12xy - 2) sin (3xy2 - y) cos (3xy2 - y) x
y
C) f = 2sin (3xy2 - y) cos (3xy2 - y); f = (12x - 2) sin (3xy2 - y) cos (3xy2 - y) x
y
D) f = 2sin (3xy2 - y) cos (3xy2 - y); f = 2sin(3xy2 - y) cos(3xy2 - y) x
y
59
Find all the local maxima, local minima, and saddle points of the function. 279) f(x, y) = x2 + 10xy + y2
279)
A) f(0, 0) = 0, saddle point; f(10, 10) = 1200, local maximim B) f(10, 0) = 100, local minimum; f(0, 10) = 100, local minimum C) f(0, 0) = 0, saddle point D) f(10, 10) = 1200, local maximim Solve the problem. 280) About how much will f(x, y, z) = -4x + 5y - 3z change if the point (x, y, z) moves from (6, 9, 4) a 1 distance of ds = unit in the direction 2i - 3j + 6k? 10
A) - 41 70
B) - 18
C) - 23
35
35
280)
D) - 51
At what points is the given function continuous? 281) f(x, y) = ex+y
70
281)
A) All (x, y) C) All (x, y) in the first quadrant
B) All (x, y) (0, 0) D) All (x, y) satisfying x + y 0
Sketch a typical level surface for the function. 282) f(x, y, z) = x2 - z
282)
A)
B)
C)
D)
60
Use implicit differentiation to find the specified derivative at the given point. 283) Find dy at the point (2, 1) for ln x + xy2 + ln y = 0. dx
A) - 3
C) 3
B) -1
10
283) D) 1
10
Solve the problem. 284) If the length, width, and height of a rectangular solid are measured to be 8, 4, and 7 inches respectively and each measurement is accurate to within 0.1 inch, estimate the maximum possible error in computing the volume of the solid. A) 1160.00 B) 1450.00 C) 1288.89 D) 966.67
285) Find the extreme values of f(x, y, z) = 2x - 3y + z subject to x2 + y2 = 1 and y2 + z 2 = 1. A) Maximum: 3 2 at
1 1 1 1 1 1 ,, ; minimum: - 3 2 at , ,2 2 2 2 2 2
B) Maximum: 2 2 at
1 1 1 1 1 1 ,,; minimum: - 2 2 at , , 2 2 2 2 2 2
C) Maximum:
1 1 1 ,, ; minimum: 2 2 2
2 at -
D) Maximum: 4 2 at
2 at
A)
285)
1 1 1 , ,2 2 2
1 1 1 1 1 1 , , ; minimum: - 4 2 at ,,2 2 2 2 2 2
286) Find the distance from the point (1, -1, 2) to the plane x + y - z = 3. 5 2
284)
B)
7 3
C)
7 2
286) D)
5 3
Provide an appropriate response. 287) Find the direction in which the function is increasing most rapidly at the point P0 .
287)
f(x, y, z) = x y2 + z 2 , P0 (1, 1, 2)
A) 30 i + j + k
B) 1
C) 30 i + j - k
D) 1
6 6
30 30
i j k + 6 30 15 30
15
i j k + + 30 6 30 15
15
Use implicit differentiation to find the specified derivative at the given point. 288) Find y at the point (1, 3, e8 ) for ln(xz)y + 2y3 = 0. x
A) 3
62
B) - 62
C) 62
3
3
61
288) D) - 3
62
Find the domain and range and describe the level curves for the function f(x,y). 2 289) f(x, y) = 8x y
289)
A) Domain: all points in the xy-plane except y = 0; range: all real numbers; level curves: parabolas y = cx2
B) Domain: all points in the xy-plane; range: real numbers z 0; level curves: parabolas y = cx2 C) Domain: all points in the xy-plane except y = 0; range: real numbers z 0 ; level curves: parabolas y = cx2
D) Domain: all points in the xy-plane; range: all real numbers; level curves: parabolas y = cx2 Find the specific function value.
290) Find f(8, 2) when f(x, y) = (x + y)3. A) 519 B) 1000
290) C) 30
D) 100
Find the extreme values of the function subject to the given constraint. 291) f(x, y) = 7x2 + 4y2 , x2 + y2 = 1
291)
A) Minimum: 4 at (0, ±1); maximum: 0 at (0, 0) B) Minimum: 4 at (±1, 0); maximum: 7 at (0, ±1) C) Minimum: 4 at (±1, 0); maximum: 0 at (0, 0) D) Minimum: 4 at (0, ±1); maximum: 7 at (±1, 0) Solve the problem. 292) Find the least squares line for the points (4, 20), (5, -25), (6, 30), (7, -35). A) y = 50 + 5x B) y = 58 - 11x C) y = 50 - 11x Find the extreme values of the function subject to the given constraint. 293) f(x, y, z) = (x - 2)2 + (y + 1)2 + (z - 4)2 , x - y + 3z = 7
A) Maximum: none; minimum: 64 at 14 , - 3 , 20 11 11 11 11 B) Maximum: none; minimum: 76 at 14 , 3 , 20 11
11 11 11
C) Maximum: none; minimum: 188 at - 14 , 3 , 20 11
11 11 11
D) Maximum: none; minimum: 36 at 14 , 3 , - 20 11 11
62
11
292) D) y = 58 + 5x
293)
Find all the second order partial derivatives of the given function. 294) f(x, y) = xy2 + yex2 + 5 2 2 2f 2f 2 A) f = yex2 (1 + 2x2 ); f = x; = = y + xex y x x y x2 y2
B) C) D)
2f
2 = 2yex ;
2f
2 = 2yex (1 + 2x2 );
2f
2 = 2yex ;
x2 x2 x2
2f
y2
2f
y2
2f
= 2x;
= 2x;
y x 2f
=
= 2x;
y2
2f
y x
=
2f
x y 2f
2 = 2y + 2xex
y x 2f
x y
294)
=
2f
x y
2 = 2y + 2xex
2 = 2xex
Find the extreme values of the function subject to the given constraint. 295) f(x, y) = 4x + 6y, x2 + y2 = 13
295)
A) Maximum: 36 at (3, 4); minimum: 0 at (0, 0) B) Maximum: 26 at (2, 3); minimum: 0 at (0, 0) C) Maximum: 26 at (2, 3); minimum: -26 at (-2, -3) D) Maximum: 36 at (3, 4); minimum: -36 at (-3, -4) Estimate the error in the quadratic approximation of the given function at the origin over the given region. 296) f(x, y) = ln(1 + 5x + 6y), x 0.1, y 0.1 A) |E(x, y)| 0.0180 B) |E(x, y)| 0.1437
C) |E(x, y)| 0.0240
296)
D) |E(x, y)| 0.0479
Solve the problem.
297) What points of the surface xy - z2 - 6y + 36 = 0 are closest to the origin? A) (2, 4, ±2) B) (-2, -4, ±2) C) (2, -4, ±2)
297) D) (-2, 4, ±2)
Find the linearization of the function at the given point. 298) f(x, y) = e2x - 9y at (0, 0)
298)
A) L(x, y) = -9x + 2y C) L(x, y) = 2x - 9y
B) L(x, y) = 2x - 9y + 1 D) L(x, y) = -9x + 2y + 1
At what points is the given function continuous? 299) f(x, y, z) = yz cos 1 x
299)
A) All (x, y, z) such that 0 1 2
B) All (x, y, z) such that x 2
C) All (x, y, z)
D) All (x, y, z) such that x 0
x
63
Solve the problem.
300) Find the derivative of the function f(x, y) = exy at the point (0, -2) in the direction in which the function decreases most rapidly. A) -4 B) -2
C) -3
300)
D) -6
301) Find the maximum value of f(x, y, z) = x + 2y + 3z subject to x - y + z = 1 and x2 + y2 = 1. A) 3 - 37 B) 3 - 29 C) 3 + 37 D) 3 + 29
301)
302) Find the equation for the tangent plane to the surface z = ln(2x2 + 7y2 + 1) at the point (0, 0, 0). A) x - y = 0 B) z = 0 C) x + y + z = 0 D) x + y = 0
302)
Write a chain rule formula for the following derivative. 303) w for w = f(x, y, z); x = g(r, s), y = h(t), z = k(r, s, t) t
303)
A) w = w z
B) w = dy + z
C) w = w + w
D) w = w dy + w z
t t
z
t
y
t
z
dt
t
t
y dt
z
t
Provide an appropriate response. 304) Which of the following space regions is (are) closed? i. The hemispherical region centered at the origin with z > 0 and radius r bounded by 0 r ro ii. The xy-plane iii. The half-space x > 0 iv. Space itself A) iv only B) ii, iii, and iv
304)
C) ii only
D) ii and iv
Solve the problem. 305) A rectangle with sides parallel to the axes is inscribed in the region bounded by the axes and the line x + 2y = 2. Find the maximum area of this rectangle. A) 1 B) 1 C) 1 D) 2 2 4 3 3
306) What is the distance from the surface xy - z 2 - 6y + 36 = 0 to the origin? A) 2 3 B) 6 3 C) 2 6
306) D) 6 2
Write a chain rule formula for the following derivative. 307) w for w = f(x, y, z); x = g(s, t), y = h(s, t), z = k(s) t
307)
A) w = w x + w y
B) w = w x + w y
C) w = w + w
D) w = w x
t t
t
x
t
t
305)
t
t
y
t
64
x x
t t
y
t
Find an upper bound for the magnitude |E| of the error in the approximation f(x, y) L(x, y) at the given point over the given region R. 308) f(x, y, z) = 10x2 + 8y2 + 8z 2 at (1, -2, 3); R: |x - 1| 0.1, |y + 2| 0.1, |z - 3| 0.1 308)
A) |E| 0.8
B) |E| 0.6
C) |E| 1.2
309) f(x, y) = e6x + 8y at (0, 0); R: |x| 0.1, |y| 0.1 A) |E| 10.3813 B) |E| 18.1673
D) |E| 0.9 309)
C) |E| 5.1907
D) |E| 9.0836
Find the extreme values of the function subject to the given constraint. 310) f(x, y) = xy, 9x2 + 4y2 = 36 2,
3 2
2 and
B) Maximum: 3 at
2,
3 2
2 ; minimum: -3 at
C) Maximum: 3 at
2,
3 2
2 ; minimum: -3 at -
D) Maximum: 3 at
2, -
A) Maximum: 3 at -
2,
3 2
-
2, -
3 2
310)
2 ; minimum: -3 at
2, -
3 2
2 and
2
3 2
2, -
2 ; minimum: -3 at -
3 2
2
2, -
3 2
2
2,
3 2
2
Find the derivative of the function at P0 in the direction of u.
311) f(x, y, z) = 4xy3 z2 , P0 (4, 64, 16), u = -2i + j - 2k A) - 738,197,504 3
311)
B) - 771,751,936
C) - 251,658,240
3
D) - 788,529,152 3
Find the absolute maxima and minima of the function on the given domain. 312) f(x, y) = 2x2 + 8y2 on the disk bounded by the circle x2 + y2 = 9
312)
A) Absolute maximum: 72 at (0, 3) and (0, -3); absolute minimum: 18 at (3, 0) and (-3, 0) B) Absolute maximum: 18 at (3, 0) and (-3, 0); absolute minimum: 0 at (0, 0) C) Absolute maximum: 90 at (3, 3); absolute minimum: 0 at (0, 0) D) Absolute maximum: 72 at (0, 3) and (0, -3); absolute minimum: 0 at (0, 0) Solve the problem.
313) Write an equation for the tangent line to the curve y2 - x = 6 at the point (1, 7). A) x - 2y + 1 = 0 B) x - 2 7y + 1 = 0 C) x - 2y + 13 = 0 D) x - 2 7y + 13 = 0
Estimate the error in the quadratic approximation of the given function at the origin over the given region. 314) f(x, y) = sin 2x sin2 3y, x 0.1, y 0.1
A) |E(x, y)| 0.0148 C) |E(x, y)| 0.0298
B) |E(x, y)| 0.0058 D) |E(x, y)| 0.0433
65
313)
314)
Find the specific function value.
315) Find f(3, 4) when f(x, y) = x - 6y .
315)
x 2 + y2
A) - 5
B) - 21
21
C) - 25
25
21
D) - 21 5
Use Taylor's formula to find the requested approximation of f(x, y) near the origin. 316) Cubic approximation to f(x, y) = ln(1 + 4x + y) A) 1 + 4x + y - 8x2 - 4xy - 1 y2 + 64 x3 + 16x2 y + 4xy2 + 1 y3 2 3 3
316)
B) 4x + y - 8x2 - 4xy - 1 y2 + 64 x3 + 16x2 y + 4xy2 + 1 y3 2
3
3
C) 4x + y - 8x2 - 4xy - 1 y2 + 64 x3 + 16 x2 y + 4 xy2 + 1 y3 2
3
3
3
3
D) 1 + 4x + y - 8x2 - 4xy - 1 y2 + 64 x3 + 16 x2 y + 4 xy2 + 1 y3 2
3
3
3
3
Solve the problem. 317) Find the equation for the tangent plane to the surface -7x + 3y + 8z = 6 at the point (1, -1, 2). A) -7x - 3y + 16z = 4 B) -7x - 3y + 16z = 6
C) -7x + 3y + 8z = 4
D) -7x + 3y + 8z = 6
318) Evaluate dw at t = 3 for the function w = ey - ln x; x = t2 , y = ln t.
318)
dt
A) - 1 3
B) 2
C) 1
3
3
D) 1
Provide an appropriate answer. 319) Find z when u = 0 and v = 5 if z(x, y) = sin x + cos y, x = u · v, and y = u + v. v 2
A) z = 2 v
317)
B) z = -1
C) z = 0
v
v
319) D) z = 1 v
Solve the problem. 320) The surface area of a hollow cylinder (tube) is given by S = 2 (R1 + R2 )(h + R 1 - R2 ), where h is the length of the cylinder and R1 and R 2 are the outer and inner radii. If h, R1 , and R2 are measured to be 3 inches, 5 inches, and 7 inches respectively, and if these measurements are accurate to within 0.1 inches, estimate the maximum possible error in computing S. A) 18.85 B) 15.71 C) 9.42 D) 12.57
66
320)
321) Find the point on the sphere x2 + y2 + z 2 = 4 that is farthest from the point (3, 1, -1). A) - 6 , - 2 , - 2
B) - 6 , 2 , 2
C) - 6 , 2 , - 2
D) - 6 , - 2 , 2
11
11
11
11
11
11
11
11
11
11
321)
11 11
322) About how much will f(x, y, z) = ln(9x + 7y - 10z) change if the point (x, y, z) moves from
322)
1 (4, 1, 6) a distance of ds = unit in the direction of 12i + 3j + 4k? 10
A) - 48
B) - 103
1105
Find the limit. lim
323)
(x, y)
4
,1
C) - 89
2210
2210
D) - 41
1105
y tan x y+ 1
A) 2
323) C) 1
B) 0
D) 4 tan 1
2
5
Write a chain rule formula for the following derivative. 324) u for u = f(v); v = h(s, t) t
A) u = du v t
dv
t
324)
B) u = dv u t
du
C) u = du
t
t
dv
D) u = v t
t
Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0). 325) f(x, y) = x + y x+y
A)
B) 0
C) 1
325) D) No limit
Solve the problem. 326) The radius r and height h of a cylinder are changing with time. At the instant in question, r = 3 cm, h = 4 cm, dr/dt = 0.01 cm/sec and dh/dt = -0.01 cm/sec. At what rate is the cylinder's volume changing at that instant? A) 0.28 cm3 /sec B) 0.47 cm3 /sec C) 1.04 cm3 /sec D) 0.09 cm3 /sec
67
326)
Provide an appropriate response. 327) For which of the following functions do both fx and fy exist?
A) II. and IV.
B) IV. only
327)
C) II., III., and IV.
Compute the gradient of the function at the given point. 328) f(x, y, z) = ln(x2 + 9y2 + 6z2 ), (9, 9, 9)
A)
328)
1 1 3 f= i+ j+ k 72 8 16
C) f =
i+
D) I. and II.
1 3 i+ j+ k 8 16
B) f =
1 1 j+ k 8 12
D) f = 1 i + 1 j + 1 k 72
8
12
Solve the problem. 329) Write parametric equations for the tangent line to the curve of intersection of the surfaces x + y + z = 6 and x - y + 2z = 5 at the point (1, 2, 3). A) x = 3t - 1, y = t + 2, z = -2t + 3 B) x = 3t + 1, y = -t + 2, z = -2t + 3
C) x = 3t + 1, y = t + 2, z = -2t + 3
329)
D) x = 3t - 1, y = -t + 2, z = -2t + 3
Use Taylor's formula to find the requested approximation of f(x, y) near the origin. 1 330) Cubic approximation to f(x, y) = 1 + 3x + y
330)
A) 1 - 3x - y + 9x2 + 6xy + y2 - 27x3 - 27x2 y - 9xy2 - y3 B) 1 - 3x - y + 9x2 + 3xy + y2 - 27x3 + 27x2 y - 9xy2 + y3 C) 1 - 3x - y + 9x2 + 3xy + y2 - 27x3 - 27x2 y - 9xy2 - y3 D) 1 - 3x - y + 9x2 + 6xy + y2 - 27x3 + 27x2 y - 9xy2 + y3 Solve the problem.
331) Evaluate u at (x, y, z) = (4, 2, 5) for the function u = p2 - q2 - r; p = xy, q = y2 , r = xz. x
A) 11
B) 27
C) 21
68
D) 37
331)
332) Find the derivative of the function f(x, y) = tan-1 y at the point (-4, 4) in the direction in which x
the function increases most rapidly. A) 3 B) 2 12 8
C)
3 8
D)
2 12
333) The Van der Waals equation provides an approximate model for the behavior of real gases. The equation is P(V, T) =
332)
333)
RT a , where P is pressure, V is volume, T is Kelvin temperature, and V - b V2
a,b , and R are constants. Find the partial derivative of the function with respect to each variable. A) PV = - 2a + RT ; PT = R B) PV = R ; PT = 2a - RT 3 2 V b V-b V (V - b) V3 (V - b)2
C) PV = 2a V3
RT
(V - b)2
; PT =
R V-b
D) PV = 2a V
RT
(V - b)2
; PT =
R V-b
Provide an appropriate answer. 334) Suppose that w = x2 + y2 + 16z + t and x + 4z + t = -2. Assuming that the independent variables
334)
w are x, y, and z, find . x
A) w = 2x + 1 x
B) w = 2x - 1
C) w = 2x + 4
x
x
D) w = 2x - 4 x
Use implicit differentiation to find the specified derivative at the given point. 335) Find y at the point (1, 1, 2) for 2 + 3 + 1 = 0. z x2 y2 z 2
B) 1
A) 24
C) - 24
24
335) D) - 1
24
Use Taylor's formula to find the requested approximation of f(x, y) near the origin. 1 336) Quadratic approximation to f(x, y) = 1 + 4x + y
A) 1 + 4x + y + 16x2 + 8xy + y2 C) 1 + 4x + y +16x2 + 4xy + y2
336)
B) 1 - 4x - y + 16x2 + 4xy + y2 D) 1 - 4x - y + 16x2 + 8xy + y2
Use implicit differentiation to find the specified derivative at the given point. 337) Find z at the point (7, 1, -1) for ln yz - exy+z 2 = 0. y x 8
A) 1 - 2e
1 - 7e8
8
8
B) 7e - 1
C) 2e - 1
1 - 2e8
1 - 7e8
69
337) 8
D) 1 - 7e
1 - 2e8
Find the domain and range and describe the level curves for the function f(x,y). 338) f(x, y) = 10x2 - 7y2
338)
A) Domain: all points in the first quadrant of the xy-plane; range: all real numbers; level curves: hyperbolas 10x2 - 7y2 = c B) Domain: all points in the xy-plane; range: all real numbers; level curves: hyperbolas 10x2 - 7y2 = c
C) Domain: all points in the xy-plane; range: all real numbers; level curves: ellipses
10x2 + 7y2 = c D) Domain: all points in the xy-plane; range: real numbers z 0; level curves: ellipses 10x2 + 7y2 = c
Solve the problem.
339) Evaluate dw at t = 5 dt
for the function w =
2
A) - 1 1 2
xy ; x = sin t, y = cost, z = t2 . z
B) 1 1
2
2
C) - 4 1
2
25
2
339) D) - 1 1 2
Find the domain and range and describe the level curves for the function f(x,y). 340) f(x, y) = (10x - 4y)3
340)
A) Domain: all points in the xy-plane; range: real numbers z 0; level curves: lines 10x - 4y = c, c 0
B) Domain: all points in the xy-plane; range: all real numbers; level curves: lines 10x - 4y = c C) Domain: all points in the xy-plane; range: real numbers z 0; level curves: lines 10x - 4y = c D) Domain: all points in the xy-plane; range: all real numbers; level curves: lines 10x - 4y = c, c 0
Find all the second order partial derivatives of the given function. 341) f(x, y) = x x+y
A) B) C) D)
2f
x2 2f
x2 2f
x2 2f
x2
===
2y
(x + y)3 2y
(x + y)3
; ;
2f
y2 2f
y2
= =
2x
(x + y)3 2x
(x + y)3
; ;
2f
y x 2f
y x
= =
2f 2y 2x =; ; x + y 3 (x + y)3 y2 (x + y)3
=-
y
(x + y)3
;
2f
y2
=
x
(x + y)3
;
2f
y x
2f
x y 2f
x y 2f
y x =
2f
= =
=
x y
=
341) y- x
(x + y)3 x-y
(x + y)3 2f
x y
=
x-y
(x + y)3
x-y
(x + y)3
Solve the problem. 342) Find the point on the line of intersection of the planes x + y + z = 1 and 3x + 2y + z = 6 that is closest to the origin. A) - 7 , 1 , 5 B) 7 , 1 , - 5 C) 7 , - 1 , 5 D) 7 , - 1 , - 5 3 3 3 3 3 3 3 3 3 3 3 3
70
342)
343) Amarillo Motors manufactures an economy car called the Citrus, which is notorious for its
343)
inability to hold a respectable resale value. The average resale value of a set of 1998 Amarillo Citrus's is summarized in the table below along with the age of the car at the time of resale and the number of cars included in the average. Fit a line of the form ln(V) = m·a + b to the data, where V is the resale value in thousands of dollars and a is the age of the car in years.. Age Average Resale Value (years) (1000's of dollars) Frequency 0 18 3 1 14 8 2 11 2 3 8 4 4 6 10 5 4 6 6 3 5 A) ln(V) = 0.299a - 2.951
B) ln(V) = 0.314a - 2.986 D) ln(V) = -0.303a + 2.953
C) ln(V) = 0.315a
Find the absolute maximum and minimum values of the function on the given curve. 344) Function: f(x, y) = x + y; curve: x2 + y2 = 9, y 0. (Use the parametric equations x = 3 cos t, y = 3 sin t.)
A) Absolute maximum: 3 2 at t = B) Absolute maximum: 3 at t =
2
2
; absolute minimum: -3 2 at t =
; absolute minimum: -3 2 at t =
C) Absolute maximum: 3 2 at t = D) Absolute maximum: 3 at t =
4
4
344)
3 4
3 4
; absolute minimum: -3 at t =
; absolute minimum: -3 at t =
Solve the problem.
345) Find an equation for the level surface of the function f(x, y, z) = n=0
through the point ( , , 1) . A) cos x+y = 2 z
(-1)n (x+y)2n that passes (2n)! z 2n
345)
B) x+y = 2 z
C) x+y = 1
D) cos x+y = 0
z
z
Determine whether the given function satisfies a Laplace equation. 346) f(x, y, z) = 7x2 - 5y2 - 2z 2
A) No
B) Yes
71
346)
Solve the problem. 347) The Redlich-Kwong equation provides an approximate model for the behavior of real gases. The RT a equation is P(V, T) = , where P is pressure, V is volume, T is Kelvin V - b T1/2V(V + b)
347)
temperature, and a,b , and R are constants. Find the partial derivative of the function with respect to each variable. a R A) PV = a(2V + b) + RT ; PT = + 2 2 1/2 2 3/2 V -b V (V + b) T (V - b) 2T V(V + b)
B) PV =
a(2V + b) RT a R ; PT = + 2 2 1/2 2 3/2 V -b 2V (V + b) T (V - b) T V(V + b)
C) PV =
a(2V + b) RT a R ; PT = + 2 2 1/2 2 3/2 V -b V (V + b) T (V - b) 2T V(V + b)
D) PV =
a(2V + b) RT a R ; PT = + V2 (V + b)2 T1/2 (V - b)2 2T3/2V(V + b) V - b
Find the equation for the level surface of the function through the given point. z x 348) f(x, y, z) = (ln + 1) d + t et dt , (5, e3 , e-3 ) y 0 A) -3e-3 - 3e3 + 4e5 + 1 = ln + 1 + t et z
B) -3e-3 - 3e3 + 4e5 - 5 = y
C) -3e-3 - 3e3 + 4e5 + 1 = y
D) -4e-3 - 2e3 + 4e5 + 1 = y
Find
z
z
(ln
x
+ 1) d + 0
(ln
+ 1) d + 0
(ln
+ 1) d +
x
x
348)
t et dt t et dt t et dt
0
f f and . x y
349) f(x, y) = ln yx A)
349)
f f = ln y; = - xln y x y
B)
C) f = ln y; f = x x
y
f f x = xln y; =x y y
D) f = 0; f = - x
y
x
y
y
Provide an appropriate answer.
350) Suppose that x2 + y2 = r2 and x = r cos , as in polar coordinates. Find y . r
A) C)
y y
= r(cos r
- sin )
B)
=0 r
D)
72
y y
= -r sin r = r cos r
350)
Solve the problem.
351) Find an equation for the level surface of the function f(x, y, z) = x + ey+z that passes through the point 5, ln(3), ln(6) .
351)
B) x + ey+z = 18 D) x + ey+z = 23
A) ln(x) + y + z = 23 C) x + ey+z = 14
352) About how much will f(x, y, z) = xy3z 2 change if the point (x, y, z) moves from (-5, -2, 10) a
352)
1 distance of ds = unit in the direction of 2i + 2j - k? 10
A) - 1840 3
B) - 2240
C) - 480
3
Find the specific function value. 353) Find f(6, 6) when f(x, y) = 6x + 7y - 7. A) 65 B) 71
D) - 1040 3
353) C) 78
D) 64
Compute the gradient of the function at the given point. 354) f(x, y) = -9x2 + 3y, (10, -7)
354)
A) f = -1800i - 21j C) f = -900i - 21j
B) f = -180i + 3j D) f = -180i - 21j
Find the requested partial derivative. 355) ( w/ y)x if w = 4x + 4y + 3z and x + y = z
A) 4
355)
B) 7
C) 11
73
D) 1
Sketch the surface z = f(x,y). 356) f(x, y) = - x2 + y2
356)
A)
B)
C)
D)
Provide an appropriate response.
357) Find the value(s) of t corresponding to the extrema of f(x, y, z) = sin(x2 + y2)cos(z) subject to the constraints x2 + y2 = 6t, 0 t
, and z =
6
357)
. Classify each extremum as a minimum or maximum.
(Hint: w = f(x, y, z) is a differentiable function of t.)
A) t =
12
C) t = -
, minimum
12
B) t =
, maximum
Find the requested partial derivative.
358) w if w = x3 + y3 + z 3 + 12xyz and x
D) t =
y 2 + x
12 12
, minimum; t = -
B) 12x2 D) 3(x2 + y2 + z 2)
74
, maximum
, minimum; t = 0, maximum
z 2 = 0. x
A) 3x2 C) 12(x2 + y2 + z 2 )
12
358)
Find an upper bound for the magnitude |E| of the error in the approximation f(x, y) L(x, y) at the given point over the given region R. 359) f(x, y, z) = tan-1 xyz at (7, 7, 7); R: |x - 7| 0.2, |y - 7| 0.2, |z - 7| 0.2 359)
A) |E| 0.00000014 C) |E| 0.00000016
B) |E| 0.00000027 D) |E| 0.00000021
Solve the problem.
360) Find an equation for the level curve of the function f(x, y) = x2 + y2 that passes through the point 4, 3 .
A) x + y = 5
B) x2 + y2 = 7
C) x2 + y2 = 5
361) Find the point on the plane x + 2y - z = 12 that is nearest the origin. A) (4, 4, 0) B) (2, 4, -2) C) (2, 4, 0)
360)
D) x2 + y2 = 25 361) D) (-2, 8, 2)
Find the absolute maximum and minimum values of the function on the given curve. 2 2 362) Function: f(x, y) = x2 + 2y2; curve: x + y = 1, x 0, y 0. (Use the parametric equations 4 64
362)
x = 2 cos t, y = 8 sin t.)
A) Absolute maximum: 128 at t = B) Absolute maximum: 128 at t = C) Absolute maximum: 64 at t = D) Absolute maximum: 64 at t =
2 2
2 2
; absolute minimum: 4 at t = 0 ; absolute minimum: 8 at t = 0
; absolute minimum: 8 at t = 0 ; absolute minimum: 4 at t = 0
Find all the local maxima, local minima, and saddle points of the function. 363) f(x, y) = 64x2 + 80xy + 100y2
363)
A) f(8, 10) = 20,496, saddle point; f(10, 8) = 19,200, saddle point B) f(80, 80) = 1,216,000, local maximum C) f(0, 0) = 0, local minimum D) f(80, 80) = 1,216,000, local maximum; f(0, 0) = 0, local minimum Find the equation for the level surface of the function through the given point. 364) f(x, y, z) = e(x2 + y2 - z), (3, 6, 2) A) x2 + y2 - z = ln(43) B) e(x2 + y2 - z) = ln(43)
C) ln(x2 + y2 - z) = 43
D) x2 + y2 - z = e43
75
364)
Solve the problem.
365) Find an equation for the level curve of the function f(x, y) = point 1, 2 .
A) x = 2y
n=0
B) x = 1 y
C) x = 2
2
y
x n that passes through the y
D) x
1 2
y
366) Find the derivative of the function f(x, y, z) = ln(xy + yz + zx) at the point (7, 14, 21) in the direction in which the function increases most rapidly. A) 5 2 B) 5 2 C) 5 2 91 77 119
Sketch a typical level surface for the function. 367) f(x, y, z) = cos (x2 + y2 + z2 )
D) 5
49
365)
366)
2
367)
A)
B)
C)
D)
76
Find the extreme values of the function subject to the given constraint. 368) f(x, y) = x2 + y2 , xy2 = 54
368)
A) Maximum: none; minimum: 0 at (0, 0) B) Maximum: 27 at (3, 3 2); minimum: -27 at (3, -3 2) C) Maximum: none; minimum: 27 at (3, ±3 2) D) Maximum: 27 at (3, ±3 2); minimum: 0 at (0, 0) Solve the problem.
369) Find parametric equations for the normal line to the surface z = -4x2 + 6y2 at the point (2, 1, -10). A) x = 2t - 8, y = t + 6, z = -t - 10
369)
B) x = -8t + 2, y = 6t + 1, z = -t - 2 D) x = -16t + 2, y = 12t + 1, z = -t - 10
C) x = 2t - 16, y = t - 8, z = -10t - 1 Find the limit.
370)
10x2 + 5y2 + 10 lim (x, y) (0, 0) 10x2 - 5y2 + 1
A) 1
370) C) 10
B) -1
Show that the function is a solution of the wave equation. 371) w(x, t) = cos (4t - 4cx), c 1 A) No
D) No limit
371) B) Yes
77
Sketch a typical level surface for the function. 2 2 2 372) f(x, y, z) = ln x + y + z 16 16 32
372)
A)
B)
C)
D)
Provide an appropriate response. 373) Find the extrema of f(x, y, z) = x + yz on the line defined by x = 6(4 + t), y = t - 6, and z = t + 4. Classify each extremum as a minimum or maximum. (Hint: w = f(x, y, z) is a differentiable function of t.) A) (12, -8, 2), minimum; (36, -4, 6), maximum
373)
B) (36, -4, 6), minimum; (0, -6, 0), minimum C) (12, -8, 2), minimum D) (36, -4, 6), minimum Provide an appropriate answer. 374) Find z when u = 3 and v = 5 if z(x) = v
A) z = 0 v
x and x = u · v. x+6
C) z =
B) z = 81 v
v
2 21
78
374) 27 2(21)3/2
D) z = v
81 2(21)3/2
Find all the second order partial derivatives of the given function. 375) f(x, y) = x2 + y - ex+y
A) B) C) D)
2f
x2 2f
x2 2f
x2 2f
x2
= 2 - ex+y; = 2 + ex+y;
2f
y2 2f
y2
= 2 - y2 ex+y; = 1 - ex+y;
2f
y2
= - ex+y; = ex+y; 2f
y2
2f
y x
x y
375)
= -ex+y
2f 2f = = ex+y y x x y
= -x2 ex+y;
= - ex+y;
=
2f
2f
y x
2f
y x =
= 2f
x y
2f
x y
= - y2 ex+y
= -ex+y
Use Taylor's formula to find the requested approximation of f(x, y) near the origin. 376) Quadratic approximation to f(x, y) = ln(1 + 4x + y) A) 4x + y - 8x2 - 2xy - 1 y2 B) 1 + 4x + y - 8x2 - 2xy - 1 y2 2 2
C) 4x + y - 8x2 - 4xy - 1 y2
376)
D) 1 + 4x + y - 8x2 - 4xy - 1 y2
2
2
Sketch the surface z = f(x,y). 377) f(x, y) = 1 - x - 2y
377)
A)
B)
C)
D)
79
Find the requested partial derivative. 378) ( w/ y)x at (x, y, z, w) = (1, 1, 2, 22) if w = 6x + 6y + 5z and x + y = z
A) 1
B) 17
C) 11
Provide an appropriate response.
2
2
378) D) 6
2
379) Define f(0, 0) in a way that extends f(x, y) = 9x - x y + 9y to be continuous at the origin. x2 + y2
A) f(0, 0) = 2
B) f(0, 0) = 18
C) f(0, 0) = 9
379)
D) f(0, 0) = 0
Solve the problem.
380) Evaluate z at (u, v) = (4, 5) for the function z = xy2 - ln x; x = eu+v, y = uv.
380)
v
A) 560e9 - 1
B) 480e9 - 1
C) 1200e9 - 1
D) -1
Find the limit.
381)
lim x ln y (x, y) (1, 4)
A) 4
381) B) ln 4
C) ln (4) - 1
D) No limit
Solve the problem. 382) The resistance R produced by wiring resistors of R 1 and R2 ohms in parallel can be calculated
382)
from the formula 1 1 1 = + . R R1 R2 If R 1 and R2 are measured to be 9 ohms and 7 ohms respectively and if these measurements are accurate to within 0.05 ohms, estimate the maximum possible error in computing R. A) 0.025 B) 0.030 C) 0.015 D) 0.020
Find the absolute maximum and minimum values of the function on the given curve. 383) Function: f(x, y) = x2 + y2 ; curve: x = 10t + 1, y = 10t - 1, 0 t 1.
383)
A) Absolute maximum: 202 at t = 1; absolute minimum: 2 at t = 0. B) Absolute maximum: 201 at t = 1; absolute minimum: 2 at t = 0. C) Absolute maximum: 201 at t = 1; absolute minimum: 11 at t = 0. D) Absolute maximum: 202 at t = 1; absolute minimum: 11 at t = 0. Find the requested partial derivative. 384) z if z 3 = z + xy - 1 and y3 = x + y - 1 y 3
A) x - 3y - y 3z 2 - 1
384)
3
3
B) x - 3y - y
C) x + 3y - y
3z 2 + 1
3z 2 - 1
80
3
D) x + 3y - y 3z 2 + 1
Solve the problem.
385) Find the point on the curve of intersection of the paraboloid x2 + y2 + 2z = 4 and the plane x - y + 2z = 0 that is closest to the origin. A) (-1, 1, 1) B) (1, 1, 1)
C) (1, 1, -1)
Provide an appropriate response. 1 386) Find F (x) if F(x) = t4 + x dt. x 1 1 A) dt - x4 + x 4 x 2 t +x 1 1 C) dt - x4 + x 4+x t x
385)
D) (1, -1, 1)
386) 1
B) x4 + x 1
D) x
Find the linearization of the function at the given point. 387) f(x, y) = 5x + 10y + 8 at (10, -8) A) L(x, y) = 50x - 80y + 64
x
1 dt 2 t4 + x
1 dt - x x4 + x t4 + x
387) B) L(x, y) = 5x + 10y + 8 D) L(x, y) = 10x - 8y + 8
C) L(x, y) = 50x - 80y + 8 At what points is the given function continuous? 388) f(x, y) = 10x + 6y A) All (x, y)
388) B) All (x, y) such that x + y 0 D) All (x, y) such that 10x + 6y 0
C) All (x, y) such that 10x + 6y 0
Answer the question. 389) You are hiking on a mountainside, following a trail that slopes downward for a short distance and then begins to climb again. At the bottom of this local "dip", what can be said about the relationship between the trail's direction and the contour of the mountainside? [Hint - Think of the trail as a constrained path, g(x, y) = c, on the mountainside's surface, altitude = f(x, y). Consider only infinitesimal displacements.] A) At the bottom of the dip, the trail is headed along the mountain's contour line which passes through that point. B) At the bottom of the dip, the trail is headed in the direction of the mountain's steepest ascent. C) The mountainside rises in all directions relative to the dip.
389)
D) At the bottom of the dip, the trail is headed perpendicular to the mountain's contour line which passes through that point.
Find the limit.
390)
P
lim 6xz + 5xy (1, - 1, 0) x2 + y2 - z 2
A) -6
390) C) - 5
B) 5
2
81
D) 6
Find the requested partial derivative. 391) ( w/ x)y if w = x3 + y3 + z 3 + 6xyz and z = x2 + y2 A) 3x(3x + 2z 2 ) + 6y(2x2 + 3z)
C) 3x(3x + 2z 2 ) + 6y(2x2 + z)
391) B) 3x(x + 2z 2) + 6y(2x2 + 3z) D) 3x(x + 2z 2) + 6y(2x2 + z)
Compute the gradient of the function at the given point. 392) f(x, y, z) = tan-1 9x , (6, 0, 0) 8y + 2z 1 k 27
B) f = - 1 j - 4 k
C) f = - 4 j - 1 k
D) f = - 1 j + 4 k
A) f =
j27
Find
392) 27
27
27
27 27
f f and . x y -x 393) f(x, y) = e
393)
x2 + y2
A) f = x
2xe-x
2 (x2 + y2 )
;
2ye-x
f = y
2 (x2 + y2 )
2 -x 2 B) f = e (x + y + 2x) ; f = x
2 (x2 + y2 )
y
2 (x2 + y2 )
2 -x 2 C) f = - e (x + y + x) ; f = x
2 (x2 + y2 )
y
2ye-x
2 (x2 + y2)
2 -x 2 D) f = - e (x + y + 2x) ; f = x
2 (x2 + y2 )
y
ye-x
2ye-x
2 (x2 + y2 )
Find the absolute maxima and minima of the function on the given domain. 394) f(x, y) = 4xy2 + 6xy on the trapezoidal region with vertices (0, 0), (1, 0), (0, 2), and (1, 1)
A) Absolute maximum: 10 at (1, 1); absolute minimum: 0 at (0, 0) B) Absolute maximum: 8 at (2, 0); absolute minimum: 4 at (1, 0) C) Absolute maximum: 10 at (1, 1); absolute minimum: 4 at (1, 0) D) Absolute maximum: 12 at (0, 2); absolute minimum: 0 at (0, 0)
82
394)
Find all the second order partial derivatives of the given function. 395) f(x, y) = xye-y2 2 2 2f 2f 2 A) f = ye-y2 ; f = 2xye-y2 (2y2 - 6); = = e-y (1 - 2y2 ) y x x y x2 y2
B) C) D)
2f
x2 2f
x2
=0; = 0;
2f
2 = 2xye-y (2y2 - 3);
2f
2 = 2xye-y (2y2 - 3);
y2 y2
2f
y x 2f
y x
2f
2 2f 2 = ye-y ; = 2xye-y (y2 - 1); x2 y2
= = 2f
y x
2f
2 = e-y (1 - y2 )
2f
2 = e-y (1 - 2y2 )
x y x y =
2f
x y
395)
2 = e-y (1 - y2 )
Determine whether the given function satisfies a Laplace equation. 396) f(x, y, z) = 5x + 5y3 z2
A) No
396)
B) Yes
Find the requested partial derivative.
397) ( w/ z)x,y at (x, y, z, w) = (1, 2, 9, 302) if w = x2 + y2 + z 2 + 12xyz A) 30 B) 54 C) 36
397) D) 42
Find fx , fy, and fz .
398) f(x, y, z) = e(yz + sin x) A) fx = (cos x + yz)e(yz + sin x); fy = ze(yz + sin x); fz = ye(yz + sin x)
398)
B) fx = e(yz + sin x); fy = ze(yz + sin x); fz = ye(yz + sin x) C) fx = (cos x)e(yz + sin x); fy = e(yz + sin x); fz = e(yz + sin x) D) fx = (cos x)e(yz + sin x); fy = ze(yz + sin x); fz = ye(yz + sin x) Use Taylor's formula to find the requested approximation of f(x, y) near the origin. 1 399) Quadratic approximation to f(x, y) = (1 + x + 3y)2
A) 1 - 2x - 6y + 3x2 - 18xy + 27y2 C) 1 + 2x + 6y + 3x2 - 18xy + 27y2
399)
B) 1 + 2x + 6y + 3x2 + 18xy + 27y2 D) 1 - 2x - 6y + 3x2 + 18xy + 27y2
Solve the problem.
400) Find the point on the curve of intersection of the paraboloid x2 + y2 + 2z = 4 and the plane x - y + 2z = 0 that is farthest from the origin. A) (-2, 2, 2) B) (2, -2, 2)
C) (-2, -2, 2)
83
D) (2, -2, -2)
400)
Find all the second order partial derivatives of the given function. 401) f(x, y) = ex/y
A) B) C) D)
2f
ex/y 2 f x2 + 2xy = = ex/y ; ; x2 y2 y2 y4 2f
x2 2f
x2 2f
x2
2f
y x
2f
=
x y
y+ x = -ex/y y3
=
ex/y 2 f x2 + 2xy = -ex/y ; ; 2 2 y y y3
2f 2f y+ x = = ex/y y x x y y3
=
ex/y 2 f x2 + 2xy = ; ; y2 y2 y4
=
=
ex/y 2 f x2 + 2xy = ex/y ; ; y2 y2 y3
2f
y x
2f
2f
x y
y x
=
=
401)
y+x y3
2f
x y
y+ x = -ex/y y3
Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0). 2 2 402) f(x, y) = x y + xy x2 + y2
A) 2
B) 0
C) 1
402) D) No limit
Solve the problem.
403) Maximize f(x, y, z) = e3x + 8y + 6z subject to x + y + z = 0, x + 2y + 3z = 0, and x + 4y + 9z = 0. A) 17 B) e C) 0 D) 1
Find the domain and range and describe the level curves for the function f(x,y). 404) f(x, y) = -5x + 3y A) Domain: all points in the xy-plane; range: real numbers z 0; level curves: lines -5x + 3y = c, c 0 B) Domain: all points in the xy-plane; range: all real numbers; level curves: lines -5x + 3y = c, c >0 C) Domain: all points in the xy-plane; range: real numbers z 0 ; level curves: lines -5x + 3y = c, c 0 D) Domain: all points in the xy-plane; range: all real numbers; level curves: lines -5x + 3y = c
403)
404)
Solve the problem.
405) Find the derivative of the function f(x, y) = exy at the point (0, -9) in the direction in which the function increases most rapidly. A) 9 B) 18
C) 10
Find the extreme values of the function subject to the given constraint. 406) f(x, y) = y2 - x2 , x2 + y2 = 4
A) Maximum: 4 at (0, ±2); minimum: -4 at (±2, 0) B) Maximum: 8 at (0, ±2 2); minimum: -4 at (±2, 0) C) Maximum: 4 at (0, ±2); minimum: -8 at (±2 2, 0) D) Maximum: 8 at (0, ±2 2); minimum: -8 at (±2 2, 0)
84
405)
D) 27
406)
Solve the problem. 407) Find the least squares line for the points (-1, 2), (-3, 3). A) y = 3 - 5 x B) y = 3 + 1 x C) y = 3 - 1 x 2 2 2 2 2 2
407) D) y = 3 + 5 x 2
2
Find an upper bound for the magnitude |E| of the error in the approximation f(x, y) L(x, y) at the given point over the given region R. 408) f(x, y, z) = e9x + 3y + 2z at (0, 0, 0); R: |x| 0.1, |y| 0.1, |z| 0.1 408)
A) |E| 17.0805
B) |E| 14.7812
C) |E| 15.7666
D) |E| 13.7958
Find the extreme values of the function subject to the given constraint. 409) f(x, y, z) = x3 + y3 + z 3 , x2 + y2 + z 2 = 4
A) Maximum: 8 at (2, 0, 0); minimum: 0 at (0, 0, 0) B) Maximum: 8 at (2, 0, 0), (0, 2, 0), (0, 0, 2); minimum: 0 at (0, 0, 0) C) Maximum: 8 at (2, 0, 0), (0, 2, 0), (0, 0, 2); minimum: -8 at (-2, 0, 0), (0, -2, 0), (0, 0, -2) D) Maximum: 8 at (2, 0, 0); minimum: -8 at (-2, 0, 0)
85
409)
Answer Key Testname: CHAPTER 14
1) B 2) A 3) A 4) A 5) D 6) C 7) D 8) B 9) Answers will vary. One possibility is Path 1: x = t, y = t ; Path 2: x = t, y = -t 10) Let = 0.04. Then if x < and y < , f(x, y) - f(0, 0) = x + y x + y < 0.04 + 0.04 = 0.08 = . 2 11) f = lim -20(2 + h) + 32 - 80 + 128 x
=
h
0
h
lim -80h - 20h 2 lim = -80 - 20h = -80 h 0 h h 0
12) Answers will vary. One possibility is Path 1: x = t, y = 0 ; Path 2: x = 0, y = t 13) Answers will vary. One possibility is Path 1: x = 0, y = t ; Path 2: x = - t2, y = t 14) Answers will vary. One possibility is Path 1: x = t, y = t ; Path 2: x = t, y = t5/2 15) Answers will vary. One possibility is Path 1: x = t, y = t ; Path 2: x = t, y = 2t 16)
lim lim 5 tan-1 xy 5 - x2 y3 = 5. Hence. by the Sandwich Theorem, = 5. (x, y) (0, 0) (x, y) (0, 0) xy
17) f = lim 6 - 16(10 + h) + 12(10 + h) + 34 x
=
h
0
h
lim -4h lim = -4 = -4 h 0 h h 0
2 18) f = lim 4 - 70(7 + h) + 14(7 + h) - 200 y
=
h
0
h
lim 126h + 14h 2 lim = 126 + 14h = 126 h 0 h h 0
19) Let = 0.02. Then if x < and y < , f(x, y) - f(0,0) =
x+y 2 x + y2 + 1
x+y
x
+ y < 0.02 + 0.02
= 0.04 = .
20) Let = 0.01. Then if x < and y < , f(x, y) - f(0, 0) = (1 + cos x)(x + y) = 0.04 = .
2 x+y
2 x + y
< 0.02 + 0.02
21) Let = 0.02. Then if x < and y < and z < , f(x, y, z) - f(0, 0, 0) = x + y + z
x + y + z < 0.02 + 0.02 + 0.02 = 0.06 = . lim lim 22) f(x,y,z) = x3 y5 z 3 = x0 3 y0 5 z 0 3 = f(x0 , y0 , z 0 ), which proves the assertion. (x, y, z) (x0 , y0 , z 0 ) (x, y, z) (x0 , y0 , z 0 )
23)
lim lim 2 2 2 2 2 2 f(x, y, z) = ex + y + z = e(0 + 0 + 0 ) = 1 = f(0, 0, 0), which proves the (x, y, z) (0 ,0, 0) (x, y, z) (0, 0, 0) assertion.
24) Answers will vary. One possibility is Path 1: x = t, y = t ; Path 2: x = 0, y = t 25) This follows from applying the chain rule. 86
Answer Key Testname: CHAPTER 14
2 26) f = lim -9(-5 + h) - 6(-5 + h) - 15 + 120 y
=
h
0
h
lim 51h - 6h 2 lim = 51 - 6h = 51 h 0 h h 0
27) cos 1
1 implies -1 cos
y
1 y
1, which allows us to obtain - sinx sin(x) cos
1 y
sin x for -
2
x
2
. As
lim lim lim 1 -sin x = 0 and = 0. sin x = 0, the Sandwich Theorem implies sin(x) cos (x, y) (0, 0) (x, y) (0, 0) (x, y) (0, 0) y
28) Answers will vary. One possibility is Path 1: x = t, y = t ; Path 2: x = 0, y = t 29) Let = 0.03. Then if x < and y < , f(x, y) - f(0, 0) =
2x + y x2 y2 + 1
2x + y
2 x + y < 0.06 + 0.03
= 0.09 = .
30) Let = <a>. Then if
x2 + y2 + z 2 < ,
f(x, y, z) - f(0, 0, 0) =
x2 + y2 + z 2 x+1
x2 + y2 + z 2 < 0.02 = .
2 31) f = lim 2(4 + h) + 40(4 + h) + 192 - 384 x
h
0
h
=
lim 2(16 + 8h + h 2 ) + 40h - 32 h 0 h
=
lim 56h + 2h 2 lim 56 + 2h = 56 = h 0 h h 0
32) Let = 0.03. Then if x < , y < , and z < , f(x, y, z) - f(0, 0, 0) = x + y - z x + y + z < 0.03 + 0.03 + 0.03 = 0.09 = . f lim 490 - 14(-5 + h) + 10(-5 + h)2 - 810 = y h 0 h
33) =
lim -114h + 10h 2 lim = -114 + 10h = -114 h 0 h h 0
34) sin 1
1 implies -1 sin
y
1 y
1, which allows us to obtain - sin x sin(x) sin
1 y
sin x for -
2
x
2
. As
lim lim lim 1 -sin x = 0 and = 0. sin x = 0, the Sandwich Theorem implies sin (x) sin (x, y) (0, 0) (x, y) (0,0) (x, y) (0, 0) y
35) f = lim 45(6 + h) - 405 - 3(6 + h) + 153 z
=
h
0
h
lim 42h lim = 42 = 42 h 0 h h 0
2 36) f = lim 360(10 + h) + 7(10 + h) + 6 - 4306 y
=
h
0
h
lim 500h + 7h 2 lim = 500 + 7h = 500 h 0 h h 0
37) Let = sin-1 0.04 . Then if x < and y < and z < , f(x, y, z) - f(0, 0, 0) = sin2x + sin2y + sin2z sin2 x + sin2 y + sin2 z < 0.04 + 0.04 + 0.04 = 0.12 = .
38) Answers will vary. One possibility is Path 1: x = t, y = t ; Path 2: x = t, y = t2 87
Answer Key Testname: CHAPTER 14
39) The sphere centered at (x0, y0, z 0) of radius circumscribes the cube centered at (xo, yo, z o) and with sides Therefore, x - x0 <
2 , y - y0 < 2
2 , and z - z 0 < 2
2 imply 2
=
(x - x0 ) 2 + (y - y0 )2 + (z - z 0 )2 < .
Likewise, the cube centered at (xo, yo, z o) with sides 2 circumscribes the sphere with radius . Thus, (x - x0 ) 2 + (y - y0 )2 + (z - z0 )2 <
implies x - xo < , y - yo < , and z - z o <
equivalent.
40) The function is not necessarily continuous at (x0 , y0 ). It is continuous only if 41) B 42) A 43) B 44) B 45) B 46) B 47) A 48) B 49) C 50) C 51) D 52) A 53) C 54) A 55) C 56) C 57) A 58) B 59) A 60) B 61) D 62) B 63) A 64) C 65) B 66) B 67) A 68) A 69) B 70) D 71) D 72) A 73) B 74) D 88
. The requirements are
lim f(x,y) = -2. (x, y) (x0 , y0 )
2 .
Answer Key Testname: CHAPTER 14
75) A 76) D 77) D 78) B 79) B 80) C 81) D 82) B 83) A 84) D 85) D 86) B 87) B 88) C 89) D 90) C 91) B 92) D 93) C 94) B 95) B 96) A 97) A 98) C 99) D 100) D 101) C 102) D 103) B 104) D 105) B 106) B 107) C 108) A 109) B 110) D 111) C 112) A 113) C 114) A 115) D 116) B 89
Answer Key Testname: CHAPTER 14
117) B 118) C 119) A 120) A 121) C 122) A 123) D 124) A 125) D 126) C 127) C 128) D 129) D 130) A 131) B 132) D 133) B 134) B 135) C 136) B 137) A 138) D 139) A 140) A 141) D 142) A 143) A 144) B 145) B 146) D 147) D 148) C 149) A 150) C 151) D 152) A 153) D 154) A 155) C 156) A 157) D 158) B 90
Answer Key Testname: CHAPTER 14
159) B 160) A 161) D 162) B 163) B 164) C 165) D 166) D 167) D 168) D 169) C 170) B 171) C 172) B 173) B 174) C 175) B 176) C 177) B 178) A 179) A 180) B 181) D 182) A 183) B 184) B 185) C 186) D 187) A 188) C 189) C 190) B 191) A 192) D 193) D 194) B 195) C 196) B 197) C 198) B 199) C 200) C 91
Answer Key Testname: CHAPTER 14
201) C 202) D 203) D 204) D 205) D 206) B 207) C 208) C 209) B 210) D 211) C 212) B 213) D 214) B 215) D 216) C 217) B 218) C 219) C 220) C 221) D 222) D 223) D 224) C 225) D 226) B 227) A 228) C 229) A 230) A 231) B 232) C 233) B 234) D 235) D 236) C 237) B 238) A 239) A 240) B 241) B 242) A 92
Answer Key Testname: CHAPTER 14
243) A 244) C 245) D 246) C 247) D 248) C 249) B 250) A 251) B 252) B 253) A 254) B 255) A 256) B 257) A 258) B 259) B 260) B 261) C 262) C 263) A 264) D 265) B 266) C 267) D 268) D 269) B 270) C 271) D 272) D 273) A 274) A 275) A 276) D 277) C 278) B 279) C 280) A 281) A 282) A 283) A 284) A 93
Answer Key Testname: CHAPTER 14
285) A 286) D 287) A 288) D 289) A 290) B 291) D 292) B 293) A 294) C 295) C 296) D 297) D 298) B 299) D 300) B 301) D 302) B 303) D 304) D 305) A 306) C 307) B 308) D 309) C 310) A 311) C 312) D 313) D 314) D 315) B 316) B 317) D 318) C 319) B 320) B 321) D 322) C 323) C 324) A 325) B 326) B 94
Answer Key Testname: CHAPTER 14
327) A 328) D 329) B 330) A 331) B 332) B 333) C 334) B 335) D 336) D 337) D 338) B 339) C 340) B 341) B 342) B 343) D 344) C 345) B 346) B 347) C 348) C 349) C 350) D 351) D 352) C 353) B 354) B 355) B 356) C 357) A 358) A 359) D 360) D 361) B 362) A 363) C 364) A 365) B 366) B 367) C 368) C 95
Answer Key Testname: CHAPTER 14
369) D 370) C 371) A 372) C 373) C 374) D 375) A 376) C 377) A 378) C 379) C 380) A 381) B 382) A 383) A 384) C 385) A 386) A 387) B 388) D 389) A 390) C 391) D 392) C 393) D 394) A 395) C 396) A 397) D 398) D 399) D 400) D 401) A 402) B 403) D 404) D 405) A 406) A 407) C 408) B 409) C 96
Chapter 15
Exam Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Write a "volume" word problem for which finding the solution would involve evaluating the double integral 2
6
1
1)
(y + x) dx dy.
4
2) Write a "volume" word problem for which finding the solution would involve evaluating
2)
the double integral 2
6
1
(y + x) dy dx.
4
3) Write an "area" word problem for which finding the solution would involve evaluating
3)
the double integral 2x
2
x2
0
dy dx.
4) Give a geometric interpretation of the triple integral 2
4)
3 - r2
3
r dzdrd . 0
0
0
5) Give a geometric interpretation of the triple integral /4
1
5)
2 sin
d d d . Describe the three-dimensional region associated 0 0 0 with this integral in detail.
6) Write an "area" word problem for which finding the solution would involve evaluating the double integral x2
5 0
dy dx.
0
1
6)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem.
7) Find the average height of the paraboloid z = x2 + y2 above the disk x2 + y2 4 in the xy-plane. A) 4
B) 6
3
C) 10
8) Find the mass of a thin plate bounded by |x| + |y| = 3 if (x, y) = x2 y2 . A) 729 40
B) 81 5
C) 729 56
D) 2
8) D) 729 25
9) Write an iterated triple integral in the order dx dy dz for the volume of the tetrahedron cut from the first octant by the plane
4
A) 0
B) 0
C) 0
D)
4
4
4
0
1 - y/6 0
0
0
1 - z/4
1 - y/6 - z/4
0 6(1 - z/4)
9(1 - y/6)
0
dx dy dz
dx dy dz
9(1 - y/6 - z/4) 0
9)
x y z + + = 1. 9 6 4
1 - y/6 - z/4 0
7)
9(1 - y/6 - z/4)
dx dy dz
dx dy dz
0
Evaluate the improper integral. dx dy 2 (x + 64)(y2 +16)
10) -
-
A)
2
32
10) B)
C)
64
32
D)
2 64
Solve the problem.
11) Find the mass of a thin circular plate bounded by x2 + y2 = 4 if (x, y) = x2 + y2 . A) 2 B) 4 C) 1 D) 8
Use a spherical coordinate integral to find the volume of the given solid. 12) the solid between the sphere = 1 and the cardioid of revolution = 10 + 4 cos A) 1.699111969e+15 B) 3.185834941e+15 2.199023256e+12 3.298534883e+12
C) 1.699111969e+15
D) 3.185834941e+15
1.099511628e+12
1.649267442e+12
2
11)
12)
Evaluate the integral. 4 8 9
13)
0
xyz dx dy dz
0 0 A) 6912
13) B) 20,736
C) 13,824
Find the area of the region specified by the integral(s). 8 y 16 16 - y 14) dx dy + dx dy 0 0 8 0 A) 128 B) 8 Evaluate the integral. 3 10
15)
2
D) 10,368
14) C) 64
D) 4
rz dr d dz
15)
8 0 1,240,798,871,945,221 3 A) 103,079,215,104
B) 3,219,370,046,128,141 3
C) 2,213,316,906,713,097 3
D) 6,572,880,510,844,955 3
824,633,720,832
137,438,953,472
824,633,720,832
Reverse the order of integration and then evaluate the integral. 27 3 1 16) dy dx 4 y 1 + 3 0 x
A) ln 3 - 1
B) ln 3
16) C) ln 82 - 1 4
D) ln 82 4
Use the given transformation to evaluate the integral. 17) x = 10u, y = 2v, z = 3w; 3 x2 y2 z 2 + + dx dy dz, 100 4 9 R where R is the interior of the ellipsoid
A) 15
17)
x2 y2 z 2 + + =1 100 4 9
B) 40
C) 20
D) 45
Evaluate the double integral over the given region.
18) R
1 dA, R: 2 x 4, 2 y 4 xy
A) ln 1 2
18)
B) ln 1
2
C) ln 2 ln 4
2
3
D) ln 2 ln 4
2
Solve the problem. 19) Find the centroid of the region in the first quadrant bounded by the curve y = 7 cos x and the axes. A) x = 7 - 14 , y = 49 B) x = - 2 , y = 7 2 8 2 8
C) x = 7 , y = 8
Evaluate the integral. 2 z
4
20)
0
-2 2
(r2 sin2
0 0 A) 112 3
D) x =
2
,y=
7 4
+ z 2) r d dr dz
20)
B) 28
C) 14
3
D) 7
3
3
Reverse the order of integration and then evaluate the integral. 3 ln 8 ln 8 2 21) ex dx dy 0
A) 27 2
19)
y/3
21) C) 21
B) 16
D) 8
2
Find the area of the region specified in polar coordinates. 22) one petal of the rose curve r = 9 cos 3 A) 27 B) 27 4 2
22) C) 81
D) 81
2
4
Solve the problem.
23) Find the center of mass of the hemisphere of constant density bounded z = 4 - x2 - y2 and the xy-plane.
A) x = 0, y = 0, z = 2
B) x = 0, y = 0, z = 1
C) x = 0, y = 0, z = 4
D) x = 0, y = 0, z = 3
3
23)
2
3
4
24) Find the mass of the solid in the first octant between the spheres x2 + y2 + z2 = 4 and
24)
x2 + y2 + z 2 = 81 if the density at any point is inversely proportional to its distance from the origin.
A) 77 k 6
C) 77 k
B) 154k
3
D) 77 k 4
25) Find the mass of the rectangular solid of density (x, y, z) = xyz defined by 0 x 3, 0 y 10, 0 z 2. A) 989,560,464,998,399 4,123,168,604,160
B) 989,560,464,998,399
C) 989,560,464,998,399
D) 989,560,464,998,399
2,199,023,255,552
3,298,534,883,328
1,649,267,441,664
4
25)
26) Let D be the region bounded below by the xy-plane, above by the sphere x2 + y2 + z 2 = 64, and on
26)
the sides by the cylinder x2 + y2 = 36. Set up the triple integral in cylindrical coordinates that gives the volume of D using the order of integration dz d dr. 6 2 8 2 64 - r2 36 - r2 A) B) dz d dr r dz d dr 0 0 0 0 0 0 6
C) 0
64 - r2
2 0
r dz d dr
8
D) 0
0
36 - r2
2 0
dz d dr
0
Find the volume of the indicated region.
27) the region bounded by the cylinder x2 + y2 = 81 and the planes z = 0 and x + z = 10 A) 900 B) 90 C) 810 D) 8100
Evaluate the improper integral. 81 36 dx dy 28) xy 0 0 A) 324
27)
28) B) 216
C) 36
D) 24
Provide an appropriate response. 29) What form do planes perpendicular to the y-axis have in spherical coordinates? A) = a sin sin B) = a csc
C) = a csc csc
29)
D) = a csc
Find the volume of the indicated region.
30) the region that lies under the paraboloid z = x2 + y2 and above the triangle enclosed by the lines x = 10, y = 0, and y = 9x A) 6300
B) 667,500
C) 63,000
D) 630,000
Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. 31) The lines x = 0, y = 7x, and y = 6 6 y/7 7 y/6 A) B) dy dx dx dy 0 0 0 0 6 y/7 7 x/6 C) D) dx dy dy dx 0 0 0 0 Solve the problem. 32) A solid of constant density is bounded below by the plane z = 0, above by the cone z = r, and on the sides by the cylinder r = 9. Find the center of mass. A) (0, 0, 27 ) B) (0, 0, 6) C) (0, 0, 45 ) D) (0, 0, 3) 8 8
5
30)
31)
32)
Evaluate the integral. /4 cos 2x
33)
0
A) 1
sin 2x dy dx
33)
0
8
B)
C) 1
8
4
D)
4
Solve the problem. 34) Find the moment of inertia about the y-axis of a thin infinite region in the first quadrant bounded by the coordinate axes and the curve y = e-2x if (x, y) = xy.
A) 3
32
B) 3
C) 5
256
256
34)
D) 3
80
35) Let D be the smaller cap cut from a solid ball of radius 4 units by a plane 2 units from the center of
35)
the sphere. Set up the triple integral for the volume of D in rectangular coordinates. 20 12 - x2 4 - x2 - y2 A) dz dy dx
B)
C)
D)
- 20 - 12 - x2 4 20 12 - x2
16 - x2 - y2
- 20 - 12 - x2 2 12 12 - x2
16 - x2 - y2
- 12 - 12 - x2 2 12 12 - x2
4 - x2 - y2
dz dy dx
dz dy dx
dz dy dx
- 12 - 12 - x2 4
Write an equivalent double integral with the order of integration reversed. 9 10 36) dy dx 0 7x/9 + 3 10 9(y - 3)/7 9 9(y - 3)/7 A) B) dx dy dx dy 9 0 3 0 10 9(y - 3)/7 10 9(y - 3)/7 C) D) dx dy dx dy 0 0 3 0
6
36)
37)
ln 6
6
ln 2
ey 6
A) C)
4y dx dy
ln x
ln 6 ln 2 6 ln x 2
37)
4y dy dx
B) 2
4y dy dx
D)
ln 6
6
ln x
6
ln 2 ln x
4y dy dx
4y dy dx
ln 2 ln 6
Reverse the order of integration and then evaluate the integral. 3 1 38) 2y2 cos xy dy dx 0 x/3 A) 7 (1 - cos 3) B) 2(1 - cos 3) C) cos 3 - 1 2
38) D) 1 (1 - cos 3) 3
Solve the problem. 39) Let Vt be the volume of the tetrahedron bounded by the coordinate planes and the plane
39)
Ve x y z x2 y2 z 2 . + + = 1, and let Ve be the volume of the ellipsoid + + = 1. Find the quotient 7 4 3 49 16 9 Vt
A) 4
C) 2
B) 8
D) 4
3
3
2
2
2
36
b2
81
40) For what value of b is the volume of the ellipsoid x + y + z = 1 equal to 5 ? A) b = 5
B) b = 5
108
C) b = 5
72
40)
D) b = 5
36
54
Change the Cartesian integral to an equivalent polar integral, and then evaluate. 3 9 - x2 2 2 41) e-(x + y ) dy dx 0 0
A)
1 - e-9 2
B)
1 - e-9 4
C)
41)
1 + e-9 2
D)
1 + e-9 4
Solve the problem.
42) Find the center of mass of the solid enclosed between the cone with equation z = 6 x2 + y2 and 7
the plane with equation z = 6 if the density at any point is proportional to the distance from that point to the axis of the cone. A) (x, y, z) = 0, 0, 4 B) (x, y, z) = 0, 0, 9 2
C) (x, y, z) = 0, 0, 15
D) (x, y, z) = 0, 0, 24
4
5
7
42)
Evaluate the spherical coordinate integral. 5 sin 2 sin d d d 43) 0 0 0 A) 125 2 B) 125 2 8 3
43) C) 125 2
D) 125 2
6
9
Use the given transformation to evaluate the integral. 44) u = x + y, v = -2x + y;
44)
2y dx dy, R where R is the parallelogram bounded by the lines y = -x + 1, y = -x + 4, y = 2x + 2, y = 2x + 5 A) 13 B) 17 C) 13 D) 17 2 2
Solve the problem. 45) Write 3
45) y
f(x, y) dx dy + 0 0 0 as a single iterated integral. 3 3 A) f(x, y) dx dy 0 0 3 y C) f(x, y) dx dy 0 -y
3
3
f(x, y) dx dy
y
B)
3
y/3
f(x, y) dx dy -y/3 3 3 f(x, y) dx dy -3 -3 0
D)
46) Find the center of mass of the region of constant density bounded by the paraboloid
46)
z = 36 - x2 - y2 and the xy-plane.
A) x = 0, y = 0, z = 3 C) x = 0, y = 0, z = 12
B) x = 0, y = 0, z = 18 D) x = 0, y = 0, z = 2
47) Write an iterated triple integral in the order dz dy dx for the volume of the region in the first octant enclosed by the cylinder x2 + y2 = 36 and the plane z = 4. 36 - x2
6
A) 0
0
36 - y2 0
dz dy dx
0 4
dz dy dx
0
8
y
36 - y2 0
dz dy dx
0
0 6
D)
0
36 - x2
6
B)
0
0 6
C)
4
4-y 0
dz dy dx
47)
Find the average value of F(x, y, z) over the given region. 48) F(x, y, z) = xyz over the rectangular solid in the first octant bounded by the coordinate planes and the planes x = 10, y = 8, z =2 A) 40 B) 10 C) 80 D) 20 3 9 Find the average value of the function f over the given region. 49) f(x, y) = ex2 over the region bounded by y = 6x, y = 10x , x = 0, and x = 1 . A) e - 1 B) 2e - 1 C) 2e - 1 6 36
48)
49) D) e - 1
Provide an appropriate response. 50) What form do planes perpendicular to the y-axis have in cylindrical coordinates? A) r = a csc B) r = a sec C) r = a sin D) r = a cos
50)
Evaluate the integral. 1 1 51) (6x - 10y) dy dx 0 0 A) 68
51) B) - 8
C) 8
D) - 2
Solve the problem. 52) Write an iterated triple integral in the order dz dy dx for the volume of the tetrahedron cut from x y z the first octant by the plane + + = 1. 5 9 3 5
A) 0
B) 0
C) 0
D) 0
5
5
5
5(1 - y/9) 0
0
0
0
3(1 - x/5 - y/9) 0 1 - x/5 - y/9
1 - x/5 0 9(1 - x/5)
dz dy dx
dz dy dx
3(1 - x/5 - y/9) 0 1 - x/5 - y/9
1 - y/9
52)
dz dy dx
dz dy dx
0
Evaluate the integral. 0 0 53) (2x + 7y) dy dx -7 -5 A) - 49 2
53) B) - 245
C) - 343
2
2
9
D) - 1715 2
Evaluate the double integral over the given region. e2x + 3y dA, R: 0 x 1, 0 y 1
54)
54)
R
A) 1 (e5 - e3 - e2 + 1)
B) 1 (e5 - e3 - e2 + 1)
C) 1 (e5 - e3 - e2 - 1)
D) 1 (e5 - e3 - e2 - 1)
4
6
6
4
Evaluate the spherical coordinate integral. 4 (1 - cos )/2 2 sin d d d 55) 0 0 (1 - sin )/2 1 A) B) 1 (15 - 16) (15 - 8) 24 48
55) C) 1 (15 - 8) 48
Evaluate the cylindrical coordinate integral. 2 10/r 56) sin dz r dr d 0 0 0 A) 20 B) 60 Solve the problem. 57) Write 3
24
56) C) 60
D) 40
57) y
6
f(x, y) dx dy +
0 0 3 as a single iterated integral. 3 6-x A) f(x, y) dy dx -3 3 3 6-x C) f(x, y) dy dx 0 0
6-y
3
f(x, y) dx dy +
0
3
0
y
3
B) 0
D)
f(x, y) dx dy
3
6-x 3
6-x
f(x, y) dy dx
f(x, y) dy dx
-3 x
Evaluate the integral by changing the order of integration in an appropriate way. 15 3 2 2 58) ze-(y + z ) dz dy dx 0 x/5 0 5 A) 1 - e-18 B) 5 1 - e-18 C) 5 1 - e-9 2 4 4
Find the Jacobian
D) 1 (15 - 16)
58) D) 5 1 - e-9 2
(x, y) (x, y, z) or (as appropriate) using the given equations. (u, v) (u, v, w)
59) x = 7u2, y = 5uv A) 70v2
59) B) 70u2
C) 35u2
10
D) 35v2
Find the average value of the function over the region. 60) f(r, , z) = r2 over the region bounded by the cylinder r = 2 between the planes z = -3 and z = 3
A) 4 3
C) 1 2
B) 24
60)
D) 2
Find the volume of the indicated region.
61) the tetrahedron cut off from the first octant by the plane x + y + z = 1 5
A) 50
B) 100
3
10
61)
2
C) 50
3
D) 25
Solve the problem. 62) Find the center of mass of a thin triangular plate bounded by the coordinate axes and the line x + y = 5 if (x, y) = x + y. A) x = 5 , y = 5 B) x = 15 , y = 15 C) x = 10 , y = 10 D) x = 25 , y = 25 3 3 8 8 3 3 12 12
63) Let D be the smaller cap cut from a solid ball of radius 10 units by a plane 7 units from the center
62)
63)
of the sphere. Set up the triple integral for the volume of D in cylindrical coordinates. 2 2 149 51 49 - r2 49 - r2 A) B) r dz dr d r dz dr d 0 0 0 0 10 10 2
C) 0
100 - r2
51 0
r dz dr d
2
D) 0
7
100 - r2
149 0
r dz dr d
7
64) What domain D in space minimizes the value of the integral 1-
64)
x2 y2 z 2 dV? 36 36 36
D A) No such maximum domain exists.
B) D = the boundary and interior of the sphere x2 + y2 + z 2 = 36. C) D = 3 D) D = the boundary of the sphere x2 + y2 + z2 = 36. Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries. 65) z = 8x + 4y + 7; 0 x 1, 1 y 3 A) 36 B) 38 C) 28 D) 26 Find the area of the region specified in polar coordinates. 66) the region enclosed by the curve r = 8 - 7 cos A) 177 B) 113 2
65)
66) C) 113 2
11
D) 81
Solve the problem. 67) Evaluate
67) 1-
x2 y2 z2 dx dy dz, + + 81 9 100
R where R is the interior of the ellipsoid
x2 y2 z2 + + = 1. Hint: Let x = 9u, y = 3v, z = 10w, and 81 9 100
then convert to spherical coordinates.
C) 135
B) 90 2
A) 90
D) 135 2
2
2
Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. 68) The lines x + y = 1, x + y = 1, and y = 0 6 10 10 6
A) 0
C)
16
10 - 5x/3
15/4
6 - 3x/5 6 - 3x/5
0
dx dy
B) 0
dx dy
D)
10 - 5x/3
16
6 - 3x/5
15/4
10 - 5x/3 10 - 5x/3
0
dx dy
dx dy
6 - 3x/5
Find the volume of the indicated region. 69) the region bounded by the coordinate planes and the planes z = x + y, z = 8 A) 256 B) 512 C) 128 3
69) D) 256 3
Find the average value of the function f over the given region. 70) f(x, y) = 1 over the square 1 x 6, 1 y 6. xy
A) ln 6 25
Find the Jacobian
70) C) ln 6
B) ln 6 36
68)
2
6
D) ln 6
2
5
(x, y) (x, y, z) or (as appropriate) using the given equations. (u, v) (u, v, w)
71) x = 4u cos 10v, y = 4u sin 10v A) 400u B) 160v
71) C) 160u
12
D) 400v
Solve the problem. 72) Evaluate
72) 9 - x2
3
5
(x2 + y2 )z dz dy dx
-3 - 9 - x2 0 by transforming to cylindrical or spherical coordinates. A) 8,906,044,184,985,601 B) 8,906,044,184,985,601 17,592,186,044,416 13,194,139,533,312
C) 225
D) 5,937,362,789,990,401
4
35,184,372,088,832
73) Find the centroid of the rectangular solid defined by 0 x 7, 0 y 6, 0 z 4. A) x = 7 , y = 3 , z = 1 4 2
B) x = 7 , y = 2, z = 4 3 3
C) x = 7 ,y = 3, z = 2
D) x = 7, y = 6, z = 4
2
73)
74) Find the centroid of the infinite region in the first quadrant bounded by the coordinate axes and the curve y = e-2x . A) x = 1 , y = 1 2 2
B) x = 1 , y = 1 4
75) Rewrite the integral 5
10(1 - x/5)
8(1 - x/5 - y/10)
B) 0
C) 0
D) 0
5
8
0
0
0
0
10(1 - x/5)
8(1 - x/5)
5(1 - z/8)
4
dz dy dx
5(1 - y/10 - z/8)
A)
5
2
4
D) x = 1 , y = 1 2
4
75)
0 0 0 in the order dx dy dz. 8 10(1 - z/8)
0
C) x = 1 , y = 1
74)
0
8(1 - x/5 - y/10)
0 10(1 - x/5 - y/10) 0
10(1 - y/10 - z/8)
dx dy dz
dx dy dz
dx dy dz
dx dy dz
0
13
Use the given transformation to evaluate the integral. 76) x = 6u, y = 4v, z = 10w;
76)
z 2 dx dy dz, R where R is the interior of the ellipsoid
A) 80
x2 y2 z2 + + =1 36 16 100
B) 64
C) 96
D) 60
Find the average value of the function over the region. 77) f( , , ) = over the "ice cream cone" cut from the solid sphere p 15 by the cone A) 6,957,847,019,519,995 B) 1 549,755,813,888 20
C) 4
4
Evaluate the improper integral. 5 8 78) ln xy dx dy 0 0
78) B) ln 40 - 2
A) 40(ln 40 - 2)
C) 40(ln 40 - 1)
40
Evaluate the integral. (1 - cos )/2
10
80)
0 A) 5 2
D) ln 40 - 1 40
(x, y) (x, y, z) or (as appropriate) using the given equations. (u, v) (u, v, w)
79) x = 6u cosh 3v, y = 6u sinh 3v, z = -7w A) 756v B) 756u
0
77)
D) 45
45
Find the Jacobian
= /3
2 sin
79) C) 378u
D) 378v
d d d
80)
0
C) 5
B) 5
3
Evaluate the cylindrical coordinate integral. 8 10 10/r 81) dz r dr d 4 8 9/r A) 72 B) 144
D) 10 3
81) C) 16
14
D) 8
Use the given transformation to evaluate the integral. 82) u = y - x, v = y + x; y- x y+ x e dx dy, R where R is the trapezoid with vertices at (5, 0), (6, 0), (0, 5), (0, 6) 11(e2 - 1) 11(e2 - 1) 11(e2 - 1)
A)
B)
3e
C)
4e
2e
82)
2
D) 11(e - 1) 6e
Evaluate the double integral over the given region. xy dA, R: 2 x 9, 2 y 8
83)
83)
R
A) 1155
B) 770
C) 2310
D) 1540
Solve the problem. 84) Evaluate
84) (4x + y - z)(z - x + 7y) dV
R where R is the parallelepiped enclosed by the planes 4x + y - z = 1, 4x + y - z = 3, -x + 7y + z = -2, -x + 7y + z = 5, x - y + 3z = -1, x - y + 3z = 6. A) 128 B) 162 C) 225 D) 3 49 49 98
Reverse the order of integration and then evaluate the integral. 2 2 sin y 85) dy dx y 0 x A) cos 2 B) 1 - cos 2 C) 1 + cos 2
D) - cos 2
Find the area of the region specified in polar coordinates. 86) the region inside both r = 6 sin and r = 6 cos A) 9 ( - 2) B) 9( - 2) 2
D) 9 ( - 1) 2
Evaluate the integral. 2 y2
z
87)
0
2 A) 23 3
85)
86) C) 9( - 1)
yz dx dz dy
87)
2
B) 8
C) - 128
3
3
D) 68 3
Solve the problem.
88) Find the average height of the part of the paraboloid z = 81 - x2 - y2 that lies above the xy-plane. A) 27
B) 243 2
C) 81 2
15
D) 405 2
88)
89) Find the centroid of the region cut from the first quadrant by the line x + y = 1 . 9
A) x = 3, y = 7
B) x = 9 , y = 7
3
4
Set up the iterated integral for evaluating
C) x = 7 , y = 9
4
4
f(r, , z) r dz dr d
89)
7
D) x = 7 , y = 3
4
3
over the given region D.
D
90) D is the right circular cylinder whose base is the circle r = 2 sin in the xy-plane and whose top lies in the plane z = 10 - y. 2 2 sin 10 - sin
f(r, , z) dz r dr d
A)
0
0 2 sin
B) 0
0
0
0
C) D) 0
2
0 10 - sin 0
2 sin
2 sin 0
f(r, , z) dz r dr d
10 - r sin
0
90)
10 - r sin
f(r, , z) dz r dr d
f(r, , z) dz r dr d
0
Write an equivalent double integral with the order of integration reversed. 7 10y/7 91) dx dy 0 0 10 7 10 10x/7 A) B) dy dx dy dx 0 7x/10 0 0 10 x/7 x 7/10 C) D) dy dx dy dx 0 0 0 0 Provide an appropriate response. 92) What does the graph of the equation = 0 look like? A) The y-axis B) The z-axis
91)
92) C) The xy-plane
D) The yz-plane
Solve the problem. 93) What region in the xy-plane minimizes the value of
93)
(49x2 + 4y2 - 196) dA ? R
A) The ellipse 4x2 + 49y2 = 196
B) The ellipse 49x2 + 4y2 = 1
2
2
4
49
D) The ellipse x + y = 1
C) The ellipse x + y = 1
16
2
2
49
4
94) The northern third of Indiana is a rectangle measuring 96 miles by 132 miles. Thus, let
94)
D = [0, 96] × [0, 132]. Assuming that the total annual snowfall (in inches), S(x,y), at (x,y) D is given by the function S(x,y) = 60e-0.001(2x + y) with (x,y) D, find the average snowfall on D.
A) 52.06 inches
B) 51.14 inches
C) 52.44 inches
D) 51.78 inches
95) Write an iterated triple integral in the order dy dz dx for the volume of the region in the first
95)
octant enclosed by the cylinder x2 + z 2 = 36 and the plane y = 7. 36 - x2
6
A) 0
0
36 - x2 0
dy dz dx
0 7
dy dz dx
0
96) Find the mass of the region of density (x, y, z) =
7
36 - z 2 0
dy dz dx
0
0 6
D)
0
36 - z 2
6
B)
0
0 6
C)
z
7-z
dy dz dx
0
1 bounded by the paraboloid 36 - x2 - y2
z = 36 - x2 - y2 and the xy-plane. A) 1296 B) 216
C) 36
D) 6
Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries. 97) z = 6x2 y; 0 x 4, 0 y 3
A) 576
B) 1256
96)
C) 2256
97)
D) 676
Evaluate the double integral over the given region. sin 5x dA, R: 0 x
98) R
A)
5
,0 y
B)
98)
10
C) 2
D)
C) 25 12
D) 25 4
5
Find the area of the region specified in polar coordinates. 99) the region enclosed by the curve r = 5 cos 3 A) 25 B) 25 2 6
5
99)
Solve the problem.
100) Find the center of mass of the hemisphere z = 25 - x2 - y2 if the density is proportional to the distance from the center.
A) 0, 0, 2
B) 0, 0, 7
C) 0, 0, 8
3
3
17
D) 0, 0, 3
100)
101) Find the moment of inertia Iy of a tetrahedron of constant density (x, y, z) = x + y + z bounded by the coordinate planes and the plane
x y z + + = 1. 7 10 4
A) 582,321,210,363,449
B) 5,240,890,893,271,041
C) 5,240,890,893,271,041
D) 582,321,210,363,449
343,597,383,680
2,199,023,255,552
3,229,815,406,592
Evaluate the integral. /2 9
/3
102)
0
3 sin
101)
206,158,430,208
cos
d d d
102)
8 0 3,387,870,203,084,813 A) 8,796,093,022,208
B) 3,387,870,203,084,813
C) 254,090,265,231,361
D) 254,090,265,231,361
17,592,186,044,416
1,099,511,627,776
2,199,023,255,552
Use the given transformation to evaluate the integral. 103) x = 10u, y = 4v, z = 9w;
103)
x2 y2 dx dy dz, R where R is the interior of the ellipsoid
A) 18
x2 y2 z 2 + + =1 100 16 81
C) 72
B) 24
D) 96
5
7
Solve the problem.
104) Find the centroid of the region cut from the fourth quadrant by the circle x2 + y2 = 49. A) x = - 3 , y = 3
B) x = 3 , y = - 3
C) x = 7 , y = - 7
D) x = 28 , y = - 28
28
3
28
28
3
Evaluate the integral. 9 7 105) (8x + 10y) dx dy 0 0 A) 511
3
104)
28
3
105) B) 4599
C) 657
Find the area of the region specified by the integral(s). 2 x(x - 2)(x - 4) 106) dy dx 0 x(x - 2) A) 12 B) 52 3
D) 73
106) C) 16 3
18
D) 76 3
Find the average value of the function over the region. 107) f(r, , z) = r over the region bounded by the cylinder r = 5 between the planes z = -8 and z = 8 A) 15 B) 10 C) 5 D) 4000 2 3 3 Find the average value of the function f over the given region. 108) f(x, y) = 1 over the region bounded by y = 1 , y = 2 , x = 3, and x = 5. xy x x
A) 5 ln 2
B) 3 ln 2
3
C) ln 2
5
107)
108) D) 15 ln 2
Solve the problem. 109) Evaluate
109) sin x dx x
0 by integrating
e-xy sin x dA. 0
A)
0
B)
3
C)
2
6
D)
4
110) Find the average height of the paraboloid z = x2 + y2 above the annular region 4 x2 + y2 36 in the xy-plane. A) 56
B) 22
C) 38
110)
D) 20
111) Find the moment of inertia about the z-axis of a thick-walled right circular cylinder bounded on
111)
the inside by the cylinder r = 1, on the outside by the cylinder r = 2, and on the top and bottom by the planes z = 8 and z = 12. A) 15 B) 30 C) 32 D) 60
Evaluate the integral. 3 y
112)
1
0 364 A) 9
x2 y2 dx dy
112) B) 350
C) 364
3
3
19
D) 350 9
Use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. 1 113) F(x, y, z) = 113) over the rectangular cube 0 x, y, z 1 3/2 5/2 2 7/2 (x2 + 4) (y2 + 4) (z + 4)
A)
11,005,853,696 1,073,741,823,999,999
5
B)
9,630,121,984 1,288,490,188,799,999
5
C)
19,260,243,968 1,288,490,188,799,999
5
D)
9,630,121,984 1,207,959,551,999,999
5
Solve the problem. 114) Find the moment of inertia about the x-axis of a thin plane of constant density x y the coordinate axes and the line + = 1. 8 10
A) 12,500
B) 10,000
3
Evaluate the integral. 10 8
115)
4
C) 6400
3
= 5 bounded by
D) 8000
3
3
8r
r d dr dz 6 0 A) 2,603,643,534,573,563 549,755,813,888
114)
115) B) 5,858,197,952,790,516 206,158,430,208
C) 176,288,364,320,085
D) 2,603,643,534,573,563
8,589,934,592
206,158,430,208
Write an equivalent double integral with the order of integration reversed. 8 x 116) dy dx 0 0 8 y 8 8 A) B) dx dy dx dy 0 8 0 y 8 y x 8 C) D) dx dy dx dy 0 0 0 -8
116)
Solve the problem. 117) Find the center of mass of the region of constant density bounded from above by the sphere x2 + y2 + z 2 = 100 and from below by the cone z = x2 + y2 .
A) x = 0, y = 0, z = 15 (2 + 2)
B) x = 0, y = 0, z = 15 (2 + 2)
C) x = 0, y = 0, z = 25 (2 + 2)
D) x = 0, y = 0, z = 35 (2 + 2)
4
8
8
8
20
117)
Reverse the order of integration and then evaluate the integral. 576 8 sin x2 118) dx dy x 0 y/9 9 A) 1 - cos 64 B) 9 (1 - cos 64) C) 9 (1 - cos 64) 2 8 4 2
118) D) 9 1 - cos 64 4
8
Evaluate the integral by changing the order of integration in an appropriate way. 2 28 7 tan x 119) dx dy dz xz 1 0 y/4 A) - 4 ln 4 ln cos 7 B) - 4 ln 2 ln cos 7
C) 4 ln 4 ln cos 7
119)
D) 4 ln 2 ln cos 7
Find the average value of the function f over the given region.
120) f(x, y) = sin 2(x + y) over the square 0 x
4
,0 y
4
B) 6
A) 4 2
.
120)
C) 8
2
2
D) 2
Provide an appropriate response.
121) What does the graph of the equation = A) The xz-plane Solve the problem. 122) Rewrite the integral 8 7(1 - z/8)
B) 0
C) 0
D) 0
6
6
0
0
0
0
7(1 - x/6)
7(1 - z/8)
7(1 - x/6)
121) C) The xy-plane
D) The z-axis
122) 6(1 - y/7 - z/8)
dx dy dz
A)
8
look like?
B) The y-axis
0 0 0 in the order dz dy dx. 8 7(1 - z/8) 6(1 - y/7 - z/8)
0
2
0
0
0
6(1 - x/6 - y/7)
8(1 - y/7 - z/8)
8(1 - x/6 - y/7)
dz dy dx
dz dy dx
dz dy dx
dz dy dx
0
21
123) Let D be the smaller cap cut from a solid ball of radius 10 units by a plane 3 units from the center
123)
of the sphere. Set up the triple integral for the volume of D in spherical coordinates. 2 tan-1 ( 91/10) 10 2 sin d d d A) 0 3 sec 0 2 tan-1 ( 91/10) 10 2 sin d d d B) 0 sec 0 -1 2 tan ( 91/3) 10 2 sin d d d C) 0 sec 0 2 tan-1 ( 91/3) 10 2 sin d d d D) 0 3 sec 0
Find the average value of F(x, y, z) over the given region. 124) F(x, y, z) = x6 y7 z 5 over the cube in the first octant bounded by the coordinate planes and the planes x = 1, y = 1, z = 1 A) 1 107
B) 1
C) 1
336
D) 1
210
317
Write an equivalent double integral with the order of integration reversed. 1 tan-1 x 125) dy dx 0 0 /2 /4 /4 1 A) B) dx dy dx dy -1 -1 0 0 tan y tan y /4
C) 0
1
dx dy
/4
D)
tan y
124)
0
/2
125)
dx dy
tan y
Solve the problem.
126) Find the average distance from a point P(x, y) in the first two quadrants of the disk x2 + y2 4 to the origin. A) 4 3
B) 1
D) 2
C) 1
2
3
Use the given transformation to evaluate the integral. 127) u = y - x, v = y + x; y- x cosh dx dy, y+ x
127)
R where R is the trapezoid with vertices at (3, 0), (10, 0), (0, 3), (0, 10) 91(e2 - 1) 91(e2 - 1) 91(e2 - 1)
A)
2e
B)
126)
C)
4e
22
3e
2
D) 91(e - 1) 6e
Evaluate the cylindrical coordinate integral. 1/r /2 8 128) z cos dz r dr d 2 0 4 1/r 128 ln 2 - 3 A) B) 64 ln 2 - 3 256 256
128) C) 64 ln 2 - 6 256
Find the area of the region specified in polar coordinates. 129) the region enclosed by the curve r = 5 sin 2 A) 25 B) 25 8 2
D) 128 ln 2 - 6 256
129) C) 25 4
D) 25 6
Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries. 130) z = x ; 0 x 1, 1 y e y
A) 1
B) 1
3
C) 1
4
2
D) 1 6
Solve the problem. 131) Evaluate
0
A)
131) 0
1
1 dx dy. 7 2 (1 + x + y2 )
B)
6
Evaluate the integral. 4 ln x
132)
C)
24
32
D)
8
ey dy dx
132)
0 A) 3.518437209e+15 2.814749767e+14
B) 9
C) 3.518437209e+15
D) 9
2
5.629499534e+14
Find the Jacobian
130)
4
(x, y) (x, y, z) or (as appropriate) using the given equations. (u, v) (u, v, w)
133) x = 5u cos 3v, y = 5u sin 3v, z = 6w A) 450v B) 270v
133) C) 450u
23
D) 270u
Solve the problem. 134) Evaluate 2
134) 4 - y2
49 - x2 - y2
dz dx dy
-2 - 4 - y2 0 by transforming to cylindrical or spherical coordinates. A) 4 49 - (45)3/2 B) 4 343 - (45)3/2 3 3
C) 2 343 - (45)3/2
D) 2 49 - (45)3/2
3
3
Find the volume of the indicated region. 135) the region that lies under the plane z = 4x + 2y and over the triangle with vertices at (1, 1), (2, 1), and (1, 2) A) 16 B) 4 C) 20 D) 8 3 3
135)
Use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. 136) F(x, y, z) = x2 + y2 + z 2 over the tetrahedron bounded by the coordinate planes and the plane 136) x y z + + =1 7 2 6
A) 1869
B) 623
16
C) 623
6
5
D) 2492 25
Use the given transformation to evaluate the integral. 137) x = 7u, y = 9v, z = 4w;
137)
x2 y2 z 2 dx dy dz, R where R is the interior of the ellipsoid
A) 8
15
x2 y2 z 2 + + =1 49 81 16
B) 16
C) 24
15
5
D) 48 5
Use a spherical coordinate integral to find the volume of the given solid. 138) the solid bounded below by the hemisphere = 1, z 0, and above by the cardioid of revolution = 8 + 4 cos A) 1.142392581e+15 B) 1.142392581e+15 2.748779069e+12 2.199023256e+12
C) 3.807975271e+14
D) 1.26932509e+14
5.497558139e+11
2.748779069e+11
24
138)
Solve the problem. 139) Evaluate
139) 2 2 e-2x - e-9x dx x
0 by writing the integrand as an integral. A) 1 ln 2 B) ln 9 2 9 2
C) ln 2
D) 1 ln 9
9
2
2
140) Find the average height of the paraboloid z = 8x2 + 2y2 above the annular region 25 x2 + y2 64 in the xy-plane. A) 801 2
C) 445
B) 267
140)
D) 445
3
2
Evaluate the integral. 49 - y2
7
141) 0
7x + 14y
dz dx dy
141)
0
0 A) 343
B) 49
C) 2401
D) 16,807
Evaluate the integral by changing the order of integration in an appropriate way. /4 21 7 tan x tan y 142) dy dz dx y 0 0 z/3 A) 3 ln 2 ln cos 7 B) 3 ln 2 ln cos 7 2
D) - 3 ln 2 ln cos 7
C) - 3 ln 2 ln cos 7 Evaluate the integral.
143) 0
0 0 A) 32 2 3
4 sin
2 sin
142)
2
d d d
143)
B) 8 2
C) 64 2
D) 64 2
C) 49 3 + 2 2 3
D) 49 3 + 4 3
9
Find the area of the region specified in polar coordinates. 144) the region inside r = 14 sin and outside r = 7 A) 49 3 + B) 49 3 + 2 2 3 4 3
3
144)
Find the volume of the indicated region.
145) the region inside the solid sphere A) 125 2
5 that lies between the cones
B) 125
C) 125
4
3
25
=
3
and
=
145)
2
D) 125 6
Find the average value of the function f over the given region. 146) f(x, y) = 2x + 7y over the rectangle 0 x 2, 0 y 9. A) 11 B) 67 2 2
146) C) 13
D) 2
2
Solve the problem. 147) Evaluate
147) 1
-
A)
8 (1 + x2 + y2 )
-
dx dy.
B)
9
C)
7
D)
16
8
148) Find the centroid of the region bounded above by the sphere x2 + y2 + z 2 = 64 and below by the plane z = 3 if the density is constant. A) x = 0, y = 0, z = 3.361033439e+14 7.036874418e+13
B) x = 0, y = 0, z = 3.361033439e+14
C) x = 0, y = 0, z = 2.381542186e+15
D) x = 0, y = 0, z = 5.172102697e+15
148)
8.796093022e+13
1.671257674e+14
6.685030697e+14
Find the volume of the indicated region.
149) the region in the first octant bounded by the coordinate planes and the surface z = 36 - x2 - y A) 2,849,934,139,195,391
B) 2,849,934,139,195,391
C) 8,549,802,417,586,173
D) 2,849,934,139,195,391
1,649,267,441,664
1,374,389,534,720
4,398,046,511,104
1,099,511,627,776
Solve the problem. 150) Evaluate
150) 1
0
A)
5 (3 + x2 + y2 )
0
dx dy.
B)
486
C)
324
D)
1296
Write an equivalent double integral with the order of integration reversed. 9 3 151) 4y dx dy 2 0 y 9
A) 0
x
0
4y dy dx
3
B) 0
0 3
C)
149)
x
4y dy dx
D) 0
0
26
151) x
4y dy dx
3 9
x 3
1944
4y dy dx
Integrate the function f over the given region. 152) f(x, y) = 1 over the region bounded by the x-axis, line x = 7, and curve y = ln x ln x
A) 1
B) 7
Evaluate the integral. 3 7
153)
0
153)
0
6 3
D) 6
(sin x + cos y) dx dy
A) 3
154)
C) 8
152)
B) 6
7
C) 7
D) 4
rz dz d dr
154)
4 0 7,592,402,667,700,225 3 A) 3,298,534,883,328
B) 8,282,621,092,036,609 3
C) 4,831,528,970,354,689 3
D) 8,282,621,092,036,609 3
13,194,139,533,312
3,298,534,883,328
16,492,674,416,640
Use the given transformation to evaluate the integral. 155) u = x + y, v = -2x + y;
155)
-5x dx dy, R where R is the parallelogram bounded by the lines y = -x + 1, y = -x + 4, y = 2x + 2, y = 2x + 5 A) 10 B) -10 C) -5 D) 5
Solve the problem. 156) Evaluate
156) (9x + y - z) dV
R where R is the parallelepiped enclosed by the planes 9x + y - z = 1, 9x + y - z = 3, -x + 2y + z = -2, -x + 2y + z = 5, x - y + 9z = -1, x - y + 9z = 6. A) 225 B) 14 C) 108 D) 88 182 13 91 91
Find the area of the region specified by the integral(s). 7 7- x 9 9 157) dy dx 0 0 A) 9 B) 7 2 2
157) C) 63 2
27
D) 42
Evaluate the improper integral. e-(5x + 7y) dy dx
158) 0
0 A) 1 70
158) B)
C) 1
70
D)
35
Evaluate the spherical coordinate integral. /2 /2 5 159) ( sin )2 d d d 0 0 4 61 A) B) 61 18 24
35
159) C) 61 2
D) 61 2
24
18
Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. 160) The curve y = ex and the lines x + y = 4 and x = 4 ex
4
A) 0
C)
x-4 4 ex - x
0
4
Solve the problem. 161) Write 5
y
5
0
dy dx
B) 0
dy dx
ex - x
4 x 4
D) 0
ex
160)
dy dx
dy dx
4-x
161)
0 10 - y
10
f(x, y) dx dy + 5
10 - y
5
f(x, y) dx dy +
0
0
5
f(x, y) dx dy +
y
f(x, y) dx dy
0 5 as a single iterated integral with the order of integration reversed. 10 10 - x 10 10 - x A) B) f(x, y) dy dx f(x, y) dy dx 0 5 5 5 10 10 - x 10 10 - x C) D) f(x, y) dy dx f(x, y) dy dx 0 0 5 0
Evaluate the improper integral. 102 101 dx dy 162) (x - 1)(y - 2) 2 1 A) 600 B) 400
162) C) 300
28
D) 500
Use a spherical coordinate integral to find the volume of the given solid. 163) the solid between the sphere = cos and the hemisphere = 2, z 0 A) 31 B) 31 C) 31 2 3 6
163) D) 31 4
Solve the problem. 164) What domain D in space maximizes the value of the integral x2 y2 z2 + + - 1 dV? 36 100 81 D
164)
2
2
2
36
100
81
A) D = the boundary and interior of the ellipsoid x + y + z = 1. 2
2
2
36
100
81
B) D = the boundary of the ellipsoid x + y + z = 1. C) D = 3 D) No such minimum domain exists. Evaluate the integral. 8 5 165) (4x2 y + 6xy) dy dx 0 0 A) 4100 B) 820 3 3
165) C) 32800
D) 6560
3
3
Provide an appropriate response. 166) What form do planes perpendicular to the z-axis have in spherical coordinates? A) = a cos B) = a sec C) = a csc D)
166) = a sin
Find the volume of the indicated region.
167) the region that lies inside the sphere x2 + y2 + z2 = 49 and outside the cylinder x2 + y2 = 4 A)
2(343 - 453/2)
B)
3
C) 3(343 - 45 2
3/2)
167)
4(343 - 453/2) 3
D) 5(343 - 45 2
3/2)
Use the given transformation to evaluate the integral. 168) u = -8x + y, v = 7x + y;
168)
(y - 8x) dx dy, R where R is the parallelogram bounded by the lines y = 8x + 6, y = 8x + 8, y = -7x + 2, y = -7x + 6 A) 112 B) 840 C) 56 D) 1680 15 15
29
Solve the problem.
169) Integrate f(x, y) = sin(x2 + y2 ) over the region 0 x2 + y2 49. A) 2 (1 - cos2 7) B) (1 - cos2 7) C) (1 - cos 49)
169) D) 2 (1 - cos 49)
Find the volume of the indicated region.
170) the region bounded by the paraboloid z = 36 - x2 - y2 and the xy-plane A) 216 B) 324 C) 432
170) D) 648
Find the average value of F(x, y, z) over the given region. 171) F(x, y, z) = 2x over the cube in the first octant bounded by the coordinate planes and the planes x = 6, y = 6, z = 6 A) 36 B) 12 C) 6 D) 72
172) F(x, y, z) = x2 y3 z 4 over the rectangular solid in the first octant bounded by the coordinate planes and the planes x = 2, y = 1, z = 3 A) 2,849,934,139,195,393 21,990,232,555,520
B) 2,849,934,139,195,393
C) 2,849,934,139,195,393
D) 2,849,934,139,195,393
171)
172)
65,970,697,666,560
197,912,092,999,680
527,765,581,332,480
Reverse the order of integration and then evaluate the integral. 4 2 173) tan-1 x2 dx dy 0 y/2
173)
A) 2 tan-1 4 - 1 ln 17
B) 4 tan-1 4 - 1 ln 17
C) 2 tan-1 4 - 1 ln 17
D) 4 tan-1 4 - 1 ln 17
2
2
Solve the problem. 174) Find the centroid of a hemispherical shell having outer radius 7 and inner radius 4 if the density varies as the square of the distance from the base. A) x = 0, y = 0, z = 11,704,954,112,704,522 B) x = 0, y = 0, z = 19,508,256,854,507,536 4,338,398,005,297,151 4,338,398,005,297,151
C) x = 0, y = 0, z = 15,606,605,483,606,029
174)
D) x = 0, y = 0, z = 23,409,908,225,409,043
3,253,798,503,972,863
4,338,398,005,297,151
Find the volume of the indicated region.
175) the solid cut from the first octant by the surface z = 49 - x2 - y A) 1.539957661e+15
B) 1.539957661e+15
C) 1.539957661e+15
D) 1.539957661e+15
4.123168604e+11
2.748779069e+11
2.061584302e+11
3.435973837e+11
30
175)
Evaluate the spherical coordinate integral. 2 /4 4 176) ( 3 cos ) 2 sin 0 0 0 A) 1024 3
d d d
176) B) 2,137,450,604,396,543 17,179,869,184
C) 4,503,599,627,370,485
D) 8,589,934,591,999,980
309,237,645,312
4,831,838,208
Reverse the order of integration and then evaluate the integral. 1 7 2 177) x4 ex y dx dy 0 7y
A) 7e49 - 350
B) e49 - 1
177) 49 C) 7e - 350 3
Use a spherical coordinate integral to find the volume of the given solid. 178) the solid enclosed by the cardioid of revolution = 6 + 2 cos A) 320 B) 640 C) 80 Solve the problem. 179) Set up the triple integral for the volume of the sphere 7 49 - x2 49 - x2 - y2 A) dz dy dx -7 - 49 - x 2 - 49 - x2 - y2
0
C) 8
D) 8
49 - x2
7
B)
49 - x2 - y2
3
178) D) 160
= 7 in rectangular coordinates.
179)
dz dy dx
- 49 - x 2 - 49 - x2 - y2 7 49 - x2 49 - x2 - y2 -7 - 49 - x2 0 7 49 - x2 49 - x2 - y2 -7 0
D) e49 - 350
dz dy dx
dz dy dx
0
Find the average value of F(x, y, z) over the given region. 180) F(x, y, z) = 6x + 7y + 4z over the rectangular solid in the first octant bounded by the coordinate planes and the planes x = 8, y = 4, z = 2 A) 42 B) 28 C) 56 D) 70 Solve the problem. 181) Find the moment of inertia IL of the rectangular solid of density (x, y, z) = 1 defined by 0 x 10, 0 y 2, 0 z 5, where L is the line through the points (1, 1, 0), (1, 2, 0). A) 11800 B) 9800 C) 3600 3 3
31
D) 6800 3
180)
181)
Evaluate the improper integral. 8 1 182) dy dx 2 1 0 x(y + 1) A) ln 8 3
182) B) (ln 8)
C) ln 8
2
D) (ln 8)
2
3
Solve the problem. 183) Let D be the region bounded below by the xy-plane, on the side by the cylinder r = 8 cos , and on top by the paraboloid z = 10r2 . Set up the triple integral in cylindrical coordinates that gives the volume of D using the order of integration dz dr d . 2 8 cos /4 8 cos 10r2 A) B) r dz dr d 0 0 0 0 0 2 /2 8 cos /2 8 cos 10r C) D) r dz dr d - /2 0 0 0 0
Evaluate the cylindrical coordinate integral. 7 10 3r 184) dz r dr d 0 0 r 7,215,545,057,280,005 A) 103,079,215,104
10r2 0
10r2
r dz dr d
r dz dr d
0
184) B) 1400 3
C) 1,202,590,842,880,001
D) 14,000
25,769,803,776
3
Find the average value of the function f over the given region. 185) f(x, y) = e6x over the square 0 x 1 , 0 y 1 . 6 6
A) e - 1
185) C) 2e - 1
B) e - 1
6
183)
36
D) 2e - 1
Use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. 186) F(x, y, z) = (x + y + z)2 over the tetrahedron bounded by the coordinate planes and the plane 186) x y z + + =1 6 4 9
A) 6669 Evaluate the integral. 1 v10
187)
0
0
A) 2
12
C) 17,784
B) 741
8
25
D) 4446 5
v du dv
187) B) 1
C) 2
11
11
32
D) 1
12
Reverse the order of integration and then evaluate the integral. 28 4 cos x 188) dx dy x 0 y/7 A) 7 sin 4 B) 7 cos 4 C) 4 cos 7 Solve the problem. 189) Write 8
188) D) 4 sin 7
189) y
f(x, y) dx dy +
16
0 0 8 as a single iterated integral. 8 16 - x A) f(x, y) dy dx -8 8 8 16 - x C) f(x, y) dy dx 0 x
16 - y
f(x, y) dx dy
0
8
B) D)
16 - x
-8 x 8 16 - x 0
f(x, y) dy dx
f(x, y) dy dx
8
190) Find the mass of a thin infinite region in the first quadrant bounded by the coordinate axes and the curve y = e-9x if (x, y) = xy. A) 1 B) 1 324 243
C) 1
D) 2
648
243
Reverse the order of integration and then evaluate the integral. 21 7 2 191) ex dx dy 0 y/3 3 49 A) e B) 3 e49 C) 3 (e49 - 1) 4 2 4 Evaluate the cylindrical coordinate integral. 8 9 r 192) (6r2 + 8z 2 ) dz r dr d 0 0 0 A) 2,197,983,873,466,369 2,147,483,648
190)
191) D) 3 (e49 - 1) 2
192) B) 2,197,983,873,466,369 2,684,354,560
C) 2,197,983,873,466,369
D) 6,593,951,620,399,107
1,610,612,736
5,368,709,120
Find the average value of the function f over the given region. 193) f(x, y) = 7x + 10y over the region bounded by the coordinate axes and the lines x + y = 8 and x + y = 10. A) 1037 B) 2074 C) 1037 D) 1037 36 27 18 27
33
193)
Write an equivalent double integral with the order of integration reversed. 3 ln x 194) 8x dy dx 1 0 3 ln 3 ln 3 3 A) B) 24x dx dy 8x dx dy y 0 0 1 e 3 ln 3 ln 3 3 C) D) 8x dx dy 24x dx dy y 0 0 1 e
194)
Use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. 195) F(x, y, z) = (1 + x + y + z)2 over the rectangular cube 0 x, y, z 10 195)
A) 2,499,670,966,272,003
B) 4,741,643,894,784,005
C) 4,827,543,240,704,005
D) 2,456,721,293,312,003
8,589,934,592
17,179,869,184
17,179,869,184
8,589,934,592
Evaluate the integral. 9 9 10r
196)
5
r d dz dr 4 0 A) 1,037,664,098,713,601 137,438,953,472
196) B) 1,037,664,098,713,601 103,079,215,104
C) 1,037,664,098,713,601
D) 1,037,664,098,713,601
206,158,430,208
274,877,906,944
Find the area of the region specified by the integral(s). 4 ex 197) dy dx 0 4-x A) e4 - 9 B) e4 - 19 3
197) C) e4 - 35
D) e4 - 5
3
Find the volume of the indicated region.
198) the tetrahedron bounded by the coordinate planes and the plane x + y + z = 1 7
A) 63 2
B) 42
C) 21
34
9
198)
2
D) 63
Solve the problem.
199) Let D be the region bounded below by the xy-plane, above by the sphere x2 + y2 + z 2 = 100, and
199)
on the sides by the cylinder x2 + y2 = 16. Set up the triple integral in cylindrical coordinates that gives the volume of D using the order of integration dz dr d . 2 4 2 100 - z 2 A) B) r dz dr d 0 0 0 0 2
C) 0
16 - z 2
10 0
r dz dr d
0
0
0
2
D)
100 - r2
4 0
16 - r2
10 0
r dz dr d
r dz dr d
0
Evaluate the integral by changing the order of integration in an appropriate way. 1 9 2 3 ey 200) dy dz dx z 0 1 x/9 A) 3 ln 2 (e - 1) B) 3 ln 3 (e - 1) C) 9 ln 2 (e - 1) 2
200) D) 9 ln 3 (e - 1) 2
Solve the problem.
201) Find the average distance from a point P(x, y) in the first quadrant of the disk x2 + y2 25 to the origin. A) 10 3
B) 5
C) 5
2
4
201)
D) 5 3
Use the given transformation to evaluate the integral. 202) u = 2x + y - z, v = -x + y + z, w = -x + y + 2z;
202)
(2x + y - z) dx dy dz, R where R is the parallelepiped bounded by the planes 2x + y - z = 6, 2x + y - z = 7, -x + y + z = 5, -x + y + z = 10, -x + y + 2z = 4, -x + y + 2z = 8 A) 520 B) 390 C) 130 D) 65 3 3
35
Set up the iterated integral for evaluating
f(r, , z) r dz dr d
over the given region D.
D
203) D is the solid right cylinder whose base is the region between the circles r = 2 cos and r = 9 cos ,
203)
and whose top lies in the plane z = 9 - y. 2 9 cos 9 - sin A) f(r, , z) r dz dr d 0 2 cos 0 9 cos 9 - sin B) f(r, , z) r dz dr d 0 2 cos 0 /2 9 cos 9 - r sin C) f(r, , z) r dz dr d 0 - /2 2 cos 2 9 cos 9 - r sin D) f(r, , z) r dz dr d 0 2 cos 0
Provide an appropriate response. 204) True or false? Consider the double integral 2
6
204)
(x2 + y) dy dx.
0 3 The first step in calculating this integral involves holding y constant. A) False B) True
Change the Cartesian integral to an equivalent polar integral, and then evaluate. ln 4 (ln 4)2 - y2 2 2 205) e x + y dx dy 0 0
A) (4 ln 4 - 3)
B) (4 ln 4 + 3)
2
2
Solve the problem.
2
C) (4 ln 4 + 3)
D) (4 ln 4 - 3)
C) 9 8
D) 3 2
4
Find the area of the region specified in polar coordinates. 206) one petal of the rose curve r = 3 sin 2 A) 9 B) 9 2 4
205)
4
206)
2
207) Integrate f(x, y) = ln(x + y ) over the region 0 x2 + y2 25.
207)
x2 + y2
A) 20 (2 ln 5 - 1)
B) 20 (ln 5 - 1)
C) 10 (2 ln 5 - 1)
36
D) 10 (ln 5 - 1)
Evaluate the cylindrical coordinate integral. 3 2 r 5 208) dz r dr d 2 2 0 r A) 25 B) 25 6 2
208) C) 25 4
D) 25 3
Evaluate the double integral over the given region. 7x sin xy dA, R: 0 x
209)
,0 y 1
209)
R
A) 7
B) 7 - 7
C)
Evaluate the integral by changing the order of integration in an appropriate way. 1 6 6 x sin z 210) dz dy dx z 0 0 y A) 1 + cos 6 B) 1 - sin 6 C) 1 + sin 6 2
D)
7
210) D) 1 - cos 6 2
Solve the problem. 211) Set up the triple integral for the volume of the sphere = 9 in spherical coordinates. 2 /2 9 2 /2 9 2 sin d d d A) B) d d d 0 0 0 0 0 0 2 9 2 9 2 sin d d d C) D) d d d 0 0 0 0 0 0 Evaluate the integral. 1 2 5
212)
(x2 + y2 + z 2 ) dx dy dz
-1 0 0 A) 133
211)
212)
B) 200
C) 23.2
Change the Cartesian integral to an equivalent polar integral, and then evaluate. 9 81 - y2 213) ln(x2 + y2 + 1) dx dy -9 - 81 - y2
A) (82 ln 82 - 81) C) (81 ln 82 + 81)
B) (82 ln 82 + 81) D) (81 ln 82 - 81)
37
D) -117
213)
Solve the problem.
214) Let D be the region bounded below by the xy-plane, above by the sphere x2 + y2 + z 2 = 81, and on
214)
the sides by the cylinder x2 + y2 = 64. Set up the triple integral in cylindrical coordinates that gives the volume of D using the order of integration dr dz d . 9 2 2 17 8 81 - z 2 A) + r dr dz d r dr dz d 0 0 0 0 17 0 2
B) 0
0
2
8
2
17 0 9 64 - z 2
+
2
17 0 8 81 - z 2
0 9
17 0
r dr dz d
0
0
0
+ 0
8
17
2
D)
r dr dz d
0
0 2
C)
9
17
r dr dz d
+
0
0
64 - z 2
r dr dz d
r dr dz d
r dr dz d
17 0
Find the average value of F(x, y, z) over the given region. 215) F(x, y, z) = xyz over the cube in the first octant bounded by the coordinate planes and the planes x = 3, y = 3, z = 3 A) 9 B) 9 C) 9 D) 27 2 4 8 Evaluate the integral. 8 6
216)
2
z
r dr d dz z
216)
2 0 4.222124651e+15 A) 7.036874418e+13
B) 1.407374884e+15
C) 4.222124651e+15
D) 1.407374884e+15
3.518437209e+13
3.518437209e+13
1.759218604e+13
Evaluate the spherical coordinate integral. /2 /3 8 sec 4 sin2 cos 217) 0 0 7 sec A) 2,193,663,136,366,587 2 1,374,389,534,720
C) 731,221,045,455,529 412,316,860,416
215)
d d d
217) B) 2,193,663,136,366,587
3
1,374,389,534,720
D) 731,221,045,455,529
2
412,316,860,416
38
3
Solve the problem. 218) Find the centroid of a hemispherical shell having outer radius 4 and inner radius 2 if the density is constant. A) x = 0, y = 0, z = 15 B) x = 0, y = 0, z = 45 14 28
C) x = 0, y = 0, z = 20
D) x = 0, y = 0, z = 10
7
7
Find the average value of F(x, y, z) over the given region. 219) F(x, y, z) = x2 + y2 + z 2 over the cube in the first octant bounded by the coordinate planes and the planes x = 6, y = 6, z = 6 A) 72
Evaluate the integral. 9 9 2r
220)
7
218)
B) 36
C) 6
219)
D) 108
r dz d dr
220)
7 0 A) 848,822,976,643,073 1,649,267,441,664
B) 848,822,976,643,073
C) 848,822,976,643,073
D) 848,822,976,643,073
3,298,534,883,328
1,099,511,627,776
2,199,023,255,552
Solve the problem.
221) Find the centroid of the region enclosed between the cone with equation z = 9 x2 + y2 and the 4
plane with equation z = 9. A) (x, y, z) = 0, 0, 36 5
B) (x, y, z) = 0, 0, 6
C) (x, y, z) = 0, 0, 27
D) (x, y, z) = 0, 0, 45
4
Evaluate the integral. /2 /2
222)
0
0 0 125 2 A) 24
5 cos
8
2 sin
d d d
222)
B) 125
C) 125 2
18
18
D) 125 24
Solve the problem. 223) Find the moment of inertia Iz of a tetrahedron of constant density (x, y, z) = 1 bounded by the coordinate planes and the plane
A) 876 5
221)
x y z + + = 1. 6 3 8
B) 1308
C) 108
5
39
D) 240
223)
Evaluate the integral. 6 9 224) 2x dy dx -8 4
224)
A) - 140
B) - 8.620171162e+15
C) 910
D) - 3.078632558e+15
8.796093022e+12 8.796093022e+12
Find the volume of the indicated region.
225) the region bounded by the paraboloid z = x2 + y2, the cylinder x2 + y2 = 4, and the xy-plane B) 16
A) 4
D) 8
C) 8
3
225)
3
Use the given transformation to evaluate the integral. 226) u = 2x + y - z, v = -x + y + z, w = -x + y + 2z;
226)
(2x + y - z)(z - x + y) dx dy dz, R where R is the parallelepiped bounded by the planes 2x + y - z = 5, 2x + y - z = 10, -x + y + z = 2, -x + y + z = 4, -x + y + 2z = 5, -x + y + 2z = 8 A) 4050 B) 225 C) 2025 D) 450
Find the volume of the indicated region.
227) the region enclosed by the paraboloids z = x2 + y2 - 7 and z = 193 - x2 - y2 A) 30,000 B) 20,000 C) 40,000
Solve the problem. 228) Find the moment of inertia about the origin of a thin plane of constant density the coordinate axes and the line x + y = 6. A) 18 B) 972 C) 648
229) Solve for a:
6
x2
a 0
A) a = 5
= 3 bounded by
228)
D) 108 229)
48a 0
227) D) 10,000
0
dy dx dz = 625
B) a = 5
C) a = 5
3
4
40
D) a = 5 2
Set up the iterated integral for evaluating
f(r, , z) r dz dr d
over the given region D.
D
230) D is the solid right cylinder whose base is the region in the xy-plane that lies inside the cardioid r = 8 + 4 cos 2
A)
0
B) 0
C) 0
D)
2
0
230)
and outside the circle r = 4, and whose top lies in the plane z = 10. 8 + 4 cos 10 f(r, , z) dz r dr d 0 0 8 + 4 cos 10 f(r, , z) dz r dr d 0 0 8 + 4 cos 10 f(r, , z) dz r dr d 4 0 8 + 4 cos 10 f(r, , z) dz r dr d 4 0
Evaluate the spherical coordinate integral. /2 /3 6 sec 3 sin cos 231) 0 0 4 sec 2,858,730,232,217,599 A) 4,398,046,511,104
d d d
231) B) 2,858,730,232,217,599 8,796,093,022,208
C) 6,860,952,557,322,237
D) 6,860,952,557,322,237
35,184,372,088,832
17,592,186,044,416
Find the volume of the indicated region.
232) the region common to the interiors of the cylinders x2 + y2 = 81 and x2 + z 2 = 81 A) 2916
B) 6.412351813e+15 1.649267442e+12
C) 6.412351813e+15
D) 6.412351813e+15
1.319413953e+13
6.597069767e+12
Find the area of the region specified by the integral(s). 2 8-x 8 8-x 233) dy dx + dy dx 0 2-x 2 0 A) 30 B) 32
233) C) 34
D) 2
Change the Cartesian integral to an equivalent polar integral, and then evaluate. 15 225 - y2 234) (x2 + y2 ) dx dy 0 0
A) 50,625 8
232)
B) 3375
C) 225
4
8
41
234) D) 3375 8
Solve the problem. 235) Find the center of mass of a tetrahedron of constant density bounded by the coordinate planes and x y z the plane + + = 1. 6 2 8
A) x = 2, y = 2 , z = 8
B) x = 3, y = 1, z = 4
C) x = 3 , y = 1 , z = 2
D) x = 4, y = 4 , z = 16
3
2
3
2
3
3
236) Find the center of mass of the thin semicircular region of constant density = 4 bounded by the x-axis and the curve y = A) x = 0, y = 44 3
237) Solve for a: a
0
B) x = 0, y = 22
C) x = 0, y = 88
3
3
D) x = 0, y = 11 3
10(1 - x/a - y/9)
dz dy dx = 15
0
C) a = 5
B) a = 1
6
D) a = 2 3
Evaluate the improper integral. 82 29 dy dx
238)
1
2
236)
237)
9(1 - x/a)
0 A) a = 6 5
121 - x2 .
235)
238)
(x - 1) (y - 2)2/3
A) 108
B) 162
C) 135
D) 81
Use a spherical coordinate integral to find the volume of the given solid. 239) the solid between the spheres = cos and = 7 A) 1.005686636e+15 B) 1.005686636e+15 2.199023256e+12 4.398046511e+12
C) 3.015960395e+15
239)
D) 3.014860883e+15
6.597069767e+12
1.319413953e+13
Find the average value of F(x, y, z) over the given region. 240) F(x, y, z) = 4x over the rectangular solid in the first octant bounded by the coordinate planes and the planes x = 3, y = 2, z = 4 A) 24 B) 72 C) 18 D) 6
240)
Find the volume of the indicated region.
241) the region enclosed by the cylinder x2 + y2 = 100 and the planes z = 0 and x + y + z = 20 A) 2000 B) 4000 C) 1000 D) 3000
42
241)
Solve the problem. 242) Find the centroid of the region cut from the first quadrant by the line x + y = 6. A) x = 3, y = 3 B) x = 4, y = 4 C) x = 3 , y = 3 D) x = 2, y = 2 2 2 Find the volume of the indicated region. 243) the region bounded below by the xy-plane, laterally by the cylinder r = 7 sin , and above by the plane z = 10 - x A) 175 B) 35 C) 245 D) 1225 2 2
244) the region bounded by the paraboloid z = x2 + y2 and the cylinder x2 + y2 = 16 B) 256
A) 384
C) 128
3
242)
243)
244) D) 1024 3
Solve the problem. 245) Given that
245)
2 e-x dx = 0 evaluate
2
,
2 e-8x dx. 0 A) 1 2
8
B) 8
C)
D) 1
8
2
8
Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. 246) The lines x + y = 1, x + y = 1, and x = 0 3 10 10 3 30/13
A)
3 - 3x/10
dy dx
B)
0
10 - 10x/3 13 3 - 3x/10 dy dx 0 10 - 10x/3
C)
13
246)
10 - 10x/3
dy dx 3 - 3x/10 30/13 10 - 10x/3 dy dx 0 3 - 3x/10 0
D)
Evaluate the integral. e-(2x + 10y + 5z) dz dx dy
247) 0
0
A) 3
100
0
247)
B) 1
C) 1
100
200
43
D) 2
25
1
248) 0
y
ex + y dx dy
0 1 2 A) (e - e)2 e
Solve the problem. 249) Evaluate
248) B) 1 (e2 - e)2
C) 1 (e - 1)2
2
D) 1 (e - 1)2
2
3
249) x x tan-1 - tan-1 6 10 x
dx
0 by writing the integrand as an integral. A) ln 5 B) ln 5 4 6 2 3
C)
Evaluate the spherical coordinate integral. 5 (1 - cos )/2 2 sin d d d 250) 0 0 0 A) 5 B) 5 3 2 Find the volume of the indicated region.
4
ln
5 3
D) ln 3 5
250) C) 5
D) 5
4
2
2
64
100
6
251) the region bounded by the paraboloid z = 1 - x - y and the xy-plane B) 80
A) 320
C) 400
3
251) D) 40
Solve the problem.
252) Find the center of mass of the solid enclosed between the cone with equation z = 1 x2 + y2 and 3
252)
the plane with equation z = 1 if the density at any point is proportional to the distance from that point to the plane z = 1. A) (x, y, z) = 0, 0, 2 B) (x, y, z) = 0, 0, 3 3 5
C) (x, y, z) = 0, 0, 7
D) (x, y, z) = 0, 0, 5
12
8
Change the Cartesian integral to an equivalent polar integral, and then evaluate. 0 0 1 253) dy dx 2 + y2 1 x + -5 - 25 - x2
A) (5 + ln 6) 2
B) (5 + ln 6)
C) (5 - ln 6)
4
4
44
253) D) (5 - ln 6) 2
Solve the problem. 254) Find the moment of inertia about the y-axis of the thin semicircular region of constant density = 9 bounded by the x-axis and the curve y = 16 - x2 .
A) 768
B) 576
C) 288
254)
D) 384
255) Let D be the region bounded below by the xy-plane, above by the sphere x2 + y2 + z 2 = 100, and
255)
on the sides by the cylinder x2 + y2 = 9. Set up the triple integral in cylindrical coordinates that gives the volume of D using the order of integration d dz dr. 2 10 3 2 100 - r2 100 - r2 A) B) r d dz dr 0 0 0 0 0 0 100 - r2
2
C) 0
0
3
r d dz dr
0
9 - r2
10
D) 0
0
2
r d dz dr
r d dz dr
0
256) Find the moment of inertia about the y-axis of the thin infinite region of constant density = 3 in the first quadrant bounded by the coordinate axes and the curve y = e-9x. A) 2 B) 1 C) 1 243 486 729
Sketch D and g(D) from the description of D and change of variables (x, y) = g(u, v). 257) x = u2 - v2, y = 2uv where D is the rectangle 0 u 1, 0 v 1
A)
45
256)
D) 1
81
257)
B) None of these. C)
D)
Solve the problem. 258) Write an iterated triple integral in the order dx dy dz for the volume of the rectangular solid in the first octant bounded by the planes x = 4, y = 8, and z = 5. 4 8-x 5-y-x 5 8-x 4-y-x A) B) dx dy dz dx dy dz 0 0 0 0 0 0 5 8 4 4 8 5 C) D) dx dy dz dx dy dz 0 0 0 0 0 0 Evaluate the integral by changing the order of integration in an appropriate way. 16 2 2 259) e-(x + z) dx dy dz 0 0 y/8 A) 4 1 - e-6 B) 2 1 - e-6 C) 2 1 - e-4
46
258)
259) D) 4 1 - e-4
Solve the problem. 260) Evaluate
260) 1
-
A)
5 (10 + x2 + y2 )
-
dx dy.
B)
60,000
C)
50,000
Find the area of the region specified in polar coordinates. 261) the region enclosed by the curve r = 7 cos A) 49 B) 49 2 4
D)
40,000
100,000
261) C) 49
D) 49
3
Provide an appropriate response.
262) What does the graph of the equation = A) A line
3
look like?
B) A cylinder
262) C) A plane
D) A cone
Integrate the function f over the given region. 263) f(x, y) = 1 over the square 4 x 10, 4 y 10 xy
A) ln 2
2
263) 2 C) ln 4
B) ln 2 5
5
D) ln 4
ln 10
ln 10
Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. 264) The coordinate axes and the line x + y = 1 8 5 8
A) 0
C) 0
5
x/8 8 8
dy dx
8
B) 0
dx dy
D)
y/5
0
5
dx dy y/8 8 5(1 - x/8)
dy dx
0
Integrate the function f over the given region. 265) f(x, y) = e2x + 3y over the rectangle 0 x 1, 0 y 1 A) 1 (e5 - e3 - e2 + 1) B) 1 (e5 - e3 - e2 - 1) 4 6
C) 1 (e5 - e3 - e2 - 1)
D) 1 (e5 - e3 - e2 + 1)
4
6
47
264)
265)
Solve the problem. 266) Find the moment of inertia Iz of the rectangular solid of density (x, y, z) = xyz defined by 0 x 10, 0 y 3, 0 z 4. A) 6,741,380,667,801,592 68,719,476,736
B) 7,174,313,371,238,392
C) 966,367,641,599,999
D) 3,092,376,453,119,997
68,719,476,736
8,589,934,592
137,438,953,472
Evaluate the cylindrical coordinate integral. 10 4 16 - r2 267) dz r dr d 0 0 - 16 - r2
A) 3200
Find the Jacobian
267) C) 1280
B) 320
3
266)
D) 640
3
3
(x, y) (x, y, z) or (as appropriate) using the given equations. (u, v) (u, v, w)
268) u = 4x + 3y, v = 5x+ 2y A) 7
268) B) - 1
C) 1
7
D) -7
7
Evaluate the integral. sin(3u + 6) du dv dw
269) 0
0 0 2 2 A) cos 6 3
269)
B) 1 cos 6
C) 2 cos 6
3
3
D) 1 2 cos 6 3
Sketch D and g(D) from the description of D and change of variables (x, y) = g(u, v).
270) x = u cos v, y = u sin v where D is the rectangle 1 u 2, 0 v
48
2
270)
A)
B)
C)
D) None of these.
49
Set up the iterated integral for evaluating
f(r, , z) r dz dr d
over the given region D.
D
271) D is the solid right cylinder whose base is the region between the circles r = 3 sin and r = 8 sin ,
271)
and whose top lies in the plane z = 10 - x - y. 2 8 sin 10 - r(cos - sin ) A) f(r, , z) dz r dr d 0 3 sin 0 8 sin 10 - r(cos + sin ) B) f(r, , z) dz r dr d 0 3 sin 0 8 sin 10 - r(cos - sin ) C) f(r, , z) dz r dr d 0 3 sin 0 2 8 sin 10 - r(cos + sin ) D) f(r, , z) dz r dr d 0 3 sin 0
Solve the problem. 272) A solid right circular cylinder is bounded by the cylinder r = 1 and the planes z = 0 and z = 3. Find the moment of inertia about the z-axis if the density is = 2z. A) 9 B) 9 C) 6 D) 9 2 4 Evaluate the integral. 5 9
273)
2
z
r dr dz d z
272)
273)
8 0 1.196268651e+15 2 A) 1.407374884e+14
B) 1.196268651e+15 2
C) 3.588805953e+15 2
D) 3.588805953e+15 2
2.111062325e+14
2.814749767e+14
5.629499534e+14
Integrate the function f over the given region. 274) f(x, y) = xy over the triangular region with vertices (0, 0), (10, 0), and (0, 8) A) 100 B) 10 C) 800 3 3 3
50
274) D) 80 3
Solve the problem. 275) Rewrite the integral 1/5 (1 - 5z)/8
275) (1 - 8y - 5z)/9
dx dy dz
0 0 0 in the order dz dy dx. 1/5 (1 - 9x)/8 (1 - 5x - 8y)/9
A)
0
B) 0
C) 0
D)
1/9
1/9
1/5
0
0
0
0
(1 - 9x)/8
(1 - 9x)/8
(1 - 5z)/8
0
0
0
0
(1 - 8x - 9y)/5
(1 - 9x - 8y)/5
(1 - 8y - 5z)/9
dz dy dx
dz dy dx
dz dy dx
dz dy dx
0
Find the area of the region specified by the integral(s). 2 y 2 2 276) dx dy + dx dy 0 0 0 y A) 8 B) 1
276) C) 2
Provide an appropriate response. 277) True or false? Consider the double integral 2
6
D) 4
277)
(x2 + y) dy dx.
0 3 The first step in calculating this integral involves integrating with respect to x. A) False B) True
51
Set up the iterated integral for evaluating
f(r, , z) r dz dr d
over the given region D.
D
278) D is the rectangular solid whose base is the triangle with vertices at (0, 0), (0, 4), and (4, 4), and
278)
whose top lies in the plane z = 10. /2 4 sec 10 A) f(r, , z) dz r dr d /4 0 0 /2 4 csc 10 B) f(r, , z) dz r dr d 0 0 0 /2 4 sec 10 C) f(r, , z) dz r dr d 0 0 0 /2 4 csc 10 D) f(r, , z) dz r dr d /4 0 0
Find the average value of the function over the region. 279) f( , , ) = over the solid ball 6 A) 6 B) 2 9
279) C) 9 2
D) 6
Solve the problem. 280) Set up the triple integral for the volume of the sphere = 2 in cylindrical coordinates. 2 2 4 - r2 2 2 4 - r2 A) B) r dz dr d dz dr d 0 0 0 0 - 4 - r2 - 4 - r2 2
C)
4 - r2
2
0
0
dz dr d
2
D) 0
0
4 - r2
2 0
280)
r dz dr d
0
Use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. 281) F(x, y, z) = 10x - 6y2 - 10z 3 over the rectangular solid 0 x 4, 0 y 2, 0 z 3 281)
A) - 1776
B) - 1332
C) - 1998
D) - 5328 5
Solve the problem. 282) For what value of a is the volume of the tetrahedron formed by the coordinate planes and the x y z plane + + = 1 equal to 8? a 7 2
A) a = 24 7
B) a = 32
C) a = 16
7
7
52
D) a = 12 7
282)
Find the area of the region specified by the integral(s). /2 2 sin x 283) dy dx /4 2 cos x A) 4 B) 1
283) C) 2
Sketch D and g(D) from the description of D and change of variables (x, y) = g(u, v). 284) x = u + v, y = 2u - v where D is the rectangle 0 u 1, -1 v 1
A) None of these. B)
C)
53
D) 2 2
284)
D)
Change the Cartesian integral to an equivalent polar integral, and then evaluate. 3 9 - x2 1 285) dy dx 2 22 -3 - 9 - x2 (1 + x + y ) A) 9 B) 9 C) 18 19 10 5
285) D) 9 5
Write an equivalent double integral with the order of integration reversed. 9 15 - y 286) dx dy 6 6 3 15 - x 9 15 - x A) B) dy dx dy dx 6 9 6 9 9 15 - x 3 15 - x C) D) dy dx dy dx 6 6 6 6 Solve the problem.
287) Find the center of mass of the region of density (x, y, z) = paraboloid z = 64 - x2 - y2 and the xy-plane. A) x = 0, y = 0, z = 1 2
1 bounded by the 64 - x2 - y2
B) x = 0, y = 0, z = 8
C) x = 0, y = 0, z = 8
D) x = 0, y = 0, z = 4
3
54
286)
287)
Find the volume of the indicated region.
288) the region bounded by the coordinate planes, the parabolic cylinder z = 49 - x2 , and the plane y = 10
A) 3.771324883e+15
B) 2.828493662e+15
C) 3.771324883e+15
D) 2.828493662e+15
5.497558139e+11
5.497558139e+11
1.649267442e+12
Evaluate the integral. 10 7
289)
7
z
288)
1.099511628e+12
r d dz dr z
289)
3 0 3.588805953e+15 A) 3.518437209e+13
B) 1.196268651e+15
C) 1.196268651e+15
D) 3.588805953e+15
1.759218604e+13
3.518437209e+13
7.036874418e+13
Solve the problem. 290) What region in the xy-plane maximizes the value of
290)
(2025 - 25x2 - 81y2 ) dA ? R
2
2
81
25
A) The ellipse x + y = 1
B) The ellipse 25x2 + 81y2 = 1
C) The ellipse 81x2 + 25y2 = 2025
D) The ellipse x + y = 1
Find the area of the region specified by the integral(s). 4y + 12 6 291) dx dy 2 -2 y A) 1.466015504e+15 8.796093022e+12
2
25
81
291) B) 6.034119813e+15 1.319413953e+13
C) 3.002399752e+15
D) 3.002399752e+15
4.398046511e+12
3.518437209e+13
Integrate the function f over the given region. 292) f(x, y) = xy over the rectangle 1 x 7, 8 y 10 A) 432 B) 864
292) C) 576
Evaluate the integral. 10 5 293) (x + y) dx dy 0 0
A) 375
2
D) 288
293) C) 15
B) 1875
2
55
D) 75 2
Use the given transformation to evaluate the integral. 294) u = -4x + y, v = 2x + y;
294)
(y - 4x)(2x + y) dx dy, R where R is the parallelogram bounded by the lines y = 4x + 3, y = 4x + 7, y = -2x + 6, y = -2x + 10 A) 3840 B) 7680 C) 640 D) 320 3 3
Use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. 295) F(x, y, z) = 8x + 3y + 9z over the tetrahedron bounded by the coordinate planes and the plane 295) x+y+z=1 A) 5 B) 20 C) 5 D) 10 6 27 9 9 Use a spherical coordinate integral to find the volume of the given solid. 296) the solid between the spheres = 2 cos and = 6 cos A) 3.659174697e+15 B) 3.659174697e+15 3.518437209e+13 7.036874418e+13
C) 1.219724899e+15
D) 1.219724899e+15
3.518437209e+13
Evaluate the integral. 1 1 1
297)
0
0 0 A) 17 6
296)
1.759218604e+13
(7x + 8y + 2z) dz dy dx
297)
B) 17
C) 17
3
2
D) 43
Use the given transformation to evaluate the integral. 298) x = 7u, y = 6v, z = 9w;
298)
(x2 + y2 + z 2 ) dx dy dz, R where R is the interior of the ellipsoid
A) 1134 5
x2 y2 z 2 + + =1 49 36 81
C) 1512
B) 252
5
D) 756 5
Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries. 299) z = x2 + y2 ; 0 x 1, 0 y 1
A) 1 3
B) 4 3
C) 8 3
56
D) 2 3
299)
Evaluate the double integral over the given region. (x + x2 y4 - 3) dA, R: 0 x 1, 0 y 2
300) R
A) 11
300)
B) 13
15
C) 14
15
15
D) - 43 15
Write an equivalent double integral with the order of integration reversed. /2 sin x 301) (9x + 3y) dy dx 0 0 1 1 sin-1 y sin-1 y A) B) (9x + 3y) dx dy (9x + 3y) dx dy 0 0 0 /2 1 /2 /2 1 C) D) (9x + 3y) dx dy (9x + 3y) dx dy -1 -1 0 0 sin y sin y Evaluate the integral by changing the order of integration in an appropriate way. 4 300 10 sin x2 302) dx dy dz xz 1 0 y/3 3 A) ln 4 (1 - cos 100) B) 3 ln 4 (1 - sin 100) 4 2
C) 3 ln 4 (1 - cos 100)
301)
302)
D) 3 ln 4 (1 - sin 100)
2
4
Solve the problem.
303) Find the average distance from a point P(x, y) in the annular region 25 x2 + y2 64 to the origin. A) 86 13
Evaluate the integral. /2 5
304)
0
B) 418 39
/3
4 sin2
2 0 680,157,892,942,233 A) 17,592,186,044,416
C) 1,133,596,488,237,055 29,686,813,949,952
cos
C) 627 52
303)
D) 387 52
d d d
304) B) 680,157,892,942,233
3
17,592,186,044,416
D) 1,133,596,488,237,055
3
29,686,813,949,952
57
2 2
Solve the problem. 305) Write an iterated triple integral in the order dz dy dx for the volume of the region enclosed by the paraboloids z = 128 - x2 - y2 and z = x2 + y2 . 8
A)
B)
C)
D)
64 - x2
128 - x2 - y2
-8 - 64 - x 2 x2 + y2 8 64 - x2 64 - x2 - y2
305)
dz dy dx
dz dy dx
-8 - 64 - x 2 x2 + y2 8 128 - x2 128 - x2 - y2 -8 - 128 - x2 x2 + y2 8 128 - x2 64 - x2 - y2
dz dy dx
dz dy dx
-8 - 128 - x2 x2 + y2
Find the average value of F(x, y, z) over the given region. 306) F(x, y, z) = 4x + 6y + 8z over the cube in the first octant bounded by the coordinate planes and the planes x = 3, y = 3, z = 3 A) 27 B) 27 C) 36 D) 27 2 4 Integrate the function f over the given region. 307) f(x, y) = x + y over the trapezoidal region bounded by the x-axis, y-axis, line x = 5, and line 5 4 y=-
B) 30
3
Evaluate the integral. ln 10 10 0
307)
4 x+8 5
A) 230
308)
306)
C) 50
ey dx dy
D) 110 3
308)
ey A) 2.849934139e+15 7.036874418e+13
B) 2.849934139e+15
C) 8.514618045e+15
D) 8.514618045e+15
1.407374884e+14
2.814749767e+14
1.407374884e+14
58
Evaluate the cylindrical coordinate integral. 5 5 9r 309) z dz r dr d 4 4 0 A) 1,026,978,220,081,151 412,316,860,416
309) B) 2,824,190,105,223,165 68,719,476,736
C) 1,026,978,220,081,151
D) 3,080,934,660,243,453
274,877,906,944
137,438,953,472
Solve the problem. 310) Find the center of mass of a thin infinite region in the first quadrant bounded by the coordinate axes and the curve y = e-2x if (x, y) = xy.
A) x = 1 , y = 8 2
27
B) x = 1 , y = 8 3
C) x = 1 , y = 2
27
3
9
D) x = 1 , y = 2 2
9
311) Find the mass of a thin triangular plate bounded by the coordinate axes and the line x + y = 6 if (x, y) = x + y. A) 144
B) 36
C) 72
310)
311)
D) 288
Use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. 3/2 1 312) F(x, y, z) = 312) over the rectangular cube 0 x, y, z 1 (x2 + 11)(y2 + 11)(z 2 + 11)
A)
12 17,424
B)
12 191,664
C)
12 2,299,968
D)
12 209,088
Evaluate the improper integral. dx dy 2 (x + 4)(y2 + 64)
313) 0
0
A)
2
96
313) B)
C)
96
2 64
D)
64
Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. 314) The curve y = ln x and the lines x = 2 and y = 0 2 ln x 2 ln y A) B) dx dy dx dy 1 0 1 -2 2 ln x 2 ln x C) D) dy dx dy dx 1 0 1 -2 Evaluate the integral by changing the order of integration in an appropriate way. 3 3 sin z sin x 315) dz dx dy x 0 y 0 A) 2(1 + sin 3) B) 2 sin 3 C) 2 cos 3
59
314)
315) D) 2(1 - cos 3)
Provide an appropriate response. 316) True or false? Consider the double integral 2
6
316)
(x2 + y) dy dx.
0 3 The first step in calculating this integral involves holding x constant. A) True B) False
Solve the problem. 317) Find the center of mass of the rectangular solid of density (x, y, z) = xyz defined by 0 x 3, 0 y 5, 0 z 10. A) x = 1, y = 5 , z = 10 B) x = 3 , y = 5 , z = 5 3 3 4 4 2
C) x = 2, y = 10 , z = 20 3
D) x = 3 , y = 5 , z = 5
3
2
2
318) Find the average distance from a point P(r, ) in the region bounded by r = 4 + 5 sin to the origin. A) 214 57
B) 428 171
C) 107 19
Change the Cartesian integral to an equivalent polar integral, and then evaluate. 5 25 - x2 319) dy dx -5 0
A) 5
B)
2
Evaluate the integral. 3 5(1 - z/3) 0
0
C) 25
2
10(1 - y/5 - z/3)
320)
2
319) D) 125 2
320) C) 75
B) 25
2
D) 75 4
Solve the problem. 321) Given that 2 e-x dx = 0 evaluate
321) 2
,
2 e-6x dx . A) 1 2
6
318)
D) 856 171
dx dy dz
0
A) 50
317)
B) 6
C)
60
6
D) 1 2
6
322) Find the volume of the parallelepiped enclosed by the planes 7x + y - z = 1, 7x + y - z = 3, -x + 3y + z = -2, -x + 3y + z = 5, x - y + 9z = -1, x - y + 9z = 6. A) 21 B) 7 C) 49 208 104 104
322)
D) 7
52
Evaluate the double integral over the given region. ( x+
323)
y) dA, R: 0 x 1, 0 y 1
R
A) 4
323)
B) 1
3
C) 2
3
D) 8
3
3
Find the volume of the indicated region.
324) the region bounded above by the sphere x2 + y2 + z2 = 9 and below by the cone z = x2 + y2 A) 27 (2 4
Find the Jacobian
B) 27 (2 4
2)
3)
C) 9 (2 - 2)
324)
D) 9 (2 - 3)
(x, y) (x, y, z) or (as appropriate) using the given equations. (u, v) (u, v, w)
325) x = -2u + 2, y = 4v - 2, z = -4w + 3 A) 12 B) -384
325) C) -12
D) 32
Use the given transformation to evaluate the integral. 326) u = x + y, v = -2x + y;
326)
(-2x - 4y) dx dy, R where R is the parallelogram bounded by the lines y = -x + 1, y = -x + 4, y = 2x + 2, y = 2x + 5 A) - 12 B) - 32 C) - 16 D) - 24
327) u = x + y, v = x - y;
327)
(x + y) sin (x - y) dx dy, R where R is the parallelogram bounded by the lines y = x, y = x - 8, y = -x, y = -x + 9 A) 81 (1 + cos 8) B) 81 (1 - cos 8) C) 81 (1 - cos 8) D) 81 (1 + cos 8) 4 4 2 2
Evaluate the integral. 5 7(1 - x/5)
8(1 - x/5 - y/7)
328)
0
0 A) 980 9
0
xyz dz dy dx
328)
B) 19600
C) 1960
81
9
61
D) 980 3
Evaluate the integral by changing the order of integration in an appropriate way. 27 3 82 1 329) dx dy dz 4 x(y + 1) 3 0 1 z ln2 82 ln2 82 ln 82
A)
B)
2
C)
2
4
329) D) ln 82 4
Solve the problem.
330) Let D be the region bounded below by the cone z = x2 + y2 and above by the sphere
330)
z = 49 - x2 - y2 . Set up the triple integral in cylindrical coordinates that gives the volume of D using the order of integration dz dr d . /2 7/ 2 2 7 49 - r2 49 - r2 A) B) r dz dr d r dz dr d 0 0 0 0 r r 2
C)
49 - r2
7/ 2
0
0
r dz dr d
/2
D) 0
r
49 - r2
7 0
r dz dr d
r
Use the given transformation to evaluate the integral. 331) u = 2x + y - z, v = -x + y + z, w = -x + y + 2z;
331)
dx dy dz, R where R is the parallelepiped bounded by the planes 2x + y - z = 6, 2x + y - z = 7, -x + y + z =4, -x + y + z = 9, -x + y + 2z = 5, -x + y + 2z = 9 A) 60 B) 40 C) 30 D) 20 3 3
Set up the iterated integral for evaluating
f(r, , z) r dz dr d
over the given region D.
D
332) D is the right circular cylinder whose base is the circle r = 4 cos in the xy-plane and whose top lies in the plane z = 5 - x. 4 cos 5 - sin
A)
0
B) 0
C) D)
2
0
/2
4 sin 0
4 cos
0
5 - r cos
0
f(r, , z) r dz dr d
5 - r cos
- /2 0 2 4 sin
0 5 - sin
0
0
0
f(r, , z) r dz dr d
f(r, , z) r dz dr d
f(r, , z) r dz dr d
62
332)
Find the Jacobian
(x, y) (x, y, z) or (as appropriate) using the given equations. (u, v) (u, v, w)
333) x = -3u - 2v, y = 5u + 5v A) -5 Evaluate the integral. 7 -1
334)
-2 A) 63
333) B) 5
C) 25
D) -25
dy dx
334)
-8
B) 9
C) 1
Find the area of the region specified in polar coordinates. 335) the region enclosed by the curve r = 7 + cos A) 99 B) 25 2
D) -23
335) C) 33
D) 50
Write an equivalent double integral with the order of integration reversed. 14 14 336) dy dx 5 19 - x 14 9 9 9 A) B) dx dy dx dy 5 14 - y 5 14 - y 14 14 9 14 C) D) dx dy dx dy 5 19 - y 5 19 - y Solve the problem. 337) Evaluate
336)
337) (5x + y - z)(z - x + 9y)(x - y + 4z) dV
R where R is the parallelepiped enclosed by the planes 5x + y - z = 1, 5x + y - z = 3, -x + 9y + z = -2, -x + 9y + z = 5, x - y + 4z = -1, x - y + 4z = 6. A) 40 B) 245 C) 338 D) 81 11 66 99 22
Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. 338) The curves y = x(x - 5) and y = x(x - 5)(x - 10) 5 x(x - 10) 10 x(x - 10) A) B) dy dx dy dx 0 x(x - 5) 5 x(x - 5) 10 x(x - 5)(x - 10) 5 x(x - 5)(x - 10) C) D) dy dx dy dx 5 x(x - 5) 0 x(x - 5)
63
338)
Solve the problem. 339) Find the mass of a tetrahedron of density (x, y, z) = x + y + z bounded by the coordinate planes x y z and the plane + + = 1. 6 3 9
A) 729
C) 972
B) 162
5
5
D) 243 2
340) Find the moment of inertia about the z-axis of a region of constant density enclosed by the paraboloid z = 49 - x2 - y2 and the xy-plane.
A) 49
B) 8,084,777,718,513,670
C) 1,005,686,635,539,115
D) 1,759,951,612,193,451
6
4,398,046,511,104
341) Find the centroid of the first-octant portion of a solid ball of radius 5 centered at the origin. A) (x, y, z) = 15 , 15 , 15 8 8 8
B) (x, y, z) = 3, 3, 3
C) (x, y, z) = 25 , 25 , 25
D) (x, y, z) = 2, 2, 2
8
8
Evaluate the integral. 1 6 342) (s + t) dt ds 0 7 A) 33 2
8
341)
342) B) 27
C) 21
D) 63 2
Provide an appropriate response. 343) True or false? Consider the double integral 6
340)
412,316,860,416
17,592,186,044,416
2
339)
343)
(x2 + y) dy dx.
0 3 The first step in calculating this integral involves integrating with respect to y. A) True B) False
Find the volume of the indicated region.
344) the region that lies under the plane x + y + z = 1 and above the square 0 x, y 20 4
5
7
9
A) 3.448068465e+17
B) 6.896136929e+16
C) 4.925812092e+16
D) 8.620171162e+16
2.849934139e+15
2.849934139e+15
2.849934139e+15
2.849934139e+15
64
344)
Evaluate the integral. 2 2x
345)
0
x2 dy dx
0 A) 16 7
345) B) 16
C) 32
15
D) 32
15
7
Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries. 346) z = 4x2 + 9y2 ; 0 x 1, 0 y 1
A) 26
B) 13
3
C) 5
3
D) 17
3
3
Use a spherical coordinate integral to find the volume of the given solid. 347) the solid bounded below by the xy-plane, on the sides by the sphere =
346)
= 3, and above by the cone
347)
3
A) 27
C) 27
B) 9
5
D) 27
4
2
Solve the problem. 348) Find the moment of inertia of a sphere of radius 6 and constant density about a diameter. A) 641,235,181,318,963 B) 2,849,934,139,195,391 137,438,953,472 687,194,767,360
C) 2,849,934,139,195,391
348)
D) 2,493,692,371,795,967
549,755,813,888
549,755,813,888
Evaluate the integral by changing the order of integration in an appropriate way. 2 8 9 z 349) dy dz dx 4 3 y +1 0 0 x 81 A) B) 81 ln 17 C) 81 ln 17 ln 9 4 8 4
349) D) 81 ln 9 8
Integrate the function f over the given region.
350) f(x, y) = sin 13x over the rectangle 0 x A)
13
B)
13
,0 y
350) C)
26
D) 2
13
Use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. 351) F(x, y, z) = -2x - 8y - 9z over the rectangular solid 0 x 4, 0 y 5, 0 z 10 351) A) - 9200 B) - 13,800 C) - 6900 D) - 41,400
65
Provide an appropriate response. 352) What does the graph of the equation = sec A) A sphere B) A plane
look like?
352)
C) A line
D) A cylinder
Find the volume of the indicated region. 353) the region that lies under the plane z = 10x + 4y and over the triangle bounded by the lines y = x, y = 2x, and x + y = 6 A) 98 B) 78 C) 90 D) 70 Solve the problem. 354) Find the mass of a sphere of radius 7 if A) 4802k B) 343k
= k , k a constant. C) 49k
Evaluate the spherical coordinate integral. /2 /3 10 3 sin d d d 355) 0 0 sec 4,122,206,531,485,699 A) 3,298,534,883,328
354) D) 2401k
355) B) 4,121,244,458,811,395 3,298,534,883,328
C) 4,122,206,531,485,699
D) 4,121,244,458,811,395
6,597,069,766,656
6,597,069,766,656
Evaluate the cylindrical coordinate integral. 1/r /2 6 356) cos dz r dr d 2 0 3 1/r
A) 3 + ln 2
353)
356)
B) 6 - ln 2
C) 6 + ln 2
D) 3 - ln 2
Find the volume of the indicated region.
357) the region bounded by the paraboloid z = 16 - x2 - y2 and the xy-plane A) 256 3
B) 32
C) 64
D) 128
C) 25 (2 + 3 3) 2
D) 25 (2 + 3 3) 4
3
Find the area of the region specified in polar coordinates. 358) the smaller loop of the curve r = 5 + 10 sin A) 25 (2 - 3 3) B) 25 (2 - 3 3) 4 2
357)
358)
Find the volume of the indicated region. 359) the region bounded below by the xy-plane, laterally by the cylinder r = 7 cos , and above by the plane z = 5 A) 245 B) 35 C) 1225 D) 175 4 4 4 4
66
359)
Integrate the function f over the given region. 360) f(x, y) = 7x sin xy over the rectangle 0 x
,0 y 1
B) 7
A)
360) C) 7 - 7
D)
7
Use the given transformation to evaluate the integral. 361) u = -5x + y, v = 8x + y;
361)
(8x + y) dx dy, R where R is the parallelogram bounded by the lines y = 5x + 5, y = 5x + 7, y = -8x + 3, y = -8x + 8 A) 715 B) 55 C) 110 D) 1430 13 13
Solve the problem.
362) Integrate f(x, y) = sin x2 + y2 over the region 0 x2 + y2 25. A) 2 (sin 5 - 5 cos 5) B) 2 (sin 5 - cos 5) C) (sin 5 - cos 5) D) (sin 5 - 5 cos 5)
362)
Use a spherical coordinate integral to find the volume of the given solid.
363) the solid bounded below by the sphere = 5 cos and above by the cone z = x2 + y2 A) 125 24
B) 125 18
C) 25 4
363)
D) 5
Solve the problem.
364) If f(x, y) = (3000ey)/(1 + x /2) represents the population density of a planar region on Earth, where
364)
x and y are measured in miles, find the number of people within the rectangle -6 x 6 and -3 y 0.
A) 3000(1 - e-3 ) ln 4 3952
B) 12,000(1 - e-3 ) ln 4 15,807
C) 6000(1 - e-3 ) ln 7 11,855
D) 6000(1 - e-3 ) ln 4 7904
2
365) Write an iterated triple integral in the order dz dy dx for the volume of the rectangular solid in the first octant bounded by the planes x = 8, y = 6, and z = 9. 8 6 9 9 6-x 8-y-x A) B) dz dy dx dz dy dx 0 0 0 0 0 0 8 6-x 9-y-x 9 6 8 C) D) dz dy dx dz dy dx 0 0 0 0 0 0
67
365)
Find the volume of the indicated region. 366) the region in the first octant bounded by the coordinate planes and the planes x + z = 9, y + 3z = 27 A) 1.202315965e+15 B) 243 3.298534883e+12 2
C) 1.352605461e+15
366)
D) 2.849934139e+15
4.123168604e+11
7.036874418e+13
Sketch D and g(D) from the description of D and change of variables (x, y) = g(u, v). 367) x = 3u, y = 1 v where D is the rectangle 0 u 2, 1 v 4 2
A)
B) None of these.
68
367)
C)
D)
Use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. 368) F(x, y, z) = x + y + z over the tetrahedron bounded by the coordinate planes and the plane 368) x y z + + =1 6 2 4
A) 64 3
B) 16
C) 32
D) 24
Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. 369) The coordinate axes and the line x + y = 6 6 6-x 6 6 A) B) dy dx dy dx -6 0 -6 x 6 x 6 6-x C) D) dy dx dy dx 0 6 0 0
69
369)
Evaluate the integral. 9 /2
/2
370)
7
0
sin
d d d
0
B) 16
A) 8 3
371) 2
370)
4
D) 16
C) 8
3
3
rz dr dz d
371)
2 0 2,005,509,209,063,425 2 A) 158,329,674,399,744
B) 2,005,509,209,063,425 2
C) 401,101,841,812,685 2
D) 2,005,509,209,063,425 2
105,553,116,266,496
26,388,279,066,624
140,737,488,355,328
Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. 372) The parabola x = y2 and the line y = x - 24 5 5
A)
C)
8
5y + 24
3 y2 5y + 24 8 -3 y2
dx dy
8
B)
5y + 24
dx dy
-3 0 dx dy
8
D) 3
5y + 24
dx dy
0
Solve the problem. 373) Find the moment of inertia Iy of the region of constant density (x, y, z) = 1 bounded by the paraboloid z = 25 - x2 - y2 and the xy-plane. A) 406,250 9
6
D) 8,933,531,975,679,995
824,633,720,832
A) 3
373)
B) 203,125
C) 5,583,457,484,799,995 Evaluate the integral. 1 1 374) dx dy 0 4y
372)
6,597,069,766,656
374) B) - 3
C) - 1
2
70
D) 5 2
Change the Cartesian integral to an equivalent polar integral, and then evaluate. 7 49 - y2 5/2 375) (x2 + y2 ) dx dy -7 - 49 - y2
A) 1,154,968,245,501,953
B) 1,154,968,245,501,953
C) 1,768,545,125,924,863
D) 1,768,545,125,924,863
68,719,476,736
34,359,738,368
15,032,385,536
Find the average value of the function over the region. 376) f( , , ) = cos over the solid lower ball A) 8 45
375)
7,516,192,768
15, /2
C) - 45
376) B) 45 16 D) - 6,957,847,019,519,995
8
549,755,813,888
Sketch D and g(D) from the description of D and change of variables (x, y) = g(u, v). 377) u = xy, v = y where D is the rectangle 1 x 4, 1 y 2 x 2
A)
71
377)
B)
C) None of these. D)
Find the average value of the function f over the given region. 378) f(x, y) = 1 over the square 1 x 9, 1 y 9. (xy)2
A) ln 9 9
B) 1
378) C) 1
81
9
Provide an appropriate response. 379) What does the graph of the equation = 0 look like? A) The x-axis B) The y-axis
72
D) ln 9 81
379) C) The xz-plane
D) The yz-plane
Solve the problem. 380) Evaluate
380) e-4x - e-10x dx x
0 by writing the integrand as an integral. A) ln 2 B) ln 5 5 2
C) ln 10 ln 4
D) ln 4
ln 10
Use a spherical coordinate integral to find the volume of the given solid. 381) the solid enclosed by the cardioid of revolution = 7 + 3 cos , z 0 A) 1.856708635e+15 B) 5.570125906e+15 4.398046511e+12 4.398046511e+12
C) 5.570125906e+15
381)
D) 1.856708635e+15
8.796093022e+12
2.199023256e+12
Integrate the function f over the given region. 382) f(x, y) = x + y over the rectangle 0 x 1, 0 y 1 A) 1 B) 2 C) 4 3 3 3
Set up the iterated integral for evaluating
f(r, , z) r dz dr d
382) D) 8 3 over the given region D.
D
383) D is the rectangular solid whose base is the triangle with vertices at (0, 0), (8, 0), and (8, 8), and
383)
whose top lies in the plane z = 10. /2 8 sec 10 A) f(r, , z) dz r dr d 0 0 0 /2 8 csc 10 B) f(r, , z) dz r dr d 0 0 0 /4 8 csc 10 C) f(r, , z) dz r dr d 0 0 0 /4 8 sec 10 D) f(r, , z) dz r dr d 0 0 0
Evaluate the improper integral. 16 16 dy dx 384) x+ y 0 0 A) 256 (1 - ln 2) 3
384) B) 128 (1 - ln 2)
C) 512 (1 - ln 2)
3
3
73
D) 64 (1 - ln 2) 3
Set up the iterated integral for evaluating
f(r, , z) r dz dr d
over the given region D.
D
385) D is the right circular cylinder whose base is the circle r = 5 cos in the xy-plane and whose top lies in the plane z = 9 - x - y. 5 cos 9 - r(cos
+ sin )
A)
0
0
B) 0
C) 0
D)
2
2
0
5 sin
0
5 cos 0
0
5 sin
0 9 - sin 0
- cos
9 - r(cos 0 9 - sin
f(r, , z) dz r dr d
f(r, , z) dz r dr d
+ sin )
- cos
f(r, , z) dz r dr d
f(r, , z) dz r dr d
0
386) D is the solid right cylinder whose base is the region in the xy-plane that lies inside the cardioid r = 6 - 2 sin
A) 0
B) 0
C) 0
D) 0
2
2
386)
and outside the circle r = 4, and whose top lies in the plane z = 3. 6 - 2 sin 3 f(r, , z) dz r dr d 4 0 6 - 2 sin 3 f(r, , z) dz r dr d 0 0 6 - 2 sin 3 f(r, , z) dz r dr d 0 0 6 - 2 sin 3 f(r, , z) dz r dr d 4 0
Reverse the order of integration and then evaluate the integral. 1 4 3 387) ey dy dx 0 x/4 4 A) (e - 1) B) 2 (e - 1) C) 4 (2e - 1) 3 3 3 Evaluate the spherical coordinate integral. /2 /2 10 388) sin d d d 0 0 6
A) 16
385)
387) D) 2 (2e - 1) 3
388) C) 32
B) 16
3
74
D) 32 3
Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. 389) The parabola y = x2 and the line y = 5x + 24 5x + 24 - x2
8
A)
-3 0 8 5x + x2
C)
0
dx dy
B)
x2
0 dy dx
5x + 24
8
5x + 24
8
D)
-3 x2
24
389)
dy dx
dy dx
Solve the problem. 390) Find the average distance from a point P(r, ) in the region bounded by r = 5 + 8 cos to the origin. A) 605 B) 605 C) 605 D) 1210 76 171 114 171
391) Find the center of mass of the upper half of a solid ball of radius 3 centered at the origin if the
390)
391)
density at any point is proportional to the distance from that point to the z-axis. A) (x, y, z) = 0, 0, 16 1 B) (x, y, z) = 0, 0, 3 5 8
C) (x, y, z) = 0, 0, 6
D) (x, y, z) = 0, 0, 3 (1 + 2)
5
8
Find the area of the region specified by the integral(s). 8 ln x 392) dy dx 0 0 A) ln 8 B) 8(ln 8 - 1) Solve the problem. 393) Write 2
2
0
392) C) ln 8 + 1
D) 8 ln 8
393) y
0 4-y
4
f(x, y) dx dy + 2
4-y
2
f(x, y) dx dy +
0
0
2
f(x, y) dx dy +
y
f(x, y) dx dy 0 2 as a single iterated integral with the same order of integration. 4 4-y 4 4-y A) B) f(x, y) dx dy f(x, y) dx dy 2 0 2 2 4 4-y 4 4-y C) D) f(x, y) dx dy f(x, y) dx dy 0 2 0 0
75
394) Find the mass of the region of density (x, y, z) =
1 bounded by the hemisphere 64 - x2 - y2
64 - x2 - y2 and the xy-plane. A) 64 B) 8
394)
z=
C) 16
Evaluate the spherical coordinate integral. /2 /2 3 cos 2 sin d d d 395) 0 0 0 3 A) B) 9 2 8
D) 32
395) C) 9 2 8
D) 3 2 2
Evaluate the double integral over the given region.
396) R
1 dA, R: 0 x 2, 0 y 1 (x + 1)(y + 1)
B) 1 ln 3
A) ln 3 ln 2 Evaluate the integral. 8 -9
397)
396) C) 2 ln 3
2
xy2 dx dy
397)
-1 -7 A) - 2736
B) - 11115
C) 2736
398) x = 2u cosh 9v, y = 2u sinh 9v A) 162v B) 36u Evaluate the integral. (1 - cos )/2
4
399)
(1 - sin )/2 A) 1 (15 - 8) 48 e6
400) 1
D) 11115
(x, y) (x, y, z) or (as appropriate) using the given equations. (u, v) (u, v, w)
Find the Jacobian
0
D) ln 6
e2 1
A) 144
1
0
2 sin
398) C) 36v
D) 162u
d d d
399)
B) 1 (15 - 16)
C) 1 (15 - 8)
48
24
D) 1 (15 - 16) 24
e4 1 dx dy dz xyz
400)
B) 16
C) 96
D) 48
Find the volume of the indicated region.
401) the region enclosed by the sphere x2 + y2 + z 2 = 324 and the cylinder (x - 9)2 + y2 = 81 A) 1215(3 - 4) B) 3888(3 - 4) C) 1296(3 - 4) D) 972(3 - 4)
76
401)
Solve the problem.
402) Let D be the region bounded below by the cone z = x2 + y2 and above by the sphere
402)
z = 4 - x2 - y2 . Set up the triple integral in cylindrical coordinates that gives the volume of D using the order of integration dr dz d . 2 2/2 z 2 2 4 - z2 A) + r dr dz d r dr dz d 0 0 0 0 2/2 0 2
B) 0
2/2
0
Find the Jacobian
r dr dz d
+
0
0 2
D)
+
0
z
2
2
2
2/ 2 0 2 8 - z2
0 z
2/ 2
0
r dr dz d
0
0 2
C)
z
2/ 2
r dr dz d
+
0
4 - z2
0
2/ 2 0
2
2
0
8 - z2
r dr dz d
r dr dz d
2/2 0
(x, y) (x, y, z) or (as appropriate) using the given equations. (u, v) (u, v, w)
403) u = 2x + 3, v = -3y - 4, w = -4z - 5 A) 1 60
404)
C) 60
404) B) 16
Solve the problem. 405) Solve for a: 2a 2a a
D) 1 24
x dy dx
0 A) 64 3
a
403)
B) 24
Evaluate the integral. 4 16 - x2 0
r dr dz d
C) 64
D) 256 3
405) 2a
dx dz dy = 216
a
A) a = 2
B) a = 12 2
C) a = 4
D) a = 6
2
406) Integrate f(x, y) = ln(x + y ) over the region 1 x2 + y2 4.
406)
x2 + y2
A) 2 ln 2
B) 2 (ln 2)2
C)
77
ln 2
D) (ln 2)2
Change the Cartesian integral to an equivalent polar integral, and then evaluate. 0 4 x2 + y2 407) dx dy 2 + y2 1 x + -4 - 16 - y2
A) (8 + 2 ln 5)
B) (8 + ln5)
4
407)
C) (8 + ln 5)
2
D) (8 + 2 ln 5)
4
2
Solve the problem.
408) Let D be the region that is bounded below by the cone =
4
and above by the sphere
the triple integral for the volume of D in spherical coordinates. 2 3 2 2 sin d d d A) B) 0 0 0 0 2 /4 3 2 2 sin d d d C) D) 0 0 0 0
Evaluate the integral. /2 9
409)
0
/2
3 sin
3 /4 0
0
3 0 3
/2
= 3. Set up
2 sin
d d d
2 sin
d d d
408)
0
d d d
409)
8 0 5.420592325e+15 A) 1.319413953e+13
B) 5.420592325e+15
C) 5.420592325e+15
D) 5.420592325e+15
1.759218604e+13
1.97912093e+13
8.796093022e+12
Use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. 410) F(x, y, z) = x2 y4 z 2 over the cylinder bounded by x2 + y2 49 and the planes z = -1, z = 3 410)
A) 5,416,169,448,144,901
B) 5,416,169,448,144,901
C) 5,416,169,448,144,901
D) 5,416,169,448,144,901
6,979,321,856
7,918,845,952
6,576,668,672
6,442,450,944
Solve the problem. 411) Evaluate 2
411) 4 - x2
2
dz dy dx 2 2 -2 - 4 - x2 x +y by transforming to cylindrical or spherical coordinates. A) 8 B) 4 C) 4 3 3
78
D) 2
Change the Cartesian integral to an equivalent polar integral, and then evaluate. 7 49 - y2 412) dx dy -7 - 49 - y2
A) 49
B) 196
412)
C) 98
Find the average value of the function f over the given region. 413) f(x, y) = 10x + 3y over the square 0 x 1, 0 y 1. A) 8 B) 23 2
D) 7
413) C) 13 2
D) 13
Solve the problem. 414) Find the center of mass of a tetrahedron of density (x, y, z) = x + y + z bounded by the coordinate x y z planes and the plane + + = 1. 10 3 8
A) x = 310 , y = 8 , z = 232
B) x = 62 , y = 24 , z = 232
C) x = 155 , y = 4 , z = 116
D) x = 155 , y = 6 , z = 58
63
7
63
7
63
21
63
42
415) Let D be the region that is bounded below by the cone =
4
35
7
414)
105 21
and above by the sphere
= 5. Set up
415)
the triple integral for the volume of D in cylindrical coordinates. 2 2 5/ 2 5/ 2 25 - r2 25 - r2 A) B) r dz dr d r dz dr d 0 0 0 0 0 r 2 5 2 5 25 - r2 25 - r2 C) D) r dz dr d r dz dr d 0 0 0 0 0 r
Evaluate the integral. 3 /2
416)
2
/2
( sin )2 d d d
416)
0 0 5.348024558e+15 A) 6.755399441e+15
B) 5.348024558e+15 2
C) 5.348024558e+15 2
D) 5.348024558e+15
6.755399441e+15
5.066549581e+15
5.066549581e+15
Solve the problem.
417) Find the average height of the paraboloid z = 8x2 + 5y2 above the disk x2 + y2 64 in the xy-plane.
A) 288
B) 416
C) 336
3
79
D) 208
417)
418) Evaluate
418) sin ax dx , a > 0 x
0 by integrating
e-xy sin ax dA. 0
A) a
0
B) a
4
C) a
3
6
D)
2
Use the given transformation to evaluate the integral. 419) u = y - x, v = y + x; y- x cos dx dy, y+ x
419)
R where R is the trapezoid with vertices at (3, 0), (4, 0), (0, 3), (0, 4) A) 7 sin 1 B) 7 sin 2 C) 7 sin 2 2 2 4
D) 7 sin 1 4
420) u = 2x + y - z, v = -x + y + z, w = -x + y + 2z;
420)
(2x + y - z)(z - x + y)(2z - x + y) dx dy dz, R where R is the parallelepiped bounded by the planes 2x + y - z = 2, 2x + y - z = 5, -x + y + z = 2, -x + y + z = 9, -x + y + 2z = 5, -x + y + 2z = 7 A) 14,553 B) 14,553 C) 1617 D) 6468 2
Evaluate the cylindrical coordinate integral. /2 cos2 9r4 421) z sin dz r dr d 0 0 4r4
A) 227 420
421)
B) 65
C) 73
576
210
D) 13 84
Integrate the function f over the given region. 422) f(x, y) = y2 ex4 over the triangular region in the first quadrant bounded by the lines x = y/6, x = 1, y=0 A) 3.799912186e+15 (e - 1) B) 216(e + 1) 2.111062325e+14
C) 3.799912186e+15 e
D) 3.799912186e+15 (e - 1)
2.111062325e+14
5.277655813e+13
80
422)
Find the average value of F(x, y, z) over the given region. 423) F(x, y, z) = x2 + y2 + z 2 over the rectangular solid in the first octant bounded by the coordinate planes and the planes x = 10, y = 6, z = 5 A) 161 B) 361 3 3
C) 233
423)
D) 211
3
3
Evaluate the improper integral.
424) 0
0 A) 1 45
dx dy (x + 9)2 (y + 5)2
424) B) 1
C) 1
180
D) 1
90
135
Find the volume of the indicated region.
425) the region enclosed by the cone z 2 = x2 + y2 between the planes z = 6 and z = 8 A) 74
B) 296 3
C) 296 3
425) D) 74
Find the average value of the function f over the given region. 426) f(x, y) = 2x + 7y over the triangle with vertices (0, 0), (6, 0), and (0, 10). A) 4 B) 82 C) 19 3 3
426) D) 22 3
Evaluate the integral. 9 9 427) 6y dx dy 4 -7
427)
A) 3120
B) - 3840
C) 480
D) - 2.638827907e+15
2.199023256e+12
Solve the problem. 428) Let D be the region bounded below by the xy-plane, on the side by the cylinder r = 4 sin , and on top by the paraboloid z = 2r2. Set up the triple integral in cylindrical coordinates that gives the volume of D using the order of integration dz dr d . 4 sin 2 4sin 2r2 A) B) r dz dr d 0 0 0 0 0 2 /4 4 sin /2 4 sin 2r C) D) r dz dr d 0 0 0 0 0
81
2r2 0
r dz dr d
2r2 0
r dz dr d
428)
Evaluate the integral. /4 4
429)
0
2
( 5 cos ) 2 sin
d d d
429)
0 0 A) 4,503,599,627,370,485 6,442,450,944
B) 1,803,473,947,459,583
C) 4096
D) 6,710,886,399,999,977
134,217,728 12,582,912
Use the given transformation to evaluate the integral. 430) x = 5u, y = 4v, z = 6w; x2 y2 z 2 + + 25 16 36
dx dy dz,
R where R is the interior of the ellipsoid
A) 480
+2
430)
x2 y2 z 2 + + =1 25 16 36
B) 480
C) 240
+3
+1
D) 480
+1
Find the volume of the indicated region.
431) the region bounded by the paraboloid z = x2 + y2 and the plane z = 64 A) 2048 3
B) 2048
C) 1024
432) the region bounded by the cylinders r = 3, r = 7 and the planes z = 6, z = 10 A) 640 B) 160 C) 320
431) D) 4096 3
432) D) 480
Solve the problem. 433) Find the moment of inertia about the x-axis of a thin triangular plate bounded by the coordinate axes and the line x + y = 4 if (x, y) = x + y. A) 1024 B) 1024 C) 1024 D) 512 9 15 5 5
433)
Find the volume of the indicated region.
434) the region under the surface z = x2 + y4, and bounded by the planes x = 0 and y = 9 and the cylinder y = x2
A) 1,271,894,435,168,257
B) 6,946,714,464,288,773
C) 1,279,548,066,889,729
D) 5,712,044,510,674,951
118,111,600,640
5,669,356,830,720
354,334,801,920
177,167,400,960
82
434)
Find the area of the region specified by the integral(s). 10 10 - x 435) dy dx 0 0 A) 100 B) 25 3
435) C) 50 3
83
D) 50
Answer Key Testname: CHAPTER 15
1) Find the volume bounded above by f(x,y) = y + x which lies over the region for which 4 x 6 and 1 y 2. 2) Find the volume bounded above by f(x,y) = y + x which lies over the region for which 1 x 2 and 4 y 6. 3) Find the area of the region between the curves y = x2 and y = 2x for x between 0 and 2. 4) This integral represents the volume of the region enclosed between the paraboloid with equation
z = 3 - r2 (or z = 3 - x2 - y2 ) and the xy-plane. 5) This integral represents the volume of a wedge cut from a sphere of radius 1 by two half-planes whose edges meet at 1 a 45° angle along a diameter of the sphere. The region resembles a section of an orange and is of the entire sphere. 8
6) Find the area of the region bounded by the curve y = x2 and the lines y = 0, x = 0, and x = 5. 7) D 8) B 9) C 10) A 11) D 12) C 13) D 14) C 15) B 16) D 17) B 18) B 19) B 20) B 21) C 22) A 23) D 24) D 25) B 26) C 27) C 28) B 29) C 30) D 31) C 32) A 33) C 34) B 35) C 36) D 37) B 38) D 39) B 84
Answer Key Testname: CHAPTER 15
40) B 41) B 42) D 43) A 44) B 45) A 46) C 47) A 48) D 49) D 50) A 51) D 52) C 53) D 54) B 55) B 56) D 57) C 58) C 59) B 60) D 61) A 62) B 63) C 64) C 65) B 66) A 67) D 68) D 69) D 70) D 71) C 72) A 73) C 74) D 75) A 76) B 77) D 78) A 79) B 80) C 81) D 85
Answer Key Testname: CHAPTER 15
82) B 83) A 84) D 85) B 86) A 87) B 88) C 89) A 90) C 91) A 92) B 93) C 94) B 95) C 96) C 97) A 98) C 99) D 100) A 101) A 102) D 103) D 104) D 105) B 106) C 107) B 108) C 109) B 110) D 111) B 112) A 113) D 114) B 115) A 116) B 117) B 118) C 119) B 120) C 121) C 122) D 123) D 86
Answer Key Testname: CHAPTER 15
124) B 125) C 126) A 127) B 128) A 129) B 130) C 131) B 132) B 133) C 134) C 135) B 136) C 137) B 138) C 139) D 140) D 141) C 142) D 143) B 144) A 145) C 146) B 147) B 148) A 149) B 150) C 151) A 152) D 153) B 154) B 155) D 156) B 157) C 158) C 159) C 160) D 161) C 162) B 163) C 164) C 165) C 87
Answer Key Testname: CHAPTER 15
166) B 167) B 168) C 169) C 170) D 171) C 172) D 173) B 174) B 175) D 176) A 177) C 178) A 179) A 180) A 181) B 182) B 183) D 184) D 185) B 186) D 187) D 188) A 189) C 190) C 191) D 192) B 193) B 194) C 195) C 196) B 197) A 198) C 199) B 200) A 201) A 202) C 203) C 204) A 205) A 206) C 207) B 88
Answer Key Testname: CHAPTER 15
208) B 209) A 210) D 211) D 212) B 213) A 214) A 215) D 216) A 217) B 218) B 219) B 220) A 221) C 222) D 223) C 224) A 225) C 226) B 227) D 228) C 229) D 230) D 231) C 232) B 233) A 234) A 235) C 236) A 237) B 238) B 239) C 240) D 241) A 242) D 243) C 244) C 245) D 246) D 247) B 248) C 249) B 89
Answer Key Testname: CHAPTER 15
250) D 251) D 252) B 253) D 254) C 255) B 256) A 257) D 258) C 259) D 260) C 261) B 262) D 263) A 264) D 265) D 266) A 267) C 268) B 269) A 270) C 271) B 272) B 273) C 274) C 275) C 276) D 277) A 278) D 279) C 280) A 281) B 282) A 283) C 284) D 285) B 286) C 287) A 288) C 289) A 290) A 291) D 90
Answer Key Testname: CHAPTER 15
292) A 293) A 294) D 295) A 296) C 297) C 298) C 299) D 300) D 301) D 302) C 303) A 304) A 305) A 306) A 307) D 308) A 309) C 310) A 311) C 312) B 313) C 314) C 315) D 316) A 317) C 318) D 319) C 320) B 321) C 322) C 323) A 324) C 325) D 326) B 327) B 328) A 329) C 330) C 331) D 332) C 333) A 91
Answer Key Testname: CHAPTER 15
334) A 335) A 336) C 337) B 338) D 339) D 340) B 341) A 342) B 343) A 344) C 345) D 346) B 347) B 348) B 349) B 350) D 351) B 352) B 353) B 354) D 355) D 356) D 357) D 358) B 359) A 360) B 361) B 362) A 363) A 364) B 365) A 366) A 367) A 368) D 369) D 370) C 371) B 372) C 373) B 374) C 375) D 92
Answer Key Testname: CHAPTER 15
376) C 377) A 378) B 379) C 380) B 381) A 382) C 383) D 384) C 385) A 386) D 387) A 388) B 389) D 390) D 391) A 392) B 393) D 394) C 395) B 396) A 397) C 398) B 399) B 400) D 401) C 402) B 403) D 404) C 405) D 406) B 407) D 408) C 409) B 410) D 411) A 412) A 413) C 414) B 415) B 416) B 417) D 93
Answer Key Testname: CHAPTER 15
418) D 419) A 420) C 421) D 422) A 423) A 424) A 425) C 426) B 427) A 428) A 429) C 430) B 431) B 432) B 433) B 434) D 435) D
94