Thomas_ Calculus Early Transcendentals, 13th Edition Test Bank by Ace Test Banks

Thomas Calculus Early Transcendentals, 13th Edition By George Thomas Jr, Maurice Weir, Joel Hass

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the domain and range of the function. 1) f(x) = -6 A) D: (-∞, ∞), R: (-∞, ∞) B) D: (-∞, -6], R: (-∞, ∞) C) D: (-∞, ∞), R: (-∞, -6] D) D: (-∞, ∞), R: [-6, ∞) 2)

F(t) = - 10 A) D: (-∞, ∞), R: [-10,∞) C) D: [0, ∞), R: (-∞, -10]

2) _______ B) D: [-100,∞), R: [-10,∞) D) D: (-∞, ∞), R: (-∞, ∞) 3) _______

3) g(z) = 2 A) D: [0,∞), R: (-∞,2] C) D: (-∞,2], R: (-∞, ∞)

B) D: (-∞, ∞), R: (-∞,2] D) D: (-∞,0], R: [2,∞) 4) _______

4) f(x) = 5 + A) D: (-∞,0], R: (-∞,5] C) D: [0,∞), R: [5,∞)

1) _______

B) D: (-∞, ∞), R: [5,∞) D) D: [0,∞), R: (-∞, ∞) 5) _______

5) F(t) = A) D: (-∞,∞), R: (-∞,∞) C) D: [0,∞), R: (-∞,∞)

B) D: (0,∞), R: (0,∞) D) D: (-∞,0), R: (-∞,0) 6) _______

6) g(z) = A) D: [1,∞), R: (-∞,∞) C) D: [0,∞), R: (-∞,∞)

B) D: (-1,∞), R: (-∞,0) D) D: (-∞,-1), R: (0,∞) 7) _______

7) f(x) = A) D: (-∞,0], R:(-∞,0]

B) D: [0,∞), R: D) D: [0,∞), R: (-∞,∞)

C) D: (-∞,∞), R:

8) _______

8) F(t) = A) D: (0,∞), R: (0,∞) C) D: (-∞,0), R: (-∞,0) 9)

g(z) = A) D: [0,∞), R: (-∞,∞) C) D: (-2,2), R: (-2,2)

Determine whether or not the graph is a graph of a function of x. 10)

B) D: [0,∞), R: [0,∞) D) D: (-∞,∞), R: (-∞,∞) 9) _______ B) D: (-∞,∞), R: (0, 2) D) D: [-2,2], R: [0,2]

___ ___

10)

A) Function

B) Not a function 11) ______

11)

A) Not a function

B) Function 12) ______

12)

A) Function 13)

B) Not a function

___ ___

13)

A) Not a function

B) Function 14) ______

14)

A) Not a function

B) Function 15) ______

15)

A) Function

B) Not a function

Find the formula for the function. 16) Express the perimeter of a square as a function of the square's side length x. A) p = 4x B) C) p = 6x

16) ______ D)

p= 17) Express the area of a square as a function of its side length x. A) B) A = 4x C) A= A= 18) Express the length d of a square's diagonal as a function of its side length x.

17) ______ D) A = 2x 18) ______

A) d = x

B) d = x

C) d = 2x

D) d = x

19) Express the perimeter of an isosceles triangle with side lengths x, 5x, and 5x as a function of the side length. A) B) p = 10x C) D) p = 11x p = 10 p = 25

19) ______

20) Express the area of a circle as a function of its radius r. A) B) A = 2πr C) A = πr A=π

20) ______

22)

V=π

A point P in the first quadrant lies on the graph of the function f(x) = line joining P to the origin as a function of x. A) m = 2x B) C) m=

23)

A=π 21) ______

21) Express the volume of a sphere as a function of its radius r. A) B) C) V=

. Express the slope of the

22) ______

D) m = x

A point P in the fourth quadrant lies on the graph of the function f(x) = of the line joining P to the origin as a function of x. A) m = -2x B) m = x C) m = -x

. Express the slope

23) ______

D) m=

Find the domain and graph the function. 24) f(x) = -6x + 8

A) D: (-∞, ∞)

24) ______

B) D: (-∞, ∞)

C) D: [0, ∞)

25) g(x) = -7 + 2x - x2

A) D: (-∞, ∞)

D) D: (-∞, ∞)

25) ______

C) D: (-∞, ∞)

26) F(x) =

A) D: (-∞, ∞)

D: (∞, 6]

D) D: (-∞, ∞)

26) ______

C) D: [0, ∞)

27) G(x) =

A) D: [0, ∞)

D: (∞, 0)

D) D: (-∞, 0]

27) ______

C) D: (-∞, 0)

D: (∞, ∞)

D) D: (-∞, 0]

28) ______

28) F(t) =

A) D: (-∞, -3) ∪ (-3>, ∞)

C) D: (-∞, 0) ∪ (0, ∞)

D: (∞, -3) ∪ (-3 , ∞)

D) D: (-∞, ∞)

29) ______

29) G(t) =

A) D: (-∞, 0) ∪ (0, ∞)

C) D: (-∞, -3) ∪ (-3, ∞)

Graph the function. 30) f(x) =

D: (∞, -3) ∪ (-3 , ∞)

D) D: (-∞, ∞)

30) ______

31) ______

31) g(x) =

32) ______

32) F(x) =

33) ______

33) G(x) =

34) ______

34) f(x) =

35) ______

35) g(x) =

Find a formula for the function graphed. 36)

37)

36) ______

___ ___

37)

B) f(x) =

f(x) = D)

f(x) =

f(x) = 38) ______

38)

39)

39) ______

B) f(x) =

f(x) = D)

f(x) =

f(x) = 40) ______

40)

B) f(x) =

f(x) = D)

f(x) =

f(x) = 41) ______

41)

B) f(x) =

D) f(x) =

42)

f(x) =

___ ___

42)

B) f(x) = f(x) =

D) f(x) = f(x) = 43) ______

43)

f(x) = C)

f(x) = D)

f(x) =

Graph the function. Determine the symmetry, if any, of the function. 44) y=

f(x) =

___ ___

44)

A) Symmetric about the origin

B) Symmetric about the origin

C) Symmetric about the y-axis

D) No symmetry

45) y=-

___ ___

45)

A) No symmetry

B) Symmetric about the y-axis

C) Symmetric about the y-axis

D) Symmetric about the y-axis

46) y=

___ ___

46)

A) No symmetry

B) Symmetric about the origin

C) Symmetric about the origin

D) Symmetric about the origin

47) y = -|x|

___ ___

47)

A) Symmetric about the y-axis

B) Symmetric about the x-axis

C) Symmetric about the y-axis

D) Symmetric about the origin

48) y =

___ ___

48)

A) No symmetry

B) No symmetry

C) Symmetric about the y-axis

D) Symmetric about the y-axis

49) y = 6

___ ___

49)

50)

A) Symmetric about the x-axis

B) No symmetry

C) No symmetry

D) Symmetric about the x-axis

___ ___

50)

51)

A) No symmetry

B) Symmetric about the y-axis

C) Symmetric about the y-axis

D) No symmetry

___ ___

51)

A) Symmetric about the y-axis

B) No symmetry

C) No symmetry

D) Symmetric about the y-axis

Graph the function. Specify the intervals over which the function is increasing and the intervals where it is decreasing. 52) y=

___ ___

52)

A) Decreasing -∞ < x < 0; Increasing 0 < x < ∞

B) Increasing -∞ < x < ∞

C) Increasing -∞ < x < 0 and 0 < x < ∞

D) Decreasing -∞ < x < 0 and 0 < x < ∞

53) y=-

___ ___

53)

A) Increasing -∞ < x < 0 Decreasing 0 < x < ∞

B) Decreasing -∞ < x < 0 Increasing 0 < x < ∞

C) Decreasing -∞ < x < 0 and 0 < x < ∞

D) Increasing -∞ < x < 0 and 0 < x < ∞

54) y=

___ ___

54)

A) Increasing -∞ < x < 0 Decreasing 0 < x < ∞

B) Increasing -∞ < x < 0 and 0 < x < ∞

C) Decreasing -∞ < x < 0 and 0 < x < ∞

D) Decreasing -∞ < x < 0 Increasing 0 < x < ∞

55) y = -|x|

___ ___

55)

A) Increasing -∞ < x < ∞

B) Decreasing -∞ < x < ∞

C) Increasing -∞ < x ≤ 0 Decreasing 0 ≤ x < ∞

D) Decreasing -∞ < x ≤ 0 Increasing 0 ≤ x < ∞

56) y =

___ ___

56)

A) Increasing -∞ < x ≤ 0

B) Decreasing -∞ < x ≤ 0

C) Decreasing -∞ < x ≤ 0 Increasing 0 ≤ x < ∞

D) Decreasing 0 ≤ x < ∞

57) y = -6

___ ___

57)

58)

A) Decreasing -∞ < x < ∞

B) Decreasing 0 ≤ x < ∞

C) Increasing 0 ≤ x < ∞

D) Increasing 0 ≤ x < ∞

___ ___

58)

59)

A) Decreasing -∞ < x < ∞

B) Decreasing -∞ < x ≤ 0

C) Increasing 0 ≤ x < ∞

D) Decreasing -∞ < x ≤ 0 Increasing 0 ≤ x < ∞

___ ___

59)

A) Increasing -∞ < x < ∞

B) Increasing -∞ < x < 0 Decreasing 0 < x < ∞

C) Decreasing -∞ < x < ∞

D) Decreasing -∞ < x ≤ 0 Increasing 0 ≤ x < ∞

Determine if the function is even, odd, or neither. 60) f(x) = 9 A) Even B) Odd

60) ______ C) Neither 61) ______

61) f(x) = 2x2 - 3 A) Even 62) f(x) = 8x4 - 5x + 7

B) Odd

C) Neither 62) ______

A) Even

B) Odd

C) Neither 63) ______

63) f(x) = -3x5 - 9x3 A) Even 64) f(x) = (x + 9)(x - 5) A) Even

B) Odd

C) Neither 64) ______

B) Odd

C) Neither 65) ______

65) f(x) = A) Even

B) Odd

C) Neither 66) ______

66) f(x) = A) Even

B) Odd

C) Neither 67) ______

67) g(x) = A) Even 68)

69)

g(x) = |2 | A) Even h(t) = A) Even

Match the equation with its graph. 70) y= A)

B) Odd

C) Neither 68) ______

B) Odd

C) Neither 69) ______

B) Odd

C) Neither

70) ______ B)

71)

71) ______

y= A)

72) y = 2x A)

72) ______

Provide an appropriate response. 73) Graph the functions f(x) =

73) ______ and g(x) = 2 +

together to identify the values of x for which

2+ . Confirm your findings algebraically. A) (-2, 6) B) (-2, 0) ∪ (6, ∞)

C) (-∞, -2) ∪ (0, 6)

D) (6, ∞) 74) ______

74) Graph the functions f(x) =

A) (-∞, -3)

and g(x) =

together to identify the values of x for which

Confirm your findings algebraically. B) (-3, ∞) C) (-1, 1) ∪ (1, ∞)

D) (-3, -1) ∪ (1, ∞)

Solve the problem. 75) The variable s is proportional to t, and s = 60 when t = 240. Determine t when s = 85. A) 330 B) 425 C) 4 D) 340 76) The kinetic energy K of a mass is proportional to the square of its velocity v. If K = 3380 joules when A) 2430 joules

what is K when v = 9 m/sec? B) 2025 joules

C) 1620 joules

pressure P decreases, so that V and P are inversely proportional. If P = 14.4 lbs/ , then what is V when P = 23 lbs/

76) ______

D) 810 joules

77) Boyle's Law says that volume V of a gas at constant temperature increases whenever the

1200

75) ______

when V =

77) ______

78) A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 15 inches by 29 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.

A) V(x) = x(15 - 2x)(29 - 2x) C) V(x) = x(15 - x)(29 - x)

B) V(x) = (15 - x)(29 - x) D) V(x) = (15 - 2x)(29 - 2x)

79) The figure shown here shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long. Express the area A of the rectangle in terms of x.

A) A(x) = 2x(1 - x)

B) A(x) = 2x(x - 1)

A(x) = 2

gives the cost of laying the cable in terms of the

C(x) = 150 + 175(5280 - x) C) C(x) = 175(750 - x) + 150(1 - x)

B) D)

79) ______

D) A(x) = x(1 - x)

80) A power plant is located on a river that is 750 feet wide. To lay a new cable from the plant to a location in a city 1 mile downstream on the opposite side costs $175 per foot across the river and $150 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

78) ______

C(x) = 175

+ 150(5280 - x)

C(x) = 175

+ 150(1 - x)

80) ______

Provide an appropriate response. 81)

81) ______

Consider the function y = A) Yes

. Can x be negative? B) No 82) ______

82) Consider the function y = A) Yes

. Can x be greater than 0, but less than 1? B) No 83) ______

83) What is the domain of the function y = A) (-∞, 0) ∪ [1, ∞) B) (-∞, 0) ∪ (1, ∞) 84)

Graph the equation

? C) (-∞, ∞)

D) (0, 1]

= x and decide whether or not the graph represents a function of x.

A) Function

B) Not a Function

85) Graph the equation |x + y| = 1 and decide whether or not the graph represents a function of x.

A) Function 86) For what values of x is A) -3 < x ≤ -2 87) For what values of x is A) 1 ≤ x < 2

84) ______

85) ______

B) Not a Function = -2? B) -2 < x ≤ -1 = 1? B) 0 ≤ x < 1

86) ______ C) -3 ≤ x < -2

D) -2 ≤ x < -1 87) ______

C) 0 < x ≤ 1

D) 1 < x ≤ 2

88) What real numbers x satisfy the equation A) {x|x ∈ integers} C) {x|x ∈ real numbers} 89) Graph the function f(x) =

? B) {x|x = 0} D) ∅

89) ______

90) Graph the function f(x) =

88) ______

___ ___

90)

1) C 2) A 3) A 4) C 5) B 6) B 7) B 8) A 9) D 10) A 11) B 12) B 13) A 14) B 15) A 16) A 17) C 18) B 19) D 20) D 21) B 22) D 23) C 24) D 25) C 26) D 27) B 28) B 29) C 30) A 31) A 32) A 33) B 34) C 35) D 36) B 37) A 38) B 39) C 40) B 41) A 42) B 43) C 44) A 45) C 46) D 47) C 48) A 49) C 50) D 51) D

52) D 53) B 54) C 55) C 56) B 57) B 58) B 59) B 60) A 61) A 62) C 63) B 64) C 65) A 66) C 67) B 68) A 69) A 70) D 71) B 72) C 73) B 74) A 75) D 76) C 77) B 78) A 79) A 80) B 81) A 82) B 83) A 84) B 85) B 86) D 87) C 88) A 89) C 90) D

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the domain and range for the indicated function. 1) f(x) = , g(x) = ; f+g A) D: x ≥ 3

B) D: x ≥ -3

R: y ≥

2) f(x) =

, A) D: x ≥ -7

g(x) =

; g-f B) D: x ≥ 7 R: y ≥ 0

R: y ≥ 3) f(x) =

, A) D: x ≥ 2 R: y ≥ 0

g(x) =

4) f(x) = 5,

g(x) = 5 + A) D: x ≥ 0 R: y ≤ 5

5) f(x) = 6,

g(x) = 6 + A) D: x ≥ 0 R: y ≤ 6

; f∙g B) D: x ≥ 2 R: -∞ < y < ∞ ;

;

f/g B) D: x ≥ 0 R: y ≥ 1 g/f B) D: x ≥ 0 R: y ≥ 1

Solve the problem. 6) If f(x) = 7x + 11 and g(x) = 5x - 1, find f(g(x)). A) 35x + 54 B) 35x + 10 7) If f(x) = -4x + 8 and g(x) = 5x + 2, find g(f(x)). A) 20x + 42 B) -20x - 38

C) D: x ≥ -3 R: y ≥ 0

1) _______

D) D: x ≥ 3 R: y ≥ 0 2) _______

C) D: x ≥ 7

D) D: x ≥ 7

R: y ≥

R: y ≥ 3) _______

C) D: x ≥ 2 R: y > 0

D) D: x > 2 R: y ≥ 0 4) _______

C) D: x ≥ -5 R: y ≥ 0

D) D: x ≥ 0 R: y ≤ 1 5) _______

C) D: x ≥ -6 R: y ≥ 0

D) D: x ≥ 0 R: y ≤ 1

6) _______ C) 35x + 18

D) 35x + 4 7) _______

C) -20x + 42

D) -20x + 16 8) _______

8) If f(x) = A) x + 4

and g(x) = 5x + 2, find g(f(x)). B) 5x + 8

D) x -

9) If f(x) = A) 2

and g(x) = 8x - 8, find f(g(x)). B) 8 -8

9) _______ C) 2

10) If f(x) = 4x2 + 4x + 5 and g(x) = 4x - 8, find g(f(x)). A) 4x2 + 4x - 3 B) 16x2 + 16x + 12 C) 4x2 + 16x + 12 11)

D) 8 10) ______ D) 16x2 + 16x + 28 11) ______

2 If f(x) = A)

and g(x) = 8x , find g(f(x)). B)

12) If f(x) = -9x + 1 and g(x) = 3x2 + 6x - 7, find g(f(-5)).

12) ______

A) 407

B) 469

C) 6617

D) -341

13) If f(x) = -2x - 5 and g(x) = 7x2 - 2x + 1, find g(f(-5)). A) 43 B) 26 C) 166

D) -377

13) ______

14) ______

14) If f(x) = A)

, g(x) =

, and h(x) = 3x+ 9, find f(g(h(x))). B) C) 3 +3

15) ______

15) If f(x) = A)

, g(x) = +2

, and h(x) = 5x+ 10, find h(g(f(x))). B) 5 C) + 10 + 10

Express the given function as a composite of functions f and g such that y = f(g(x)). 16) y= A)

16) ______

B) f(x) =

, g(x) = x - 8

f(x) =

, g(x) =

f(x) =

, g(x) = -

-8

D) f(x) =

, g(x) =

-8

17) ______

17) y = A) f(x) = x, g(x) = 10x + 2

B) f(x) =

, g(x) = 10x - 2

C) f(x) = -

D) f(x) =

, g(x) = 10x + 2

18) ______

18) y= A)

+9 B) f(x) =

, g(x) =

f(x) =

, g(x) = 9

D) f(x) = x, g(x) =

f(x) = x + 9, g(x) = 19) ______

19) y= A) f(x) =

, g(x) = 8

B) f(x) =

D) f(x) = 8, g(x) = f(x) =

20)

y= A) C)

, g(x) = 4x + 6

, g(x) = 4x + 6 20) ______

f(x) = (-4

, g(x) = -12

f(x) = -4x - 12, g(x) =

Solve the problem. 21)

B) D)

f(x) =

, g(x) = -4x - 12

f(x) = -4

, g(x) = x - 12

Let f(x) =

___ ___

21) . Find a function y = g(x) so that (f ∘ g)(x) = x. A)

B) g(x) = x(x - 9)

g(x) = 22)

D) g(x) =

Let f(x) = . Find a function y = g(x) so that (f ∘ g)(x) = A) g(x) = 2x B) C) g(x) = g(x) =

23) Let g(x) =

. Find a function y = f(x) so that (f ∘ g)(x) = A) f(x) = x B) C)

g(x) = 22) ______

. -2

g(x) =

+2 23) ______

. f(x) =

f(x) =

24) Let g(x) = x + 1. Find a function y = f(x) so that (f ∘ g)(x) = 2x + 2 A) f(x) = 2x - 1 B) f(x) = 2x C) f(x) = 2(x + 1)

D) f(x) = 2x + 1

25)

The accompanying figure shows the graph of y = equation for the new graph.

A) 26)

B) y = x2 + 3

The accompanying figure shows the graph of y = equation for the new graph.

24) ______

shifted to a new position. Write the

D) y = x2 - 3

shifted to a new position. Write the

25) ______

___ ___

26)

A) y = x2 + 6 27)

The accompanying figure shows the graph of y = equation for the new graph.

A) y = x2 - 4 29)

The accompanying figure shows the graph of y = equation for the new graph.

A) y = x2 - 4 28)

The accompanying figure shows the graph of y = equation for the new graph.

D) y = x2 - 6

shifted to a new position. Write the

D) y = x2 + 4

shifted to a new position. Write the

27) ______

D) y = x2 + 4

shifted to a new position. Write the

28) ______

___ ___

29)

A) 30)

y=-

B) y = -x2 - 6

The accompanying figure shows the graph of y = equation for the new graph.

A) 32)

B) y = -x2 + 2

The accompanying figure shows the graph of y = equation for the new graph.

A) 31)

y=-

The accompanying figure shows the graph of y = equation for the new graph.

y=-

D) y = -x2 - 2

shifted to a new position. Write the

C) y = -x2 + 6

y=-

shifted to a new position. Write the

C) y = -x2 - 6

30) ______

D) y = -x2 + 6

shifted to a new position. Write the

31) ______

___ ___

32)

A) 33)

y=-

The accompanying figure shows the graph of y = equation for the new graph.

A) y = (x - 4)2 - 5 34)

B) y = -x2 + 6

B) y = (x - 5)2 + 4

The accompanying figure shows the graph of y = equation for the new graph.

A) y = -(x - 4)2 + 6

B) y = -(x + 4)2 - 6

C) y = -x2 - 6

y=-

shifted to a new position. Write the

C) y = (x - 5)2 - 4

D) y = (x + 5)2 + 4

shifted to a new position. Write the

C) y = -(x - 4)2 - 6

33) ______

34) ______

D) y = -(x - 6)2 + 4

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. 35)

+ 25

___ ___

35)

Up 2, right 5

36)

A) (x + 5)2 + (y - 2)2 = 25

B) (x - 5)2 + (y + 2)2 = 25

C) (x + 5)2 + (y + 2)2 = 25

D) (x - 5)2 + (y - 2)2 = 25

Down 6, left 1

___ ___

36)

37) y = -

y-6=

y+6=

Left 4

y-6=

y+6=

___ ___

37)

A) y =

C) y =

B) y =

38) y=

Down 5, right 6

D) y =

-4

___ ___

38)

B) y-5=

y+5=

D) y-5=

Graph the function. 39) y =

y+5=

___ ___

39)

40) y = |x - 6| - 6

___ ___

40)

41) y = 5 -

___ ___

41)

42)

___ ___

42)

43)

y+1=

___ ___

43)

44) y=

___ ___

44)

45) y=

___ ___

45)

46) y=

-2

___ ___

46)

47) y=

___ ___

47)

48) y =

___ ___

48)

49)

___ ___

49)

50) y=

-2

___ ___

50)

51) y=

-3

___ ___

51)

52)

___ ___

52)

53)

___ ___

53)

54) y =

___ ___

54)

Using the graph below, find the domain and range of the given function, and sketch the graph.

55) ______

55) y = -f(x)

A) D: [-7.5, 1]; R: [-1, 5]

C) D: [-7, 2]; R: [-4, 2]

B) D: [-5, 4]; R: [-3, 3]

D: [-4 , 5]; R: [-3 , 3]

56) ______

56) y = f(-x)

A) D: [-4, 5]; R: [-3, 3]

C) D: [-4, 5]; R: [-3, 3]

B) D: [-4, 5]; R: [-3, 4]

D: [-5 , 4]; R: [-3 , 3]

57) ______

57) y = f(x + 2) -1

A) D: [-7, 2]; R: [-4, 2]

C) D: [-4, 5]; R: [-3, 3]

B) D: [-8, 1]; R: [-2.5, 5]

D: [-5 , 4]; R: [-3 , 3]

58) ______

58) y = f(-x - 2) + 1

A) D: [-6, 3]; R: [-2, 4]

C) D: [-7, 2]; R: [-2, 4]

B) D: [-3, 6]; R: [-2, 4]

D: [-7 , 2]; R: [-2 , 4]

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. 59) 59) ______ y= +1 stretched horizontally by a factor of 2 A) B) C) D) y=4 +1 y=2 +2 y= 60)

y= A)

60) ______

compressed vertically by a factor of 4 B) C) +4 y=4 + 16

+4 y = 16

D) y=

+1 61) ______

61) y=1+ A)

stretched vertically by a factor of 5 B) C)

y=1+

D) y=1+

y=5+ 62) ______

62) y=1+ A)

compressed horizontally by a factor of 3 B) C)

y=1+ 63) y = A) y =

y=1+

y=3+

compressed vertically by a factor of 5 B) C) y =

D) y=

+ 63) ______

D) y = 5

y= 64)

stretched vertically by a factor of 2

64) ______

y=8

C) y=

y=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. 65) fg 65) ______ A) Even B) Odd 66) ______

66) f/g A) Even

B) Odd 67) ______

67) A) Even

B) Odd 68) ______

68) g ∘ f A) Even

B) Odd 69) ______

69) f ∘ f A) Even

B) Odd 70) ______

70) g ∘ g A) Even

B) Odd

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 71) Graph the functions f(x) = 71) _____________ and g(x) = together with their sum, product, two differences, and two quotients.

72)

Let f(x) = x - 1 and g(x) =

. Graph f and g together with f ∘ g and g ∘ f.

72)

___ ___ ___ ___ _

1) A 2) D 3) A 4) D 5) B 6) D 7) C 8) D 9) C 10) B 11) D 12) C 13) C 14) A 15) C 16) C 17) D 18) D 19) B 20) B 21) A 22) B 23) C 24) B 25) D 26) A 27) C 28) C 29) D 30) C 31) B 32) D 33) B 34) C 35) D 36) C 37) D 38) D 39) D 40) D 41) A 42) A 43) C 44) B 45) A 46) D 47) B 48) A 49) B 50) B 51) A

52) B 53) A 54) C 55) B 56) C 57) A 58) A 59) B 60) D 61) D 62) A 63) B 64) D 65) B 66) B 67) A 68) A 69) A 70) B 71)

72)

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) 1) _______ f(x) = -8 + 6x A) [-5, 5] by [-25, 15] B) [-10, 15] by [-5, 5] C) [-5, 5] by [-10, 15] D) [-25, 15] by [-5, 5] 2)

f(x) = -2 - 3x + 19 A) [-5, 25] by [-5, 5] C) [-2, 2] by [-10, 10] f(x) = 10 + 5x A) [-10, 10] by [-10, 5] C) [-4, 5] by [-5, 5] f(x) = A) [-10, 10] by [-10, 5] C) [-4, 5] by [-5, 5] f(x) = (3 - x) A) [0, 6] by [-10, 10] C) [-2, 2] by [-15, 15] f(x) = | - 4| A) [-5, 5] by [-2, 10] C) [-10, 10] by [-15, 15]

2) _______ B) [-20, 20] by [-100, 100] D) [-5, 5] by [-5, 25] 3) _______ B) [-10, 20] by [-50, 50] D) [-4, 5] by [-15, 25] 4) _______ B) [-10, 20] by [-50, 50] D) [-4, 5] by [-15, 25] 5) _______ B) [-4, 6] by [-10, 10] D) [-4, 0] by [-5, 5] 6) _______ B) [-5, 5] by [-15, 15] D) [0, 5] by [-2, 10] 7) _______

7) f(x) = A) [-10, 10] by [-10, 10] C) [-5, 5] by [-15, 15]

B) [-1, 1] by [-2, 2] D) [-10, 10] by [-2, 2] 8) _______

8) f(x) = A) [-5, 0] by [-10, 10] C) [-5, 5] by [-10, 10]

B) [-2, 2] by [-10, 10] D) [0, 5] by [-10, 10]

9) f(x) = 2 cos 70x A) [-0.2, 0.2] by [-4, 4] C) [-1, 1] by [-4, 4]

B) [-10, 10] by [-10, 10] D) [-0.2, 0.2] by [-1, 1]

9) _______

10) ______

10) f(x) = + cos 50x A) [-10, 10] by [-10, 10] C) [-0.1, 0.1] by [-0.1, 0.1]

B) [-0.6, 0.6] by [-0.1, 0.6] D) [-2, 2] by [-1, 1]

Determine an appropriate viewing window for the given function and use it to display its graph. 11) f(x) = -0.6 +5 -4 -7 +x-3

12)

f(x) =

11) ______

___ ___

12)

13)

f(x) =

-3

+ 15

+ x - 18

___ ___

13)

14) f(x) =

___ ___

14)

15) y = 16

___ ___

15)

16) f(x) =

___ ___

16)

17) f(x) =

___ ___

17)

18) f(x) =

___ ___

18)

19) f(x) =

___ ___

19)

20) f(x) =

___ ___

20)

Graph the function. 21) Graph the upper half of the circle defined by the equation x 2 + y2 - 10x - 10y + 34 = 0.

___ ___

21)

22)

Graph the lower branch of the hyperbola

-4

= 1.

___ ___

22)

23) Graph five periods of the function f(x) = tan 4x.

___ ___

23)

24) Graph two periods of the function f(x) = -cot

+ 1.

___ ___

24)

25) Graph the function f(x) = sin 4x + cos 2x.

___ ___

25)

26)

Graph the function f(x) = 2

___ ___

26)

Solve the problem. 27) The table shows the average weight for women of medium frame based on height. Find a regression line for the data.

27)

______ A) y = 3.2848x - 78.5727 C) y = 3.8428x - 73.7191

B) y = 3.9941x - 68.5636 D) y = 4.5781x - 52.6742

28) The table shows the amount of yeast cells (measured as biomass) growing over a 7-hour period in a nutrient. (a) Find a regression quadratic for the data. (b) Use the regression quadratic to predict the biomass of yeast in the nutrient after 10 hours.

A) C)

(a) y = 4.5432 (b) 542.662

+ 6.8417x + 19.925

(a) y = 5.7661 (b) 508.337

- 8.1557x + 13.284

B) D)

(a) y = 5.4215 (b) 486.406

- 7.1512x + 15.768

(a) y = 5.9631 (b) 538.723

- 7.1512x + 13.925

28) ______

1) A 2) D 3) D 4) D 5) B 6) A 7) D 8) C 9) A 10) B 11) A 12) C 13) B 14) A 15) C 16) A 17) B 18) C 19) C 20) B 21) D 22) D 23) D 24) A 25) A 26) B 27) A 28) D

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1)

1) _______

On a circle of radius 18 meters, how long is an arc that subtends a central angle of radians? A) 15 m B) 15π m C) 3π m D) 90 m 2) _______

2) On a circle of radius 3 meters, how long is an arc that subtends a central angle of 115°? A) 345 m B) 345π m C) D) m

3) You want to make an angle measuring 35° by marking an arc on the perimeter of a disk with a diameter of 16 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) 2.4 in. B) 9.8 in. C) 19.5 in. D) 4.9 in.

3) _______

4) If you roll a 1-m-diameter wheel forward 61 centimeters over level ground, through what angle (to the nearest degree) will the wheel turn? A) 122° B) 1° C) 35° D) 70°

4) _______

Find the exact value of the trigonometric function. Do not use a calculator or tables. 5) tan A)

B) 1

5) _______ D)

6) _______

6) sec A)

D) 2

7) _______

7) sin A)

8) _______

8) sin A)

C) -

D) 9) _______

9) cos A)

C) -

10)

D) tan

___ ___

10) A) 1

D) -1

C) -

11) ______

11) tan A) -1

B) 1

C) 0

D) Undefined 12) ______

12) csc (- π ) A) -1

B) 1

C) 0

D) Undefined

One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. 13) sin x = A)

13) ______

x in B)

cos x =

, tan x = -

cos x = -

, tan x = -

cos x = -

, tan x = -

D) cos x =

, tan x =

14) ______

14) cos x = A)

x in B)

sin x =

, tan x = -

sin x = -

, tan x = -

sin x = -

, tan x = -

D) sin x =

, tan x =

15) ______

15) tan x = A)

x in B)

sin x =

, cos x =

sin x = -

, cos x = -

sin x =

, cos x =

D) sin x = -

, cos x = -

16) ______

16) sin x = A)

x in B)

cos x = -

, tan x = -

cos x =

, tan x = -

cos x = -

, tan x =

D) cos x =

, tan x =

17) ______

17) cos x = A)

sin x = C)

x in B) , tan x = -2

sin x = si n

, tan x = 2

D) x=-

, tan

x = -2

si n x=

, ta n x= 2

18) ______

18) tan x = A)

x in B)

sin x =

, cos x =

sin x = -

, cos x =

sin x =

, cos x = -

D) sin x =

, cos x = -

19) ______

19) sin x = A)

x in B)

cos x = -

, tan x = -

cos x =

, tan x = -

cos x = -

, tan x =

D) cos x =

, tan x =

20) ______

20) cos x = A)

x in B)

sin x =

, tan x = 1

sin x = -

, tan x = 1

sin x = -

, tan x = -1

D) sin x =

, tan x = -1

21) ______

21) tan x = 1, A)

x in B)

sin x = -

, cos x = -

sin x =

, cos x =

sin x =

, cos x = -

D) sin x = -

, cos x =

22) ______

22) sin x = A)

x in B) cos x =

cos x = -2, tan x = C)

, tan x = -

D) sx=-

, tan x

co sx = , ta n x=

State the period of the function and graph. 23) -sin 8x

23) ______

B) Period

C) Period

Period

Pe rio d

24) ______

24) -cos

A) Period 4

C) Period 4

B) Period 4

Pe rio d 4

25) ______

25) sin

A) Period 2π

B) Period 2π

C) Period 2π

D) Period 2π

26) cos

___ ___

26)

A) Period 2π

B) Period 2π

C) Period 2π

D) Period 2π

27) sin

___ ___

27)

A) Period 2π

B) Period 2π

C) Period 2π

D) Period 2π

28) cos

-3

___ ___

28)

A) Period 2π

B) Period 2π

C) Period 2π

D) Period 2π

Graph the function in the ts-plane (t-axis horizontal, s-axis vertical). State the period and symmetry of the function. 29) s = -tan 4t

___ ___

29)

B) Period

, symmetric about the s-axis

C) Period π, symmetric about the origin

30) s = -sec

Period

, symmetric about the origin

Period

, symmetric about the origin

___ ___

30)

A) Period 4π, symmetric about the s-axis

B) Period 4π, symmetric about the t-axis

C) Period 4π, symmetric about the s-axis

D) Period 4π, symmetric about the t-axis

31) s = csc

___ ___

31)

A) Period 4, symmetric about the s-axis

B) Period 4, symmetric about the s-axis

C) Period 4, symmetric about the origin

D) Period 4, symmetric about the origin

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the addition formulas to derive the identity. 32) 32) _____________ cos

= sin x 33) _____________

33) cos

= -sin x 34) _____________

34) sin

= cos x

35) _____________

35) sin

= -cos x

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Express the given quantity in terms of sin x or cos x. 36) cos (6π + x) A) -cos x B) -sin x C) cos x - sin x D) cos x 37) cos (3π + x) A) cos x

36) ______

37) ______ B) sin x - cos x

C) -cos x

D) -sin x 38) ______

38) sin (3π + x) A) cos x - sin x

B) cos x + sin x

C) sin x

D) -sin x

39) sin (6π + x) A) -sin x

B) cos x - sin x

C) cos x + sin x

D) sin x

39) ______

40) ______

40) cos A) sin x

B) -sin x

C) cos x

D) cos x + sin x 41) ______

41) sin A) cos x 42) sin (6π - x) A) -sin x 43) cos (4π - x) A) -cos x

B) -cos x

C) -cos x - sin x

D) sin x - cos x 42) ______

B) cos x - sin x

C) sin x

D) sin (-x) 43) ______

B) cos x - sin x

C) cos x

D) cos x + sin x 44) ______

44) sin A) -cos x

B) -cos x - sin x

C) -cos (-x)

D) cos x 45) ______

45) cos A) sin (-x)

B) -sin x

C) sin x

D) cos x + sin x

Use the appropriate addition formula to find the exact value of the expression. 46) sin A)

46) ______ D)

47) ______

47) sin A)

D) -

48) ______

48) cos A)

49) ______

49) cos A)

50) ______

50) tan A)

B) -2 -

C) 2 +

51) ______

51) tan A)

B) -2 -

C) 2 +

52) ______

52) tan A)

C) 2 -

D) 2 +

53) ______

53) sin A)

54) ______

54) cos A)

B) -

55) ______

55) sin A)

Find the function value. 56) A)

56) ______ B) 2 -

57) ______

57) A)

B) 2 -

Solve for the angle θ, where 0 ≤ θ ≤ 2π 58) θ= A)

58) ______ B)

θ=

θ= , , D) θ = 0, π, 2π

59) ______

59) θ= A)

B) θ = 0, π, 2π

θ=

D) θ=

θ=

, 60) ______

60) θ= A) θ = 0, π, 2π

θ=

D) θ=

61) ______

61) θ= A) θ = 0, π, 2π

θ=

D) θ=

62) ______

62) sin 2θ - cos θ = 0 A) 0,

, π,

, 2π

θ=

D) ,

63) ______

63) sin 2θ + cos θ = 0 A) θ=

B) ,

θ=

D) ,

Find the requested information using the law of cosines and/or the law of sines. Round to three decimal places.

64) A triangle has sides a = 4 and b = 5 and angle C = 30°. Find the length of side c. A) 4.866 B) 4.583 C) 6.359 D) 2.522

64) ______

65) A triangle has sides a = 2 and b = 3 and angle C = 50°. Find the sine of B. A) 0.111 B) 0.333 C) 1

65) ______ D) 0.666 66) ______

66) A triangle has side c = 3 and angles A = A) 2.196 B) 0.244

and B =

. Find the length b of the side opposite B. C) 2.69 D) 0.732

For f(x) = A sin + D, identify either A, B, C, or D as indicated for the sine function. 67) y = sin A) 2π

Find A. B)

67) ______

C) 1

D) 2

68) ______

68) y= -2 cos A) 2π

Find B. B) 2

69) y = -2 cos

Find C. B)

A) -

D) 4

69) ______ C) π

D) -

π 70) ______

70) y = 4 - cos A) 4

Find D. B) π

C) -4

D) 1 71) ______

71) y = 5 cos A) 2

Find A. B)

C) 5

D) 10

72) ______

72) y = 2 sin A) 2

Find A. B)

C) 8

D) 4

73) ______

73) y = 3 sin A) π

74)

Find B. B)

C) 3

D) 4

y = cos 2

___ ___

74) F ind B. A)

C) 2

D) π

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 75) Use the angle sum formulas to derive sin (A - B) = sin A cos B - cos A sin B. 75) _____________ 76) The standard formula for the tangent of the difference of two angles is

76) _____________

Derive the formula. 77) _____________

77) Graph y = cos 2x and y = sec 2x together for ≤x≤ of sec 2x in relation to the signs and values of cos 2x.

. Comment on the behavior

78) _____________

78) Graph y = sin csc

and y = csc

together for -2π ≤ x ≤ 2π. Comment on the behavior of

in relation to the signs and values of sin

79) What happens if you set B = -2π in the angle sum formulas for the sine and cosine functions? Do the results agree with something you already know?

79) _____________

1) B 2) D 3) D 4) D 5) C 6) D 7) D 8) C 9) D 10) D 11) D 12) D 13) B 14) D 15) B 16) B 17) B 18) C 19) B 20) C 21) A 22) B 23) B 24) B 25) D 26) C 27) C 28) D 29) B 30) A 31) C 32) cos

= cos x cos - sin x sin = cos x (0) - sin x (-1) = 0 + sin x = sin x

cos

= cos x cos - sin x sin = cos x (0) - sin x (1) = 0 - sin x = -sin x

sin

= sin x cos + cos x sin = sin x (0) + cos x (1) = 0 + cos x = cos x

sin

= sin x cos + cos x sin = sin x (0) + cos x (-1) = 0 - cos x

33)

34)

35)

= -cos x 36) D 37) C 38) D 39) D 40) A 41) B 42) A 43) C 44) A 45) B 46) A 47) C 48) B 49) D 50) B 51) C 52) C 53) B 54) D 55) A 56) D 57) C 58) A 59) D 60) D 61) D 62) D 63) A 64) D 65) C 66) A 67) B 68) A 69) C 70) A 71) C 72) A 73) B 74) A 75) sin (A - B) = sin (A + (-B)) = sin A cos (-B) + cos A sin (-B) = sin A cos B - cos A sin B 76) tan (A - B) =

= . 77) When y = cos 2x is at a maximum point, which is at any multiple of π, y = sec 2x is a minimum point. Similarly, when cos (2x) is at a minimum point, which is at any odd multiple of

, y = sec 2x is a at maximum point.

78) When y = sin

is at a maximum point, which is at x = (4n + 1)π for all integers n, y = csc

is at a minimum point.

Similarly, when y = sin is at minimum point, , which is at x = (4n - 1)π for all integers n, y = csc is at a maximum point. 79) If B = -2π, then cos (A + B) = cos A and sin (A + B) = sin A. Because the period of both of the sine and cosine functions is 2π, if B is replaced by a multiple of 2π the angle sum formulas must produce the same value as the sine or cosine function.

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Graph the function. 1) f(x) =

2) f(x) =

1) _______

__ __ __ _

f(x) =

3) _______

4) _______

f(x) =

5) _______

f(x) =

f(x

__ __ __ _

6) )= +1

f(x) =

-2

_______

f(x) =

8) _______

10)

f(x) =

9) _______

f(x)

= 2

___ ___

10)

Use the laws of exponents to simplify. Do not use negative exponents in your answer. 11) ∙ A) B) C) 12)

D) 13) ______

∙ A)

14)

D) 12) ______

∙ A)

13)

11) ______

∙7∙

D) 14) ______

15) ______

15) A)

16) ______

16) A)

D) 17) ______

17) A)

B) 4

18) ______

18) A)

C) 1

D) 19) ______

19) A)

20) ______

20) A) 3

B) 32

State the domain and range of the function. 21) f(x) = -4 A) domain: (-∞, ∞); range: (-∞, -4] C) domain: [0, ∞); range: (-∞, ∞) 22)

23)

24)

f(x) = +2 A) domain: (-∞, ∞); range: (-∞, ∞) C) domain: (-∞, 0) ∪ (0, ∞); range: (2, ∞) f(x) = -3 A) domain: (-∞, 0) ∪ (0, ∞); range: (-3, ∞) C) domain: (-∞, ∞); range: [-3, ∞) f(x) = -5 A) domain: (-∞, ∞) ∪ (-∞, ∞); range: (-∞, -5) C) domain: (-∞, ∞); range: (-∞, -5)

D) 2

21) ______ B) domain: (-∞, ∞); range: (-∞, -4) D) domain: (-∞, 0) ∪ (0, ∞); range: (-∞, -4) 22) ______ B) domain: (-∞, ∞); range: (2, ∞) D) domain: (-∞, ∞); range: [2, ∞) 23) ______ B) domain: (-∞, ∞); range: (-∞, ∞) D) domain: (-∞, ∞); range: (-3, ∞) 24) ______ B) domain: (-∞, ∞); range: (-∞, ∞) D) domain: (-∞, ∞); range: (-∞, -4)

25)

26)

27)

28)

f(x) = +2 A) domain: (-∞, ∞) ∪ (-∞, ∞); range: (-∞, 2) C) domain: (-∞, ∞); range: (2, ∞) f(x) = -5 A) domain: (-∞, ∞); range: (-∞, -5) C) domain: (-∞, ∞); range: (-5, ∞) f(x) = 5 -3 A) domain: (-∞, ∞); range: (-∞, ∞) C) domain: (-∞, ∞); range: (-∞, 2] f(x) = -3 A) domain: (-∞, ∞); range: (-∞, ∞) C) domain: (-∞, ∞) ∪ (-∞, ∞); range: (-∞, -3)

25) ______ B) domain: (-∞, ∞); range: (-∞, 2] D) domain: (-∞, ∞); range: (-∞, 2) 26) ______ B) domain: (-∞, ∞); range: (-∞, ∞) D) domain: (-∞, ∞); range: (-∞, 5] 27) ______ B) domain: (-∞, ∞); range: (-3, ∞) D) domain: (-∞, 0) ; range: (-∞, -3) 28) ______ B) domain: (-∞, ∞); range: (-∞, -3) D) domain: (-∞, ∞); range: (-∞, -3] 29) ______

29) f(x) = A) domain: (0, ∞); range: (0, ∞) C)

B) domain: (-∞, ∞); range: (0, 5) D) domain: (-∞, ∞); range: (-∞, ∞)

domain: (-∞, ∞); range: 30)

f(x) = A) domain: (-∞, ∞); range: (-∞, ∞) C) domain: (0, ∞); range: (-∞, ∞)

30) ______ B) domain: (-∞, ∞); range: (4, ∞) D) domain: (-∞, ∞); range: (0, 4)

Use a graph to find an approximate solution to the equation. Round to the nearest thousandth. 31) = 24 A) 4.800 B) 1.569 C) 1.975 D) 0.506 32)

= 24 A) 7.000

33)

34)

35)

36)

37)

= 18 A) 1.723 6

= 36 A) 1.132 = A) 32.514 = 27 1.412 A) = A) 2.113

31) ______

32) ______ B) 3.292

C) 2.792

D) 1.292 33) ______

B) 1.066

C) 1.057

D) 3.333 34) ______

B) -0.535

C) -0.734

D) 5.167 35) ______

B) -33.644

C) -31.384

D) -32.514 36) ______

B) 3.860

C) 1.837

D) -1.412 37) ______

B) -9.827

C) 0.590

D) 1.113

38)

39)

40)

=2 A) -1.006 = -0.127 A) 170 = 340 A) 5.780

38) ______ B) -2.994

C) -4.246

D) -2.626 39) ______

B) 1.324

C) 2.413

D) 1.143 40) ______

B) 12.473

C) 9.673

D) 11.073

Solve the problem. 41) In the formula N = I , N is the number of items in terms of an initial population I at a given time t and k is a growth constant equal to the percent of growth per unit time. How long will it take for the population of a certain country to double if its annual growth rate is 1.5%? Round to the nearest year. A) 20 yr B) 46 yr C) 133 yr D) 1 yr 42)

43)

44)

45)

In the formula N = I , N is the number of items in terms of an initial population I at a given time t and k is a growth constant equal to the percent of growth per unit time. How long will it take for the population of a certain country to triple if its annual growth rate is 6.8%? Round to the nearest year. A) 16 yr B) 7 yr C) 44 yr D) 1 yr In the formula N = I , N is the number of items in terms of an initial population I at a given time t and k is a growth constant equal to the percent of growth per unit time. There are currently 63 million cars in a certain country, increasing by 3.7% annually. How many years will it take for this country to have 92 million cars? Round to the nearest year. A) 8 yr B) 10 yr C) 4 yr D) 91 yr The number of acres in a landfill decreases according to the function B = 7900 , where t is measured in years. How many acres will the landfill have after 10 years? A) 12,715 acres B) 4985 acres C) 5522 acres D) 6468 acres A bacteria colony doubles in 5 hr. How long does it take the colony to triple? Use N =

41) ______

42) ______

43) ______

44) ______

45) ______

where is the initial number of bacteria and T is the time in hours it takes the colony to double. (Round to the nearest hundredth, as necessary.) A) 7.92 hr B) 15 hr C) 2.03 hr D) 7.5 hr 46)

47)

The population of a small country increases according to the function B = 1,800,000 t is measured in years. How many people will the country have after 4 years? A) 2,852,808 people B) 1,258,146 people C) 2,896,988 people D) 2,198,525 people

t, where

Use the formula P = I , where P is the resulting population, I is the initial population, and t is measured in hours. A bacterial culture has an initial population of 10,000. If its population declines to 5000 in 6 hours, what will it be at the end of 8 hours? A) 3969 B) 1985 C) 2500 D) 4353

46) ______

47) ______

48) The amount of particulate matter left in solution during a filtering process decreases by the

48) ______

equation where n is the number of filtering steps. Find the amounts left for n = 0 and n = 5. (Round to the nearest whole number.) A) 400, 25 B) 400, 13 C) 400, 6400 D) 800, 25 49)

The decay of 575 mg of an isotope is given by A(t)= 575 , where t is time in years. Find the amount left after 66 years. A) 35 mg B) 67 mg C) 70 mg D) 557 mg

49) ______

50) Find the amount of interest earned on the following deposit: $1,000 at 11% compounded annually for 3 years A) $232.10 B) $367.63 C) $1367.63 D) $518.07

50) ______

51) Find the amount of interest earned on the following deposit: $1,000 at 10% compounded annually for 7 years A) $1143.59 B) $948.72 C) $1948.72 D) $771.56

51) ______

52) Find the amount of interest earned on the following deposit: $1,000 at 11% compounded annually for 7 years A) $2076.16 B) $870.41 C) $1076.16 D) $1304.54

52) ______

53) Suppose the consumption of electricity grows at 6.9% per year, compounded continuously. Find the number of years before the use of electricity has tripled. Round the answer to the nearest hundredth. A) 43.48 yr B) 0.16 yr C) 1.59 yr D) 15.92 yr

53) ______

54) An economist predicts that the buying power B(x) of a dollar x years from now will decrease

54) ______

according to the formula B(x) = . How much will today's dollar be worth in 5 years? Round the answer to the nearest cent. A) $2.46 B) $0.75 C) $2.80 D) $0.06 55) The purchasing power of a dollar is decreasing at the rate of 4.2% annually, compounded continuously. How long will it take for the purchasing power of $1.00 to be worth $0.51? Round answers to the nearest hundredth. A) 12.14 yr B) 1.60 yr C) 16.03 yr D) 0.16 yr

55) ______

1) B 2) C 3) D 4) D 5) A 6) D 7) C 8) A 9) B 10) A 11) B 12) C 13) C 14) D 15) B 16) D 17) D 18) B 19) D 20) D 21) B 22) B 23) D 24) C 25) C 26) C 27) B 28) B 29) C 30) B 31) C 32) B 33) A 34) B 35) D 36) C 37) C 38) D 39) D 40) D 41) B 42) A 43) B 44) D 45) A 46) D 47) A 48) A 49) C 50) B 51) B

52) C 53) D 54) D 55) C

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Is the function graphed below one-to-one? 1)

A) No

B) Yes 2) _______

A) No

B) Yes 3) _______

A) Yes 4)

1) _______

B) No

__ __ __ _

A) No

B) Yes 5) _______

A) No

B) Yes 6) _______

A) No 7)

B) Yes

__ __ __ _

A) No

B) Yes

Determine from its graph if the function is one-to-one. 8) f(x) = A) Yes

8) _______ B) No 9) _______

9) f(x) = A) Yes

B) No 10) ______

10) f(x) = A) Yes

B) No 11) ______

11)

f(x) = A) Yes

B) No 12) ______

12) f(x) = A) Yes

B) No

Graph the inverse of the function plotted, on the same set of axes. Use a dashed curve for the inverse. 13)

13) ______

14) ______

14)

15) ______

15)

16)

___ ___

16)

17) ______

17)

A) Function is its own inverse.

18) ______

18)

19) ______

19)

Find the inverse of the function.

20) ______

20) f(x) = 4x - 5 A)

f-1(x) = C) Not a one-to-one function

f-1(x) = D) f-1(x) =

21)

f(x) = A)

+5 21) ______

f-1(x) = C) Not a one-to-one function

B) D)

f-1(x) =

-2

f-1(x) = 22) ______

22) f(x) = 2x3 - 5 A)

f-1(x) = C) Not a one-to-one function

f-1(x) =

D) f-1(x) = 23) ______

23) f(x) =

, x ≥ -5 A) f-1(x) = -x2 + 5, x ≥ 0

B) Not a one-to-one function

C) f-1(x) = x2 - 5, x ≥ 0

D) f-1(x) = x2 - 25, x ≥ 0

24) f(x) =

24) ______

- 3, x ≥ 0

A) -1 f (x) = C) -1 f (x) =

,x≥3

B) f-1(x) = x + 3, x ≥ 3

,x≥3

D) -1 f (x) = 25) ______

25) f(x) =

,x≥9 A) f-1(x) = x2 + 9, x ≥ 0

B) f-1(x) = x + 9, x ≥ 0

C) Not a one-to-one function

D) f-1(x) = x2 - 9, x ≥ 0 26) ______

26) f(x) = A) Not a one-to-one function

B) f-1(x) =

D) f-1(x) =

f-1(x) = 27) ______

27) f(x) =

28)

A) f-1(x) = (x - 4)2

B) Not a one-to-one function

C) f-1(x) = x2 + 4, x ≥ 0

D) f-1(x) =

f(x) = A) C)

28) ______

, x ≥ 10 (x) =

+ 10, x ≥ 0

B) Not a one-to-one function

(x) = x≥0

+ 10,

D) (x) = ,x ≥ 10

Find the domain and range of the inverse of the given function. 29) f(x) = 3.9 - 2.12x A) Domain: all real numbers; range: [3.9, ∞) B) Domain: [3.9, ∞); range: all real numbers C) Domain: all real numbers; range: (-∞, 3.9] D) Domain and range: all real numbers

29) ______

30) ______

30) f(x) = x - 8 A) Domain: (-∞, 8) ∪ (8, ∞); range: (-∞, 4) ∪ (4, ∞) B) Domain and range: all real numbers C) Domain and range: (-∞, 4) ∪ (4, ∞) D) Domain: (-∞, 4) ∪ (4, ∞); range: all real numbers 31)

f(x) = -3 A) Domain: [-3, ∞); range: all real numbers C) Domain: all real numbers; range: [-3, ∞)

31) ______ B) Domain: [0, ∞); range: [0, ∞) D) Domain and range: all real numbers 32) ______

32) f(x) = A) Domain: [4, ∞); range: [0, ∞) C) Domain and range: all real numbers

B) Domain: [4, ∞); range: [4, ∞) D) Domain: [0, ∞); range: [4, ∞) 33) ______

33) f(x) = A) Domain: (-∞, 0) ∪ (0, ∞); range: (-∞, 0) C) Domain: (0, ∞); range: (-∞, 0) 34)

35)

f(x) = A) Domain: [0, ∞); range: all real numbers C) Domain: [4, ∞); range: [0, ∞) f(x) = + 9, x ≥ 0 A) Domain: [0, ∞); range: [9, ∞) C) Domain: (-∞, 0]; range: (-∞, 9]

B) Domain and range: all real numbers D) Domain and range: (-∞, 0) ∪ (0, ∞) 34) ______ B) Domain: [3, ∞); range: [0, ∞) D) Domain and range: all real numbers 35) ______ B) Domain and range: all real numbers D) Domain: [9, ∞); range: [0, ∞) 36) ______

36) f(x) = ,x≥0 A) Domain: (0, 3]; range: [0, ∞) C) Domain: [0, ∞); range: (0, 3] Express the following logarithm as specified. 37) ln 4.5 in terms of ln 2 and ln 3

B) Domain and range: [0, ∞) D) Domain: (-∞, 0}; range: [-3, 0)

37) ______

A) 2 ln 3 + ln 2 38)

B) 2 ln 3 - 2 ln 2

C) 2 ln 3 - ln 2

D) ln 3 + 2 ln 2 38) ______

in terms of ln 3 A)

ln 9 39) ln

ln 3

40) ln (1/625) in terms of ln 5 A) -4 ln 5

ln 3 39) ______

in terms of ln 3 and ln 2 B)

D) 6 ln 3

C) 4 ln 3

C) 4 ln 625

40) ______ B) ln 5

41) ln 2

in terms of ln 2 and ln 3 B)

ln 3 +

41) ______ C)

ln 2 +

42) ln 36 in terms of ln 2 and ln 3 A) -2 ln 2 - 2 ln 3 B) 4 ln 2

ln 5

ln 2 42) ______ C) 2 ln 2 + 2 ln 3

D) 2 ln 2 - 2 ln 3 43) ______

43) in terms of ln 2 and ln 3 B)

Express as a single logarithm and, if possible, simplify. 44) ln ( - 36) - ln (x + 6) A) ln (x + 6) B) ln (x - 36)

C) 1

44) ______ C) ln (x - 6)

ln (

- 6) 45) ______

45) ln cos θ - ln A)

C) ln cos θ

D) ln 6

46) ______

46) ln

+ ln A)

B) ln (x - 6)

ln C) ln (x - 2)

ln 47) ______

47) ln A)

- ln 5 ln

ln 5

D) ln

48) ______

48) ln (6 sec θ) + ln (2 cos θ) A) ln

B) ln (6 sec θ + 2 cos θ)

C) ln (12) 49) ln (8x + 4) - 2 ln 2 A) ln (2x + 2)

D) ln (12 cot θ) 49) ______ B) ln (8x)

C) ln (2x + 1)

D) ln (16(2x + 1))

Simplify the expression. 50) A) 2.05

50) ______ B) 2440.59

C) 7.8

D) 21.20 51) ______

51) A) 7

52) ______

52) A) ln x + ln y

C) xy

D) x+y 53) ______

53) A)

D) 10x ln

54)

55)

56)

54) ______

3 ln A)

B) 3

D) 1

55) ______

ln( ) A) 8

ln( A)

C) e

D) 8x

56) ______

) ln

Solve for t or y, as appropriate. 57) ln y = 5x + 8 A) 5x 58) ln(y - 36) = 2x A)

59) ln(9 - 4y) = x A)

60) ln(y - 9) - ln 3 = x + ln x A) + 3x + 9

B) ln 9

C) 9

B) 13

D) ln(5x + 8)

57) ______

58) ______ B) 2x + 36

+ 36

D) ln(2x) + 36

59) ______ B)

-5

60) ______ B)

C) 2x + 12

(x + 3)

61)

61) ______

=9 A)

B) ln 9

C) ln 3

ln 62)

100 A)

62) ______

= 700 B)

ln 63)

=k A) 990 ln k

64)

63) ______ B) ln 990k

990

64) ______

= 0.3 A)

D) ln

65)

65) ______

= A) (ln

66)

)

66) ______

= A)

ln(

+ 7x + 8)

+ 7x + 8

- 7x - 8

Simplify the expression. 67) log 100 10 A) 100

67) ______ B) 20

C) 10

D) 2 68) ______

68) log 3 A) 3 69)

B) -1

C) 1

D) 0 69) ______

ln A)

e 70) ______

70) A) -6 71)

10 A) 10

B) 6

C) 2

D) -2 71) ______

B) 0

C) -1

D) 1

72) ______

72) A) 5

B) 9

D) 45 73) ______

73) A) 2

D) - 2 74) ______

74) A)

B) 1

C) 2x

D) 9 75) ______

75) A)

B) 31

C) log

D) 76) ______

76) A)

B) 2

Rewrite the ratio as a ratio of natural logarithms and simplify. 77) A)

77) ______ C)

78) ______

78) A) 3

C) ln 3

79) ______

79) A)

C) ln 5

80) ______

80)

Find the exact function value. 81) (0) A) π

81) ______ B)

C) 0

82) ______

82) A)

C) 1

D) 0

83) ______

83) sin-1 A)

84) ______

84) arcsin A)

C) -

D) 85) ______

85) cos-1 A)

87) tan-1 (1) A)

(

86) ______

86) arccos(0) A)

88)

C) 0

87) ______ B)

C) 1

D) 0

88) ______

)

± 2πn,

89) arccsc(-2) A)

± 2πn

89) ______ B)

D) -

90)

90) ______

(-1) A)

B) -

Solve the problem. 91) How long will it take for prices in the economy to double at a 6% annual inflation rate? Round the answer to the nearest hundredth. A) 18.85 yr B) 10.24 yr C) 23.45 yr D) 11.9 yr 92) Suppose the consumption of electricity grows at 4.2% per year, compounded continuously. Find the number of years before the use of electricity has tripled. Round the answer to the nearest hundredth. A) 0.26 yr B) 26.16 yr C) 2.62 yr D) 71.43 yr

91) ______

92) ______

93)

94)

95)

In the formula N = I , N is the number of items in terms of an initial population I at a given time t and k is a growth constant equal to the percent of growth per unit time. How long will it take for the population of a certain country to double if its annual growth rate is 6.9%? Round to the nearest year. A) 29 yr B) 10 yr C) 4 yr D) 1 yr In the formula N = I , N is the number of items in terms of an initial population I at a given time t and k is a growth constant equal to the percent of growth per unit time. There are currently 77 million cars in a certain country, increasing by 1% annually. How many years will it take for this country to have 92 million cars? Round to the nearest year. A) 5 yr B) 271 yr C) 18 yr D) 15 yr The population of a small country increases according to the function B = 1,300,000 is measured in years. How many people will the country have after 4 years? A) 1,465,746 people B) 2,756,343 people C) 1,197,064 people D) 1,713,734 people

, where t

96) 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 A) 750 yr 97)

In how many years will there be 85% of the isotope left? B) 234 yr C) 1500 yr

93) ______

94) ______

95) ______

96) ______

D) 813 yr

In the formula A = I , A is the amount of radioactive material remaining from an initial amount I at a given time t and k is a negative constant determined by the nature of the material. A certain radioactive isotope has a half-life of approximately 1200 years. How many years would be required for a given amount of this isotope to decay to 20% of that amount? A) 386 yr B) 960 yr C) 2766 yr D) 2786 yr

Provide an appropriate response. 98) Consider the graph of f(x) = , 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 not its own

97) ______

98) ______

inverse because C) The graph of f is symmetric with respect to the y-axis. The function f is its own inverse because D) The graph of f is symmetric with respect to the line y = x. The function f is its own inverse because 99) Consider a linear function that is perpendicular to the line y = x. Will this function be its own inverse? Explain. A) No it won't be its own inverse. The slope will be the same but the y-intercept will be different. 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) Yes it will be its own inverse. All perpendicular lines are their own inverses. D) No it won't be its own inverse. Its inverse will be some other line that is perpendicular to it.

99) ______

100) 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 y-axis. It will be one-to-one. B) There is not enough information to determine whether g(x) is one-to-one. C) g(x) is a reflection of f(x) across the line y = x. It will not be one-to-one. D) g(x) is a reflection of f(x) across the x-axis. It will be one-to-one.

100) _____

101)

101) _____

Find the inverse of the function f(x) = x + 5. How is the graph of related to the graph of f? A) f(x) = -x - 5. The graph of is a line perpendicular to the graph of f at x = 5. B) f(x) = + 5. The graph of equidistant from the y-axis.

is a curve intersecting the graph of f at two points

f(x) = x + . The graph of is a line parallel to the graph of f. The graphs of f and on the same side of the line y = x.

lie

f(x) = x - 5. The graph of is a line parallel to the graph of f. The graphs of f and on opposite sides of the line y = x and are equidistant from that line.

lie

102) If f(x) is one-to-one, can anything be said about h(x) = 6f(x) + 6? Is it also one-to-one? Give reasons for your answer. A) 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. B) No, h(x) will not be one-to-one. The function h(x) does not pass the horizontal line test. C) Yes, h(x) will be one-to-one. For every distinct value of f(x) there is one distinct value of h(x). D) Yes, h(x) will be one-to-one. The inverse of f(x) is h(x) and is therefore one-to-one.

102) _____

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 103) 103) ____________ Derive the identity (-x) = π x by combining the following two equations: (-x) = π x=

x (1/x)

1) A 2) B 3) B 4) A 5) B 6) B 7) A 8) B 9) B 10) A 11) A 12) B 13) A 14) A 15) A 16) C 17) A 18) A 19) D 20) A 21) D 22) A 23) C 24) C 25) A 26) B 27) C 28) A 29) D 30) B 31) D 32) D 33) D 34) D 35) D 36) A 37) C 38) C 39) A 40) A 41) C 42) C 43) D 44) C 45) D 46) C 47) D 48) C 49) C 50) C 51) D

52) C 53) A 54) D 55) D 56) A 57) C 58) C 59) C 60) B 61) C 62) D 63) A 64) D 65) C 66) C 67) D 68) B 69) A 70) D 71) D 72) B 73) C 74) C 75) D 76) C 77) D 78) A 79) A 80) B 81) C 82) D 83) C 84) B 85) B 86) A 87) B 88) A 89) C 90) A 91) D 92) B 93) B 94) C 95) A 96) D 97) D 98) D 99) B 100) A 101) D 102) C 103)

(-x) =

(-1/x) = π -

(1/x) = π -

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the average rate of change of the function over the given interval. 1) y= + 2x, [3, 7] A) B) 9 C) 12 D)

y=4 A)

2) _______

- 3, [-6, 2] B) 352

3) y =

C) 88

3) _______

, [2, 8]

A) 2

1) _______

C) 7

4) _______

4) y= A)

, [4, 7] B) 2

C) 7

D) 5) _______

5) y=4 , A)

B) 2

C) 7

y = -3 - x, [5, 6] A) -34

6) _______ B) -2

D) 7) _______

7) h(t) = sin (3t), A)

D) 8) _______

8) g(t) = 3 + tan t, A)

C) 0

Find the slope of the curve at the given point P and an equation of the tangent line at P. 9) y = x2 + 5x, P(4, 36) A) slope is -39; y = -39x - 80 B) C) slope is 13; y = 13x - 16

slope is -

;y=-

slope is

;y=

9) _______ +

D) +

10) ______

10) y = x2 + 11x - 15, P(1, -3)

11)

12)

13)

A) slope is 13; y = 13x - 16

C) slope is -39; y = -39x - 80

slope is

;y=

slope is -

;y=-

+ + 11) ______

y= - 9x, P(1, -8) A) slope is 3; y = 3x - 7 C) slope is 3; y = 3x - 11

B) slope is -6; y = -6x - 2 D) slope is -6; y = -6x 12) ______

y= - 3 + 4, P(3, 4) A) slope is 1; y = x - 23 C) slope is 9; y = 9x + 4

B) slope is 9; y = 9x - 23 D) slope is 0; y = -23 13) ______

y = -3 - , (1, -4) A) slope is -1; y = -x - 1 C) slope is -3; y = -3x - 1

B) slope is 0; y = -1 D) slope is 3; y = 3x - 1

Use the slopes of UQ, UR, US, and UT to estimate the rate of change of y at the specified value of x. 14) x = 5

A) 0 15) x = 3

B) 2

C) 1

14) ______

D) 5 15)

______ A) 4

B) 2

C) 6

D) 0 16) ______

16) x = 5

D) 0

17) ______

17) x = 2

A) 0 18) x = 2.5

B) 6

C) 3

D) 4 18)

______ A) 0

B) 1.25

C) 7.5

D) 3.75

Use the table to estimate the rate of change of y at the specified value of x. 19) x = 1.

A) 2

B) 1

C) 1.5

19) ______

D) 0.5 20) ______

20) x = 1.

A) 2

B) 1.5

C) 0.5

D) 1 21) ______

21) x = 1.

A) 4 22) x = 2.

B) 8

C) 6

D) 2 22) ______

A) 8

B) 4

C) -8

D) 0 23) ______

23) x = 1.

A) 0

B) 0.5

C) 1

D) -0.5

Solve the problem. 24) 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:

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.

24) ______

37.5 minutes 2.6 da ys D)

0.1 day

5.8 days

25) 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.

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.

25)

______ A)

-0.14 day per minute C)

5.6 days D)

38 minutes

-6.7 days per minute

26) The graph below shows the number of tuberculosis deaths in the United States from 1989 to 1998.

Estimate the average rate of change in tuberculosis deaths from 1991 to 1993. A) About -30 deaths per year B) About -0.4 deaths per year C) About -80 deaths per year D) About -45 deaths per year

26) ______

1) C 2) C 3) D 4) D 5) C 6) A 7) B 8) D 9) C 10) A 11) B 12) B 13) C 14) B 15) A 16) B 17) B 18) D 19) B 20) C 21) C 22) D 23) B 24) D 25) A 26) D

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the graph to evaluate the limit. 1)

1) _______

f(x)

A) ∞

C) -1

D) 2) _______

2) f(x)

A) 0 3)

B) -3

C) does not exist

D) 3 3)

f(x)

_______ A) -3

B) 3

C) does not exist

D) 0 4) _______

4) f(x)

A) 0

B) 6

C) does not exist

D) -1 5) _______

5) f(x)

A) ∞ 6)

B) 1

C) -1

D) does not exist 6)

f(x)

_______ A) 1

B) does not exist

C) ∞

D) -1 7) _______

7) f(x)

A) 2

B) does not exist

C) 0

D) -2 8) _______

8) f(x)

A) 1 9)

B) does not exist

C) 0

D) -2 9)

f(x)

_______ A) -2

B) -1

C) 2

D) does not exist 10) ______

10) f(x)

A) 1

B) does not exist

C) 0

D) -1

Find the limit. 11)

11) ______

(4x + 6) A) 34

B) -22

C) 10

D) 6 12) ______

12) ( A) 0

+ 8x - 2) B) -18

C) 18

D) does not exist 13) ______

13) ( - 5) A) does not exist

B) 5

C) -5

D) 0 14) ______

14) ( - 2) A) -2

B) does not exist

C) 2

D) 0 15) ______

15) ( +5 A) 29

- 7x + 1) B) does not exist

C) 15

D) 0 16) ______

16) (2 A) 41

-2

-4

+ 5) B) 9

C) -119

D) -55 17) ______

17) A) 8 18)

B) ±8

C) 64

D) does not exist 18) ______

A) 0

C) does not exist

D) 1

Find the limit if it exists. 19) A) 3

19) ______ B)

C) 15

D) 20) ______

20) (10x - 2) A) -8

B) 8

C) 12

D) -12 21) ______

21) (5 - 10x) -125 A)

B) 115

C) 125

D) -115 22) ______

22) (10 A) 34

- 2x - 10) B) 46

C) 26

D) 54 23) ______

23) 8x(x + 8)(x - 4) A) 2240

B) -20,160

C) -960

D) -2240 24) ______

24) 8x A)

B) -

D) -

25) ______

25) A) 5

C) 25

26) ______

26) A) 1

B) 25

C) -3125

D) -125 27) ______

27) A)

B) -17

C) 17

D) 28) ______

28) (x - 123 A) -3125

B) 625

C) -625

D) -125

Find the limit, if it exists. 29) A) 12

29) ______ B) Does not exist

C) 24

D) 0

30) ______

30) A) Does not exist

B) -4

C) 0

D) 4 31) ______

31) A)

B) -

C) Does not exist

32) ______

32) A) Does not exist

B) 0

D) -

33) ______

33) A) 6

B) Does not exist

C) 0

D) -6 34) ______

34) A) 5

B) -8

C) Does not exist

D) 0 35) ______

35) A) 1/2

B) 2

C) 1

D) Does not exist 36) ______

36) A)

B) Does not exist

D) 17x

37) ______

37) A) 0

B) 1/2

C) 1/4

D) Does not exist 38) ______

38) A) 1/3

B) 3

C) 0

D) Does not exist 39) ______

39) A) -1

B) 5

C) Does not exist

D) 0 40) ______

40) A) 4

B) Does not exist

C) 2

D) 0

41) ______

41) A) 12

B) Does not exist

C) 1

D) 6 42) ______

42) A) 6

B) 36

C) 0

D) Does not exist 43) ______

43) A) Does not exist

B) 7

C) 3

D) 0 44) ______

44) A) 0

C) Does not exist

D) 45) ______

45) A)

C) 0

D) Does not exist

46) ______

46) A)

B) Does not exist

C) - 1

47) ______

47) A) 3x2

B) Does not exist

C) 3x2 + 3xh + h2

D) 0 48) ______

48) A) 1

B) Does not exist

C) 0

D) -1

Find the limit. 49)

49) ______

(4 sin x - 1) A) 4

B) 4 - 1

C) -1

D) 0 50) ______

50) cos(x + π) A) 0

C) -

D) 1 51) ______

51) A) 4

B) 15

C) 16

Provide an appropriate response. 52)

52) ______

Suppose f(x) = 1 and g(x) = -3. Name the limit rules that are used to accomplish steps (a), (b), and (c) of the following calculation.

= A) (a) (b) (c) B) (a) (b) (c) C) (a) (b) (c) D) (a) (b) (c)

= Quotient Rule Difference Rule, Power Rule Constant Multiple Rule and Sum Rule Quotient Rule Difference Rule, Sum Rule Constant Multiple Rule and Power Rule Difference Rule Power Rule Sum Rule Quotient Rule Difference Rule Constant Multiple Rule 53) ______

53) Let f(x) = -9 and A) -9

g(x) = 6. Find B) -3

[f(x) - g(x)]. C) -15

D) 6 54) ______

54) Let A) -5

f(x) = 9 and

g(x) = 2. Find B) 11

[f(x) ∙ g(x)]. C) 18

D) 2 55) ______

55) Let A) 5

f(x) = -1 and

g(x) = -5. Find B)

. C) 3

D) 4

56) ______

56) Let f(x) = 32. Find A) 25

f(x). B)

C) -8

D) 5

57) ______

57) Let f(x) = 49. Find A) 49

. B) 1

C) 2.6458

D) 7

58) ______

58) Let f(x) = -5 and A) 74

g(x) = -7. Find B) 2

. C) -12

D) 144 59) ______

59) Let f(x) = 2. Find A) -4

. C) 256

B) 2

D) 16 60) ______

60) Let A) 2

f(x) = 32. Find

. B) 7

C) 32

D) 5 61) ______

61) Let A)

f(x) = 6 and

g(x) = -9. Find B) 9

. D) - 15

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form occur frequently in calculus. Evaluate this limit for the given value of x and function f. 62)

63)

f(x) = 4 , x = 8 A) 256 f(x) = 4 - 2, x = -1 A) -10

64) f(x) = 2x + 5, x = 4 A) 2

62) ______ B) Does not exist

C) 32

D) 64 63) ______

B) 4

C) Does not exist

D) -8

C) Does not exist

D) 13

64) ______ B) 8

65) ______

65) f(x) = + 1, x = 6 A) Does not exist

C) 2

D) 3

66) ______

66) f(x) = , x = 8 A) Does not exist

D) -16

C) -

67) ______

67) f(x) = 5

, x=4 A) Does not exist

68) f(x) =

C) 10

68) ______

, x=3

69) f(x) = 3

B) 5

+ 5, x = 16

C) Does not exist

69) ______

A) 24

B) 6

C) Does not exist

Provide an appropriate response. 70)

70) ______

It can be shown that the inequalities -x ≤ x cos

≤ x hold for all values of x ≥ 0.

Find x cos if it exists. A) does not exist B) 0

C) 1

D) 0.0007 71) ______

71) The inequality 1Find A) 0

< 1 holds when x is measured in radians and

< 1.

if it exists. B) does not exist

C) 0.0007

D) 1 72) ______

72) If

≤ f(x) ≤ x for x in [-1,1], find A) 1 B) -1

f(x) if it exists. C) 0

D) does not exist

Use the table of values of f to estimate the limit. 73) Let f(x) = x2 + 8x - 2, find f(x).

73) ______

A) ; limit = 18.0 B) ; limit = 5.40 C) ; limit = ∞ D) ; limit = 17.70 74) ______

74) Let f(x) =

, find

f(x).

A) ; limit = 4.0 B)

; limit = ∞ C) ; limit = 5.10 D) ; limit = 1.20 75) ______

75) Let f(x) = x2 - 5, find

f(x).

A) ; limit = ∞ B) ; limit = -5.0 C) ; limit = -15.0 D) ; limit = -3.0 76) ______

76) Let f(x) =

, find

f(x).

A) ; limit = 0.2 B) ; limit = 0.3 C) ; limit = 0.1 D) ; limit = -0.2 77)

77) Let f(x) =

, find

f(x).

______ A) ; limit = -2 B) ; limit = -1.9 C) ; limit = -2.1 D) ; limit = 0.5714 78) ______

78) Let f(x) =

, find

f(x).

A) limit = 4 C) limit = 3.5

B) limit = 0 D) limit does not exist 79) ______

79) Let f(θ) =

, find

f(θ).

A) limit = 6.9670671 C) limit = 0

B) limit does not exist D) limit = 8

Find the limit. 80) If

80) ______ = 2, find

A) 5

f(x). B) 3

C) 6

D) Does not exist 81) ______

81) If

= 3, find A) 2

f(x). B) 3

C) 6

D) Does not exist 82) ______

82) If

= 4, find A) 4

. B) 2

C) 8

D) 16 83) ______

83) If

= 1, find A) 2

84)

f(x). B) 1

C) 0

D) Does not exist If

___ ___

84) = 1, find . A) 0

B) 2

C) 1

D) Does not exist 85) ______

85) If

= 2, find A) 1

f(x). B) 3

Use a CAS to plot the function near the point 86) A)

B) 0

C) 2

D) Does not exist

being approached. From your plot guess the value of the limit. 86) ______ C) 8

87) ______

87) A)

B) 1

C) 0

D) 2

88) ______

88) A)

B) 0

D) 3

89) ______

89) A)

B) 9

D) 18

90) ______

90) A)

C) 81

91) ______

91) A) 0

92) ______

92) A) 0

93)

C) 6

___ ___

93)

A) 3

C) 4

D) 8

94) ______

94)

A) 4

B) 1

C) 2

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 95) 95) _____________ It can be shown that the inequalities 1 -

< 1 hold for all values of x

close to zero. What, if anything, does this tell you about

Explain.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 96) Write the formal notation for the principle "the limit of a quotient is the quotient of the limits" and include a statement of any restrictions on the principle. A) =

96) ______

, provided that

and

then

provided that

and

then

provided that

D) =

97) What conditions, when present, are sufficient to conclude that a function f(x) has a limit as x approaches some value of a? A) The limit of as from the left exists, the limit of as from the right exists, and these two limits are the same. B) Either the limit of

from the left exists or the limit of

from the right

exists C) f(a) exists, the limit of

from the left exists, and the limit of

the right exists. D) The limit of as from the left exists, the limit of and at least one of these limits is the same as f(a).

from

from the right exists,

97) ______

98) Provide a short sentence that summarizes the general limit principle given by the formal notation

given that

98) ______

and

A) The limit of a sum or a difference is the sum or the difference of the functions. B) The sum or the difference of two functions is continuous. C) The limit of a sum or a difference is the sum or the difference of the limits. D) The sum or the difference of two functions is the sum of two limits. 99) The statement "the limit of a constant times a function is the constant times the limit" follows from a combination of two fundamental limit principles. What are they? A) The limit of a product is the product of the limits, and the limit of a quotient is the quotient of the limits. B) The limit of a function is a constant times a limit, and the limit of a constant is the constant. C) The limit of a constant is the constant, and the limit of a product is the product of the limits. D) The limit of a product is the product of the limits, and a constant is continuous.

99) ______

1) B 2) C 3) D 4) A 5) D 6) B 7) D 8) D 9) A 10) A 11) A 12) C 13) C 14) A 15) C 16) D 17) A 18) D 19) D 20) D 21) D 22) A 23) D 24) A 25) A 26) D 27) A 28) B 29) B 30) B 31) B 32) D 33) B 34) D 35) A 36) A 37) B 38) A 39) A 40) A 41) A 42) C 43) B 44) B 45) B 46) D 47) A 48) B 49) D 50) B 51) A

52) A 53) C 54) C 55) B 56) D 57) D 58) D 59) D 60) A 61) A 62) D 63) D 64) A 65) B 66) C 67) D 68) D 69) D 70) B 71) D 72) C 73) A 74) A 75) B 76) A 77) A 78) A 79) B 80) A 81) C 82) C 83) C 84) A 85) B 86) D 87) A 88) C 89) A 90) B 91) C 92) A 93) D 94) A 95) Answers may vary. One possibility: , which is squeezed between

96) C 97) A

1 = 1. According to the squeeze theorem, the function

and 1, must also approach 1 as x approaches 0. Thus,

98) C 99) C

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Given the interval (a, b) on the x-axis with the point c inside, find the greatest value for δ > 0 such that for all x, ⇒ a < x < b. 1) a = -10, b = 0, c = -8 A) δ = 8 B) δ = 2 C) δ = 1 D) δ = 4

2) _______

2) a= ,b= A) δ = 2

,c= B)

C) δ=

3) a = 1.373, b = 2.751, c = 1.859 A) δ = 0.486 B) δ = 1.378 Use the graph to find a δ > 0 such that for all x, 0 < 4)

A) δ = 0.4 5)

1) _______

B) δ = 4

D) δ=

δ= 3) _______

C) δ = 0.892 <δ ⇒

D) δ = 1

< ε. 4) _______

C) δ = 0.1

D) δ = 0.2

__ __ __ _

A) δ = 0.4

B) δ = 0.04

C) δ = 0.08

D) δ = 6 6) _______

A) δ = 0.5

B) δ = 0.05

C) δ = -0.05

D) δ = 11

7) _______

A) δ = 0.2

B) δ = -0.1

C) δ = 9

D) δ = 0.1 8) _______

A) δ = 3 9)

B) δ = 0.1

C) δ = -0.2

D) δ = 0.2

__ __ __ _

A) δ = 0.2

B) δ = -0.2

C) δ = 5.5

D) δ = 0.1 10) ______

10)

A) δ = 0.865 11)

B) δ = 0.46

C) δ = 0.4169

D) δ = 0.4481

______ A) δ = 0.4375

11)

B) δ = 3

C) δ = 0.5625

D) δ = 1 12) ______

12)

A) δ = 0.13

B) δ = 0.25

C) δ = 0.12

D) δ = 6 13) ______

13)

A) δ = 0.24

B) δ = 0.51

C) δ = 0.27

D) δ = 0

A function f(x), a point c, the limit of f(x) as x approaches c, and a positive number ε is given. Find a number for all x, 0 < <δ ⇒ < ε. 14) f(x) = 5x + 9, L = 29, c = 4, and ε = 0.01 A) δ = 0.01 B) δ = 0.0025 C) δ = 0.004 D) δ = 0.002 15) f(x) = 10x - 1, L = 29, c = 3, and ε = 0.01 A) δ = 0.003333 B) δ = 0.0005 16) f(x) = -7x + 10, L = -18, c = 4, and ε = 0.01 A) δ = 0.002857 B) δ = -0.0025

such that 14) ______

15) ______ C) δ = 0.002

D) δ = 0.001 16) ______

C) δ = 0.005714

D) δ = 0.001429

17) f(x) = -9x - 1, L = -19, c = 2, and ε = 0.01 A) δ = -0.005 B) δ = 0.002222 18) f(x) = A) δ = 3 19) f(x) = A) δ = 4 20)

17) ______ C) δ = 0.001111

D) δ = 0.000556 18) ______

, L = 2, c = 2, and ε = 1 B) δ = 1

C) δ = 9

D) δ = 5 19) ______

, L = 2, c = 3, and ε = 1 B) δ = 3

C) δ = -5

f(x) = 7 , L =567, c = 9, and ε = 0.4 A) δ = 0.00318 B) δ = 8.99682

D) δ = 6 20) ______

C) δ = 9.00317

D) δ = 0.00317 21) ______

21) f(x) = , L = , c = 9, and ε = 0.1 A) δ = 4.2632 B) δ = 0.4737 22) f(x) = mx, m > 0, L = 6m, c = 6, and ε = 0.05 A) B) δ=6+

C) δ = 810

D) δ = 81 22) ______

C) δ = 0.05

D) δ = 6 - m

δ= 23) ______

23) f(x) = mx + b, m > 0, L = A) δ=

+ b, c = B)

, and ε = c > 0

δ=

D) δ=

δ=

Find the limit L for the given function f, the point c, and the positive number ε. Then find a number δ > 0 such that, for all x, 24) ______

24) f(x) = 6x - 2, c = -3, ε = 0.12 A) L = -20; δ = 0.03 C) L = -16; δ = 0.02

B) L = 16; δ = 0.03 D) L = -20; δ = 0.02 25) ______

25) f(x) = , c = -10, ε = 0.03 A) L = -16; δ = 0.04 C) L = 2; δ = 0.04

B) L = 0; δ = 0.03 D) L = -18; δ = 0.03 26) ______

26) f(x) =

, c = -4, ε = 0.5 A) L = j-3; δ = 0.88 C) L = 4; δ = 1.88

B) L = 4; δ = 2.13 D) L = 5; δ = 1.88 27) ______

27) f(x) = , c = 9, ε = 0.2 A) L = 4; δ = 0.43

B) L = 4; δ = 0.95

C) L = 4; δ = 4.74

D) L = 4; δ = 0.47

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Prove the limit statement 28) 28) _____________ (2x - 3) = -1

29) _____________

29) =6

30) _____________

30) =7

31) _____________

31) =

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 32) You are asked to make some circular cylinders, each with a cross-sectional area of To do this, you need to know how much deviation from the ideal cylinder diameter of you can allow and still have the area come within

of the required

let and look for the interval in which you must hold x to make interval do you find? A) (4.8580, 4.9396) B) (1.9381, 1.9706) C) (2.7408, 2.7869)

cm To find out, What

D) (0.5642, 0.5642) 33) ______

33) 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 10 volts and I is to be

amperes. In what interval

does R have to lie for I to be within 0.1 amps of the target value A) B) C)

34) The cross-sectional area of a cylinder is given by A = π Find the tolerance range of D such that A) = 3.567, = 3.570 C)

= 3.567,

/4, where D is the cylinder diameter.

= 3.558,

= 3.570

35) 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 12Ω resistor's tolerance for the current to be within 1 ± 0.01 amp? A) 1% B) 0.01% C) 10% D) 0.1% Provide an appropriate response. 36) δ > 0, such that for all x, 0 < A) B) >δ

35) ______

36) ______ f(x) = L, means if given any number ε > 0, there exists a number < δ implies

. C)

<ε

>ε

<δ

37) Identify the incorrect statements about limits. I. The number L is the limit of f(x) as x approaches c if f(x) gets closer to L as x approaches x 0. II. The number L is the limit of f(x) as x approaches c if, for any ε > 0, there corresponds a δ > 0 such that

34) ______

< 0.01 as long as <D< . B) = 3.558, = 3.578

= 3.578

The definition of the limit,

32) ______

< ε whenever 0 <

< δ.

r L is III. the The limit of nu f(x) as mbe x

___ ___

approach 37) es c if, given any ε > 0, there exists a value of x for which < ε. A) II and III

B) I and II

C) I and III

D) I, II, and III

1) B 2) D 3) A 4) C 5) B 6) B 7) D 8) B 9) D 10) C 11) A 12) C 13) A 14) D 15) D 16) D 17) C 18) A 19) B 20) D 21) A 22) B 23) B 24) D 25) D 26) C 27) D 28) Let ε > 0 be given. Choose δ = ε/2. Then 0 <

< δ implies that

Thus, 0 < < δ implies that 29) Let ε > 0 be given. Choose δ = ε. Then 0 <

<ε

Thus, 0 < < δ implies that 30) Let ε > 0 be given. Choose δ = ε/3. Then 0 <

<ε

Thus, 0 <

< δ implies that

<ε

31) Let ε > 0 be given. Choose δ = min{7/2, 49ε/2}. Then 0 <

Thus, 0 < 32) C 33) D 34) A 35) A 36) B 37) C

< δ implies that

<ε

< δ implies that

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the graph to estimate the specified limit. 1) Find

f(x) and

A) -7; -5

1) _______

f(x)

B) -5; -2

C) -2; -7

D) -7; -2 2) _______

2) Find

f(x)

A) 0

B) 1

C) does not exist

D) -1 3) _______

3) Find

f(x)

A) does not exist

B) 2

C) -2

D) 0 4) _______

4) Find

f(x)

B) -3

5) _______

5) Find

f(x) and

A) 4; 3

f(x)

B) π; π

D) 3; 4 ;

6) Find

f(x) and

f(x)

_______ A) -1; -1

B) -1; 1

C) 1; -1

D) 1; 1 7) _______

7) Find

f(x) and

f(x)

A) does not exist; does not exist C) -2; 4

B) 4; -2 D) 1; 1 8) _______

8) Find

f(x)

A) -1

B) does not exist

C) 0

D) 6

Determine the limit by sketching an appropriate graph. 9)

9) _______

f(x), where f(x) =

A) -5

B) -3

C) -10

D) -4

10) ______

10) f(x), where f(x) =

A) 17

B) -24

C) -2

D) -3 11) ______

11) f(x), where f(x) =

A) 16

B) 0

C) 13

D) 19 12) ______

12)

A) Does not exist 13)

B) 4

C) 1

D) 0

___ ___

13)

A) -7

B) Does not exist

C) 6

D) -0

Find the limit. 14) A)

14) ______ B)

D) Does not exist

15) ______

15) A)

B) Does not exist

D) 7 16) ______

16) A)

B) Does not exist

17) ______

17) A) Does not exist

18) ______

18) A)

C) Does not exist

19) ______

19) A) Does not exist

B) 2

C) 4

D) -2 20) ______

20) A) -3

B) 7

C) 3

D) Does not exist 21) ______

21) A) -

B) Does not exist

C) 0

22) ______

22) A)

B) -

Use the graph of the greatest integer function y = 23) A) 1

B) 7

C) 0

D) Does not exist

to find the limit. 23) ______ C) -7

D) 0 24) ______

24) (x A) 2

Find the limit using 25)

) B) 0

C) -4

D) 4

= 1.

A) 1

25) ______ B) 5

D) does not exist

26) ______

26) A) 1

B) 3

C) does not exist

27) ______

27) A)

B) 1

C) does not exist

D) 4

28) ______

28) A)

B) 0

D) does not exist

29) ______

29) A)

C) 0

D) does not exist

30) ______

30) A) 0

B) does not exist

31) ______

31) 6 (cot 3x)(csc 2x) A) 1

B) does not exist

32) ______

32) A) 0

B) does not exist

C) 1

D) -1 33) ______

33) A) does not exist

B) 1

C) -1

D) 0 34) ______

34) A) does not exist

C) 0

Provide an appropriate response. 35) Given

f(x) =

35) ______ f(x) =

f(x) =

II.

f(x) =

III.

f(x) does not exist. B) III

A) II

, and

≠

, which of the following statements is true?

C) I

D) none 36) ______

36) Given

f(x) =

II.

f(x) =

III. A) none

f(x) =

, and

f(x) does not exist. B) III

, which of the following statements is false?

C) II

D) I 37) ______

37) If

f(x) = L, which of the following expressions are true? I.

f(x) does not exist.

II.

f(x) does not exist.

III.

f(x) = L

IV. f(x) = L A) I and II only

B) I and IV only

C) II and III only

D) III and IV only

38) ______

38) If

f(x) = 1 and f(x) is an odd function, which of the following statements are true? I.

f(x) = 1

II.

f(x) = -1

III. f(x) does not exist. A) I and II only B) II and III only

C) I, II, and III

D) I and III only 39) ______

39) If true?

f(x) = 1,

f(x) = -1, and f(x) is an even function, which of the following statements are

f(x) = -1

II.

f(x) = -1

III. f(x) does not exist. A) I and III only B) II and III only

C) I, II, and III

D) I and II only

40) Given ε > 0, find an interval I = (6, 6 + δ), δ > 0, such that if x lies in I, then is being verified and what is its value? A)

40) ______

< ε. What limit

41) ______

=0 C)

< ε. What limit

=0 D)

41) Given ε > 0, find an interval I = (1 - δ, 1), δ > 0, such that if x lies in I, then is being verified and what is its value? A)

=0 C)

=0 D)

1) C 2) C 3) D 4) C 5) A 6) C 7) B 8) C 9) C 10) A 11) D 12) D 13) A 14) C 15) A 16) D 17) C 18) B 19) B 20) A 21) A 22) A 23) A 24) B 25) B 26) D 27) D 28) C 29) A 30) C 31) A 32) D 33) B 34) C 35) B 36) B 37) D 38) B 39) A 40) C 41) D

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find all points where the function is discontinuous. 1)

A) x = 2

B) x = 4

C) None

D) x = 4, x = 2 2) _______

A) x = 1

B) x = -2

C) x = -2, x = 1

D) None 3) _______

A) x = -2, x = 0 C) x = 0, x = 2

B) x = 2 D) x = -2, x = 0, x = 2 4) _______

A) None

B) x = 6

C) x = -2, x = 6

D) x = -2 5) _______

A) None C) x = 1, x = 5 6)

1) _______

B) x = 1, x = 4, x = 5 D) x = 4

_______ A) x = 0

B) x = 0, x = 1

C) x = 1

D) None 7) _______

A) x = 0

B) x = 0, x = 3

C) None

D) x = 3 8) _______

A) None

B) x = -2, x = 2

C) x = -2

D) x = 2 9) _______

A) x = -2, x = 0, x = 2 C) x = -2, x = 2

B) x = 0 D) None

Answer the question. 10) Does

10) ______ f(x) exist?

A) No

B) Yes

11) Does

f(x) = f(-1)?

___ ___

11)

A) Yes

B) No 12) ______

12) Does

f(x) exist?

A) No 13) Is f continuous at f(1)?

B) Yes

___ ___

13)

A) No

B) Yes 14) ______

14) Is f continuous at f(3)?

A) No

B) Yes

15) Does

f(x) exist?

___ ___

15)

A) No

B) Yes 16) ______

16) Does

f(x) = f(2)?

A) Yes 17) Is f continuous at x = 0?

B) No

___ ___

17)

A) Yes

B) No 18) ______

18) Is f continuous at x = 4?

A) Yes 19) Is f continuous on (-2, 4]?

B) No

___ ___

19)

A) No

B) Yes

Solve the problem. 20) To what new value should f(1) be changed to remove the discontinuity?

20) ______

f(x) = A) 2

B) 1

C) 3

D) 4 21) ______

21) To what new value should f(2) be changed to remove the discontinuity?

f(x) = A) 0

B) -8

C) -7

D) -1

Find the intervals on which the function is continuous. 22) y= - 4x A) continuous everywhere C) discontinuous only when x = -9

22) ______ B) discontinuous only when x = -5 D) discontinuous only when x = 5 23) ______

23) y= A) discontinuous only when x = 8

B) continuous everywhere

C) discontinuous only when x = -16

D) discontinuous only when x = -2 24) ______

24) y= A) discontinuous only when x = -7 or x = 1 C) discontinuous only when x = 1 or x = 7

B) discontinuous only when x = 1 D) discontinuous only when x = -1 or x = 7 25) ______

25) y= A) discontinuous only when x = -3 C) discontinuous only when x = 9

B) discontinuous only when x = -3 or x = 3 D) discontinuous only when x = -9 or x = 9 26) ______

26) y= A) discontinuous only when x = -7 or x = -3 C) discontinuous only when x = -3

B) continuous everywhere D) discontinuous only when x = -10 27) ______

27) y= A) discontinuous only when θ = π C)

B) continuous everywhere D) discontinuous only when θ = 0

discontinuous only when θ = 28) ______

28) y= A) continuous everywhere C)

B) discontinuous only when θ = -8 D) discontinuous only when θ = 8

discontinuous only when θ = 29) ______

29) y = A)

B) continuous on the interval

continuous on the interval D)

continuous on the interval 30) ______

30) y= A)

B) continuous on the interval

continuous on the interval D)

continuous on the interval 31)

continuous on the interval 31) ______

y= A) continuous everywhere B) continuous on the interval [

, ∞) C) continuous on the intervals (-∞, - ] and [ D)

, ∞)

continuous on [the interval

]

Find the limit and determine if the function is continuous at the point being approached. 32) sin(4x - sin 4x) A) 0; no C) 0; yes

32) ______

B) does not exist; yes D) does not exist; no 33) ______

33) cos(5x - cos 5x) A) does not exist; yes C) 0; no

B) 0; yes D) does not exist; no 34) ______

34) sin A) does not exist; no C) 1; yes

B) 1; no D) does not exist; yes 35) ______

35) cos A) 1; no C) does not exist; no

B) does not exist; yes D) 1; yes 36) ______

36) sec(x x-x x - 1) A) does not exist; no C) sec 8; yes

B) sec 8; no D) csc 8; yes 37) ______

37) sin(x x+x x + 2) A) sin 8; no C) does not exist; no

B) sin -4; yes D) sin 8; yes 38) ______

38) tan A) 0; yes C) does not exist; no

B) -1; no D) -1; yes 39) ______

39) tan(sin(-π cos(sin θ))) A) 0; no C) 1; yes

B) 0; yes D) does not exist; no 40) ______

40) cos A)

B) 1; yes

; no C) does not exist; no

D) ; yes

41) ______

41) A) does not exist; no

B) ; yes

D) ; no

; yes

Determine if the given function can be extended to a continuous function at

If so, approximate the extended function's

value at (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 Otherwise write "no continuous extension." 42) 42) ______ f(x) = A) f(0) = 0 C) f(0) = 0 only from the left

B) No continuous extension D) f(0) = 0 only from the right 43) ______

43) f(x) = A) f(0) = 2 C) No continuous extension 44)

f(x) = A) f(0) = 2.7183 C) No continuous extension

B) f(0) = 2 only from the left D) f(0) = 2 only from the right 44) ______ B) f(0) = 5.4366 D) f(0) = 7.3891 45) ______

45) f(x) = A) f(0) = 1 C) f(0) = 1 only from the right

B) No continuous extension D) f(0) = 1 only from the left

Find numbers a and b, or k, so that f is continuous at every point. 46)

A) a = -4, b = -40

B) a = -4, b = -8

46) ______

C) a = 8, b = -24

D) Impossible 47) ______

47)

A) a = -8, b = -15

B) a = 8, b = 15

C) a = 8, b = -15

D) Impossible 48) ______

48)

A) k = - 2 49)

B) k = 2

C) k = 9

D) k = 7

___ ___

49)

A) k = 30

B) k = 20

C) k = -5

D) Impossible 50) ______

50)

A) k = 100

C) k = 10

D) Impossible

k= SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 51) 51) _____________ Use the Intermediate Value Theorem to prove that 5 -8 + 4x - 3 = 0 has a solution between 1 and 2. 52)

53)

Use the Intermediate Value Theorem to prove that 10 solution between -1 and 0. Use the Intermediate Value Theorem to prove that x and 4.

- 9x - 6 = 0 has a

= 3 has a solution between 2

52) _____________

53) _____________

54) _____________

54) Use the Intermediate Value Theorem to prove that 3 sin x = x has a solution between and π.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 55) 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?

55) ______

f(x) = A) continuous extension exists at origin; f(0) ≈ 1.9556 B) continuous extension exists from the left; f(0) ≈ 1.9556 C) continuous extension exists from the right; f(0) ≈ 1.9556 D) continuous extension exists at origin; f(0) = 0 56) 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

exte ction's nde value(s d ) fun should

be?

___ ___

56) f(

x) =

A) continuous extension exists at origin; f(0) = 0 B) continuous extension exists at origin; f(0) = 7 C) continuous extension exists from the right; f(0) = 1 continuous extension exists from the left; f(0) = -1 D) continuous extension exists from the right; f(0) = 7 continuous extension exists from the left; f(0) = -7 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 57) A function y = f(x) is continuous on [1, 2]. It is known to be positive at 57) _____________ and negative at What, if anything, does this indicate about the equation Illustrate with a sketch.

58) Explain why the following five statements ask for the same information. (a) Find the roots of f(x) = 2

58) _____________

- 1x - 3.

(b) Find the x-coordinate of the points where the curve y = 2 (c) Find all the values of x for which 2

crosses the line

- 1x = 3.

(d) Find the x-coordinates of the points where the cubic curve y = 2

- 1x crosses the

line (e) Solve the equation 2 59)

If f(x) =

- 1x - 3 = 0.

- 5x + 5, show that there is at least one value of c for which f(x) equals

59) _____________ 60) _____________

60) If functions f

and g

are continuous for 0 ≤ x ≤ 2, could

discontinuous at a point of

possibly be

Provide an example.

61) Give an example of a function f(x) that is continuous at all values of x except at

, where it has a removable discontinuity. Explain how you know that f is discontinuous at

61) _____________

and how you know the discontinuity is removable.

62) Give an example of a function f(x) that is continuous for all values of x except where it has a nonremovable discontinuity. Explain how you know that f is

dis uous at con and why the tin discontinuity is

nonremo 62) vable.

___ ___ ___ ___ _

1) B 2) A 3) D 4) B 5) A 6) D 7) D 8) B 9) B 10) B 11) A 12) B 13) A 14) A 15) B 16) B 17) A 18) A 19) A 20) C 21) A 22) B 23) B 24) C 25) B 26) B 27) D 28) B 29) D 30) B 31) C 32) C 33) B 34) C 35) C 36) C 37) D 38) D 39) B 40) D 41) B 42) A 43) C 44) D 45) A 46) B 47) C 48) D 49) B 50) C 51) Let f(x) = 5

-8

+ 4x - 3 and let

= 0. f(1) = -2 and f(2) = 13. Since f is continuous on

and since

= 0 is

betw 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. een Such a c is a solution to the equation 52) Let f(x) = 10 +6 - 9x - 6 and let = 0. f(-1) = 7 and f(0) = -6. Since f is continuous on and since =0 is between f(-1) and f(0), by the Intermediate Value Theorem, there exists a c in the interval (-1, 0) with the property that f(c) = 0. Such a c is a solution to the equation 53)

Let f(x) = x and

and let

= 3.

and

Since f is continuous on

and since

by the Intermediate Value Theorem, there exists a c in the interval

is between

with the property that

Such a c is a solution to the equation 54) Let f(x) = between f

and let

. f

≈ 0.6366 and f(π) = 0. Since f is continuous on

and since

and f(π), by the Intermediate Value Theorem, there exists a c in the interval

that Such a c is a solution to the equation 3 sin x = x. 55) A 56) D 57) The Intermediate Value Theorem implies that there is at least one solution to

, with the property

on the interval

Possible graph:

58) The roots of f(x) are the solutions to the equation f(x) = 0. Statement (b) is asking for the solution to the equation Statement (d) is asking for the solution to the equation These three equations are equivalent to the equations in statements (c) and (e). As five equations are equivalent, their solutions are the same. 59) 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 60) Yes, if f(x) = 1 and g(x) = x - 1, then h(x) =

is discontinuous at x = 1.

61) Let f(x) =

be defined for all x ≠ 10. The function f is continuous for all

defined at

because division by zero is undefined; hence f is not continuous at

removable because

(We can extend the function to

The function is not . This discontinuity is

by defining its value to be 1.)

62) Let f(x) =

, for all x ≠ 4. The function f is continuous for all

unbounded as x approaches 4, f is discontinuous at

and

As f is

, and, moreover, this discontinuity is nonremovable.

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. For the function f whose graph is given, determine the limit. 1) Find

f(x)

and

A) -2; -7

1) _______

f(x).

B) -7; -2

C) -7; -5

D) -5; -2 2) _______

2) Find

f(x)

A) 3; -3 C) -3; 3

and

f(x).

B) 1; 1 D) does not exist; does not exist

3) Find

f(x).

_______ A) -1

B) 4

C) 2.3

D) 1.3 4) _______

4) Find

f(x).

B) 2

C) does not exist

D) -1

5) _______

5) Find

f(x).

A) 3

C) does not exist

D) 4

6) Find

f(x).

_______ A) -2

B) 2

C) 0

D) does not exist 7) _______

7) Find

f(x).

A) -2

B) 2

C) 0

D) does not exist 8) _______

8) Find

f(x).

A) 2

B) 0

C) -2

D) does not exist 9) _______

9) Find

f(x).

A) does not exist

B) -1

10) ______

10) Find

f(x).

A) -2

B) -∞

C) 0

D) does not exist

Find the limit. 11)

11) ______ -4

A) -5

B) -3

C) 4

D) -4 12) ______

12) A) -∞

B) 1

C) 4

D) 8 13) ______

13) A) -∞

B) ∞

D) 14) ______

14) A)

B) 1

C) ∞

D) 0

15) ______

15) A)

B) 1

C) ∞

16) ______

16) A) ∞

B) 0

D) -

17) ______

17) A) -7

B) ∞

D) 7

18) ______

18) A)

B) -∞

C) ∞

D) 4

19) ______

19) A) 0

B) 4

C) 1

D) -∞

Divide numerator and denominator by the highest power of x in the denominator to find the limit. 20)

B) does not exist

20) ______

21) ______

21) A) 4

B) ∞

C) 0

D) 16 22) ______

22) A) ∞

B) 1

C) 0

23) ______

23) A) ∞

B) -∞

C) 1

D) 0 24) ______

24)

A) 0

C) -

D) -∞ 25) ______

25) A) 343

B) 7

C) 49

D) does not exist 26) ______

26) A) 6

B) 216

C) does not exist

D) 36

27) ______

27) A)

B) ∞

D) 0

Find the limit. 28)

28) ______

A) Does not exist

B) ∞

C) 1/2

D) -∞ 29) ______

29) A) ∞

B) -∞

C) 0

D) -1 30) ______

30) B) -∞

A) 0

C) ∞

D) -1 31) ______

31) A) ∞

B) -∞

C) -1

D) 0 32) ______

32) A) -∞

B) ∞

C) 0

D) -1 33) ______

33) A) 1

B) 0

C) ∞

D) -∞ 34) ______

34) A) ∞

B) -∞

C) 0

D) 1 35) ______

35) A) -∞

B) ∞

C) 0

D) 2/3 36) ______

36) tan x A) 0

B) ∞

C) -∞

D) 1 37) ______

37) sec x A) 0

B) -∞

C) 1

D) ∞ 38) ______

38) (1 + csc x) A) 1

B) ∞

C) 0

D) Does not exist

39) ______

39) (1 - cot x) A) -∞

B) 0

C) ∞

D) Does not exist 40) ______

40) A) Does not exist

B) 0

C) ∞

D) -∞ 41) ______

41) A) ∞

B) -∞

C) 0

2 42) ______

42) A) 0

B) ∞

C) -∞

D) 43) ______

43) A) -∞

B) Does not exist

C) 0

D) ∞ 44) ______

44) A) ∞

B) -∞

C) Does not exist

D) 9 45) ______

45) A) ∞

B) 1

C) -∞

D) Does not exist 46) ______

46)

A) 0

B) -∞

C) Does not exist

D) ∞ 47) ______

47) A) ∞

B) -∞

C) Does not exist

Graph the rational function. Include the graphs and equations of the asymptotes. 48) y=

D) 0

___ ___

48)

A) asymptotes: x = -1, y = 0

B) asymptotes: x = -1, y = 1

C) asymptotes: x = 1, y = 1

D) asymptotes: x = 1, y = 0

49) y=

___ ___

49)

A) asymptote: y = 0

B) asymptote: y = 1

C) asymptotes: x = 2, x = -2

D) asymptote: y = 0

50) y=

___ ___

50)

A) asymptotes: x = 0, y = 0

B) asymptote: y = 0

C) asymptotes: x = 0, y = 0

D) asymptotes: x = 0, y = 0

51) y=

___ ___

51)

A) asymptotes: x = 1, y = 0

B) asymptotes: x = 1, y = 0

C) asymptotes: x = -1, y = 0

D) asymptotes: x = -1, y = 0

52) y=

___ ___

52)

A) asymptotes: x = 1, y = -1

B) asymptotes: x = 1, y = 1

C) asymptotes: x = -1, y = -1

D) asymptotes: x = -1, y = 1

53) y=

___ ___

53)

A) asymptotes: x = -2, y = 0

B) asymptotes: x = 2, y = 0

C) asymptotes: x = -2, y = 0

D) asymptotes: x = 2, y = 0

54) y=

___ ___

54)

A) asymptotes: x = -2, x = 2, y = 0

B) asymptotes: x = -2, x = 2, y = -2

C) asymptotes: x = -2, x = 2, y = 2

D) asymptotes: x = -2, x = 2, y = 0

55) y=

___ ___

55)

A) asymptotes: x = -2, y = -x + 2

asymptotes: x = -2, y = -

asymptotes: x = -4, y = -

D) asymptotes: x = -2, y = -

56)

x+1

___ ___

56)

A) asymptotes: x = 0, y = -x + 2

B) asymptotes: x = 0, y = -x - 4

C) asymptotes: x = 0, y = -x

D) asymptotes: x = 0, y = -x - 2

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. 57) f(0) = 0, f(1) = 6, f(-1) = -6,

f(x) = -5,

f(x) = 5.

___ ___ ___ ___ _

57)

58) _____________

58) f(0) = 0, f(1) = 4, f(-1) = 4,

f(x) = -4.

59) _____________

59) f(0) = 5, f(1) = -5, f(-1) = -5,

f(x) = 0.

60) f(0) = 0,

f(x) = 0,

f(x) = -∞,

f(x) = ∞,

___ ___ ___ ___ _

60)

Find a function that satisfies the given conditions and sketch its graph. 61) f(x) = 0,

f(x) = ∞,

61) _____________

f(x) = ∞.

62) _____________

62) f(x) = 0,

f(x) = ∞,

f(x) = ∞.

g(x) = -4,

g(x) = 4,

63) g(x) = -4.

_____________

63)

64) _____________

64) f(x) = 0,

f(x) = ∞.

65) _____________

65) f(x) = 1,

f(x) = -1,

f(x) = 1

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit. 66) (4x -

66) ______

) B) -12

C) 0

D) -∞

67) ______

67) -x A) 12

B) 0

C) 6

D) ∞ 68) ______

68) ( A) ∞

) B)

C) 0

69) ______

69) A)

Provide an appropriate response. 70)

B) 13

C) does not exist

Which of

the whenever following statemen ts defines f(x) = ∞? I. For every positive real number B there exists a correspo nding δ > 0 such that f(x) >B wheneve r

-δ<

x< + δ. II. For every positive real number B there exists a correspo nding δ > 0 such that f(x) >B wheneve r

< + δ. III. For every positive real number B there exists a correspo nding δ > 0 such that f(x) >B

-δ<x<

70) ______

A) III

B) II

C) I

D) None 71) ______

71) Which of the following statements defines f(x) = ∞? I. For every positive real number B there exists a corresponding δ > 0 such that f(x) > B whenever -δ<x< + δ. II. For every positive real number B there exists a corresponding δ > 0 such that f(x) > B whenever <x< + δ. III. For every positive real number B there exists a corresponding δ > 0 such that f(x) > B whenever A) II

-δ<x<

. B) I

C) III

D) None 72) ______

72) Which of the following statements defines f(x) = ∞? I. For every positive real number B there exists a corresponding δ > 0 such that f(x) > B whenever -δ<x< + δ. II. For every positive real number B there exists a corresponding δ > 0 such that f(x) > B whenever <x< + δ. III. For every positive real number B there exists a corresponding δ > 0 such that f(x) > B whenever A) III

-δ<x<

. B) I

C) II

D) None 73) ______

73) Which of the following statements defines f(x) = -∞? I. For every negative real number B there exists a corresponding δ > 0 such that f(x) 0 such that f(x) 0 such that f(x) < B whenever A) III

-δ<x<

. B) II

C) I

D) None 74) ______

74) Which of the following statements defines f(x) = -∞? I. For every negative real number B there exists a corresponding δ > 0 such that f(x) 0 such that f(x) 0 such that f(x) < B whenever A) II

-δ<x<

. B) III

C) I

D) None

75) Which of the following statements defines f(x) = -∞? I. For every negative real number B there exists a corresponding δ > 0 such that f(x) < B

whe δ < x < nev + δ. er II. For - every

___ ___

negative 75) real number B there exists a correspo nding δ > 0 such that f(x) <B wheneve r

< + δ. III. For every negative real number B there exists a correspo nding δ > 0 such that f(x) <B wheneve r

-δ<

. A) III

B) II

C) I

D) None 76) ______

76) Which of the following statements defines f(x) = ∞? 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) N A) II B) III C) IV D) I

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 77) 77) _____________ Use the formal definitions of limits to prove

=∞ 78) _____________

78) Use the formal definitions of limits to prove

=∞

1) A 2) A 3) D 4) B 5) A 6) D 7) C 8) D 9) C 10) A 11) D 12) B 13) C 14) D 15) D 16) D 17) A 18) C 19) A 20) A 21) A 22) C 23) A 24) C 25) B 26) A 27) A 28) A 29) B 30) B 31) A 32) B 33) C 34) B 35) B 36) C 37) D 38) B 39) D 40) D 41) C 42) D 43) D 44) A 45) A 46) B 47) A 48) C 49) D 50) C 51) C

52) D 53) A 54) B 55) C 56) D 57) Answers may vary. One possible answer:

58) Answers may vary. One possible answer:

59) Answers may vary. One possible answer:

60) Answers may vary. One possible answer:

61) (Answers may vary.) Possible answer: f(x) =

62) (Answers may vary.) Possible answer: f(x) =

63) (Answers may vary.) Possible answer:

f(x) =

64) (Answers may vary.) Possible answer: f(x) =

65) (Answers may vary.) Possible answer: f(x) =

66) A 67) C 68) C 69) D 70) C 71) C 72) C 73) C 74) A 75) A 76) A 77) Given B > 0, we want to find δ > 0 such that 0 <

< δ implies

> B.

Now, > B if and only if < . Thus, choosing δ = 7/B (or any smaller positive number), we see that < δ implies

≥ B.

Therefore, by definition

=∞

78) Given B > 0, we want to find δ > 0 such that Now,

> B if and only if x <

We know x < δ implies

<x<

+ δ implies

> B.

= 0. Thus, choosing δ = 8/B (or any smaller positive number), we see that >

≥ B.

Ther efore , by defin ition

= ∞

Exam Name___________________________________ 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. 1)

A) -1

B) 0

C) 1

1) _______

D) Undefined 2) _______

A) 0

B) 1

C) -1

D) Undefined 3) _______

A) 0

B) 1

C) -1

D) Undefined 4) _______

A) 1

B) Undefined

C) 0

D) -1 5) _______

A) 1

B) Undefined

C) 0

D) -1 6) _______

B) 2

C) -2

D) -

7) _______

A) -2

B) 2

D) 8) _______

B) 2

C) -2

D) -

Find an equation for the tangent to the curve at the given point. 9) y= , (8, 128) A) y = 256x + 48 10)

11)

12)

13)

14)

y= - 3, (2, 1) A) y = 2x - 7 y= + 3, (3, 12) A) y = 6x - 12 y= - x, (-4, 20) A) y = -9x - 12 y = x - , (2, -2) A) y = -3x + 4 h(x) = - 9t - 2, (3, -2) A) y = 16t - 56

15) f(x) = 2

B) y = 48x - 256

C) y = 16x - 256

D) y = 16x + 256 10) ______

B) y = 4x - 11

C) y = 4x - 14

D) y = 4x - 7 11) ______

B) y = 6x - 6

C) y = 3x - 6

D) y = 6x - 15 12) ______

B) y = -9x - 16

C) y = -9x + 12

D) y = -9x + 16 13) ______

B) y = 5x + 4

C) y = 3x + 4

D) y = 5x - 4 14) ______

B) y = 18t - 2

C) y = 18t - 56

D) y = -2 15) ______

- x + 9, (4, 9)

B) y=

9) _______

x - 11

C) y=-

x + 11

D) y = 9 y=-

x+9

Graph the equation and its tangent. 16) Graph y = -3 and the tangent to the curve at the point whose x-coordinate is 1.

___ ___

16)

17)

Graph y = 4

and the tangent to the curve at the point whose x-coordinate is -2.

17) ______

18) Graph y = x2 - 2x - 7 and the tangent to the curve at the point whose x-coordinate is 1.

18) ______

19) Graph y = x2 - 3x - 8 and the tangent to the curve at the point whose x-coordinate is -2.

19) ______

20)

Graph y =

- 5 and the tangent to the curve at the point whose x-coordinate is 0.

Find the slope of the curve at the indicated point. 21) y = 2x + 8, x = -3 A) m = -2 B) m = 2 22)

23)

y= + 5x - 2, x = -1 A) m = -4 y=

+ 8x, x = 7 A) m = 14

20) ______

21) ______ C) m = 6

D) m = -6 22) ______

B) m = -2

C) m = 1

D) m = 3 23) ______

B) m = 15

C) m = 105

D) m = 22 24) ______

24) y= A)

,x=4 m

m =

m =-

Solve the problem. 25) Find the points where the graph of the function have horizontal tangents. f(x) = 4 A)

25) ______

+ 5x + 2 B) (0, -2)

D) (-13, 769)

26) ______

26) Find the points where the graph of the function have horizontal tangents. f(x) = A) (-

- 9x ,6

), (

C) (

, -18

)

, -6

B) (-

)

, 18 D) (-3, 0), (

), (0,0), (

, -18

)

, 0) 27) ______

27) Find equations of all tangents to the curve f(x) = A) y = -x + 8, y = -x - 8 C) y = -x + 8

that have slope -1. B) y = x - 8 D) y = x + 8, y = x - 8 28) ______

28) Find equations of all tangents to the curve f(x) = A) y = -x + 18, y = -x - 14 C) y = -x - 14, y = -x - 18

that have slope -1. B) y = -x + 18 D) y = -x - 14 29) ______

29) Find an equation of the tangent to the curve f(x) = A) B) y=30)

Find an equation of the tangent to the curve f(x) = A) y = 2x + 2 B) y = 2x

that has slope C)

. D)

- 2x + 1 that has slope 2. C) y = 2x - 1 D) y = 2x + 1

31) For a motorcycle traveling at speed v (in mph) when the brakes are applied, the distance d (in

30) ______

31) ______

feet) required to stop the motorcycle may be approximated by the formula Find the instantaneous rate of change of distance with respect to velocity when the speed is 44 mph. A) 4.4 mph B) 10.8 mph C) 5.4 mph D) 45 mph 32)

The power P (in W) generated by a particular windmill is given by where V is the velocity of the wind (in mph). Find the instantaneous rate of change of power with respect to velocity when the velocity is A) 12.2 W/mph B) 2.5 W/mph

C) 0.3 W/mph

33) The velocity of water ft/s at the point of discharge is given by pressure

32) ______

D) 5.5 W/mph where P is the

of the water at the point of discharge. Find the rate of change of the velocity

wit ect to h pressur resp e if the

___ ___

pressure 33) is

A) 22.96 ft/s per lb/in2

B) 2.2958 ft/s per lb/in2

C) 0.7260 ft/s per lb/in2

D) 4.59 ft/s per lb/in2

34) A balloon used in surgical procedures is cylindrical in shape. As it expands outward, assume that the length remains a constant to radius when the radius is A) 140.28π /mm C) 70.28π /mm

. Find the rate of change of surface area with respect (Answer can be left in terms of π). B) 70.14π /mm D) 140.0π /mm

35) 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 and A) C)

1.7 x

/°C

3.4 x

/°C

34) ______

and the length is B) D)

2.9 x 1x

35) ______

/°C /°C

36) A cubic salt crystal expands by accumulation on all sides. As it expands outward find the rate of

36) ______

change of its volume with respect to the length of an edge when the edge is A) B) 63.48 /mm 0.29 /mm C) D) 6.35 /mm 0.6348 /mm 37)

The equation for free fall at the surface of Planet X is

with t in seconds. Assume a

rock is dropped from the top of a cliff. Find the speed of the rock at A) 28.02 m/sec B) 14.01 m/sec C) 13.01 m/sec 38)

Assume that a watermelon dropped from a tall building falls

37) ______

D) 29.02 m/sec

in t sec. Find the

38) ______

watermelon's average speed during the first 4 sec of fall and the speed at the instant A) 32 ft/sec; 64 ft/sec B) 65 ft/sec; 130 ft/sec C) 64 ft/sec; 128 ft/sec D) 128 ft/sec; 65 ft/sec Provide an appropriate response. 39) Find the slope of the tangent to the curve y = A)

B) m=

m=-

at the point where x = 9. C) m=-

39) ______ D) m=

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 40) 40) _____________ Does the graph of f(x) = for your answer. 41)

have a tangent at the origin? Give reasons

Do the graph of f(x) es =

___ ___ ___ ___ _

41)

have a tangent at the origin? Give reasons for your answer.

42) _____________

42) Does the graph of f(x) = reasons for your answer.

have a vertical tangent at the origin? Give

43) _____________

43) Does the graph of f(x) = reasons for your answer.

have a vertical tangent at the point (0, 5)? Give

1) B 2) B 3) C 4) D 5) B 6) A 7) D 8) B 9) B 10) D 11) B 12) B 13) A 14) C 15) B 16) A 17) C 18) A 19) A 20) D 21) B 22) D 23) D 24) C 25) A 26) A 27) A 28) C 29) D 30) C 31) C 32) B 33) B 34) A 35) C 36) D 37) A 38) C 39) C 40) The function f is differentiable at x = o and hence has a tangent at the origin. 41) The function f is not differentiable at x = 0 and hence does not have a tangent at the origin. 42) The function is not differentiable at the x = 0 and hence does not have a tangent at the origin. 43) 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.

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Calculate the derivative of the function. Then find the value of the derivative as specified. 1) f(x) = 5x + 9; (2) A) (x) = 9; (2) = 9 B) (x) = 5; (2) = 5 C)

(x) = 5x;

(2) = 10

2) g(x) = 3x2 - 4x;

(x) = 0;

(2) = 0 2) _______

(3)

(x) = 3x - 4;

(x) = 6x;

(3) = 5 (3) = 18

3) f(x) = x2 + 7x - 2;

(x) = 6x - 4;

(3) = 14

(x) = 2x- 4;

(3) = 2 3) _______

(0)

(x) = 2x - 2;

(0) = - 2

(x) = 2x;

(x) = 2x + 7;

(0) = 7

(x) = x + 7;

4) g(x) = x3 + 5x; A) C)

(0) = 0 (0) = 7 4) _______

(1)

(x) = 3

+ 5x;

(x) = 3

;

1) _______

(1) = 8

(1) = 3

B) D)

(x) = 3x2 + 5;

(1) = 8

(x) =

(1) = 6

+ 5;

5) _______

5) f(x) = A)

;

(-1)

(x) = 8;

(-1) = 8

(x) = C)

(x) = - 8

;

(-1) = - 8

;

(-1) = 8

D) (x) = -

;

(-1) = -8 6) _______

6) g(x) = - ; (-2) A) (x) = - 2; (- 2) = - 2 C)

(x) = - 2

;

(- 2) = - 8

D) (x) = -

;

(x) =

(- 2) = -

;

(-2) = 7) _______

7) f(x) = A)

;

(0) B)

(x) = -

;

D) ;

;

(x) = 8;

(0) = 8

(0) = - 32

(0) = -2

C) (x) =

(x) = - 8

(0) = 2 8) _______

8) if s =

-t

B) = t - 1;

= 2 - t;

D) = 2t - 1;

= 2t + 1;

=5 9) _______

9) if r = A)

B) =-

;

D) =

;

Find the indicated derivative. 10) if A)

10) ______

y=6 B) 18x

3 11) ______

11) if

B) -

12) ______

12) if

v=t+

C) 1-

D) 1+

113) ______

13) if p = A)

B) -

D) -

Differentiate the function and find the slope of the tangent line at the given value of the independent variable. 14) f(x) = 9x + A)

, x=4 B)

15) ______

15) g(x) = A)

, x=8 B)

C) -

16)

14) ______

s=5

+ 4 , t = -1

D) 16) ______

B) -8

A) 32

C) 8

D) -32

Find an equation of the tangent line at the indicated point on the graph of the function. 17) y = f(x) = 2 - x + 5, (x, y) = (4, 5) A)

B) y=

x-7

18) w = g(z) = 1 + A) w=

y=-

f(x) = A)

w=-

x+5

y=-

C) z+6

D) w=

z-6

w=-

z-6

to find the derivative of the function. 19) ______ B)

C) -

D) 20) ______

20) f(x) = 3x2 + 5x + 5 A) 6x

x+7 18) ______

, (z, w) = (8, 2) B)

z+6

Use the formula f'(x) = 19)

C) y = 5

17) ______

B) 6x + 5

+ 5x

D) 3x + 5 21) ______

21) g(x) = A)

22) ______

22) g(x) = 9x + A)

B) 9+

The graph of a function is given. Choose the answer that represents the graph of its derivative. 23)

23) ______

24) ______

24)

25) ______

25)

26)

___ ___

26)

27) ______

27)

28) ______

28)

Given the graph of f, find any values of x at which 29)

A) x = 2

B) x = 1

is not defined. 29) ______

C) x = 0

D) x = -1 30) ______

30)

A) x = -2, 2

B) x = -3, 3

C) x = -2, 0, 2

D) x = -3, 0, 3 31) ______

31)

A) x = 0

B) x = 2

C) x = -2, 0, 2

D) x = -2, 2 32) ______

32)

A) x = 0, 1, 2 33)

B) x = 2

C) x = 1

D) x = 0

___ ___

33)

A) x = 1, 3 C) x = 1, 2, 3

B) x = 2 D) Defined for all values of x 34) ______

34)

A) x = 2, 5 C) x = 2

B) x = 5 D) Defined for all values of x 35) ______

35)

A) x = -1, 1 C) x = -1, 0, 1

B) x = 0 D) Defined for all values of x 36) ______

36)

A) x = 0 C) x = -2, 0, 2

B) x = -2, 2 D) Defined for all values of x 37) ______

37)

A) x = 0, 3 C) x = 0

B) x = 3 D) Defined for all values of x

Solve the problem. 38) The graph of y = f(x) in the accompanying figure is made of line segments joined end to end. Graph the derivative of f.

___ ___

38)

39) 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.

39) ______

Compare the right-hand and left-hand derivatives to determine whether or not the function is differentiable at the point whose coordinates are given. 40) 40) ______

y=x A) B) C) D) 41)

y = 2x

Since

(x) = 1 while

(x) = 2, f(x) is not differentiable at x = 0.

Since

(x) = 2 while

(x) = 1, f(x) is not differentiable at x = 0.

Since

(x) = -2 while

(x) = -1, f(x) is not differentiable at x = 0.

Since

(x) = 1 while

(x) = 1, f(x) is differentiable at x = 0.

___ ___

41)

y = y = -1 A) B) C) D)

Since

(x) = 0 while

(x) = 1, f(x) is not differentiable at

Since

(x) = 0 while

(x) = -1, f(x) is not differentiable at

Since

(x) = -1 while

(x) = 0, f(x) is not differentiable at

Since

(x) = 0 while

(x) = 0, f(x) is differentiable at 42) ______

42)

A) Since

(x) = 2 while

(x) =

, f(x) is not differentiable at

Since

(x) =

while

(x) = 1, f(x) is not differentiable at

Since

(x) =

while

(x) = 2, f(x) is not differentiable at

Since

(x) = 2 while

B) C) D)

(x) = 2, f(x) is differentiable at

Determine if the piecewise defined function is differentiable at the origin. 43) f(x) = A) Differentiable

43) ______

B) Not differentiable 44) ______

44) g(x) = A) Not differentiable

B) Differentiable

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? 45) x = 0 45) ______

A) Differentiable B) Continuous but not differentiable C) Neither continuous nor differentiable 46) x = 0

A) Differentiable B) Continuous but not differentiable C) Neither continuous nor differentiable 47) x = -1

46) ______

___ ___

47)

A) Differentiable B) Continuous but not differentiable C) Neither continuous nor differentiable 48) ______

48) x = 1

A) Differentiable B) Continuous but not differentiable C) Neither continuous nor differentiable SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 49) 49) _____________ What is the range of values of the slope of the curve y = + 5x - 2? 50)

Over what intervals of x-values, if any, does the function y = 2 For what values of x, if any, is

increase as x increases?

50) _____________

positive? How are your answers related? 51) _____________

51) Over what intervals of x-values, if any, does the function y = increases? For what values of x, if any, is 52) Does the curve y = answer.

decrease as x

negative? How are your answers related?

ever have a negative slope? If so, where? Give reasons for your

52) _____________

53)

54)

Does the curve y = your answer.

have any horizontal tangents? If so, where? Give reasons for

Does the curve y = + 4x - 10 have a tangent whose slope is -2? If so, find an equation for the line and the point of tangency. If not, why not?

53) _____________

54) _____________

55) Can a tangent line to a graph intersect the graph at more than one point? If not, why not. If so, give an example.

55) _____________

56) If g(x) = 2f(x) + 3, find

(4) given that

(4) = 5.

56) _____________

57) If g(x) = - f(x) - 3, find

(4) given that

(4) = 5.

57) _____________

58) Is there any difference between finding the derivative of f(x) at slope of the line tangent to f(x) at

Explain.

and finding the

58) _____________

1) B 2) B 3) C 4) B 5) D 6) D 7) A 8) C 9) C 10) C 11) B 12) D 13) A 14) D 15) B 16) B 17) D 18) B 19) B 20) B 21) D 22) A 23) B 24) C 25) A 26) A 27) C 28) C 29) C 30) A 31) A 32) D 33) D 34) C 35) B 36) C 37) A 38) C 39) A 40) B 41) B 42) C 43) B 44) A 45) A 46) C 47) B 48) C 49) [5, ∞) 50) The function y = 2 increases (as x increases) over the interval 0 < x < ∞. Its derivative The function increases (as x increases) wherever its derivative is positive.

= 4x is positive for x > 0.

51) The function y = decreases (as x increases) over no intervals of x-values. Its derivative no values of x. The function decreases (as x increases) wherever its derivative is negative.

is negative for

52) The curve y =

never has a negative slope. The derivative of the curve is

, which is never negative. A

curve only has a negative slope where its derivative is negative. Since the derivative of y = the curve never has a negative slope. 53) 54) 55)

is never negative,

The curve y = has a horizontal tangent at x = -3. The derivative of the curve is zero at x = -3. A curve has a horizontal tangent wherever its derivative equals zero. The curve has no tangent whose slope is -2. The derivative of the curve, never equals -2.

+ 4, is always positive and thus

Yes, a tangent line to a graph can intersect the graph at more than one point. For example, the graph has a horizontal tangent at x = 0. It intersects the graph at both (0, 0) and (2, 0).

56)

(4) = 10

57)

(4) = -5

58) There is no difference at all.

which equals

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the derivative. 1) y = 7 - 3x2 A) -6x B) 7 - 3x C) 7 - 6x D) -6

2) _______

2) y = 8 - 3x3 A) 8 - 9x2

B) -6x2

C) -9x

D) -9x2 3) _______

3) y = 4x4 + 5x3 + 4 A) 16x3 + 15x2

B) 4x3 + 3x2 - 7

C) 4x3 + 3x2

D) 16x3 + 15x2 - 7 4) _______

4) s = 5t2 + 4t + 1 A) 10t2 + 4

B) 10t + 4

C) 5t + 4

D) 5t2 + 4 5) _______

5) y = 17x-2 - 4x3 + 13x

A) -34x-1 - 12x2

B) -34x-3 - 12x2 + 13

C) -34x-3 - 12x2

D) -34x-1 - 12x2 + 13

y=3 A)

6) _______

+ 10x + 4 6x + 10 - 12

6x + 10 + 12

6x - 12

3x + 4 7) _______

7) w= A)

B) -

-4

8) _______

8) r= A)

B) -

C) -

D) -

9) _______

9) y= A)

+ B) -

10)

1) _______

y=2 A)

C) -

D) -

+ 10) ______

+ 7x + 3 4x + 7 +

Find the second derivative. 11) y = 9x2 + 8x - 8 A) 18x + 8

2x + 3

4x + 7 + 3

4x + 3

11) ______ B) 9

C) 0

D) 18

12) ______

12) y = 8x4 - 2x2 + 8 A) 32x2 - 4x

B) 32x2 - 4

C) 96x2 - 4

D) 96x2 - 4x 13) ______

13) y = 5x3 - 6x2 + 4 A) 12x - 30

B) 30x - 12

C) 12x - 20

D) 20x - 12 14) ______

14) s= + 13 A) 13t

C) 26t

D) 26t + 13 15) ______

15) y= -2 A) 17x - 2

D) 17x x

16)

y=3 A)

16) ______

+ 11x + 4 B)

6 - 48

6 + 48

6x + 11 - 12

6 + 48 17) ______

17) w= A)

B) -3

C) 12

D) 12

-3

+ 18) ______

18) r= A)

D) -

+ 19) ______

19) y= A)

+ B) -

20)

Find

y = 8x3 - 5x2 + 5 A) 10x - 32 + 5

. 21) y = (4x - 5)(6x + 1) A) 48x - 34

C) -

D) +

20) ______

32x - 10 + 5

48x - 10 + 5

10x - 48 + 5

21) ______ B) 48x - 26

C) 24x - 26

D) 48x - 13 22) ______

22) y = (4x - 5)(4x3 - x2 + 1) A) 48x3 + 72x2 - 24x + 4

B) 16x3 + 24x2 - 72x + 4

C) 64x3 - 72x2 + 10x + 4

D) 64x3 - 24x2 + 72x + 4

23) y = (x2 - 4x + 2)(5x3 - x2 + 4)

23) ______

A) 5x4 - 84x3 + 42x2 + 4x - 16

B) 5x4 - 80x3 + 42x2 + 4x - 16

C) 25x4 - 84x3 + 42x2 + 4x - 16

D) 25x4 - 80x3 + 42x2 + 4x - 16 24) ______

24) y = (3x3 + 6)(5x7 - 8) A) 150x9 + 210x6 - 72x2

B) 12x9 + 210x6 - 72x

C) 12x9 + 210x6 - 72x2

D) 150x9 + 210x6 - 72x 25) ______

25) y= A)

B) 2x +

C) 2x -

D) 2x +

2x + 26) ______

26) y= A)

B) -

+ 2x

C) -

- 2x

+ 2x

- 2x 27) ______

27) y= A)

B) + 4x

C) -

- 4x

D) -

+ 4x

+ 4x 28) ______

28) y= A)

B) +6

C) +6

D) -

-6

Find the derivative of the function. 29) y= A)

-6

29) ______

B) =

= D)

= 30) ______

30) y= A)

B) =

= D)

= 31)

= g(x) =

______ A)

31)

B) (x) =

(x) = D)

(x) =

(x) = 32) ______

32) y= A)

B) =

= D)

= 33) ______

33) y= A)

B) =

= D)

= 34)

f(t) = (5 - t) A)

= 34) ______ B)

(t) = C)

(t) = D)

(t) =

(t) = 35) ______

35) r= A)

B) =-

= C)

D) =

= 36) ______

36) y= A)

B) =

C) = Find the derivative.

= D) =

37)

37) ______

y= A)

-7/8 x

38)

9/8

8 x-1/

-9/8 -

x 38) ______

y= A)

x1/ 39) ______

39) s= A)

40)

41)

y=9 A)

40) ______ 9x

(x + 2)

9x (2 - x)

(2 - x)

18x

(1 - x) 41) ______

w= A)

(4 - e)

(5 - e)

42) ______

42) y= A)

+ B) +e

D) +e

+e 43) ______

43) y= A)

+ B) +

D) +

+ 44) ______

44) w= A) C)

+8 2.2

+ 24.8

2.2

+ 24.8

B) D)

2.2

+ 24.8

2.2

+ 3.1 45) ______

45) y= A)

B) -3.4

-2.4

+ +π

-2.4

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the derivatives of all orders of the function. 46) 46) _____________ y=

+ 5x - 13 47) _____________

47) y= 48)

y = (x + 1)(

48) _____________

- 4x + 7)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the derivative of the function. 49) s= A)

49) ______

B) =

D) =5

= 12

+ 12t 50) ______

50) y= A)

B) =x-

D) =

= 28 +

51) ______

51) r= A)

(7 - θ) B) =

-1

D) =-

-1

+7 52) ______

52) p= A)

B) =

+ 15

D) =

+ 16

+ 27

+ 53) ______

53) y= A)

B) = -18

+ 14

-3

D) =

= -

+ +

54)

54) ______

z=8 A)

B) = 8x

= 16x

-8

= 8x

+ 16

D) = 16x

Find the second derivative of the function. 55) y= A)

55) ______

B) =2-

C) = 2x -

D) =1+

=2+ 56) ______

56) s= A)

B) =7

= 42

D) = 42

57) ______

57) y= A)

B) =

D) =-

+ 58) ______

58) r= A)

(4 - θ)

B) =

-θ =

D) =-

-1

-1 59) ______

59) y= A)

B) =-

D) =

60) ______

60) p= A)

B) =-

D) =

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. 61) u(1) = 5, 61) ______ (1) = -6, v(1) = 6, (1) = -2. (uv) at x = 1 A) 26 62) u(1) = 2,

B) 46

(1) = -5, v(1) = 6,

C) -46

D) -42 62) ______

(1) = -4.

at x = 1 A)

B) -

63) u(1) = 5,

C) -

(1) = -5, v(1) = 6,

D) -

63) ______

(1) = -4.

at x = 1 A) - 2

C) 2

64) u(1) = 5,

(1) = -6, v(1) = 7,

(1) = -2.

(3v - u) at x = 1

______ A) 26

64)

B) 16

65) u(1) = 5,

(1) = -6, v(1) = 6,

(2u - 4v) at x = 1 A) -28 66) u(2) = 7,

67) u(2) = 7,

D) -12 65) ______

(1) = -4.

B) -14

(2) = 2, v(2) = -2,

(uv) at x = 2 A) -39

C) 0

C) 4

D) 34 66) ______

(2) = -5.

B) 39

(2) = 3, v(2) = -3,

C) 24

D) -31 67) ______

(2) = -4.

at x = 2 A)

C) -

68) u(2) = 10,

(2) = 3, v(2) = -3,

D) 68) ______

(2) = -5.

at x = 2 A)

B) -

69) u(2) = 9,

(2) = 2, v(2) = -1,

(3v - u) at x = 2 A) -13 70) u(2) = 9,

(2) = 4, v(2) = -1,

(2u - 4v) at x = 2 A) 22

D) 69) ______

(2) = -5.

B) -17

C) -12

D) 6 70) ______

(2) = -5.

B) 28

C) -12

D) 14

Provide an appropriate response. 71) Find an equation for the tangent to the curve y = A) y = x + 4 B) y = 4

71) ______ at the point (1, 4). C) y = 4x

D) y = 0 72) ______

72) Find an equation for the tangent to the curve y = A) y = 8x + 4 B) y = -8x + 20 73)

at the point (1, 12). C) y = -8

D) y = -4x + 16

The curve y = a + bx + c passes through the point (2, 10) and is tangent to the line origin. Find a, b, and c. A) a = 0, b = 1, c = 3 B) a = 3, b = 0, c = 1

at the

73) ______

C) a = 1, b = 3, c = 0 74)

75)

D) a = 2, b = 0, c = 0

The curves y = a + b and y = 2 and c. A) a = 1, b = -1, c = 2 C) a = -1, b = 1, c = -2

+ cx have a common tangent line at the point (-1, 0). Find a, b, B) a = 1, b = 0, c = 2 D) a - 2, b = 1, c = -1

Find all points (x, y) on the graph of f(x) = 2 A) (0, 0), (3, 9)

74) ______

B) (3, 9)

75) ______

-3x with tangent lines parallel to the line C) (3, 18)

D) (6, 9) 76) ______

76) Find all points (x, y) on the graph of y = A) (6, 2)

with tangent lines perpendicular to the line

B) (0, 0)

C) (0, 0), (3, 2)

D) (0, 0), (6, 2)

Solve the problem. 77) A charged particle of mass m and charge q moving in an electric field E has an acceleration a given by

where q and E are constants. Find A) B) =-

. C) = qEm

D) =-

78) A charged particle of mass m and charge q moving in an electric field E has an acceleration a given by

where q and E are constants. Find A) B) =

78) ______

. C)

79) 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

where n, the volume density of the gas, is a constant. Find A) B) = C)

77) ______

79) ______

. =

D) =-

80) Under standard conditions, molecules of a gas collide billions of times per second. If each

mol ecule

___ ___

has 80) diameter t, the average distance between collisions is given by

where n, the volume density of the gas, is a constant. Find

. A)

B) = -

= -

D) =

= -

81) A heat engine is a device that converts thermal energy into other forms. The thermal efficiency, e, of a heat engine is defined by

where

is the heat absorbed in one cycle and

cycle, is a constant. Find A)

, the heat released into a reservoir in one

. B) =

=C)

D) =

82) A heat engine is a device that converts thermal energy into other forms. The thermal efficiency, e, of a heat engine is defined by

where

81) ______

is the heat absorbed in one cycle and

, the heat released into a reservoir in one

cycl e, is a con stan t. Fin d

______ A)

82)

Provide an appropriate response. 83) Evaluate A)

83) ______

= by first converting it to a derivative at a particular x-value. B) 2 C) 1 D) 3

84) ______

84) Evaluate A)

= by first converting it to a derivative at a particular x-value. B) C) D) -

85) ______

85) Find the value of a that makes the following function differentiable for all x-values. A) 0

B) 25

C) -5

D) 5

1) A 2) D 3) A 4) B 5) B 6) A 7) B 8) B 9) C 10) C 11) D 12) C 13) B 14) C 15) D 16) B 17) B 18) A 19) C 20) C 21) B 22) C 23) C 24) A 25) C 26) B 27) A 28) A 29) B 30) D 31) B 32) D 33) C 34) D 35) D 36) B 37) A 38) C 39) C 40) C 41) D 42) B 43) D 44) B 45) D 46)

+ 3x + 5,

= 16x + 3,

= 16,

= 0 for n ≥ 4

47) 48)

y' = 3 49) B

- 6x + 3; y'' = 6x -6; y''' = 6;

= 0 for n ≥ 4

= 0 for n ≥ 8

50) C 51) C 52) B 53) B 54) C 55) D 56) C 57) B 58) B 59) B 60) C 61) C 62) D 63) D 64) C 65) C 66) A 67) D 68) D 69) B 70) B 71) B 72) B 73) C 74) A 75) B 76) D 77) C 78) C 79) D 80) C 81) C 82) C 83) B 84) D 85) C

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The function s = f(t) gives the position of a body moving on a coordinate line, with s in meters and t in seconds. 1) s = 5 + 3t + 3, 0 ≤ t ≤ 2 Find the body's displacement and average velocity for the given time interval. A) 26 m, 13 m/sec B) 26 m, 26 m/sec C) 32 m, 16 m/sec D) 16 m, 23 m/sec 2)

s = 7t - , 0 ≤ t ≤ 7 Find the body's displacement and average velocity for the given time interval. A) 98 m, 14 m/sec B) 98 m, -7 m/sec C) 0 m, 0 m/sec D) -98 m, -14 m/sec s=+ 8 - 8t, 0 ≤ t ≤ 8 Find the body's displacement and average velocity for the given time interval. A) -64 m, -8 m/sec B) 64 m, 8 m/sec C) 960 m, 120 m/sec D) -64 m, -16 m/sec s = 4 + 4t + 7, 0 ≤ t ≤ 2 Find the body's speed and acceleration at the end of the time interval. A) B) 20 m/sec, 8 m/ 12 m/sec, 2 m/ C) D) 27 m/sec, 8 m/ 20 m/sec, 16 m/ s = 8t - , 0 ≤ t ≤ 8 Find the body's speed and acceleration at the end of the time interval. A) B) 24 m/sec, -16 m/ 8 m/sec, -16 m/ C) D) 8 m/sec, - 2 m/ -8 m/sec, - 2 m/ s=+ 7 - 7t, 0 ≤ t ≤ 7 Find the body's speed and acceleration at the end of the time interval. A) B) 56 m/sec, -28 m/ -56 m/sec, -28 m/ C) D) 7 m/sec, 0 m/ 56 m/sec, -7 m/

1) _______

2) _______

3) _______

4) _______

5) _______

6) _______

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. 7) v (ft/sec)

When is the body's accelerat ion

equal to zero?

__ __ __ _

A) 0 < t < 2, 6 < t < 7 C) t = 0, t = 4, t = 7 8)

C) 3 ft/sec

D) 2 ft/sec 9) _______

v (ft/sec)

When does the body reverse direction? A) t = 7 sec C) t = 4, t = 7 sec 10)

8) _______

v (ft/sec)

What is the body's greatest velocity? A) 4 ft/sec B) 5 ft/sec 9)

B) t = 2, t = 3, t = 5, t = 6 D) 2 < t < 3, 5 < t < 6

v (ft/sec)

B) t = 2, t = 3, t = 5, t = 6, t = 7 sec D) t = 4 sec

___ ___

10)

What is the body's accelerati on when t = 8 sec? A) 11)

-2 ft/

0 ft/

1 ft/

1.67 ft/ 11) ______

v (ft/sec)

What is the body's speed when t = 2 sec? A) -2 ft/sec B) 3 ft/sec 12)

v (ft/sec)

C) 2 ft/sec

D) 0 ft/sec

___ ___

12)

When is the body moving backwar d? A) 4 < t < 7 C) 0 < t < 2, 6 < t < 7 13)

13) ______

s (m)

When is the body moving forward? A) 0 < t < 8 C) 0 < t < 1, 3 < t < 4, 5 < t < 7, 9 < t < 10 14)

B) 7 < t < 10 D) 0 < t < 4

s (m)

B) 0 < t < 1, 3 < t < 4, 5 < t < 7 D) 0 < t < 3, 3 < t < 5, 5 < t < 8

___ ___

14)

When is the body standing still? A) 1 < t < 2 C) 2 < t < 3, 4 < t < 5, 7 < t < 9 15)

15) ______

s (m)

When is the body moving backward? A) 2 < t < 3, 4 < t < 5, 7 < t < 9 C) 8 < t ≤ 10 16)

B) t = 3, t = 5, t = 8 D) 8 < t ≤ 10

s (m)

B) 2 < t < 3, 4 < t < 5, 7 < t < 8 D) 8 < t < 9

___ ___

16)

What is the body's velocity when t = 8 sec? A) 4 m/sec

B) 0 m/sec

C) -4 m/sec

The graphs show the position s, velocity v = ds/dt, and acceleration a = functions of time t. Which graph is which? 17)

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 18)

D) 3 m/sec of a body moving along a coordinate line as 17) ______

18)

___ ___

A) A = position, C = velocity, B = acceleration B) B = position, A = velocity, C = acceleration C) C = position, B = velocity, A = acceleration D) B = position, C = velocity, A = acceleration 19) ______

19)

A) A = position, B = velocity, C = acceleration B) C = position, A = velocity, B = acceleration C) A = position, C = velocity, B = acceleration D) B = position, C = velocity, A = acceleration 20) ______

20)

A) C = position, A = velocity, B = acceleration B) B = position, A = velocity, C = acceleration C) A = position, C = velocity, B = acceleration D) A = position, B = velocity, C = acceleration

21) ______

21)

A) B = position, C = velocity, A = acceleration B) B = position, A = velocity, C = acceleration C) C = position, B = velocity, A = acceleration D) C = position, A = velocity, B = acceleration 22) ______

22)

A) A = position, C = velocity, B = acceleration B) C = position, A = velocity, B = acceleration C) B = position, A = velocity, C = acceleration D) A = position, B = velocity, C = acceleration Solve the problem. 23) The position of a body moving on a coordinate line is given by

with s in meters

23) ______

and t in seconds. When, if ever, during the interval does the body change direction? A) t = 10 sec B) no change in direction C) t = 5 sec D) t = 2.5 sec 24)

25)

26)

At time t, the position of a body moving along the s-axis is body's acceleration each time the velocity is zero. A) B) a(10) = 0 m/ , a(4) = 0 m/ a(20) = 120 m/ C) D) a(10) = -18 m/ , a(4) = 18 m/ a(10) = 18 m/

m. Find the

24) ______

, a(8) = 20 m/ , a(4) = -18 m/

At time t, the position of a body moving along the s-axis is

Find the total

distance traveled by the body from A) 22 m B) 38 m

D) 18 m

25) ______

to C) 54 m

At time t ≥ 0, the velocity of a body moving along the s-axis is

When is the body

mo ving

___ ___

backwar 26) d? A) 0 ≤ t < 7 27)

28)

B) 0 ≤ t < 1

C) 1 < t < 7

At time t ≥ 0, the velocity of a body moving along the s-axis is body's velocity increasing? A) t < 3 B) t < 5 C) t > 3

D) t > 7 When is the

27) ______

D) t > 5

A ball dropped from the top of a building has a height of s = 144 - 16 meters after t seconds. How long does it take the ball to reach the ground? What is the ball's velocity at the moment of impact? A) 9 sec, -288 m/sec B) 3 sec, -96 m/sec C) 6 sec, -48 m/sec D) 3 sec, 96 m/sec

29) A rock is thrown vertically upward from the surface of an airless planet. It reaches a height of

28) ______

29) ______

meters in t seconds. How high does the rock go? How long does it take the rock to reach its highest point? A) 1800 m, 30 sec B) 1785 m, 15 sec C) 900 m, 15 sec D) 3480 m, 30 sec 30) 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 3 miles? Round your answer to four decimal places.

A) 0.0649 miles/min C) 0.0862 miles/min 31)

32)

30) ______

B) 14.3266 miles/min D) 0.0698 miles/min

Suppose that the dollar cost of producing x radios is cost when 35 radios are produced. A) $34 B) $6 C) $855

Find the marginal

Suppose that the dollar cost of producing x radios is cost per radio of producing the first 45 radios. A) $1545.00 B) $1495.00 C) $945.00

Find the average

31) ______

D) -$855 32) ______

D) $34.33 33) ______

33) Suppose that the revenue from selling x radios is r(x) = 85x dollars. Use the function (x) to estimate the increase in revenue that will result from increasing production from 125 radios to 126 radios per week. A) $60.00 B) $110.00 C) $72.50 D) $59.80 34)

The area A = π

of a circular oil spill changes with the radius. At what rate does the area

change with respect to the radius when A) B) 6π /ft 36π /ft 35)

/ft

12π

34) ______

/ft

The area A = π 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 5 ft to 5.1 ft?

35) ______

2.0

3.1

31.4

15.7

36) The number of gallons of water in a swimming pool t minutes after the pool has started to drain is A) 300 gal/min

36) ______

How fast is the water running out at the end of 14 minutes? B) 900 gal/min C) 1800 gal/min D) 600 gal/min

37) The driver of a car traveling at 36 ft/sec suddenly applies the brakes. The position of the car is

37) ______

t seconds after the driver applies the brakes. How far does the car go before coming to a stop? A) 432 ft B) 216 ft C) 6 ft D) 108 ft 38) The driver of a car traveling at 60 ft/sec suddenly applies the brakes. The position of the car is

38) ______

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) 60 sec B) 10 sec C) 30 sec D) 20 sec 39)

The size of a population of mice after t months is P = 100(1 + 0.2t + 0.02 ). Find the growth rate at

40)

months. A) 136 mice/month C) 34 mice/month

B) 168 mice/month D) 68 mice/month

The size of a population of lions after t months is P = 100 (1 + 0.2t + 0.02 ). Find the growth rate when A) 180 lions/month C) 140 lions/month

39) ______

B) 10,020 lions/month D) 160 lions/month

40) ______

1) A 2) C 3) A 4) A 5) C 6) A 7) D 8) B 9) D 10) D 11) C 12) D 13) C 14) A 15) A 16) C 17) C 18) D 19) C 20) D 21) C 22) B 23) D 24) D 25) A 26) C 27) C 28) B 29) C 30) D 31) B 32) D 33) A 34) D 35) B 36) D 37) D 38) B 39) D 40) C

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the derivative. 1) y= + 3 sec x A)

- 3 csc x

C) - 3 sec x tan x

+ 3 sec x tan x

x 2) _______

s= tan t A)

B) sec t tan t + 4

tan t -

t+4

D) =

t+4

tan t -

tan t + 3) _______

3) y = (csc x + cot x)(csc x - cot x) A) =0 C)

=D)

= 2)

B) D)

= - csc x cot x =-

x 4) _______

4) y= A) C) 5)

1) _______

+ = 4 csc x cot x = 4 cos x -

B) D)

= 4 csc x cot x -

= - 4 csc x cot x +

x 5) _______

s= cos t - 13t sin t - 13 cos t A) =

sin t - 7

cos t + 13t cos t

B) =-7

sin t - 13 cos t + 13 sin t

sin t + 7

cos t - 13t cos t

sin t + 7

cos t - 13t cos t - 26 sin t

C) D)

6) _______

6) y= A)

+ B) =

D) =

- csc t + 6

+ 7) _______

B) =5

+ csc t cot t

D) =5

- csc t cot t

t+6 8) _______

8) p= A)

B) =

D) =-

r=6A)

=9) _______

cos θ B) =6

cos θ -

sin θ

=-6

cos θ +

D) =6

sin θ -

cos θ

sin θ 10) ______

10) p= A)

B) = - csc q cot q

D) =

11)

= sec q tan q 11) ______

s = sin t A)

B) = -cost -

= cos t +

D) = -cos t +

12)

= cost 12) ______

s= - cos t + 7 A)

= 2t - sin t + 7

= 2t + sin t - 7

D) = t + sin t + 7

Find the indicated derivative. 13) Find if y = 5 sin x. A) = 5 sin x 14)

15)

Find A)

q+1

= 2t + sin t + 7

13) ______ B)

= 25 sin x

= 5 cos x

= - 5 sin x 14) ______

if y = 6 sin x. = 6 cos x

= 6 sin x

= - 6 sin x

= - 6 cos x Fin

d if 15) y = -7 cos x. A) 16)

___ ___

= 7 sin x

Find A)

if y = -8 cos x.

17) Find

if y = 7x sin x.

= 8 sin x

= -7 sin x

= 7 cos x

16) ______ B)

= -8 sin x

= -8 cos x

= 8 cos x 17) ______

= 14 cos x - 7x sin x

= 7 cos x - 14x sin x

= - 7x sin x

= - 14 cos x + 7x sin x

18) Find

= -7 cos x

18) ______

if y = 3x sin x.

= 3x cos x + 9 sin x

= - 3x cos x - 9 sin x

= - 3x cos x + 9 sin x

= 6 cos x - 3x sin 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. 19) 19) ______ y = cos(x), -2π ≤ x ≤ 2π, x =

20) ______

20) y = cot(x), -2π ≤ x ≤ 2π, x =

Find the limit. 21)

21) ______ cos

B) 1

C) -1

D) 0

22) ______

22) A) 1

C) 3

D) 0 23) ______

23) 8 cos A) 1

B) 8

D) - 8

C) 0

24) ______

24) 2 sin A) -2

B) 0

C) 4

D) 2 25) ______

25) sec A) 1

B) 0

D) -1 -

Solve the problem. 26)

26) ______

Find the tangent to y = cos x at x = A) y = 1 B)

. C)

y=-x+

D) y=-x-

y=x+ 27) ______

27) Find the tangent to y = cot x at x = . A) B) y = 2x + 1

y = -2x + 28) Find the tangent to y = 2 - sin x at x = π. A) y = - x + π - 2 B) y = x - π + 2

D) y = -2x +

y = 2x -

+1 28) ______

C) y = x - 2

D) y = - x + 2

29) Does the graph of the function y = tan x - x have any horizontal tangents in the interval If so, where? A) Yes, at x = 0, x = π, x = 2π

C) No

Yes, at x = , x = D) Yes, at x = π

30) Does the graph of the function y = 5x + 10 sin x have any horizontal tangents in the interval If so, where? A)

Yes, at x = C) No

Yes, at x =

,x=

29) ______

30) ______

Yes, at x =

, x=

31) Does the graph of the function y = 12x + 6 sin x have any horizontal tangents in the interval

31) ______

If so, where? A)

Yes, at x = C) No

,x=

Yes, at x =

,x=

D) Yes, at x =

32) Find all points on the curve y = sin x, 0 ≤ x ≤ 2π, where the tangent line is parallel to the line

B) ,

D) ,

33) Find all points on the curve y = cos x, 0 ≤ x ≤ 2π, where the tangent line is parallel to the line A) (2π, 1)

C) y = -

x+1+ x-1-

33) ______

D) (π, - 1)

34) ______

34) Find an equation for the tangent to the curve at P.

A) y =

32) ______

B) y=-

-1-

+1+

35) Find an equation for the horizontal tangent to the curve at P.

___ ___

35)

A) y = 1

B) y =

D) y = 0 y=

The equation gives the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds). 36) s = 6 + 7 cos t Find the body's velocity at time t = π/3 sec. A) B) C) D) m/sec

m/sec

37) s = -4 + 11 cos t Find the body's speed at time t = π/3 sec. A) B) m/sec

C) m/sec

m/sec 38) ______

D) -

m/ 39) ______

m/sec

D) -

39) s = 1 + 11 cos t Find the body's jerk at time t = π/3 sec. A)

37) ______

m/sec

38) s = 6 + 3 cos t Find the body's acceleration at time t = π/3 sec. A) B)

m/sec

36) ______

D) -

40) s = 10 sin t - cos t Find the body's velocity at time t = π/4 sec. A) B) m/sec

40) ______ C)

m/sec

41) s = 10 sin t - cos t Find the body's acceleration at time t = π/4 sec. A)

m/sec

41) ______ m/

B) m/ C)

D) -

42) s = 5 sin t - cos t Find the body's jerk at time t = π/4 sec. A) B) 2 m/ -3 m/

m/ 42) ______

-2

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 43) Find a value of c that will make 43) _____________

f(x) = continuous at x = 0. 44) 45) 46)

Find

(cos x).

Find

(sin x).

Find

(sin x).

44) _____________ 45) _____________ 46) _____________ 47) _____________

47) Graph y = -tan x and its derivative together on . Does the graph of appear to have the largest slope? If so, what is it? If not, explain.

48) Graph y = -tan x and its derivative together on . Does the graph of appear to have the smallest slope? If so, what is it? If not, explain.

___ ___ ___ ___ _

48)

49) _____________

49) Graph y = -tan x and its derivative together on

. Is the slope of the graph of

ever positive? Explain.

50) Rewrite tan x and use the product rule to verify the derivative formula for tan x.

50) _____________

1) B 2) C 3) A 4) D 5) C 6) D 7) B 8) A 9) D 10) C 11) B 12) D 13) D 14) B 15) C 16) C 17) A 18) B 19) C 20) C 21) B 22) C 23) D 24) A 25) A 26) B 27) C 28) B 29) A 30) D 31) C 32) B 33) B 34) D 35) B 36) D 37) D 38) B 39) D 40) C 41) D 42) B 43) c = 16 44) - cos x 45) - sin x 46) cos x 47)

Yes, the graph of

does appear to have a largest slope. It occurs at x = 0, when the slope equals -1.

48)

No, the graph of does not appear to have a smallest slope. The slope of the graph of y = - tan x decreases without bound as x approaches π/2 or -π/2. 49)

No, the slope of the graph of . Since positive. 50) tan x =

= sin x sec x

is never positive. The slope at any point is equal to the derivative, which is

x is never negative,

is never positive, and the slope of the graph is never

(tan x) =

(sin x sec x) = sin x sec x tan x+ sec x cos x =

∙ tan x+1 = = x

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Given y = f(u) and u = g(x), find dy/dx = (g(x)) (x). 1) y= , u = 3x - 4 A) 18x - 24 B) 9x - 12 C) 24x - 12 D) 18x

2) _______

2) y= A)

, u = 8x - 5 B) -

y = u(u - 1), u = A) 2 + 4x

3) _______

+x B)

- 2x

+ 4x + 1

-1

B) - 10 cos (10x + 19) D) - cos (10x + 19) 5) _______

5) y = tan u, u = -9x + 13 A) -9 (-9x + 13) C) (-9x + 13)

B) -9 sec (-9x + 13) tan (-9x + 13) D)

(-9x + 13) 6) _______

y = csc u, u = + 9x A) -(6 + 9) ( + 9x) C) -(6 + 9) csc x cot x

B) D)

- csc ( -(6

+ 9x) cot ( + 9) csc (

+ 9x)

+ 9x) cot (

+ 9x) 7) _______

7) y = sin u, u = cos x A) - cos(cos x) sin x C) sin(cos x) sin x

B) - cos x sin x D) cos x sin x

Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x. 8) y= A) B) y = 5u + 10; u =

4) _______

4) y = sin u, u = 10x + 19 A) cos (10x + 19) C) 10 cos (10x + 19)

1) _______

;

= -10

8) _______

; u = -2x + 10;

= -10

D) y=

; u = -2x + 10;

= -2

9) _______

9) y= A) y=

;u=6

- x;

;u=6

- x;

B) C)

y=6

10)

- u; u = D)

;

= 12

;u=6

- x;

y= A)

10) ______

x B) y=

; u = cos x;

=-4

x sin x

y = cos u; u =

;

= - sin(

)

y = cos u; u =

;

=-4

sin(

D) y=

; u = cos x;

x sin x

) 11) ______

11) y = csc(cot x) A) y = csc u; u = cot x;

x cot x

y = csc u; u = cot x;

= - csc(cot x) cot(cot x)

y = cot u; u = csc x;

y = csc u; u = cot x;

= csc(cot x) cot(cot x)

B) C) (csc x) csc x cot x

D) x 12) ______

12) y = tan A) y = tan u; u = π -

;

y = tan u; u = π -

;

y = tan u; u = π -

;

y = tan u; u = π -

;

B) C) sec

tan

13) ______

13) y = cot(6x - 6) A) y = cot u; u = 6x - 6;

= - 6 cot(6x - 6) csc(6x - 6)

y = cot u; u = 6x - 6;

y = 6u - 6; u = cot x;

= - 6 cot x

y = cot u; u = 6x - 6;

=-6

B) (6x - 6)

C) x

14)

y= A)

(6x - 6) 14) ______ y =

; u = 7x/2; =

y = ;u = 7x /2;

D) y=

; u = 7x/2;

Find the derivative of the function. 15) q= A)

; u = 7x/2;

15) ______ B)

16) ______

16) s = sin A)

- cos B)

cos

+ sin

C) 17)

cos

sin

cos

sin

D) cos

sin

y= cos x - 15x sin x - 15 cos x A) sin x - 4 cos x + 15x cos x C) -4 sin x - 15 cos x + 15 sin x

17) ______ B) D)

sin x + 4

cos x - 15x cos x

sin x + 4

cos x - 15x cos x - 30 sin x 18) ______

18) y= A)

+ B) -

+ D)

19) ______

19) y = (x + 1)2(x2 + 1)-3 A)

20) ______

20) h(x) = A)

D) -5

21)

r= A) C)

22)

23)

y = (5 A)

y= A) C)

Find dy/dt. 25) y= A) C) 26)

21) ______ -2

(

θ + sec θ tan θ)

y= A) C)

-2

22) ______

y = (1 + 6x) A) -6(1 + 6x)

24)

-2 (15 (15

-36x

-36 23) ______

+ 8x - 6) -6

+ 24

-6

+ 24

+ 15 -18

- 4x + 8) )

B) D)

(15

-6

+ 24

(15

- 4x + 8)

-3

- 4x + 8)

24) ______

cos 6θ -6

sin 6θ

-56

(4 cos 6θ - 2 cos 6θ + 6 cos 6θ)

sin 6θ

(4 cos 6θ - 2θ cos 6θ - 6θ sin 6θ)

25) ______

(πt - 8) -5π

(πt - 8)

-5

(πt - 8) sin(πt - 8)

B) D)

-5π 5

(πt - 8) sin(πt - 8) (πt - 8) 26) ______

-6 -6

27) y = cos(

cos 8t

-48

cos 8t

-48 27) ______

)

B) -

C) -sin(

sin( )

)

sin( D) -sin

)

28)

28) ______

y= A) C)

29)

y = 5t A) C)

30)

(22

- 42)

(22

- 42)

B) D)

(15

- 6) 29) ______

(5t + 3)

(8t + 3) 30) ______

y= A)

B) cos

D) sin

Find

105

sin

. 31)

31) ______ y= A)

B) -

D) 12

32) ______

32) y = A)

C) -

33)

D) -

33) ______

y= A)

34) y = 6 sin(4x + 11) A) -96 sin(4x + 11) C) -96 cos(4x + 11)

34) ______ B) -24 sin(4x + 11) D) 24 cos(4x + 11) 35) ______

35) y = 2 cot A)

B) -4 csc

cot C)

D) -

cot

36) ______

36) y= tan(-9x - 5) A)

(-9x - 5) tan(-9x - 5) C)

(-9x - 5) tan(-9x - 5) D)

sec(-9x - 5) 37)

38)

(-9x - 5)

37) ______

y = -2 A)

-1792

+ 1344

-208

+ 384

- 240 - 60

-16

+ 32

-208

+ 224

y = sin(3 ) A) 6x cos (3 )+3 cos (3 ) B) 3( + 4x + 2) cos(3 ) - 9x ( + 4 + 4x) sin(3 C) ( + 4x + 2) cos(3 ) + 3x ( +4 + 4x) cos(3 D) -6 sin(3 ) - 6x sin(3 )

Find the value of at the given value of x. 39) f(u) = , u = g(x) = + 2, x = 1 A) -30 B) 15

- 12 - 48 38) ______

) )

39) ______ C) 6

D) 30 40) ______

40) f(u) = A)

, u = g(x) =

, x = 49 B)

41) ______

41) f(u) = , u = g(x) = 18 A)

,x=0 B) 12

C) - 12

D) 18 42) ______

42) f(u) = A)

, u = g(x) = 3x -

,x=1 B) 1

D) - 1

C) -

43) ______

43) f(u) = A) 8

+ 15, u = g(x) =

,x=8 B) 64

C) 16

D) -31 44) ______

44) f(u) = A)

, u = g(x) = 4

+ x + 4, x = 0 B)

D) -

45)

f(u) = tan

___ ___

45) ,u= g(x) = x=8

, A) 16 46)

f(u) = A) 0

B) -8π πu + u, u = g(x) = -x, x = 10 B) -10

C) -64π

D) 8π 46) ______

C) -1

D) 1 47) ______

47) f(u) = A) -π

- u, u = g(x) = πx, x = 8 B) 2 - π

C) 8π

48) Assume that f'(3) = 8 , g'(4) = -6, g(4) = 3, and y = f(x). What is y' at x = 4? A) -48 B) -30 C) 24

D) -8π 48) ______ D) 8

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. 49) 49) ______

f(g(x)), x = 4 A) -12

B) 6

C) -36

D) 18 50) ______

50)

,x=3 A)

D) 51) ______

51)

1/ (x), x = 4 A)

D) -

52) ______

52)

,x=3 A)

B) -

53) ______

53)

(x) ∙ g(x), x = 3 A) 31

B) 15

C) 55

D) 84 54) ______

54)

g(x + f(x)), x = 3 A) -6

B) -48

C) -54

D) 27 55) ______

55)

(x), x = 4 A)

D) -

Solve the problem. 56) The position of a particle moving along a coordinate line is s = seconds. Find the particle's velocity at A) 2 m/sec B) -

C) 1 m/sec

seconds. Find the particle's acceleration at A) B)

58)

56) ______

m/sec

57) The position of a particle moving along a coordinate line is s =

, with s in meters and t in

with s in meters and t in

57) ______

D) m/

Suppose that the velocity of a falling body is v = k (k a constant) at the instant the body has fallen s meters from its starting point. Find the body's acceleration as a function of s. A) B) C) D) a = 2ks a = 2k a = 2k a=2

58) ______

59) ______

59) Suppose that the velocity of a falling body is v = (k a constant) at the instant the body has fallen s meters from its starting point. Find the body's acceleration as a function of s. A) B) C) D) a=-

a=-

60) The position(in feet) of an object oscillating up and down at the end of a spring is given by at time t (in seconds). The value of A is the amplitude of the motion, k is a measure of the stiffness of the spring, and m is the mass of the object. Find the object's velocity at time t.

60) ______

B) v = A cos

ft/sec

v=A

cos

ft/sec

v=-A

cos

ft/sec

D) v=A

cos

ft/sec

61) The position (in centimeters) of an object oscillating up and down at the end of a spring is given

61) ______

by at time t (in seconds). The value of A is the amplitude of the motion, k is a 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) C) D) cm/sec

cm/sec

62) The position (in centimeters) of an object oscillating up and down at the end of a spring is given

62) ______

by at time t (in seconds). The value of A is the amplitude of the motion, k is a measure of the stiffness of the spring, and m is the mass of the object. Find the object's acceleration at time t. A) B) a=-

sin

cm/

cos

cm/

D) a = - A sin

cm/

a=-A

sin

cm/

63) The position (in centimeters) of an object oscillating up and down at the end of a spring is given

63) ______

by at time t (in seconds). The value of A is the amplitude of the motion, k is a 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) B) C) D) A cm/ cm/ cm/ A cm/ SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 64) Find ds/dt when θ = π/4 if s = sin θ and dθ/dt = 10. 64) _____________ 65)

Find dy/dt when x = 5 if y = 2

- 6x + 7 and dx/dt = 1/2.

65) _____________ 66) _____________

66) Find the tangent to the curve y = 4 cot

at x =

. What is the largest value the slope

of the curve can ever have on the interval 67) Find equations for the tangents to the curves y = tan 2x and y = -tan(x/2) at the origin. How are the tangents related?

67) _____________

1) A 2) D 3) D 4) C 5) A 6) D 7) A 8) D 9) A 10) A 11) D 12) B 13) D 14) A 15) A 16) B 17) B 18) A 19) D 20) B 21) D 22) C 23) B 24) D 25) B 26) B 27) B 28) C 29) D 30) D 31) C 32) D 33) A 34) A 35) A 36) A 37) A 38) B 39) D 40) A 41) A 42) C 43) C 44) D 45) D 46) C 47) A 48) A 49) C 50) D 51) A

52) C 53) C 54) C 55) B 56) C 57) D 58) D 59) D 60) B 61) C 62) A 63) D 64) 5 65) 7 66) y = - 4πx + 2π + 4; - 2π 67) y = 2x and y = -

x; The tangents are perpendicular.

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use implicit differentiation to find dy/dx. 1) 2xy - y2 = 1 A) B) C) D)

2) _______

2) x3 + 3x2y + y3 = 8 A)

1) _______

D) 3) _______

3) = x 2 + y2 A)

4) y

4) _______

=4 A)

B) -

5) _______

5) xy + x = 2 A)

D) -

6) _______

6) xy + x + y = x2y2 A)

7) _______

= cot y A)

cos xy + A)

D) 8) _______

= B)

9) _______

9) y cos A)

= 7x + 7y B)

10) ______

10) x = sec(4y) A) 4 sec(4y) tan(4y)

B) sec(4y) tan(4y) D) cos(4y) cot(4y)

C) cos(4y) cot(4y) 11)

11) ______

= sin(x + 8y) A)

D) =

Find dr/dθ. 12) θ4/3 + r4/3 = 1 A)

12) ______ B)

13) ______

13) θ1/3 - r1/3 = 1 A)

14) r

14) ______

=4 A)

15)

cos (rθ)+ A)

15) ______

= B)

Use implicit differentiation to find dy/dx and 16) xy - x + y = 6 A) =

;

. 16) ______ B) =

;

D) =-

;

= 17) ______

17) 2y - x + xy = 7 A)

B) ;

;

D) =-

y/d

18)

;

18) ______

B) =-

;

C) = 19)

+ A)

;

20) ______

=6 B) =

;

D) = +

;

21) ______

B) =

;

D) =-

;

= 22) ______

- y = 2x A)

B) =

- 2;

C) ;

23) xy + 3 = y, at the point (4, -1) A) = 3;

;

= 23) ______

= - 24

;

D) =

+ A)

= D)

24)

22) 4

19) ______

21)

;

20)

= D)

;

= 24) ______

= 5, at the point (2, 1) B) = - 2;

= 2;

D) = - 2;

= -5

;

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. 25) 25) ______ + = + 9x, slope at (0, 1) A) B) C) D) 26)

26) ______

= 16, slope at (2, 1) B) -8

C) 2

27)

27) ______

y - π cos y = 3π, slope at (1, π) A) 0 B)

C) -2π

D) π

28)

+ A)

= y=-

29)

x-1

y=-

31)

C) x+2

= y=

32)

y=-

D) y = 16x - 1 y=

C) y = πx

D) y = -2πx + 3π

x+ 31) ______

+ 12x, normal at (0, 1) B) x+1

x+1

30) ______

y - π cos y = 3π, tangent at (1, π) A) y = -2πx + π B)

+ A)

29) ______

y=-

C) y = 4x + 1

x+1

y=-

x 32) ______

= 64, normal at (2, 1) A)

B) y = 2x - 3 y=-

33)

C) y = 3x + 1

= 32, tangent at (2, 1) A) y = 16x + 1 B) y=-

30)

28) ______

+ 9x, tangent at (0, 1) B)

x+2

D) y = -2x + 5 y=

x 33) ______

y - π cos y = 6π, normal at (1, π) A)

y=- x+ +π C) y = -2πx + 3π

+π

Provide an appropriate response. 34) Find the slope of the curve x A) B) -

+π

34) ______

= -4 at (-1, 2). C)

D) -

35)

36)

37)

38)

At the two points where the curve + 2xy + = 9 crosses the x-axis, the tangents to the curve are parallel. What is the common slope of these tangents? A) 1 B) -1 C) -3 D)

At the two points where the curve - xy + = 36 crosses the x-axis, the tangents to the curve are parallel. What is the common slope of these tangents? A) -1 B) 1 C) -2 D) 2 The line that is normal to the curve - xy + point? A) (-8, -8) B) (-4, -4)

= 16 at (4, 4) intersects the curve at what other C) (0, -4)

Find the normal to the curve + = 2x + 2y that is parallel to the line A) y = x + 2 B) y = -x - 2 C) y = x - 2

35) ______

36) ______

37) ______

D) (-4, 0) 38) ______ D) y = -x + 2

1) D 2) A 3) D 4) B 5) D 6) B 7) C 8) C 9) A 10) C 11) A 12) D 13) D 14) D 15) C 16) A 17) B 18) D 19) D 20) C 21) D 22) B 23) C 24) C 25) B 26) A 27) C 28) C 29) B 30) D 31) B 32) B 33) D 34) C 35) B 36) D 37) B 38) D

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the formula for 1) f(x) = 27 A)

/dx. 1) _______ B)

2) _______

2) f(x) = A)

x+ B) 7x -

f(x) = A)

D) 7

x3) _______ B)

-3(

4) _______

f(x) = A)

Find the value of /dx at x = f(a). 5) f(x) = 4x + 9, a = -1 A)

B) 9

C) 4

5) _______

6) _______

6) f(x) = A)

x + 10, a = -2 B) 10

f(x) = 2 , x ≥ 0, a = 5 A) 20

f(x) = -3 A) 72

C) 3

7) _______ B)

8) _______

- 3, x ≥ 2, a = 6 B)

f(x) = A) 2

9) _______

- 6x + 1; a = 4 B)

Find the derivative of y with respect to x, t, or θ, as appropriate. 10) y = ln 2x A)

10) ______

11) ______

11) y = ln(x - 7) A)

12) ______

12) y = ln 8x2 A)

13) ______

13) y= A)

14) ______

14) y= ln x A) ln x C) 5 -

ln x

15) ______

15) y = ln(ln 9x) A)

16) ______

16) y = ln A)

D) ln 17) ______

17) y = ln A)

18) y = ln(cos(ln θ)) A) tan(ln θ)

18) ______ B) -tan(ln θ)

D) -

Use logarithmic differentiation to find the derivative of y. 19) y = A) C)

(2x + 3)

19) ______ B) D)

20) ______

20) y= A)

D) (lnx - ln(x + 7)) 21) ______

21) y = cos x A)

cos x cos x

cos x 22) ______

22) y = x(x - 8)(x + 5) A) +

B) 1

D) x(x - 8)(x + 5)(lnx + ln(x - 8) + ln(x + 5)) x(x - 8)(x + 5)

23)

23) ______

y= A) B) C)

(4ln(

+ 1) + 3ln(x - 1) + 4ln x)

D) +

+ 24) ______

24) y= A)

D) + cot x 25) ______

25) y= A)

B) ln x +

ln(

+ 4) -

ln(x + 5) D)

26) ______

26) y= A)

B) 4

D) (ln x + ln(x - 2) - ln(

+ 7)) 27) ______

27) y= A) (ln(4x + 1) + 2ln(x + 4) - ln(

+ 6) - ln(x + 7))

B) 5 C) +

Find the derivative of y with respect to x, t, or θ, as appropriate. 28) y= A) B) C) 3 -6 29)

y= A) C)

30)

31)

32)

33)

) ln (2

y=( A)

- 2x + 6) + 4)

C) 7x

y = 4 (sin θ - cos θ) A) 0 (sin θ - cos θ) + 4 )

+ 14 31) ______

B) D)

(2x - 2)

y = ln (2θ

B) D)

)

(

)

30) ______

-7

D) -6 ln (3 - 6x) 29) ______

y = 7x A) 7

28) ______

(

+ 4x + 4) 32) ______

B) D)

sin θ

(sin θ - cos θ) 33) ______

ln (2

(1-θ))

-1 34)

34) ______

y = sin A) cos (-5 C) (-5

)

) cos

cos 5

cos 35) ______

35) y = ln A)

D) ln

36)

y= A)

(ln

36) ______

+ 6) B)

D) (cos t)(ln

Find

. 37)

ln y = A)

+ 6) +

37) ______

cos 4x -4y

sin 4x

38)

cos 4x - 4

sin 4x

38) ______

= sin (x + 7y) A) ln sin (x + 7y)

-1

39)

40)

39) ______

ln 8xy = A)

tan y = A)

40) ______

+ ln 8x B) +

y 41)

= sin x

41) ______

B) 5 + 5 - cos y

42) ______

42) sin y = 5x + 5y A)

43)

43) ______

= cos (4x +y) A)

B) ln cos (4x + y)

44)

44) ______

= A)

Find the derivative of y with respect to the independent variable. 45) y= A) B) x ln 5 C) 46)

y= A) -

y= A) C)

48)

49)

y= A)

ln 5

ln x 46) ______

cos θ sin θ

C) 47)

45) ______

sin θ

sin θ 47) ______

ln 10 sin πθ

-π

ln 10 sin πθ

ln 10

D) 48) ______

(6 - e)

(5 - e)

49) ______ B)

ln π

50) ______

50) y= A)

51)

51) ______

y= A)

52) ______

52) y= A)

53) ______

53) y= A)

(cos θ - sin θ -

)

B) D)

(cot θ - tan θ - ln 7 -1) 54)

y= A)

(sec θ csc θ - ln 7 -1) 54) ______

ln 3

Use logarithmic differentiation to find the derivative of y with respect to the independent variable. 55) y= A) x ln(x + 3) B) C)

55) ______

D) ln(x + 3) +

56)

56) ______

y= A)

D) 3t

57)

y= A) C)

58)

57) ______ (ln cos x - x tan x) (ln cos x + x cot x)

B) D) ln cos x - x tan x 58) ______

y= A)

B) 8 cos x ln x +

C) 8 sin x ln x

59)

60)

y= A)

59) ______ ln x

y= A) sin x ln (x + 6) C)

ln x

60) ______ B) D)

cos x ln (x + 6) + 61)

y= A) ln x ln (ln x) C)

62)

y= A)

ln 2x C) x(2 ln 2x +1)

63)

y= A) x ln (3x + 3)

61) ______ B) D)

62) ______ B) D)

(2 ln 2x ) (2 ln 2x + 1) 63) ______

B) ln (3x + 3) +

64)

y= A) cos x cot x - sin x ln(sin x) C) cos x ln ( sin x)

64) ______ B)

(cos x cot x - sin x ln (sin x)) D) cos x cot x - ln (sin x)

1) C 2) D 3) B 4) D 5) D 6) C 7) C 8) D 9) D 10) D 11) D 12) A 13) C 14) B 15) D 16) B 17) D 18) D 19) D 20) B 21) B 22) C 23) A 24) C 25) D 26) D 27) D 28) C 29) D 30) B 31) A 32) B 33) B 34) C 35) A 36) A 37) C 38) D 39) A 40) D 41) A 42) C 43) D 44) D 45) C 46) B 47) C 48) C 49) A 50) A 51) A

52) A 53) C 54) C 55) C 56) C 57) A 58) B 59) A 60) B 61) B 62) D 63) D 64) B

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use a reference triangle in an appropriate quadrant to find the angle. 1) A)

1) _______

2) _______

2) sin-1 A)

3) _______

3) cos-1 A)

4) tan-1 (1) A) 0

(

4) _______ B)

D) 1

5) _______

)

B) ± 2πn,

± 2πn

6) _______

(-1) A)

C) -

D) 7) _______

0 A)

B) 0

C) π

Evaluate exactly. 8) cos A)

8) _______ B)

9) _______

9) sec A)

D) -

10) ______

10) sin A)

D) 1

11) ______

11) sec A) -

12)

13)

tan( A)

1) + cos(

csc( A) -

)) B)

C) -1

12) ______ C)

D) Undefined

13) ______

1) B) 0

14) ______

14) A)

D) -

Find the limit. 15)

15) ______ x

A) 1

B) -1

D) 16) ______

16) x A) -1

B) 1

C) 0

D) π 17) ______

17) x B) ∞

D) 0 18) ______

18) x B) -∞

A) 0

D) 19) ______

19) x A) 0

B) ∞

20) ______

20) x B) -∞

C) 0

D) π

Find the derivative of y with respect to x. 21) y= A)

22)

y= A)

21) ______ C)

22) ______

- 2) B)

23) ______

23) y= A)

B) -

24) ______

24) y= A)

25) ______

25) y= A)

26)

27)

y=3 A)

y= A)

26) ______

) B)

27) ______ B)

28)

y=2 A)

28) ______

x B) +8

D) x

+8 29)

30)

y= A)

(ln 4x)

y= A)

(

29) ______ B)

30) ______

) B)

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 31) Which of the following items is undefined and why? 31) _____________ 8 or

8 32) _____________

32) Which of the following items is undefined and why? or 33)

Derive the identity

(-x) = π -

(-x) = π x= 34)

35)

x by combining the following two equations:

x (1/x)

Consider the graphs of y = x and y = x. Does it make sense that the derivatives of these functions are opposites? Explain. Graph y =

(sin x). Explain why the graph looks like it does.

34) _____________

35) _____________ 36) _____________

36) Graph f(x) = does. 37)

33) _____________

Graph f(x) =

and g(x) =

. Explain why the graph looks like it

x together with its first derivative. Comment on the behavior of f and

the shape of its graph in relation to the signs and values of

37)

___ ___ ___ ___ _

1) D 2) D 3) D 4) C 5) C 6) B 7) B 8) C 9) D 10) C 11) B 12) B 13) D 14) B 15) C 16) C 17) A 18) D 19) A 20) D 21) D 22) A 23) C 24) A 25) B 26) D 27) D 28) B 29) D 30) D 31)

8, There is no angle whose cosine is 8.

32) , There is no angle whose cosecant is

33)

(-x) = (-1/x) = π (1/x) = π x 34) 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. 35) When plugging in angles such that -

≤x≤

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. 36) When x is positive these graphs are identical because they are both giving the same angle.

When x is negative both functions are still referring to the same angle. However, inverse cosine gives values between π/2 and π while inverse tangent gives values between -π/2 and 0. 37)

The graph of f(x) =

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

approaches its vertex at x = 0. To

the right of the y-axi s, the grap h of quick ly desce nds and f(x) beco mes more steep .

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Suppose that the radius r and the circumference C = 2πr of a circle are differentiable functions of t. Write an equation that relates dC/dt to dr/dt. A) B) C) D) = 2πr

= 2π

1) _______

= 2π 2) _______

2) Suppose that the radius r and volume V = π Write an equation that relates dV/dt to dr/dt. A) B) = 4π

of a sphere are differentiable functions of t. C)

D) = 4π

3) The area of the base B and the height h of a pyramid are related to the pyramid's volume V by the formula A)

How is dV/dt related to dh/dt if B is constant? B) C)

4) The volume of a square pyramid is related to the length of a side of the base s and the height h by the formula A)

How is dV/dt related to ds/dt if h is constant? B) C)

3) _______

4) _______

D) =

5) If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse,

5) _______

How is dc/dt related to da/dt and db/dt? B)

= 2a

+ 2b

= D)

+ 6) _______

6) The kinetic energy K of an object with mass m and velocity v is related to dv/dt if K is constant? A) B) =C)

How is dm/dt

= - 2m D)

7) The range R of a projectile is related to the initial velocity v and projection angle θ by the equation A)

where g is a constant. How is

related to

if v is constant?

7) _______

B) =

D) =

=8) _______

8) The range R of a projectile is related to the initial velocity v and projection angle θ by the equation A)

where g is a constant. How is B)

related to

if θ is constant?

D) =

9) _______

9) The range R of a projectile is related to the initial velocity v and projection angle θ by the equation where g is a constant. How is neither v nor θ is constant? A) B)

related to

and

D) =

Provide an appropriate response. 10) If xy + x = 12 and dx/dt = -3, then what is dy/dt when x = 2 and y = 5? A) -9 B) -3 C) 3

10) ______ D) 9 11) ______

11) If x3 + y3 = 9 and dx/dt = -3, then what is dy/dt when x = 1 and y = 2? A)

C) -

D) 12) ______

12) If xy2 = 4 and dx/dt = -5, then what is dy/dt when x = 4 and y = 1? A)

B) -

D) 13) ______

13) If A)

= x2 + y2 and dx/dt = 12, then what is dy/dt when x = 1 and y = 0? B) -12 C)

D) 12

14) If y A)

= 12 and dx/dt = 8, then what is dy/dt when x = 15 and y = 3? -

14) ______

Solve the problem. 15) A company knows that the unit cost C and the unit revenue R from the production and sale of x units are related by cost is changing by A) $297.50/unit

Find the rate of change of unit revenue when the unit and the unit revenue is $4000. B) $636.10/unit C) $974.70/unit

D) $280.00/unit

16) Water is falling on a surface, wetting a circular area that is expanding at a rate of fast is the radius of the wetted area expanding when the radius is answer to four decimal places.) A) 0.0043 mm/s B) 0.0268 mm/s C) 234.0485 mm/s

How

16) ______

(Round your D) 0.0085 mm/s

17) A wheel with radius 2 m rolls at 18 rad/s. How fast is a point on the rim of the wheel rising when the point is decimal place.) A) 72.0 m/s

15) ______

17) ______

radians above the horizontal (and rising)? (Round your answer to one B) 9.0 m/s

C) 18.0 m/s

18) Assume that the profit generated by a product is given by

D) 36.0 m/s where x is the number of

18) ______

units sold. If the profit keeps changing at a rate of per month, then how fast are the sales changing when the number of units sold is 1200? (Round your answer to the nearest dollar per month.) A) $207,846/month B) $4157/month C) $8314/month D) $7/month 19) 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

19) ______

then how fast are the people moving when they are from the right angle? (Round your answer to two decimal places.) A) 0.80 m/s B) 12.50 m/s C) 0.20 m/s D) 0.40 m/s Solve the problem. Round your answer, if appropriate. 20) Water is discharged from a pipeline at a velocity v (in ft/sec) given by the pressure (in psi). If the water pressure is changing at a rate of acceleration (dv/dt) of the water when A) 2030 ft/sec2 B) 33.9 ft/sec2

C) 81.6 ft/sec2

where p is

20) ______

find the D) 49.0 ft/sec2

21) As the zoom lens in a camera moves in and out, the size of the rectangular image changes. Assume that the current image is 6 cm × 4 cm. Find the rate at which the area of the image is changing (dA/df) if the length of the image is changing at 0.8 cm/s and the width of the image is changing at 0.4 cm/s. A) B) C) D) 11.2 /sec 5.6 /sec 12.8 /sec 6.4 /sec

21) ______

22) One airplane is approaching an airport from the north at 163 km/hr. A second airplane approaches from the east at 261 km/hr. Find the rate at which the distance between the planes

cha s when nge the

___ ___

southbou 22) nd plane is 31 km away from the airport and the westbou nd plane is 18 km from the airport. A) -544 km/hr

B) -136 km/hr

C) -408 km/hr

D) -272 km/hr

23) Water is being drained from a container which has the shape of an inverted right circular cone. The container has a radius of 4.00 inches at the top and a height of 7.00 inches. At the instant when the water in the container is 4.00 inches deep, the surface level is falling at a rate of 1 in./sec. Find the rate at which water is being drained from the container. A) 21.5 in.3s B) 15.7 in.3/s C) 16.4 in.3/s D) 24.6 in.3/s

23) ______

24) A man 6 ft tall walks at a rate of 7 ft/sec away from a lamppost that is 14 ft high. At what rate is the length of his shadow changing when he is 30 ft away from the lamppost? (Do not round your answer) A) B) 35 ft/sec C) D)

24) ______

ft/sec

25) Boyle's law states that if the temperature of a gas remains constant, then PV = c, where ,

25) ______

, and c is a constant. Given a quantity of gas at constant temperature, if

V is decreasing at a rate of 10 in. 3/sec, at what rate is P increasing when P = 90 lb/in.2 and V = 80 in.3? (Do not round your answer.) A)

720

per sec

per sec C)

D) per sec

per sec

26) 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 8 A/s while the voltage remains constant at 14 V, at what rate is the resistance increasing (in Ω/sec) when the current is 48 A? (Do not round your answer.) A) B) C) D) Ω/sec

Ω/sec

27) The volume of a sphere is increasing at a rate of 7 cm3/sec. Find the rate of change of its surface area when its volume is A) 14 cm2/sec

cm3. (Do not round your answer.) B) C) 14π cm2/sec cm2/sec

26) ______

27) ______

D) cm2/sec

28) The volume of a rectangular box with a square base remains constant at 600 cm 3 as the area of

the base

___ ___

increases 28) at a rate of 2 2/sec. Find the rate at which the height of the box is decreasin g when each side of the base is 19 cm long. (Do not round your answer.) A)

B) cm/sec

C) cm/sec

29) The radius of a right circular cylinder is increasing at the rate of

D) cm/sec

cm/sec , while the height is

decreasing at the rate of . At what rate is the volume of the cylinder changing when the radius is 11 in. and the height is 19 in.? A) 1309π in.3/sec B) 1309 in.3/sec C) 2981π in.3/sec D) -271 in.3/sec

29) ______

1) D 2) A 3) B 4) C 5) B 6) A 7) A 8) B 9) D 10) D 11) A 12) D 13) D 14) C 15) A 16) A 17) C 18) C 19) D 20) B 21) B 22) D 23) C 24) A 25) C 26) D 27) A 28) C 29) C

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) No absolute extrema. B) Absolute minimum and absolute maximum. C) Absolute maximum only. D) Absolute minimum only. 2) _______

A) No absolute extrema. B) Absolute minimum only. C) Absolute maximum only. D) Absolute minimum and absolute maximum. 3) _______

A) No absolute extrema. B) Absolute minimum only.

C) Absolute maximum only. D) Absolute minimum and absolute maximum. 4) _______

A) Absolute maximum only. B) Absolute minimum and absolute maximum. C) Absolute minimum only. D) No absolute extrema. 5) _______

A) Absolute maximum only. B) No absolute extrema. C) Absolute minimum only. D) Absolute minimum and absolute maximum. Find the location of the indicated absolute extremum for the function. 6) Minimum

__ __ __ _

A) x = 5

B) x = 3

C) x = -5

D) x = -3 7) _______

7) Maximum

A) No maximum

B) x = 1

C) x = 4

D) x = -4 8) _______

8) Maximum

A) x = 3 9) Minimum

B) x = 5

C) No maximum

D) x = 0

__ __ __ _

A) x = 2

B) x = -2

C) x = 1

D) x = -1 10) ______

10) Minimum

A) x = 2

B) x = -4

C) x = -3

D) x = 0 11) ______

11) Maximum

A) x = 4 12) Minimum

B) x = -1

C) x = 1

D) No maximum

___ ___

12)

A) No minimum

B) x = -1

C) x = 2

D) x = 1 13) ______

13) Maximum

A) x = -4

C) x = 0

D) No maximum

x= 14) ______

14) Maximum

A) x = 0

B) No maximum

Solve the problem. 15) Find the graph that matches the given table.

C) x = 2

D) x = -1

___ ___

15)

16) ______

16) Find the graph that matches the given table.

17) ______

17) Find the table that matches the given graph.

18) Find the table that matches the graph below.

18)

______ A)

Sketch the graph of the function and determine whether it has any absolute extreme values on its domain. 19) y = ∣x - 4∣, 0 < x ≤ 10

A) absolute maximum at x = 10; absolute minimum at x = 4

C) absolute maximum at x = 10; no absolute minimum

19) ______

B) absolute maximum at x = 10; absolute minimum at x = 4

absolute maximum at x = 10; no absolute minimum

20) ______

20) y = 4 cos x, 0 < x < 2π

A) absolute maximum at x = 0 and x = 2π; absolute minimum at x = π

B) no absolute maximum; no absolute minimum

C) absolute maximum at x = 0; absolute minimum at x = π

D) no absolute maximum; absolute minimum at x = π

Find the absolute extreme values of the function on the interval.

21) f(x) = 3x - 5, -3 ≤ x ≤ 4 A) absolute maximum is 7 at x = -3; absolute minimum is - 4 at x = 4 B) absolute maximum is 17 at x = -4; absolute minimum is - 14 at x = 3 C) absolute maximum is 7 at x = 4; absolute minimum is - 14 at x = -3 D) absolute maximum is 17 at x = 4; absolute minimum is - 4 at x = -3

21) ______

22)

22) ______

g(x) = A)

+ 11x - 28, 4 ≤ x ≤ 7

absolute maximum is

at x =

; absolute minimum is 0 at 7 and 0 at x = 4

B) absolute maximum is

at x =

; absolute minimum is 0 at 7 and 0 at x = 4

C) absolute maximum is

at x =

; absolute minimum is 0 at 7 and 0 at x = 4

absolute maximum is

at x =

; absolute minimum is 0 at 7 and 0 at x = 4

23) ______

23) f(θ) = sin A)

,0≤θ≤

absolute maximum is 1 at θ =

π; absolute minimum is -1 at θ =

absolute maximum is 1 at θ = π; absolute minimum is -1 at θ = C) absolute maximum is 1 at θ = 0; absolute minimum is -1 at θ = π D)

π,

absolute maximum is 1 at θ =

π; absolute minimum is -1 at θ =

π, 24) ______

24) f(x) = csc x, ≤x≤ A) absolute maximum is 1 at x = π; absolute minimum is -1 at x = π B) absolute maximum is 0 at x = -π; absolute minimum is -1 at x = π C) absolute maximum is -1 at x = π; absolute minimum is 1 at x = 0 D) absolute maximum does not exist; absolute minimum does not exist

25) ______

25) F(x) = A)

, 0.5 ≤ x ≤ 3

absolute maximum is -

at x = 3; absolute minimum is -4 at x =

absolute maximum is -

at x =

; absolute minimum is -4 at x = -3

absolute maximum is

at x =

; absolute minimum is -4 at x =3

absolute maximum is -

at x = 3; absolute minimum is -4 at x = -

B) C) D)

26) ______

26) F(x) = , -2 ≤ x ≤ 27 A) absolute maximum is 3 at x = -27; absolute minimum is 0 at x =0

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 27) ______

27) h(x) = A)

x + 5, -2 ≤ x ≤ 3

absolute maximum is 7 at x = 3; absolute minimum is at x = -2 B) absolute maximum is - 3 at x = -3; absolute minimum is -3 at x = 2 C) absolute maximum is - 3 at x = -2; absolute minimum is

at x = 3

absolute maximum is - 3 at x = 3; absolute minimum is

at x = -2

28)

28) ______

g(x) = 9 - 7 , -3 ≤ x ≤ 4 A) absolute maximum is 7 at x = 0; absolute minimum is -121 at x = 4 B) absolute maximum is 63 at x = 0; absolute minimum is -54 at x = -3 C) absolute maximum is 18 at x = 0; absolute minimum is -54 at x = 4 D) absolute maximum is 9 at x = 0; absolute minimum is -103 at x = 4

Find the absolute extreme values of the function on the interval. 29) f(x) = tan x, A)

29) ______

≤x≤

absolute maximum is

at x =

; absolute minimum is -

at x = -

absolute maximum is

at x =

; absolute minimum is -

at x = -

absolute maximum is

at x =

and -

absolute maximum is -

at x =

; absolute minimum is

B) C) ; absolute minimum does not exist

D) at x = -

30) f(x) = |x - 7|, 5 ≤ x ≤ 12 A) absolute maximum is -2 at x = 5; absolute minimum is 5 at x = 12 B) absolute maximum is 2 at x = 5; absolute minimum is 0 at x = 7 C) absolute maximum is 5 at x = 12; absolute minimum is 2 at x = 5 D) absolute maximum is 5 at x = 12; absolute minimum is 0 at x = 7

30) ______

31)

31) ______

32)

f(x) = , -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 does not exist C) absolute maximum is 256 at x = 8; absolute minimum is 0 at x = 01 D) absolute maximum is 256 at x = 8; absolute minimum is 1 at x = -1 f(x) = 4 , -27 ≤ x ≤ 1 A) absolute maximum is 324 at x = -27 ; absolute minimum is 4 at x = 1 B) absolute maximum is 81 at x = -27 ; absolute minimum is 0 at x = 0

32) ______

C) absolute maximum is 4 at x = 1 ; absolute minimum is 0 at x = 0 D) absolute maximum is 324 at x = -27 ; absolute minimum is 0 at x = 0 33) ______

33) f(x) = ln(x + 2) + A)

, 1 ≤ x ≤ 10

Minimum value is -1 at x = -1; maximum value is ln 12 +

at x = 10

B) Minimum value is ln 4 +

at x = 2; maximum value is ln 3 + 1 at x = 1

Minimum value is ln 4 +

at x = 2; maximum value is ln 12 +

C) at x = 10

D) Minimum value is ln 3 + 1 at x = 1; maximum value is ln 12 + 34)

at x = 10 34) ______

f(x) = - 6 , -∞ < x < ∞ A) No minimum value and no maximum value B) Minimum value is - 6 at x = 0; no maximum value C) Minimum value is - 6 at x = 0; maximum value is D) Maximum value is - 6 at x = 0; minimum value

at x = 1

35) f(x) = ln (-x), -7 ≤ x ≤ -1 A) Minimum value is 0 at x = -1; no maximum value B) Maximum value is 0 at x = -1; minimum value is -ln 7 at x = -7 C) No minimum value; no maximum value D) Minimum value is 0 at x = -1; maximum value is ln 7 at x = -7

35) ______

36)

36) ______

f(x) = A)

- x, -4 ≤ x ≤ 2

Minimum value is + 4 at x = -4; maximum value is - 2 at x = 2 B) Minimum value is 1 at x = 0; no maximum value C) Minimum value is 1 at x = 0; maximum value is - 2 at x = 2 D) Minimum value is 1 at x = 0; maximum value is + 4 at x = -4

Determine all critical points for the function. 37) f(x) = + 4x + 4 A) x = 2 B) x = 0 38)

39)

40)

f(x) = - 12x + 4 A) x = -2 and x = 2 C) x = -2, x = 0, and x = 2 f(x) = -3 +3 A) x = 0 and x = 1 C) x = 0 and x = 2 f(x) = 80 -3 A) x = -4

37) ______ C) x = -2

D) x = -4 38) ______

B) x = 2 D) x = -2 39) ______ B) x = -1 and x = 1 D) x = 0 40) ______ B) x = -4 and x = 4

C) x = 0, x = -4, and x =

D) x = 4 41) ______

41) f(x) = A) x = 0 and x = 6 C) x = -6 42)

43)

f(x) = A) x = 0 and x = 1 C) x = 0, x = 1, and x = 5 y=4 - 128 A) x = 4 C) x = 0

Find the extreme values of the function and where they occur. 44) y = x2 + 2x - 3 A) The minimum is 1 at x = 4. C) The minimum is -1 at x = 4.

B) x = -24 and x = 0 D) x = 6 42) ______ B) x = 1 and x = 5 D) x = 1 43) ______ B) x = 0 and x = 4 D) x = 0, x = 4, and x = -4

44) ______ B) The minimum is -4 at x = -1. D) The minimum is 1 at x = -4.

45) y = x3 - 3x2 + 1

45) ______

A) Local maximum at (0, 1), local minimum at (2, -3). B) Local maximum at (0, 1). C) None D) Local minimum at (2, -3). 46) y = x3 - 12x + 2

46) ______

A) Local maximum at (2, -14), local minimum at (-2, 18). B) Local maximum at (-2, 18), local minimum at (2, -14). C) Local maximum at (0, 0). D) None 47) ______

47) y= A) Local maximum at (1, 0), local minimum at (-1, 0). B) None C) Local maximum at (0, -1). D) Local maximum at (-1, 0), local minimum at (1,0).

48) ______

48) y= 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. C) The minimum value is -1 at x = 0.5. D) The maximum value is 1 at x = 0.5.

49) ______

49) y= A) The minimum value is 0 at x = 0.

B) The minimum value is 0 at x = 1. The maximum value is 0 at x = -1. C) The maximum value is 0 at x = 0. D) The minimum value is - 1 at x = -1. The maximum value is 1at x = 1. 50)

51)

50) ______

y= A) The minimum value is 0 at x = 2. C) The minimum value is 0 at x = -2.

B) There are no definable extrema. D) The maximum value is 0 at x = -2. 51) ______

y= - 3 + 6x - 8 A) The maximum is 4 at x = 1. C) None

B) The minimum is 4 at x = -1. D) The maximum is 4 at x = 2. 52) ______

52) y= A) The maximum is 4 at x = 2. C) The minimum is 4 at x = 0.

B) The minimum is 0 at x = 1. D) The maximum is 4 at x = -2. 53) ______

53) y= A) None B) The maximum is 2 at x = 0; the minimum is

at x = -2.

C) The maximum is -

at x = 0; the minimum is

at x = -2.

The maximum is

at x = 0; the minimum is -

at x = -2.

54)

54) ______

y= A)

Minimum value is 4 at x = -2; no maximum value. B) Minimum value is 0 at x = 0; no maximum value. C) Minimum value is 0 at x = 0, maximum value is 4 at x = -2. D) None 55) ______

55) y= A) Minimum value is

at x =

; no maximum value.

Maximum value is

at x =

; minimum value is 0 at x = 1.

Maximum value is D) None

at x =

; no minimum value.

B) C)

56)

y= A) B)

56) ______

+ 2x Maximum value is 2

(1 +

) at x = ; no minimum value.

Minimum value is

2 C)

(1 -

) at x = -

Minimum value is 2

; no maximum value. (1 -

) at x = -

; maximum value is 2

(1 +

) at x =

. D) None Find the derivative at each critical point and determine the local extreme values. 57) y= ( - 1); x ≥ 0 A) B)

58)

y = x(16 A)

57) ______

)

58) ______

59)

y= A)

59) ______

60) ______

60) y= A)

61) ______

61) y= A)

62) ______

62) y= A)

63) ______

63) y= A)

Graph the function, then find the extreme values of the function on the interval and indicate where they occur. 64) y = + on the interval -5 < x < 5

A) Absolute minimum is 4 on the interval [-3, 1]

64) ______

C) Absolute minimum is 5 on the interval [-1, 3]

65) y =

on the interval -2 < x < 7

A bs ol ut e mi ni m u m is 5 on th e int er va l [-3 , 1]

D) Absolute minimum is 4 on the interval [-1, 3]

___ ___

65)

A) Absolute maximum is 4 on the interval [6, 7); Absolute minimum is -4 on the interval (-2, 2]

B) Absolute maximum is 5 on the interval [6, 7); Absolute minimum is -3 on the interval (-2, 2]

C) Absolute maximum is -4 on the interval [6, 7); Absolute minimum is 4 on the interval (-2, 2]

D) Absolute maximum is 4 on the interval [6, 7); Absolute minimum is -3 on the interval (-2, 3]

66) y =

on the interval -∞ < x < ∞

___ ___

66)

A) Absolute maximum is 7 on the interval [5, ∞); Absolute minimum is -7 on the interval (-∞,3]

B) Absolute maximum is 9 on the interval [5, ∞); Absolute minimum is -9 on the interval (-∞,3]

C) Absolute maximum is 8 on the interval [5, ∞); Absolute minimum is -8 on the interval (-∞, 3]

D) Absolute maximum is 8 on the interval [6, ∞); Absolute minimum is -8 on the interval (-∞,2]

67) y =

on the interval -∞ < x < ∞

___ ___

67)

A) Absolute minimum is 5 on the interval (-4, 2]

B) Absolute minimum is 6 on the interval (-3, 3]

C) Absolute minimum is 6 on the interval (-4, 4]

D) Absolute minimum is 7 on the interval (-3, 3]

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Answer the problem. 68) Use the following function and a graphing calculator to answer the questions. f(x) =

-5

+ 4x + 2, [-0.5, 1.8]

a). Plot the function over the interval to see its general behavior there. Sketch the graph below.

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.

b). Find the interior points where = 0 (you may need to use the numerica l equation solver to approxi mate a solution). You may wish to plot as well. List the points as ordered pairs (x, y).

c). Find the interior points where does not exist. List the

68) _____________

69) Use the following function and a graphing calculator to answer the questions. f(x) =

69) _____________

+ 0.9 sin x, [0, 2π]

a). Plot the function over the interval to see its general behavior there. Sketch the graph below.

b). Find the interior points where

= 0 (you may need to use the numerical equation

solver to approximate a solution). You may wish to plot ordered pairs (x, y).

c). Find the interior points where

as well. List the points as

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.

Provide an appropriate response. 70) Let f(x) =

70) _____________

(a) Does (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) = Give reasons for your answers.

71)

71) _____________

Let f(x) = (a) Does

(0) exist?

(b) Does

(4) exist?

72) The function P(x) = 2x +

, 0 < x < ∞ models the perimeter of a rectangle of

dimensions x by . (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). 73) If the derivative of an even function f(x) is zero at x = c, can anything be said about the value of

Give reasons for your answer.

74) If the derivative of an odd function g(x) is zero at x = c, can anything be said about the value of

74) _____________

Give reasons for you answer.

75) Imagine there is a function for which

(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.

76)

73) _____________

Consider the quartic function f(x) = a

+ dx + e, a ≠ 0. Must this function

have at least one critical point? Give reasons for your answer. (Hint: Must some x?) How many local extreme values can f have?

75) _____________

76) _____________

for

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 77) 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) 8 hundred thousand B) 3 hundred thousand C) 10 hundred thousand D) 5 hundred thousand

77) ______

1) C 2) A 3) C 4) B 5) B 6) B 7) B 8) D 9) B 10) C 11) B 12) A 13) D 14) C 15) B 16) A 17) A 18) B 19) A 20) D 21) C 22) A 23) C 24) D 25) A 26) D 27) A 28) D 29) B 30) D 31) C 32) D 33) C 34) B 35) D 36) C 37) C 38) A 39) C 40) C 41) D 42) D 43) B 44) B 45) A 46) B 47) C 48) B 49) D 50) A 51) C

52) C 53) D 54) B 55) C 56) B 57) C 58) D 59) C 60) D 61) D 62) D 63) D 64) A 65) A 66) C 67) B 68) a).

Solid line: f(x); dashed line: b). See figure above.

(x)

(x) = 0 at x = 0.4323 and x = 1.3200.

Critical points of f(x) are

and

c). (x) is defined on the entire interval. d). Endpoints are (-0.5, -1.1875) and (1.8, 3.4976). e). Absolute minimum: (-0.5, -1.1875); absolute maximum (1.8, 3.4976). 69) a).

Solid line: f(x); dashed line:

(x)

b). (x) = 0 at x = 0.45 and x = 2.7650. See Critical points of f(x) are and figur c). (x) is undefined at the endpoint x = 0. e d). Endpoints are (0, 0) and (2π, 3.2984). abov e). Absolute minimum: (0, 0); absolute maximum (2π, 3.2984). e. 70)

(a) No, since (x) = , which is undefined at x = 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. 71) (a) No (b) No (c) No (d) Minimum: 0 at x = ± 4 and x = 0; local maximum: 24.6336115 at x = ± 2.30940108 72) (a) The only critical point in the interval (0, ∞) at x = 10.0. The minimum value of P(x) is at (b) The smallest possible perimeter of the rectangle is rectangle a 10.0 by 10.0 square. 73) Yes. The value of

units and it occurs at

which makes the

at x = -c must also be zero since even functions are symmetrical with respect to the y-axis.

74) Yes. The value of

at x = -c is also zero since odd functions are symmetric with respect to the origin. 75) 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 76)

definition points where

But none of these points correspond to extreme values.

Yes. The derivative is cubic: 4a

+ 3b

+ 2cx + d. The derivative approaches -∞ as x approaches

approaches ∞ as x approaches ∞. By the Intermediate Value Theorem, the interval -∞ ≤ x ≤ ∞. 77) A

and it

(x) must equal zero at at least one point on

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the value or values of c that satisfy the equation function and interval. 1) f(x) = + 3x + 3, [-1, 2] A) -1, 2 B)

D) -

, 2) _______

2) f(x) = x + A) 0, 5

, B) 5

C) -5

D) 3, 25 3) _______

3) f(x) = x, [A) ±0.320 4) f(x) = ln (x - 2), [3, 6] A) 5.164

]

Round to the nearest thousandth. B) 0.320 C) -0.320, 0, 0.320 Round to the nearest thousandth. B) 4.164 C) ±4.164

D) 0, 0.320 4) _______ D) 4.731

Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval. 5) f(x) = , A) Yes B) No 6)

6) _______

g(x) = , A) No

7) s(t) = A) No

5) _______

B) Yes 7) _______

, B) Yes

8) _______

8) f(θ) = A) No

B) Yes

Plot the zeros of the given polynomial on the number line together with the zeros of the first derivative. 9) y= + 4x + 3

9) _______

10)

10) ______

y = (x - 5)

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Show that the function has exactly one zero in the given interval. 11) 11) _____________ f(x) =

+ 7, (-∞, 0). 12) _____________

12) r(θ) = 2 cot θ +

+ 8, (0, π).

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find all possible functions with the given derivative. 13) = 13 15 A) B) C) D) 15 13 14x + C 15x + C x +C x +C 14)

= A)

15)

14) ______

- 5x B) -

C) -

15) ______

=4 +1 A) 8x + C

=7 A)

- 3x

x3 -

D) x + C x3 + C 16) ______

B) x2 + C

x2 +

x+C

x3 -

x2 + C

D) x3 + C

17)

D) 3x2 - 5 + C + 5x2 + C

x3 + x + C 16)

13) ______

17) ______

=6 A)

B) +C

C) +C

D) +C

+C 18) ______

18) =2A)

2x -

+ C

19) ______

19) = A)

B) 5

20) ______

20) = 5t A) 5 -4

-2

D) 5

-4

+C 21) ______

21) =6+ A)

r = 6θ + 22)

= A)

r = 6θ -

r = 6θ 22) ______

3θ B) -3 cot 3θ + C C

3θ +

D) csc 3θ + C

Find the function with the given derivative whose graph passes through the point P. 23) (x) = x - 9, P(3, -15) A) B) f(x) = - 9x C)

24)

r = 6θ +

f(x) =

- 9x + 3

f(x) =

- 9x + 24) ______

+ 9x + 29

- 9x +

23) ______

(x) = + 9, P(0, 29) A) f(x) = +9 + 29 f(x) =

f(x) =

cot 3θ + C

f(x) =

+ 9x

f(x) =

+ 9x + 29 25) ______

25) (x) = A)

+ 2x , P(-4, 4)

g(x) = C)

B) -

g(x) = g(

D) x) = -

g( x) =

+ -

26) ______

26) (θ) = 7 A)

θ, P B)

r(θ) = 7θ - cotθ -

π-2

C) r(θ) = 7θ - cotθ 27)

r(θ) = 7θ + cotθ -

π+1

r(θ) = 7θ + cotθ -

π-1

D) π+2

27) ______

(t) = t - 4, P(0, 0) A) r(t) = sec t tan t - 4t -1 C) r(t) = sec t - 4t - 4

B) r(t) = sec t - t - 6 D) r(t) = tan t - 4t 28) ______

28) (x) = + 6x, P(-1, 1) A) g(x) = -4 +3 C) g(x) = -4 +3 -6 29)

(θ) = 4 + A) (θ) = C)

r(θ) =

B) D)

g(x) = 4

+ 6x - 6

-6

29) ______

θ, P(0, 0) B)

+ tan θ

r(θ) = 4θ + D) r(θ) = 4θ + tan θ

+ tan θ

30) ______

30) (θ) = csc θ cot θ - 3, P A) r(θ) = csc θ - 3θ + C)

B) r(θ) = -csc θ - 3θ + D) r(θ) = -csc θ - 3θ +

r(θ) = -csc θ -

+ 31) ______

31) (x) = A)

f(x) =

f(x) = 2

f(x) =

D) f(x) =

Solve the problem. 32) 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 = -19t + 7, s(0) = 8 A) B) s=-

+ 7t + 8

D) s=-

+ 7t - 8

s = -19

+ 7t + 8

+ 7t - 8

33) 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 A)

33) ______

t, s(0) = 1 B)

s= sin C) s = sin t +1

t +1

s = 2π sin

t+π

D) sin

34) 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= A)

32) ______

sin

, s(

34) ______

)=2 B)

s = 2 cos

s = -2 cos

+ 8.2134

D) s = -2 cos

+ 3.3073

35) 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) = 7, s(0) = 7 A) B) C) D) s = 8 + 7t s = -8 - 7t + 7 s = 16 + 7t + s = 8 + 7t + 7 7

35) ______

36)

36) ______

37)

On our moon, the acceleration of gravity is 1.6 m/ . 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 A rocket lifts off the surface of Earth with a constant acceleration of 30 m/ the rocket be going 2.5 minutes later? A) 3375 m/sec B) 4500 m/sec C) 2250 m/sec

. How fast will D) -4500 m/sec

38) At about what velocity do you enter the water if you jump from a 15 meter cliff? (Use A) 2 m/sec 39)

B) -17 m/sec

C) 17 m/sec

The acceleration of gravity near the surface of Mars is 3.72 m/ from the surface with an initial velocity of (Hint: When is velocity zero?)

(about

37) ______

38) ______

D) -8.5 m/sec . If a rock is blasted straight up how high does it go?

39) ______

A) Approximately 1942.25 meters C) Approximately 971.1 meters 40)

B) Approximately 170 meters D) Approximately 22.85 meters

A particle moves on a coordinate line with acceleration conditions that position s in terms of t. A)

and

when

subject to the

Find the velocity

v = 10

-6

+ 13; s =

+ 13t +

v = 10

- 13; s =

- 13t +

v = 10

- 13; s =

- 13t +

40) ______

in terms of t and the

B) C) D) - 13t +

; s = 10

- 13

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Answer the question. 41) It took 26 seconds for the temperature to rise from 10° F to 193° F when a thermometer 41) _____________ 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 time, the temperature was increasing at a rate of

° F/sec. Explain.

42) 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?

42) _____________

43) A marathoner ran the 26.2 mile New York City Marathon in 2.6 hrs. Did the runner ever exceed a speed of 9 miles per hour?

43) _____________

Provide an appropriate response. 44)

44) _____________

The function f(x) = is zero at x = 0 and x = 1 and differentiable on (0, 1), but its derivative on (0,1) is never zero. Does this example contradict Rolle's Theorem? 45) Suppose that g(0) = -3 and that

45) _____________

(t) = 2 for all t. Must g(t) = 2t - 3 for all t?

46) _____________

46) Decide if the statement is true or false. If false, explain. The points (-1, -1) and (1, 1) lie on the graph of f(x) =

. Therefore, the Mean Value

Theorem says that there exists some value x = c on (-1, 1) for which

(x) =

47) Let f have a derivative on an interval I. f' has successive distinct zeros at

= 1. and

47) _____________

Prove that there can be at most one zero of f on the interval MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 48) Suppose that f'(x) = 2x for all x. Find f(2) if f(-2) = 2. A) 0 B) 6 C) 2 D) 4

48) ______

1) D 2) B 3) A 4) B 5) B 6) B 7) A 8) A 9) B 10) C 11) 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 (x) = 3 , which is everywhere positive on (-∞, 0). Thus, f(x) has a single root on (-∞, 0). 12) The function r(θ) is continuous on the open interval (0, π). Also, r(θ) approaches ∞ as θ approaches 0 from the right, and r(θ) approaches -∞ as θ approaches π from the left. Since r(θ) is continuous and changes sign along the interval, it must have at least one root on the interval.

The first derivative of r(θ) is root on (0, π). 13) A 14) A 15) B 16) D 17) D 18) A 19) B 20) D 21) B 22) D 23) D 24) D 25) C 26) D 27) D 28) C 29) D 30) D 31) C 32) A 33) A 34) B 35) D 36) B 37) B 38) B

(θ) = -2

θ-

, which is everywhere negative on (0, π). Thus,

has a single

39) C 40) B 41) The average rate of temperature change is

° F/sec during the 26 seconds. Therefore, the Mean Value Theorem

implies that at sometime during this time period, the temperature was changing at a rate of ° F/sec. 42) 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. 43) Yes, the Mean Value Theorem implies that the runner attained a speed of 10.1 mph, which was her average speed throughout the marathon. 44) 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. 45) Yes, all antiderivatives of are of the form G(t) = 2t + C, where C is a constant. The only such function to satisfy the initial condition g(0) = -3 is g(t) = 2t - 3. 46) False. The function has a non-removable discontinuity at x = 0. The mean value theorem does not apply. 47) Assume there are 2 (or more) zeros a and b on the interval where By Rolle's theorem there is at least one number c in (a, b) such that

and

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 48) C

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Using the derivative of f(x) given below, determine the critical points of f(x). 1) (x) = (x + 6)(x + 10) A) -10, -6 B) 0, -10, -6 C) 6, 10 D) -16 2)

(x) = A) -5, 2

2) _______

(x + 2) B) -2, 5

C) -5, -2, 5

D) -5, 0, 2 3) _______

(x) = (x - 4) A) 5

B) 4

C) -5

D) -4 4) _______

(x) = A) -2, 0

B) -2

C) 2

D) 0

Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing. 5) (x) = (5 - x)(6 - x) A) Decreasing on (-∞, 5); increasing on (6, ∞) B) Decreasing on (-∞, 5) ∪ (6, ∞); increasing on (5, 6) C) Decreasing on (-∞, -5) ∪ (-6, ∞); increasing on (-5, -6) D) Decreasing on (5, 6); increasing on (-∞, 5) ∪ (6, ∞) 6)

1) _______

(x) = (x - 5) A) Decreasing on (0, 5); increasing on (5, ∞) B) Increasing on (0, ∞) C) Decreasing on (0, 5); increasing on (-∞, 0) ∪ (5, ∞) D) Decreasing on (-∞, 0) ∪ (5, ∞); increasing on (0, 5) (x) = (x - 1) A) Decreasing on (-∞, -1); increasing on (-1, ∞) B) Increasing on (-∞, -1); decreasing on (-1, ∞) C) Decreasing on (-∞, 1); increasing on (1, ∞) D) Increasing on (-∞, 1); decreasing on (1, ∞) (x) = A) Never increasing; decreasing on (-∞, -2) ∪ (-2, ∞) B) Decreasing on (-∞, -2); increasing on (-2, ∞) C) Never decreasing; increasing on (-∞, 2) ∪ (2, ∞) D) Never decreasing; increasing on (-∞, -2) ∪ (-2, ∞)

5) _______

6) _______

7) _______

8) _______

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. 9) 9) _______ f(x) = 0.1 -15 - 43x + 55 A) Approximate local maximum at -101.413; approximate local minimum at 1.413 B) Approximate local maximum at -1.413; approximate local minimum at 101.413 C) Approximate local minimum at -101.413; approximate local maximum at 1.413 D) Approximate local minimum at -1.413; approximate local maximum at 101.413 10)

f(x) =

- 15

+ 59x + 7

10) ______

A) Approximate local maximum at 1.735; approximate local minima at -6.777 and 12.542 B) Approximate local maximum at 1.677; approximate local minima at -6.746 and 12.623 C) Approximate local maximum at 1.786; approximate local minima at -6.776 and 12.46 D) Approximate local maximum at 1.736; approximate local minima at -6.864 and 12.491 11)

12)

13)

14)

15)

f(x) = - 21 + 74x - 60 A) Approximate local maximum at 1.668; approximate local minima at -3.122 and 3.83 B) Approximate local maximum at 1.534; approximate local minima at -3.145 and 3.717 C) Approximate local maximum at 1.604; approximate local minima at -3.089 and 3.735 D) Approximate local maximum at 1.628; approximate local minima at -3.136 and 3.714 f(x) = - 4 - 53 - 86x + 87 A) Approximate local maximum at 1.039; approximate local minima at -3.183 and 7.082 B) Approximate local maximum at 0.864; approximate local minima at -3.097 and 7.206 C) Approximate local maximum at 0.921; approximate local minima at -3.218 and 7.121 D) Approximate local maximum at -0.944; approximate local minima at -3.192 and 7.136 f(x) = - 15 - 172 + 135x - 0.094 A) Approximate local maximum at 0.325; approximate local minima at -0.558 and -12.612 B) Approximate local maximum at 0.379; approximate local minima at -0.472 and 12.565 C) Approximate local maximum at 0.379; approximate local minimum at 12.565 D) Approximate local maximum at 0.377; approximate local minimum at -12.639 f(x) = 0.1 +5 - 15 - 6x - 82 A) Approximate local maxima at -41.062 and -0.18; approximate local minima at -0.592 and 2.045 B) Approximate local maxima at -41.097 and -0.328; approximate local minima at -0.604 and 2.009 C) Approximate local maxima at -41.222 and -0.234; approximate local minima at -0.558 and 2.042 D) Approximate local maxima at -41.132 and -0.273; approximate local minima at -0.547 and 1.952 f(x) = 0.01 + +8 - 7x - 51 A) Approximate local maxima at -1.829 and 2.179; approximate local minima at 0.327 and 79.16 B) Approximate local maxima at -1.861 and 2.247; approximate local minimum at 0.423 C) Approximate local maxima at -1.845 and 2.22; approximate local minima at 0.522 and 79.165 D) Approximate local maxima at -1.861 and 2.247; approximate local minima at 0.423 and 79.192

11) ______

12) ______

13) ______

14) ______

15) ______

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. 16)

16)

___ ___

A) increasing on (-3, 3); decreasing on (-6, -3) and (3, 6); absolute maximum at (3, 3); absolute minimum at (-3, -3) B) increasing on (-3, 3); decreasing on (-6, 0); absolute maximum at (3, 3); absolute minimum at (-3, -3) 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 (0, 6); absolute maximum at (3, 3); absolute minimum at (-3, -3) 17) ______

17)

A) increasing on (-3, 0); decreasing on (-5, -3) and (2, 5) absolute maximum at (-5, 2); local minimum at (-3, -5) and (5, -5) B) increasing on (-3, 0); decreasing on [-5, -3) and (2, 5] absolute maximum at (-5, 2); absolute minimum at (5, -5) C) increasing on (-3, 1); decreasing on (-5, -3) and (0, 5) absolute maximum at (-5, 2); no absolute minimum D) increasing on (-3, 0); decreasing on (-5, -3) and (2, 5) absolute maximum at (-5, 2); local maximum at (0, -2) and (2, -2); absolute minimum at (5, -5) 18)

___ ___

18)

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); 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) and(0,2); absolute minimum at (-2, 0) and (2, 0) Find the largest open interval where the function is changing as requested. 19) Increasing y = 7x - 5 A) (-5, ∞) B) (-∞, 7) C) (-∞, ∞)

19) ______ D) (-5, 7) 20) ______

20) Increasing A) (1, ∞) 21) Increasing A) (-∞, 1) 22) Increasing

f(x) =

x2 -

x B) (-1, 1)

f(x) = x2 - 2x + 1 B) (1, ∞)

C) (-∞, -1)

D) (-∞, ∞) 21) ______

C) (0, ∞)

D) (-∞, 0) 22) ______

y = (x2 - 9)2

A) (3, ∞)

B) (-∞, 0)

C) (-3, 3)

D) (-3, 0) 23) ______

23) Increasing A) (0, ∞)

f(x) =

24) Decreasing

f(x) =

B) (-∞, 1)

D) (1, ∞) 24) ______

A) (-4, ∞) 25) Decreasing A) (8, ∞)

C) (-∞, 0)

B) (-∞, 4)

C) (-∞, -4)

D) (4, ∞)

B) (-∞, -8)

C) (-8, ∞)

D) (-∞, 8)

25) ______

f(x) = ∣x - 8∣

26) ______

26) Decreasing A) (-7, 0)

+7 B) (0, ∞)

C) (7, ∞)

D) (-7, 7)

27) Decreasing A) (-∞, 3) 28) Decreasing

27) ______

f(x) = B) (3, ∞)

C) (-3, ∞)

28) ______

f(x) = x3 - 4x

Identify the function's local and absolute extreme values, if any, saying where they occur. 29) f(x) = - - 9 - 24x + 3 A) local maximum at x = 4; local minimum at x = 2 B) local maximum at x = -2; local minimum at x = -4 C) local maximum at x = -4; local minimum at x = -2 D) local maximum at x = 2; local minimum at x = 4 30)

f(x) = A)

+ 3.5

D) (-∞, -3)

29) ______

30) ______

+ 2x - 1

local maximum at x = -

; local minimum at x = -1

local maximum at x =

; local minimum at x = 2

B) C) local maximum at x = 1; local minimum at x = D) local maximum at x = -2; local minimum at x = 31)

32)

f(x) = + 12 + 48x + 3 A) no local extrema B) local maximum at x = -4; local minimum at x = 4 C) local maximum at x = -4 D) local minimum at x = -4 f(r) = A) local minimum: x = 0 B) local minimum: x = 5 C) local minimum: x = 0; local maximum: x = 5 D) no local extrema

31) ______

32) ______

33) ______

33) g(x) = + + - 6x - 2 A) local maxima at x = 3 and x = -1; local minimum at x = -2 B) local maximum at x = -3; local minimum at x = 1 C) local maximum at x = -2; local minima at x = -3 and x = 1 D) local maxima at x = -3 and x = 1; local minimum at x = -2

34) ______

34) h(x) = A) local minimum at x = -2; local maximum at x = 6 B) local minimum at x = -5; local maximum at x = 6 C) no local extrema

D) local minimum at x = -2; no local maxima 35)

35) ______

f(x) = A) no local extrema B) relative minimum: 3 at x = -3; relative maximum: -3 at x = 3 C) absolute maximum: 3 at x = -3 D) absolute minimum: 3 at x = -3

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. 36) 36) ______ f(x) = , -∞ < x ≤ 0 A) local and absolute maximum: 64 at x = 0; local and absolute minimum: 0 at x = -8 B) local and absolute minimum: 0 at x = -8 C) no local extrema; no absolute extrema D) local maximum: 64 at x = 0; local and absolute absolute minimum: 0 at x = -8 37)

37) ______

f(x) = - 8x, -∞ < x ≤ 8 A) local minimum: -16 at x = 4; local and absolute maximum: 0 at x = 8 B) local minimum: 0 at x = 8; local and absolute maximum:-16 at x = 4 C) local and absolute minimum: -16 at x = 4; local and absolute maximum: 0 at x = 8 D) local and absolute minimum: -16 at x = 4; local maximum: 0 at x = 8

38) ______

38) g(t) = A)

+ 18t, 0 ≤ t < ∞

local minimum:

at x =2; local maximum: 0 at x = 0; absolute maximum: -

at x = 9

B) local and absolute minimum: -

at x = 9; local maximum:

at x =2

at x =2; local and absolute maximum: -

at x = 9

C) local minimum: D) local minimum: 0 at x = 0; local and absolute minimum: at x =2 39)

40)

h(x) = +3 + 5x + 7, -∞ < x ≤ 0 A) local and absolute maximum: 7 at x = 0; B) local and absolute maximum: 7 at x = 0; local and absolute minimum: 6 at x = -1 C) no local extrema; no absolute extrema D) local and absolute minimum: 6 at x = -1; f(x) = , -5 ≤ x < 5 A) no local extrema; no absolute extrema B) local and absolute minimum: 0 at x = -5 and x = 5; local and absolute maximum: 5 at x = 0 C) local and absolute maximum: 0 at x = -5;

at x = 9; local maximum:

39) ______

40) ______

local and

absolute minimum: 5 at x = 0 D) local and absolute minimum: 0 at x = -5; local and absolute maximum: 5 at x = 0

Find the extrema of the function on the given interval, and say where they occur. 41) sin 4x, 0 ≤ x ≤ A) local maxima: 1 at x =

41) ______

B) and 0 at x =

local maxima: 1 at x =

; local minimum: 0 at x = 0

and 0 at x =

; local minimum: -1 at x =

D) local maxima: 1 at x =

and 0 at x =

local maxima: 1 at x =

;

and 0 at x =

;

local minima: 0 at x = 0 and -1 at x =

local minima: 0 at x = 0 and -1 at x = 42) ______

42) sin x + cos x, 0 ≤ x ≤ 2π A) local maxima: 1 at x = 2π and

at x =

local minima: 1 at x = 0 and -

at x =

local maxima: 1 at x = 0 and -

at x =

local minima: 1 at x = 2π and

at x =

local maxima: 1 at x = 2π and

at x =

local minima: 1 at x = 0 and -

at x =

local maxima: 1 at x = 0 and -

at x =

local minima: 1 at x = 2π and

at x =

;

B) ;

C) ;

43)

x + 2 cot x, 0 < x < π A)

;

43) ______ B)

local maximum: 0 at x = C)

local minimum: 0 at x = D)

local minimum: 0 at x =

local maximum: 0 at x =

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 44) As x moves from

___ ___ ___ ___ _

left to 44) right though the point c = 5, is the graph of f(x) = x + rising, or is it falling? Give reasons for your answer. 45) 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)

45) _____________

iii). k(x) = iv). p(x) = v). r(x) = f(g(x)) = (f ∘ g)(x) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 46) Determine the values of constants a and b so that f(x) = a + bx has an absolute maximum at the point A) a = -1, b = 4

B) a = 1, b = 4

C) a = 1, b = 2

46) ______

D) a = -1, b = 2 47) ______

47) Find the absolute maximum and minimum values of f(x) = ln (sin x) on A)

Maximum = 0 at x = 0, minimum = -ln 2 at x = B) Maximum = 0 at x =

, minimum = ln

at x =

Maximum = 0 at x =

, minimum = -ln 2 at x =

Maximum = 0 at x =

, minimum = -ln 2 at x =

C) D)

48)

Find the absolute maximum and minimum values of f(x) = 2x A) Maximum = ln 4 - 2 at x = ln 2, minimum = -1 at x = 0 B) Maximum = ln 2 - 2 at x = ln 2, minimum = -1 at x = 0 C) Maximum = 0 at x = ln 2, minimum = 1 at x = 0 D) Maximum = ln 4 - 2 at x = ln 2, minimum = 1 at x = 0

on [0, 1].

48) ______

1) A 2) B 3) B 4) B 5) D 6) C 7) C 8) D 9) B 10) A 11) C 12) D 13) C 14) D 15) D 16) A 17) B 18) A 19) C 20) A 21) B 22) A 23) C 24) B 25) D 26) B 27) C 28) C 29) B 30) D 31) A 32) D 33) C 34) A 35) D 36) D 37) D 38) D 39) A 40) D 41) C 42) A 43) B 44) The derivative of the function is is increasing at c = 5.

(x) = 1 -

, which is positive for all x > 1. By the first derivative test, the function

45) i).

(x) = (x) + (x) . The signs of the terms are (+) + (-), therefore (x) may be positive or negative. Without more information, we cannot determine where h(x) is increasing or decreasing. ii). (x) = (x)g(x) + f(x) (x) . The signs of the terms are (+)(+) + (+)(-) , therefore (x) may be positive or negative. Without more information, we cannot determine where j(x) is increasing or decreasing.

iii). (x) =

= (-). The quotient is continuous and differentiable for all x, and So, according to the first derivative test, k(x) is everywhere decreasing. iv).

(x) = (g(x))(

(x) is everywhere negative.

)( (x) ). The signs of the factors are (+)(+)(-) = (-). The function

everywhere continuous and differentiable, and (x) is everywhere negative. So, according to the first derivative . The test, (x) is everywhere decreasing. signs v). (x) = (g(x)) (x). The signs of the factors are (+)(-) = (-). The function (f∘g)(x) is everywhere continuous and of the differentiable, and (x) is everywhere negative. So, according to the first derivative test, r(x) is everywhere term decreasing. s are 46) A 47) C 48) A

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 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. 1) 1) _______

A) Local minimum at on (-∞, 0) B) Local minimum at C) Local minimum at on (-∞, 0) D) Local minimum at

x = 1; local maximum at x = -1; concave down on (0, ∞); concave up x = 1; local maximum at x = -1; concave up on (-∞, ∞) x = 1; local maximum at x = -1; concave up on (0, ∞); concave down x = 1; local maximum at x = -1; concave down on (-∞, ∞) 2) _______

A) Local minimum at x = 1; local maximum at x =-1; concave up on (0, ∞); concave down on (-∞, 0) B) Local minimum at x = 1; local maximum at x =-1; concave down on (0, ∞); concave up 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) 3)

__ __ __ _

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 up on (0, ∞); concave down 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 down on (0, ∞); concave up on (-∞, 0) 4) _______

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) 5)

__ __ __ _

A) Local minimum at (-∞, 0) B) Local minimum at (-∞, 0) C) Local minimum at (-∞, 0) D) Local minimum at (-∞, 0)

x = 2; local maximum at x = 0; concave up on (0, ∞); concave down on x = 0; local maximum at x = 2; concave down on (0, ∞); concave up on x = 2; local maximum at x = 0; concave down on (0, ∞); concave up on x = 0; local maximum at x = 2; concave up on (0, ∞); concave down on

Graph the equation. Include the coordinates of any local and absolute extreme points and inflection points. 6) y = 7x2 + 70x

A) absolute minimum: (5,-175) no inflection points

C) absolute minimum: (-10,-70) no inflection points

B) absolute minimum: (10,-70) no inflection points

6) _______

ab sol ut e mi ni m u m: (-5 ,-1 75 ) no inf lec tio n po int s

7) _______

7) y=

A) local minimum: local maximum:

inf int: (0,0) lec tio n po

lo cal mi ni m u m: (4, -1) lo cal m ax im u m: (-4 , 1) inf lec tio n po int : (0, 0)

D) absolute maximum: no inflection point

local minimum: (-4, -1) local maximum: (4, 1) inflection points: (0, 0), (-4

, -2

)

8) _______

8) y = 2x3 + 12x2 + 18x

A) local minimum: (2, 10) no inflection point

B) local minimum: (-7, 343) local maximum: (0, 0) inflection point: (-3.5, 171.5)

C) no extrema inflection point: (0, 0)

D) local minimum: (-1, -8) local maximum: (-3, 0) inflection point:

9) _______

9) y = x1/3(x2 - 252)

A) local minimum: (0, 0) no inflection points

B) local minimum: local maximum: inflection point: (0, 0)

C) no extrema inflection point: (0, 0)

D) local minimum: local maximum: (0, 0) inflection points:

10) ______

10) y=

A) local minimum: (0, 0) inflection points:

C) local minimum: no inflection points

B) local minimum: (0, 0) no inflection points ,

lo cal mi ni m u m:

no inf lec tio n po int s

11) y = x + cos 2x, 0 ≤ x ≤ π

A) local minimum: (1.444, -0.246) local maximum: (0.126, 1.031) inflection points: (0.785, 0.393), (2.356, 1.178)

11) ______

B) no local extrema inflection point:

C) local minimum: local maximum: inflection point:

D) local minimum:

local maximum:

inflection points:

12)

12) ______

y=x

B) local maximum: no inflection point.

local minimum: local maximum: inflection point: (0, 0)

C) local minimum: local maximum:

inf int: (0, 0) lec tio n po

lo cal m ax im u m: (0, ) no inf lec tio n po int s.

Sketch the graph and show all local extrema and inflection points. 13) y=- +4 -9

A) Absolute maxima: (-

, -5), (

Local minimum: (0, -9) Inflection points:

, -5)

13) ______

A bs ol ut e mi ni m a: (, 5), ( , 5) Lo cal m ax im u m: (0, 9) Inf lec tio n po int :

bsolute

maxima: (-

-5), ( , -5) Inflection points: ,

A bs ol ut e m ax im a: (, -5) , ( , -5) Lo cal mi ni m u m: (0, -9) N o inf lec tio n po int s

14) y=x

___ ___

14)

A) Absolute maximum:

B) Local maximum:

Absolute minimum: (10, 0)

Local minimum: (10, 0)

Inflection points:

Inflection point:

C) Absolute minimum: (10, 0) No inflection points

D) Local maximum: Local minimum: (10, 0) Inflection points:

15) y = x - sin x, 0 ≤ x ≤ 2π

, (10, 0)

___ ___

15)

16)

A) Local minimum: (0, 0) Local maximum: (2π, 2π) Inflection point: (π, π)

B) Local minimum: (0, 0) Local maximum: (2π, 2π) No inflection points

C) Local minimum: (0, 0) Local maximum: (2π, 2π) Inflection point: (π, π)

D) Local minimum: (0, 0) Local maximum: (2π, 2π) No inflection points

y=|

- 5x|

___ ___

16)

B) Local minima: (0, 0), (5, 0) Local maximum: Inflection points: (0, 0), (5, 0)

C) Local minima: (0, 0), (5, 0) Local maximum: No inflection points

17)

y = ln (3 -

)

Local maximum: Inflection points: (0, 0), (5, 0)

D) Local minimum: No inflection points

___ ___

17)

18)

A) Local minimum (0, -ln 3) No inflection point

B) Local maximum (0, ln 3) No inflection point

C) Local minimum (0, ln 3) No inflection point

D) No extrema No inflection point

-4

- 5x

18)

______ A) Local maximum (0, -3) Local minimum (ln 4, 3 - 5 ln 4)

B) Local maximum (0, -3) Local minimum (ln 4, 3 - 5 ln 4) No inflection point

Inflection point

C) Local minimum (2, -2) No inflection point

For the given expression 19) y' =

-1

D) Local minimum No inflection point

, find y'' and sketch the general shape of the graph of y = f(x). 19) ______

20)

(5 - x)

20) ______

21) y' = cos x, 0 ≤ x ≤ 2π

21) ______

22)

(x - 4)

22) ______

23) ______

23) y' = 5|x| =

Solve the problem. 24) 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.

24) ______

[NOTE: Graph vertical scales may vary from graph to graph.]

[N O TE : Gr ap h ve rti cal sc ale s m ay va ry fro m gr ap h to gr ap h.] D)

[NOTE: Graph vertical scales may vary from graph to graph.]

25) The graphs below show the first and second derivatives of a function y = f(x). Select a possible

gra ph f

___ ___

that 25) passes through the point P.

[NOTE: Graph vertical scales may vary from graph to graph.] C)

[NOTE: Graph vertical scales may vary from graph to graph.] [N ales O may vary from graph to graph.] TE : Gr ap h ve rti cal sc

[NOTE: Graph vertical scales may vary from graph to graph.] 26) The graphs below show the first and second derivatives of a function

. Select a possible

graph f that passes through the point P.

[NOTE: Graph vertical scales may vary from graph to graph.] C)

[NOTE: Graph vertical scales may vary from graph to graph.] [NOTE: Graph vertical scales may vary from graph to graph.]

26) ______

[NOTE: Graph vertical scales may vary from graph to graph.] 27) The graphs below show the first and second derivatives of a function

. Select a possible

graph f that passes through the point P.

[NOTE: Graph vertical scales may vary from graph to graph.] C)

[NOTE: Graph vertical scales may vary from graph to graph.] [NOTE: Graph vertical scales may vary from graph to graph.]

27) ______

[NOTE: Graph vertical scales may vary from graph to graph.] 28) The graphs below show the first and second derivatives of a function

. Select a possible

graph of f that passes through point P.

[NOTE: Graph vertical scales may vary from graph to graph.] C)

[NOTE: Graph vertical scales may vary from graph to graph.] [NOTE: Graph vertical scales may vary from graph to graph.]

28) ______

[NOTE: Graph vertical scales may vary from graph to graph.] 29) The graphs below show the first and second derivatives of a function

. Select a possible

graph f that passes through the point P.

[NOTE: Graph vertical scales may vary from graph to graph.] C)

[NOTE: Graph vertical scales may vary from graph to graph.] [NOTE: Graph vertical scales may vary from graph to graph.]

29) ______

[NOTE: Graph vertical scales may vary from graph to graph.] Graph the rational function. 30) y=

30) ______

31) ______

31) y=

32) ______

32) y=

33) ______

33) y=

34)

34) y=

______ A)

35) ______

35) y=

36) ______

36) y=

37) ______

37) y=

38)

38) y=

______ A)

Solve the problem. 39) Using the following properties of a twice-differentiable function y = f(x), select a possible graph of f.

39) ______

40) Select an appropriate graph of a twice-differentiable function y = f(x) that passes through the points following sign patterns.

, (0, 0),

and (

40) ______

, 1), and whose first two derivatives have the

: A)

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 41) Find the inflection points (if any) on the graph of the function and the coordinates of the sim usly. Add to points on the graph where the function has a local maximum or local minimum value. ultayour picture the Then graph the function in a region large enough to show all these points neo graphs of the

___ ___ ___ ___ _

function' 41) s first and second derivativ es. y=

- 200

42) 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. y=

42) _____________

- 15

Provide an appropriate response. 43) If f(x) is a differentiable function and

43) _____________

for all x in the domain, must f have a local minimum at x = c? Explain. 44) _____________

44) Sketch a smooth curve through the origin with the following properties: (x) > 0 for

for

(x) approaches 0 as x approaches -∞;

and (x) approaches 0 as x approaches ∞. 45) _____________

45) Sketch a continuous curve y = f(x) with the following properties: f(2) = 3;

(x) > 0 for x > 4; and

(x) < 0 for x < 4 .

46) 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.

46) _____________

47) What can you say about the inflection points of the quartic curve

47) _____________

Give reasons for your answer. 48) _____________

48) For x > 0, sketch a curve y = f(x) that has f(1) = 0 and (x) = - . Can anything be said about the concavity of such a curve? Give reasons for your answer. 49) 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 as positive, negative, or zero.

and

49)

50) The graph below shows the position s = f(t) of a body moving back and forth on a

___ ___ ___ ___ _

50) _____________

coordinate line. 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?

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 51) Suppose the derivative of the function y = f(x) is y' = (x + 4). At what points, if any, does the graph of f have a local minimum or local maximum? A) no local minimum or local maximum B) local minimum at x = -4 C) local minimum at x = 3 D) local maximum at x = -4 52) Suppose that the second derivative of the function y = f(x) is y'' = (x - 3)(x + 9). For what x-values does the graph of f have an inflection point? A) 3, -9 B) -3, -9 C) -3, 9 D) 3, 9

51) ______

52) ______

1) C 2) A 3) B 4) A 5) A 6) D 7) D 8) D 9) B 10) A 11) D 12) C 13) A 14) B 15) A 16) C 17) B 18) A 19) C 20) A 21) B 22) C 23) C 24) B 25) D 26) D 27) D 28) A 29) A 30) B 31) A 32) D 33) B 34) C 35) D 36) D 37) A 38) D 39) C 40) A 41) The zeros of

= 0 and

= 0 are extrema and points of inflection, respectively. Inflection at

maximum at x = 0, local minimum at x =

local

42) The zeros of

= 0 and = 0 are extrema and points of inflection, respectively. Inflection at x = 5, local maximum at x = 0, local minimum at x = 10.

43) Yes. The point x = c is either a local maximum, a local minimum, or an inflection point. But, since

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. 44) Answers will vary. A general shape is indicated below:

45) Answers will vary. A general shape is indicated below:

46) The graph will be a straight line. Since

= 0, this means there is no change in

, which is the slope of y. A

constant slope implies a straight line. 47)

The curve can have 0 or 2 inflection points. The second derivative is quadratic, quadratics may have 0, 1, or 2 roots, depending on the value of the discriminant. If no real roots and y has no inflection points. If inflection point. Finally, if

then

and then

has

has exactly one real root and y has a single

has two real roots and y has exactly two inflection points.

48)

Since

(x) =

49) a: both

and

> 0 for all x > 0, then the function is everywhere concave up. are undefined.

=0 and

> 0 and

= 0 and

> 0 and

= 0 and

g: < 0 and =0 50) (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

(2.3, 4), (5.1, 7), and (8.5, 10). Acceleration is negative on (0, 2.3), (4, 5.1), and (7, 8.5). 51) B 52) A

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use l'Hopital's Rule to evaluate the limit. 1) A) 2

B) -2

C) 1

1) _______

D) -1 2) _______

2) A) 14

B) -5

C) -2

D) 6 3) _______

3) A) 13

B) 20

C) -17

D) 23 4) _______

4) A) 0

D) 5) _______

D) -

6) _______

6) A)

B) -1

C) 0

D) 1

7) _______

7) A) 0

B) -5

C) 1

D) 5 8) _______

8) A) -1

B) 0

C) 1

D) ∞ 9) _______

9) A) 1

D) -

10)

___ ___

10)

A) ∞

C) 1

Use l'H pital's rule to find the limit. 11) A)

11) ______ B) 0

C) 1

D) ∞

12) ______

12) A)

D) 0

13) ______

13) A) 1

D) 0

14) ______

14) A) 1

B) ∞

C) 0

D) -∞ 15) ______

15) x sin A)

B) 1

C) 20

D) 0

16) ______

16) A) 4

C) 2

B) 0

D) - 2 17) ______

17) B) 3

C) 1

D) 0

18) ______

18) A) ∞

C) 8

D) 1

Find the limit. 19) A) 1

19) ______ B) 0

C) ∞

E) 3

20) ______

20) A) 0

B) 1

C) 2

D) ∞ 21) ______

21) A) -5

22) ______

22) A) 0

B) 1

C) 4

D) 23) ______

23) ln x A) 0

B) -1

C) 6

D) 1 24) ______

24) 10x csc x A) ∞

B) 10

C) 0

D) 1 25) ______

25) x cot B) -1

C) 0

D) 1

L'Hopital's rule does not help with the given limit. Find the limit some other way. 26) A) 0

B) 25

C) 5

26) ______ D) ∞ 27) ______

27) A) -1

B) 0

C) 1

D) ∞ 28) ______

28) A) 0

B) ∞

C) -1

D) 1 29) ______

29) A) -1

B) 0

C) 1

D) ∞ 30) ______

30) A) -1 31)

B) 0

C) 1

D) ∞ 31) ______

A) ∞

B) 0

C) 1

D) -1 32) ______

32) A) 0

B) ∞

C) 1

D) -1

Find a value of c that makes the function continuous at the given value of x. If it is impossible, state this. 33) f(x) = A) -5

;

x = -5 B) Impossible

C) 5

33) ______

D) -4 34) ______

34) f(x) = A) -2

;

x=0 B) -4

C) 2

D) Impossible 35) ______

35) f(x) =

;

x= 5 B) Impossible

A) -5

C) 25

D) 5 36) ______

36) f(x) = A) Impossible

;

x=0 B) 4

C) 1

D) -1 37) ______

37)

f(x) = A) Impossible

;

x=0 B) -3

C) 9

D) 3 38) ______

38) f(x) = A)

; x=0 B) 16

C) 0

39) ______

39) f(x) = A) -1

;

x=0 B) Impossible

C) 1

D) 0

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 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. 40) 40) _____________

41)

41) _____________

42)

42) _____________

43)

43) _____________

44)

44) _____________

45)

45) _____________

46)

46) _____________ -

Provide an appropriate response. 47) A student attempted to use l'Hôpital's Rule as follows. Identify the student's error.

47) _____________

= =

=1 48) _____________

48) Which one is correct, and which one is wrong? Give reasons for your answers. (a)

(b)

= =0 49) _____________

49) Which one is correct, and which one is wrong? Give reasons for your answers. (a)

(b)

==0 50) _____________

50) Give an example of two differentiable functions f and g with that satisfy

f(x) =

=∞

= 4. 51) _____________

51) Give an example of two differentiable functions f and g with that satisfy

f(x) =

g(x) = 0

52) _____________

52) If f(x) =

and g(x) =

, show that

53) Find the error in the following incorrect application of L'Hôpital's Rule. =

= 0.

54) Find the error in the following incorrect application of L'Hôpital's Rule.

53) _____________

54) _____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 55) Given that x > 0, find the maximum value, if any, of . A) ∞ B) C) 1 D)

55) ______

1) B 2) C 3) C 4) D 5) B 6) D 7) D 8) B 9) D 10) D 11) B 12) B 13) D 14) C 15) C 16) C 17) B 18) B 19) D 20) B 21) C 22) B 23) A 24) B 25) D 26) C 27) C 28) D 29) D 30) D 31) B 32) A 33) B 34) D 35) A 36) D 37) A 38) A 39) C 40)

Usin estimate the limit to be approximately 5.4. g the grap Using l'Hopital's rule: h, stude = nts shoul = = d 41)

Using the graph, students should estimate the limit to be 0. Using l'Hopital's rule: = =

= 0

42)

Using the graph, students should estimate the limit to be -1. Using l'Hopital's rule: = = 43)

= -1

estimate the limit to be 1. Using l'Hopital's rule: the limit leads to the indeterminate form ln f(x) =

ln x

ln f(x)

so let f(x) =

= =

Usin g the grap h, stude nts So shoul d 44)

and take logarithms of both sides

differentiate

-2

Using the graph, students should estimate the limit to be 0. Using l'Hopital's rule: =

45)

Using the graph, students should estimate the limit to be 1. Using l'Hopital's rule: the limit leads to the indeterminate form

so let f(x) =

and take logarithms of both sides

ln f(x) = f(x) (1/x) ln

= =

differentiate

46)

Using the graph, students should find the limit to be ∞. Using l'Hopital's rule: the limit leads to the indeterminate form ∞ - ∞ so combine the fractions and apply l'Hopital's rule:

= = = =

=∞

47) L'Hôpital's Rule cannot be applied to form.

because it corresponds to

, which is not an indeterminate

48) Choice (a) is incorrect. L'Hôpital's rule cannot be applied to an indeterminate form. Choice (b) is correct.

because it corresponds to

which is not

49) Choice (a) is correct. L'Hôpital's rule can be applied to form 50) 51)

. Choice (b) is incorrect because

Two such functions are f(x) = 4

+ 4 and g(x) =

not -4. + 1.

since it corresponds to the indeterminate

Two such ions are f(x) = funct 52) =

and g(x) =

. =0

53) L'Hôpital's Rule cannot be applied to

because it corresponds to

which is not an indeterminate form.

54) L'Hôpital's Rule cannot be applied to indeterminate form. 55) B

because it corresponds to -

which is not an

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) From a thin piece of cardboard 10 in. by 10 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) B) 5 in. × 5 in. × 2.5 in.; 62.5 6.7 in. × 6.7 in. × 1.7 in.; 74.1 C) D) 3.3 in. × 3.3 in. × 3.3 in.; 37 6.7 in. × 6.7 in. × 3.3 in.; 148.1 2) A company is constructing an open-top, square-based, rectangular metal tank that will have a

1) _______

2) _______

volume of What dimensions yield the minimum surface area? Round to the nearest tenth, if necessary. A) 3.5 ft × 3.5 ft × 3.5 ft B) 9.3 ft × 9.3 ft × 0.5 ft C) 4.4 ft × 4.4 ft × 2.2 ft D) 5.1 ft × 5.1 ft × 1.7 ft SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 3) You are planning to close off a corner of the first quadrant with a line segment 19 units 3) _____________ long running from (x, 0) to (0, y). Show that the area of the triangle enclosed by the segment is largest when

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 4) 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 120 in. What dimensions will give a box with a square end the largest possible volume?

_______ A) 20 in. × 20 in. × 100 in. C) 20 in. × 40 in. × 40 in.

B) 40 in. × 40 in. × 40 in. D) 20 in. × 20 in. × 40 in.

5) 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. Suppose you want to mail a box with square sides so that its dimensions are will give the box its largest volume? A) 36 in. × 18 in. × 36 in. C) 18 in. × 18 in. × 36 in.

and it's girth is

What dimensions

B) 18 in. × 18 in. × 90 in. D) 24 in. × 24 in. × 18 in.

6) 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-fifth 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.

5) _______

6) _______

B) =

D) =

7) A trough is to be made with an end of the dimensions shown. The length of the trough is to be

7) _______

long. Only the angle θ can be varied. What value of θ will maximize the trough's volume?

A) 46°

B) 32°

C) 30°

D) 14°

8) A rectangular sheet of perimeter 39 cm and dimensions x cm by y cm is to be rolled into a

cyl inder as

__ __ __ _

shown in 8) part (a) of the figure. What values of x and y give the largest volume?

B) x = 13 cm; y =

x = 12 cm; y =

x = 15 cm; y =

D) x = 14 cm; y =

9) 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) 4 in. B) 3 in. C) 6 in. D) 4 in. on one side, 5 in. on the other

9) _______

10) Determine the dimensions of the rectangle of largest area that can be inscribed in a semicircle of radius 3. A) h = B) ,w= 3

10) ______

C) h = 3

,w=

,w=3

h=3

,w=

D) ,

11) The 9 ft wall shown here stands 30 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.

___ ___

11)

A) 52.3 ft

B) 39 ft

C) 51.3 ft

D) 53.3 ft

12) The strength S of a rectangular wooden beam is proportional to its width times the square of its

12) ______

depth. Find the dimensions of the strongest beam that can be cut from a cylindrical log. (Round answers to the nearest tenth.)

A) w = 8.7 in.; d = 12.2 in. C) w = 7.7 in.; d = 13.2 in.

B) w = 9.7 in.; d = 13.2 in. D) w = 9.7 in.; d = 11.2 in.

13) 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 (Round answers to the nearest tenth.)

A) w = 6.0 in.; d = 10.4 in. C) w = 7.0 in.; d = 11.4 in.

cylindrical log.

B) w = 7.0 in.; d = 9.4 in. D) w = 5.0 in.; d = 11.4 in.

14) The positions of two particles on the s-axis are meters and t in seconds.

13) ______

= sin t and

with

and

t At time(s) wha in the

interval 0 14) ≤ t ≤ 2π do the particles meet? A) t = π sec and t = 2π sec C)

___ ___

B) t=

π sec and t =

π sec

π sec and t =

π sec

D) t=

π sec and t =

π sec

15) 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

15) ______

. What is the cart's maximum speed? When is the cart moving that fast? What is the magnitude of of the acceleration then? A) 10π ≈ 31.42 cm/sec; t = 0.5 sec, 1.5 sec, 2.5 sec, 3.5 sec; acceleration is 0 cm/ B) π ≈ 3.14 cm/sec; t = 0.5 sec, 1.5 sec, 2.5 sec, 3.5 sec; acceleration is 0 cm/ C) 10π ≈ 31.42 cm/sec; t = 0 sec, 1 sec, 2 sec, 3 sec; acceleration is 0 cm/ D) 10π ≈ 31.42 cm/sec; t = 0.5 sec, 2.5 sec; acceleration is 1 cm/ 16) 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

16) ______

was sailing east at and continued to do so all day. The visibility was 5 nautical miles. Did the ships ever sight each other? A) No. The closest they ever got to each other was 4.6 nautical miles. B) Yes. They were within 3 nautical miles of each other. C) Yes. They were within 4 nautical miles of each other. D) No. The closest they ever got to each other was 5.6 nautical miles. 17) If the price charged for a candy bar is p(x) cents, then x thousand candy bars will be sold in a certain city, where p(x) = 142 A) 3408 candy bars C) 3408 thousand candy bars

. How many candy bars must be sold to maximize revenue? B) 1704 candy bars D) 1704 thousand candy bars

18) A rectangular field is to be enclosed on four sides with a fence. Fencing costs $6 per foot for two opposite sides, and $2 per foot for the other two sides. Find the dimensions of the field of area 760 ft2 that would be the cheapest to enclose. A) 15.9 ft @ $6 by 47.7 ft @ $2 C) 47.7 ft @ $6 by 15.9 ft @ $2

17) ______

18) ______

B) 82.7 ft @ $6 by 9.2 ft @ $2 D) 9.2 ft @ $6 by 82.7 ft @ $2

19) Find the optimum number of batches (to the nearest whole number) of an item that should be produced annually (in order to minimize cost) if 250,000 units are to be made, it costs $1 to store a unit for one year, and it costs $560 to set up the factory to produce each batch. A) 11 batches B) 17 batches C) 13 batches D) 15 batches

19) ______

20) 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:

20) ______

R(x) = 60x - 0.5 C(x) = 3x + 7.

A) 57 units

B) 63 units

C) 64 units

D) 58 units

21) 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 C

22)

23)

21) ______

= 3x = 0.001 + 1.1x + 80. A) 4100 units B) 1900 units

C) 950 units

D) 2050 units

Suppose c(x) = - 22 + 10,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items. A) 12 items B) 13 items C) 11 items D) 10 items Suppose that c(x) = 5 - 21 + 6263x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items. A) There is not a production level that will minimize average cost. B) -57 items C) 66 items D) 6263 items

24) Suppose a business can sell x gadgets for p = 250 - 0.01x dollars apiece, and it costs the business

22) ______

23) ______

24) ______

dollars to produce the x gadgets. Determine the production level and cost per gadget required to maximize profit. A) 111 gadgets at $248.89 each B) 11,250 gadgets at $137.50 each C) 10,000 gadgets at $150.00 each D) 13,750 gadgets at $112.50 each SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 25) A manufacturer uses raw materials to produce p products each day. Suppose that each 25) _____________ 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? MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 26) How close does the curve y = come to the point the distance, you can avoid square roots.) A) The distance is minimized when x =

26) ______

? (Hint: If you minimize the square of

; the minimum distance is

units.

B) The distance is minimized when x = 2; the minimum distance is

units.

C) The distance is minimized when x = 1; the minimum distance is

units.

The distance is minimized when x =

units.

D) ; the minimum distance is

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 27) 27) _____________ The function y = cot x < π. Find it.

csc x has an absolute maximum value on the interval 0 < x

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 28) How close does the semicircle y = come to the point (1, )? A) x = 1.55 B) x = 0.45 C) x = 2.00 D) x = 2.45

28) ______

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 29) 29) _____________ Show that g(x) = 30)

Let c(x) = t(

- p)

is an increasing function of x. where t and

are constants. Show that c(x) is greatest when

30) _____________

1) B 2) C 3) If x , y represent the legs of the triangle, then

Solving for y, y = A(x) = xy = x

A'(x) = -

Solving A'(x) = 0, x = ± Substitute and solve for y: ( + = 361 ; y = ∴ x = y. 4) D 5) D 6) C 7) C 8) A 9) B 10) B 11) A 12) A 13) A 14) C 15) A 16) A 17) D 18) A 19) D 20) A 21) C 22) C 23) A 24) B 25) 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 storage for each cycle is approximately Cost per cycle = delivery costs + storage costs

. Thus, the cost of delivery and

Cost per cycle = d + ∙x We compute the average daily cost c(x) by dividing the cost per cycle by the number of days x in the cycle. c(x) = + We find the critical points by determining where the derivative is equal to zero. (x) = -

x=±

Therefore, an absolute minimum occurs at

days.

26) C 27) = -

(x) +

=0⇒

csc(x)cot(x) = csc(x)

⇒ cos(x) =

= 0 ⇒

⇒x=

28) A 29)

30)

Notice that

(x) =

We calculate

(x) = 3t

(x) < 0 for p >

. As

is positive for all values of x. Therefore g is increasing everywhere. - 4t

, and find that the only critical point is and

(x) > 0 for p <

, the absolute maximum of c(x) occurs at

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use Newton's method to estimate the requested solution of the equation. Start with given value of estimated solution. 1) 3 + 2x - 1 = 0; = 1; Find the right-hand solution. A) 0.85 B) 0.33 C) 0.50 D) 0.35 2)

= 0; Find the left-hand solution. B) 0.23

+ 5x + 2 = 0; A) -0.39

= -1; Find the one real solution. B) -0.38

- 4x + 2 = 0; A) 0.51

= 0; Find the left-hand solution. B) 0.5

- 6x + 3 = 0; A) 1.604

= 2; Find the right-hand solution. B) 1.607 C) 1.602 = 1; Find the negative solution. B) 1.32

as the

1) _______

2) _______

+ 4x -1 = 0; A) 0.14

- 3 = 0; A) 1.31

and then give

C) -0.33

D) 0.25 3) _______

C) -0.64

D) -0.44 4) _______

C) 0.57

D) 0.52 5) _______ D) 1.600 6) _______

C) 1.29

D) 1.33

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 7) 7) _____________ Use Newton's method to estimate the solutions of the equation . Start with

for the right-hand solution and with

Then, in each case find 8)

for the right-hand solution and with

Then, in each case find

11)

. 10) _____________

Start with

Then, in each case find

11) _____________

Use Newton's method to estimate the solutions of the equation with

9) _____________

Start with

Use Newton's method to estimate the one real solution of the equation Start with

12)

Use Newton's method to estimate the one real solution of and then find

8) _____________

. Start

= -2 for the solution on the left.

Use Newton's method to estimate the one real solution of and then find

10)

Use Newton's method to estimate the solutions of the equation with

= -2 for the solution on the left.

for the right-hand solution and with

. Start

= -1.5 for the solution on the left.

The h case find n, in eac

_____________

12)

13) Use Newton's method to estimate the solution of the equation with

Then, in each case find

13) _____________

. Start

. 14) _____________

14) Use Newton's method to find the positive fourth root of 3 by solving the equation Start with

and find

. 15) _____________

15) Use Newton's method to find the negative fourth root of 5 by solving the equation Start with 16)

and find

Use Newton's method to estimate the solutions of the equation with

for the right-hand solution and with

Then, in each case find

16) _____________

. Start

for the solution on the left.

. 17) _____________

17) The curve y = tan x crosses the line y = 3x between x = 0 and x = . Use Newton's method to find where the line and the curve cross. (Round your answer to two decimal places.)

18) _____________

18) Use Newton's method to find the two real solutions of the equation . 19)

How many solutions does the equation cos 4x = 0.95 -

19) _____________

have?

20) _____________

20) Write down the first four approximations to the solution of the equation using Newton's method with an initial estimate of 21)

= 1.

Use Newton's method to find the four real zeros of the function

21) _____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 22) The graphs of y = and y = 5 intersect at one point x = r. Use Newton's method to estimate the value of r to four decimal places. A) r ≈ 1.9208 B) r ≈ 2.9028 C) r ≈ 2.9208 D) r ≈ 1.9028

22) ______

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 23) Find the approximate values of 23) _____________ through in the factorization 6

- 12

-7

+ 13x - 1 = 6(x -

)(x -

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 24) Use Newton's method to find the value of x for which Find the answer correct to five decimal places. A) -2.99288 B) -2.99107 C) -2.99294 D) -2.99263 25)

Use Newton's method to find the value of x for which to five decimal places. A) 0.45065 B) 0.45081 C) 0.45076

Find the answer correct D) 0.45071

24) ______

25) ______

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 26) Marcus Tool and Die Company produces a specialized milling tool designed specifically 26) _____________ 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) = 301 + 29 . Use Newton's method to find the break-even 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. 27) 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

27) _____________

equation where V is in volts and t is the time of operation in 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) = 1.6 volts). Use t = 7 hours as your initial guess and show all your work. Provide an appropriate response. 28) Explain why the following four statements ask for the same information. (i) Find the roots of f(x) = 3

28) _____________

- 4x - 1

(ii) Find the x-coordinates of the intersections of the curve y =

with the line y = 4x + 1.

(iii) Find the x-coordinates of the points where the curve y =

- 4x crosses the

horizontal line

(iv) Find the values of x where the derivative of g(x) =

-2

- x + 5 equals zero.

29) You plan to estimate π to five decimal places by using Newton's method to solve the equation answer.

29) _____________

. Does it matter what your starting value is? Give reasons for your

30) Suppose Newton's Method is used with an initial guess What happens to

that lies at a critical point

30) _____________

and later approximations? Give reasons for your

answer. 31) _____________

31) Apply Newton's method to f(x) = initial guess

, a > 0, and write an expression for

is greater than or equal to 1, what happens to

32) Show that if h > 0, applying Newton's method to f(x) = leads to

= h if

= h and to

= -h if

= -h when 0 < 5 < h.

. If the

as n → ∞? 32) _____________

1) D 2) B 3) D 4) D 5) D 6) D 7) f(x) = 4

+ 2x - 3,

(x) = 8x + 2

Right-hand solution: = 0.5 =

therefore

= = 0.6667

Left-hand solution: = -2 =

therefore 8)

f(x) = -3

= = -1.3571

- 2x + 1,

(x) = -6x - 2

Right-hand solution: = 0.5 =

therefore

= = 0.3500

Left-hand solution: = -2 =

therefore 9) 10)

= = -1.3000

= 1.3288 f(x) = -2

- 3x - 4,

(x) = -6(

= -0.5

= therefore 11)

= = -1.0000

)-3

f(x) = 4 - 3x 2,

-3 =1

(x) =

therefore 12)

= = 1.0588

f(x) =

(x) = 12

Right-hand solution: =1

therefore

= = 1.0714

Left-hand solution: = -1.5

therefore = -1.3393 13) f(x) = 2sin x - 4x + 3 (x) = 2 cos x- 4 = 1.5 = therefore 14)

f(x) =

must be 1.23951692 - 3, f '(x) =

therefore 15)

f(x) =

= = 1.5000

- 5, f '(x) =

= -1

there fore 16)

= -2.0000 f(x) = 2x - 3

+ 5, f '(x) = 2 - 6x

Right-hand solution: = 1.5 =

therefore

= = 1.6786

Left-hand solution: = -1 =

therefore = -1.0000 17) The curves cross at x = 1.32. 18) f(x) = -3 -3 - 3x + 4 , f '(x) = 4

19) 4 solutions 20)

21)

f(x) = 3

-6

+ 2,

(x) = 12

- 12x

-9

- 6x - 3

22) D 23)

= 2.0783 = 0.0809 = 0.9191 = -1.0783

24) A 25) C 26) Find the root of f(x) = C(x) - R(x) = 301 + 29

(x) =

- 4x.

-4

= 370

= 370 -

= 364.68 -

= 364.68

= 364.68 -

= 364.67

The break-even point is x = 364.67 tools. 27)

Find the root of f(x) = -0.0306 (x) = -0.0918

+ 0.373

- 2.16t + 15.1 - 1.6.

+ 0.746t - 2.16

=7-

= 12.40 -

=7-

= 12.40

= 12.40 -

= 10.60

The useful working time is t = 10.60 hours. 28)

(i) The roots of the function y = 3 -1, -0.26373525, and 1.26376686.

- 4x - 1 can be calculated by using Newton's method. Resulting in three roots:

(ii) The x-coordinate of the intersection of y =

and y= 4x + 1 is the root of

(iii) The curve y = - 4x crosses the horizontal line y = 1 at the solution of the functions in part (i) and (ii).

which is the same as

(iv) The values of x where the derivative of g(x) equals zero are the same as the functions in part (i) and (ii) 29) Yes. Since cos x = 0 at all points of π + kπ, if you choosing your starting value too large or too small, it will converge to a different solution. 30) At the critical point = 0.This puts and all later approximations out at -∞ or ∞. 31) f(x) =

;

(x) =

= (1-a)

= as n → ∞ ,

→∞

32) f(x) =

(x) = Let

= h. Then,

=h-

= h - 2(h - 5) = 10 - h

= 10 - h + Likewise, let

=-h+

= 10 + h -

= 10 - h + 2(h - 5) = h = - h. Then,

= - h + 2(5 + h) = 10 + h

= 10 + h - 2(h + 5) = - h

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find an antiderivative of the given function. 1) A)

2) _______

+ 14x - 9 A)

3) 4

1) _______

- 9x

- 8x

- 9x 3) _______

-2 A)

- 2x

-2

-2 4) _______

4) A)

5) _______

5) + A)

C) -

D) -

+ 6) _______

6) A)

B) -

7) 4 cos 2x A) 4 sin 2x

C) 4

D) -

+ 7) _______

B) sin 2x

C) 2 sin 2x

D) - 8 sin 2x 8) _______

8) cos πx + 3 sin A)

sin πx - cos C)

-π sin πx + cos D)

-sin πx - 9 cos

sin πx - 9 cos 9) _______

9) A)

B) 6 cot

cot -6

D) cot co t

10) ______

10) 6 csc 4x cot 4x A)

csc 4x

C) -6 csc 4x -

cot 4x

csc 4x 11) ______

11) A)

D) -

12)

12) ______

+ A)

Find the most general antiderivative. 13)

13) ______

dt A)

B) 18t +

C) 27

D) 3

+t+C

+C 14) ______

14) dx A)

-6+C

+ 8x + C

- 12

-6

+ 8x + C

-3

+ 8x + C

15) ______

15) dx A)

-4

D) -

16) ______

16) dt A)

B) -

D) -

+C 17) ______

17) dy A)

B) -6

D) +

+C 18) ______

18) dx A) C

D) -

-2

+C 19) ______

19) dt A) -9sin t + C

C) -

D) -9 cos t + C -

+C 20) ______

20) dx A)

B) 3 cot x + C

C) -3 tan x + C

D) -3 cot x + C

+C 21) ______

21) dθ A) sin θ + C C) sin θ + θ + C

B) csc θ + cos θ + C D) cos θ + C 22) ______

22) dθ A)

B) cot θ + C

θ+C

C) -cot θ + C

D) θ + tan θ + C 23) ______

23) dx A)

-7

D) +7

24) ______

24) dx A) C)

x - ln |x| + C

B) 2

x + 7 ln |x| + C

x - 7 ln |x| + C

+ C 25) ______

25) dx A)

B) -

x - ln |x| + C

x - 8 ln |x| + C

+C D)

x + 8 ln |x| + C

Use differentiation to determine whether the integral formula is correct. 26) dx =

26) ______

A) Yes

B) No 27) ______

27) dx = -

A) Yes

B) No 28) ______

28) dx = -5 cot

A) Yes

B) No 29) ______

29) dx = -

A) No

B) Yes 30) ______

30) dx = -x cos x + sin x + C A) Yes

B) No 31) ______

31) dx =

sin x + C

A) No

B) Yes 32) ______

32) dx =

A) No

B) Yes 33) ______

33) dx = A) Yes

+C B) No

34) ______

34) dx =

A) No

B) Yes 35) ______

35) dx = A) Yes

+C B) No

Which of the graphs shows the solution of the given initial value problem? 36) = 6x, y = 1 when x = -1 A)

36) ______

37) ______

37) = -2x, y = 0 when x = -1 A)

Solve the initial value problem. 38) = A)

38) ______

+ x, x > 0; y(2) = 0 B) y=-

D) y=-

39) ______

39) =4 , y(1) = 5 A) y = 16 + 80 C) y =-

B) D)

y=4

y = 16

- 11

40) ______

40) = A)

+ 3, y(9) = 1 B) y = y=

+ 3x y

+ 3x + 31

D) =

+ 3x -

y = + 3x 29 41) ______

41) = cos t - sin t, s =8 A) s = sin t - cos t + 7 C) s = 2sin t + 6

B) s = sin t + cos t + 7 D) s = sin t + cos t + 9 42) ______

42) =A)

cos

θ, r(0) = -10 B)

r = cos

θ - 11

r = sin

θ - 10

r=-

sin

D) r = -sin

θ - 10

θ - 10 43) ______

43) = A)

csc t cot t, v

= B)

C) v= -

v = csc t -

D) v= -

v= -

44) ______

44) = 4t + t, r(-π) = 3 A) r = 4 + tan t + 3 - 4 C) r = 4 + tan t - 1

B) D)

r=2

+ cot t + 3 - 2

r=2

+ tan t + 3 - 2 45) ______

45) = 3 - 9x, A) y = 2

(0) = 8, y(0) = 2 B)

y=3

+ 8x + 2

D) y=

+ 8x + 2

- 8x - 2 46) ______

46) =

;

= 2,

r(1) = 5 B) r = 2t + 4t + 11

+ 4t + 11

D) r=

47)

t-1

+ 4t - 1

___ ___

47) = 4; (0) = -5, (0) = 2, y(0) =5 A)

y= C) y = 5

+ 2x + 5

48) ______

48) = , y(-7) = 3 A) y = ln(x + 8) + 3

y=+4 D) y = ln(x + 8) + 3 - ln 2 y=-

+3 49) ______

49) = A) C)

y(0) = 6

y= 3

x+4

x+3

y= 3

x+4

x+6

B) D)

y= 3

x+4

x+6

y= 3

x+4

x+3

Find the curve y = f(x) in the xy-plane that has the given properties. 50) f(x) has a slope at each point given by A) B) y=-

50) ______

and passes through the point C) y=

D) y=

+5 51) ______

51) = 36x and the graph of y passes through the point (0, 11) and has a horizontal tangent there. A) B) C) D) y = 18 + 11 y=6 - 11 y = 18 + 11 y=6 + 11

The graph below shows solution curves of a differential equation. Find an equation for the curve through the given point. 52)

___ ___

52)

=1A)

B) y=x+

y=1-

y=x-

D) y=x-

53) ______

53)

= sin x + cos x A) y = sin x - cos x C) y = sin x - cos x -2

B) y = sin x - cos x -1 D) y = sin x - cos x +1

Solve the problem. 54) 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 = -16t + 6, s(0) = 5 A) B) s = 8 + 6t - 5 s = - 8 + 6t + 5 C) D) s = -16 + 6t + 5 s = - 8 + 6t - 5 55) 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.

54) ______

v= cos

t, s(0)

___ ___

55) A)

B) s = sin t s=

sin

D) s=

sin

t+π

s = 2π sin

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. v= A)

sin

, s(

56) ______

)=2 B)

s = 2 cos

s = -2 cos

+ 3.3073

D) s = -2 cos

+ 8.2134

57) 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 = 9.8, v(0) = 14, s(0) = -4 A) B) s = 4.9 + 14t - 4 s = 4.9 + 14t C) D) s = -4.9 - 14t - 4 s = 9.8 + 14t - 4

57) ______

58) 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 = 32 cos 2t, v(0) = -8, s(0) = 8 A) s = 8 sin 2t - 8t + 8 B) s = - 8 sin 2t - 8t + 8 C) s = - 8 cos 2t - 8t + 8 D) s = 8 cos 2t + 8t + 8

58) ______

59)

59) ______

60)

On our moon, the acceleration of gravity is 1.6 m/ . 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) -36 m/sec C) 3240 m/sec D) 72 m/sec A rocket lifts off the surface of Earth with a constant acceleration of 30 m/ the rocket be going 2.5 minutes later? A) -75 m/sec B) 187.5 m/sec C) 37.5 m/sec

. How fast will D) 75 m/sec

61) At about what velocity do you enter the water is you jump from a 15 meter cliff? (Use g = 9.8 m/ .) A) 17 m/sec

B) -17 m/sec

C) -8.5 m/sec

C) 2.06 sec

62) ______

7.26 ft/

63) An object is dropped from 11 ft above the surface of the moon. How long will it take the object to hit the surface of the moon if A) 1.45 sec B) 1.06 sec

61) ______

D) 2 m/sec

62) You are driving along a highway at a steady 81 ft/sec when you see a deer ahead and slam on the brakes. What constant deceleration is required to stop your car in A) B) C) 29.03 ft/ 14.52 ft/ 0.07 ft/

60) ______

D) 4.23 sec

63) ______

Answer each question appropriately. 64) 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:

= a,

ii.

when t = 0, and

iii. s =

when t = 0,

where t is time, A) s= C)

s=a

is the initial position, and

+ +

is the initial velocity. B)

65) 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)/se s=- g

64) ______

65) ______

is given by the equation:

where s is the height above the earth,

is the initial velocity,

and is 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. A) B) =g,

(0) =

s(0) =

= -g D)

= -gt , s(0) =

= -g ,

(0) =

s(0) = 66) ______

66) Suppose the velocity of a body moving along the s-axis is

= 9.8t - 4.

Find the body's displacement over the time interval from

A) 212

B) 39.2

given that

C) 4

when

D) 180 67) ______

67) Suppose the velocity of a body moving along the s-axis is

= 9.8t - 5.

Find the body's displacement over the time interval from

A) -10 C) 48.8

given that

when

B) 28.8 D) Not enough information is given. 68) ______

68) Suppose the velocity of a body moving along the s-axis is = 9.8t - 1. 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) Yes, integration is not possible without knowing the initial position. 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) No, displacement has nothing to do with the position of the body. 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. Use a computer algebra system (CAS) to solve the given initial value problem. 69) =3 sin x, y(0) = 5 A) B) y = -3( - 2) cos x + 6x sin x - 1 y = -3 cos +8 C) D) y = -3x cos x + 3 sin x + 5 y = -3x sin x cos x + 3 x+3 +5

69) ______

70) ______

70) = A)

, y(0) = 2 y = 22

x - 11x

B) y = 11 ln

y = 12

x-x+2

- 11x + 2

y = 22

x - 11x + 2

Answer each question appropriately. 71) Find a curve y = f(x) with the following properties:

71) ______

i. = 9x ii. The graph passes through the point (0,1) and has a horizontal tangent at that point. B) y = 9x + 1 C) D) y=

y= 1

+x+

72) How many curves y = f(x) have the following properties?

72) ______

i. = 4x ii. The graph passes through the point (0,3) and has a horizontal tangent at that point. A) 2 B) Infinitely many C) 1 D) 0 73) If differentiable functions y = F(x) and y = G(x) both solve the initial value problem on an interval I, must for every x in I? Justify the answer. 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).

73) ______

1) C 2) A 3) A 4) C 5) C 6) D 7) C 8) D 9) C 10) D 11) C 12) C 13) C 14) D 15) C 16) A 17) D 18) D 19) A 20) C 21) C 22) C 23) B 24) C 25) D 26) A 27) A 28) B 29) B 30) A 31) A 32) B 33) A 34) A 35) A 36) D 37) B 38) A 39) D 40) D 41) B 42) C 43) C 44) D 45) C 46) D 47) A 48) A 49) B 50) C 51) D

52) C 53) B 54) B 55) A 56) B 57) A 58) C 59) A 60) D 61) B 62) B 63) C 64) A 65) D 66) D 67) C 68) B 69) A 70) D 71) C 72) C 73) C

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. 1) 1) _______ f(x) = between x = 0 and x = 4 using a lower sum with two rectangles of equal width. A) 38.75 B) 8 C) 40 D) 20 2)

f(x) = A) 1

between x = 0 and x = 2 using an upper sum with two rectangles of equal width. B) 2.5 C) 5 D) 4.5

f(x) = between x = 0 and x = 3 using the "midpoint rule" with two rectangles of equal width. A) 16.875 B) 8.4375 C) 3.375 D) 12.5

f(x) = A)

between x = 2 and x = 8 using a lower sum with two rectangles of equal width. B) C) D)

5) _______

5) f(x) = A)

between x = 1 and x = 9 using an upper sum with two rectangles of equal width. B) C) D)

6) _______

6) f(x) = A)

10)

3) _______

4) _______

2) _______

between x = 1 and x = 7 using the "midpoint rule" with two rectangles of equal width. B) C) D)

f(x) = between x = 2 and x = 6 using a lower sum with four rectangles of equal width. A) 86 B) 54 C) 62 D) 69 f(x) = between x = 3 and x = 7 using an upper sum with four rectangles of equal width. A) 126 B) 86 C) 117 D) 105 f(x) = between x = 4 and x = 8 using the "midpoint rule" with four rectangles of equal width. A) 149 B) 126 C) 174 D) 165 f(x) = 25 width. A) 62.5

between x = -5 and x = 5 using the "midpoint rule" with two rectangles of equal B) 187.5

C) 10

7) _______

8) _______

9) _______

10) ______

D) 93.75

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. 11) 11) ______ f(x) = 5 on [1, 3] divided into 4 subintervals A) 293 B) 173 C) 58.6 D) 1172 12)

f(x)

___ ___

12) =

divided into 5 subinterv als A)

13)

f(x) = -7 A) 12

14) f(t) = 3 + sin πt A)

[-3, 7] divided into 5 subintervals B) 25 C) 5

[0, 2] divided into 4 subintervals B) 0 C)

13) ______ D)

14) ______ D) 3

15) ______

15) f(t) = 2 A) 0

[0, 4] divided into 4 subintervals B) 2 C)

Estimate the value of the quantity. 16) The table gives dye concentrations for a cardiac-output determination. The amount of dye

16)

injected was 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.

Time (sec

Dye concentration (mg/L)

______ A) 0.2 L/min

B) 397.5 L/min

C) 18.1 L/min

D) 9.1 L/min

17) The table gives dye concentrations for a cardiac-output determination. The amount of dye

17) ______

injected was 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.

Dye concentration (mg/L)

Time (sec) A) 6.3 L/min

B) 0.1 L/min

C) 285.7 L/min

D) 12.6 L/min

18) 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.

A) 129 in.

B) 114 in.

C) 248 in.

18) ______

D) 124 in.

19) The table shows the velocity of a remote controlled race car moving along a dirt path for 8

seconds.

___ ___

Estimate 19) the distance traveled by the car using 8 subinterv als of length 1 with right-end point values.

A) 152 in.

B) 158 in.

C) 142 in.

D) 148 in.

20) 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.

A) 5400 ft

B) 3710 ft

C) 361 ft

D) 3610 ft

21) 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

20) ______

___ ___

21)

A) 224 ft

B) 244 ft

C) 221 ft

D) 122 ft

22) 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.

A) 427.4 m

B) 854.8 m

C) 815.6 m

D) 407.8 m

23) 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

A) 38.03 ft/sec

B) 192 ft/sec

C) 69.14 ft/sec

22) ______

D) 32.06 ft/sec

24) A swimming pool has a leak. The leak is getting worse. The following table gives the leakage rate every 6 hours.

23) ______

___ ___

24)

Give the upper estimate for the number of gallons lost. A) 32.9 gallons

B) 145.2 gallons

C) 257.4 gallons

D) 197.4 gallons

25) A swimming pool has a leak. The leak is getting worse. The following table gives the leak rate every 6 hours.

25) ______

Suppose it keeps leaking 6.6 gallons every 6 hours. After losing 3500 gallons the leak is fixed. Approximately how long did the leak last? Use the right endpoints to estimate the first 48 hours. A) 34.39 hours B) -81.40 hours C) 28.39 hours D) 55.45 hours Write the sum without sigma notation and evaluate it. 26)

26) ______

B) +

D) +

27) ______

27)

B) +

∙

D) +

= 28) ______

28)

A) C)

-2

-3

= -46

= -11

B) D)

= -20 +

= 11 29) ______

29)

B) +

∙

D) +

= 15

= 30) ______

30)

A) 2 sin π + 2 sin

+ 2 sin

=2+

2 sin π + 2 sin

+ 2 sin

=6+

2 sin π + 2 sin

+ 2 sin

=1+

B) C) D)

31) ______

31)

A) 2 cos π + 2 cos

+ 2 cos

= -2 +

2 cos π + 2 cos

= -2 +

2 cos π + 2 cos

+ 2 cos

= -1 +

2 cos π + 2 cos

+ 2 cos

=3+

B) C) D)

32) ______

32)

B) -sin

- sin

-sin -si n

+ sin

- sin

= -1

D) + sin sin

-si n

+ si n

si n

= 0 33) ______

33)

A) cos 4π + cos 12π = 2 C) cos 4π - cos 8π + cos 12π = - 1

B) -cos 4π + cos 8π - cos 12π = -1 D) cos 4π - cos 8π + cos 12π = 1 34) ______

34)

A) 2 + 4 + 8 = 14

2 sin

+ 4 sin

+ 8 sin

2 sin

+ 4 sin

+ 8 sin

D) 2 sin

+ 8 sin

35) ______

35)

A) 3 cos π + 9 cos π + 27 cos π = -39 C) 3 cos π + 9 cos 2π + 27 cos 3π = 21

B) 3 cos π + 9 cos 2π + 27 cos 3π = -21 D) 3 cos π + 27 cos 3π = -30

Provide an appropriate response. 36) Which of the following express 1 + 5 + 25 + 125 + 625 in sigma notation?

I. A) II only

II. B) I, II, and III

36) ______

III. C) I and II

D) II and III 37) ______

37) Which of the following express -4 + 16 - 64 + 256 - 1024 in sigma notation?

I. A) I and II

II. B) II only

III. C) I, II, and III

D) III only

38) Which formula is not equivalent to the other two? I.

___ ___

38)

II .

II I.

A) II C) I

B) III D) All are equivalent. 39) ______

39) Which formula is not equivalent to the other two?

II.

A) I C) II Express the sum in sigma notation. 40) 1 - 3 + 9 - 27 + 81 A)

III. B) All are equivalent. D) III

40) ______ B)

41) 7 + 8 + 9 + 10 + 11 + 12 A)

41) ______ B)

D) k 42) ______

42) + A)

+ B)

43) ______

43) 6 + 12 + 18 + 24 + 30 A)

44) ______

44) -

+ A)

Find the value of the specified finite sum. 45) Given A) -32

= -4 and

45) ______

= 8, find B) -4

. C) 32

D) 4 46) ______

46) Given A) 35

= -7 and

= 5, find B) 12

. C) -12

D) -14 47) ______

47) Given A) 1

= -7, find

. B) -70

C) 70

D) -1 48) ______

48) Given A) -1

= 6, find

. B) 0

C) 36

D) 1 49) ______

49) Given A) 14

= 2 and

= 6, find B) -10

. C) 10

D) -14

Evaluate the sum. 50)

A) 33

50) ______

B) 11

C) 66

D) 132 51) ______

51)

A) 1000

B) 1210

C) 385

D) 3025 52) ______

52)

A) 30 53)

B) 12

C) 14

D) 26 53) ______

A) 140

B) 161

C) 158

D) 147

Graph the function f(x) over the given interval. Partition the interval into 4 subintervals of equal length. Then add to your

sketch the rectangles associated with the Riemann sum

, using the indicated point in the kth subinterval for

. 54) ______

54) f(x) = 2x + 4, [0, 2], left-hand endpoint

55) f(x) = -2x - 1, [0, 2], left-hand endpoint

55) ______

56)

f(x) =

56) ______

- 1, [0, 8], midpoint

57)

f(x) =

57) ______

- 2, [0, 8], left-hand endpoint

58)

f(x) =

58) ______

- 1, [0, 8], right-hand endpoint

59)

f(x) = -3

59) ______

, [0, 4], midpoint

60)

f(x) = -2

60) ______

, [0, 4], left-hand endpoint

61) ______

61) f(x) = cos x + 4, [0, 2π], left-hand endpoint

62) f(x) = cos x + 4, [0, 2π], right-hand endpoint

___ ___

62)

63) f(x) = cos x + 4, [0, 2π], midpoint

63) ______

Find the norm of the partition P. 64) P = {1, 2.3, 2.8, 3.4, 5.2, 6} A) 1.8 65) P = {-3, -2, -0.7, 0, 0.2, 1} A) 0.7

64) ______ B) 1.3

C) 0.5

D) 0.6 65) ______

B) 1.3

C) 0.2

D) 0.3

Find a formula for the Riemann sum obtained by dividing the interval [a, b] into n equal subintervals and using the right-hand endpoint for each . Then take a limit of these sums as 66) f(x) = 3 + 2 over the interval [0, 3]. A) B) 6+

; Area = 27

to calculate the area under the curve over 66) ______

; Area = 33

D) 6+

; Area = 6

67) f(x) = 3x + 1 over the interval [0, 3]. A) 3+

67) ______ B)

; Area =

; Area = 15

D) 3-

; Area = -

Express the limit as a definite integral. 68)

___ ___

68) △ , where P is a partition of [-3, 4] A)

B) dx

D) dx

dx 69) ______

69) △

, where P is a partition of [-2, 4] B) C)

70) ______

70) △

, where P is a partition of [1, 7] B) C)

71) ______

71) △

, where P is a partition of [2, 3] B) C)

A) dx

D) dx

dx 72) ______

72) △ A)

, where P is a partition of [8, 10] B)

D) dx

dx 73) ______

73) △ A)

, where P is a partition of [-2, 4] B)

74) ______

74) △

, where P is a partition of [-π/4, 0] B) C)

A) dx

dx 75) ______

75) △ A)

, where P is a partition of [0, 2π] B)

D) dx

dx 76) ______

76) △ A)

, where P is a partition of [-4π, 4π] B)

D) dx

dx 77) ______

77) △ A)

, where P is a partition of [0, 5π] B)

D) dx

Solve the problem. 78) Suppose that A) 0; -4

78) ______ = 4. Find B) 4; -4

and

. C) 4; 4

D) 0; 4 79) ______

79) Suppose that A)

= -4. Find B) -8; 4

and C) -2; -4

. D) 2; 4

-8; 80)

Sup pose

that

___ ___

80)

= -11. Find

and

. A) 0; -11

B) -1; 11

C) 1; 11

D) 1; -11 81) ______

81) Suppose that f and g are continuous and that

A) 13

B) 1

and

C) 27

D) 21 82) ______

82) Suppose that f and g are continuous and that

A) -12

B) 23

and

C) -48

D) -33 83) ______

83) Suppose that f and g are continuous and that

A) -7

B) 13

and

C) 7

D) -13 84) ______

84) Suppose that h is continuous and that

and A) -13; 13

B) 13; -13

= 5 and

C) 3; -3

Find

D) -3; 3

85)

85) Suppose that g is continuous and that

and

Find .

______ A) 22

B) -22

C) -12

D) 12 86) ______

86) Suppose that f is continuous and that A) -16 B) 8

and C) 16

Find D) -8

87) ______

87) Suppose that f is continuous and that A) -6 B) -3

and C) 18

Find D) -18

Graph the integrand and use areas to evaluate the integral. 88)

A) 7

B) 21

88) ______

C) 42

D) 30 89) ______

89)

A) 16

B) 32

C) 128

D) 64 90) ______

90)

A) 112

B) 8

C) 28

D) 56 91) ______

91)

A) 8

B) 128

C) 64

D) 32 92) ______

92)

A) 100

B) 20

C) 200

D) 50 93) ______

93)

A) 36

B) 48

C) 12

D) 72 94) ______

94)

A) 41

D) 9

95) ______

95)

A) 3

B) 6

C) 9

D) 5

96) ______

96)

A) 18

B) 36

C) 72

D) 108 97) ______

97)

A) 16

B) 8π

C) 4π

D) 16π

Evaluate the integral. 98)

98) ______

-3

B) 2

C) 4

D) - 2 99) ______

99)

B) π2

100) _____

100)

A) 5184

D) -

101) _____

101)

B) 121

102) _____

102)

103) _____

103)

A) -46

B) 40

C) 48

D) 0 104) _____

104)

A) 27

B) 15

C) -6

D) 6

105) _____

105)

A) -

C) -

D) -

106) _____

106)

B) 1168

C) 78

D) 592

107) _____

107)

D) -

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]. 108) 108) _____ y = 27 A) 54b B) -54b C) D) 9 -9 109)

109) _____

y = 24π A) 48πb

-8π

C) -48πb

D) -8b

8π 110) _____

110) y = 8x A) 8b

-4

111) _____

111) y= A)

+2 B)

C) -

+ 2b

-2

Find the average value of the function over the given interval. 112) f(x) = 10x on [7, 9] A) 160 B) 40 113) f(x) = 6 - x on [0, 6] A) 108 114) f(x) = -2x + 10 on [-10, 5] A) 5 115) f(x) = 2x + 6 on [-3, 3] A) 12 116) f(x) = A) 36

D) +2

- 2b

112) _____ C) 80

D) 320

C) 18

D) 3

113) _____ B) 6

114) _____ B) 225

C) 15

D) 30 115) _____

B) 3

C) 36

D) 6 116) _____

on [-6, 6] B)

117) f(x) =

117) _____ B) 45

118) f(x) = 7 -

D) 15

118) _____

on [-7, 7]

A) 49

120)

+ 6 on [-3, 3]

119)

y= A)

- 3x + 6; [0, 9]

y=5A)

; [-1, 3]

D) 7

119) _____ B)

D) 60

120) _____ B)

Compute the definite integral as the limit of Riemann sums. 121)

A) p(r - q)

B) r - q

121) _____

D) p p 122) _____

122) , a<b A)

B) 27

D) 9(n - m)

3 123) _____

123)

A) - 3

C) -

D) 0 124) _____

124) dx A) 27

C) 0

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 125) 125) ____________ Integrate using the substitution u = cot x and using the substitution Show the results are the same.

126) ____________

126) What values of a and b maximize the value of

dx? 127) ____________

127) What values of a and b minimize the value of

dx? 128) ____________

128) Explain why the rule

= b - a holds for any constants a and b.

129) Use the max-min inequality to show that if f is integrable and f(x) ≤ 0 on [a, b], then

129) ____________

130)

130) ____________ Use the max-min inequality to find upper and lower bounds for the value of

dx. 131) ____________

131) Use the max-min inequality to find upper and lower bounds for

. Add these to arrive at an estimate of

and

. 132) ____________

132) Use the max-min inequality to show that 1 ≤

≤ 2.

133) A plane averaged 300 mph on a 600 mile trip and then returned over the same 600 miles

133) ____________

at the rate of What was the plane's average speed for the entire trip? Give reasons for your answer. 134) Derive a formula for the area of a circumscribed regular n-sided polygon for a circle of radius r.

134) ____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the integral. 135)

135) _____

dx A) 10

C) 60

D) 40

136) _____

136) dx A) 1215

B) -45,927

C) 275,562

D) 45,927

137) _____

137)

A) 99

D) 2145

138) _____

138) dx A)

139) _____

139)

D) 32

140) _____

140)

A) 1

B) 14

C) 0

D) -14 141) _____

141)

A) 32

B) 16

C) -16

D) -32 142) _____

142)

A) -18

B) 9

C) 0

D) -9 143) _____

143)

A) -9

B) 0

C) 18

D) -18 144) _____

144)

B) -

+ 145) _____

145)

Find the derivative.

B) 3

C) 4

D) 2

146) _____

146)

-cos (

)-1

sin (

)

D) sin (

sin (

)

) 147) _____

147)

-3

9 148) _____

148)

149) _____

149)

A) C)

θ cot θ -

θ cot θ

(cot θ) -

(cot θ) 150) _____

150) y= A)

151) _____

151) y= A)

152) _____

152) y= A)

cos (

)-1

sin (

)

cos (

)

cos (

) 153) _____

153) y= A)

B) cos (

cos (

)

) )

7 cos (

)-7

sin (

)

154) _____

154) y= A)

sec x

x 155) _____

155) y= A)

-8

cos (

)

cos (

)

1 - cos (

)

-sin (

) 156) _____

156) y= A)

-1

157) _____

157) y= A) C)

ln 7

Find the total area of the region between the curve and the x-axis. 158) y = 2x + 7; 1 ≤ x ≤ 5 A) 26 B) 18 C) 52 159)

160)

y= A)

- 6x + 9; 2 ≤ x ≤ 4

y = 2x A)

; 0≤x≤2

158) _____ D) 9 159) _____

160) _____ B)

161) _____

161) y= ; 1≤x≤3 A)

162)

163)

y=A)

+ 9; 0 ≤ x ≤ 5

+ 1; 0 ≤ x ≤ 1

C) 3

162) _____ B)

163) _____

164)

y= A)

164) _____

; 0≤x≤2 B)

165) _____

165) y= ; 1≤x≤4 A) 4

166)

y= A)

; 0≤x≤2

167) y = (x + 4)

; 1≤x≤9

D) 2

166) _____ B) 136

C) 544

D) 48

167) _____ B) 346

C) 761

Find the area of the shaded region. 168)

A) 7.5

168) _____

B) 5

C) 12.5

D) 10 169) _____

169)

170)

170) _____

B) 3

D) 5

171) _____

171)

172) _____

172)

A) 36

B) 45

C) 18

D) 27 173) _____

173)

174) _____

174)

A) 1.69 175)

B) 1.39

C) 1.25

D) 1.50

____ _

175)

176) _____

176)

A) 2 + 2π

B) 2

C) 2π

D) π 177) _____

177)

A) π - 2

B) π

C) π + 2

D) 2

Solve the initial value problem. 178)

178) _____

= csc x, y(4) = -5 A)

y=-

-5

D) y=

-5

179) _____

179) = cot x, y(-3) = 7

B) y=

y=-

+ -3

D) y=

180) _____

180) = 15 A)

x cos x, y(0) = 2 B) y = 30 sin x cos x + 2

y= C)

y=5

x+2 D)

x+2

y=-5

x-2 181) _____

181) = sin (3x + π), y(0) = 3 A)

y=cos (3x + π) + 3 C) y = 3 cos (3x + π) + 3

y=cos (3x + π) + D) y = - cos (3x + π) + 2 182) _____

182) =x A)

, y(0) = 0

- 625

y= D)

183) _____

183) = A)

, y(0) = 6 B) y=

y=-

D) y=

Solve the problem. 184) A certain company has found that its expenditure rate per day (in hundreds of dollars) on a

184) _____

certain type of job is given by where x is the number of days since the start of the job. Find the expenditure if the job takes 7 days. A) $52 B) $5200 C) $217 D) $21,700 185) After a new firm starts in business, it finds that its rate of profits (in hundreds of dollars per year) after t years of operation is given by operation. A) $3400 B) $1600

Find the profit in year 2 of the C) $2600

D) $3300

186) In a certain memory experiment, subject A is able to memorize words at a rate given by

185) _____

____ _

186)

In the same memory experime nt, subject B is able to memoriz e at the rate given by

How many more words does subject B memoriz e from t = 0 to t = 14 (during the first 14 minutes) ? A) -5

B) 28

C) -16

D) 34

Provide an appropriate response. 187) Suppose that A) 5 + 7x - 4 C) +

187) _____ =5

+ 7x - 4. Find f(x). B) 10x + 7 D)

- 4x

- 4x - 8 188) _____

188) Find f(π) if A) 2π

= x sin 2x. B) 0

C) π

D) 2 189) _____

189) Find the linearization of f(x) = 2 +

at x = 0.

A) 2x + 1

B) 2

D) x + 2

x+2

190) Suppose that f has a positive derivative for all values of x and that f(2) = 0. Which of the

following statements must be true of the function g(x) = A) The graph of g crosses the x-axis at x = 2. B) The function g has a local minimum at x = 2. C) The function g has a local maximum at x = 2. D) The graph of g has an inflection point at x = 2.

190) _____

191) _____

191) What definite integral is represented by evaluate the integral. A)

? Use the Fundamental Theorem to B)

= 21

D) =

= 64

Evaluate the integral using the given substitution. 192) dx, u=9 A)

192) _____ B)

sin (u) + C C)

sin(9

sin (9

)+C

sin (9

)+C

) +C

193) _____

193) dt,

u = 2 - sin

B) sin

D) +C

+C 194) _____

194) dx ,

u = 5x - 2

B) +C

195) _____

195) dx , A) +C

-5 B)

C +

-5+C

D) + C 196) _____

196) , A)

u=1B)

-14

+C C)

-14

197) _____

197) dy ,

+4 B)

+C C)

+C D)

(10

+ 6y) + C 198) _____

198) dx ,

-4 B)

sin (

- 4) + C

(

- 4) +

sin 2(

- 4) + C

D) -4+

sin 2(

- 4)+ C

(

- 4) + C 199) _____

199) dx ,

u=-

B) -

sin

D) -

sin

+C 200) _____

200) ,

u = cot 5θ

B) -

5θ + C

D) 5θ

5θ + C

θ+C 201) _____

201) ,

u = 7x + 9

B) 2

+C C)

D) +C

Evaluate the integral. 202)

202) _____

B) -

(7x2 + 3)-6 + C

(7x2 + 3)-4 + C

(7x2 + 3)-6 + C

D) -

(7x2 + 3)-4 + C

203) _____

203) dx A)

+C 204) _____

204) dx A)

B) -

+C 205) _____

205) A)

D) ln

+C 206) _____

206) dx A)

-2

+C C)

+C 207) _____

207) A)

cos (2x - 6) + C C) -cos (2x - 6) + C

cos (2x - 6) + C D) 2 cos (2x - 6) + C 208) _____

208) A) 12 csc (6θ + 7) cot (6θ + 7) + C C) 6 cot (6θ + 7) + C

B) -cot (6θ + 7) + C D) -

cot (6θ + 7) + C 209) _____

209) dz A)

B) csc +C

cs c

+ C D)

C) -cot

-csc

+C 210) _____

210) A)

B) +C

D) +C

+C 211) _____

211) dt A)

B) 4 cos

cos

D) -cos

cos

+C 212) _____

212) A) C) ln

+C 213) _____

213) A)

B) +C

C) +C

D) +C

+C 214) _____

214) A)

B) +C

x +C

D) +C

215) _____

215) A)

B) + C

+ C

D) + C

+ C 216) _____

216)

-2

( ) +C

( ) +C 217) _____

217)

D) +C

+C 218) _____

218) dx A)

+C 219) _____

219) dx A)

220) _____

220) dt A)

-5

+C 221) _____

221) A)

t+C

B) ln

D) +C

ln + C

Evaluate the integral by using multiple substitutions. 222)

222) _____

dr A)

B) sin

-sin

D) sin

223) _____

223) dy A)

B) +C

+C C) 2 sin

+C 224) _____

224) dx A)

B) +C

+C C)

D) +C

+C 225) _____

225) dt A)

C) -

D) +C

+C 226) _____

226) dx A)

B) +C

+C C)

D) +C

+C 227) _____

227) dx A)

+C C)

D) +C

+C 228) _____

228) dx A)

B) (3

)+C

(

- 4x) + C

(

- 4x)+ C

- 4x) + C

229) _____

229) dy A)

+C C)

D) +C

Solve the initial value problem. 230)

230) _____

= x cos (7 ), y(0) = 5 A) y = sin (7 ) + 5 C)

B) y=

sin(7

)+5

sin (u)

D) y=

sin(7

)+5

231) _____

231) = 12t A)

s(1) = 4 B)

- 12

-4

D) s=

232) _____

232) =6 A)

y(0) = 1 B)

y=-

D) y=-

+1 -5

+1 233) _____

233) = A)

y(1) = 5 B)

-3+

D) y=

+ 234) _____

234) = A)

7θ cot 7θ,

=3 B)

r=-

7θ +

7θ

θ+

7θ + 3

D) r=-

7θ +

235) _____

235) = (14t + 3) sin (7 + 3t), A) s = sin (7 + 3t) - 6 C) s = -cos

s(0) = -6 B) D)

s = cos (7

+ 3t) - 7

s = -cos (7

+ 3t) - 5

236) _____

236) = -128cos (8t), (0) = 7, s(0) = 8 A) s = 2 cos (8t) + 7t + 6 C) s = 16 cos (8t) + 7t - 8

B) s = -16 sin (8t) + 7 D) s = 2 cos (8t) + 6 237) _____

237) = -36sin A)

(0) = 13, s(0) = -2 B)

s = sin

+ 13t - 3

s = 6cos

+ 13

s = sin

-3

D) s = sin

+ 12t - 2

Solve the problem. 238) 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 = -20t + 2, s(0) = 15 A) B) s = -20 + 2t + 15 s= 10 + 2t - 15 C) D) s = -10 + 2t + 15 s = -10 + 2t - 15 239) 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 A) s=

238) _____

239) _____

t, s(0) = 1 B) sin

t+π

C) s = 2π sin

s= sin D) s = sin t

240) 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= sin

____ _

240) , s( =2

) A)

B) s = -2 cos

s = -2 cos

D) s = 2 cos

241) 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 = 10, v(0) = 9, s(0) = 1 A) B) C) D) s = 10 + 9t + s = 5 + 9t s = 5 + 9t + 1 s = -5 - 9t + 1 1

241) _____

242) 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) = -7, s(0) = -12 A) B)

242) _____

sin 5t - 7t - 12

s=-

sin 5t - 7t - 12

s=-

cos 5t - 7t - 12

D) s=

cos 5t + 7t - 12

Use the substitution formula to evaluate the integral. 243)

243) _____

dx A) 5

-8

D) -

244) _____

244)

D) -

245) _____

245)

-3

B) - 2

C) 2

-6

D) 246) _____

246)

A) 9

247)

B) 12

D) 4

247) _____

B) 11

D) 22

248) _____

248)

B) -

C) -

D) 249) _____

249)

250) _____

250)

251) _____

251)

B) -5

A) 10

C) 5

D) 20 252) _____

252)

D) -

253) _____

253) 2x dx B)

C) e

D) 2e

254) _____

254)

A) -ln 2

B) 0

C) ln 3

D) -ln 3 255) _____

255) dx A)

256)

____ _

256)

B) ln 2

D) 257) _____

257)

A) 10 ln 3

B) -5 ln 3

C) -10 ln 3

D) 5 ln 3 258) _____

258)

A) 3 ln 2

B) 3 ln 3

C) 6 ln 2

D) -3 ln 2 259) _____

259)

B) -

260) _____

260)

9 261) _____

261)

B) -

Find the area of the shaded region. 262) f(x) =

C) -

- 6x

_____ A)

262)

263)

f(x) = -

263) _____

+ 16x

264)

264) _____

- 4x + 3

D) 3

265) _____

265)

y=-

266)

y=2

+x-6

C) 2

266) _____

-4

267) _____

267)

268)

B) 32

268) _____

- 32

y=A)

269) _____

269)

A) 8

D) 4

4+ 270) _____

270)

B) 2 2-

271)

D) 2+

A) 1 +

C) 2 + π

271) _____

Find the area enclosed by the given curves. 272) y = 2x - x2, y = 2x - 4 A) B)

272) _____ C)

273) _____

273) y = x3, y = 4x A) 2

B) 4

C) 8

D) 16 274) _____

274) y = x, y = x2 A)

275) _____

275) y= x2, y = -x2 + 6 A) 16

B) 8

C) 4

D) 32 276) _____

276) y = - 4sin x, y = sin 2x, 0 ≤ x ≤ π A) 8 B)

C) 4

D) 16

277) _____

277) y = sin x, y = A)

≤x≤ B)

C) +

D) -

278) _____

278) y= A) π

x, y =

x, x =

, and x = B)

279) Find the area of the region in the first quadrant bounded by the line y = 8x, the line x = 1, the curve A)

and the x-axis. B) 6

279) _____

280) _____

280) Find the area of the region in the first quadrant bounded on the left by the line y = right by the curves y = A) 0.5858

x and y = B) 0.4126

and on the

x. (Round to four decimal places.) C) 0.3094 D) 4.3094

281) Find the area of the region in the first quadrant bounded on the left by the y-axis, below by the

line above left by

____ _

y = x + 4, 281) and above right by y=+ 10. A)

C) 15

282) Find the area between the curves y = ln x and y = ln 2x from x = 1 to x = 5. A) ln 2 B) ln 16 - 8 C) ln 16

282) _____ D) ln 32

283) Find the area of the "triangular" region in the first quadrant that is bounded above by the curve A)

284)

285)

, below by the curve y = , and on the right by the line x = ln 5. B) 19 C) 5 ln 5

D) 8

Find the area of the region between the curve y = 6x/(1 + ) and the interval of the x-axis. A) 0 B) C) 6 ln 10 D) ln 10 6 Find the area of the region between the curve y = A) 343 B) 336 ln 7

283) _____

and the interval 0 ≤ x ≤ 2 on the x-axis. C) D)

Answer each question appropriately. 286)

284) _____

285) _____

286) _____

Suppose the velocity of a body moving along the s-axis is = 9.8t - 9. 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) 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. B) No, displacement has nothing to do with the position of the body. 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, integration is not possible without knowing the initial position. 287) If differentiable functions y = F(x) and y = G(x) both solve the initial value problem on an interval I, must for every x in I? Justify the answer. A) F(x) = G(x) for every x in I because integrating f(x) results in one unique function. B) There is not enough information given to determine if F(x) = G(x). C) 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. 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.

287) _____

288) 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) -44 square units B) 55 square units C) 44 square units D) 66 square units

288) _____

289) Suppose the area of the region between the graph of a positive continuous function f and the

289) _____

x-axis from x = a to x = b is 7 square units. Find the area between the curves A) 35 square units C) -21 square units

and

B) -35 square units D) 21 square units 290) _____

290) Which of the following integrals, if any, calculates the area of the shaded region?

B) dx

C) dx

D) dx

dx 291) _____

291) Suppose that A) 12

= 12. Find B) 60

, if f is odd. C) -60

D) -12 292) _____

292) Suppose that A) 8

= 8. Find B) 32

, if f is even. C) -32

D) -8

1) B 2) C 3) B 4) A 5) D 6) C 7) B 8) A 9) A 10) B 11) A 12) D 13) C 14) A 15) C 16) D 17) A 18) D 19) A 20) D 21) B 22) D 23) C 24) D 25) C 26) B 27) D 28) C 29) C 30) A 31) C 32) C 33) D 34) D 35) B 36) B 37) A 38) B 39) C 40) B 41) D 42) A 43) C 44) C 45) D 46) C 47) B 48) D 49) B 50) C 51) D

52) C 53) D 54) D 55) D 56) D 57) A 58) A 59) C 60) B 61) A 62) D 63) B 64) A 65) B 66) B 67) A 68) B 69) A 70) C 71) B 72) C 73) A 74) A 75) B 76) D 77) D 78) A 79) B 80) D 81) B 82) D 83) D 84) D 85) C 86) D 87) D 88) C 89) D 90) D 91) C 92) A 93) A 94) B 95) D 96) B 97) B 98) B 99) C 100) B 101) C 102) A 103) B

104) B 105) B 106) D 107) C 108) C 109) D 110) B 111) A 112) C 113) D 114) C 115) D 116) D 117) A 118) B 119) A 120) C 121) A 122) B 123) A 124) C 125) If u = cot x and du = -

x dx, then

then

and

from

126) a = 0 and b = 2 127) a = -2 and b = 2 128) =

= b - a. The integral represents the area of the rectangle under

which equals 129) If f(x) ≤ 0 on [a, b], then max f ≤ 0. So

≤ max f ∙ (b - a) ≤ 0.

130) If f(x) =

on [1, 3], then max f = 1 and min f =

So min f ∙ (3 - 1) ≤ lower bound is

dx ≤ max f ∙ (3 - 1) and

and

, and the upper bound is 2.

131) If f(x) =

on [1, 2], then max f = 1 and min f =

So min f ∙ (2 - 1) ≤

dx ≤ max f ∙ (2 - 1) or

≤

dx ≤ 1.

So the

If f(x) and min f = . = on [2, 3], So min f ∙ (3 - 2) ≤ then max So f= 132)

≤

If f(x) =

dx ≤ max f ∙ (3 - 2) or

dx ≤ 1 +

on [0, 1], then max f =

So min f ∙ (1 - 0) ≤

≤

dx ≤

and min f = 1.

≤ max f ∙ (1 - 0) or 1 ≤

≤

≤ 2.

133) Since the plane averaged 300 mph on the first 600 miles, the first part of the trip took trip took

134) A=n 135) B 136) D 137) B 138) C 139) C 140) B 141) D 142) A 143) D 144) C 145) A 146) D 147) D 148) D 149) D 150) A 151) D 152) C 153) A 154) B 155) A 156) C 157) A 158) C 159) A 160) B 161) A 162) A 163) D

tan

The entire trip of 1200 miles took

So the average speed was

The return

164) D 165) D 166) B 167) A 168) D 169) C 170) A 171) B 172) A 173) C 174) B 175) A 176) C 177) A 178) B 179) C 180) C 181) B 182) D 183) D 184) D 185) A 186) A 187) B 188) A 189) D 190) B 191) C 192) D 193) D 194) A 195) D 196) C 197) A 198) B 199) A 200) B 201) C 202) C 203) D 204) B 205) D 206) D 207) B 208) D 209) D 210) B 211) B 212) D 213) C 214) C 215) C

216) B 217) B 218) C 219) B 220) C 221) B 222) A 223) B 224) D 225) C 226) C 227) D 228) D 229) D 230) B 231) A 232) B 233) D 234) A 235) D 236) A 237) A 238) C 239) B 240) B 241) C 242) D 243) D 244) C 245) C 246) A 247) B 248) B 249) A 250) A 251) A 252) C 253) A 254) A 255) D 256) A 257) D 258) A 259) C 260) C 261) A 262) B 263) C 264) A 265) D 266) B 267) D

268) D 269) D 270) D 271) D 272) A 273) C 274) C 275) A 276) A 277) C 278) C 279) D 280) C 281) D 282) C 283) D 284) C 285) C 286) C 287) D 288) C 289) A 290) D 291) D 292) A

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the volume of the described solid. 1) The solid lies between planes perpendicular to the x-axis at and . The cross sections perpendicular to the x-axis between these planes are squares whose bases run from the parabola to the parabola A) 640

. B) 162

C) 648

D) 324 2) _______

2) The solid lies between planes perpendicular to the x-axis at

and . The cross sections perpendicular to the x-axis between these planes are squares whose bases run from the semicircle A) 18

to the semicircle B) 36

The base of the solid is the disk between

and B)

. C) 144

D) 72 3) _______

≤ 4. The cross sections by planes perpendicular to the

are isosceles right triangles with one leg in the disk. C) D)

4) _______

4) 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 semicircles whose diameters run from A)

B) π

C) π

to D)

π 5) _______

5) 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 = the parabola y = 32 A)

1) _______

. B)

6) The base of a solid is the region between the curve y = 4cos x and the x-axis from x = 0 to

6) _______

The cross sections perpendicular to the x-axis are squares with bases running from the x-axis to the curve. A) 2π B) 8π C) D) 4π π 7) The base of a solid is the region between the curve y = 3cos x and the x-axis from x = 0 to x = π/2. The cross sections perpendicular to the x-axis are isosceles right triangles with one leg on the base of the solid. A) B) C) D) 2π π

7) _______

8) The base of a solid is the region between the curve y = 6cos x and the x-axis from x = 0 to x = π/2. The cross sections perpendicular to the x-axis are squares with diagonals running from the x-axis to the curve. A)

8) _______

3π

9π

π π 9) _______

9) The solid lies between planes perpendicular to the x-axis at x = π/6 to x = π/2. The cross sections perpendicular to the x-axis are circular disks with diameters running from the curve to the curve A)

B) -

(

- 1) π +

- 2) π -

D) -

10) 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 the curve A) 81

to D) 81π

Find the volume of the solid generated by revolving the shaded region about the given axis. 11) About the x-axis

B) 30π

11) ______

D) π

π 12) ______

12) About the x-axis

B) π

10) ______

C) 18π π

D) π

13) ______

13) About the x-axis

A) 36π

B) 21π

C) 6π

D) 18π 14) ______

14) About the y-axis

A) 42π

B) 49π

D) 21π π 15) ______

15) About the y-axis

A) 50π

C) 625π π

16) About the y-axis

D) 25π

___ ___

16)

28π - 7

C) 28 - 7π

7π + 7

14π 17) ______

17) About the y-axis

A) 28π

- 14π

- 28π

+ 7π 18) ______

18) About the y-axis

B) 18π π

D) π

19) ______

19) About the x-axis

B) π

π 20) ______

20) About the x-axis

- 3π

- 9π

+ 9π

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the 21) y = x, y = 0, x = 1, x = 5 A) 13π B) 12π C) D) π 22) y =

22) ______ B)

π 23)

C) 27π

23) ______ B) 9π

25)

D) π

D) 9π

y= , y = 0, x = 0, x = 3 A)

24) y =

, y = 0, x = 0, x = 9

. 21) ______

, y = 0, x = 0, x = 1 B) 4π

π 24) ______

C) π

D) 2π

___ ___

25) , y = 0, x = 1, x = 5 A) πln 5

π 26) ______

26) y = x + 1, y = 0, x = -1, x = 3 A) 2π B)

C) 16π

π 27)

y= A)

, y = 0, x = 0, x = 5 B) 100π

π 27) ______ C)

D) 10π

π 28) ______

28) y= , y = 0, 0 ≤ x ≤ A) 10π

C) 5π

D) 2π

π 29) ______

29) y = 4csc x, y = 0, x = A) 48π

,x= B) 32π

30) y = 8cos πx, y = 0, x = -0.5, x = 0.5 A) 32π B) 128π

C) 8π

D) 16π 30) ______

D) 64π π 31) ______

31) y= A)

, y = 0, x = 1, x = 5 B) π(ln 5)

(ln 5) 32) y = 4x, y = 4, x = 0 A)

32) ______ B) 12π

C) 2π

π 33) y = - 6x + 12, y = 6x, x = 0 A) 216π

π 33) ______ B) 36π

C) 72π

D) 12π 34) ______

34) y =

, y = 2, x = 0 A) 4π

B) 8π

D) 2π π

35)

y= , y = 16, x = 0 A) π

35) ______ B)

D) π π 36)

36) ______

y= + 2, y = 4x + 2 A) 64π

π 37) ______

37) y= ,y=-x+9 A) 243π

C) 56π

π 38) ______

38) y= A)

, y = 1, x = 0 to x = B)

+π

-π 39) ______

39) y = 7csc x, y = 7 , A) 49 - 98π

≤x≤ B)

+ 14π

+ 98π

- 49π 40) ______

40) y = 7cos (πx), y = 7, x = -0.5, x = 0.5 A) 49π B) 98π

D) π

41) ______

41) y = sec x, y = tan x, x = 0, x = A) B)

Find the volume of the solid generated by revolving the region about the given line. 42) The region bounded above by the line , below by the curve the line A)

, about the line B) π

C) π

, and on the right by

42) ______

D) π

π 43) ______

43) The region in the first quadrant bounded above by the line y = 7, below by the line y = on the left by the y-axis, about the line y = 7 A) B) C) D) π

44) The region in the first quadrant bounded above by the line y = 3, below by the curve y = and on the left by the y-axis, about the line y = 3 A)

, and

44) ______

3π

π π

π 45) ______

45) The region in the first quadrant bounded above by the line y = 1, below by the curve and on the left by the y-axis, about the line y = 1 B) C)

46) The region bounded above by the line

, below by the curve

line , and on the right by the line x = 0.5, about the line A) 75π - 50 B) C) π

π - 100

47) The region in the first quadrant bounded above by the line y = 2, below by the curve y = and on the left by the y-axis, about the line x = -1 A) B) π

46) ______

, on the left by the

47) ______

D) π

48) The region in the first quadrant bounded above by the line 3x + y = 6, below by the x-axis, and on the left by the y-axis, about the line x = -2 A) 24π B) 5π C) 32π D) 64π

48) ______

49)

49) ______

50)

The region in the second quadrant bounded above by the curve y = 4 and on the right by the y-axis, about the line x = 1 A) 8π B) C)

, below by the x-axis,

The region in the first quadrant bounded above by the line y = 3 right by the line x = 1, about the line y = - 1 A) B) C) π

51) The region in the first quadrant bounded above by the line y = 1, below by the curve y = and on the left by the y-axis, about the line y = - 1 A) B) -

π-

The region enclosed by x = , x = 0, y = - 2, y = 2 A) B) C)

53)

The region enclosed by x = , x = 0, y = 8 A) 32π B)

51) ______

D) +

Find the volume of the solid generated by revolving the region about the y-axis. 52)

50) ______

, below by x-axis, and on the

52) ______ D)

π 53) ______

16 π

12 π 54) ______

54) The region enclosed by x = , x = 0, y = 1, y = 6 A) B) π

π 55) ______

55) The region enclosed by x = 2tan , x = 0, y = A) B) 20 - 5π

20π - 5

5π + 5

10π 56) ______

56) The region enclosed by x = ,0≤y≤ A) B) 6π

,x=0 C)

D) 3π

57) ______

57) The region enclosed by the triangle with vertices (4, 0), (4, 2), (6, 2) A) B) C) π

π 58) ______

58) The region enclosed by the triangle with vertices (0, 0), (5, 0), (5, 2) A) 100π B) C) π 59)

60)

The region in the first quadrant bounded on the left by the circle line , and above by the line y = 2 A) 8π B) 2π

= 4, on the right by the

59) ______

D) π

The region in the first quadrant bounded on the left by y = below by the x-axis A) 64π B) C)

, on the right by the line

, and

60) ______

D) π

π 61) ______

61) The region in the first quadrant bounded on the left by y = above by the line y = 2 A) B) 3π C) 9π

, on the right by the line

π Solve the problem. 62) The disk

(Hint:

, and

D) π

≤ 9 is revolved about the y-axis to generate a torus. Find its volume.

π, since it is the area of a semicircle of radius 3.)

62) ______

C) 54π

108

63) The hemispherical bowl of radius 5 contains water to a depth 2. Find the volume of water in the bowl. A) B) C) 59π D) π 64)

A water tank is formed by revolving the curve y = 7 about the y-axis. Find the volume of water in the tank as a function of the water depth, y. A) B) V(y) =

64) ______

V(y) =

D) V(y) =

65)

63) ______

V(y) =

A water tank is formed by revolving the curve y = 7 about the y-axis. Water drains from the 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 A) B) =

.) C)

D) =

66) A right circular cylinder is obtained by revolving the region enclosed by the line x = r, the x-axis, and the line A) 2π h

65) ______

about the y-axis. Find the volume of the cylinder. B) πrh C) π h

πr

67) A frustum of a right circular cone has a height of 10 m, a base of radius 5m, and a top of radius 4m. Find its volume. A) 610π B) 61π C) D) π

66) ______

67) ______

68) An auxiliary fuel tank for a helicopter is shaped like the surface generated by revolving the

68) ______

curve - 3 ≤ x ≤ 3, about the x-axis (dimensions are in feet). How many cubic feet of fuel will the tank hold to the nearest cubic foot? A) 5 B) 3 C) 13 D) 10 69) An auxiliary fuel tank for a helicopter is shaped like the surface generated by revolving the

69) ______

curve - 3 ≤ x ≤ 3, about the x-axis (dimensions are in feet). If a cubic foot holds 7.481 gallons 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) 56 B) 226 C) 75 D) 113 70) Find the volume that remains after a hole of radius 1 is bored through the center of a solid sphere of radius 3. A) 36π B) C) 18π D) π

70) ______

71) Find the volume that remains after a hole of radius 1 is bored through the center of a solid cylinder of radius 2 and height 4. A) 12π B) 16π C) 6π D) 4π

71) ______

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. 72) About the x-axis 72) ______

B) 50π

D) π

π 73) ______

73) About the y-axis x=3

B) π

74) About the x-axis y=4

C) 12π

x=4

D) π

74)

______ A) 64π

C) π

D) π

π 75) ______

75) About the y-axis x=5

A) 50π

C) 75π

π 76) ______

76) About the y-axis x=4

A) 40π

B) 20π

C) 48π

D) 32π 77) ______

77) About the y-axis x=

A) 19π

C) 18π π

D) π

78) ______

78) About the y-axis

B) π

C) 32π

D) 64π

π 79) ______

79) About the x-axis

B) 8π

C) 12π

π 80) ______

80) About the y-axis

A) 3π 81) About the y-axis

B) 12π

C) 9π

D) 6π

___ ___

81)

A) 12π

B) 9π

C) 6π

D) 15π

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. 82) 82) ______ y = 6x, y = A)

,x=1 B)

C) 37π

83) y = 3x, y = 6x, x = 3 A) 54π

π 83) ______

B) 27π

C) 18π

D) π

84)

84) ______

y=9 ,y=9 A)

B) π

π 85)

y = 50 - , y = A) 625π

C) π

D) π

π 86) ______

,x=0 B)

C) π

87)

85) ______

y= , y = 5 + 4x, for x ≥ 0 A) B) π

86)

y = 8 , y = 8x, for x ≥ 0 A) π

D) 1250π π 87) ______

C) π

D) π

π 88) ______

88) y=

, y = 0, x = 1, x = 9

A) 112π 89) y = 8

B) 108π

C) 52π

D) 104π 89) ______

, y = 0, x = 1

B) 40π

D) 16π

π 90) ______

90) y = , y = 0, x = 5, x = 7 A) 12π 91)

y= A)

B) 24π

C) 8π

D) 16π 91) ______

- 3, y = 2x, x = 0, for x ≥ 0 B) 18π

C) 45π

π 92)

y=5 A)

B) 10

93)

92) ______

, y = 0, x = 0, x = 1 π

y=5 , y = 0, x = 0, x = 1 A) 10(e - 1) π

C) 5(e - 1) π 10

π 93) ______

B) 10e π

D) 5(e - 1) π (e - 1) π

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. 94) x = 4 , x = - 4y, y = 1 94) ______ A)

B) 8π

π 95)

x = 4 , x = - 4y, y = 4 A)

x = 3y A)

π 96) ______

C) π

D) 27π π 97) ______

,y=8 B) π

98) y = 3x, y = 6x, y = 3 A) π

,x=0

99) y =

π 97) y = 8

π 95) ______

π 96)

, y = 0, y = x - 6

π 98) ______

B) 9π

D) 3π π 99) ______

27 π

D) π

100)

100) _____

y=5 ,y=5 A)

π 101)

x=8- ,x= A) 8π

π 101) _____

,y=0 B) 16π

C) 32π

D) 4π 102) _____

102) x=4 ,x=4 A)

π 103)

D) 6π

π 103) _____

y = 7 , y = 7x, for x ≥ 0 A)

104) _____

104) x= A)

, y = 0, x = 0, y = 1 B) 7π(e - 1)

C) π(e - 1)

πe 105)

x=5 A)

(e - 1) 105) _____

, y = 0, x = 0, y = 1 B) 10

C) 5

D) 5(e - 1) π 10

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. 106) y = 4x, 106) _____ y = 0, x = 2; revolve about the x-axis A)

D) π

π 107)

y = 3x, A)

; revolve about the y-axis B)

π 108)

, A) 625π

107) _____ C)

π y = 25,

D) -

y = 0, A) - 36π

x = 3; revolve about the line x = -3 B) 180π C) 90π

π 108) _____

x = 5; revolve about the line B) C) π

109) y = 4x,

D) π

π 109) _____ D) 90

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line.

110) _____

110) About the line y = 6

A) 216π

B) 432π

C) 108π

D) 36π 111) _____

111) About the line y = 5

B) π

C) π

D) π

π 112) _____

112) About the line y = -1

B) π

C) π

D) π

113) _____

113) About the line y = 1

x=5

(solid)

B) π

C) π

D) π

π 114) _____

114) About the line y = -1

x=4

(solid)

B) π

C) π

D) π

π 115) _____

115) About the line y = 2

x=6

(solid)

B) π

C) 6π π

D) π

116) _____

116) About the line y = - 6 x=y+6

D) 192π π 117) _____

117) About the line y = 3 x=y+6

A) 99π

C) π

D) π

π 118) _____

118) About the line y = 4 x=

(solid)

B) π

119) About the line y = -1

C) π

D) π

____ _

119)

B) π

Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. 120) The triangle with vertices (0, 0), (0, 2), and (4, 2) about the line x = 4 120) _____ A) B) C) D) π

π 121) _____

121) The region bounded by x = 5

, x = - 5y, and y = 1 about the line y = 1 B) C) 5π

A) π

π 122) _____

122) The region bounded by y = 3

, y = 3, and x = 0 about the line y = 3 B) C)

A) π

D) 3π

π 123) _____

123) The region bounded by y = 4

, y = 4, and x = 0 about the line x = 1 B) C)

A) π

π 124) _____

124) The region bounded by y = 8

, y = 8, and x = 0 about the y-axis B) C)

A) π 125)

The region bounded by y = 3x A) 2π B)

π and y = x about the y-axis C)

D) π 125) _____ D) 4π

126)

127)

The region bounded by y = 7x and y = x about the line x = 6 A) 324π B) 108π C) 216π

129)

D) 162π 127) _____

The region in the first quadrant bounded by x = 6y and the y-axis about the y-axis A) B) C) D) π

128)

126) _____

π 128) _____

The region in the first quadrant bounded by x = 6y and the y-axis about the x-axis A) 108π B) 162π C) 324π D) 216π

129) _____

The region in the first quadrant bounded by x = 3y and the y-axis about the line x = -1 A) B) C) D) π

Solve the problem. 130) A bead is formed from a sphere of radius 3 by drilling through a diameter of the sphere with a drill bit of radius 1. Find the volume of the bead. A) 36π B) 18π C) D) π

130) _____

131) A water noodle is formed from a cylinder of radius 4 and height 8 by drilling through the diameter of the cylinder with a drill bit of radius 1. Find the volume of the water noodle. A) 8π B) 60π C) 128π D) 120π

131) _____

132) An auxiliary fuel tank for a helicopter is shaped like the surface generated by revolving the

132) _____

curve - 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 D) 5 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 133) The region shown here is to be revolved about the x-axis to generate a solid. Which of 133) ____________ 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=-

134) The region shown here is to be revolved about the line x = 3 to generate a solid. Which

the methods

(disk, 134) washer, shell) could you use to find the volume of the solid? How many integrals would be required in each case?

____ ____ ____

x=-

135) 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?

135) ____________

136) 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?

136) ____________

y = 4x137) 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?

137) ____________

138) 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?

138) ____________

139) The region bounded by the lines x = 2, x = 6, y = -2, and y = 1 is revolved about the y-axis to form a solid. Explain how you could use elementary geometry formulas to verify the volume of this solid.

139) ____________

140)

The first-quadrant

____ ____ ____

region 140) bounded by y = , the x-axis, and the y-axis is revolved about the y-axis to form a solid. Explain how you could use elementa ry geometry formulas to verify the volume of the solid.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the length of the curve. 141) y=3 A)

142)

141) _____

from x = 0 to x = B)

y= A)

from x = 1 to x = 27 B) 72

C) 1

142) _____ C) 24

D) 36

143) _____

143) y= A)

from x = 1 to x = 2 B)

144) _____

144) y= ( A)

-2

) from x = 1 to x = 8 B)

D) 23

145) _____

145) x=

from y = 1 to y = 3

146) _____

146) x= A) 7

from y = 1 to y = 4 B) 1

147) _____

147) x= A)

from x = 16 to x = 25 B) 32

C) 20

148) _____

148) y= A) 18

,3≤x≤6 B) 27

C) 9

149) _____

149) x= A)

,1≤y≤9 B) 20

150) _____

150) y= A)

,0≤x≤ B) 5

Set up an integral for the length of the curve. 151) y= ,0≤x≤1 A)

C) 25

151) _____ B)

152) _____

152) y= A)

≤x≤ B)

153)

,0≤y≤2

153) _____

154)

x= A)

154) _____

+ 6y, 0 ≤ y ≤ 6 B)

155) _____

155) y = 2 cot x, A)

≤x≤

156) y = 2 cos x, 0 ≤ x ≤ π A)

157) x = sin 2y, - π ≤ y ≤ 0 A)

156) _____ B)

157) _____ B)

158) _____

158) x = 6 tan y, 0 ≤ y ≤ A)

159)

+ 4y = 4x - 1, 1 ≤ y ≤ 2 A)

159) _____ B)

160) _____

160) y= A)

≤x≤ B)

Solve the problem. 161)

161) _____

Find a curve through the point A) B) y=5 y=5

whose length integral, 1 ≤ x ≤ 2, is C)

y= 162) _____

162) Find a curve through the point (-6, 1) whose length integral, 1 ≤ y ≤ 2, is L = A) B) x = -6 C) x = -6 x=

D) x= 163) _____

163) Find a curve through the point (0, 6) whose length integral, 0 ≤ x ≤ 1, is L = A) y = x B) C) y= y= +6

D) y = 2x + 6

Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. 164) y = cotx, 0 ≤ x ≤ π/4; x-axis A) B) 2π

dx D)

dx 165)

164) _____

y= , 0 ≤ x ≤ 2; x-axis A) 2π

2π

165) _____ B)

D) 2π

166) xy = 5, 3 ≤ y ≤ 4; y-axis A)

166) _____ 5π

10 π

dx D)

C) 10π

5π

dx 167) _____

167) x= A)

dt, 0 ≤ y ≤ π/3; y-axis B) csc y dy

-2π C)

D) π

csc y dy

Use your grapher to find the surface's area numerically. 168) y = sinx, 0 ≤ x ≤ π/6; x-axis A) 7.65 B) 1.15 169)

cot y dy

-2π

+ = 3, 2 ≤ x ≤ 4; x-axis A) 56.93 B) 54.52

cot y dy

168) _____ C) 13.39

D) 5.1 169) _____

C) 54.87

D) 65.94

Solve. 170) Find the lateral (side) surface area of the cone generated by revolving the line segment y = x/3, 0 ≤ x ≤ 4, about the x-axis. A) B) C) D) π

170) _____

171) Find the lateral surface area of the cone generated by revolving the line segment

171) _____

about the A) 30π

B) 180π

C) 30π

D) 180π 172) _____

172) Find the surface area of the cone frustum generated by revolving the line segment , A)

, about the x-axis. B) 7π

D) π

173) Find the surface area of the cone frustum generated by revolving the line segment , A) 27π

, about the y-axis. B) 31π

C) 28π

Find the area of the surface generated by revolving the curve about the indicated axis. 174) y= /9, 0 ≤ x ≤ 2; x-axis

173) _____

D) 26π

174) _____

B) π

175)

C) 98π

x= /14, 0 ≤ y ≤ 4; y-axis A) B) 4528π

π 175) _____ C)

π 176) y =

176) _____

, 3/2 ≤ x ≤ 9/2; x-axis

B) π

D) π

177)

y= A) 2π

178) x = 3

, 0.5 ≤ x ≤ 1.5; x-axis B) 3π , 0 ≤ y ≤ 15/4; y-axis B) 5π

π 177) _____ C) π

D) 4π 178) _____

D) π

π 179) _____

179) y= A)

, 0 ≤ x ≤ ln 7; x-axis B) π ln 7 π

D) π

Solve the problem. 180) A company applies a clear glaze finish on the outside of the ceramic bowls it produces. The bowl

180) _____

corresponds to the bottom half of a sphere which is created by rotating the circle 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 to calculate how many liters of finish are required. Assume that all specifications for the bowl are in cm. A) 1.51 L of finish B) 150.8 L of finish C) 60.32 L of finish D) 30.16 L of finish 181) 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) B) C) D) 94 942 9 9425

181) _____

182) The spring of a spring balance is 6.0 in. long when there is no weight on the balance, and it is 9.6 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 11.8 in.? A) 7.6 in.-lb B) 37 in.-lb C) 6.4 in.-lb D) 110 in.-lb

182) _____

183) 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 8 in.? A) 6300 in.-lb B) 0.045 in.-lb C) 23,000 in.-lb D) 11,000 in.-lb

183) _____

184) It took 1930 J of work to stretch a spring from its natural length of 4 m to a length of the spring's force constant. A) 1447.5 N/m B) 965 N/m 185) A spring has a natural length of

. A force of

far beyond its natural length will a A) 4 in. B) 5 in. 186) It takes a force of compressed height of A) 950 in.-lb

C) 3860 N/m

. Find

D) 482.5 N/m

stretches the spring to 40 in. How

force stretch the spring? C) 40 in.

to compress a spring from its free height of

184) _____

185) _____

D) 2.5 in. . to its fully

186) _____

. How much work does it take to compress the spring the first inch? B) 190,000 in.-lb C) 1900 in.-lb D) 3800 in.-lb 187) _____

187) A bathroom scale is compressed in. when a 190 lb person stands on it. Assuming that the scale behaves like a spring that obeys Hooke's law, how much does someone who compresses the scale in. weigh? A) 95 lb 188) A force of

B) 47.5 lb

will stretch a rubber band

work is done on the rubber band by a A) 2700 J B) 0.03 J

C) 380 lb

D) 142.5 lb

. Assuming Hooke's law applies, how much

188) _____

force? C) 0.27 J

189) Find the work done in winding up a 175-ft cable that weighs 3.00 lb/ft. A) 45,900 ft∙lb B) 15,300 ft∙lb C) 788 ft∙lb

D) 0.09 J 189) _____ D) 138,000 ft∙lb

190) At lift-off, a rocket weighs 40.3 tons, including the weight of fuel. It is fired vertically, and the fuel is consumed at the rate of 2.09 tons per 1,000 ft of ascent. How much work is done in lifting the rocket to an altitude of 14,000 ft? A) B) C) D) 4.62 x 4.28 x 7.69 x 3.59 x ft∙ton ft∙ton ft∙ton ft∙ton

190) _____

191)

191) _____

The gravitational force (in lb) of attraction between two objects is given by where x is the distance between the objects. If the objects are 10 ft apart, find the work required to separate them until they are 50 ft apart. Express the result in terms of k. A) B) C) D) k

k 192) _____

192) How much work is done by a 120-lb woman as she walks up 12 steps, each with a -ft rise? A) 1440 ft ∙ lb B) 2880 ft ∙ lb C) 360 ft ∙ lb D) 720 ft ∙ lb 193) 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 40 ft above the water? A) 10,400 ft ∙ lb B) 7280 ft ∙ lb C) 8800 ft ∙ lb D) 1780 ft ∙ lb

193) _____

194) A fisherman is about to reel in a 17-lb fish located 11 ft directly below him. If the fishing line

weig hs 1

oz per 194) foot, how much work will it take to reel in the fish? Round your answer to the nearest tenth, if necessar y. A) 190.8 ft ∙ lb

____ _

B) 194.6 ft ∙ lb

C) 247.5 ft ∙ lb

195) A construction crane lifts a bucket of sand originally weighing lost from the bucket at a constant rate of

D) 198 ft ∙ lb at a constant rate. Sand is

195) _____

. How much work is done in lifting the sand

? (Neglect the weight of the bucket.) A) 9600 ft ∙ lb B) 11,200 ft ∙ lb

C) 12,800 ft ∙ lb

D) 14,400 ft ∙ lb

196) A construction crane lifts a 100-lb bucket originally containing 150 lb of sand at a constant rate. The sand leaks out at a constant rate so that there is only 75 lb of sand left when the crane reaches a height of 90 feet. How much work is done by the crane? A) 10,125 ft ∙ lb B) 22,500 ft ∙ lb C) 3600 ft ∙ lb D) 19,125 ft ∙ lb

196) _____

197) A vertical right circular cylindrical tank measures

197) _____

high and

diameter. It is full of

oil weighing . 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) 2,412,743 ft ∙ lb B) 40,212 ft ∙ lb C) 4,825,486 ft ∙ lb D) 9,650,973 ft ∙ lb 198) A vertical right circular cylindrical tank measures

high and

in diameter. It is full of

oil weighing How much work does it take to pump the oil to a level top of the tank? Give your answer to the nearest ft ∙ lb. A) 4,728,976.59 ft ∙ lb B) 5,404,544.67 ft ∙ lb C) 4,503,787.23 ft ∙ lb D) 10,809,089.3 ft ∙ lb

above the

199) A vertical right circular cylindrical tank measures 20 ft high and 8 ft in diameter. It is full of oil weighing

198) _____

199) _____

How long will it take a (1/2)-horsepower (hp) motor (work output

to pump the oil to the level of the top of the tank? Give your answer to the nearest minute. A) 12 min B) 37 min C) 73 min D) 2193 min 200) The base of a rectangular tank measures 9 ft by 18 ft. The tank is 15 ft tall, and its top is 10 ft below ground level. The tank is full of water weighing 62.4 lb/ . 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) 1,137,390 ft ∙ lb B) 1,137,240 ft ∙ lb C) 4,928,040 ft ∙ lb D) 2,653,560 ft ∙ lb

200) _____

201) A swimming pool has a rectangular base 10 ft long and 20 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/ answer to the nearest ft ∙ lb. A) 305,760 ft ∙ lb B) 152,880 ft ∙ lb C) 114,660 ft ∙ lb

201) _____

. Give your D) 229,320 ft ∙ lb

202) A swimming pool has a rectangular base 13 ft long and 26 ft wide. The sides are 7 ft high, and the pool is full of water. How much work will it take to lower the water level 2 feet by

202) _____

pumping the water out over the top of the pool? Assume that the water weighs Give your answer to the nearest ft ∙ lb. A) 21,091 ft ∙ lb B) 1,033,469 ft ∙ lb C) 42,182 ft ∙ lb D) 516,734 ft ∙ lb 203) A conical tank is resting on its apex. The height of the tank is

, and the radius of its top is

203) _____

. The tank is full of gasoline weighing How much work will it take to pump the gasoline to the top? Give your answer to the nearest ft ∙ lb. A) 49,412,980 ft ∙ lb B) 1508 ft ∙ lb C) 96,510 ft ∙ lb D) 12,064 ft ∙ lb 204) A conical tank is resting on its apex. The height of the tank is . The tank is full of gasoline weighing gasoline to a level A) 24,127 ft ∙ lb 205)

, and the radius of its top is

204) _____

How much work will it take to pump the

above the cone's top? Give your answer to the nearest ft ∙ lb. B) 12,064 ft ∙ lb C) 60,319 ft ∙ lb D) 112,595 ft ∙ lb

A tank is designed by revolving the parabola y = 2

, 0 ≤ x ≤ 2, about the y-axis. The tank, with

205) _____

dimensions in meters, is filled with water weighing 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) 656,802 J B) 3,940,814 J C) 41,050 J D) 1,313,605 J 206) A spherical tank of water has a radius of

, with the center of the tank above the ground. How much work will it take to fill the tank by pumping water up from ground level? Assume the water weighs A) 22,583,276 ft ∙ lb C) 28,229,095 ft ∙ lb

Give your answer to the nearest ft ∙ lb. B) 7,188,480 ft ∙ lb D) 11,291,638 ft ∙ lb

A variable force of magnitude F(x) moves a body of mass m along the x-axis from

moving the body from

and

206) _____

where

. The net work done by the force in

are the body's velocities at

. Knowing that the work done by the force equals the change in the body's kinetic energy, solve the problem. 207) A 2-oz tennis ball was served at 140 ft/sec about (95 mph). How much work was done on the ball 207) _____ to make it go this fast? (To find the ball's mass from its weight, express the weight in pounds and divide by A) 38 ft-lb

the acceleration of gravity.) B) 36 ft-lb

C) 18 ft-lb

D) 77 ft-lb

208) How many foot-pounds of work does it take to throw a baseball 30 mph? A baseball weighs 5 oz, or 0.3125 lb. A) 141 ft-lb B) 9 ft-lb C) 303 ft-lb D) 4 ft-lb

208) _____

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) _____

B) 125w

D) w

w 210) _____

210)

B) w

C) w

D) 343w w 211) _____

211)

A) 108w

B) 648w

C) 324w

D) 1296w 212) _____

212)

A) 864w

B) 324w

C) 432w

D) 648w

Solve the problem. 213) 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 A) 33,700 lb

deep? The density of water is B) 2810 lb

. C) 16,800 lb

D) 8420 lb

213) _____

214) 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/ . A) 43,000 lb

B) 38,100 lb

C) 86,000 lb

214) _____

D) 54,200 lb

215) A right triangular plate of base 8 m and height 4 m is submerged vertically, as shown below.

215) _____

Find the force on one side of the plate.

8m A) 630,000 N

B) 100,000 N

C) 420,000 N

D) 310,000 N

216) 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.

216) _____

6m A) 210,000 N

B) 410,000 N

C) 180,000 N

D) 150,000 N

217) 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/ A) 8300 N B) 1.7 N 218) A tank truck hauls oil in a density of the oil is tank is half full? A) 19,440 lb

217) _____

. C) 8.3 N

D) 330 N

horizontal right circular cylindrical tank. If the

218) _____

, how much force does the oil exert on each end of the tank when the B) 12,960 lb

C) 8640 lb

D) 4320 lb

219) A semicircular plate

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) 21,299.2 lb B) 31,948.8 lb C) 15,974.4 lb D) 47,923.2 lb

219) _____

220) An isosceles triangular plate is submerged vertically in seawater, with its base on the bottom.

220) _____

The base is

long, and the height of the triangle is

of the plate if the water level is

. Find the force exerted on one face

above the base of the triangle. Seawater weighs

Round your answer to one decimal place if necessary. A) 426.7 lb B) 2176 lb C) 4352 lb

D) 1450.7 lb

221) A rectangular swimming pool has a parabolic drain plate at the bottom of the pool. The drain

plate is

____ _

shaped 221) like the region between

and the line

from to The pool is 10 ft by 20 ft and 8 ft deep. If the pool is being filled at a rate of what is the force on the drain plate after 2 hours of filling? Round your answer to two decimal places if necessar y. A) 83.2 lb

B) 237.12 lb

C) 35.36 lb

D) 70.72 lb

222) A rectangular swimming pool has a parabolic drain plate at the bottom of the pool. The drain

222) _____

plate is shaped like the region between y = and the line y = from x = -1 to x = 1. The pool 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 1 in. B) 5 ft 6 in. C) 5 ft 9 in. D) 4 ft 11 in. Find the center of mass of a thin plate of constant density covering the given region. 223) The region bounded by y = and y = 9 A) =

223) _____ 0,

B) =

= 0,

= 15

224) _____

224) The region bounded by y = 6 - x and the axes A) = 2, = 2 B) = 18, = 18 225)

The region bounded by y = A) =

= 36,

= 36 225) _____

The region bounded by the parabola y = 81 A) =

226) _____

and the x-axis B)

= 0,

D) = 0,

228)

D) =

227)

, x = 5, and the x-axis

226)

= 6,

The region enclosed by the parabolas y = A) = 0, = 20 B) = 0, = 25 The region bounded by the parabola x = A) B) =

227) _____

+ 50 and y = C) = 25,

= 0,

= 50 228) _____

and the line x = 1 C) = 1,

= 0,

= π,

= 229) _____

229) The region bounded by the x-axis and the curve y = 7sin x, 0 ≤ x ≤ π A) B) C) =

230) _____

230) The region between the x-axis and the curve y = 6 A) B) = 231)

x, C)

D) =

The region cut from the first quadrant by the circle A) B) =

≤x≤ ,

=6 231) _____

= 49 C)

D) = 0, 232)

= 232) _____

The region bounded by the x-axis and the semicircle y = A) B) C) = 0,

= 0,

D) =

233) _____

233) The region in the first and fourth quadrants enclosed by the curves and by the lines A) =

and

and B) ,

D) =

= 0,

= 234) _____

234) The region between the curve y = A) =

ln 9,

and the x-axis from B)

D) =

to ,

= 52,

ln 9

= 9 ln 9

ln 9

Find the center of mass of a thin plate covering the given region with the given density function. 235) The region bounded below by the parabola and above by the line , with density A)

B) =

236)

C) =

The region bounded by the parabola y = 4 A) B) = 0,

= 0,

D) =

236) _____

and the x-axis, with density δ(x) = 2 C) D)

= 0,

= 237) _____

237) The region between the x-axis and the curve y = A) B) = 238)

235) _____

, 1 ≤ x ≤ 2, with density δ(x) = C) D)

The region enclosed by the parabolas y = 8 A) = 0, = 4 B) = 0, = 8

= and y = C)

, with density δ(x) = D) = 0,

= 238) _____

= 4,

239)

The region bounded by x = and the line x = 16, with density δ(x) = A) B) C) =

= 0,

239) _____ D)

240) The triangular region cut from the first quadrant by the line y = -x + 4, with density δ(x) = x A) = 2, = 1 B) = 1, = 1 C) = 1, = 2 D) = 2, = 2

240) _____

241)

241) _____ The region between the curve y = and the x-axis from x = 1 to x = 9, with density δ(x) = A) B) = 5, = 2 C) D) =

= 24

=1 242) _____

242) The region bounded by the curves y = ± A) = 2, = 0 B) = 3, = 0

and the lines x = 1 and x = 9, with density δ(x) = C) = 5, = 0 D) = 4, = 0

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. 243) (-7, 0), (7, 0), (0, 5) 243) _____ A) (0, 5) B) C) D) (0,

)

(0,

)

(0,

) 244) _____

244) (0, 0), (9, 0), ( , 4) A) (3, 2)

C) ( ,

)

D) (0,

)

( ,

)

Find the moment or center of mass of the wire, as indicated. 245) Find the moment about the x-axis of a wire of constant density that lies along the first-quadrant portion of the circle A)

= 25. B) 25

D) 5

246) Find the center of mass of a wire of constant density that lies along the first-quadrant portion of the circle A)

= 0,

C) =

= 3,

C) =

248)

= 3,

from x = 0 to D)

= 1,

247) _____

Find the moment about the y-axis of a wire of constant density that lies along the curve y = from

246) _____

247) Find the center of mass of a wire of constant density that lies along the line A)

245) _____

248) _____

D) 2

249) _____

249) Find the moment about the x-axis of a wire of constant density that lies along the curve y = 2 from A) 21

. B)

D) 14

250) _____

250) Find the moment about the x-axis of a wire of constant density that lies along the curve y = from A)

to B)

251) _____

251) Find the center of mass of a wire that lies along the first-quadrant portion of the circle A)

= 2π,

if the density of the wire is B) = 2π =

π,

D) = 0,

π,

π 252) _____

252) Find the center of mass of a wire that lies along the first-quadrant portion of the circle A)

if the density of the wire is B) = 1, = 1 =

253)

D) =

Find the center of mass of a wire that lies along the semicircle y = wire is A)

B) =

π,

if the density of the

C) =

π 254)

253) _____

D) = 0,

= 0,

Find the center of mass of a wire that lies along the semicircle y = wire is A) = 2,

if the density of the

254) _____

. =2

= 0,

= 3,

= 0,

Find the centroid of the thin plate bounded by the graphs of the given functions. Use δ = 1 and M = area of the region covered by the plate. 255) 255) _____ g(x) = and f(x) = x + 2 A) B) C) D) =

= 2,

256)

g(x) = A)

, =

Solve. 257)

f(x) = 4, ,

256) _____

and x = 0 B)

C) = 4,

D) =

Find the volume of the torus generated by revolving the circle A) B) C) 27 36 9

258) Locate the centroid of a semicircular region. A) =

= 1 about the y-axis. D) 18

257) _____

258) _____

D) =

1) C 2) C 3) C 4) B 5) C 6) D 7) A 8) D 9) C 10) C 11) D 12) B 13) A 14) D 15) D 16) B 17) C 18) C 19) D 20) C 21) D 22) A 23) A 24) B 25) C 26) D 27) C 28) B 29) B 30) A 31) B 32) A 33) C 34) A 35) D 36) B 37) B 38) C 39) A 40) D 41) D 42) D 43) B 44) A 45) D 46) C 47) C 48) C 49) D 50) D 51) C

52) A 53) B 54) D 55) C 56) A 57) B 58) B 59) C 60) B 61) A 62) B 63) B 64) D 65) B 66) C 67) C 68) D 69) B 70) D 71) A 72) D 73) B 74) C 75) D 76) A 77) D 78) B 79) D 80) D 81) A 82) A 83) A 84) C 85) C 86) A 87) C 88) D 89) A 90) D 91) D 92) D 93) D 94) A 95) D 96) A 97) A 98) D 99) A 100) B 101) B 102) B 103) B

104) D 105) B 106) D 107) A 108) A 109) B 110) A 111) A 112) A 113) A 114) C 115) B 116) B 117) B 118) D 119) A 120) B 121) D 122) B 123) A 124) A 125) C 126) C 127) B 128) D 129) D 130) D 131) D 132) A 133) The volume can be found using the washer method with 2 integrals or using the shell method with 1 integral. 134) The volume can be found using the washer method with 1 integral or using the shell method with 2 integrals. 135) The volume can be found using the washer method with 2 integrals or using the shell method with 1 integral. 136) The volume can only be found using the shell method. Only 1 integral is required. Since y = 4xcannot be solved for x, the washer method cannot be used. 137) The volume can be found using the disk method with 1 integral. Since x = 2y cannot be solved for y, the shell method cannot be used. 138) The volume can be found using the shell method with 1 integral. Since x = 2y cannot be solved for y, the washer method cannot be used. 139) The resulting solid is a cylindrical shell whose volume is given by V = π h, where is the outer radius, is the inner radius, and h is the height. Since

= 6,

= 2 and h = 3, V = π

3 = 96π.

140) The resulting solid is a hemisphere of radius 2. The volume is given by V = 141) A 142) D 143) D 144) A 145) B 146) C 147) D 148) D

149) D 150) B 151) D 152) C 153) A 154) A 155) B 156) A 157) A 158) A 159) B 160) A 161) C 162) C 163) C 164) D 165) A 166) C 167) A 168) B 169) C 170) A 171) D 172) A 173) C 174) A 175) C 176) C 177) B 178) D 179) A 180) C 181) B 182) B 183) D 184) B 185) D 186) C 187) A 188) C 189) A 190) D 191) D 192) D 193) C 194) A 195) B 196) D 197) A 198) B 199) B 200) D

201) D 202) C 203) D 204) C 205) D 206) A 207) A 208) B 209) C 210) C 211) C 212) D 213) C 214) A 215) C 216) B 217) C 218) C 219) A 220) D 221) D 222) A 223) C 224) A 225) D 226) C 227) B 228) B 229) A 230) A 231) A 232) A 233) B 234) B 235) D 236) D 237) C 238) A 239) C 240) A 241) D 242) B 243) C 244) B 245) B 246) D 247) B 248) A 249) C 250) D 251) B 252) A

253) D 254) B 255) A 256) D 257) D 258) B

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the integral. 1) A) ln

D) -6 ln -

1) _______

+C +C

+C 2) _______

B) ln

ln 3) _______

3) A)

B) 5 sin x + C

ln C) 5 ln

+C D) ln

+C 4) _______

D) ln

ln 5) _______

5) A)

B) ln

D) ln

6) _______

6) dx A)

7) _______

C) ln 2

D) -

__ __ __ _

A) -7 ln 3

B) 7 ln 3

C) -14 ln 3

D) 14 ln 3 9) _______

A) 3 ln 3

B) 6 ln 2

C) 3 ln 2

D) -3 ln 2 10) ______

10) dx A) 3 ln

C) 3 ln

B) ln

D) -ln

+C 11) ______

11) A)

B) +C

C) -

+C 12) ______

12) dx A)

B) ln

C) 6 ln

D) +C 13) ______

13) dx A)

+C 14) ______

14) A)

-5

D) -

+C 15) ______

15) A)

B) (e2x + e-2x) + 2x + C

(e2x - e-2x) + 2x + C

D) (e2x + e-2x) + C

(e2x - e-2x) + C 16) ______

16) dx A) 3

B) 4

C) -3

D) 7

17) ______

17) A)

D) -

+C 18) ______

18) 5x dx B)

A) 5e

C) e

D) 19) ______

19)

dx A) 1

B) 1 - cos 1

C) 1 + cos 1

D) -1 20) ______

20) dθ A)

ln (1 +

)+C

+C C)

4 ln (1 +

)+C

+C 21) ______

21)

dx A)

D) 1

22) ______

22) dx A)

C) 589,788

+C 23) ______

23) sin t dt A)

C) 4

24) ______

24) dx

25) ______

25) A)

D) +C

+C 26) ______

26) A)

D) 1

+C 27) ______

27) dx A)

-1

28) ______

28) dt A) 6 ln 3

C) 18

D) 3

29) ______

29) dx A)

C) 10

+C 30) ______

30) A)

ln 3 + C

+C 31) ______

31)

A) 2 ln 4

B) ln 4 - 1

Solve the initial value problem. 32) = A) C)

sin ( y = sin y=

32) ______

- 6), y(ln 6) = 0 - sin 2

cos (

- 6) - 6

B) D)

y = cos (

- 6) - 1

y = -cos (

- 6) + 1

33) ______

33) = A)

(

), y(-ln 7) = B)

y= C)

y = cot (

)+2

y= 34) ______

34) = A)

cos

, y(0) = 0 B)

sin x

sin

y=-

sin

D) y=

sin

sin 1

sin 1 35) ______

35) = A) C)

sec

tan

y = -8 sec

y = 8 tan

, y(0) = 8 sec 1 + 4 B) y = -8 sec x + 1 D)

y = 8 sec

+4 36) ______

36)

A) C)

, y(0) = 1,

y=9

+ 9x - 8

y=9

- 9x + 10

(0) = 0 B) D)

y = -9

y=9

+1 37) ______

37)

A) C)

, y(0) = -6,

y=7

y = -7

+ 7x - 13

(0) = 0 B) D)

y = -7

-6

y = -7

- 7x + 1 38) ______

38) =

+ 9 sin t, y(0) = 0,

(0) = 4

B) y=

- 9 sin t +

- 9 sin t + 4t -

- 9 sin t + 12t -

D) y=

- 9 sin t

39) ______

39) =3-

, y(1) =

(0) = -4 B)

y= C)

- 5t + y =

- 4t

D) -

y =

+ 5t + 2-

Solve the problem. 40) Find the volume of the solid that is generated by revolving the area bounded by the

curve A)

and B)

about the x-axis. C)

π ln

the

40) ______

D) 9π ln

π ln 41) ______

41) Find the volume of the solid that is generated by revolving the area bounded by and A)

about the x-axis. B)

π ln (8)

π ln (33)

D) π ln (8)

π ln (33) 42) ______

42) The region between the curve y = and the x-axis from to y-axis to generate a solid. Find the volume of the solid. A) 4π ln 6 B) π ln 6 - π C) 2π ln 6 - π

is revolved about the D) 2π ln 6 43) ______

43) Find the length of the curve y = ln x, 2 ≤ x ≤ 4. A) B) 3 + ln 2 C) 2 + ln 3

D) 3+ 44) ______

44) Find the length of the curve x = A) 8 + 4 ln 2 B)

- 4 ln

, 8 ≤ y ≤ 16. C) 6 + 4 ln 2

6 + 4 ln

8 + 4 ln

Determine if the given function y = f(x) is a solution of the accompanying differential equation. 45) =y= A) Yes

B) No

45) ______

46)

46) ______

-y=5 y= + 5x A) Yes

47)

B) No 47) ______

-y=4 y= + 4x A) Yes

48)

B) No 48) ______

+y= y=9 + A) Yes

49)

B) No 49) ______

+y= y=3 + A) Yes

B) No 50) ______

50) y=

+ xy = A) Yes 51)

B) No

Differential equation: = Initial condition: y(- 4) = 0 Solution candidate: y = (x + 4) A) No

52)

Differential equation: = Initial condition: y(0) = 0

B) Yes

B) No

Solution candidate: y = A) No

53) ______

+ 6y = cos x

Solution candidate: y = A) Yes 54) Differential equation: 8x Initial condition: y(π) = 0

52) ______

- 7y

Solution candidate: y = 7x A) Yes 53) Differential equation: 6x Initial condition: y(π) = 0

51) ______

- 4y

B) No 54) ______

+ 8y = cos x

B) Yes

55) ______

55) Differential equation: x + y = Initial condition: y(3) = 2 Solution candidate: y = A) No

+ B) Yes

Solve the differential equation. 56) = A)

y = ln (

56) ______ B)

+ C)

D) y=

ln (

y=2

y = 2ln (

+ C)

+ C) 57) ______

57) = A) y = 9ln x + C

B) y = - 3ln x + C

D) y=

y= 58) ______

58) = 15 A) y=5 + C) y = 25 +C

y=5

y= 59) ______

59) = 5y A)

y=C

D) y=-

y=C

+C 60) ______

60) =5 A) y = ln (

+ C)

y = C ln (

)

y = ln (

+ C) 61) ______

61) =3 y A) y= ( + C) C) y = tan ( + C)

B) D)

(

+ C)

62) ______

62) = 6 sec y A) y = sin ( + C) C) y= ( + C)

B) D)

(

+ C) 63) ______

63) =6

B) y=

D) y=

+1 y=

+C 64) ______

64) = 6x A) y = 3 sin (3 + C) C) y = sin (3 + C)

B) D)

y=3 y=

(3 (3

+ C) + C) 65) ______

65) = 7 cos x sec y A) y= (7 sin x + C) C) y = sin (7 sin x + C)

y= (7 cos x + C) D) y = 7 sin x + C

Solve the problem. 66) A certain radioactive isotope decays at a rate of 3% per 400 years. If t represents time in years and y represents the amount of the isotope left, use the condition that value of k in the equation A) -0.03046

B) 0.00659

to find the

C) -0.00008

D) 0.00008

67) 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 A) 363 years

In how many years will there be 93% of the isotope left? B) 350 years C) 253 years

68) A loaf of bread is removed from an oven at . If the bread cools to

66) ______

67) ______

D) 700 years

and cooled in a room whose temperature is

68) ______

in 20 minutes, how much longer will it take the bread to cool

to A) 23 min

B) 14 min

C) 13 min

D) 33 min 69) ______

69) In a chemical reaction, the rate at which the amount of a reactant changes with time is proportional to the amount present, such that

, when t is measured in hours. If there

are of reactant present when t = 0, how many grams will be left after answer to the nearest tenth of a gram. A) 9.5 g B) 19 g C) 0.2 g

? Give your D) 28.5 g 70) ______

70) Find the half-life of the radioactive element radium, assuming that its decay constant is A) 1600 years

with time measured in years. B) 2308 years

C) 1400 years

D) 800 years

71) The charcoal from a tree killed in a volcanic eruption contained 62.7% of the living matter. How old is the tree, to the nearest year? Use A) 3839 years

B) 2661 years

C) 5700 years

found in

for the half-life of D) 1844 years

71) ______

72) The intensity L(x) of light x ft beneath the surface of a lake satisfies the differential equation

72) ______

. At what depth, to the nearest foot, is the intensity one tenth the intensity at the surface? A) 26 ft

B) 38 ft

C) 13 ft

D) 17 ft

73) The barometric pressure p at an altitude of h miles above sea level satisfies the differential equation pressure at 17,000 ft. A) 1 in.

73) ______

If the pressure at sea level is 29.92 inches of mercury, find the barometric B) 15.71 in.

C) 56.97 in.

D) 7.86 in.

74) Suppose that the amount of oil pumped from a well decreases at the continuous rate of

per year. When, to the nearest year, will the well's output fall to one-eighth of its present value? A) 35 years B) 23 years C) 2 years D) 15 years

74) ______

75) The amount of alcohol in the bloodstream, A, declines at a rate proportional to the amount, that

75) ______

is, . If for a particular person, how long will it take for his alcohol concentration to decrease from 0.10% to 0.05%? Give your answer to the nearest tenth of an hour. A) 1.7 hr B) 0.3 hr C) 2.6 hr D) 3.5 hr A value of sinh x or cosh x is given. Use the definitions and the identity indicated hyperbolic function. 76) sinh x = A)

to find the value of the other 76) ______

, cosh x = B)

77) ______

77) sinh x = A)

, tanh x = B)

C) -

D) 78) ______

78) sinh x = A)

, coth x = B)

79) ______

79) sinh x = A)

, sech x = B)

80) ______

80) sinh x = A)

, csch x =

81) ______

81) cosh x = A)

, x > 0, sinh x = B)

82) ______

82) cosh x = A)

, x > 0, tanh x = B)

D) 83) ______

83) cosh x = A)

, x < 0, coth x = B)

C) -

D) 84) ______

84) cosh x = A)

, x < 0, sech x = B)

85) ______

85) cosh x = A)

, x < 0, csch x = B)

C) -

D) -

Rewrite the expression in terms of exponentials and simplify the results. 86) 10 cosh (ln x) A) 5x B) C) 0 5( + )

86) ______ D) 5

87) sinh (4 ln x) A) 2x

87) ______ B)

2 88) cosh 5x + sinh 5x A) 89) cosh 5x - sinh 5x A)

88) ______ B)

C) 5x

89) ______

D) -5x 90) ______

90) A)

91) ______

91) ln (cosh 5x - sinh 5x) A) -5x

B) -10x

C) -5

92) sinh (2 ln 3x) A)

B) 6x

ln (

92) ______

93) ______

93) 4 cosh (ln x) + 14 sinh (ln x) A) 0 B)

C) 9x

9 94) ln

9x -

+ ln (cosh 5x + sinh 5x) B) ln ( )

A) 12x

94) ______ D) -2

C) -2x

95) ______

95) A)

Find the derivative of y. 96) y = cosh A) sinh

96) ______ 5

-sinh x

-5

sinh

sinh 97) ______

97) y = ln(sech (9x + 7)) A) -9 tanh (9x + 7)

98)

)

C) -tanh (9x + 7)

D) tanh (9x + 7)

98) ______

y= 3x A) 2 cosh 3x C) 2 sinh 3x cosh 3x

B) 6 sinh 3x cosh 3x D) 6 cosh 3x 99) ______

99) y = csch A)

csch

coth

-csch

coth

D) -

csch

coth

csch

coth 100) _____

100) y = ln (sinh 10x) A) 10 csch 10x

B) coth 10x

D) 10 coth 10x

101) _____

101) y= A)

tanh B) -12

tanh

-12 -1

tanh

-8

D) 2

-1 2

tanh ta nh

+ 8 sech

+ 4

102) y = sech (-4θ) (1 - ln sech(-4θ)) A) sech (-4θ) tanh (-4θ) ln sech (-4θ) C) -4 sech (-4θ) tanh (-4θ) ln sech (-4θ)

102) _____ B) 1 + 4 sech (-4θ) tanh (-4θ) ln sech (-4θ) D) -4 sech (-4θ) tanh (-4θ) ln (-4θ)

Find the derivative of y with respect to the appropriate variable. 103) y= A) B) C)

104)

y= A)

104) _____

2 B)

105)

103) _____

y = (7 - 7θ) A)

105) _____

θ B) -

106)

y=( A)

+ 5θ)

θ 106) _____

(θ + 4) B)

(2θ + 5) C)

(2θ + 5) -

(θ + 4) D)

107)

y = (1 - 5t) A)

(2θ + 5)

(θ + 4) 107) _____

B) t 5

(1 - 5t)

-5 108)

y = 7 ln x + A)

108) _____

x B)

7 ln x -

D) -

109) _____

109) y= A)

110)

111)

112)

y= A)

110) _____

(cos x) B)

y= 3 A)

(ln x)

y = 10 A)

(cos x)

D) - sin x

111) _____ B)

112) _____ B)

D) ln

Verify the integration formula. 113)

113) _____ (sinh 3x) + C

A) Yes

sin x

B) No

114) _____

114) x+

A) Yes

B) No 115) _____

115) x

ln (1 -

)+ C

A) Yes

B) No 116) _____

116) (1 -

)+C

A) Yes

B) No 117) _____

117) ln (cosh

)+C

A) No

B) Yes 118) _____

118) (x - 1)

A) Yes

B) No 119) _____

119) x+

A) Yes

B) No 120) _____

120) ln x

(ln x) +

A) No

ln (1 -

)+ C B) Yes 121) _____

121) ln

A) No

B) Yes 122) _____

122) ln

6x + C

A) Yes

B) No

Evaluate the integral. 123) A)

123) _____ 5x + C

B) cosh 5x + C

D) cosh 5x + C

cosh 5x +

124) _____

124) A)

B) +C

125)

C) sinh

D) -6 sinh

6 sinh

____ _

125)

B) 3 sinh

6 sinh

2 sinh

D) sinh

126) _____

126) A) 5 cosh (4x - ln 8) + C

C) 20 cosh (4x - ln 8) + C

cosh (4x - ln 8) + C cosh 4x + C 127) _____

127) A)

B) 6 ln

D) 6

6 ln

+C 128) _____

128) A)

B) 5x + C

D) ln 5 ln

+C +C

+C 129) _____

129) A) tanh (6x - 9) + C

tanh (6x - 9) + C 2

(6x - 9) + C

coth (6x - 9) + C 130) _____

130) A)

B) 6 coth

6 tanh

D) -coth

131) _____

131) A)

B) +C

4 sech

- 4 sech

8 csch

132) _____

132) A)

csch (ln x) + C C) x csch (ln x) + C

csch (ln x) + C D) 10 sech (ln x) + C 133) _____

133)

D) ln 2 ln 134) _____

134)

135) _____

135)

B) ln

ln 2

ln 136) _____

136)

B) 5 + 2 ln

C) 13 + 2 ln 6

D) 5 + ln 1

2 + 2 ln 137) _____

137)

B) ln

C) +

D) +2

138) _____

138)

A) C)

B) 9(

- 1)

139) _____

139)

B) 10 10

10(

)

D) 0

140) _____

140)

B) -14 28

14(

- e)

14 141) _____

141)

B) (

)

142) _____

142)

(

D) 5

)

Express the value of the inverse hyperbolic function in terms of natural logarithms. 143) A)

B) ln

C) ln 2

143) _____ D) ln (4)

ln 144) _____

144) A) ln 5

C) ln 4

ln 145) _____

145) A)

ln 1

ln -21

ln 21

ln 231 146) _____

146) A)

C) 0

ln -17

ln 153

ln 17 147) _____

147) A) Undefined

C) ln

148)

D) ln

ln 148) _____

B) ln

ln 149)

149) _____

(2) A) ln 4

150)

B) ln (2 +

C) ln (2 -

)

D) ln (2 +

)

) 150) _____

7 A) ln (7 +

B) ln (7 +

)

C) ln (7 -

)

D) ln (14)

)

Evaluate the integral in terms of an inverse hyperbolic function. 151)

151) _____

B) (

(

D) (

152) _____

152)

B) -

D) -

Evaluate the integral in terms of natural logarithms. 153)

A) ln (

+ 3)

B) ln (

153) _____

)

D) ln (

+ 2)

ln 154) _____

154)

A) 4 ln (2

+ 3)

B) ln 11

C) 4 ln (2

D) 16 ln (2

+ 3)

) 155) _____

155)

A) ln 2

C) ln

D) ln

156) _____

156)

A) ln -6

D) ln 29

ln 15 157) _____

157)

B) ln

ln D) ln 5

C) ln

-1 158) _____

158)

B) -

D) -ln

159) _____

159)

A) 0

B) -2

D) ln 2 ln 160) _____

160)

A) 5 ln (1 +

)

B) 5 ln (3 +

)

C) 3

D) ln (3 +

)

Solve the problem. 161) The velocity of a body of mass m falling from rest under the action of gravity is given by the

161) _____

equation where k is a constant that depends on the body's aerodynamic properties and the density of the air, g is the gravitational constant, and t is the number of seconds into the fall. Find the limiting velocity, 0.006. A) There is no limiting speed. C) 60.55 ft/sec

v, of a 220 lb. skydiver (mg = 220) when k = B) 0.01 ft/sec D) 191.49 ft/sec

162) The velocity of a body of mass m falling from rest under the action of gravity is given by the equation where k is a constant that depends on the body's aerodynamic properties and the density of the air, g is the gravitational constant, and t is the number of

seconfall. ds Find into the the limiti

____ _

ng 162) velocity, v, of a 400 lb. skydiver (mg = 400) when k = 0.006. A) There is no limiting speed. C) 0.01 ft/sec

B) 81.65 ft/sec D) 258.20 ft/sec 163) _____

163) Consider the area of the region in the first quadrant enclosed by the curve y = cosh 10x, the 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 A) B)

What is the height of the rectangle? C) 10 D) sinh 100

sinh 100 164) A region in the first quadrant is bounded above by the curve y = cosh x, below by the curve on the left by the y-axis, and on the right by the line solid generated by revolving the region about the x-axis. A) 2π B) 0 C) 9π

Find the volume of the D)

165) A region in the first quadrant is bounded above by the curve y = tanh x, below by the on the left by the and on the right by the line generated by revolving the region about the x-axis. A) 2π B) C) -

164) _____

165) _____

Find the volume of the solid D) 0 π 166) _____

166) Find the length of the segment of the curve y = A) B) 6

cosh 2x from x=0 to x = ln C)

. D)

Find the slowest growing and the fastest growing functions as 167) y = x + 10

167) _____

y= y= y= A)

x B) Slowest: y = x + 10

Slowest: y =

Fastest: y = + C) Slowest: y = x + 10 Fastest: y =

Fastest: y = D) Slowest: y = x + 10 Fastest: y =

168)

y=2

168) _____

+ 10x

y= y= y= A)

/5 x Slowest: y =

Fastest: 2

+ 10x

Slowest: 2

+ 10x

Fastest: y = C) Slowest: y =

Fastest: y =

169)

D) Slowest: y =

Fastest: y =

and y =

/5 grow at the same rate. 169) _____

y=5 y= y= y=x A)

Slowest: y = 5 Fastest: y = x

Slowest: y = x Fastest: y =

Slowest: y = 5 Fastest: y =

and y =

grow at the same rate.

Slowest: y = Fastest: y = x

170)

170) _____

+ 8x

y= y= y=6 A)

Slowest: y =

and y = 6

grow at the same rate.

Fastest: y = B)

Slowest: y = Fastest: y = 6

Slowest: y =

Fastest: y = + 8x D) They all grow at the same rate. 171) y = ln 6x

y = 2 ln x

____ _

171) y= y= A) Slowest: y = Fastest: y = 2 ln x B) Slowest: y = Fastest: y = C) Slowest: y = Fastest: y = ln 6x and y = 2 ln x grow at the same rate. D) Slowest: y = ln 6x and y = 2 ln x grow at the same rate. Fastest: y = 172)

172) _____

y= y= y= y= A)

Slowest: y =

and y =

grow at the same rate.

and y =

grow at the same rate.

Fastest: y = B)

Slowest: y = Fastest: y =

TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false. Determine if the statement is true or false as x→∞. 173) x = o(x + 10) 174)

= o(

)

173) _____ 174) _____

175) ln x = o(ln 5x)

175) _____

176) x = O(x + 9)

176) _____

177) x = O(10x)

177) _____

178) 9x + ln x = O(x)

178) _____

179)

179) _____

+ 5x = O( )

180) 9 + cos x = O(9)

180) _____

181) 182)

+ cos x = O(5

ln x = o(ln(

181) _____

)

182) _____

+ 10))

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 183) A polynomial f(x) has a degree smaller than or equal to another polynomial g(x). Does 183) ____________ and does 184) ____________

184)

185)

Show that = Explain why this is the case.

Show that y =

grow at the same rate as x→∞ by showing

and y =

185) ____________

that they both grow at the same rate as 186) Suppose you are looking for an item in an ordered list one million items long. Which would be better, a sequential search or a binary search? Why?

186) ____________

1) B 2) D 3) A 4) C 5) C 6) A 7) B 8) B 9) C 10) B 11) B 12) D 13) B 14) D 15) B 16) C 17) C 18) B 19) C 20) C 21) C 22) D 23) D 24) B 25) D 26) B 27) A 28) C 29) D 30) A 31) C 32) D 33) D 34) C 35) D 36) A 37) D 38) A 39) A 40) D 41) B 42) A 43) D 44) C 45) A 46) A 47) B 48) A 49) B 50) A 51) B

52) B 53) A 54) A 55) B 56) C 57) C 58) D 59) D 60) D 61) A 62) C 63) A 64) A 65) A 66) C 67) A 68) C 69) B 70) A 71) A 72) A 73) B 74) B 75) A 76) B 77) C 78) B 79) B 80) C 81) D 82) B 83) B 84) D 85) C 86) D 87) D 88) A 89) A 90) B 91) A 92) C 93) D 94) C 95) C 96) D 97) A 98) B 99) C 100) D 101) A 102) C 103) C

104) D 105) D 106) D 107) C 108) A 109) B 110) A 111) B 112) B 113) A 114) A 115) A 116) B 117) B 118) A 119) B 120) B 121) B 122) B 123) C 124) D 125) B 126) B 127) D 128) B 129) B 130) A 131) C 132) B 133) A 134) C 135) A 136) A 137) D 138) B 139) D 140) D 141) C 142) A 143) A 144) A 145) C 146) D 147) C 148) C 149) B 150) B 151) A 152) C 153) D 154) A 155) C

156) C 157) B 158) B 159) C 160) B 161) D 162) D 163) A 164) C 165) C 166) C 167) B 168) D 169) A 170) D 171) B 172) B 173) FALSE 174) TRUE 175) FALSE 176) TRUE 177) TRUE 178) TRUE 179) TRUE 180) TRUE 181) FALSE 182) FALSE 183) f = O(g) but g ≠ O(f) except in the case where their degrees are equal. 184) =

= = =1 y = ln x, y = ln(x+1), and y = ln (x+9985) all grow at the same rate. 185) =

= = =1 Therefore both functions grow at the same rate as x→∞. 186) The binary search. The sequential search could take up to a million steps. The binary search would take at most 20 steps.

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the integral. 1)

1) _______

dx A)

B) 6

(x + 6) + C

+C C) 6

(x - 6) + C

+C 2) _______

2) A)

ln(ln

)+C

D) ln(ln

)+C

+C 3) _______

3) A) x + 14 ln

B) x + 6 ln

D) -

/2 + 10x) ln

+C 4) _______

4) A)

B) +C

+C C)

+C 5) _______

5) A)

C 6) _______

6) A)

B) θ+C

C) -

θ+C

D) C

θ+

θ+C

7) _______

7) A)

B) x

+x+C

+x+C x

+ 5x + C

D) x

+x+C 8) _______

8) A) tan θ - sec θ + C C)

θ + sec θ + C D) tan θ + sec θ + C

θ - tan θ + C

9) _______

9) A)

B) +C

D) +C

+C 10) ______

10) A)

B) 8

-2

D) 8

+C 11) ______

11)

A) 972

C) 108

108 12) ______

12) A)

B) +C

(2x + 6) ln

+C D)

Solve the problem. 13) Find the length of the curve y = ln(csc x), π/3 ≤ x ≤ π/2 A) ln( B) ln(2 ) C) 1 - ln( + 1) 14) Find the length of the curve y = ln(sin x), π/3 ≤ x ≤ π/2 A) 1 - ln( ) B) ln(2 ) C) ln(

13) ______ )

D) ln(

) 14) ______

)

D) ln(

+ 1)

15) Find the volume generated by revolving the curve y = cos 5x about the x-axis, 0 ≤ x ≤ π/12 A) B) C) D) +

15) ______

1) B 2) C 3) B 4) B 5) C 6) B 7) A 8) D 9) B 10) C 11) C 12) B 13) D 14) C 15) B

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the integral. 1) x dx A) B) x x+ +C C)

x-2

1) _______

2) _______

2) 4x dx A)

B) - x cot 4x + ln -

x cot 4x +

D) - 4x cot 4x + 16 ln x cot 4x -

+C 3) _______

B) -

cos 4x - 5x sin 4x + C

cos 4x -

x sin 5x + C

cos 4x -

sin 4x + C

D) -

cos 4x -

x sin 4x + C

4) _______

A) 7 sin x + 7x cos x + C C) 7 sin x - 7 cos x + C

B) 7 sin x - x cos x + C D) 7 sin x - 7x cos x + C 5) _______

B) 24x sin

x - 48 cos

x+C

96 sin

x - 48x cos

x+C

24 sin

x + 48x cos

x+C

D) 48x sin

x + 96 cos

x+C

6) _______

A) 1

D) Diverges

7) _______

7) A) -

cos 7y +

y sin 7y +

cos 7y + C

B) -

sin 7y +

y cos 7y +

sin 7y + C

cos 7y -

y sin 7y -

cos 7y + C

cos 7y +

y sin 7y +

cos 7y + C

C) D) -

8) _______

8) A) cos 8x +

sin 8x -

x cos 8x -

sin 8x + C

sin 8x +

cos 8x -

x sin 8x -

cos 8x + C

sin 8x +

cos 8x -

sin 8x -

cos 8x +

B) C) x sin 8x -

cos 8x + C

D) x sin 8x +

cos 8x + C 9) _______

9) (Give your answer in exact form.) B) C)

A) -

10) ______

10) A)

B) [8 sin 8x + 3 cos 8x] + C

[8 sin 8x - 3 cos 8x] + C D)

sin 8x + 3 cos 8x] + C

[ sin 8x + cos 8x] + C 11) ______

11) A) 2ex - ex + C

B) 2ex - 2xex + C

C) 2xex - 2ex + C

D) xex - 2ex + C 12) ______

12) A)

B) x2e2x - xe2x +

e2x + C

x2e2x -

xe2x +

e2x + C

x2e2x -

xe2x + C

D) x2e2x -

xe2x +

e2x + C

13) ______

13) ln 3x dx A)

B) ln 3x -

ln 3x -

D) ln 3x ln 3x +

+ C 14) ______

14)

A) 6.70

B) 33.5

C) 7.9

D) 45.5 15) ______

15)

B) -0.51

A) 3.49

C) 7.8

D) 7.49 16) ______

16) ln 3x dx A) 391.88

B) -153.54

C) 462.20

D) 384.07 17) ______

17) A)

B) ln 6x -

ln 6x - x2 + x + C

+x+C

D) ln 6x -

+ 2x + C

ln 6x -

+x+C 18) ______

18) A)

B) -

-21x

- 75

- x

19) ______

19) A)

[

- 8x + 8] + C

B) -3

[

- 6x + 6] + C

[

- 8x - 8] + C 20) ______

20) A)

+ C D)

C) -

Evaluate the integral by using a substitution prior to integration by parts. 21)

A) -2.27

B) 0.39

21) ______

C) -1.33

D) -0.94 22) ______

22) A)

B) [

(6x + 7)

- 6] + C

D) [

- 1] + C 23) ______

23) A)

B) +C

+C C)

D) +C

+C 24) ______

24) A)

B) + 10 ln (x +

)+C

- 10 ln (x +

)+C

- 10 ln (x +

)+C

D) + 10 ln (x +

)+C

25) ______

25) A) C)

x x

-x + 2x

+x+C - 2x + C

B) D)

- 2x

+ 2x + C 26) ______

26) A)

B) x cos (ln x) + sin (ln x)+ C [cos (ln x) - sin (ln x)] + C

D) x[cos (ln x) + sin (ln x)] + C [cos (ln x) + sin (ln x)] + C

27) ______

27) A) C)

ln (5x + 5 xln (5x + 5

) + ln (x +1) + C

)+C

ln (5x + 5

) - 2x + C

xln (5x + 5

) + ln (x +1) - 2x + C 28) ______

28) dx A)

B) -

D) -

Evaluate the integral. The integral may not require integration by parts. 29) dx A) B) 3 ln +C sec C) D) ln

29) ______ +C

+C 30) ______

30) dx A)

B) -

ln x -

D) -

ln x -

+C 31) ______

31) dx A)

Solve the problem. 32) Find the area between y = (x - 4)ex and the x-axis from x = 4 to x = 8. A) 8 4 B) 8 C) 8 4 e -e

3e + e

33) Find the area between y = ln x and the x-axis from x = 1 to x = 5. A) 5 ln 5 - 4 B) 5 ln 5 - 5 C)

32) ______ 4 D) 8 e +e 33) ______ D) ln 5

34) Find the area of the region enclosed by y = 4x sin x and the x-axis for 0 ≤ x ≤ π. A) 4π B) 2π C) π D) 8π

34) ______

35) Find the area of the region enclosed by the curve y = x sin x and the x-axis for 7π ≤ x ≤ 8π. A) 15π B) 0 C) 14π D) 14

35) ______

36)

Fin d the

___ ___

area of 36) the region enclosed by the curve y = x cos x and the x-axis for π ≤x ≤

π. A) 7

B) 9π

C) 8π

D) 7π 37) ______

37) Find the volume of the solid generated by revolving the region bounded by the curve the A) π e

and the vertical line about the x-axis. B) π(e - 1) C) 2π(

- 1)

π(

- 1)

38) Find the volume of the solid generated by revolving the region bounded by the curve and the A) 16π

about the x-axis. B) 16

39) Find the volume of the solid generated by revolving the region in the first quadrant bounded by and the A) 14πln 7

from to B) 2π(7ln 7 - 6)

about the C) 2π(7ln 7 - 7)

38) ______

39) ______

D) 7πln 7

40) Find the volume of the solid generated by revolving the region in the first quadrant bounded by

40) ______

the coordinate axes, the curve y = , and the line x = ln 3 about the line x = ln 3. A) 2π(2 - ln 3) B) 2π(3 - ln 3) C) 2π(3 - ln 4) D) 2π(2 + ln 3) 41) Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = A) π(1 - 20

41) ______

, and the line x = 7 about the y-axis. B)

)

π(1 - 22

)

π(1 - 21

)

D) -

π(1 + 22

)

42) 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 8x, 0 ≤ x ≤ π/8 about the line x = π/8. A) B) C) D) π

42) ______

-π

43) 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 ≤ π/2 about the y-axis. A)

43) ______

- 8π 4π

+ 2

4π

4π Use integration by parts to establish a reduction formula for the integral. 44)

44) ______

A) =

-n

B) C) D)

45) ______

45) A) =-

=-a

+ na

B) C) D)

46) ______

46) A) =-2

- 2(n - 1)

B) =-

C) =-

+ 2n

47) ______

47) A) =-

x cos x +

x cos x -

B) C)

= + (n - 1) x cos x D) =

x cos x - (n - 1) 48) ______

48) A) =

x sin x - (n - 1)

x sin x + (n - 1)

x sin x +

x sin x -

B) C) D)

49) ______

49) ,n≠1 A) =

x tan x + (n - 2)

x tan x -

x tan x - (n - 2)

x tan x +

B) C) D)

50) ______

50) ,n≠1 A) =

x cot x + (n - 2)

x cot x -

x cot x +

x cot x -

B) C) D)

51) ______

51) ,n≠1 A) =

B) C) = D)

x+ 52) ______

52) ,n≠1 A) =

B) C) D) = -

x+ 53) ______

53) A) =

= ax

- an

-n

B) C) D)

Evaluate the integral. 54) Use the formula

A) C)

x + sin(

x - sin x + C

54) ______ to evaluate the integral.

x) + C

B) D)

x-x+C

x - sin(

x) + C 55) ______

55) Use the formula

A) C)

x x

to evaluate the integral.

x - ln sin x + C x + ln sin(

x) + C

x-x+C

x - ln sin(

x) + C 56) ______

56) Use the formula

to evaluate the integral.

B) x

x+x+C

57) ______

57) Use the formula

A) C)

to evaluate the integral.

x - ln

x + ln

B) D)

x+x+C

x + ln x + C

Solve the problem. 58) The charge q (in coulombs) delivered by a current i (in amperes) is given by

58) ______ where t

is the time (in seconds). A damped-out periodic wave form has current given by Find a formula for the charge delivered over time t. A) B) +C C)

+C D) +C

+C 59) The voltage v (in volts) induced in a tape head is given by

59) ______

60) The rate of water usage for a business, in gallons per hour, is given by

60) ______

where t is the time (in seconds). Find the average value of v over the interval from t = 0 to t = 4. Round to the nearest volt. A) 30 volts B) 183,853 volts C) 28,130 volts D) 2,559,851 volts

where t is the number of hours since midnight. Find the average rate of water usage over the interval A) 5.91 gallons per hour C) 6.82 gallons per hour

B) 5.68 gallons per hour D) 0.57 gallons per hour

1) D 2) A 3) C 4) D 5) C 6) C 7) D 8) B 9) D 10) A 11) C 12) B 13) B 14) B 15) A 16) D 17) D 18) B 19) A 20) C 21) B 22) D 23) D 24) B 25) D 26) C 27) D 28) C 29) D 30) B 31) B 32) C 33) A 34) A 35) A 36) C 37) C 38) D 39) B 40) A 41) B 42) C 43) B 44) A 45) B 46) C 47) A 48) C 49) D 50) C 51) A

52) C 53) D 54) D 55) D 56) B 57) C 58) C 59) B 60) B

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the integral. 1)

C) 0

1) _______

2) _______

2) dx A)

B) 8 sin 2x -

2x + C

4 sin 2x -

2x + C

4 sin 2x +

2x + C

D) 4 sin 2x -

2x + C

3) _______

A) 0

D) 4) _______

D) 5) _______

5) A)

B) x+

sin 4x +

sin 8x + C

sin 2x +

sin 4x + C

sin 2x +

sin 8x + C

D) x+

sin 4x +

sin 8x + C

6) _______

D) -

7) _______

__ __ __ _

A) 2

B) 1

C) 0

D) 4 9) _______

B) 2

C) 2

10) ______

10) dx A) 0

B) 2

11) ______

11)

A) 2 -

B) 2 +

C) 1

D) 2 12) ______

12) dx A)

B) -

x+C D)

x sec x + C

x + ln

+C 13) ______

13) A) C)

+C -2

x+C

6 tan x + 2 2

x+C

x+C 14) ______

14) A)

B) -

2x +

cot 2x + x + C

2x +

cot 2x + C

D) -

2x + cot 2x + x + C

2x + x + C 15) ______

15)

B) -

16)

C) -

D) -

16) ______

A) -

csc 8t cot 8t -

B) csc 8t

8t -

csc 8t cot 8t -

csc 8t cot 8t +

C) D) ln

+C 17) ______

17) A)

B) 9t -

tan 9t + t + C

9t -

9t +

tan 9t + t + C

D) 9t - tan 9t + t + C

9t +

tan 9t + C 18) ______

18)

C) -

D) -

19) ______

19) A) sec 7x tan 7x -

B) sec 7x tan 7x +

C) 7x tan 7x +

sec 7x tan 7x +

20) ______

20)

Give your answer in exact form. A) +

ln(7 + 4

)

ln(7 + 4

)

D) +

ln 4

ln(2

) 21) ______

21) A) 4x B)

4x - ln

4x -

4x +

C) -

4x +

4x -

D) 4x -

4x -

+C 22) ______

22) A)

B) -

cos 13x -

cos 3x + C

cos 13x -

sin 13 x + C

D) sin 3x +

sin 13x+ C

sin 3x -

sin 13x+ C 23) ______

23) A)

B) sin 4t +

sin 10t + C

sin 7t -

sin 3t + C

sin 4t -

sin 10t + C

D) sin 4t -

cos 10t + C

24) ______

24) A)

B) sin 3x -

sin 11x + C

sin 3x +

sin 11x + C

sin7x +

sin 4x + C

D) cos 3x +

cos 11x + C

25) ______

25)

26) ______

26)

27) ______

27)

28) ______

28)

Use various trigonometric identities to simplify the expression then integrate. 29) cos 5θ dθ A) B) sin 3θ -

sin 5θ -

sin 7θ + C

29) ______

sin 5θ -

sin 3θ -

sin 5θ -

sin 3θ -

sin 7θ + C

D) sin 5θ -

sin 7θ -

sin 3θ + C

sin 7θ + C 30) ______

30) A)

B) θ+C

C) C

θ+

θ+C

31) ______

31) dθ A)

B) cos 2θ -

cos 6θ + C

cos 4θ + C D)

cos 2θ -

cos 6θ + C

Use any method to evaluate the integral. 32)

32) ______

B) x cot x + C

x+C

D) -

x+C

33) ______

33) dx A) -csc x + C

B) -sec x + C

C) sec x + C

D) csc x + C 34) ______

34) A)

B) sin x - sec x + C

+C C) cos x + sec x + C

D) cos x + C 35) ______

35) A)

B) +

sin 2x -

cos 2x + C

D) -

sin 2x -

cos 2x + C

Solve the problem. 36) Find the length of the curve y = ln(csc x), π/4 ≤ x ≤ π/2 A) ln(2 ) B) ln( ) C) 1 - ln(

sin 2x -

cos 2x + C

36) ______ + 1)

D) ln(

+ 1) 37) ______

37) Find the length of the curve y = ln(sin x), π/6 ≤ x ≤ π/2 A) ln( ) B) 1 - ln( C) ln( + 2)

+ 1)

D) ln(

+ 2)

38) Find the volume generated by revolving the curve y = cos 3x about the x-axis, 0 ≤ x ≤ π/36 A) B) C) D) +

38) ______

1) B 2) B 3) D 4) B 5) C 6) B 7) A 8) D 9) C 10) B 11) A 12) A 13) B 14) A 15) D 16) C 17) A 18) C 19) D 20) A 21) D 22) A 23) D 24) B 25) D 26) C 27) D 28) C 29) B 30) D 31) C 32) C 33) A 34) C 35) C 36) D 37) D 38) B

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Integrate the function. 1) A) C)

x + ln

+ C

9 ln

1) _______

+ C

+ C 2) _______

D) 9 3) _______

2 4) _______

4) dx A)

B) ln

x + ln C

C C)

D) ln

ln C

5) _______

5) A)

B) 2

D) 2 ln

+C 6) _______

6) A)

B) +C

D) +

7) _______

7) A)

B) +

D) +

+C 8) _______

8) A)

C 9) _______

9) ,x> A)

B) ln

D) ln

+C 10) ______

10) ,x>5 A)

+C C)

D) ln

11) ______

11) dx A)

B) +

-4

+C 12) ______

12) dt A)

B) -

7t + C

D) +C

13)

___ ___

13)

-5

14) ______

14) dy A)

B) +C

D) -

y+C 15) ______

15) dt A)

B) -

D) -

16) ______

16) dt A)

B) -

+C 17) ______

17) A)

B) + C

+ C D)

+ C

+ C 18) ______

18) ,x>5 A)

B) +C

D) +C

19) ______

19) dx, x < 3 A)

B) +C

D) +C

20) ______

20) A)

B) +C

4x +C

D) 4x +

Use a trigonometric substitution to evaluate the integral. 21) A)

21) ______ B)

x+C

+C C)

D) +C

+C 22) ______

22)

A) 0.029

B) 0.234

D) -0.029

C) 0.017

23) ______

23)

A) 0.029

B) 0.025

C) 0.456

D) 0.203 24) ______

24)

B) -0.511

A) 0.061

C) 0.064

D) 0.245 25) ______

25)

A) C)

( ) +C -2

B) D)

( ) +C ( ) +C 26) ______

26) A)

B) (9 ln x) + C

(9 ln x) + C D)

C) (81

x) + C

ln ( 1 + 81

x) + C 27) ______

27) A)

B) +C

+C D)

+C 28) ______

28)

5C)

D) -

529) ______

29) A)

+C C)

D) ln

+C 30) ______

30)

A) ln 6 - ln(1 +

)

B) ln(3 + D)

ln (

) - ln(1 + +

)

ln Solve the initial value problem for y as a function of x. 31) (

+ 64) A) y= C)

31) ______

= 1, y(8) = 0 B) -

y= y

y =

-1 32) ______

32) (4 - ) A) y=

= 1, y(0) = 3 B) ln

D) y=

33) ______

33) = 1, x < 8, y(0) = 12 A)

B) y=

C) y = ln

D) y = ln

+ 12 + 12 34) ______

34) = x, x > 5, y(10) = 0 A)

-5

y= C)

y= 35) ______

35) = 1, x > 3, y(6) = ln (2 + 3

)

A) y= ln B) y = ln C) D)

+ ln 3 + ln (2 + 3

)

y = ln (x +

) + ln (2 + 3

) - ln (6 + 3

y = ln (x +

) + ln (2 + 3

)

36) ______

36) x

= 1, x > 5, y(10) = 0 A)

B) y=

D) y=

y= 37) ______

37) x

= A)

, x ≥ 9, y(9) = 0 y =

y =9 ln

D) y=

-x+9 38) ______

38) =

, y(0) = 6

B) y=

D) y=

Solve the problem. 39) Find the area bounded by y(4 + 25 A) B)

) = 5, x = 0, y = 0, and x = 5. C)

39) ______ D)

5 40) ______

40) Find the area bounded by y = A) B)

, x = 0, y = 0, and x = 2. C)

41) ______

41) Find the average value of the function y = A) B) π

over the interval from D) π

to π

42) An oil storage tank can be described as the volume generated by revolving the area bounded by

44.3

about the x-axis. Find the volume C) 1.85 1320

42) ______

of the tank. D) 177 43) ______

43) During each cycle, the velocity v (in ft/s) of a robotic welding device is given by where t is the time (in s). Find the expression for the displacement s (in ft) as a function of t if for A)

B) s=

-9

D) s=

44)

Fin d the

___ ___

length of 44) the curve between x = 0 and x = 2. A) π

C) π

D) π

1) B 2) A 3) B 4) C 5) B 6) D 7) A 8) C 9) C 10) D 11) C 12) B 13) D 14) B 15) C 16) D 17) C 18) D 19) D 20) D 21) D 22) A 23) A 24) C 25) B 26) A 27) D 28) B 29) A 30) B 31) B 32) A 33) B 34) A 35) C 36) B 37) C 38) A 39) A 40) B 41) D 42) D 43) A 44) B

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Expand the quotient by partial fractions. 1) A)

1) _______

B) +

D) +

+ 2) _______

2) A)

B) +

C) +

D) +

+ 3) _______

3) A)

B) +

C) +

D) +

+ 4) _______

4) A)

B) +

+ D)

+ 5) _______

5) A)

B) +

+ D)

+ 6) _______

6) A)

B) +

+ C)

D) +

+ 7) _______

7) A)

B) +

D) +

+ 8) _______

8) A)

B) +

D) +

+ 9) _______

9) A)

B) +

D) 1+

+ 10) ______

10) A)

B) 1+

D) 1+

Express the integrand as a sum of partial fractions and evaluate the integral. 11)

11) ______

dx A)

B) ln

+ C

D) ln

+ C

12) ______

12)

A) 3.199

B) -0.560

C) 0.560

D) 6.560

13) ______

13) A)

B) ln

3 ln

D) ln

14) ______

14) A) 2ln

+ 4ln

C) 3ln

- 4ln

B) 2ln

D) ln

+ 4ln

+C +C 15) ______

15) A) 8 ln

- 4ln

B) -8 ln

+ 4ln

D) -

+ 4ln

-8 ln

+ C 16) ______

16) A)

+ 10ln + 5ln C) -5 ln + 10ln - 5ln

-15 ln

+ 10ln

D) -15 ln

+ 10ln

+ 5ln

+C +C

+C 17) ______

17) dx A) ln

- ln

C) -3 ln

+ ln

+ 4 ln

B) 2 ln

- 3 ln

+ 4 ln

D) 2 ln

+ 4 ln

- 3 ln

+C 18) ______

18) A)

B) - 9x + 9 ln

- 9x + 27 ln

- 9x - 27 ln

27 ln

+C 19) ______

19)

B) 12 ln 5 -

48 ln

64 ln

D) 48 ln 5 - 12ln 4 +

20)

___ ___

20) dx A)

B) 4 ln

2 ln

4 ln

D) 4 ln

21) ______

21) A)

B) +

D) +

22) ______

22) dx A) ln

B) C) D) +C 23) ______

23)

D) -

24) ______

24) dx A) 4 ln

4 ln

x + C

B) ln

C) ln

D) ln 25)

+ C 25) ______

B) 5 ln

5 ln

D) ln

+C 26) ______

26)

A) 1.07

B) 2.14

C) 1.74

D) 4.28 27) ______

27)

A) 5.12

C) -2.56

B) 3.12

D) 2.56 28) ______

28) dx A)

B) 2 ln

D) -

+C 29) ______

29) A)

B) + ln

D) +

+C 30) ______

30) dx A)

B) 3 ln

6 ln

+ ln

3 ln

+ ln

3 ln

+ ln

31) ______

31) dx A) 5 ln

+ ln

+C +C

B) 5 ln

+ ln

5 ln

- ln

C) D)

7 ln

+ ln +

+C 32) ______

32) dx A) ln

B) -

C) ln

D) +C 33) ______

33) dx A)

B) (25x) +

(5x) -

D) (5x) -

Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions. 34) 34) ______ A) 7 ln

- 7 ln

+ 7 ln

- 7 ln

+ 7 ln

D) + 7 ln

- 7 ln

35) ______

35) A) + 8 ln

- 8 ln

B) + 16x - 32 ln

+ 32 ln

+ 16x + 32 ln

- 32 ln

+ 16x + 32 ln

- 32 ln

C) D)

36)

+C 36) ______

B) - 10x + 15ln

75ln

D) - 10x - 75ln

- 10x + 75ln

+C 37) ______

37) A)

B) 2x + 7ln

+ 8ln

2x + 6ln

+ 8ln

D) + 6ln

+ 8ln

2x + 6ln

+ 8ln

+C 38) ______

38) A)

B) +

+ ln

D) + ln

+C 39) ______

39) A)

B) + ln

- x ln

+ ln

D) + ln

+C 40) ______

40) A) + ln

+ ln

+ 3 ln

+ ln

- ln

+ 3 ln

+ ln

+ (2y + 1) ln

B) C) D) +C

Evaluate the integral. 41)

41) ______

B) ln

D) ln

42) ______

42) dt A)

B) 3 ln

3 ln

D) 3 ln

4 ln

+C 43) ______

43) A) ln

( )+C

B) C) D) ln

t+C 44) ______

44) A)

B) +t-

+t-

- t + ln

D) + ln

45) ______

45) A)

B) ln ln

- ln

D) ln

+C 46) ______

46) A)

B) 4 ln

3 ln

D) 3 ln

3 ln

+C 47) ______

47) A) B)

- ln

-5

(sin t) + C

ln -ln

-3

t+C

+ C) -ln

-ln

-3

D) (sin t) + C 48) ______

48) dx A)

B) +

+ ln

D) + ln

5x + ln

+C 49) ______

49) dx A) - ln

ln(

+ 1) + C

- ln

-5

x + ln(

- ln

-3

B) + 1) + C

C) ln(

+ 1) + C

D) ln(

+ 1) + C

Solve the initial value problem for x as a function of t. 50) (

- 5t + 6) A)

x = -ln C) x = -ln

50) ______

(t > 3), x(4) = 0 B) x = ln

- ln

+ ln 2

D) x = -ln

+ ln

+ 2

+ ln 2

+ ln

+ ln 2

51) ______

51) (t + 4) = +1, t > -4 , x(2) = tan 1 A) x = tan [ ln - ln 6] C) x=

B) x = tan [ ln D)

- ln 6 + 1] [ ln

- ln 6 + 1]

52) ______

52) (

+ 4t) A) x=

= 4x + 8, x(1) = 1 B) -2

C) x = 15t + ln

x = 15 ln -2

-2

-2 53) ______

53) (2

-2 A)

+ t - 1)

= 3, x(2) = 0

x = ln

t -

x = ln

t - ln

x = ln

t +

ln 4.5

B) C)

+ ln 4.5

D) x = ln

ln 4.5

Solve the problem by integration. 54)

54) ______

Find the first-quadrant area bounded by A) B) 12 ln

and C)

D) ln

ln 55) ______

55) Find the area bounded by hundredth. A) 0.51 B) 4.09

y = 0, x = 1, and x = 3. Round to the nearest C) 1.28

D) 10.22 56) ______

56) Find the volume generated by rotating the area bounded by

and

about the A)

B) π ln

57)

C) π ln

D) π ln

π ln

Find the x-coordinate of the centroid of the area bounded by A) 1.96

B) -1.96

C) 1.42

and

57) ______

D) 7.11 58) ______

58) The general expression for the slope of a curve is passes through (1, 0). A) y = ln

Find the equation of the curve if it B) y = ln

D) y = ln

y = ln

59) The current i (in A) as a function of the time t (in s) in a certain electric circuit is given by Find the total charge that passes a given point in the circuit during the first

thre 59) e seco nds.

______ A) 11.675 C

B) 3.871 C

C) 1.297 C

D) 3.892 C 60) ______

60) The force F (in N) applied by a stamping machine in making a certain computer part is where x is the distance (in cm) through which the force acts. Find the work done by the force from A) 0.0826 N∙ cm

to B) 7.5223 N∙ cm

C) 0.0054 N∙ cm

D) 0.0163 N∙ cm

61) Under specified conditions, the time t (in min) required to form x grams of a substance during a chemical reaction is given by

61) ______

Find the equation relating t and x if x = 0 g

when A)

B) t=

D) t=

62) ______

62) Find the first-quadrant area bounded by A) 6 ln 280 B)

and x = 6. C) 3 ln 70

D) 6 ln 70

6 ln 63) ______

63) Find the volume generated by revolving the first-quadrant area bounded by and x = 5 about the y-axis. A) B) 5π ln 10,044 C) D) π ln 10,044

5π ln

10π ln 64) ______

64) Find the volume generated by revolving the first-quadrant area bounded by

and

about the A)

B) π

C) π

D) π

65) Under certain conditions, the velocity v (in m/s) of an object moving along a straight line as a

function of the time t (in s) is given by object during the first 3 s. A) 2.244 m B) 2.886 m

65) ______

Find the distance traveled by the C) 1.277 m

D) 10.581 m

66) By a computer analysis, the electric current i (in A) in a certain circuit is given by

where t is the time (in s). Find the total charge that passes a point in the circuit in the first 0.4 s. A) 0.0003 C B) 0.0016 C C) 0.0013 C D) 0.0001 C

66) ______

67) ______

67) Find the x-coordinate of the centroid of the first-quadrant area bounded by and A) -1.815

B) 0.789

D) -0.908

C) 1.578

68) ______

68) The slope of a curve is given by through A) y=-

Find the equation of the curve if it passes

. B) +

D) y=-

1) D 2) B 3) D 4) D 5) D 6) C 7) D 8) D 9) C 10) D 11) D 12) C 13) A 14) A 15) B 16) D 17) B 18) B 19) B 20) C 21) B 22) C 23) A 24) B 25) B 26) B 27) D 28) C 29) C 30) D 31) B 32) A 33) C 34) D 35) C 36) D 37) D 38) A 39) B 40) A 41) C 42) A 43) B 44) A 45) C 46) B 47) D 48) B 49) C 50) C 51) B

52) D 53) A 54) D 55) C 56) A 57) D 58) D 59) D 60) D 61) A 62) D 63) C 64) A 65) B 66) A 67) C 68) A

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the integral. 1) A)

1) _______

B) ln

D) ln

+C 2) _______

2) A)

B) 2

+ 2 ln

C) 2

- 2 ln

3) _______

3) A)

B) -

D) -

+C 4) _______

4) A)

B) +

D) +

+C 5) _______

5) A)

B) -5

D) ln

-5

+C 6) _______

6) A)

B) + 4 ln

D) -

- 4 ln

+C 7) _______

B) 4

- 2 ln

D) -2

-x

+C 8) _______

8) A)

B) +C

- 2 ln

D) +2

+C 9) _______

9) dx A)

B) -

D) -

+C 10) ______

10) A)

B) -

- 4 ln

D) -

+C 11) ______

11) A)

B) ln

D) +C

+C 12) ______

12) A)

B) -

ln -

ln ln +C + C 13) ______

13) dx A)

B) -

D) -

+C 14) ______

14) dθ A)

B) sin

θ-

sin

θ+C

sin

θ+

sin

θ+C

D) cos

θ +

cos

θ+ C

sin 3θ +

sin 7θ + C 15) ______

15) dt A)

B) sin

sin

t+C

cos

t+C

D) sin

sin

t+C

sin t -

sin 7t + C 16) ______

16) A)

B) 4

C) 4

4 D) 4

+C +C

+C 17) ______

17)

A) Diverges

D) 1

18) ______

18)

C) Diverges

D) 1

19) ______

19) dx A)

B) x-

C C)

D) x-

Evaluate the integral by making a substitution and then using a table of integrals. 20) A)

20) ______

B) +

C C)

D) +

+C 21) ______

21) A)

B) 16

D) - 4 ln

+C 22) ______

22) A)

B) +C

D) +C

+C 23) ______

23) A)

B) -

+ ln

D) -

+C 24) ______

24) A)

B) -

ln(5x + 3) + C

D) ln(5

+ 3x)

25) ______

25) A)

B) ln

D) ln

26) ______

26) A) ln

+ x - 6 ln

C) ln

- 6 ln

B) - 6 ln

D) ln

+C - 6 ln

+C 27) ______

27) dx A) (x + 2)

(x + 3)

B) C) (2x + 3) D) (x + 3) tan-1

+C +C

28) ______

28) A) ln

- sin t + C

B) ln

C) ln

+ sin t + C

D) ln

- sin t + C + cos t + C 29) ______

29) dθ A)

B) -

D) -

Use reduction formulas to evaluate the integral. 30) A)

B) x-

x+C

D) x-

31)

30) ______

x+C

(

x) + C 31) ______

B) sin 4x -

4x + C

2 sin 4x -

4x + C

sin 4x +

4x + C

D) sin 4x -

4x + C

32) ______

32) A) -

3x cos 3x -

cos 3x + C

3x sin 3x -

cos 3x + C

3x cos 3x -

sin 6x +

3x cos 3x -

cos 3x + C

B) C) +C

33) ______

33) A)

B) -

sin 2x

2x +

cos 4x + C

sin 2x

2x +

sin 4x + C

sin 2x

2x +

sin 4x + C

D) -

sin 2x

2x +

sin 4x + C

34) ______

34) A) 5 ln

+ 5 sin x + C

5 ln

sin x + C

D) 5 ln

x+C

x+C 35) ______

35) A)

B) 4x -

4x +

4x -

4x - ln

4x -

D) C

4x +

4x -

36) ______

36) A)

B) -

3x +

cot 3x + C

3x -

cot 3x + x + C

3x +

cot 3x + x + C

D) -

3x +

cot 3x - x + C

37) ______

37) A)

B) +C + C D)

C) +C

+C 38) ______

38) A) sec 3x tan 3x +

sec 3x tan 3x -

B) C) 3x tan 3x +

D) sec 3x tan 3x +

+C 39) ______

39) A)

B) -

cot

csc

D) -

cot

Evaluate the integral by making a substitution (possibly trigonometric) and then applying a reduction formula. 40)

40) ______

A) -

[csc(

- 2) cot(

- 2) + ln

]+C

[csc(

- 2) cot(

- 2) - ln

]+C

B) C) [csc(

- 2) cot(

- 2) + ln

]+C

[csc(

- 2) cot(

- 2) - ln

]+C

41) ______

41)

A) 5[ C)

- ln( + ln(

+ 1)]

+ 1)

[ D) 5[

+ ln(

+ 1)]

+ ln(

+ 1)] 42) ______

42) dθ A) sec

tan

+ ln

sec

tan

- ln C) -sec

+C tan

+ ln

D) -sec

tan

+ ln

C 43) ______

43) A)

B) +C

D) +C

Find the surface area or volume. 44) Use an integral table and a calculator to find to two decimal places the area of the surface generated by revolving the curve A) 34.93 B) 24.84

about the x-axis. C) 30.48

D) 42.69

45) Use an integral table and a calculator to find to two decimal places the area of the surface generated by revolving the curve A) 372.65 B) 186.32

about the x-axis. C) 261.32

D) 7.21

about the x-axis. C) 10,148.47

47) ______

about the D) 14.42

48) Use substitution and a table of integrals to find, to two decimal places, the area of the surface generated by revolving the curve A) 12,238.78 B) 11,342.93

46) ______

about the

47) 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 x-axis. A) 1.48 B) 9.29 C) 7.21

45) ______

D) 319.41

46) 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 x-axis. A) 2.29 B) 14.42 C) 1.15

44) ______

48) ______

D) 9376.82

49) The region between the curve y = sinx, 0 ≤ x ≤ 1.3, 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) 1.56 B) 1.73 C) 1.67 D) 1.64

49) ______

50)

50) ______ The region between the curve y = , -1.4 ≤ x ≤ 1.3, and the x-axis is revolved about the x-axis to generate a solid. Use a table of integrals to find the volume of the solid generated to two decimal places. A) 5.93 B) 5.03 C) 5.29 D) 6.41

Use a CAS to perform the integration.

51) ______

51) A)

B) ln x -

ln x +

ln x -

D) ln x -

52) ______

52) A)

52) D

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the Trapezoidal Rule with n = 4 steps to estimate the integral. 1)

A) 16

B) 32

C) 64

D) 40 2) _______

A) 72

B) 36

D) 18

3) _______

A) 30

B) 44

D) 22

4) _______

5) _______

6) _______

7) _______

8) _______

1) _______

__ __ __ _

B) -(1 + -

)π

D) -

10) ______

10)

B) -

C) -1 -

Use Simpson's Rule with n = 4 steps to estimate the integral. 11)

A) 15

B) 9

11) ______

C) 18

D) 36 12) ______

12)

A) 116

B) 58

C) 29

13) ______

13)

14) ______

14)

15) ______

15)

16) ______

16)

17)

17) ______

18) ______

18)

19) ______

19)

A) -(1 + 2

)π

C) -

D) -

π 20) ______

20)

A) -1 - 2

C) -

D) -

Solve the problem. 21) Find an upper bound for A)

21) ______ in estimating B) 0

with C)

steps. D)

22) ______

22) Find an upper bound for A)

in estimating B)

with n = 9 steps. D)

23) ______

23) Find an upper bound for A)

in estimating B)

with n = 8 steps. D)

24) ______

24) Find an upper bound for A)

in estimating B)

with n = 4 steps. C)

25) ______

25) Find an upper bound for in estimating with n = 9 steps. Give your answer as a decimal rounded to four decimal places. A) 0.3828 B) 0.0638 C) 0.1914 D) 0.6013

26) ______

26) Find an upper bound for A)

in estimating B)

with n = 4 steps. D) 0

27) ______

27) Find an upper bound for A)

in estimating B)

with n = 6 steps. D) 1

28) ______

28) Find an upper bound for A)

in estimating B)

with n = 10 steps. D)

29) ______

29) Find an upper bound for A)

in estimating B)

with n = 4 steps. C)

30) ______

30) Find an upper bound for in estimating with n = 4 steps. Give your answer as a decimal rounded to five decimal places. A) 0.13282 B) 0.13955 C) 0.16603 D) 0.10432 31) Estimate the minimum number of subintervals needed to approximate the integral

with an error of magnitude less than A) 0

B) 1

using the Trapezoidal Rule. C) 2

D) 3

32) Estimate the minimum number of subintervals needed to approximate the integral

with an error of magnitude less than A) 92 B) 949

using the Trapezoidal Rule. C) 475

using the Trapezoidal Rule.

32) ______

D) 238

33) Estimate the minimum number of subintervals needed to approximate the integral

with an error of magnitude less than

31) ______

33) ______

A) 174

B) 317

C) 142

D) 347

34) Estimate the minimum number of subintervals needed to approximate the integral

with an error of magnitude less than A) 8 B) 9

using the Trapezoidal Rule. C) 71

D) 18

35) Estimate the minimum number of subintervals needed to approximate the integral

with an error of magnitude less than A) 50 B) 197

using the Trapezoidal Rule. C) 279

using Simpson's Rule. C) 2

using Simpson's Rule. C) 12

using Simpson's Rule. C) 1

using Simpson's Rule. C) 570

using Simpson's Rule. C) 18

39) ______

D) 6

40) Estimate the minimum number of subintervals needed to approximate the integral

with an error of magnitude less than A) 20 B) 22

38) ______

D) 4

39) Estimate the minimum number of subintervals needed to approximate the integral

with an error of magnitude less than A) 48 B) 24

37) ______

D) 14

38) Estimate the minimum number of subintervals needed to approximate the integral

with an error of magnitude less than A) 3 B) 2

36) ______

D) 0

37) Estimate the minimum number of subintervals needed to approximate the integral

with an error of magnitude less than A) 10 B) 22

35) ______

D) 99

36) Estimate the minimum number of subintervals needed to approximate the integral

with an error of magnitude less than A) 1 B) 10

34) ______

D) 21

40) ______

41) 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.

A) 169.5 ft

B) 257.5 ft

C) 189 ft

D) 339 ft

42) 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.

A) 170.33 ft

B) 126 ft

C) 169.5 ft

B) 79.1°F

C) 65.92°F

42) ______

D) 168.33 ft

43) 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.

A) 65.94°F

41) ______

D) 76.67°F

44) 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.

43) ______

___ ___

44)

A) 9975 gal

B) 9983 gal

C) 11,350 gal

D) 9050 gal

45) 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.

A) 9983 gal

B) 11,475 gal

C) 9050 gal

D) 9975 gal

46) 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.

28,500

29,000

34,500

8700

7700

10,200

46) ______

28,000

47) A rectangular swimming pool is being constructed, 18 feet long and 100 feet wide. The depth of 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.

45) ______

5800

48) A rectangular swimming pool is being constructed, 18 feet long and 100 feet wide. The depth of 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.

47) ______

___ ___

48)

7700

10,200

5800

8700 49) ______

49) The growth rate of a certain tree (in feet) is given by y = + , where t is time in years. 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) 3.41 ft B) 2.34 ft C) 3.68 ft D) 5.11 ft 50) 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.]

50) ______

Circumferences

101.552

62.876

61.856

105.479 51) ______

51) The length of one arch of the curve y = 3 sin 2x is given by

L= Estimate L by the Trapezoidal Rule with n = 6. A) 6.1995 B) 5.8169

C) 6.2807

D) 6.4115 52) ______

52) The length of the ellipse x = a cos t, y = b sin t, 0 ≤ t ≤ 2π is

Length = 4a where e is the ellipse's eccentricity. Use Simpson's Rule with n = 6 to estimate the length of the ellipse when a = 2 and e = . A) 12.2097 B) 11.2546 C) 11.8956 D) 12.1751 53) Estimate the area of the surface generated by revolving the curve y = cos 2x, 0 ≤ x ≤

about the

x-ax the is. Trapez Use oidal

Rule 53) with n = 6. A) 4.652 54)

___ ___ B) 4.606

C) 5.108

Estimate the area of the surface generated by revolving the curve y = 2 x-axis. Use Simpson's Rule with n = 6. A) 996.028 B) 1021.107 C) 1024.885

D) 7.091 , 0 ≤ x ≤ 3 about the

54) ______

D) 1007.254

Provide an appropriate response. 55) Given that we know the Fundamental Theorem of Calculus, why would we want to develop numerical methods for definite integrals?

55) ______

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. B) Only iii is correct. C) Only i is correct. D) Only ii is correct. 56) 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 C) D) None of the above is correct.

56) ______

57) 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) either an even or odd number D) an odd number

57) ______

58) The "Simpson" sum is based upon the area under a ? . A) rectangle B) triangle C) trapezoid

58) ______ D) parabola 59) ______

59) 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) ii and iv

B) i and iii

C) i, iii, and iv

60) The error formula for the Trapezoidal Rule depends upon i) f(x). ii) f'(x). iii) f'' (x). iv) the number of steps A) i and iii B) ii and iv C) i, iii, and iv

D) iii and iv 60) ______

D) iii and iv

1) B 2) B 3) D 4) C 5) C 6) C 7) B 8) B 9) C 10) B 11) C 12) B 13) A 14) B 15) D 16) C 17) C 18) D 19) D 20) D 21) B 22) A 23) B 24) D 25) A 26) D 27) A 28) A 29) C 30) B 31) B 32) C 33) D 34) D 35) C 36) C 37) A 38) B 39) B 40) A 41) A 42) A 43) C 44) A 45) A 46) D 47) A 48) D 49) A 50) C 51) C

52) A 53) B 54) C 55) A 56) C 57) B 58) D 59) D 60) D

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the improper integral or state that it is divergent. 1)

A) 1.569

B) -1.569

C) Divergent

1) _______

D) 0.482 2) _______

B) ln 6

C) -

ln 11

ln 11 3) _______

B) ln 5

C) ln 4

D) 4 ln 5 -

ln 5 4) _______

B) 0

D) π + 8

5) _______

A) 2

B) -1

C) 1

D) Divergent 6) _______

A) 0

B) 9

C) 18

D) Divergent 7) _______

B) ln 2

C) ln 4

D) Divergent

8) _______

D) Divergent -

9) _______

A) Divergent

10) ______

10)

A) ln 6

C) ln 7

D) ln 2 -

ln 2 11) ______

11)

B) 0

D) -5π 12) ______

12)

A) 0

B) -1

C) Divergent

D) 1 13) ______

13)

B) 0

D) Divergent 14) ______

14)

A) 0.3333

B) Divergent

C) -4.667

D) -0.5556 15) ______

15)

25 ln

π 16) ______

16)

A) -9

D) 0 17) ______

17)

C) 25 ln 2 25 ln

18)

D) 25 ln 18) ______

A) 2.6667

B) 1.6667

C) 1.3333

D) Divergent 19) ______

19)

A) 2

B) 4

C) Divergent

D) -2 20) ______

20)

A) Divergent

B) 25

C) 0

D) -25

Evaluate the improper integral. 21)

A) -8

21) ______

B) 0

C) 4

D) 8 22) ______

22)

A) 3

B) 0

C) 5

D) 15 23) ______

23)

C) 8

D) 1

24) ______

24)

25) ______

25)

A) -2

B) 5

C) 10

D) 4 26) ______

26)

A) 8 ln5 - 4

C) 8 ln20 - 4 -

D) 8 ln20 - 8

ln20 + 4 27) ______

27)

A) 64

B) 8

Determine whether the improper integral converges or diverges. 28)

C) -64

D) -8

___ ___

28)

A) Converges

B) Diverges 29) ______

29)

A) Diverges

B) Converges 30) ______

30) dx A) Converges

B) Diverges 31) ______

31)

A) Converges

B) Diverges 32) ______

32)

A) Diverges

B) Converges 33) ______

33)

A) Converges

B) Diverges 34) ______

34)

A) Converges

B) Diverges 35) ______

35)

A) Diverges

B) Converges 36) ______

36)

A) Diverges

B) Converges 37) ______

37)

A) Converges 38)

B) Diverges 38) ______

A) Diverges

B) Converges 39) ______

39)

A) Diverges

B) Converges 40) ______

40)

A) Converges

B) Diverges 41) ______

41)

A) Converges

B) Diverges 42) ______

42)

A) Diverges

B) Converges 43) ______

43)

A) Diverges

B) Converges 44) ______

44)

A) Diverges

B) Converges 45) ______

45)

A) Diverges

B) Converges 46) ______

46)

A) Diverges

B) Converges 47) ______

47)

A) Converges

B) Diverges 48) ______

48)

A) Diverges 49)

B) Converges

___ ___

49)

A) Converges

B) Diverges 50) ______

50)

A) Diverges

B) Converges 51) ______

51)

A) Diverges

B) Converges 52) ______

52)

A) Converges

B) Diverges 53) ______

53)

A) Diverges

B) Converges 54) ______

54)

A) Converges

B) Diverges 55) ______

55)

A) Converges

B) Diverges 56) ______

56) + )dx A) Converges

B) Diverges 57) ______

57)

A) Converges

B) Diverges 58) ______

58)

A) Diverges

B) Converges

Find the area or volume. 59) Find the area of the region in the first quadrant between the curve y = A)

and the x-axis.

59) ______

e 60)

Find the area of the region bounded by the curve y = 10 , the x-axis, and on the left by x = 1. A) 100 B) 5 C) 20 D) 10

60) ______

61) ______

61) Find the area under the curve y = A) B) 8

bounded on the left by x = 15. C)

62) ______

62) Find the area under y = A)

in the first quadrant. B) 5π C) 10π

π 63) ______

63) Find the area between the graph of y = A) 14 B) 28

and the x-axis, for -∞ < x ≤ 0. C) 7 D) 1 64) ______

64) Find the volume of the solid generated by revolving the region under the curve to A) 4π

from

about the B) 8π

C) 4

D) π

65)

Find the volume of the solid generated by revolving the region under the curve first quadrant about the y-axis. A) 64π B) 8π C) 4π D)

in the 8

66) Find the volume of the solid generated by revolving the region in the first quadrant under the curve A) 49π

bounded on the left by x = 1, about the x-axis. B) 7π C) π

67)

68)

65) ______

66) ______

D) π

Find the volume of the solid generated by revolving the area under y = 8 about the y-axis. A) 1 B) 32π C) 8π

in the first quadrant

Find the volume of the solid generated by revolving the area under y = 8 about the x-axis A) 64π B) 128π C) 32π

in the first quadrant

67) ______

D) 16π

D) 8π

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response.

68) ______

69) _____________

69) A student wishes to find the integral limit

of a function that has the property

Why can this not be done? 70) _____________

70) A student wishes to take the integral over all real numbers of f(x) = , if x < 0 x > 0 and claims this is zero because -∞ + ∞ equals zero. What is wrong with this thinking?

, if

71) _____________

71) A student claims that always exists, as long as a and b are both positive. Refute this by giving an example of a function for which this is not true.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 72) A student knows that A) Yes, -37

= 37. Can

72) ______

be found, and if so, what is it? B) No 73) ______

73) A student knows that A) No

diverges, but needs to investigate

, where

Does this integral necessarily also diverge? B) Yes 74) ______

74) A student knows that A) Yes

converges. Does

also necessarily converge? B) No

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 75) 75) _____________ A student needs why?

. Is this integral the same as 2

, and if so,

76) _____________

76) i) Show that

diverges, and hence

ii) Show that

diverges.

= 0. 77) _____________

77) Show that

78) Here is an argument that ln 2 = ∞ - ∞. Where does the argument go wrong?

ln 2 = ln 1 + ln 2 = ln

=∞-∞ 1 - ln =

ln -

ln =

dx =

dx -

d x =

78) _____________

79) _____________

79) (a)

Show that

converges.

(b)

Show that

≤ 0.0002.

(c)

Suppose is approximated by . Based on your answer to part (b), what is the maximum possible error?

(d)

Use a numerical method to estimate the value of

(e)

Determine whether converges or diverges, and justify your answer. If it converges, estimate its value to an accuracy of at least two decimal places.

80) _____________

80) Let f(x) = , x ≥ 1. (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.

81) _____________

81) (a)

Express

(b)

Evaluate

Evaluate . Find a solution to the initial value problem: =

as a sum of partial fractions. .

y(4) = 12. 82) _____________

82) (a) Find the values of p for which

dx converges.

(b) Find the values of p for which

dx diverges.

83) _____________

83) (a) Show that

dx =

< 0.0092 and hence that

(b) Explain why this means that dx can be replaced by introducing an error of magnitude greater than 0.0092.

dx < 0.0092.

dx without

84) _____________

84) The Cauchy density function, f(x) =

, occurs in probability theory. Show that

85) _____________

85) The standard normal probability density function is defined by f(x) =

(a) Show that dx = . (b) Use the result in part (a) to show that the standard normal probability density function has mean 0. 86) _____________

86) The standard normal probability density function is defined by f(x) =

(a) Use the fact that dx = 1 to show that dx = . (b) Use the result in part (a) to show that the standard normal probability density function has variance 1.

1) D 2) D 3) A 4) C 5) C 6) A 7) A 8) C 9) C 10) D 11) C 12) D 13) B 14) D 15) C 16) C 17) C 18) D 19) C 20) A 21) D 22) D 23) A 24) C 25) C 26) C 27) B 28) A 29) B 30) B 31) B 32) A 33) A 34) B 35) A 36) A 37) A 38) B 39) B 40) A 41) A 42) A 43) B 44) B 45) B 46) A 47) A 48) A 49) A 50) A 51) A

52) A 53) B 54) A 55) A 56) A 57) B 58) B 59) C 60) D 61) C 62) D 63) A 64) A 65) B 66) C 67) D 68) C 69) The only way the limit of the integral can exist is if the limit of the function is zero. 70) Infinity cannot be added like this. 71) Answers will vary, but f(x) = where a < c < b and d a positive integer is a family of examples. 72) B 73) B 74) B 75) Yes, the function is symmetric about the y-axis. 76) i)

ii)

ln (

+ 1)

ln (

+ 1)

(ln (

=∞

+1) - ln (

+1)) =

0=0

77) +

78) The mistake is in the second to last step: 79) (a) Possible answer:

Note that

Since

is a proper integral and

by direct comparison to

(b) (c)

≤ 0.0002

= 0.0002.

= ln 2

converges

converges.

(d)

1.209 (e) 80) (a)

diverges 1

(b)

(c)

2π

converges by using the limit comparison

(d)

test with the function y = 7.61

81) (a)

(b)

+ 2ln

(c)

2x + ln

+ 2ln

(d) y = 2(x + 2) 82) (a) The integral converges if p <1. (b) The integral diverges if p ≥ 1. 83) (a)

dx =

(b) Since 0 <

dx =

≤

dx =

for x ≥ 2,

dx +

dx ≤

dx ≈

< 0.0092

dx . So

dx < 0.0092. Therefore,

dx, with an error of magnitude no greater than 0.0092.

84) dx =

dx =

function,

dx =

(a)

dx =

and

dx =

. Since f(x) is an even

= 1.

85)

(b) Since y =

dx =

is an odd function,

dx = -

mean of the distribution is equal to +

dx =

dx = -

dx +

dx = -

= 0.

86) (a) Since

dx = 1 and f(x) is an even function,

dx =

. The

. Then

dx = . (b) The varia nce of the distri butio n is equal to

dx =

dx +

dx = + = 1.

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the function is a probability density function over the given interval. 1) f(x) = x, 0 ≤ x ≤ 8 A) Yes

1) _______

B) No 2) _______

2) f(x) = x, 4 ≤ x ≤ 7 A) No

B) Yes 3) _______

3) f(x) = over A) Yes

B) No 4) _______

4) f(x) = A) No

B) Yes 5) _______

5) f(x) = 9 sin 9x over A) Yes

B) No

Find the indicated probability. 6) f(x) = e-x; [0, ∞), P(x ≥ 1) A) 0.3679 B) 0 C) 0.6321 D) The function f(x) is not a probability density function.

6) _______

7) _______

7) f(x) = ; [0, ∞), P(1 ≤ x ≤ 5) A) 0.0820 B) 0.1231

C) 0.4923

D) 0.0615 8) _______

8) f(x) = cos x over A) 0.365

,P B) 0.658

C) -0.233

D) 0.707 9) _______

9) f(x) = 3x A) 0.936

over [0, 1], P(0.6 < x) B) 0.856

C) 0.396

D) 0.496

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. 10) f(x) = k(11 - x) over [0, 11] 10) ______ A) B) C) 11 D) 121

11) ______

11) f(x) = over [1, 14] A) ln 14

12)

f(x) = k A)

D) 1 - ln 14

12) ______

over [0, 16] B)

-1

13) ______

13) f(x) = x over [4, k] A) 2 14)

f(x) = 7 A)

B) 3

C) 20

D) 2 14) ______

over [0, k] ln 7

C) ln

D) -

ln 7

For the given probability density function, over the stated interval, find the requested value. 15) f(x) = x, over [1, 2]; Find the mean. A) 0 B)

15) ______

16) ______

16) f(x) = A)

, over [-2, 4]; Find the mean. B)

D) 20

17) ______

17) f(x) = A)

; Find the median. B)

D) 8 ln 18) ______

18) f(x) = A)

x over [0,

]; Find the median. B)

Solve the problem. 19) The number of new mini-vans sold by a particular salesperson during the month of March is exponentially distributed with a mean of 3. What is the probability that the salesperson will sell between 3 and 7 mini-vans in March? A) 0.126 B) 0.295 C) 0.159 D) 0.271 20) 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

19) ______

greathan 6 ter years.

20)

______ A) 0.7769

B) 0.1942

C) 0.2231

D) 0.0558

21) The time between major earthquakes in the Alaska panhandle region can be modeled with an exponential distribution having a mean of 620 days. Find the probability that the time between a major earthquake and the next one is less than 300 days. A) 0.6164 B) 0.0010 C) 0.3836 D) 0.0006

21) ______

22) The lifetime of an appliance that costs $300 is exponentially distributed with a mean of 3 years. The manufacturer gives a full refund if the appliance fails in the first year following its purchase. If the manufacturer sells 500 printers, how much should it expect to pay in refunds? A) $107,480 B) $425 C) $42,520 D) $38,268

22) ______

23) The amount of time that goes by between a new driver getting a license and the moment the driver is involved in an accident is exponentially distributed. An insurance company observes a sample of new drivers and finds that 80% are involved in an accident during the first 3 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 4.5 years after receiving their license? A) ≈ 18 drivers B) ≈ 191 drivers C) ≈ 155 drivers D) ≈ 182 drivers

23) ______

24) 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 450 seconds and a standard deviation of 50 seconds. Find the probability that a randomly selected boy in secondary school can run the mile in less than 335 seconds. A) 0.5107 B) 0.4893 C) 0.9893 D) 0.0107

24) ______

25) 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 440 seconds and a standard deviation of 40 seconds. Find the probability that a randomly selected boy in secondary school will take longer than 348 seconds to run the mile. A) 0.5107 B) 0.4893 C) 0.9893 D) 0.0107

25) ______

26) 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

26) ______

and a standard deviation of 0.04 ounce. Find the probability that the bottle contains fewer than A) 0.0062

of beer. B) 0.4938

C) 0.5062

D) 0.9938

27) 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 12.14 onces and a standard deviation of 0.04 ounce. Find the probability that the bottle contains more than 12.14 ounces of beer. A) 1 B) 0.5 C) 0.4 D) 0

27) ______

28) 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

28) ______

and a standard deviation of 0.04 ounce. Find the probability that the bottle contains between 12.15 and 12.21 ounces. A) 0.8351 B) 0.1649 C) 0.8475 D) 0.1525 29) The length of time it takes college students to find a parking spot in the library parking lot

foll ows a

normal 29) distributi on with a mean of 6.0 minutes and a standard deviation of 1 minute. Find the probabili ty that a randoml y selected college student will find a parking spot in the library parking lot in less than 5.5 minutes. A) 0.1915

___ ___

B) 0.3551

C) 0.3085

D) 0.2674

30) 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 5.5 minutes and a standard deviation of 1 minute. Find the probability that a randomly selected college student will take between 4.0 and 6.5 minutes to find a parking spot in the library lot. A) 0.7745 B) 0.2255 C) 0.4938 D) 0.0919

30) ______

31) The pulse rate for a species of small mammal follows a normal distribution with a mean

31) ______

beats per minute and standard deviation beats per minute. In a population of 900 of these mammals, how many would you expect to have a pulse rate between 195 and 243? A) ≈ 621 of these mammals B) ≈ 486 of these mammals C) ≈ 450 of these mammals D) ≈ 432 of these mammals 32) The lifespan for a pair of walking shoes is normally distributed with a mean of

32) ______

steps and standard deviation of steps. If a manufacture produces 5000 pairs of these shoes, how many can be expected to last for at least 762,800 steps? A) ≈ 1094 pairs B) ≈ 1059 pairs C) ≈ 1159 pairs D) ≈ 1004 pairs 33) There are 3 balls in a hat; one with the number 3 on it, one with the number 5 on it, and one with the number 9 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) {3 5 9 H, 3 5 9 T} B) {3 H T, 5 H T, 9 H T} C) {3 H, 3 T, 5 H, 5 T, 9 H, 9 T} D) {3 H, 5 H, 9 H}

33) ______

34) There are 3 balls in a hat; one with the number 1 on it, one with the number 6 on it, and one with the number 7 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) B) C) D)

34) ______

35) 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 = 4? A) B) C) D) 3

35) ______

36) 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 > 10? A) B) C) D) 3

36) ______

37) A spinner marked off in five equal-sized regions, numbered 3, 4, 5, 6, 7, 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) {6, 7, 8, 9, 10, 11, 12, 13, 14} (b)

37) ______

B) (a) {6, 7, 8, 9, 10, 11, 12, 13, 14} (b)

C) (a) {6, 7, 8, 9, 10, 11, 12, 13, 14} (b)

D) (a) {6, 7, 8, 9, 10} (b)

1) A 2) A 3) A 4) A 5) A 6) A 7) C 8) A 9) D 10) B 11) B 12) C 13) A 14) D 15) D 16) D 17) B 18) A 19) D 20) C 21) C 22) C 23) D 24) D 25) C 26) A 27) B 28) D 29) C 30) A 31) B 32) B 33) C 34) B 35) A 36) B 37) B

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) =x-y A)

1) _______

=y+2 A)

2) _______

3) _______

3) = A)

= A)

4) _______

= (y + 3)(y - 3) A)

5) _______

= y(y + 2)( y - 2) A)

6) _______

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. 7) 7) _______ = 1 + , y(4) = 0, dx = 0.5 A) = 0.7500, = 1.2667, C) = 1.0000, = 2.1111, 8)

= -x(1 - y), y(3) = 1, dx = 0.2

= 1.9933

= 0.5000,

= 1.0556,

= 1.6611

= 6.6444

= 1.0000,

= 1.5833,

= 3.3222 8) _______

10)

= 0.1000,

= 0.5000,

= 0.5000

= 1.0000, 10.0000

= 10.0000,

= 1.0000,

= 1.0000

= 0.4000,

= 2.0000,

= 2.0000

= 39.7958

= 9.0000,

= 12.6900,

= 112.6872

= 60.0998

= 15.0000,

= 22.8420,

= 75.1248

= A) C)

11)

12)

9) _______

= 2xy - 2y, y(3) = 6, dx = 0.2 A) = 10.8000, = 20.3040, = 3.0000,

= 20.3040,

10) ______

(1 - 2x), y(-1) = -1, dx = 0.5 = 0.5,

= 0.75,

= 1.03125

= 0.4,

= 0.63,

= 0.7472

= 0.7,

= 0.99,

= 1.9136

= 0.7,

= 0.99,

= 1.2656

= 2x , y(1) = 2, dx = 0.1 A) = 2.7980, = 3.5358, C) = 2.5437, = 3.2814, =y- 1, y(2) = 2, dx = 0.5 A) = -0.6945, = -7.0830, C) = -1.1945, = -8.3830,

11) ______ = 4.5488

= 2.2893,

= 3.0271,

= 4.0400

= 4.2944

= 2.0349,

= 2.7727,

= 3.7857 12) ______

= -22.7673

= -1.6945,

= -9.6830,

= -23.4673

= -23.1173

= -0.1945,

= -5.7830,

= -22.4173

Solve. Round your results to four decimal places. 13) Use the Euler method with dx = 0.2 to estimate y(1) if value of A) y ≈ 0.2277, exact value is C) y ≈ 0.3277, exact value is

15)

13) ______

B) y ≈ 0.0277, exact value is -2e D) y ≈ 0.1277, exact value is -e

14) Use the Euler method with dx = 0.2 to estimate y(3) if value of A) y ≈ 0.6429, exact value is 0.5000 C) y ≈ 0.5789, exact value is 1.0000

= -y and y(0) = 1. What is the exact

= -y/x and y(2) = 1. What is the exact

14) ______

B) y ≈ 0.3684, exact value is 2.0000 D) y ≈ 0.2632, exact value is 6.0000

Use the Euler method with dx = 0.5 to estimate y(5) if = / and y(4) = -1. What is the exact value of y(5)? A) y ≈ -0.7103, exact value is -0.1907 B) y ≈ -0.9118, exact value is -0.1867 C) y ≈ -1.0126, exact value is -0.1705 D) y ≈ -0.7607, exact value is -0.1988

15) ______

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Obtain a slope field and add to its graphs of the solution curves passing through the given points. 16) 16) _____________ = -y with (0, 2) 17) y' = 3(y - 1) with (2, 0)

17) _____________

18)

18) _____________

19)

= y(1 - y) with (0, -1) =

with (0, 1)

19) _____________

20) _____________

20) =

with (-2, 0) 21) _____________

21) =

with (0, -1)

Solve the problem. 22) 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(3 - y), y(0) = 0 ≤ x ≤ 5, 0 ≤ y ≤ 4

;

23) 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(2 - lny), y(0) = 0 ≤ x ≤ 4, 0 ≤ y ≤ 3

22) _____________

23) _____________

;

24) 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.

24) _____________

= ; -3 ≤ x ≤ 3, -3 ≤ y ≤ 3 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use Euler's method with the specified step size to estimate the value of the solution at the given point exact solution at . 25) = 5x , y(0) = 1, dx = 0.1, =1 A) Euler's method gives y ≈ 2.7183; the exact solution is 3.2619 B) Euler's method gives y ≈ 4.2526; the exact solution is 2.9901 C) Euler's method gives y ≈ 3.4794; the exact solution is 2.4465 D) Euler's method gives y ≈ 3.8660; the exact solution is 2.7183 26)

27)

=2 / , y(1) = -1, dx = 0.5, =5 A) Euler's method gives y ≈ -0.1949; the exact solution is -0.2594 B) Euler's method gives y ≈ 0.0821; the exact solution is -0.1601 C) Euler's method gives y ≈ -0.1482; the exact solution is -0.2224 D) Euler's method gives y ≈ -0.0879; the exact solution is -0.1893 =y, y(0) = 2, dx = 1/3, =2 A) Euler's method gives y ≈ 9.4866; the exact solution is 13.3004 B) Euler's method gives y ≈ 8.9605; the exact solution is 12.3212 C) Euler's method gives y ≈ 9.2983; the exact solution is 12.9315 D) Euler's method gives y ≈ 8.2740; the exact solution is 11.1513

. Find the value of the 25) ______

26) ______

27) ______

Solve the differential equation. 28)

28) ______

cos x + y sin x = sin x cos x A) y = cot x + C cos x C) y = cos x ln

+ C cos x

B) y = sin x ln D) y = cos x ln

+ C sin x + C cos x 29) ______

29) A)

+ Cx

B) y=

+ Cx

D) y=

+ Cx

+ Cx 30) ______

30) x

=y+ A)

B) y=

-4

- 4 + Cx

y= D)

- 4x -

- 8x ln x+ Cx -

- 4 + Cx

+C 31) ______

31) A) C)

y = 4, x > 0

,x>0

B) D)

y= y=

+C +C

,x>0 ,x>0 32) ______

32) x

+ 4y = A)

,x>0 B)

,x>0

C) y= 33)

,x>0

D) ,x>0

+ y tan x = cos x, -π/2 < x < π/2 A) y = x cos x + C sin x, -π/2 < x < π/2 C) y = x sin x + C cos x, -π/2 < x < π/2

33) ______ B) y = x cos x + C cos x, -π/2 < x < π/2 D) y = x sin x + C sin x, -π/2 < x < π/2 34) ______

34) x

+ 2y = 5 -

,x>0

B) y=

,x>0

C) y= 35)

,x>0

= A)

,x>0

35) ______

+y y

y =

D) y=

36)

+ 4y A) C)

36) ______

= 4x

y=2

y=4

y=2

+C 37) ______

37) x A) C)

- 3y, x > 0

(sin x + C), x > 0

(cos x + C), x > 0

Solve the initial value problem. 38) + y = 2 ; y(0) = 12 A) y=2 +9

B) D)

(cos x + C), x > 0

(sin x + C), x > 0

38) ______ B)

+ 11

y = 12

y=4

+ 20 39) ______

39) 2

- 4xy = 8x; y(0) = 10 A) C)

y = -2 + 12

y = -1 + 11

y = -2 + 12 y = 2 + 10 40) ______

40) + 5y = 3; y(0) = 1 A) y=

C) y=

D) y=

+ 41) ______

41) t

+ 6y =

; t > 0, y(2) = 1

B) y=

,t>0

D) y=

,t>0

42) ______

42) θ

+ y = cos θ; θ > 0, y(π) = 1 A)

B) y=

,θ>0

y= y =

,θ>0

D) ,θ

y =

>0 ,θ >0 43) ______

43) - 3θy =

sec θ tan θ; θ > 0, y(π) = 0

B) y=

,θ>0

D) y=-

,θ>0

y=-

,θ>0 44) ______

44) (x + 4) A)

- 2(

+ 4x)y =

; x > -4, y(0) = 0 B)

y=-

, x > -4

y=-

, x > -4

D) y=

, x > -4

45) ______

45) + xy = 3x; y(0) = -4 A) y= +3 C) y= -7

B) D)

-7

Determine which of the following equations is correct. 46) x

46) ______

dx = A)

B) 2x ln

D) ln

+C 47) ______

47) dx = A) cos x + C Solve the differential equation. 48) +y= A) 1-C

49)

- y = -x A)

B) -cot x + C

C) 1/sin x + C

D) sin x + C

48) ______ B)

1+C

49) ______ B) x - 1 + C

C) x + 1 + C

50)

-x

50) ______

-y= A) =1-C

51)

=1-C

= -1 + C 51) ______

B) y=

52)

= -1 + C

C) y=

D) y=

52) ______

- 2xy = A)

B) =

D) =

Solve the problem. 53) dy/dt = ky + f(t) is a population model where y is the population at time t and f(t) is some

53) ______

function to describe the net effect on the population. Assume k = .02 and when t = 0. Solve the differential equation of y when f(t) = 15t. A) y = -750t - 37,500 + 47,500e-.02t B) y = 750t + 37,500 + 47,500e-.02t C) y = -750t - 37,500 + 47,500e.02t D) y = 750t - 37,500 + 47,500e-.02t 54) dy/dt = ky + f(t) is a population model where y is the population at time t and f(t) is some

54) ______

function to describe the net effect on the population. Assume k = .02 and when t = 0. Solve the differential equation of y when f(t) = -19t. A) y = 950t + 47,500 - 37,500e-.02t B) y = -950t + 47,500 - 37,500e-.02t C) y = 950t + 47,500 - 37,500e.02t D) y = -950t - 47,500 - 37,500e.02t 55) 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 5 gal/min. The mixture is kept uniform by stirring and flows out of the tank at the rate of 10 gal/min. Write, in standard form, the differential equation that models the mixing process. A) B) +

y = 10

y=5

55) ______

D) =5-

56) A tank initially contains 100 gal of brine in which 40 lb of salt are dissolved. A brine containing

56) ______

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) B) y = 4(50 + t) C)

y = 4(50 + t) D)

y = 2(100 + t) -

57) 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 When

___ ___

the tank 57) is full, how many pounds of concentr ate will it contain? A) 200 pounds

B) 120 pounds

C) 150 pounds

D) 100 pounds

58) 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 A) 187.5 pounds

58) ______

When the tank is full, how many pounds of concentrate will it contain? B) 150 pounds C) 200 pounds D) 175 pounds

59) 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

59) ______

of Find the maximum amount of fertilizer in the tank and the time required to reach the maximum. A) 48 pounds, 40 minutes B) 48 pounds, 60 minutes C) 50 pounds, 50 minutes D) 60 pounds, 40 minutes 60)

60) ______

An office contains 1000 of air initially free of carbon monoxide. Starting at time = 0, cigarette smoke containing 4% carbon monoxide is blown into the room at the rate of 0.5 /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 0.01%. A) 8.01 min

/min. Find the time when the concentration of carbon monoxide reaches B) 5.01 min

C) 6.01 min

D) 7.01 min

61) How many seconds after the switch in an RL circuit is closed will it take the current i to reach

61) ______

of its steady state value? Express answer in terms of R and L and round coefficient to the nearest hundredth. A) 0.36 L/R seconds B) 1.20 L/R seconds C) 1.40 L/R seconds D) 0.56 L/R seconds 62) If the switch is thrown open after the current in an RL circuit has built up to its steady-state value, the decaying current obeys the equation thrown open will it take the current to fall to A) 1.61 L/R seconds C) 1.81 L/R seconds

62) ______

How long after the switch is of its original value? B) -4.40 L/R seconds D) -4.60 L/R seconds

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. 63) y = -mx 63) _____________ 64)

Show that the curves are orthogonal.

64) _____________

65) 66)

= 5 and

= 1 and xy = 2

65) _____________

66) _____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve. 67) A 55-kg skateboarder on a 3-kg board starts coasting on level ground at . Let k = 3.2 kg/sec. About how far will the skater coast before reaching a complete stop? A) 257.81 m B) 90.63 m C) 105.60 m D) 2640.00 m

67) ______

68) A 62-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 2 m/sec? A) 13.43 sec B) -13.65 sec C) 0.22 sec D) 13.65 sec

68) ______

69) A local pond can only hold up to 32 geese. Six geese are introduced into the pond. Assume that the rate of growth of the population is

69) ______

= (0.0015)(32 - P)P where t is time in weeks. Find a formula for the goose population in terms of t. A) B) P(t) =

P(t) =

D) P(t) =

P(t) =

70) A local pond can only hold up to 32 geese. Eight geese are introduced into the pond. Assume that the rate of growth of the population is = (0.0015)(32 - P)P where t is time in weeks. How long will it take for the goose population to be 13? A) 58.30 weeks B) 4.12 weeks C) 48.65 weeks

, P(0) = B)

P(t) = C)

D) 14.98 weeks 71) ______

71) Solve the initial value problem = 2k A)

70) ______

P(t) = D)

P(t) =

Identify equilibrium values and determine which are stable and which are unstable. 72) = (y + 3)(y + 2) A) y = -2 is a stable equilibrium value and y = 3 is an unstable equilibrium. B) y = 2 is a stable equilibrium value and y = -3 is an unstable equilibrium. C) y = 3 is a stable equilibrium value and y = 2 is an unstable equilibrium. D) y = -3 is a stable equilibrium value and y = 2 is an unstable equilibrium.

72) ______

73) ______

73) = - 16 A) y = -4 and y = 5 are stable equilibrium values. B) y = 4 is a stable equilibrium value and y = -4 is an unstable equilibrium. C) y = -4 is a stable equilibrium value and y = 4 is an unstable equilibrium. D) There are no equilibrium values.

74) ______

74) = - 4y A) y = 0 and y = 4 are unstable equilibrium values. B) y = 0 is a stable equilibrium value and y = 4 is an unstable equilibrium. C) y = 4 is a stable equilibrium value and y = 0 is an unstable equilibrium. D) There are no equilibrium values. 75)

76)

= , y>0 A) y = 0 is a stable equilibrium value. C) y = 0 is an unstable equilibrium value.

75) ______ B) y = 5 is an unstable equilibrium value. D) There are no equilibrium values.

= (y - 1)(y - 3)(y - 4) A) y = 5, y = 1 and y = 4 are unstable equilibria. B) y = 4 is a stable equilibrium value and y = 3 and y = 1 are unstable equilibria. C) y = 3 is a stable equilibrium value and y = 1 and y = 4 are unstable equilibria. D) y = 1 is a stable equilibrium value and y = 3 and y = 4 are unstable equilibria.

Construct a phase line. Identify signs of 77) = (y + 1)(y - 2) A)

and

76) ______

. 77) ______

78) ______

78) = A)

- 25

79)

= A)

, y>0

79) ______

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Sketch several solution curves. 80) 80) _____________ =

-1 81) _____________

81) = 82) 83)

- 3y

=y-2 =

82) _____________ 83) _____________

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? 84) 84) _____________ = 1 - 3P 85) _____________

85) = P(P - 5)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 86) If a body of mass m falling from rest under the action of gravity encounters an air resistance proportional to three times the square root of velocity, then the body's velocity t seconds into the fall satisfies the equation: m = mg - 3k 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 150 lb skydiver (mg = 150) with k = 0.005. A) Equilibrium: v =

= 100 ft/sec B) Equilibrium: v =

= 100 ft/sec C) Equilibrium: v =

86) ______

= 100 ft/sec

D) Equilibrium: v =

= 100 ft/sec 87) 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

= 0 or v = 20 ft/sec

The maximum velocity occurs when

= 25 or v = 0 ft/sec

The maximum velocity occurs when

= 20 or v = 0 ft/sec

The maximum velocity occurs when

= 0 or v = 25 ft/sec

87) ______

B) C) D)

Solve the problem. 88)

88) ______

The system of equations = (-7 + 8y)x and = (-7 + x)y describes the growth rates of two 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) B) (x, y) = (0, 0) and (x, y) = C)

(x, y) = (0, 0) and (x, y) = D)

(x, y) = (0, 0) and (x, y) =

(x, y) =

89) ______

89) The system of equations = (-2 + 3y)x and = (-2 + x)y describes the growth rates of two 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) They are both at a maximum. B) They are both at a minimum. C) Both populations equal zero. D) They both remain constant over all time.

90) ______

90) The system of equations = (-4 + 5y)x and = (-4 + x)y describes the growth rates of two 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 could either increase or decrease. B) The rhinoceros population decreases. C) The rhinoceros population increases. D) The rhinoceros population stays the same.

1) D 2) B 3) B 4) B 5) D 6) A 7) B 8) C 9) A 10) A 11) C 12) C 13) C 14) A 15) A 16)

17)

18)

19)

20)

21)

22)

lny +

23)

24)

25) D 26) A 27) D 28) D 29) B 30) A 31) C 32) B 33) B 34) A 35) C 36) D 37) A 38) B 39) B 40) A 41) B 42) B 43) C 44) C 45) D 46) C 47) B 48) C 49) D 50) B 51) B 52) A 53) C 54) C 55) B 56) C 57) C 58) A 59) C 60) B 61) A 62) A 63) y=

x+C

64)

65)

66)

y=C

67) B 68) D 69) B 70) D 71) C 72) D 73) C 74) B 75) D 76) C 77) C 78) C 79) D 80)

81)

82)

83)

84) = 1- 3P has stable equilibrium at P =

85)

= -3(1 - 3P)

= P(P - 5) has a stable equilibrium at P = 0 and an unstable equilibrium at P = 5;

86) B 87) D 88) B 89) D 90) B

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write the first four elements of the sequence. 1) A)

B) 0,

C) -1, 1,

1) _______

, 2) _______

2) A)

B) 0,

ln 2,

D) ,

, 3) _______

3) A)

B) ,-

D) ,

, 4) _______

4) A)

B) 1,

5) sin (nπ) A) 0, 1, 0, -1

C) 0, 1,

D) 2,

0, 2,

, 5) _______

B) 1, 0, -1, 0

C) 1, 1, 1, 1

D) 0, 0, 0, 0

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. 6) 6) _______ = 1, =4 A) 4, 5, 6, 7, 8 C) 4, 16, 64, 256, 1024 7)

= 1, = A) 1, 6, 36, 216, 1296 C) 1, 6, 36, 216, 1296, 7776 = 1, = +5 A) 1, 6, 11, 16, 21 C) 1, 5, 25, 125, 625, 3125 = 5, =A) 5, 0, -5, -10, -15 C) -5, 5, -5, 5, -5

B) 1, 4, 16, 64, 256 D) 1, 5, 9, 13, 17 7) _______ B) 1, 7, 13, 19, 25 D) 1, 1, 1, 1, 1 8) _______ B) 6, 11, 16, 21, 26 D) 1, 6, 11, 16, 21, 26 9) _______ B) 5, -25, 125, -625, 3125 D) 5, -5, 5, -5, 5

10)

10) ______

= 2, = A) 2, 2, -2, -2, 2

B) 2, -2, -2, 2, 2

C) -2, 2, -2, 2, -2

D) 2, -2, 2, -2, 2 11) ______

11) = 5, A)

= B)

5, -

, -5,

-5,

, 5, -

, -5

5, -

, 5, -

D) 5,

, -5, -

12) ______

12) = 1, A)

, 56,

B) , 5040

D) 1,

, 13) ______

13) = 1, A) 1,

= B) ,

C) 1, 14)

15)

D) ,

= 1, = 3, A) 1, 3, 4, 8, 12

= 1, = 2, A) 1, -2, 3, -4, 5

+ B) 1, 3, 4, 5, 6 B) 1, 2, 1, -1, -2

Find a formula for the nth term of the sequence. 16) 3, 4, 5, 6, 7 (integers beginning with 3) A) B) =n+3 =n-3

, 14) ______

C) 1, 1, 2, 3, 5

D) 1, 3, 4, 7, 11 15) ______

C) 1, 2, 1, 0, -1

D) 1, -1, 2, -3, 5

16) ______ C)

=n+4

=n+2 17) ______

17) -3, -2, -1, 0, 1 (integers beginning with -3) A) B) =n-3 =n+3

18) 6, -6, 6, -6, 6 (6's with alternating signs) A) B) =6 =6

=n-2

=n-4 18) ______

=6 19) ______

19) 1, - , A)

(reciprocals of squares with alternating signs) B) C) =

D) =

20) 7, 9, 11, 13, 15 (every other integer starting with 7) A) B) C) = 2n + 5 = n + 12 21) 0, 2, 0, 2, 0 (alternating 0's and 2's) A) =1+ C) =1+

20) ______ = 2n + 6

= n + 13 21) ______

B) D)

=1=1+ 22) ______

22) 0,

, 0, A)

, 0 (alternating 0's and

's) B)

D) =

23) 0, 0, 2, 2, 0, 0, 2, 2 (alternating 0's and 2's in pairs) A) = C) =1+ 24) 0, 2, 2, 2, 0, 2, 2, 2 (0, 2, 2, 2 repeated) A) =1+ C) =1+ 25) 0, -1, 0, 1, 0, -1, 0, 1 (0, -1, 0, 1 repeated) A) B) = sin(nπ)

23) ______ B) D)

=1=1+ 24) ______

B) D)

=1+ =1+ 25) ______

= cos(nπ)

= sin

D) = cos

Find the limit of the sequence if it converges; otherwise indicate divergence. 26) =2+ A) 2 B) 2.3 C) 3

26) ______ D) Diverges 27) ______

27) = A) 52

B) 6

D) Diverges

28) ______

28) = A) 3

B) ±3

C) 0

D) Diverges 29) ______

29) = A) 5

B) 1

C) 0

D) Diverges 30) ______

30) =

D) Diverges

31) ______

31) = A)

32)

= ln(7n + 1) - ln(4n - 7) A)

D) Diverges

32) ______ B) ln 3

D) Diverges ln 33) ______

33) = A) ln 5

C) 0

D) Diverges 34) ______

34) = A) 1

35)

=1+ A) 0

C) 0

D) Diverges

35) ______

+ B) 1

C) 3

D) Diverges 36) ______

36) = A) 0

B) ln 2

C) 1

D) Diverges 37) ______

37) = A) 1

C) e

D) Diverges 38) ______

38) = A) ln 5

B) 0

C) 1

D) Diverges 39) ______

39) = A)

B) 1

C) 0

D) Diverges 40) ______

40) = A) 0

B) 1

C) ln 2

D) Diverges 41) ______

41) = A) 7

B) 1

C) 0

D) Diverges

42) ______

42) = A)

B) 0

C) 6

D) Diverges 43) ______

43) = ln A) 6

B) ln 6

C) 0

D) Diverges 44) ______

44) = A) 0

45)

B) 1

D) Diverges

45) ______

=nA) 0

D) Diverges

Assume that the sequence converges and find its limit. 46)

47)

48)

= 5, A) 4

= -2, A) 4

= 5, A) 5

B) 5

46) ______ C) 8

D) 6 47) ______

B) -6

C) -2

D) 8 48) ______

B) 25

C) 1

D) -5

Determine if the sequence is monotonic and if it is bounded. 49) = A) nondecreasing; bounded C) decreasing; unbounded

49) ______ B) not monotonic; bounded D) decreasing; bounded 50) ______

50) = A) nondecreasing; unbounded C) not monotonic; unbounded

B) decreasing; unbounded D) nondecreasing; bounded 51) ______

51) A) nonincreasing; unbounded C) nondecreasing; unbounded

B) not monotonic; unbounded D) nondecreasing; bounded 52) ______

52) =

A) increasing; bounded C) nondecreasing; bounded

B) nondecreasing; unbounded D) not monotonic; unbounded 53) ______

53) = A) not monotonic; bounded C) increasing; unbounded

B) decreasing; bounded D) nondecreasing; bounded 54) ______

54) = A) increasing; unbounded C) nondecreasing; bounded

B) decreasing; bounded D) not monotonic; bounded 55) ______

55) = A) increasing; unbounded C) not monotonic; unbounded

B) nondecreasing; bounded D) decreasing; bounded 56) ______

56) =5A) increasing; bounded C) nonincreasing; bounded

B) not monotonic; bounded D) decreasing; unbounded

Determine whether the nonincreasing sequence converges or diverges. 57) = A) Converges

57) ______

B) Diverges 58) ______

58) = A) Diverges

B) Converges 59) ______

59) = A) Converges 60)

= 1, =4 A) Diverges

B) Diverges 60) ______

-7 B) Converges

Find the smallest value of N that will make the inequality hold for all n > N. 61) < A) 124 B) 117 C) 122

61) ______ D) 120 62) ______

62) < A) 1612 63)

B) 1609

C) 1605

D) 1607

___ ___

63) < A) 733 64)

B) 737

C) 731

D) 750 64) ______

< A) 9899

B) 9883

C) 9891

D) 9907 65) ______

65) < A) 18

B) 16

C) 15

D) 17 66) ______

66) < A) 17

B) 18

C) 19

D) 20 67) ______

67) < A) 13

B) 15

C) 16

D) 14 68) ______

68) < A) 20

B) 21

C) 30

D) 31

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 69) = A) L = ln 2, N = 394 C) L = 1, N = 652 70)

= cos n A) L = 0, N = 394

for

otherwise indicate divergence. 69) ______ B) L = 1, N = 394 D) diverges 70) ______

B) L = 1, N = 394

C) L = 0, N = 237

D) diverges 71) ______

71) = n tan A) L = 0, N = 28

B) L = 1, N = 28

C) L = 1, N = 6

D) diverges 72) ______

72) = A) L = 0, N = 10

B) L = 0, N = 13

C) L = 0, N = 8

D) diverges 73) ______

73) = n cos A) L = 0, N = 37 74)

B) L = 1, N = 28

C) L = 0, N = 28

D) diverges 74) ______

A) L = 1, N = 1759 75)

B) L = 0, N = 1759

C) L = 1, N = 751

D) diverges 75) ______

= A) L ≈ 1.1653, N = 2 C) L ≈ 1.1653, N = 6

B) L ≈ 1.153, N = 6 D) diverges 76) ______

76) = A) L = ln 4, N = 8

B) L = ln 2, N = 6

C) L = 0, N = 8

D) diverges 77) ______

77) = A) L = ln 4, N = 8

B) L = ln 2, N = 6

C) L = 0, N = 7

D) diverges 78) ______

78) = A)

B) L=

,N=6

C) L = 0, N = 5 L=

D) diverges

,N=8

Provide an appropriate response. 79)

79) ______

A sequence of rational numbers { } is defined by

Find . A) 2

C) 3

B) 50

, and if

then D) 49 80) ______

80) A sequence of rational numbers { } is defined by Find A) 2

, and if

then

Hint: Compute the square of several terms of the sequence on a calculator. B) C) D) 81) ______

81) It can be shown that

≈

for large values of n. Find the smallest value of N such that

for all A) 22

B) 33

C) 29

D) 27 82) ______

82) It can be shown that all if A) 10

= 0 for c > 0. Find the smallest value of N such that

for

and B) 11

C) 12

D) 13 83) ______

83) It can be shown that for all

= 0 for c > 0. Find the smallest value of N such that and

A) 28 84)

B) 25

C) 31

D) 35

Let f(n) = . What is f(n) approximately equal to as n gets large? Hint: Compute various examples on your calculator. A) B) C) D)

84) ______

1) D 2) B 3) C 4) C 5) D 6) B 7) D 8) A 9) D 10) B 11) C 12) D 13) A 14) D 15) B 16) D 17) D 18) A 19) A 20) A 21) C 22) C 23) A 24) C 25) D 26) A 27) C 28) C 29) D 30) C 31) D 32) C 33) C 34) D 35) D 36) D 37) B 38) C 39) D 40) B 41) A 42) D 43) A 44) A 45) B 46) C 47) A 48) A 49) A 50) A 51) C

52) B 53) B 54) B 55) D 56) A 57) A 58) A 59) A 60) A 61) D 62) B 63) A 64) A 65) C 66) C 67) B 68) D 69) C 70) D 71) C 72) A 73) D 74) C 75) A 76) C 77) C 78) C 79) C 80) D 81) D 82) D 83) B 84) D

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. 1) 2+ + A)

+ ... +

+ ... B)

;

5 + 40 + 320 + ... + 5 ∙ A)

;

2) _______

+ ... B)

; series diverges

;

D) ;

; series diverges 3) _______

3) 7+ A)

+ ... +

+ ... B)

;

; 4)

1) _______

;

4 - 12 + 36 - 108 + ... + A)

4∙

4) _______

+ ... B)

; series diverges

D) ;1

; series diverges 5) _______

5) + A)

+ ... +

+ ... B)

C) ;4

D) ;4

;4 6) _______

6) + A)

+ ... +

+ ...

;4 ;4

;8 7) _______

7) + A)

+ ... +

+ ... B)

D) ;4

;4 8) _______

8) + A)

+ ... +

+ ... B)

;

D) ;

; 9) _______

9) +

+ ... +

+ ...

B) ;3

D) ;3

;3 10) ______

10) +

+ ... +

+ ...

B) ;2

D) ;1

Find the sum of the series. 11)

A) 1

11) ______

D) 5

12) ______

12)

B) 3

C) 1

13) ______

13)

14) ______

14)

C) 8

D) 1

15) ______

15)

16) ______

16)

17) ______

17)

A) - 2

D) 2

18) ______

18)

B) - 3

C) 15

19) ______

19)

20) ______

20)

Determine if the geometric series converges or diverges. If it converges, find its sum. 21) 1+ A)

22)

+... B)

converges, 1 + (-7) + + A) converges, 70

21) ______

C) converges, 2

D) diverges

converges, +

+... B) converges, 49

22) ______ C) diverges

D) converges, 7

23) ______

23) + A)

+... B)

converges,

25) 0.626262 . . . A)

26) 0.9595959 . . . A)

27) 0.229229 . . . A)

28) 0.363636 . . . A)

converges,

Express the number as the ratio of two integers. 24) 0.88888 . . . A) B)

D) diverges

24) ______ C)

25) ______ B)

26) ______ B)

27) ______ B)

28) ______ B)

Use the nth-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. 29)

A) inconclusive

B) converges, 7

C) diverges

29) ______

D) converges, 30) ______

30)

A) converges, 1

B) diverges

C) inconclusive

D) converges, 9 31) ______

31)

A) converges; ln 10 C) inconclusive

B) diverges D) converges; ln

Determine if the series converges or diverges. If the series converges, find its sum. 32)

32) ______

di ve rg es

ges;

co nv er ge s;

33) ______

33)

B) converges;

C) diverges

D) converges; 1

converges; 34) ______

34)

A) diverges

D) converges;

converges;

converges; 35) ______

35)

B) converges;

D) converges;

converges;

converges; 36) ______

36)

B) converges;

C) converges;

D) diverges converges; 37) ______

37)

B) converges;

converges;

converges; D) diverges

38) ______

38)

A) converges; 1

converges; D) diverges converges;

39)

___ ___

39)

A) diverges

C) converges;1

converges; -

converges; 40) ______

40)

A) converges; ln 2 C)

B) diverges D)

converges;

converges; 41) ______

41)

B) converges; π

C) diverges

converges; -

converges;

Determine if the series converges or diverges; if the series converges, find its sum. 42)

A) Converges; 1 -

B) Converges;

C) Converges;

D) Diverges

-1

42) ______

43) ______

43)

B) Converges;

Converges; D) Diverges

C) Converges;

44) ______

44)

B) Converges;

C) Converges;

D) Diverges Converges; 45) ______

45)

B) Converges;

C) Converges;

Converges; D) Diverges

46) ______

46)

B) Converges; ln 8

Converges; ln C) Converges; 1

D) Diverges 47) ______

47)

B) Converges;

Converges; D) Diverges

Converges;

48) ______

48)

A) Converges; 1

C) Converges; e

D) Diverges

converges; 49) ______

49)

B) Converges;

Converges; D) Diverges

C) Converges;

50) ______

50)

B) Converges; 1

C) Converges; 2

D) Diverges

Converges; 51) ______

51)

B) Converges; 4 Converges;

D) Diverges Converges;

Find the values of x for which the geometric series converges. 52)

A) |x| < 20

B) |x| < 10

52) ______

C) |x| < 1

D) |x| <

53)

53) ______

A) |x| < 1

B) |x| < 7

C) |x| < 14

D) |x| < 54) ______

54)

A) |x| < 4

B) |x| < 2

D) |x| < 1 |x| < 55) ______

55)

A) 1 < x < 2

B) -2 < x < 2

C) 2 < x < 3

D) 1 < x < 3 56) ______

56)

A) -9 < x < -8

B) -10 < x < -9

C) -10 < x < -8

D) -9 < x < 9 57) ______

57)

B) -

<x<

D) 0<x<

0<x<

<x<0 58) ______

58)

A) -1 < x < 11

B) -11 < x < 11

C) 7 < x < 17

D) -7 < x < 17 59) ______

59)

A) 3 < x < 7

B) -7 < x < 7

C) -1 < x < 9

D) 1 < x < 9 60) ______

60)

A) diverges for all x C)

B) -∞ < x < ∞ D) 61) ______

61)

A) -∞ < x < ∞ C)

B) D) diverges for all x

Find the sum of the geometric series for those x for which the series converges. 62)

62) ______

63) ______

63)

64) ______

64)

65) ______

65)

66) ______

66)

67) ______

67)

D) 68) ______

68)

69) ______

69)

70) ______

70)

71) ______

71)

Solve the problem. 72) 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) 30,000 customers B) The sum diverges to infinity. C) 15,000 customers D) 20,000 customers

72) ______

73) 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

73) ______

74) 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) 50.0 m B) It will roll an infinite distance. C) 12.2 m D) 20.0 m

74) ______

75) 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) 24 m B) 25 m C) 60 m D) The child will travel an infinite distance.

75) ______

76) 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) 250 m D) 500 m

76) ______

77) A pendulum is released and swings until it stops. If it passes through an arc of 15 inches the first

77) ______

pass, and if on each successive pass it travels it travel before stopping? A) 90 in. B) 135 in.

the distance of the preceding pass, how far will C) 60 in.

D) 75 in. 78) ______

78) A ball is dropped from a height of 21 m and always rebounds of the height of the previous drop. How far does it travel (up and down) before coming to rest? A) 10.5 m B) 63 m C) 42 m D) 31.5 m 79) A child on a swing sweeps out a distance of 16 ft on the first pass. If she is allowed to continue swinging until she stops, and if on each pass she sweeps out a distance

of the previous pass,

ho s the w child far travel? doe

79)

______ A) 37.33 ft

B) 21.33 ft

C) 10.67 ft

D) 5.33 ft

80) Mari drops a ball from a height of 29 meters and notices that on each bounce the ball returns to

80) ______

about of its previous height. About how far will ball travel before it comes to rest? A) 87 m B) 145 m C) 174 m D) 72.5 m 81) Mari drops a ball from a height of 21 meters and notices that on each bounce the ball returns to

81) ______

about of its previous height. About how far will ball travel before it comes to rest? A) 357 m B) 44.6 m C) 378 m D) 189 m 82) ______

82) Find the value of b for which 1+

+ ... = 10. B)

A) ln

D) ln

ln 83) ______

83) Find the value of b for which 1A) ln 4

+ ... = . B) ln 6

D) ln 5 ln 84) ______

84) For what value of r does the infinite series 1 + 5r + converge? A) |r| < 1

+ ...

C) |r| < 5

|r| <

|r| < 85) ______

85) Find the sum of the infinite series 1 + 3r + +3 + +3 + + ... for those values of r for which it converges. A) B)

86) ______

86) For what value of r does the infinite series 1 + 7r + 6 converge? A)

B) |r| < 1

+ ... C)

|r| <

D) |r| <

87) ______

87) Find the sum of the infinite series 1 + 8r + 5 + 8 + +8 +5 +8 for those values of r for which it converges. A)

|r| <

... B)

1) C 2) A 3) A 4) B 5) A 6) A 7) A 8) B 9) A 10) C 11) C 12) C 13) D 14) A 15) C 16) A 17) B 18) C 19) A 20) D 21) A 22) C 23) A 24) C 25) C 26) B 27) D 28) A 29) C 30) B 31) B 32) D 33) B 34) D 35) C 36) A 37) C 38) A 39) D 40) D 41) D 42) D 43) A 44) B 45) B 46) D 47) D 48) D 49) B 50) A 51) B

52) D 53) D 54) C 55) D 56) C 57) D 58) A 59) A 60) B 61) C 62) C 63) C 64) D 65) C 66) A 67) C 68) B 69) C 70) C 71) A 72) D 73) D 74) A 75) C 76) A 77) D 78) C 79) B 80) B 81) A 82) A 83) A 84) A 85) B 86) B 87) D

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the integral test to determine whether the series converges. 1)

A) diverges

1) _______

B) converges 2) _______

A) diverges

B) converges 3) _______

A) converges

B) diverges 4) _______

A) converges

B) diverges 5) _______

A) diverges

B) converges 6) _______

A) diverges

B) converges 7) _______

A) converges

B) diverges 8) _______

A) diverges

B) converges 9) _______

A) diverges

B) converges 10) ______

10)

A) diverges

B) converges

Provide an appropriate response. 11) Which of the following is not a condition for applying the integral test to the sequence {

11) ______

where I. f(x) is everywhere positive II. f(x) is decreasing for x ≥ N III. f(x) is continuous for x ≥ N A) II only B) I only C) III only D) All of these are conditions for applying the integral test. 12) ______

12) Which of the following statements is false? A) If

and f(n) satisfy the requirements of the Integral Test, and if

converges,

then = dx. B) The integral test does not apply to divergent sequences. C) converges if p >1 and diverges if p ≤ 1. D) converges if p > 1. 13) Which of the following sequences do not meet the conditions of the Integral Test? I.

= n(sin n + 1)

II.

III. = A) II and III

B) I, II, and III

C) I and III

D) I only

13) ______

1) B 2) A 3) B 4) B 5) B 6) B 7) A 8) B 9) A 10) A 11) B 12) A 13) D

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the Comparison Test to determine if the series converges or diverges. 1)

A) converges

1) _______

B) diverges 2) _______

A) diverges

B) converges 3) _______

A) converges

B) diverges 4) _______

A) diverges

B) converges 5) _______

A) converges

B) diverges 6) _______

A) converges

B) diverges 7) _______

A) diverges

B) converges 8) _______

A) converges

B) diverges 9) _______

A) converges

B) diverges

Use the limit comparison test to determine if the series converges or diverges. 10)

A) Converges

B) Diverges

10) ______

11) ______

11)

A) Converges

B) Diverges 12) ______

12)

A) Converges

B) Diverges 13) ______

13)

A) Converges

B) Diverges 14) ______

14)

A) Diverges

B) Converges 15) ______

15)

A) Converges

B) Diverges 16) ______

16)

A) Diverges

B) converges 17) ______

17)

A) Diverges

B) Converges 18) ______

18)

A) Converges

B) Diverges 19) ______

19)

A) Diverges

B) Converges

Provide an appropriate response. 20) Which of the following statements is false? A) If {

} and {

} meet the conditions of the Limit Comparison test, then, if

converges, then B)

The series

20) ______

= 0 and

converges.

must have no negative terms in order for the Direct Comparison test to be

applicable. C) All of these are true. D) The sequences { } and {

} must be positive for all n to apply the Limit Comparison Test. 21) ______

21) Suppose that

> 0 and

> 0 for all n ≥ N (N an integer). If

conclude about the convergence of ? A) diverges if diverges, and converges if B) diverges if diverges C) The convergence of cannot be determined. D)

converges if

converges

= ∞, what can you

converges

1) B 2) A 3) A 4) B 5) A 6) A 7) B 8) A 9) A 10) B 11) B 12) A 13) A 14) B 15) B 16) A 17) A 18) B 19) B 20) D 21) B

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the Ratio Test to determine if the series converges or diverges. 1)

A) Diverges

1) _______

B) Converges 2) _______

A) Converges

B) Diverges 3) _______

A) Diverges

B) Converges 4) _______

A) Converges

B) Diverges 5) _______

A) Diverges

B) Converges 6) _______

A) Converges

B) Diverges 7) _______

A) Converges

B) Diverges

Use the Root Test to determine if the series converges or diverges. 8)

A) Diverges

8) _______

B) Converges 9) _______

A) Converges

B) Diverges 10) ______

10)

A) Diverges

B) Converges

11) ______

11)

A) Diverges

B) Converges 12) ______

12)

A) Converges

B) Diverges 13) ______

13)

A) Diverges

B) Converges 14) ______

14)

A) Diverges

B) Converges 15) ______

15)

A) Converges

B) Diverges 16) ______

16)

A) Diverges

B) Converges 17) ______

17)

A) Diverges

B) Converges

Determine convergence or divergence of the series. 18)

A) Diverges

18) ______

B) Converges 19) ______

19)

A) Converges

B) Diverges 20) ______

20)

A) Diverges 21)

B) Converges

___ ___

21)

A) Diverges

B) Converges 22) ______

22)

A) Converges

B) Diverges 23) ______

23)

A) Converges

B) Diverges 24) ______

24)

A) Converges

B) Diverges 25) ______

25)

A) Diverges

B) Converges 26) ______

26)

A) Diverges

B) Converges 27) ______

27)

A) Diverges

Determine if the series 28) = 4, = A) Converges

B) Converges

defined by the formula converges or diverges. 28) ______ B) Diverges 29) ______

29) = 8, = A) Converges

B) Diverges 30) ______

30) = 7, = A) Converges 31)

B) Diverges = 4,

___ ___

31)

A) Diverges

B) Converges 32) ______

32) = 6, = A) Converges

B) Diverges 33) ______

33) = 5, = A) Converges

B) Diverges 34) ______

34) = , = A) Diverges

B) Converges 35) ______

35) = , = A) Diverges

B) Converges

Solve the problem. 36) If is a convergent series of nonnegative terms, what can be said about positive integer? A) Always diverges B) May converge or diverge C) Always converges

, where k is a

37) ______

37) If is a convergent series of nonnegative terms, what can be said about positive integer? A) Always converges B) May converge or diverge C) Always diverges 38)

39)

40)

36) ______

is a convergent series of nonnegative terms, what can be said about A) Always converges B) May converge or diverge C) Always diverges

is a convergent series of nonnegative terms, what can be said about A) Always diverges B) Always converges C) May converge or diverge

If is a convergent series of nonnegative terms, what can be said about a positive integer? A) Always diverges

, where k is a

38) ______

39) ______

, where k is

40) ______

B) May converge or diverge C) Always converges 41) If p > 0 and q > 0, what can be said about the convergence of

41) ______

? A) May converge or diverge B) Always converges C) Always diverges 42) If p > 0 and q > 1, what can be said about the convergence of

42) ______

? A) Always converges B) May converge or diverge C) Always diverges 43) If p > 1 and q > 0, what can be said about the convergence of

43) ______

? A) Always diverges B) May converge or diverge C) Always converges 44) If p > 1 and q > 1, what can be said about the convergence of

44) ______

? A) May converge or diverge B) Always converges C) Always diverges 45) If p > 1 and q > 1, what can be said about the convergence of

? A) Always converges B) Always diverges C) May converge or diverge

45) ______

1) A 2) A 3) B 4) B 5) B 6) A 7) B 8) B 9) B 10) B 11) B 12) A 13) B 14) B 15) B 16) A 17) B 18) A 19) A 20) B 21) B 22) A 23) A 24) A 25) A 26) A 27) B 28) A 29) A 30) B 31) A 32) A 33) B 34) A 35) B 36) C 37) B 38) A 39) C 40) B 41) A 42) B 43) C 44) B 45) B

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine convergence or divergence of the alternating series. 1)

A) Diverges

1) _______

B) Converges 2) _______

A) Converges

B) Diverges 3) _______

A) Converges

B) Diverges 4) _______

A) Converges

B) Diverges 5) _______

A) Converges

B) Diverges 6) _______

A) Converges

B) Diverges 7) _______

A) Diverges

B) Converges

Determine if the series converges absolutely, converges, or diverges. 8)

8) _______

A) Converges conditionally B) Diverges C) Converges absolutely 9) _______

A) converges conditionally 10)

B) Converges absolutely

C) Diverges

___ ___

10)

A) diverges B) Converges conditionally C) Converges absolutely 11) ______

11)

A) Diverges B) Converges conditionally C) converges absolutely 12) ______

12)

A) Diverges

B) Converges absolutely

C) converges conditionally 13) ______

13)

A) converges conditionally

B) diverges

C) converges absolutely

14) ______

14)

A) converges conditionally

B) converges absolutely

C) diverges

15) ______

15)

A) diverges

B) converges absolutely

C) converges conditionally 16) ______

16)

A) diverges

B) converges absolutely

C) converges conditionally

Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series. 17) 17) ______

A) C) 18)

< 7.78 × 10-2 < 3.24 × 10-2

B) D)

< 7.78 × 10-3 < 1.56 × 10-2

___ ___

18)

A) C)

< -2.85 × 10-2 < -3.14 × 10-1

< 2.35 × 10-2 < -4.30 × 10-2 19) ______

19)

< 2.14 × 10-5

< 1.29 × 10-4

< C)

< 7.72 × 10-4

20) ______

20) , -1 < t ≤ 1 A)

< 0.20

Determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001. 21) 21) ______

A) n ≥ 15

B) n ≥ 14

C) n ≥ 16

D) n ≥ 31 22) ______

22)

A) n ≥ 33

B) n ≥ 35

C) n ≥ 36

D) n ≥ 34

Find the infinite sum accurate to three decimal places. 23)

A) 3.000

B) 0.500

23) ______

C) 1.000

D) 1.500 24) ______

24)

A) 0.200

B) 0.143

C) -0.571

D) 0.800 25) ______

25)

A) -0.859

B) 0.859

C) -0.320

D) 0.460 26) ______

26)

A) 1.036 27)

B) -0.972

C) 0.972

D) -1.036

___ ___

27)

A) 0.010

B) -0.010

C) 0.007

D) -0.007

Provide an appropriate response. 28)

28) ______

For an alternating series where it is not true that ≥ for sufficiently large n, what can be said about the convergence or divergence of the series? A) The series always diverges. B) The series may or may not converge. C) The series always converges. 29) ______

29) For an alternating series where convergence or divergence of the series? A) The series always diverges. B) The series may or may not converge. C) The series always converges. 30) Let

≠ 0, what can be said about the

denote the kth partial sum of the alternating harmonic series. Compute . Which of these is closest to the exact sum B)

, and

of the alternating harmonic series? C)

31) ______

31) Let

denote the kth partial sum of the alternating harmonic series. Compute

, and harmonic series? A)

32) Let

34)

. Which of these is closest to the exact sum (ln 2) of the alternating B)

denote the kth partial sum of the alternating harmonic series. If e(13) denotes the

absolute value of the error in approximating ln 2 by , compute floor floor(x) denotes the integer floor (or greatest integer) function. A) 25 B) 27 C) 28 D) 26 33)

30) ______

converges, what can be said about A) The series always diverges. B) The series always converges. C) The series may converge or diverge.

32) ______

, where

, where k is an integer greater than 1?

33) ______

___ ___

converge 34) s, what can be said about ? A) The series always converges. B) The series may converge or diverge. C) The series always diverges. 35)

36)

37)

converges conditionally, what can be said about A) The series always diverges. B) The series may converge or diverge. C) The series always converges.

converges conditionally, what can be said about A) The series may converge or diverge. B) The series always diverges. C) The series always converges.

and both converge conditionally, what can be said about A) The series may converge or diverge. B) The series always diverges. C) The series always converges.

35) ______

36) ______

37) ______

1) B 2) B 3) B 4) A 5) B 6) A 7) B 8) B 9) C 10) B 11) B 12) B 13) C 14) C 15) C 16) A 17) D 18) A 19) D 20) A 21) A 22) D 23) B 24) A 25) D 26) C 27) D 28) B 29) A 30) B 31) B 32) C 33) C 34) B 35) A 36) C 37) A

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the series' radius of convergence. 1)

A) ∞, for all x

B) 2

C) 0

1) _______

D) 1 2) _______

A) 2

B) 0

C) ∞, for all x

D) 1 3) _______

A) 0

B) 2

C) ∞, for all x

D) 1 4) _______

A) 2

B) 0

C) ∞, for all x

D) 4 5) _______

A) ∞, for all x

B) 2

C) 1

D) 0 6) _______

A) ∞, for all x

B) 0

C) 2

D) 1 7) _______

A) 3

B) ∞, for all x

C) 6

D) 1 8) _______

A) 3

B) ∞, for all x

C) 6

D) 1 9) _______

A) ∞, for all x

B) 0

C) 1

D) 2

Find the interval of convergence of the series. 10)

A) 3 < x < 5

B) x < 5

10) ______

C) 3 ≤ x < 5

D) -5 < x < 5

11) ______

11)

A) x < 3

B) -1 ≤ x < 5

C) 1 ≤ x < 3

D) 1 < x < 3 12) ______

12)

A) -3 ≤ x ≤ 7

B) 1 ≤ x < 3

C) -3 < x < 7

D) <x< 13) ______

13)

A) 5 ≤ x ≤ 9

B) x < 9

C) -9 < x < 9

D) 6 ≤ x ≤ 8 14) ______

14)

A) 3 ≤ x < 5

B) -∞ < x < ∞

C) x < 5

D) 3 < x < 5 15) ______

15)

A) -∞ < x < ∞

B) x ≤ 10

C) 8 ≤ x ≤ 10

D) 7 ≤ x ≤ 11 16) ______

16)

A) 0 < x < 2

B) -2 < x < 4

C) x < 4

D) -4 < x < 4

For what values of x does the series converge absolutely? 17)

A) -8 < x ≤ -6

B) -8 < x < -6

17) ______

C) -8 ≤ x < -6

D) -8 ≤ x ≤ -6 18) ______

18)

B) -

<x<-1

C) -

<x≤-1

D) -1≤x<-

-1<x<19) ______

19)

B) 0<x<

20)

C) -

<x<

D) None x> 20) ______

A) 2 < x ≤ 4

B) 2 ≤ x ≤ 4

C) 2 < x < 4

D) 2 ≤ x < 4 21) ______

21)

A) 0 < x < ∞

B) -∞ < x < ∞

C) -∞ < x < 0

D) 0 ≤ x < ∞ 22) ______

22)

B) -1 < x < 1 -

<x<

D) -8 < x < 8 -

<x<

23) ______

23)

A) -1 ≤ x < 1

B) -1 < x ≤ 1

C) -1 ≤ x ≤ 1

D) -1 < x < 1 24) ______

24)

A) 0 < x < 3

B) -7 ≤ x < 3

C) 0 < x ≤ 3

D) -7 < x < 3 25) ______

25)

A) x = -8

B) -9 < x < -7

C) x = ±8

D) x = 8 26) ______

26)

A) 0 ≤ x < ∞

B) -1 ≤ x ≤ 1

C) x = 0

D) -1 < x < 1

For what values of x does the series converge conditionally? 27)

A) x = -3

B) x = -1

27) ______

C) x = -2

D) None 28) ______

28)

A) x = -10

C) x=-

D) None x=29) ______

29)

B) x = 0 x=

D) None x=-

30) ______

30)

A) x = -4, x = -2

B) x = -4

C) x = -2

D) None 31) ______

31)

B) x = 0

x=-

D) None x= 32) ______

32)

A) x = -1

B) x = 1

C) x = ±1

D) None 33) ______

33)

A) x = 1

B) x = -1

C) x = ±1

D) None 34) ______

34)

A) x = -3

B) x = -9, x = -3

C) x = -9

D) None 35) ______

35)

A) x = -2, x = 0

B) x = -2

C) x = 0

D) None 36) ______

36)

A) x = 1

B) x = -1

C) x = ±1

D) None

Find the sum of the series as a function of x. 37)

B) -

37) ______

38) ______

38)

C) -

D) 39) ______

39)

40) ______

40)

B) -

C) -

D) -

41) ______

41)

D) 42) ______

42)

Provide an appropriate response. 43) If the series series converge? A) -8 < x ≤ -6

43) ______ is integrated term by term, for what value(s) of x does the new B) -8 < x < -6

C) -8 ≤ x < -6

D) -8 ≤ x ≤ -6 44) ______

44) If the series is integrated term by term, for what value(s) of x (if any) does the new series converge and for which the given series does not converge? A) x = -10, x = -8 B) x = -8 C) x = -10 D) none

45) ______

45) If the series new series converge? A) 5 < x ≤ 7

is integrated twice term by term, for what value(s) of x does the B) 5 < x < 7

C) 5 ≤ x ≤ 7

D) 5 ≤ x < 7 46) ______

46) If the series is integrated twice term by term, for what value(s) of x (if any) does the new series converge and for which the given series does not converge? A) x = 6 B) none C) x = 4 D) x = 4, x = 6 47) Find the sum of the series

by expressing

as a geometric series, differentiating

bot s of the h resultin side g

___ ___

equation 47) with respect to x, and replacing x by

. A)

48) ______

48) Find the sum of the series

by expressing

as a geometric series,

differentiating both sides of the resulting equation with respect to x, and replacing x by A) B) C) D)

49) ______

49) Find the sum of the series by expressing as a geometric series, differentiating both sides of the resulting equation with respect to x, multiplying both sides by x, differentiating again, and replacing x by A)

. B)

50) ______

50) Find the sum of the series by expressing as a geometric series, differentiating both sides of the resulting equation with respect to x, multiplying both sides by x, differentiating again, and replacing x by A) B)

. C)

51) ______

51) Find the sum of the series

[Hint: Write the series as 1 + A) 3(e - 1) B) 4e - 5

=1+

+ C) e + 3

.] D) 2e 52) ______

52) Find the sum of the series

[Hint: Write the series as 1 + 4 +

=5+

A) 9e - 11

B) 3e + 5

C) 8(e - 1)

D) 5e

1) D 2) D 3) D 4) A 5) C 6) A 7) A 8) A 9) C 10) A 11) C 12) B 13) A 14) A 15) A 16) B 17) B 18) A 19) B 20) C 21) B 22) B 23) C 24) D 25) B 26) B 27) D 28) D 29) D 30) C 31) D 32) A 33) D 34) A 35) B 36) D 37) A 38) C 39) D 40) C 41) C 42) D 43) A 44) B 45) C 46) D 47) A 48) C 49) C 50) D 51) D

52) D

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the Taylor polynomial of order 3 generated by f at a. 1) f(x) = A)

1) _______

,a=0 B) (x) =

(x) =

D) (x) =

2) _______

2) f(x) = A)

,a=0 B) (x) =

(x) =

D) (x) =

3) _______

3) f(x) = A)

,a=1 B) (x) =

(x) =

D) (x) =

(x) =

4) _______

4) f(x) = A)

,a=1 B) (x) =

(x) =

D) (x) =

5) _______

5) f(x) = ln x, a = 5 A)

(x) = ln 5 -

(x) = ln 5 +

D) (x) =

6) _______

6) f(x) = ln(x + 1), a = 6 A) (x) = ln 5 +

(x) =

B) +

(x) = ln 7 + + C)

D) (x) = ln 7 -

f(x) = A)

(x) = ln 5 -

7) _______ B) -

(x) = 1 - 7x +

B) C) D) 9)

f(x) = A) B) C) D)

10)

f(x) = A) B) C) D)

D) (x) = 1 - 7x +

f(x) = A)

,a=0 (x) = 1 - 7x +

(x) = 1 - 49x +

8) _______

,a=4 (x) = 1 + 8(x - 4) + 12

+ 16

(x) = 16 + 8(x - 4) + 12

+ 16

(x) = 1 + 32(x - 4) + 192

+ 1024

(x) = 16 + 8(x - 4) + 9) _______

,a=3 (x) = 6 + 3(x - 3) +

(x) = 27 + 27(x - 3) + 9

(x) = 108 + 27(x - 3) + 6

(x) = 27 + 9(x - 3) + 9

+ 10) ______

+ x + 1, a = 9 (x) = 91 + 19(x - 9) + (x) = 10 + 19(x - 9) + 28 (x) = 1 + 3(x - 9) + 3

(x) = 91 + 19(x - 9) + 19

+ 91

Find the Maclaurin series for the given function. 11)

11) ______

12) ______

12) A)

13) ______

13) A)

14) ______

14) A)

15) ______

15) sin 5x A)

16) ______

16) cos 6x A)

17) ______

17) cos (-3x) A)

D) -

18) ______

18) A)

Find the Taylor series generated by f at x = a.

19) ______

19) f(x) = 3x - 6, a = -3 A) 3(x + 3) - 3 20)

f(x) = A)

f(x) = A) C)

22)

f(x) = A)

C) 3(x - 3) - 15

20) ______

+ 2(x + 2) - 3

+ 2(x - 2) - 3

+ 2(x + 2) - 13

+ 2(x - 2) - 13 21) ______

- 7x + 8, a = 4

- 20 + 20

+ 105(x - 4) - 172

- 20

+ 119(x - 4) - 172

+ 105(x - 4) + 172

+ 20

+ 119(x - 4) + 172 22) ______

- 5x + 7, a = -3

+ 10

- 12

+ 64

- 173(x + 3) + 193

- 12

+ 64

- 163(x + 3) + 1

+ 12

+ 64

- 163(x + 3) + 1

+ 12

+ 64

- 173(x + 3) + 193 23) ______

23) f(x) = A)

,a=6 B)

24) ______

24) f(x) = A)

,a=8 B)

25) ______

25) f(x) = A)

,a=3 B)

26)

D) 3(x - 3) - 3

+ 6x + 5, a = -2

C) 21)

B) 3(x + 3) - 15

f(x) = A)

26) ______

,a=2 B)

27)

f(x) = A)

27) ______

,a=8 B)

28)

f(x) = A)

28) ______

, a = 10 B)

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. 29) 29) ______ f(x) = cos x A)

-3 - 4x -

- ..., -1 < x < 1

-3 - x -

- ..., -1 < x < 1

D) -3 - 4x +

- ..., -1 < x < 1

+ ..., -1 < x < 1 30) ______

30) f(x) = (cos x) ln(1 + x) A) x-

-3 + 4x +

+ ..., -1 < x < 1

D) x-

+ ..., -1 < x < 1

Find the quadratic approximation of f at x = 0. 31) f(x) = 7x A) Q(x) = 7x B)

31) ______ C)

Q(x) = 1 - 7

Q(x) = 1 + 7

Q(x) 1 + 32) f(x) = tan 10x A) Q(x) = 1 + 5 33)

32) ______ B)

Q(x) = 1 + 10

C) Q(x) = 10x

Q(x) = 1 - 5 33) ______

f(x) = A)

B) Q(x) = 9 +

34) f(x) = ln(cos 4x)

C) Q(x) = 9 -

D) Q(x) = 9 +

Q(x) = 9 34) ______

Q(x) = 1 + 8

Q(x) = 1 - 8

B) D)

Q(x) = 8

Q(x) = 1 + 10x + 50 Q(x) = 1 - 10x + 50 36) ______

f(x) = A)

B) Q(x) = 5x -

Q(x) = 1 + 5x +

D) Q(x) = 1 - 5x +

37)

Q(x) = - 8

35) ______

35) f(x) = ln(1 + sin 10x) A) Q(x) = 10x + 50 C) Q(x) = 10x - 50 36)

Q(x) = 5x + 37) ______

f(x) = A)

B) Q(x) = 9x Q(x) = 9x -

D) Q(x) = 9x +

38)

f(x) = x A) Q(x) = 10x

Q(x) = 1 + 9x + 38) ______ B)

Q(x) = 100

C) Q(x) = 1 + 10x

D) Q(x) = 1 - 10x 39) ______

39) f(x) = A)

B) Q(x) = 7x

Q(x) = 1 + 40) f(x) = sin ln(6x + 1) A) Q(x) = 6x + 18 C) Q(x) = 1 - 6x + 18

D) Q(x) = 1 + 7x Q(x) = 40) ______

B) D)

Q(x) = 1 + 6x + 18 Q(x) = 6x - 18

Provide an appropriate response. 41) For which of the following is the corresponding Taylor series a finite polynomial of degree 3? A) B) C) 3ln(x) D) 5 +2 - 12 sin x 42) 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) I, II, and III B) III only C) I and III D) I and II

41) ______

42) ______

1) C 2) A 3) B 4) A 5) B 6) B 7) C 8) D 9) B 10) A 11) D 12) B 13) B 14) D 15) B 16) B 17) A 18) C 19) B 20) A 21) C 22) A 23) A 24) D 25) B 26) D 27) D 28) C 29) A 30) A 31) A 32) C 33) D 34) C 35) C 36) B 37) B 38) A 39) C 40) D 41) A 42) C

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use substitution to find the Taylor series at x = 0 of the given function. 1)

ln(1 + A)

1) _______

2) _______

) B)

3) _______

3) ln A)

B) 2

D) 2

Use power series operations to find the Taylor series at x = 0 for the given function. 4) A)

5) _______

4) _______

6) _______

sin x A)

__ __ __ _

cos πx

8) _______

(4x) A)

B) +

+ D)

+ 9) _______

9) A)

10) ______

10)

11)

11) ______

ln(1 + 2x) A)

12)

12) ______

(8x) A)

13) ______

13) A)

Find the first four nonzero terms in the Maclaurin series for the function. 14) A)

14) ______

B) 1 + 6x +

+...

D) 1 + 6x +

x + 6x +

+...

6x + 216

+ 7776

+ 279,936

+...

+... 15) ______

15) A)

B) 1-

+...

D) 2+

16)

ln(1 + A)

+...

+... 16) ______

) B) -

+...

D) +

+...

17) ______

17) x sin(2x) A) 2

B) +

+...

D) 2x - 2

+...

+... 18) ______

18) sin x cos x A) x+

2x -

B) -

+...

1+x-

+...

D) x-

+...

Solve the problem. 19) Use a graphical method to determine the approximate interval for which the second order Taylor polynomial for more than 0.04. A) -0.5525 ≤ x ≤ 0.2640 C) -0.5525 ≤ x ≤ 0.5525 20)

approximates

19) ______

with an absolute error of no

B) -0.4310 ≤ x ≤ 0.5525 D) -0.1928 ≤ x ≤ 0.7063 The polyno

mial 1 +

___ ___

20)

7x + 21 is used to approxi mate on the interval . Use the Remaind er Estimatio n Theorem to estimate the maximu m absolute error. A) 21)

≈ 2.061 x

≈ 3.642 x

You plan to estimate e by evaluating the Maclaurin series for at How many terms of the series would you have to add to be sure the estimate is good to 4 decimal places? A) 9 B) 11 C) 10 D) 12

21) ______

22) ______

22) For what values of x can we replace cos x by A) -0.182 ≤ x ≤ 0.182 C) -0.251 ≤ x ≤ 0.251

with an error of magnitude no greater than B) -0.288 ≤ x ≤ 0.288 D) -0.394 ≤ x ≤ 0.394

23) Use the Alternating Series Estimation Theorem to estimate the error that results from replacing by A) 0.00133

when B) 0.00027

C) -0.00133

D) 0.00267

Provide an appropriate response. 24)

24) ______

For approximately what values of x can sin x be replaced by no greater than A) |x| < 0.04949

23) ______

with an error of magnitude

? B) |x| < 0.09029

C) |x| < 0.03310

D) |x| < 0.06544 25) ______

25) For approximately what values of x can sin x be replaced by magnitude no greater than

with an error of

A) |x| < 0.59235

B) |x| < 0.63837

C) |x| < 0.82127

D) |x| < 0.79476 26) ______

26) For approximately what values of x can cos x be replaced by no greater than A) |x| < 0.58857

with an error of magnitude

? B) |x| < 0.49324

C) |x| < 0.31072

D) |x| < 0.41618 27) ______

27) For approximately what values of x can cos x be replaced by magnitude no greater than ? A) |x| < 0.26672 B) |x| < 0.19786

C) |x| < 0.14310

with an error of D) |x| < 0.20477 28) ______

28) For approximately what values of x can magnitude no greater than ? A) |x| < 0.18086 B) |x| < 0.17627

x be replaced by C) |x| < 0.23091

with an error of D) |x| < 0.22588 29) ______

29) If sin x is replaced by A) |E| < 0.00010125 C) |E| < 0.000020250

and |x| < 0.3, what estimate can be made of the error? B) |E| < 0.00033750 D) |E| < 0.000067500 30) ______

30) If sin x is replaced by A) |E| < 0.000021701 C) |E| < 0.000001550

and |x| < 0.5, what estimate can be made of the error? B) |E| < 0.000003100 D) |E| < 0.000010851 31) ______

31) If cos x is replaced by A) |E| < 0.00033750 C) |E| < 0.0011250

and |x| < 0.3, what estimate can be made of the error? B) |E| < 0.0045000 D) |E| < 0.0013500 32) ______

32) If cos x is replaced by A) |E| < 0.00016340 C) |E| < 0.00023343

and |x| < 0.7, what estimate can be made of the error? B) |E| < 0.00098041 D) |E| < 0.0014006 33) ______

33) If

x is replaced by A) |E| < 0.0066651 C) |E| < 0.0077760

and |x| < 0.6, what estimate can be made of the error? B) |E| < 0.0046656 D) |E| < 0.0039991

1) C 2) B 3) B 4) C 5) B 6) D 7) D 8) A 9) C 10) C 11) C 12) B 13) D 14) A 15) D 16) B 17) B 18) D 19) B 20) B 21) C 22) D 23) A 24) B 25) C 26) A 27) A 28) C 29) C 30) C 31) A 32) A 33) D

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the first four terms of the binomial series for the given function. 1) A) C)

1 + 2x - 2 1 - 2x + 2

-2

1 - 2x + 2

-4

1 + 2x - 2

1) _______

2) _______

2) A) C)

1 - 4x - 8

- 16

1 + 4x + 8

- 16

1 - 4x + 8

- 32

1 - 4x - 8

- 32 3) _______

3) A)

B) 1+

D) 1+

4) _______

4) A)

B) 1-

D) 1-

5) _______

5) A)

B) 1-

D) 1-

6) _______

6) A)

B) 1-

D) 1-

7) _______

7) A)

B) 1+

D) 1-

8) _______

8) A)

B) 1+

D) 1-

9) _______

9) A)

B) 1+

D) 1+

10) ______

10) A) C)

1+6 1+6

+ 18 + 12

+ 54

1+6

+ 20

1+6

+ 56 +

Solve the problem. 11) Use series to estimate the integral's value to within an error of magnitude less than

A) 0.8427 12)

D) 0.6499

B) 0.002487

C) 0.001313

B) 0.6007

C) 1.411

B) 0.7287

C) 0.3866

13) ______

D) 0.6223

Use series to estimate the integral's value to within an error of magnitude less than

A) 1.204

12) ______

D) 0.01056

Use series to estimate the integral's value to within an error of magnitude less than

A) 0.3257 14)

C) 0.7758

Use series to estimate the integral's value to within an error of magnitude less than

A) 0.002635 13)

B) 0.3748

11) ______

14) ______

D) 0.7121

Answer the question. 15) Estimate the error if A)

15) ______ is approximated by 1 - 5 + B) C)

in the integral of D)

dx.

16) ______

16) Estimate the error if sin A)

is approximated by B)

in the integral of D)

dx.

Find the polynomial of lowest degree that will approximate F(x) throughout the given interval with an error of magnitude less than than 17)

. 17) ______ F(x) = A)

[0, 1] B)

18) ______

18) F(x) = A) x+

[0, 1] B)

C) x-

D) x-

+ 19) ______

19) F(x) = A) x-

[0, 0.5] B)

C) x-

D) x-

+ 20) ______

20) F(x) = A) x-

[0, 1] B)

D) x-

+ 21) ______

21) F(x) = A) x-

[0, 0.5] B)

D) x-

22) ______

22) F(x) = A)

[0, 0.75]

B) -

+ -

23) ______

23) F(x) = A)

[0, 0.5] B)

24) ______

24) F(x) = A)

[0, 0.5] B)

D) x-

+ 25) ______

25) F(x) = A)

, -

[0, 0.6] B)

C) -

26) ______

26) F(x) = A)

[0, 0.75] B)

C) x+

D) x+

Use series to evaluate the limit. 27) A) 5

27) ______ B) 1

C) 0

D) -5 28) ______

28) A) 9

29)

B) 0

29) ______

A) -1

B) 6

C) -6

D) 1 30) ______

30) A) -10

B) -100

C) 100

D) 10 31) ______

31) A) -7

C) -

D) -

32) ______

32) A) - 243

B) - 27

C) - 486

D) 33) ______

33) A)

D) -

34) ______

34) A)

35) ______

35) A)

B) - 500

C) - 50

D) - 5

36) ______

36) A) 3

B) -3

C) -1

D) 1

Find the sum of the series. 37) 1+5+ A)

37) ______ +

+ ... B) ln 6

D) sin 38) ______

38) A)

+ cos

+ ... B) sin

D) ln

sin

39)

40)

+ A)

39) ______

+ ... B)

40) ______

+ ... B)

Solve the problem. 41)

41) ______

Using the Maclaurin series for ln(1 + x), obtain a series for ln A) B)

2 C)

D) 2 42) ______

42) Using the Maclaurin series for ln(1 + x), obtain a series for A) B)

43) ______

43) Using the Maclaurin series for A)

x, obtain a series for B)

Provide an appropriate response. 44) Use the fact that x. A) -x+ B) C)

44) ______ x=x+

for |x| < 1 to find the series for

-x+ D) -x45) ______

45) Use the fact that A)

for |x| < 1 to find the series for B)

D) 46) ______

46) Use the fact that tan x = x + the series for ln(cos x). A)

+ ... for |x| <

to find the first four terms of

D) +

+ ...

+ ... 47) ______

47) Use the fact that cot x = series for ln(sin x). A)

for |x| < π to find the first four terms of the B) +

+ ...

D) ln|x| -

- ln|x| +

+ ... 48) ______

48) Use the fact that csc x =

the series for 2 csc(2x) A) ln|x| -

+ ... (for |x| < π) to find the first four terms of

and thus for ln tan x. B)

+ ...

ln|x| +

ln|x| -

+ ...

D) ln|x| +

+ ...

+ ... 49) ______

49) Derive the series for A)

for x > 1 by first writing B) C)

. D)

50) ______

50) Derive the series for A)

for x > 1 by first writing B) C)

. D)

51) ______

51) Derive a series for ln(1 + x) for x > 1 by first finding the series for

and then integrating.

B) ln x +

ln x +

D) ln x +

ln x + 52) ______

52) Derive the series for A)

for x > 1 by first writing B) C)

. D)

53) ______

53) Derive a series for ln(1 +

) for x > 1 by first finding the series for

and then integrating.

B) 2 ln x +

2 ln x +

D) ln x +

ln x + 54) ______

54) Obtain the first two terms of the Maclaurin series for sin(tan x). A) B) C) x-

x+ 55) ______

55) Obtain the first nonzero term of the Maclaurin series for sin x - tan x. A) B) C)

56)

Obtain the first nonzero term of the Maclaurin series for sin x A) B) C)

56) ______

x. D)

57)

Obtain the first nonzero term of the Maclaurin series for

57) ______

58)

Obtain the first nonzero term of the Maclaurin series for A) B) C) -

. D)

59) ______

59) Obtain the first nonzero term of the Maclaurin series for sin(tan x) - tan(sin x). A) B) C) D) 60)

58) ______

Obtain the first nonzero term of the Maclaurin series for A) B) C) -

60) ______

. D) -

1) A 2) D 3) D 4) B 5) A 6) C 7) B 8) B 9) A 10) C 11) D 12) A 13) B 14) B 15) A 16) B 17) B 18) C 19) C 20) D 21) C 22) B 23) C 24) D 25) B 26) B 27) A 28) C 29) C 30) B 31) C 32) C 33) A 34) C 35) B 36) A 37) A 38) B 39) D 40) B 41) D 42) C 43) D 44) D 45) B 46) A 47) C 48) C 49) C 50) D 51) B

52) D 53) B 54) D 55) B 56) C 57) C 58) B 59) B 60) C

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 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. 1) x = 2 cos t, y = 2 sin t, π ≤ t ≤ 2π 1) _______

+ = 4; Counterclockwise from (2, 0) to (2, 0), one rotation

= 4; Clockwise from (2, 0) to (-2, 0)

+ = 1; Clockwise from (1, 0) to (1, 0), one rotation

+ = 4; Counterclockwise from (-2, 0) to (2, 0)

2) x = 4 sin t, y = 5 cos t, 0 ≤ t ≤ 2π

A) + = 1; Counterclockwise from (5, 0) to (5, 0), one rotation

2) _______

B) +

= 1; Co un ter cl oc k wi se fro m (0, 4) to (0, 4), on e rot ati on

C) + = 1; Counterclockwise from (0, 5) to (0, 5), one rotation

D) +

= 1; Co un ter cl oc k wi se fro m (4, 0) to (4, 0), on e rot ati on

x = 4 , y = 2t, -∞ ≤ t ≤ ∞

__ __ __ _

x= ; Entire parabola, top to bottom (from 1st quadrant to origin to 4th quadrant)

x= ; Entire parabola, bottom to top (from 4th quadrant to origin to 1st quadrant)

y= ; Entire parabola, right to left (from 1st quadrant to origin to 2nd quadrant)

y= ; Entire parabola, left to right (from 2nd quadrant to origin to 1st quadrant)

4) x = 5t + 2, y = 15t + 14, -∞ ≤ t ≤ ∞

A) y = 3x + 8; Entire line, from right to left

4) _______

C) y = 3x - 8; Entire line, from right to left

y = 3x + 8; En tir e lin e, fro m lef t to rig ht

D) y = 3x - 8; Entire line, from left to right

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find a parametrization for the curve. 5) The line segment with endpoints (7, -6) and (10, 3) 5) _____________ 6)

The upper half of the parabola x - 8 =

6) _____________

7) _____________

7) The ray (half line) with initial point (-4, 6) that passes through the point (4, 0)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 8) Find the point on the parabola x = , y = t, -∞ < t < ∞, closest to the point Minimize the square of the distance as a function of t.) A) B) (9, 3) C) (36, 6)

8) _______

. (Hint: D) (0, 0)

9) _______

9) Find the point on the curve x = 5 sin t, y = cos t, ≤t≤ Minimize the square of the distance as a function of t.) A) (5, 0) B) (-5, 0) C)

, closest to the point

Graph the pair of parametric equations with the aid of a graphing calculator. 10) x = 4(t - sin t), y= 4(1 - cos t), 0 ≤ t ≤ 4π

. (Hint:

10) ______

11) ______

11) x = t - sin t, y = 1 - cos t, -4π ≤ t ≤ 4π

12) ______

12) x = 7t - 7 sin t, y = 7 - 7 cos t, -2π ≤ t ≤ 2π

13) ______

13) x = cos 5t, y = sin 3t, 0 ≤ t ≤ 2π

14) ______

14) x = 3 cos t + cos 3t, y = 3 sin t - sin 3t, 0 ≤ t ≤ 2π

15) ______

15) x = 6 cos t + 2 cos 3t, y = 6 sin t - 2 sin3t, 0 ≤ t ≤ 2π

16) ______

16) x = cos t + 4 cos 3t, y = 5 cos t - 4 sin 3t, 0 ≤ t ≤ 2π

17) ______

17) x = 5 cos 2t + 4 cos 6t, y = 5 sin 2t - 4 sin 6t, 0 ≤ t ≤ π

18) ______

18) x = 4 sin t - sin 4t, y = 4 cos t - cos 4t, 0 ≤ t ≤ 2π

19) ______

19) x = 5 tan t, y = 3 sec t; -π/2 ≤ t ≤ π/2

1) D 2) C 3) B 4) B 5) Answers will vary. Possible answer: x = 7 + 3t, y = -6 + 9t, 0 ≤ t ≤ 1 6) Answers will vary. Possible answer: x = - 8, y = t, t ≥ 0 7) Answers will vary. Possible answer: x = -4 + 8t, y = 6 - 6t, t ≥ 0 8) B 9) C 10) C 11) D 12) D 13) C 14) C 15) A 16) D 17) D 18) B 19) B

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find an equation for the line tangent to the curve at the point defined by the given value of t. 1) x = 7 sin t, y = 7 cos t, t = A) y = - x + 7

B) y = x - 7

C) y = 7x + 7

D) y = 7

1) _______

x+1 2) _______

2) x = sin t, y = 10 sin t, t = A) y = -10x + 10

B) y = 10x

C) y = 10x - 10

D) y = 10x + 3) _______

3) x = t + cos t, y = 2 - sin t, t = A) y=-

π+2

y=-

D) y=-

π+3

π 4) _______

4) x = csc t, y = 18 cot t, t = A) y = 36x - 18

B) y = 6

C) y = -36x + 18 5) x = t, y =

D) y = 36x + 6 5) _______

,t=8

B) y=-

x=8 A)

- 5, y =

C) y=

x+2

xy/

D) y=

y=-

x-2 6) _______

,t=1 B)

Find the value of 7)

x - 36

C) y = 8x -

D) y=

x+1

at the point defined by the given value of t.

x = 7 sin t, y = 7 cos t, t = A) -2

7) _______ B) 2

D) 8) _______

8) x = sin t, y = 9 sin t, t = A) -18 9)

B) 18

C) 0

D) -9 x = t + cos t,

__ __ __ _

y = 2 - sin 9) t, t = A) -

B) -2

D) -4

10) ______

10) x = tan t, y = 9 sec t, t = A)

B) 9

D) 11) ______

11) x = csc t, y = 2 cot t, t = A) 2 12) x = t, y =

B) -4

12) ______ B)

x=8 A)

D) 4

,t=3

13)

C) -6

C) -

- 5, y =

D) 13) ______

,t=1 B)

D) -

14) x =

14) ______

, y = - t, t = 22

B) 10

D) -2

Assuming that the equations define x and y implicitly as differentiable functions at the given value of t. 15) 2x + 4 = + t, y(t + 1) - 4t = 36, t = 0 A) - 6 B) 24 C) -24 16)

x(t + 1) - 4t A)

= 9, 2y + 4

= B) 6

find the slope of the curve 15) ______ D) -12 16) ______

+ t, t = 0 C) 3

17) ______

17) x cos t + x = 2t, y = t sin t + t, t = A) 2 + π B)

D) 2

18) ______

18) x = t sin t + t, y cos t + y = 2t, t = A) B)

C) 2 + π

D) -

19)

20)

2x - t = 0, 2ty + 6 A) - 4 2tx + 2 = 2, 2y A) - 3

= 6, t = 1 B) - 6 - t = 0, t = 1 B)

19) ______ C) 9

D) - 9 20) ______

D) -

Find the area. 21) Find the area under one arch of the cycloid A) 75π B) 50π

21) ______ C)

D) 15π

22) Find the area enclosed by the ellipse x = 3 cos t, y = 12 sin t, 0 ≤ t ≤ 2π. A) B) 72π C) 9 + 144π 36

22) ______ D) 36π 23) ______

23) Find the area of the region between the curve x = A) 15 B) 8

,y= C) 16

and the x-axis, 0 ≤ t ≤ ln 9. D) 9

Find the length of the curve. 24) x = 5 cos t, y = 5 sin t, 0 ≤ t ≤ π A) π B) 10π

C) 25π

D) 5π

25) x = cos 2t, y = sin 2t, 0 ≤ t ≤ 2π A) π B) 2π

C) 6π

D) 4π

26) x = 3 sin t + 3t, y = 3cos t, 0 ≤ t ≤ π A) 6 B) 18

C) 12

D) 3π

24) ______

25) ______

26) ______

27) ______

27) x = 5 sin t - 5t cos t, y = 5cos t + 5t sin t, 0 ≤ t ≤ A) B)

D) π

28)

29)

x=3 A) 18

t, y = 3

t, 0 ≤ t ≤ π B) 6

x= ,y=2 ,0≤t≤1 A)

28) ______ C) 9

D) 3 29) ______

30) ______

30) x= ( A)

- 3t), y =

+ 5, 0 ≤ t ≤ 1 B) 2

31) ______

31) x=

,y=

,0≤t≤1

A) 3

B) 2

D) 1

32) ______

32) x = 3t, y = A) 49

, 0 ≤ t ≤ 16 B)

33) ______

33) x= A) 2

, y = 4t, 0 ≤ t ≤ 1 B) 14

Find the area of the surface generated by revolving the curves about the indicated axis. 34) x = sin t, y = 6 + cos t, 0 ≤ t ≤ 2π; x-axis A) B) C) 6 18 24

34) ______ D)

36 35) ______

35) x=t+ , y= A) 1330π

t, ≤t≤ B) 532π

; y-axis C)

D) π

Find the coordinates of the centroid of the curve. 36) Find the coordinates of the centroid of the curve x = 3 cos t + 3t sin t, y = 3 sin t - 3t cos t, A) B) ( , )= C)

36) ______

( , )= D)

( , )=

( , )= 37) ______

37) Find the coordinates of the centroid of the curve x = cos 4t, y = A) B) ( , )= C)

( , )= D)

( , )=

sin 4t, 0 ≤ t ≤

1) B 2) B 3) C 4) A 5) B 6) D 7) C 8) C 9) D 10) C 11) C 12) C 13) B 14) D 15) C 16) A 17) C 18) A 19) A 20) C 21) A 22) D 23) B 24) D 25) D 26) C 27) D 28) C 29) A 30) A 31) C 32) C 33) D 34) C 35) D 36) C 37) B

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine if the given polar coordinates represent the same point. 1) (10, π/4), (10, 5π/4) A) Yes B) No 2) (7, π/4), (-7, π/4) A) No

2) _______ B) Yes 3) _______

3) (3, π/4), (-3, -π/4) A) No

B) Yes

4) (9, π/2), (-9, 3π/2) A) No

B) Yes

5) (8, π/3), (-8, -2π/3) A) No

B) Yes

6) (5, π/6), (5, 7π/6) A) No

B) Yes

7) (8, π/6), (-8, 7π/6) A) No 8) (4, -4π/3), (-4, 5π/3) A) Yes 9) (r, θ), (-r, θ + π) A) No Find all the polar coordinates of the point. 10) (3, π/4) A) (3, π/4 + 2nπ), (-3, -π/4 + 2nπ) C) (3, π/4 + 2nπ), (-3, -3π/4 + 2nπ) 11) (-10, π/3) A) (10, π/3 + 2nπ), (10, 4π/3 + 2nπ) C) (10, 4π/3 + 2nπ), (-10, 2π/3 + 2nπ) Plot the point whose polar coordinates are given. 12) (2, -5π/4)

1) _______

4) _______

5) _______

6) _______

7) _______ B) Yes 8) _______ B) No 9) _______ B) Yes

10) ______ B) (3, π/4 + 2nπ), (3, -3π/4 + 2nπ) D) (3, π/4 + 2nπ), (-3, 3π/4 + 2nπ) 11) ______ B) (10, 4π/3 + 2nπ), (-10, π/3 + 2nπ) D) (10, 4π/3 + 2nπ), (-10, -π/3 + 2nπ)

___ ___

12)

13) (4, π/2)

___ ___

13)

14) (2, 0)

___ ___

14)

15) (-4, 2π/3)

___ ___

15)

Find the Cartesian coordinates of the given point. 16) A) (0, 5) 17) (

, π/6) A)

18) (-4, -π/3)

B) (-5, 0)

16) ______ C) (5, 0)

D) (0, -5) 17) ______

18) ______

A) (-2

B) (-2, 2

, 2)

)

C) (2, 2

D) (-2

)

, -2) 19) ______

19) (-8, π) A) (-8, 0)

B) (0, -8)

C) (0, 8)

D) (8, 0)

20) (-3, 0) A) (0, -3)

B) (0, 3)

C) (3, 0)

D) (-3, 0)

20) ______

21) ______

21) A) (9

B) (18, 18)

C) (9

)

D) (18

, 9)

, 18

) 22) ______

22)

23) ______

23)

24) ______

24) A) (-12 C) (6

, -12

)

, -6)

B) (6

, -6

)

D) (-6

, -6

) 25) ______

25) A) (-6, -6

)

B) (6

, -6)

Find the polar coordinates, 26) (3, 3) A)

and

, of the point given in Cartesian coordinates.

28) (-2, 0) A) (-2, π)

29) (0, -4) A) (4, 0)

)

D) (-6

, -6)

26) ______ B)

27) ______

27) (4, -4) A)

30) (0,

C) (6, -6

28) ______ B)

C) (2, π)

29) ______ B)

C) (4, π)

30) ______

) B)

31) (-

32) (

,A)

31) ______

) B)

32) ______

) B)

( 33) ______

33) A)

34) ______

34) A)

35) ______

35) A)

Graph the set of points whose polar coordinates satisfy the given equation or inequality. 36) r = 3

36) ______

37) ______

37) r ≤ 4

38) 2 ≤ r ≤ 5

___ ___

38)

39) 0 ≤ θ ≤ π, r ≤ 3

39) ______

40) ______

40) π≤θ≤

, r = -4

41) ______

41) 0≤θ≤

,0≤r≤4

Describe the graph of the polar equation. 42) r cos θ = 7 A) Circle centered at origin with radius 7 B) Line with slope 7 passing through the origin C) Vertical line through (7, 0) D) Horizontal line through (0, 7) 43) r = -11 csc θ A) Line through origin with slope -11 C) Horizontal line through (0, -11)

42) ______

43) ______ B) Vertical line through (-11, 0) D) Circle of radius 11 centered at origin

44) 7r cos θ + r sin θ = 9 A) Parabola with vertex (7, 9) opening upward B) Line with slope -7 and y-intercept (0, 9) C) Line with slope 9 and y-intercept (0, 7) D) Vertical line passing through (7, 0)

44) ______

45)

45) ______ r= A) Ellipse centered at origin with intercepts (±2, 0) and (0, ±1) B) Line with slope 4 and y-intercept (0, -2) C) Line with slope 2 and y-intercept (0, -4) D) Hyperbola passing through (4, 2)

46) r = 6cot θ csc θ A) Circle with radius 6 centered at the origin B) Parabola with vertex at origin opening upward C) Line with slope 6 passing through origin D) Parabola with vertex at origin opening to the right

46) ______

47)

47) ______

48)

sin 2θ = 24 A) Hyperbola with the x- and y-axes as asymptotes B) Parabola centered at the origin opening upward C) Parabola centered at the origin opening to the right D) Hyperbola with asymptotes y = ±12x + 2 sin θ cos θ = 441 A) Two straight lines of slope -1, y-intercepts (0, ±21) B) Parabola opening upward with vertex at (0, 21) C) Parabola opening to the right with vertex at (21, 0) D) Two horizontal lines passing through (0, ±21)

48) ______

49) r = 32 sin θ A) Horizontal line passing through (0, 32) B) Circle with radius 16 and center (0, 16) C) Parabola opening upward with vertex at (0, 16) D) Ellipse centered at the origin with intercepts (±1, 0) and (0, ±16)

49) ______

50)

50) ______

= 24r cos θ A) Circle of radius 12 and center (0, 12) C) Circle of radius 12 and center (12, 0)

B) Vertical line passing through (24, 0) D) Horizontal line passing through (0, 24)

51)

51) ______

= 50 rcos θ - 6r sin θ - 9 A) Parabola opening upward with vertex (50, -6) B) Circle with radius 25 and center (25, -3) C) Circle with radius 3 and center (50, -6) D) Parabola opening to the right with vertex (50, -6)

Replace the polar equation with an equivalent Cartesian equation. 52) r cos θ = 12 A) x = 12 B) 12x = 1 C) 12y = 1

52) ______ D) y = 12 53) ______

53) r = -13 csc θ A) x = -13

B) -13x = 1

C) -13y = 1

D) y = -13 54) ______

54) 7r cos θ + 2r sin θ = 1 A) 7y + 2x = 1

C) 7x + 2y = 1

7x + 2y =

+ 55) ______

55) r= A)

B) 9x - 6y = 1 +

D) 9y - 6x = 1

=1 56) ______

56) r = 4 cot θ csc θ A)

= 4x

C) y = 4x

y=4

y= 57)

57) ______

sin 2θ = 34 A)

B) y = 17x

y = 34

y= 58)

+ 2 sin θ cos θ = 625 A) x + y = 25

58) ______ B) x + y = ±25

61)

= 289

= 625

= 289

= 289 60) ______

= 40r cos θ A) x = 40

= 400

= 400 61) ______

= 48r cos θ - 6r sin θ - 9 A) 48x - 6y = 9 C)

= 625

59) ______

59) r = 34 sin θ A) y = 34

60)

B) =9

= 576

Replace the Cartesian equation with an equivalent polar equation. 62) + = 144 A) r = 144 B) r = 12 sin θ C) r = 12 cos θ D) r = 12 or r = -12 63) y = 11 A) r = 11 sec θ 64) x = 11 A) r = 11 cos θ

62) ______

63) ______ B) r = 11 cos θ

C) r = 11 csc θ

D) r = 11 sin θ 64) ______

B) r = 11 csc θ

C) r = 11 sin θ

D) r = 11 sec θ 65) ______

65) 2x + 3y = 6 A) 2 cos θ + 3 sin θ = 6r C) r(2 sin θ + 3 cos θ) = 6

B) r(2 cos θ + 3 sin θ) = 6 D) 2 sin θ + 3 cos θ = 6r 66) ______

66) x2 + y2 - 4x = 0 A) r = 4 cos θ C) r sin2θ = 4 cos θ

B) r = 4 sin θ D) r cos2θ = 4 sin θ 67) ______

67) x2 + 4y2 = 4 A) 4 cos2θ + sin2θ = 4r

B) r2(4 cos2θ + sin2θ) = 4

C) cos2θ + 4 sin2θ = 4r

D) r2(cos2θ + 4 sin2θ) = 4 68) ______

68) x2 - y2 = 4 A) cos2θ - sin2θ = 4

B) r cos 2θ = 4

C) cos2θ - sin2θ = 4r

D) r2 cos 2θ = 4 69) ______

69) xy = 1 A) 2r sin θ cos θ = 1

B) 2r2 sin θ cos θ = 1

C) r2 sin 2θ = 2 70)

71)

+ = 256 A) r = 32 sin θ

D) r sin 2θ = 2 70) ______ B)

= 32 cos θ

C) r = 32 cos θ

71) ______

+ = 144 A) r = 24 cos θ - 6 sin θ - 9

= -12r cos θ + 3r sin θ + 144

D) r = 16 sin θ

= 24r sin θ - 6r cos θ - 9 = 24r cos θ - 6r sin θ - 9

1) B 2) A 3) A 4) B 5) B 6) A 7) B 8) A 9) B 10) C 11) B 12) C 13) D 14) B 15) D 16) A 17) D 18) B 19) D 20) D 21) B 22) D 23) C 24) B 25) C 26) A 27) D 28) C 29) D 30) B 31) A 32) D 33) C 34) B 35) D 36) D 37) B 38) C 39) C 40) A 41) B 42) C 43) C 44) B 45) C 46) D 47) A 48) A 49) B 50) C 51) B

52) A 53) D 54) C 55) B 56) B 57) D 58) B 59) D 60) D 61) B 62) D 63) C 64) D 65) B 66) A 67) D 68) D 69) C 70) A 71) D

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine the symmetries of the curve. 1) r = 5 + 5 cos θ A) x-axis only B) y-axis only C) No symmetry D) Origin only 2) r = -5 + 3 sin θ A) No symmetry

2) _______ B) x-axis only

C) Origin only

D) y-axis only 3) _______

3) r = 7θ A) No symmetry C) y-axis only

B) x-axis only D) x-axis, y-axis, origin 4) _______

4) r= ,θ>0 A) No symmetry 5) r = -2 sin θ - 5 cos θ A) x-axis only C) x-axis, y-axis, origin 6) r = 4 cos 5θ A) x-axis, y-axis, origin C) Origin only 7)

10)

1) _______

= 6 cos 4θ A) y-axis only C) Origin only = 5 sin 6θ A) Origin only C) y-axis only = -3 cos 5θ A) x-axis, y-axis, origin C) x-axis only = 4 sin 5θ A) x-axis only C) Origin only

Graph the polar equation. 11) r = 2(1 + 3 sin θ)

B) Origin only

C) x-axis only

D) y-axis only 5) _______

B) y-axis only D) No symmetry 6) _______ B) x-axis only D) y-axis only 7) _______ B) x-axis only D) x-axis, y-axis, origin 8) _______ B) x-axis, y-axis, origin D) x-axis only 9) _______ B) No symmetry D) Origin only 10) ______ B) x-axis, y-axis, origin D) y-axis only

___ ___

11)

12) r = 5 - cos θ

12) ______

13) ______

13) r = 5 sin 3θ

14) ______

14) r = 6 cos 7θ

15)

16)

15) ______

= 63 cos 4θ

= 10 sin 3θ

16) ______

17) ______

17) r = 5(1 -sin θ)

18) ______

18) r = 3θ

19) ______

19) r=-

- sin θ

20) r=-

+ cos θ

___ ___

20)

Find the slope of the polar curve at the indicated point. 21) r = 1 - sin θ, θ = A) 2

-1

21) ______ C)

22) ______

22) r = 5 cos 3θ, θ = A) -

D) 23) ______

23) r = 3 sin 4θ, θ = A) 1 + 24) r = -7 - 5 sin θ, θ = 0 A)

C) -1

D) undefined 24) ______

25) ______

25) r = 5 - 3 cos θ, θ = A)

D) 26) ______

26) r = 9(1 + cosθ), θ = A) -2

B) 6

D) -1

27) ______

27) r = 4, θ = A) -1

C) -

D) 1

28) ______

28) r = , θ = 2π A) -2π

B) 0

29) r = -7 cos θ + 9 sin θ, θ = π A) B) -

C) 2π

D) Undefined

29) ______ 30) ______

30) r = -4 csc θ, θ = A) 0

B) Undefined

Sketch the region which is defined by the given conditions. 31) 0 ≤ θ ≤ π, r ≤ 3

C) -1

D) 1

31) ______

32) ______

32) π≤θ≤

, r = -4

33) ______

33) 0≤θ≤

,0≤r≤4

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 34) Graph the equation 34) _____________ for several values of a in order to get an idea of what these curves look like. (Try both

and

inner loop for every a such that inner loop.

and find the values of θ that correspond to the

35) How are the graphs of of

Show that the graph will have an

and

related to the graph

35) _____________

In general, how is the graph of r = f(θ - α) related to the graph of

36) _____________

36) Consider the curves corresponding to polar equations of the form

where

Explain how the graph changes as a changes. Identify any values of a for which the basic shape of the curve changes. 37) _____________

37) Find any horizontal or vertical asymptotes for the spiral Express the curve in parametric form happens to x and y as θ varies.)

(Suggestion: and investigate what

38) _____________

38) Find any horizontal or vertical asymptotes for the curve (Suggestion: Express the curve in parametric form investigate what happens to x and y as θ varies.)

and

39) _____________

39) Find the left most point on the cardioid 40) Show that if a and b are not both 0, the graph of the equation

is a

40) _____________

circle. Find the radius and the rectangular coordinates of the center of this circle. 41)

Determine whether or not the point (1, π) lies on the curve

Explain.

41) _____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 42) Which of the statements below is true? A: If ( ,

) and ( ,

B: If ( , ) and ( , some integer n. A) A only C) Neither A nor B

) represent the same point in polar coordinates, then ) represent the same point in polar coordinates, then B) B only D) Both A and B

for

42) ______

1) A 2) D 3) A 4) A 5) D 6) B 7) D 8) A 9) A 10) B 11) B 12) A 13) D 14) B 15) D 16) C 17) B 18) D 19) B 20) B 21) B 22) C 23) D 24) D 25) C 26) D 27) A 28) A 29) A 30) A 31) A 32) B 33) D 34) For any a, the graph will be traced out as θ varies from 0 to 2π. The loop will occur when r is negative. For for all θ so there is no loop. For

when

For

when

and 35) They are rotated counterclockwise about the origin through angles of 36)

For 0 < a < 1, the curve is an oval. As a →

, and α, respectively.

, the oval develops a dimple. When

the graph resembles a

parabola with a dimple at the vertex. When 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. 37)

As θ → ∞, x → 0 and y → 0. As θ → horizontal asymptote.

, x → +∞ and y → 1 (by L'Hospital's Rule). Thus, the line

is a

38) As θ →

, x → 0 and y → 0. As θ →

, x → 4 and y → +∞. As

and

Thus the lines

and

are vertical asymptotes.

39) Express curve in parametric form, differentiate x(θ), find critical values, test to see which gives minimum. The left most point on the cardioid is

in polar coordinates or

40) Radius: ; center: 41) Yes; (1, π) = (-1, 0) and the pair r = -1, θ = 0 satisfies the equation. 42) A

in rectangular coordinates.

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the area of the specified region. 1) Inside the limacon r = 9 + 4 sin θ A) 89π B) 89π + 72 C) 178π - 72 D) 178π 2) Inside the cardioid r = α(1 + sin θ), α > 0 A) B) +2

1) _______

2) _______ C)

D) 2απ

-2

3) Inside one leaf of the four-leaved rose r = 3 sin 2θ A) B)

3) _______ -

Inside the lemniscate A)

= B)

C) 0

Inside one loop of the lemniscate A) 7 B)

Inside the six-leaved rose A)

4) _______

sin 2θ, a > 0 D)

5) _______

= 7 cos 2θ C) 14

= 5 cos 3θ B)

6) _______ C) 10

D) 5

7) _______

7) Inside the circle r = 8 cos θ + 9 sin θ A) B)

D) 36π

8) Inside the smaller loop of the limacon r = 5 + 10 sin θ A) B) 25π C)

D) 5(3

+ 36

(2π - 3

8) _______ - π)

)

9) Inside the graph of r = 5 + cos 3θ A) B) 51π

9) _______ C)

51π +

10) Inside the three-leaved rose r = 8 cos 3θ A) 32π B)

10) ______ C) 8π

D) 16π

π 11) Inside the circle r = A)

sin θ and outside the cardioid r = 1 + cos θ B) C)

11) ______ D)

12) Shared by the circles r = 7 cos θ and r = 7 sin θ A) B) (2 -

)

12) ______ C)

(π - 2)

(4π - 3

13) ______

13) Shared by the circles r = 3 and r = 6 sin θ A) 3π B)

D) (4π - 3

)

(2π + 3

(5π + 8)

D) 0 (5π - 8) 15) ______

15) Shared by the cardioids r = 4(1 + sin θ) and r = 4(1 - sin θ) A) 4(5π + 8) B) 8(3π - 8) C) 1616 Inside the lemniscate A)

D) 8(3π + 8) 16) ______

= 4 sin 2θ and outside the circle r = B) C) (2π + 3

)

- π)

17) Inside the circle r = a sinθ and outside the cardioid r = a(1 - sin θ), a > 0 A) B) C) (6 - π)

(2π - 3

)

(4π - 3

)

- π + 6) 17) ______

)

- π) 18) ______

18) Inside the circle r = -8 cos θ and outside the circle r = 4 A) B) C) 8π (4π - 3

(2π + 3

19) Inside the outer loop and outside the inner loop of the limacon r = 6 cos θ - 3 A) B) C) D) 9(π + 3 (2π - 3

)

(4π - 3

(4π + 3

)

20) Inside the circle r = 6 and to the right of the line r = 3 sec θ A) 6π B) 3(4π - 3 ) C) 3(2π + 3 Find the length of the curve. 21) The spiral r = ,0≤θ≤2 A) ((

) 19) ______ )

) 20) ______ )

D) 6(2π - 3

)

21) ______ B)

- 1)

22)

) 14) ______

14) Shared by the circle r = 3 and the cardioid r = 3(1 + sin θ) A) 9 B) C)

16)

)

The spiral r = A) (

22) ______

,0≤θ≤π B) - 1)

23) The cardioid r = 9(1 - sin θ) A)

C) (

- 1)

- 1) 23) ______

72 π

24) ______

24) The curve r = a A)

, 0 ≤ θ ≤ π, a > 0 B)

C) 2πa

D) 2a

25) ______

25) The curve r = 2 A) (2π - 3

,0≤θ≤ B) )

C) (3

- π)

(5π - 8)

(5π - 9

) 26) ______

26) The circle r = 9 cos θ A) 18π

C) 81π

D) 9π

π 27) ______

27) The spiral r = 6θ, 0 ≤ θ ≤ π A) 36 C)

- 4)

16(

3(π

+ ln (π +

))

D) 6 28) ______

28) The line segment r = 6 sec θ, 0 ≤ θ ≤ A)

B) 3(

+ ln(

+ 1))

6 ln C)

D) 6

29) ______

29) The parabolic segment r = A)

,0≤θ≤ B) 4 (

- ln(

- 1))

π C) 4 (

+ ln(

+ 1))

30) ______

30) The curve r = 8 A) 8(π + 2) C) 2π

,0≤θ≤ B) 8(π + ln(π +

))

D) 2π

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 31) Show that the arc length of one petal of the rose is given by 2

___ ___ ___ ___ _

and use 31) this formula to help make a conjectur e about the limit of such arc lengths as

32)

Does the spiral r = explain why not.

, 0 ≤ θ < +∞, have finite length? If yes, give its length. If no,

32) _____________

33) _____________

33) Find the error in the following "proof" that the area inside the lemniscate and then find the correct area.

34) A radial line is drawn from the origin to the spiral of Archimedes r = aθ, 0 ≤ θ, 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.

34) _____________

35) The area of the surface formed by revolving the graph of

35) _____________

from

about the line is The area of the surface formed by revolving the graph of r = f(θ) from θ = α to θ = β about the line

is Use the idea underlying these two integral formulas to find the surface area of the torus formed by revolving the circle r = a about the line

where

36) Find the area of the region enclosed by the rose 37) Assuming r = f(θ) is continuous for

for

and what can be said about the relative areas between the origin and the polar curves r = f(θ), α≤θ≤β and r = 2f(θ), α≤θ≤β? Give reasons for your answer.

36) _____________ 37) _____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 38) True or false? If n > 0 is an even integer, then the area of the region enclosed by is twice the area of the region enclosed by

38) ______

A) False

B) True

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 39) 39) _____________ (a) Graph the curves r = and r = . (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.

40)

Consider the inequality + < 3x + y. (a) Find an equivalent inequality in polar coordinates. (b) Use an integral with respect to θ 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).

40) _____________

1) A 2) C 3) C 4) D 5) D 6) C 7) A 8) A 9) C 10) D 11) A 12) C 13) C 14) C 15) B 16) C 17) D 18) D 19) D 20) B 21) C 22) A 23) D 24) D 25) A 26) D 27) B 28) D 29) B 30) C 31) L= As n → +∞, L → 2.

;

32) Yes; L = = 33) The error is in the limits of integration. There are no points on the curve corresponding to values of θ such that or 34) 35)

since

is negative on these intervals. The correct area is

8 4

36) For n odd, A = . For n even, A = . 37) The area between the origin and the curve r = 2f(θ) is four times greater than the area between the origin and the

curve 38) B 39) (a)

since

(b) The curves intersect in two points. In polar coordinates, these points may be represented by

and

. (c) For r = , = -cot . For r = , = tan . Thus the slopes of the tangent lines to the curves at both of the points of intersection are negative reciprocals of each other so the tangent lines are perpendicular.

40) (a) r < 3 cos θ + sin θ (b) (c) π (d) -

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)

= 8y

= -8y

= 8x 2) _______

= 8y

= -8x

= -8y

-8

=y 3) _______

A) 4)

1) _______

= -y

= 8y

__ __ __ _

= 6y

= -6x

= 6x 5) _______

= -4y

= 4x

-4

= -4x 6) _______

B) +

C) -

D) +

=1 7) _______

C) +

D) +

=1 8) _______

B) -

C) -

D) -

=1 9) _______

B) -

C) -

D) -

=1 10) ______

10)

A) 16x2 - 9y2 = 144

B) 9x2 + 16y2 = 144

C) 9x2 - 16y2 = 144

D) 16x2 + 9y2 = 144 11) ______

11)

A) 4y2 - 25x2 = 100

B) 4x2 - 25y2 = 100

C) 25x2 + 4y2 = 100

D) 25x2 - 4y2 = 100

Find the focus and directrix of the parabola. 12) = 32y A) (0, -8); x = - 8 B) (8, 0); x = 8

12) ______ C) (0, 8); y = -8

D) (8, 0); y = 8 13) ______

13) -

14)

=y A) (0, -4); y = -4

C) (-8, 0); x = 4

C) ,x=-

B) ,x=D) ,y=-

,y= 16) ______

y = -9 A)

B) ,y=-

, x=

D) ,y=

18)

19)

20)

= 28x A) (7, 7); x = 7 = -32x A) (-8, 0); x = 8 = -8y A) (0, -2); y = -2 - 36x = 0 A) (0, 9); y = -9

Graph the parabola. 21) y = 3x2

,x= 15) ______

y = 10 A) C)

17)

D) ,y=-

,x=-

16)

D) (0, -4); y = 4 14) ______

x=2 A) ,x=-

15)

B) (0, 4); y = -4

,x= 17) ______ B) (7, 0); x = -7

C) (7, 0); x = 7

D) (0, 7); y = -7 18) ______

B) (8, 0); x = -8

C) (0, -8); y = 8

D) (-8, 0); y = 8 19) ______

B) (0, -2); x = 2

C) (0, -2); y = 2

D) (-2, 0); y = 2 20) ______

B) (9, 0); y = -9

C) (9, 0); x = -9

D) (9, 0); x = 9

___ ___

21)

22) x = 4y2

22) ______

23)

= 2y

23) ______

24)

24) ______

= -2x

Graph. 25)

25) ______ +

26)

27)

26) ______

= 56

___ ___

27)

+ 12 = 192

28)

+ 36

= 36

28)

______ A)

Find the vertices and foci of the ellipse. 29)

29) ______

+ =1 A) Vertices: (±20, 0); Foci: (±16, 0) C) Vertices: (0, ±20); Foci: (0, ±16)

B) Vertices: (±20, 0); Foci: (±12, 0) D) Vertices: (0, ±20); Foci: (0, ±12) 30) ______

30) + =1 A) Vertices: (±12, 0); Foci: ( ±9, 0) C) Vertices: (0, ±15); Foci: (0, ±12) 31)

+ 100

+ 16

31) ______

= 4900

A) Vertices: (±7, 0); Foci: (± C) Vertices: (±10, 0); Foci: (± 32)

, 0) , 0)

B) Vertices: (0, ±10); Foci (0, ± D) Vertices: (0, ±7); Foci (0, ±

) ) 32) ______

= 1296

A) Vertices: (±4, 0); Foci: (± , 0) C) Vertices: (0, ±4); Foci: (0, ± ) 33)

B) Vertices: (±15, 0); Foci: ( ±12, 0) D) Vertices: (0, ±15); Foci: (0, ±9)

144 + 49 = 7056 A) Vertices: (0, ±7); Foci: (0, ± ) C) Vertices: (±7, 0); Foci: (± , 0)

B) Vertices: (±9, 0); Foci: (± , 0) D) Vertices: (0, ±9); Foci: (0, ± ) 33) ______ B) Vertices: (0, ±12); Foci: (0, ± ) D) Vertices: (±12, 0); Foci: (± , 0)

34)

34) ______

+ 100 = 3600 A) Vertices: (±36, 0); Foci: (±8, 0) C) Vertices: (0, ±36); Foci: (0, ±10)

B) Vertices: (0, ±10); Foci: (0, ±8) D) Vertices: (±10, 0); Foci: (±8, 0)

Find the standard-form equation of the ellipse centered at the origin and satisfying the given conditions. 35) An ellipse with intercepts (±7, 0) and (0, ±3) A) B) C) D) +

35) ______ =1 36) ______

36) An ellipse with length of major axis 18 and y-intercepts A)

B) +

C) +

B) +

Graph. 40)

D) +

=1 39) ______

. C)

38) ______

39) An ellipse with vertices (0, ±9) and foci at

D) +

38) An ellipse with vertices (±8, 0) and foci at (±3, 0) A) B) +

37) ______

37) An ellipse with vertices (0, ±9) and foci (0, ±5) A) B) +

D) +

40) ______ -

41) ______

41) -

42) ______

42) -

43)

44)

-9

43) ______

= 324

-4

= 36

44) ______

45)

45) ______

46) ______

46) -

Solve the problem. 47) Find the foci and asymptotes of the following hyperbola: A)

47) ______

=1 Foci: (0, -6), (0, 6); Asymptotes: y =

x, y = -

Foci: (0, -10), (0, 10); Asymptotes: y =

x, y = -

Foci: (-10, 0), (10, 0); Asymptotes: y =

x, y = -

Foci: (-10, 0), (10, 0); Asymptotes: y =

x, y = -

B) C) D)

48) ______

48) Find the foci and asymptotes of the following hyperbola: -

A) Foci: (0, -15), (0, 15); Asymptotes: y =

x, y = -

Foci: (-15, 0), (15, 0); Asymptotes: y =

x, y = -

Foci: (-25, 0), (25, 0); Asymptotes: y =

x, y = -

Foci: (0, -25), (0, 25); Asymptotes: y =

x, y = -

B) C) D)

49) ______

49) Find the foci and asymptotes of the following hyperbola: A)

Foci: ( B) Foci: (0,

, 0), (), (0, -

, 0); Asymptotes: y =

x, y = -

); Asymptotes: y = 3x, y = -3x

C) Foci: (3, 0), (-3, 0); Asymptotes: y = x, y = - x D) Foci: ( , 0), (, 0); Asymptotes: y = 3x, y = -3x 50) ______

50) Find the foci and asymptotes of the following hyperbola: 16

= 16

A) Foci: ( B) Foci: ( C) Foci: (0,

, 0), (-

, 0); Asymptotes: y =

, 0), (-

, 0); Asymptotes: y = 4x, y = -4x

), (0, -

x, y = -

); Asymptotes: y = 4x, y = -4x

D) Foci: (4, 0), (-4, 0); Asymptotes: y =

x, y = -

51) Find the foci and asymptotes of the following hyperbola:

___ ___

51)

16 A) Foci: (0, 17), (0, -17); Asymptotes: y = 4x, y = -4x B) Foci: (0,

), (0, -

); Asymptotes: y =

x, y = -

, 0); Asymptotes: y =

x, y = -

C) Foci: ( D) Foci: (0,

, 0), (), (0, -

); Asymptotes: y = 4x, y = -4x 52) ______

52) Find the foci and asymptotes of the following hyperbola: = 162 A) Foci: (9, 0), (-9, 0); Asymptotes: y = x, y = - x B) Foci: (18, 0), (-18, 0); Asymptotes: y =2x, y = - 2x C) Foci: (18, 0), (-18, 0); Asymptotes: y = x, y = - x D) Foci: (0, 18), (0, -18); Asymptotes: y = x, y = - x

53) ______

53) Find the vertices and foci of the following hyperbola: =1 A) Vertices: (2, 0), (-2, 0); Foci: (9, 0), (-9, 0) C) Vertices: (4, 0), (-4, 0); Foci: (9, 0), (-9, 0)

B) Vertices: (0, 9), (0, -9); Foci: (0, 2), (0, -2) D) Vertices: (0, 2), (0, -2); Foci: (0, 9), (0, -9) 54) ______

54) Find the vertices and asymptotes of the following hyperbola: A)

=1 Vertices: (4, 0), (-4, 0); Asymptotes: y = ±

B) Vertices: (0, 16), (0, -16); Asymptotes: y = ±

C) Vertices: (0, 4), (0, -4); Asymptotes: y = ± x D) Vertices: (0, 4), (0, -4); Asymptotes: y = ± 4x 55) ______

55) Find the foci and asymptotes of the following hyperbola: 4

- 25 A)

= 100

Foci: (

, 0), (-

, 0); Asymptotes: y =

x, y = -

Foci: (

, 0), (-

, 0); Asymptotes: y =

x, y = -

B) x

C) Foci: (0,

), (0, -

); Asymptotes: y =

x, y = -

Foci: (0,

), (0, -

); Asymptotes: y =

x, y = -

Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. 56) Vertices at (2, 0) and (-2, 0); foci at (6, 0) and (-6, 0) A)

56) ______

B) -

=1 -

= 1

= 1 57) ______

57) Vertices at (0, 5) and (0, -5); foci at (0, 8) and (0, -8) A) B) -

D) -

=1 58) ______

58) Vertices at (0, 6) and (0, -6); asymptotes y = A) B) -

x and y = C)

x D) -

=1 59) ______

59) Asymptotes y = A) -

x, y = -

x; one vertex is (12, 0) B)

= 12

= 3600

D) -

60) ______

60) Asymptotes y = A) -

x, y = -

x; one vertex is (0, 14) B) C)

=1 61) ______

61) Foci at (-5, 0), (5, 0); asymptotes: y = A) B) -

x, y = -

x C)

D) -

=1 62) ______

62) Foci at (0, -25), (0, 25); asymptotes: y = A) B) 63) Foci at ( A)

=1 , 0), (-

Solve the problem. 64) The parabola

x, y = -

x C)

=1 63) ______

, 0); asymptotes y = 4x, y = -4x B) C) -

D) -

= 14x is shifted up 7 units and left 4 units. Find an equation for the new

par abola

and find 64) the new vertex. A) C) 65)

66)

67)

68)

69)

70)

___ ___

= 14(x - 4); vertex; (-4, 7) = 14(x + 4); vertex; (-4, 7)

B) D)

= 14(x + 7); vertex; (7, -4) = 14(x - 7); vertex; (7, -4)

The parabola = -12y is shifted down 6 units and right 8 units. Find an equation for the new parabola and find the new vertex. A) B) = -12(y + 6); vertex; (8, -6) = -12(y - 8); vertex; (-6, 8) C) D) = -12(y - 6); vertex; (8, -6) = -12(x - 8); vertex; (-6, 8) The parabola = -16x is shifted down 8 units and left 3 units. Find an equation for the new parabola and find the new vertex. A) B) = -16(x + 3); vertex; (-3, -8) = -16(x - 3); vertex; (-3, -8) C) D) = -16(x + 8); vertex; (-8, -3) = -16(x - 8); vertex; (-8, -3) The parabola = -32y is shifted 2 units up and 7 units to the left. Find the focus and directrix of the new parabola. A) Focus: (-7, 10); Directrix: y = 10 B) Focus: (-7, -6); Directrix: y = -6 C) Focus: (-7, -6); Directrix: y = 10 D) Focus: (7, -10); Directrix: y = -10 The parabola = -24x is shifted 7 units down and 3 units to the right. Find the focus and directrix of the new parabola. A) Focus: (-7, -3); Directrix: y = 9 B) Focus: (-3, -7); Directrix: x = -3 C) Focus: (9, -7); Directrix: x = 9 D) Focus: (-3, -7); Directrix: x = 9 The parabola = -28y is shifted 4 units up and 6 units to the right. Find the focus and directrix of the new parabola. A) Focus: (6, -3); Directrix: y = 11 B) Focus: (-3, 6); Directrix: x = 11 C) Focus: (6, 11); Directrix: y = 11 D) Focus: (6, -3); Directrix: y = -11 The parabola parabola.

= -20x is shifted 7 units down and 4 units to the right. Graph the new

65) ______

66) ______

67) ______

68) ______

69) ______

70) ______

71)

The parabola

= -24y is shifted 4 units up and 6 units to the left. Graph the new parabola.

71) ______

72) ______

72) The ellipse + = 1 is shifted up 3 units and left 8 units. Find an equation for the new ellipse and find the new vertices. A) +

= 1; vertices: (-8 ±

, 3)

= 1; vertices: (-8 , 3 ±

= 1; vertices: (-8 ±

, 3)

= 1; vertices: (-8 , 3 ±

)

B) )

C) D)

73) ______

73) The ellipse + = 1 is shifted down 8 units and right 3 units. Find an equation for the new ellipse and find the new vertices. A) +

= 1; vertices: (3, -8 ±

)

B) +

= 1; vertices: (3 ±

, -8 )

= 1; vertices: (3 ±

, -8 )

= 1; vertices: (3, -8 ±

C) D) ) 74) ______

74) The ellipse + = 1 is shifted down 7 units and right 6 units. Find an equation for the new ellipse and find the new center. A) B) +

= 1; center: (7, -6)

= 1; center: (6, -7)

D) +

= 1; center: (-7, 6)

75) ______

75) The ellipse + = 1 is shifted up 7 units and left 7 units. Find an equation for the new ellipse and find the new center. A) B) +

= 1; center: (-7, 7)

= 1; center: (7, -7)

= 1; center: (7, 7)

D) +

= 1; center: (-7, 7)

76) ______

76) The ellipse

= 1 is shifted horizontally and vertically to obtain the ellipse

+ = 1. Find the center and vertices of the new ellipse. A) Center: (1, 3); Vertices: (1, -2), (1, 8) C) Center: (3, 1); Vertices: (1, -2), (1, 8)

B) Center: (3, 1); Vertices: (-2, 1), (8, 1) D) Center: (1, 3); Vertices: (-2, 1), (8, 1) 77) ______

77) The ellipse

= 1 is shifted horizontally and vertically to obtain the ellipse

+ = 1. Find the foci of the new ellipse. A) (-1, -13), (-1, 5) B) (-1, -16), (-1, 8)

C) (-16, -1), (8, -1)

D) (-13, -1), (5, -1)

78) The ellipse

= 1 is shifted horizontally and vertically to obtain the ellipse +

= 1. Find the foci

of the new ellipse.

___ ___

78)

A) (-11, 4), (19, 4)

B) (-16, 4), (24, 4)

C) (4, -11), (4, 19)

D) (4, -16), (4, 24) 79) ______

79) The ellipse

= 1 is shifted horizontally and vertically to obtain the ellipse = 1. Graph the new ellipse.

80) ______

80) The hyperbola = 1 is shifted up 7 units and left 3 units. Find an equation for the new hyperbola and find the new vertices. A) -

= 1; vertices: (-3 ±

, 7)

= 1; vertices: (-3 , 7 ±

)

C) -

= 1; vertices: (-3 ±

= 1; vertices: (-3 , 7 ±

, 7)

D) ) 81) ______

81) The hyperbola = 1 is shifted down 3 units and right 7 units. Find an equation for the new hyperbola and find the new vertices. A) -

= 1; vertices: (7, -3 ±

= 1; vertices: (7 ±

= 1; vertices: (7, -3 ±

= 1; vertices: (7 ±

)

B) , -3 )

C) )

D) , -3 ) 82) ______

82) The hyperbola = 1 is shifted down 8 units and right 3 units. Find an equation for the new hyperbola and find the new center. A) B) -

= 1; center: (3, -8)

= 1; center: (-8, 3)

= 1; center: (3, -8)

D) -

= 1; center: (8, -3)

83) ______

83) The hyperbola

= 1 is shifted horizontally and vertically to obtain the hyperbola

= 1. Find the center and vertices of the new hyperbola. A) Center: (-2, 1); Vertices: (-14, -2), (16, -2) C) Center: (1, -2); Vertices: (-14, -2), (16, -2)

B) Center: (-2, 1); Vertices: (-2, -14), (-2, 16) D) Center: (1, -2); Vertices: (-2, -14), (-2, 16) 84) ______

84) The hyperbola

= 1 is shifted horizontally and vertically to obtain the hyperbola

=1 Find the foci of the new hyperbola. A) (3, -19), (3, 11) B) (3, -13), (3, 5)

C) (-19, 3), (11, 3)

D) (-13, 3), (5, 3)

85) The hyperbola

= 1 is shifted horizontally and vertically to obtain the hyperbola -

___ ___

Find the 85) foci of the new hyperbol a. A) (-2, -14), (-2, 10)

B) (-14, -2), (10, -2)

C) (-3, 18), (-3, -22)

D) (-2, -22), (-2, 18) 86) ______

86) The hyperbola

= 1 is shifted horizontally and vertically to obtain the hyperbola

=1 Find the asymptotes of the new hyperbola. A) y+3=±

(x - 5)

y=±

D) y+3=±

(x - 5)

y+3=±

(x - 5) 87) ______

87) The hyperbola

= 1 is shifted horizontally and vertically to obtain the hyperbola

=1 Find the asymptotes of the new hyperbola. A) y-3=±

(x + 1)

y-3=±

(x + 1)

D) y=±

y=±

x 88) ______

88) The hyperbola

= 1 is shifted horizontally and vertically to obtain the hyperbola Graph the new hyperbola.

89) ______

89) The hyperbola

= 1 is shifted horizontally and vertically to obtain the hyperbola Graph the new hyperbola.

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. 90) 90) ______ - 4y + 8x - 48 = 0 A) C: (-2, 2); F: (6 , 0), (- 6 , 0); A: y = x, y = -x B) C: (4, 2); F: (4, 2 + 6

), (4, 2 - 6

); A: y - 2 = x - 4, y - 2 = -(x - 4)

C) C: (4, -4); F: (6

, 0), (- 6

D) C: (4, 2); F: (0, 2 + 6 91)

), (0, 2 - 6

- 48x + 10y - 145 = 0 A) C: (-2, 5); F: (6 , 0), (-6 B) C: (4, 5); F: (4 + 6

, 0); A: y = x, y = -

); A: y - 2 = x - 4, y - 2 = -(x - 4) 91) ______

, 0); A: y = 6x, y = -6x

, 5), (4 - 6

, 5); A: y - 5 =

(x - 4), y - 5 = -

(x - 4)

C) C: (4, 5); F: (6

, 0), (- 6 , 0); A: y - 5 = (x - 4), y - 5 = - (x - 4) D) C: (4, 5); F: (4 + 6 , 5), (4 - 6 , 5); A: y - 5 = 6(x - 4), y - 5 = -6(x - 4) 92)

93)

+ 90x + 25 + 150y + 225 = 0 A) C: (-5, -3); V: (-10, -3), (0, -3); F: (-8, -3), (-2, -3) B) C: (-5, -3); V: (-3, -10), (-3, 0); F: (-3, -9), (-3, -1) C) C: (-5, -3); V: (-3, -10), (-3, 0); F: (-3, -8), (-3, -2) D) C: (-5, -3); V: (-10, -3), (0, -3); F: (-9, -3), (-1, -3) +9

- 400x - 54y + 1456 = 0

92) ______

93) ______

A) C: (8,3); V: (8, 6), (8, 0); F: (8, 8.8), (8, -2.8) B) C: (8, 3); V: (13, 3), (3, 3); F: (12.0, 3), (4.0, 3) C) C: (8, 3); V: (11, 3), (5, 3); F: (13.8, 3), (2.2, 3) D) C: (8, 3); V: (8, 8), (8, -2); F: (8, 7.0), (8, -1.0) 94)

95)

+ - 8x + 14y + 65 = 9 A) (-4, 7); r = 9 B) (7, -4); r = 9

96)

+ - 14x - 6y = -42 A) (3, 7); r = 4

97)

98)

94) ______

+9 - 40x + 162y + 793 = 0 A) C: (5, -9); V: (5, -6), (5, -12); F: (5, -6.8), (5, -11.2) B) C: (5, -9); V: (8, -9), (2, -9); F: (7.2, -9), (2.8, -9) C) C: (5,-9); V: (5, -7), (5, -11); F: (5, -5.4), (5, -12.6) D) C: (5, -9); V: (7, -9), (3, -9); F: (8.6, -9), (1.4, -9)

95) ______ C) (-7, 4); r = 3

D) (4, -7); r = 3 96) ______

B) (-7, -3); r = 16

C) (-3, -7); r = 16

D) (7, 3); r = 4 97) ______

- 8x - 8y + 40 = 0 A) F: (4, 11); D: y = 5

C) F: (4, 1); D: y = 5

F: ; D: y = D) F: (4, 5); D: y = 1 98) ______

+ 8x + 10y + 1 = 0 A) F: (-5, 3); D: y = -11 C) F: (1, -5); D: x = 5

B) F: (5, 3); D: y = 1 D) F:

; D: y =

Solve the problem. 99) Find the volume of the solid generated by revolving the region enclosed by the ellipse 4 25

= 100 about the x-axis. A)

100) Find the volume generated by revolving about the x-axis the region bounded by the following graph: A)

B) π

C) 14π

-9 A)

100) _____

D) 196π

101) Find the volume generated by revolving about the y-axis the area bounded by the curves:

101) _____

= 9, x = 8 B)

C) π

D) π

102) Find the volume generated by revolving about the y-axis the area bounded by the curves 2

99) ______

= 50, x = 8 A) 26π

B) 52π

C) 52π

D) 52π

102) _____

103)

Find the coordinates of the centroid of the area bounded by y = A) B) C)

103) _____

and y = 9 D)

1) B 2) C 3) A 4) D 5) D 6) D 7) C 8) A 9) B 10) D 11) D 12) C 13) D 14) B 15) C 16) C 17) B 18) A 19) C 20) C 21) B 22) D 23) B 24) B 25) D 26) C 27) B 28) D 29) B 30) D 31) C 32) D 33) B 34) D 35) D 36) C 37) C 38) D 39) D 40) D 41) D 42) A 43) C 44) B 45) D 46) B 47) D 48) D 49) D 50) B 51) B

52) C 53) A 54) C 55) D 56) D 57) D 58) C 59) D 60) B 61) A 62) B 63) B 64) C 65) A 66) A 67) C 68) D 69) A 70) D 71) A 72) C 73) D 74) D 75) C 76) B 77) C 78) C 79) A 80) A 81) A 82) A 83) C 84) C 85) C 86) C 87) A 88) B 89) A 90) B 91) B 92) D 93) D 94) B 95) D 96) D 97) D 98) C 99) C 100) B 101) B 102) C 103) D

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the eccentricity of the ellipse. 1) 16 + 9 = 144 A) B) C) D)

+ 16 A)

= 144

= 256

= 16

= 36

4) _______ B)

3) _______ B)

+ 16 A)

2) _______ B)

5) _______ B)

6) _______

6) x2 + 2y2 = 6 A)

7) _______

7) 4x2 + y2 = 36 A)

8) Find the eccentricity of an ellipse centered at the origin having a focus of (0, corresponding directrix y = A) B)

) and

Find the foci of the ellipse. 10) 25 + 36 = 900

8) _______

9) Find the eccentricity of an ellipse centered at the origin having a focus of corresponding directrix A)

1) _______

and

9) _______

10) ______

A) (± 11)

13)

B) (±5, 0)

D) y=±

C) y=±

y=±

D) y=±

x=± 17) ______

= 36 B)

C) y=±

D) x=±

x=± 18) ______

=4 B)

x=±

16) ______

y=±

Graph. 19)

+ A)

D) (±9, 0)

= 64

+ 36 A)

C) x=±

= 56

)

15) ______

x=±

D) (0, ±4

y=±

x=±

18)

C) (0, ±4)

)

14) ______

x=±

17)

C) (0, ±7)

= 1296

, 0)

x=±

+ 81

D) (0, ±

13) ______

y=±

16)

C) (0, ±8)

12) ______ B) (±4

+ 25 = 225 A) (±4, 0)

)

= 11,025

Find the directrices of the ellipse. 14) 64 + 25 = 1600 A)

15)

D) (0, ±

11) ______ B) (0, ±3)

, 0)

225 + 49 A) (0, ±15)

C) (±5, 0)

= 576

A) (± 12)

B) (±6, 0)

, 0)

D) y=±

y=±

19) ______

20)

+ 27

= 432

20) ______

21)

+ 49

= 49

21) ______

22)

+ 49

= 441

22) ______

23)

+ 16

= 1024

23) ______

Find the standard-form equation for an ellipse which satisfies the given conditions. 24)

24) ______

An ellipse centered at the origin having focus at (2, 0) and eccentricity equal to A) B) C) D) +

+ y2 = 1

25) ______

25) An ellipse centered at the origin having vertex at (0, -3) and eccentricity equal to A) B) C) D) -

=1 26) ______

26) An ellipse centered at the origin having focus ( A) B) +

, 0) and directrix x = C) +

=1 27) ______

27) An ellipse centered at the origin having focus (0, -2 ) and directrix y = A) B) C) +

D) +

28) ______

28) An ellipse centered at the origin having focus ( A) B) +

Find the eccentricity of the hyperbola. 29) =2 A) 2 B)

30)

31)

- 144 A)

- 16

29) ______ C) 0

31) ______

= 1296 C)

32) ______

= 3136

33)

30) ______

- 49

= 144

32)

, 0) and directrix x = C)

- 576

33) ______

= 28,224

34) ______

34) x2 - 2y2 = 6 A)

35) ______

35) 8x2 - 9y2 = 72 A)

Find the foci of the hyperbola. 36) 9 =9 A) ( , 0), (, 0)

B) (0, 3), (0, -3)

C) (0,

D) (3, 0), (-3, 0)

37)

38)

), (0, -

36) ______

)

37) ______

=4 A) (0, ), (0, - ) C) ( , 0), (- , 0) = 50 A) (5, 0), (-5, 0)

B) (0, 2), (0, -2) D) (2, 0), (-2, 0) 38) ______ B) (0, 5), (0, -5)

C) (10, 0), (-10, 0)

D) (0, 10), (0, -10)

39)

- 25 A) (

39) ______

= 100 , 0), (-

C) (0,

), (0, -

B) (0,

, 0)

D) (

)

Find the directrices of the hyperbola. 40) = 64 A)

-4 A)

- 49

- 25

-9

x=±

C) x=±

y=±

D) y=±

y=± 43) ______

= 1600 B)

- 16 A)

42) ______ B)

C) y=±

D) x=±

y=± 44) ______

= 144 B)

y=± Graph. 45)

= 3136

y=± 9

x=±

y=±

44)

41) ______

x=± 64

C) y = ± 8

43)

, 0)

y=±

y=± 42)

, 0), (-

)

40) ______

x=± 41)

), (0, -

C) x=±

= 36

D) x=±

y=±

45) ______

46)

-4

46) ______

= 64

47)

47) ______

Find the standard-form equation for a hyperbola which satisfies the given conditions. 48) A hyperbola centered at the origin having vertex at (0, -5) and eccentricity equal to 2 A) B) C) D) -

48) ______ +

=1 49) ______

49) A hyperbola centered at the origin having focus at (3, 0) and eccentricity equal to 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. 50) e = 5, x = 9 50) ______ A) B) C) D) r= 51) e = 5, y = 6 A)

C) r=

D) r=

r= 52) ______

r= 53) e = 3, y = -7 A)

r= 51) ______

r= 52) e = 3, x = -6 A)

C) r=

D) r=

r= 53) ______

C) r=

D) r=

r= 54) ______

54) e= ,x=2 A)

C) r=

D) r=

r= 55) ______

55) e= ,y=2 A)

C) r=

D) r=

r= 56) ______

56) e= , x = -7 A)

C) r=

D) r=

r= 57) ______

57) e = , y = -9 A) r=

C) r=

D) r=

Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. 58) r=

59) r=

58) ______

___ ___

59)

60) r=

60)

______ A)

61) ______

61) r=

62) ______

62) r=

63) ______

63) r=

64) r=

___ ___

64)

65) ______

65) r=

Find a Cartesian equation for the line whose polar equation is given. 66) r cos A) y =

66) ______

=5 x - 10

B) y = x + 5

C) y = x + 5

D) y = x - 10 67) ______

67) r cos =2 A) y = 2 x

B) y = 4 -

D) y = y=4-

x+2

x 68) ______

68) r cos =5 A) y = x + 5

B) y = x + 10

C) y = x - 5

D) y =

x - 10 69) ______

69) r cos A) y =

=2 x+2

B) y =

x+4

D) y=

x+4

x-2 70) ______

70) r cos A) y =

=5 x-5

B) y =

Find a polar equation in the form r cos(θ 71) xy=4

x + 10 )=

C) y =

x-5

D) y = x - 10

for the given line. 71) ______

B) r cos

C) r cos 72)

r cos

=2 72) ______

x+y=8 A)

r cos

D) r cos

73) x -

r cos D)

73) ______

y=4

B) r cos

r cos

D) r cos

74) ______

74) x = -8 A)

B) r cos θ = 8 r cos

D) r cos r cos

=8 75) ______

75) y = -8 A)

B) r cos

r cos C) r cos θ = 8

=8 D) r cos

The polar equation of a circle is given. Give polar coordinates for the center of the circle and identify its radius. 76) r = 10 cos θ A) B) C(5, 0), radius = 5 C , radius = 5 C) C(10, 0), radius = 10

D) C

, radius = 10 77) ______

77) r = -4 cos θ A)

B) C(-4, 0), radius = 4

C , radius = 2 C) C(-2, 0), radius = 2

D) C

, radius = 4 78) ______

78) r = 8 sin θ A) C

B) C(4, 0), radius = 4 , radius = 4

D) C(4, π), radius = 4 C

76) ______

, radius = 4

79) ______

79) r = -4 sin θ A) C

B) , radius = 2

C) , radius = 4

Find a polar equation for the circle. 80) + = 64 A) r = 16 cos θ

82)

+ = 25 A) r = 10 cos θ +

84)

85)

, radius = 4

80) ______ B) r = - 16 cos θ

C) r = 16 sin θ

D) r = 8 cos θ 81) ______

B) r = -10 cos θ

C) r = -10 sin θ

D) r cos θ = -10 82) ______

= 100

A) r = -20 sin θ 83)

, radius = 2

D) C

81)

+ = 36 A) r = -12 cos θ - 18x + =0 A) r = -18 sin θ + - 12y = 0 A) r = -12 sin θ

B) r = 20 cos θ

C) r = 20 sin θ

D) r cos θ = -20 83) ______

B) r = 6 sin θ

C) r = 6 cos θ

D) r = -12 sin θ 84) ______

B) r = 18 sin θ

C) r = -9 cos θ

D) r = 18 cos θ 85) ______

B) r = 12 sin θ

C) r = -12 cos θ

D) r = -6 sin θ 86) ______

86) + A)

=0 B)

cos θ

C) r=

sin θ

D) r=-

cos θ

r=-

sin θ

1) C 2) B 3) B 4) D 5) B 6) C 7) C 8) A 9) D 10) A 11) D 12) D 13) A 14) C 15) A 16) B 17) D 18) D 19) B 20) D 21) D 22) D 23) A 24) B 25) C 26) B 27) A 28) A 29) B 30) D 31) D 32) A 33) D 34) D 35) C 36) A 37) A 38) C 39) C 40) D 41) B 42) B 43) D 44) C 45) A 46) C 47) D 48) A 49) C 50) C 51) C

52) C 53) C 54) A 55) D 56) D 57) A 58) C 59) A 60) A 61) C 62) C 63) D 64) B 65) D 66) D 67) B 68) B 69) C 70) B 71) B 72) B 73) A 74) D 75) A 76) B 77) C 78) A 79) A 80) A 81) B 82) C 83) D 84) D 85) B 86) C

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Give a geometric description of the set of points whose coordinates satisfy the given conditions. 1) x = 8, z = -10 A) All points in the x-z plane B) The point (8, -10) C) The line through the point (8, 0, -10) and parallel to the y-axis D) The line through the point (8, -10, 0) and parallel to the z-axis 2)

+ A)

= 4, x = -2

+ A)

1) _______

2) _______

The line tangent to the circle + = 4 at the point x = -2 B) All points more than 2 units from the origin C) The circle + = 4 in the plane x = -2 D) The cylinder with the radius 2 along the x-axis

B) C) D)

3) _______

= 64, z = 1

All points on the sphere The circle

The sphere

= 64 and above the plane z = 1

= 63 in the plane z = 1 +

All points within the sphere

= 64 and above the plane z = 1

4) 4 ≤ z ≤ 5 A) The line from z = 4 to z = 5 B) The slab between the planes z = 4 and z = 5 (including planes) C) No set of points satisfies the given relation D) All points between z = 4 and z = 5 in the x-y plane

4) _______

5) 0 ≤ y ≤ 1, 0 ≤ z ≤ 1 A) The cube located in the first quadrant and with sides 0 units in length B) The square with corners at (0, 0, 0), (0, 0, 1), (0, 1, 0), and (0, 1, 1) C) The infinitely long square prism parallel to the x-axis D) The line between the points (0, 0, 0) and (0, 1, 1)

5) _______

6) _______

+ ≤ 4, z = 8 A) All points on the cylinder with radius 2 along the z-axis B) All points on or outside of the circle + = 4 and in the plane z = 8 C) All points within the parabola + = 4 in the plane z = 8 D) All points on or within the circle + = 4 and in the plane z = 8 + + > 25 A) All points outside the cylinder with radius 5 B) All points in space C) All points outside the sphere of radius 5 D) All points on the surface of the cylinder with radius 5 +

7) _______

8) _______

A) All points outside the lower hemisphere centered at (6, 3, 10) B) All points within the lower hemisphere centered at (6, 3, 10) C) No set of points satisfy the given relations. D) All points on the lower hemisphere centered at (6, 3, 10) Describe the given set of points with a single equation or with a pair of equations. 9) The plane perpendicular to the y-axis and passing through the point (-5, -9, -3) A) -5x - 3z = 0 and y = -9 B) y = -9 C) x - 9 + z = 0 D) x = -5 and z = -3

9) _______

10) The plane through the point (-8, 5, -6) and parallel to the yz-plane A) y = 5 and z = -6 B) x = -8 C) -8 + y + z = 0 D) 5y - 6z = 0 and x = -8

10) ______

11) The plane through the point (-10, 9, -9) and parallel to the xy-plane A) x + y - 9 = 0 B) x = -10 and y = 9 C) z = -9 D) x + y = -1

11) ______

12) The plane through the point (-4, 4, 8) and perpendicular to the x-axis A) y = 4 and z = 8 B) -4 + y + z = 0 C) x = -4

12) ______ D) y + z = 12

13) The circle of radius 9 centered at the point (4, -6, 81) and lying in a plane parallel to the xy-plane A) B) + = 81 and x + y = -2 + C)

D) +

13) ______

+ 1 = 81 +

= 81 and z = 81

14) The circle of radius 5 centered at the point (5, -1, 6) and lying in a plane perpendicular to the A)

= 25 and y + z = 5

B) +

D) +

14) ______

+ 1 = 25 +

= 25 and x = 5

15) The set of points equidistant from the points (0, 0, -7) and (0, 0, -4) A) x + y = 0 and -7 < z < -4 B) z > -7 and z < -4 C) z = -5.5 D) x + y = -5.5

15) ______

16) The set of points equidistant from the points (-1, 0, 0) and (8, 0, 0) A) y + z = 3.5 B) x > -1 and x < 8 C) x = 3.5 D) y + z = 0 and -1 < x < 8

16) ______

17) The circle in which the plane through the point (-10, 12, -4) perpendicular to the y-axis meets the sphere of radius 15 centered at the origin. A) B) + + = 225 and y = 12 + + = 144 and y = 12 C) D) + + = 63 and y = 12 + + = 81 and y = 12

17) ______

18) The circle in which the plane through the point (10, -5, 5) perpendicular to the x-axis meets the sphere of radius 26 centered at the origin.

18) ______

A) C)

= 676 and z = 10

= 476 and y = 10

B) D)

= 576 and y = 10

= 676 and x = 10

Write one or more inequalities that describe the set of points. 19) The slab bounded by the planes x = -10 and x = 11 (planes included) A) x ≤ -10 and x ≥ 11 B) -∞ < y < ∞ and -∞ < z < ∞ C) x = -10 and x = 11 D) -10 ≤ x ≤ 11

19) ______

20) The rectangular solid in the first octant bounded by the planes x = 1, x = 8, y = 1, y = 10,

20) ______

and (planes excluded) A) x < 1, x > 8; y < 1, y > 10; z < 4, z > 9 B) 1 < x < 8; 1 < y < 10; 4 < z < 9 C) x < y < z D) The given planes do not form a rectangular solid 21)

The interior of the sphere A) + + ≤ 36 C) + + <6

21) ______

= 36 B) D)

< 36

≤6 22) ______

22) The half-space consisting of the points on and behind the yz-plane A) y ≤ 0, z ≤ 0 B) x ≤ 0 C) x > 0

D) y > 0, z > 0

23) The closed region bounded by the spheres of radius 4 and 9, both centered at the origin, and the planes A) 16 < C) 4≤

and + +

+ +

< 81 and 1 < x < 4 ≤ 9 and x = 1 and x = 4

B) D)

16 ≤ 4<

+ +

24) The exterior of the sphere of radius 5 centered at the point (3, -3, 2) A) B) + + < 25 + C) D) + + > 25 + Find the distance between points and 25) (-6, -5, 2) and (2, -1, 10) A) 12 26)

27)

28)

23) ______

(-1, -13, 2) B) 7

(1, 3, 3) and A) 10

(6, -1, 6)

(4, 7, -3) and A) 2

(5, 8, -4)

≤ 81 and 1 ≤ x ≤ 4 < 9 and x = 1 and x = 4 24) ______ +

≥ 25

< 25

B) 14

(-3, -7, 5) and A) 6

25) ______ C) 10

D) 18 26) ______

C) 5

D) 27) ______

B) 50

C) 25

D) 5

; 28) ______

B) 3

Find the center and radius of the sphere. 29) + + = 49

D) 9

29) ______

A) C(0, -1, 9), a = 49 C) C(0, -1, 9), a = 7

B) C(0, 1, -9), a = 7 D) C(0, 1, -9), a = 49 30) ______

30) +

B) C(2, 9, 7), a =

C(2, 9, 7), a = D)

C(-2, -9, -7), a = 31)

32)

33)

+ + - 8x - 14y + 16z = -120 A) C(4, 7, -8), a = 9 C) C(4, 7, -8), a = 3 + + - 14x - 6y - 14z = -91 A) C(-7, -3, -7), a = 4 C) C(7, 3, 7), a = 4 +2 A)

31) ______ B) C(4, 7, 8), a = 3 D) C(-4, -7, 8), a = 3 32) ______ B) C(-7, -3, -7), a = 16 D) C(7, 3, 7), a = 16 33) ______

-x+y-z=9 B)

,a=

D) C

34)

C(-2, -9, -7), a =

+3 A) C

,a= +3

,a= 34) ______

- 2x + 2y = 9 B) ,a=

,a=

D) C

,a=

Find an equation for the sphere with the given center and radius. 35) Center (0, -6, -1), radius = 2 A) B) + +2 + 12y - 2z = -33 C) D) + + + 12y - 2z = -33 36) Center (-1, 3, 0), radius = 9 A) + + - 2x + 6y = 71 C) + + + 2x - 6y = 71 37) Center (0, 0, 8), radius = 5 A) + + - 16z = -39 C) + + + 16z = -39 38) Center (-7, 0, 0), radius = 1 A) + + + 14x = 1

35) ______ +

+ 12y + 2z = -33

+ 12y + 2z = -33 36) ______

B) D)

+ 2x + 6y = 71

- 2x - 6y = 71 37) ______

B) D)

- 16z = -39

+ 16z = -39 38) ______

- 14x = -48

+ 14x = -48

- 14x = 1

Solve the problem. 39) Find a formula for the distance from the point P(x, y, z) to the z-axis. A) B) C)

39) ______ D)

40) Find a formula for the distance from the point P(x, y, z) to the xy plane. A) z B) y C) x

40) ______ D)

41) Find the perimeter of the triangle with vertices A(5, 1, 3), B(2, -2, 6), and A) 3 C) 3

+ +

42) Show that the point P(-3, 2, 6) is equidistant from the points

) and B(-2, 4, 5).

41) ______

42) ______

A) The distance between P and A is

; the distance between P and B is B) The distance between P and A is 2; the distance between P and B is 2 C) The distance between P and A is ; the distance between P and B is D) The distance between P and A is 3; the distance between P and B is 3

Find the indicated vector. 43) Let u = ,v= A) 44) Let u =

,v=

D) 47) ______

. Find 4u. B)

D) 48) ______

. Find -5u. B)

D) 49) ______

. Find -9u.

A) 50) Let u =

45) ______

A) 49) Let u =

46) ______

A) 48) Let u =

. Find 2u.

A) 47) Let u =

D) 44) ______

. Find v - u. B)

A) 46) Let u =

. Find u - v. B)

A) 45) Let u =

43) ______

. Find u + v. B)

B) ,v=

D) 50) ______

. Find -2u - 3v. B)

D) 51) ______

51) Let u = A)

,v=

. Find B)

v. C)

52) ______

52) Let u = A)

Find the magnitude. 53) Let u = A) -6 54) Let u = A) -20 55) Let u = A) 185

,v=

. Find B)

v. C)

. Find the magnitude (length) of the vector: 6u. B) C) 6 . Find the magnitude (length) of the vector: -4u. B) 100 C) -20 and v =

53) ______ D) 6 54) ______ D) 20

. Find the magnitude (length) of the vector: -4u - v. B) 155 C) D)

Find the component form of the specified vector. 56) The vector , where P = (1, -10) and Q = (10, 7) A) B) 57) The vector from the point A(3, 3) to the origin A) B)

Express the vector in the form v = i + j + k. 59) if is the point (-1, 3, -3) and A) v = -2i - 3j + 4k C) v = -2i + 3j + 4k

D) 57) ______

D) 58) ______ D)

59) ______

is the point (1, 0, -7) B) v = -2i + 3j - 4k D) v = 2i - 3j - 4k

if A is the point (-1, -9, 1) and B is the point (4, -16, 4) A) v = 5i + 7j + 3k B) v = 5i + 7j - 3k C) v = 5i - 7j - 3k

60) ______ D) v = 5i - 7j + 3k 61) ______

61) 9u - 6v if u =

and v = A) v = -9i + 9j - 6k C) v = 9i + 9j - 6k

55) ______

56) ______

58) The unit vector that makes an angle 4π/3 with the positive x-axis A) B) C)

60)

B) v = -9i + 15j - 6k D) v = 27i + 9j - 6k

Use the vectors u, v, w, and z head to tail as needed to sketch the indicated vector.

62) ______

62) u + z

63) 3w

___ ___

63)

64) v - w

64) ______

65) ______

65) z - v

66) ______

66) 2u - z - w

Express the vector as a product of its length and direction. 67) 3i + 6j + 6k A) 9 C) 9(3i + 6j + 6k)

67) ______ B) 9 D) 9(i + j + k)

68) ______

68) i - 3j - 2k A)

D) (i - j - k) 69) ______

69) -6i + 6j + 6k A)

6 C) 6( -i + j + k)

D) 6 70) ______

70) -2i - 8j + A)

k B) (-i - j + k)

71) ______

71) 2j - k A)

(j - k) 72) ______

72) -

i - 4j A)

( -i - j)

73) ______

73) 3j + k A)

(j + k) 74) ______

74) 5j A) 5(5j)

B) 5j

D) 5

j 75) ______

75) i+ A) C)

k B)

76) ______

76) i+ A)

k B)

D) 1

Solve the problem. 77) Find a vector of magnitude 9 in the direction of v = 8i - 15k. A) 9(i - k) B) C) (8i - 15k)

77) ______ D) 9(8i - 15k)

(8i - 15k) 78) ______

78) Find a vector of magnitude 9 in the direction opposite to the direction of v = A) B) 12

Calculate the direction of 79) (1, 4, -10) and

and the midpoint of line segment

. 79) ______

(11, 14, -5)

B) i+

D) i+

(2, 1, 6) and A)

80) ______

(8, 3, 9) B)

D) i+

(-6, -5, 6) and A) i+

k; 81) ______

(-3, -1, 1) B) k;

D) i+

82)

D) 12

81)

80)

(8, 1, -7) and

(9, 0, -6)

i +

k; 82) ______

B) i-

D) i+

Solve the problem. 83) Let u = 7i + j, v = i + j, and w = i - j. Find scalars a and b such that u = av + bw. A) 7v + w B) 4.000 v + 3.000w C) 0.2500 v + 0.3333 w D) 7v - w 84) Let u = -2i + 9j, v = 2i + 3j, and w = i - j. Write u = is parallel to w. A) = 1.400 v = -4.800w

= 3v = 2w

where

= 0.7143 v = -0.2083w

83) ______

is parallel to v and D)

84) ______

= -4v = -27w

85) A force of magnitude 17 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) -16.19i - 5.182j B) 14.72i + 8.500j C) 0.5000i + + 0.8660j D) 8.500i + 14.72 j

85) ______

86) A garden hose is spraying a stream of water at a box on the ground with a force of 1.5 pounds. The water stream makes an angle of 60 degrees with the ground. What is the horizontal component of the force? A) 0.5000 B) -1.429 C) 0.7500 D) 1.299

86) ______

87) An airplane is flying in the direction 69° west of north at 828 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) B) C) D)

87) ______

88) An airplane is flying in the direction 28° east of south at 692 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) B) C) D)

88) ______

89) A bird flies from its nest 6 km in the direction 20° north of east, where it stops to rest on a tree. It then flies 9 km in the direction 25° 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) (2.448, 5.478) B) (0.9397, 0.3420) C) (2.052, 5.638) D) (5.638, 2.052)

89) ______

90) A bird flies from its nest 4 km in the direction 1° north of east, where it stops to rest on a tree. It then flies 7 km in the direction 1° 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) (3.999, 0.06981) B) (-3.000, -0.05236) C) (-1.621, -0.4162) D) (-6.999, -0.1222)

90) ______

91) For the triangle with vertices located at A(5, 3, 5), B(2, 2, 5), and C(1, 1, 1) , find a vector from

vert ex C to

___ ___

the 91) midpoint of side AB. A)

B) i+

C) i+

j + 6k

D) i+

j + 4k

j + 5k

92) Find the vector from the origin to the center of mass of a thin triangular plate (uniform density) whose vertices are A(4, 8, 7), B(9, 7, 1), and C(9, 3, 9). A) B) C) D) -

i + 6j +

j - 1k

i + 3j -

Find v ∙ u. 93) v = 2i + 2j and u = 9i + 4j A) 26

B) 10

C) 18i + 8j

D) 11i + 6j

94) v = -7i + 2j and u = 5i + 5j A) -35i + 10j

B) -25

C) -45

D) -2i + 7j

95) v = 5i - 9j and u = -8i + 5j A) 5

B) -3i - 4j

C) -85

D) -40i - 45j

96) v = 9i - 6j and u = -2i - 5j A) 7i - 11j

B) -18i + 30j

C) 12

D) -48

97) v = 2i - 9j and u = -7 A) -27

+ 14

92) ______

93) ______

94) ______

95) ______

96) ______

97) ______

i + 3j B) -27 - 14

C) -27

i - 14j

D) (2 - 7

)i - 6j 98) ______

98) v= A)

and u = B)

C) 0 i-

D) i-

j 99) ______

99) v= A) 0

and u = B)

C) i-

D) i

Find the angle between u and v in radians. 100) u = 9i + 4j + 5k, v = 2i + 7j + 2k A) 0.83 B) 1.48

C) 0.74

D) 1.25

101) u = -8i + 9j - 8k, v = 7i + 9j - 6k A) 1.17 B) 1.57

C) 0.40

D) 1.37

102) u = 7i - 5j - 2k, v = 8i + 7j - 4k A) 0.29 B) 1.28

C) 1.57

D) 1.43

103) u = -10i - 5j, v = 6i + 10j + 6k A) 1.58 B) -0.85

100) _____

101) _____

102) _____

103) _____ C) 2.42

D) 1.95

104) _____

104) u = 3j - 9k, v = 5i - 6j - 4k A) 1.46

B) 1.35

C) 1.57

D) 0.22 105) _____

105) u = 7i, v = 9i - 6j A) 0.59

B) 1.18

C) 0.98

106) _____

106) u = -5j and v = 5i - 8k A) 1.57

B) 1.68

C) 0.11

Find the vector u. 107) v = 3i - j + 3k, u = 2i + 10j + 11k A) i+

D) 1.56

D) 0.00

107) _____ B)

D) i-

108) _____

108) v = i + j + k, u = 4i + 12j + 3k A) i+

D) i+

109) _____

109) v = k, u = 9i + 2j + 6k A)

B) 6k

k C)

D) i+

k 110) _____

110) v = 7i - 3j + k, u = -4j + 3k A) -

B) -

D) i-

111) _____

111) v = 2i - 2j - 4k, u = 5i - 12k A) i-

B) i-

D) i -

112) v = 3j, u = 4i + 3k A) j

i -

k 112) _____

C) i +

j +

D) 0 i +

k Find an equation for the line that passes through the given point and satisfies the given conditions.

113) P = (9, 3); perpendicular to v = 6i + 9j A) 6x + 9y = 117 B) y - 3 = - 2(x - 6) 114) P = (-9, 8); perpendicular to v = -2i - 6j A) -2x - 6y = 40 B) -2x - 6y = -30

D) 6x + 9y = 81 114) _____

C) y - 8 = - 2(x + 2)

D) -6x + 2y = 70 115) _____

115) P = (-9, -2); perpendicular to v = -5i + 9j A) -5x + 9y = 106 C) y+2=

113) _____ C) 9x - 6y = 63

B) 9x + 5y = -91 D) -5x + 9y = 27

(x + 5) 116) _____

116) P = (10, 8); perpendicular to v = 8i - 5j A) -5x - 8y = -114 B) y-8= 8)

C) 8x - 5y = 40

D) 8x - 5y = 89

(x -

117) P = (5, 8); parallel to v = 4i + 5j A) y - 8 = 3(x - 4) B) 4x + 5y = 60

117) _____ C) 4x + 5y = 41

118) _____

118) P = (-7, 9); parallel to v = -4i - 4j A)

B) -4x + 4y = 64

y-9=(x + 4) C) -4x - 4y = -8

D) -4x - 4y = 32

119) P = (-5, -5); parallel to v = -3i + 8j A) 8x + 3y = -55 C) -3x + 8y = 73

B) -3x + 8y = -25 D)

119) _____

y+5= 120) P = (9, 6); parallel to v = 3i - 3j A) 3x - 3y = 18 B) 3x - 3y = 9

D) 5x - 4y = -7

(x + 3) 120) _____

D) -3x - 3y = -45 y-6=

(x - 3)

Solve the problem. 121) A bullet is fired with a muzzle velocity of 1161 ft/sec from a gun aimed at an angle of 11° above the horizontal. Find the horizontal component of the velocity. A) 225.7 ft/sec B) 221.5 ft/sec C) 5.138 ft/sec D) 1140 ft/sec

121) _____

122) A bullet is fired with a muzzle velocity of 1282 ft/sec from a gun aimed at an angle of 24° above the horizontal. Find the vertical component of the velocity. A) 1171 ft/sec B) 543.8 ft/sec C) 521.4 ft/sec D) 570.8 ft/sec

122) _____

123) A bullet is fired with a muzzle velocity of 1126 ft/sec from a gun aimed at an angle of 25° above the horizontal. Find the horizontal component of the velocity. A) 1021 ft/sec B) 1116 ft/sec C) 475.9 ft/sec D) 525.1 ft/sec

123) _____

124) A ramp leading to the entrance of a building is inclined upward at an angle of 7°. A suitcase is to be pulled up the ramp by a handle that makes an angle of 34° 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) -44.42 lbs B) 55.79 lbs C) 44.55 lbs D) 59.86 lbs

124) _____

125) Find the work done by a force of 6i (newtons) in moving as object along a line from the origin to the point A) 60 joules

(distance in meters). B) 6 joules

C) 42.43 joules

D) 60

joules

126) Find the work done by a force of 13i (newtons) in moving an object along a line from the origin to the point (distance in meters). A) 57.99 joules B) 16.65 joules

C) 83.24 joules

125) _____

126) _____

D) 65.00 joules

127) How much work does it take to slide a box 28 meters along the ground by pulling it with a 155 N force at an angle of 30° from the horizontal? A) 2170 B) 4340 joules C) D) 4340 joules joules

127) _____

joules 128) How much work does it take to slide a box 31 meters along the ground by pulling it with a 246 N force at an angle of 37° from the horizontal? A) 6090 joules B) 5837 joules C) 7626 joules D) 196.5 joules Find the acute angle, in degrees, between the lines. 129) 3x - y = 7 and 2x + y = -3 A) 30° B) 60° 130) y =

x - 9 and y = A) 75°

131) x -

y = 2 and A) 45°

132) (1 +

) x + (1 A) 45°

128) _____

129) _____ C) 75°

D) 45° 130) _____

x + 19 B) 60°

C) 45°

D) 30° 131) _____

x - y = -2 B) 60°

C) 75°

D) 30° 132) _____

) y = -1 and x + y = -7 B) 75°

C) 30°

Find the acute angle, in radians, between the lines. 133) 7x + 2y = 3 and 3x + 3y = -8 A) 0.5070 radians B) 1.064 radians

D) 60°

133) _____ C) 0.8742 radians

D) 2.078 radians 134) _____

134) 7x - 9y = -6 and 5x - 8y = 5 A) 0.9948 radians B) 0.1020 radians

C) 1.469 radians

D) 1.673 radians 135) _____

135) -3x - 6y = 8 and 6x - 6y = 8 A) 0.3162 radians B) 1.249 radians

C) 0.3217 radians

D) 2.820 radians

136) 3x - 6y = -10 and 8x + 8y = 6 A) 1.249 radians B) 0.3217 radians

C) 0.3162 radians

D) 2.820 radians

136) _____

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 137) The unit vectors u and v are combined to produce two new vectors 137) ____________ and Show that a and b are orthogonal. Assume 138) Show that the vectors

a and

a are orthogonal.

138) ____________

139) Show that the midpoint of the hypotenuse of a right triangle is equidistant from all three vertices. [Hint: See the figure below. Show that

139) ____________

140) ____________

140) Show that A = au + bv is orthogonal to B = bu - av, where u and v are orthogonal unit vectors.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the length and direction (when defined) of u × v. 141) u = 6i, v = 8j A) 48; -k B) 48; 48k C) 6; 8k D) 48; k

142) _____

142) u = 2i + 4j , v = i - j A) 2; -2k

B) 6; -k

C) 2; 2k

; (5i - 5j + 3k) C) 0; 5i- 5j + 3k

; (5i - 5j + 3k) D) 0; no direction 144) _____

144) u = 4i + 2j + 8k, v = -i - 2j - 2k A) i+

D) 6; -6k 143) _____

143) u = -5i + 5j - 3k, v = 10i - 10j + 6k A)

180;

141) _____

;

D) 6

;

k 145) _____

145) u = 2i + 2j - k, v = -i + k A) 9;

180;

C) 3;

D) 9;

k 146) _____

146) u=- i+ A) 8;

j + k, v = i + j + 2k B) i-

;

D) 2

147) _____

147) u = -4i + 7i - 2k, v = 0 A) ;

B) 0; no direction

(-4i + 7i - 2k)

D) 0; -4i + 7i - 2k 0;

(-4i + 7i - 2k)

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. 148) u = i, v = k 148) ____________ 149) u = i - j, v = k

149) ____________

150) u = 2i + j, v = i - 2j

150) ____________

151) u = i + k, v = i - k

151) ____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 152) Find the area of the triangle determined by the points P(1, 1, 1), Q(-4, -3, -6), and A) B) C) D) 3

153) _____

153) Find the area of the triangle determined by the points P(-3, 6, -4), Q(-10, -4, 6), and A)

154) _____

154) Find the area of the parallelogram determined by the points P(7, -5, 5), Q(-7, 2, -2), and A)

B) 7

C) 7

155) 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) B) C) D) (j + k)

(i + j)

(i - j)

(-i - 2j - k)

155) _____

(j - k)

156) 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) B) C) D) (i - 2j + k)

152) _____

(i - 2j - 2k)

156) _____

(i - 2j - k)

Find the triple scalar product (u x v) ∙ w of the given vectors. 157) u = i + j + k; v = 7i + 4j + 6k; w = 5i + 3j + 10k A) 63 B) -1

C) -17

D) -19

158) u = -5i - 3j + 4k; v = 10i - 3j + 3k; w = 4i - 7j - 10k A) -319 B) 57

C) -151

D) -823

159) u = 2i - 4j + 3k; v = -9i - 7j + 4k; w = 10i - 8j + 3k A) 180 B) -48

C) 220

D) 716

157) _____

158) _____

159) _____

160) u = 4i + 2j - k; v = 7i + 5j - 3k; w = 8i + 5j - 7k A) -25 B) -125

160) _____ C) 0

D) -129

Solve the problem. 161) Find the magnitude of the torque in foot-pounds at point P for the following lever:

= 3 in. and

161) _____

= 5 lb

B) ft-lb

C) ft-lb

D) ft-lb

ft-lb 162) _____

162) Find the magnitude of the torque in foot-pounds at point P for the following lever:

= 4 in. and A) 3.52 ft-lb

= 25 lb B) 12.26 ft-lb

163) For the vectors u and v with magnitudes which makes A) 60.00°

C) 100 ft-lb = 4 and

D) 7.55 ft-lb

= 7, find the angle θ between u and v

163) _____

=2 B) 55.15°

C) 30.00°

D) 73.40°

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Determine whether the following is always true or not always true. Given reasons for your answers. 164) |u| = 164) ____________ 165) u × 0 = 0

165) ____________

166) u × (v + w) = u × v + u × w

166) ____________

167) (u × v) ∙ w = u ∙ (w × v)

167) ____________

168) u × v = -(v × u)

168) ____________

169) (u × v) ∙ v = 0

169) ____________

170) (u × v) ∙ v = u ∙ (u × v)

170) ____________

171) c(u ∙ v) = cu ∙ cv (any number c)

171) ____________

172) ____________

172) c(u × v) = cu × cv (any number c)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find parametric equations for the line described below. 173) The line through the point P(-5, -3, 4) parallel to the vector A) x = -3t + 5, y =2t + 3, z = -5t - 4 B) x = -3t - 5, y = 2t - 3, z = -5t + 4 C) x = 3t - 5, y = 2t - 3, z = -5t + 4 D) x = 3t + 5, y = 2t + 3, z = -5t - 4

174) _____

174) The line through the point P(-4, 1, -3) parallel to the vector A) x = 3t + 4, y = -4t - 1, z = -2t + 3 C) x = -3t- 4, y = -4t + 1, z = 2t - 3

173) _____

B) x = -3t + 4, y = 4t - 1, z = -2t + 3 D) x = 3t - 4, y = -4t + 1, z = -2t - 3

175) The line through the points P(-1, -1, 1) and Q(-3, 3, -7) A) x = -2t - 1, y = 4t - 1, z = -8t + 1 B) x = t + 2, y = t - 4, z = 1t + 8 C) x = t - 2, y = t + 4, z = 1t - 8 D) x = -2t + 1, y = 4t + 1, z = -8t - 1

175) _____

176) The line through the point P(-3, 0, 2) and parallel to the line x = 3t + 2, A) x = 3t - 3, y = 2t, z = 3t + 2 B) x = 2t - 3, y = -3t, z = 2 C) x = -3, y = 3t, z = -2t + 2 D) x = 3 + 3, y = 2t, z = 3t - 2

176) _____

177) The line through the point P(3, 1, -4) and parallel to the line x = 2t - 4, A) x = 2t - 3, y =3t - 1, z = 4t + 4 B) x = 3t + 3, y = -2t, z = -4 C) x = 2t + 3, y = 3t + 1, z = 4t - 4 D) x = 3, y = 4t + 1, z = -3t - 4

177) _____

178) The line through the point P(-1, 5, -6) and perpendicular to the plane A) x = 3t + 1, y = -3t - 5, z = -7t + 6 B) x = -3t - 1, y = -3t + 5, z = -6 C) x = -3t - 1, y = 3t + 5, z = 7t - 6 D) x = -3t + 1, y = 3t - 5, z = 7t + 6

178) _____

179) The line through the point P(-4, 6, 0) and perpendicular to the plane A) x = 3t + 4, y = 7t - 6, z = 6t B) x = -7t - 4, y = -7t - 4, z = 0 C) x = -3t + 4, y = -7t - 6, z = -6t D) x = 3t - 4, y = 7t + 6, z = 6t

179) _____

180) The line through the point P(-3, 2, 7) and perpendicular to the vectors A) x = 6t - 3, y = -24t + 2, z = 0t + 7 C) x = 6t - 3, y = 24t + 2, z = -7t + 7 Find a parametrization for the line segment joining the points. 181) (-4, 0, 3), (0, -4, 0) A) x = 5t, y = 5t - 4, z = -2t, 0 ≤ t ≤ 1 C) x = 4t - 4, y = -4t, z = -3t + 3, 0 ≤ t ≤ 1 182) (-4, -6, -3), (0, -6, -5) A) x = 4t - 4, y = -6, z = -2t - 3, 0 ≤ t ≤ 1 C) x = 4t - 4, y = -6t, z = -2t - 3, 0 ≤ t ≤ 1

and

180) _____

B) x = 6t + 3, y = 24t - 2, z = -7t - 7 D) x = 6t - 3, y = 24t + 2, z = 0t + 7

181) _____ B) x = 5t - 4, y = 5t, z = -2t + 3, 0 ≤ t ≤ 1 D) x = -4t, y = 4t - 4, z = 3t, 0 ≤ t ≤ 1 182) _____ B) x = -4t, y = -6, z = 2t - 5, 0 ≤ t ≤ 1 D) x = -4t, y = -6t, z = 2t - 5, 0 ≤ t ≤ 1 183) _____

183) (5, 4, -5), A) x = -5t + 5, y = 4, z = C)

B) t - 5, 0 ≤ t ≤ 1

x = -5t + 5, y = 4t, z = x=

t - 5, 0 ≤ t ≤ 1

5t, y = 4t, z = t + 5, 0 ≤ t ≤ 1

x= 5t, y = 4, z= -

t+ 5, 0≤ t≤ 1 184) (0, 0, 0), (2, 3, -3) A) x = 2t + 2, y = 3t + 3, z = -3t - 3 C) x = 2t, y = 3t, z = -3t

184) _____ B) x = -2t, y = -3t, z = 3t D) x = 2t - 2, y = 3t - 3, z = -3t + 3

Write the equation for the plane. 185) The plane through the point P(-4, -7, -3) and normal to n = 5i + 8j + 6k. A) -5x - 8y - 6z = -94 B) -4x - 7y - 3z = -94 C) 4x + 7y + 3z = -94 D) 5x + 8y + 6z = -94

185) _____

186) The plane through the point P(-7, -6, 3) and normal to n = -2i - 7j + 5k. A) -2x - 7y + 5z = 71 B) 7x + 6y + 3z = -43 C) -7x - 6y - 3z = -43 D) 2x + 7y - 5z = -43

186) _____

187) The plane through the point P(-4, 7, -3) and parallel to the plane

187) _____

A) 5x + 5y + 3z = -6 C) 5x + 5y + 3z = 6

B) -4x + 7y - 3z = 6 D) 5x + 3y + 5z = 6 188) _____

188) The plane through the point P(1, -3, -1) and parallel to the plane A) -5x - 6y + 7z = -20 C) -5x - 6y + 7z = 6

B) 4y = 6 D) -5x - 6y + 7z = -6

189) The plane through the points P(5, -1, -27) , Q(-3, -3, 17) and R(-1, 8, -15). A) 5x + 2y + z = 4 B) 2x + y + 5z = -4 C) 5x + 2y + z = -4 190) The plane through the points P(-1, -6, 48) , Q(2, 4, -10) and A) 4x + 7y +z = 10 C) 4x - 7y - z = 10

190) _____

191) _____

B) 5x + 8y + 8z = 11 D) 5x + 8y + 8z = 21

192) The plane through the point P(3, 8, -7) and perpendicular to the line A) 8x + 3y - z = -55 C) 8x + 3y - z = 4

D) 2x + y + 5z = 4

B) 4x - 7y - z = -10 D) 4x + 7y + z = -10

191) The plane through the point P(3, 4, 4) and perpendicular to the line A) 5x + 8y + 8z = -79 C) 5x + 8y + 8z = 79

189) _____

B) 8x + 3y + z = 55 D) 8x + 3y - z = 55

192) _____

193) The plane through the point A(8, 9, 10) perpendicular to the vector from the origin to A. A) 8x + 9y + 10z = B) 8x + 9y + 10z = 27 C) 8x + 9y + 10z = -245

D) 8x + 9y + 10z = 245

194) The plane through the point A(-3, -7, 6) perpendicular to the vector from the origin to A. A) -3x - 7y + 6z = -94 B) 3x + 7y - 6z = C) 3x - 7y - 6z = 4 D) 3x + 7y - 6z = -94 Calculate the requested distance. 195) The distance from the point S(-7, -6, -9) to the line x = 10 + 12t, y = 8 + 3t, A)

196) _____ D)

197) _____

197) The distance from the point S(10, 4, 3) to the line x = -2 + 9t, y = 6 + 2t, A)

198) _____

198) The distance from the point S(5, -2, -4) to the line x = 8 + 2t, y = 9 + 2t, A)

199) The distance from the point S(-4, -8, -7) to the plane 2x + 2y + z = 7 A) B) C)

200) The distance from the point S(1, 5, -9) to the plane 11x + 2y + 10z = 1 A) B) C)

201) The distance from the point S(-9, 3, 7) to the plane -9x + 2y + 6z = 3 A) B) C)

202) The distance from the point S(-2, -8, 7) to the plane 3x + 4y = 8 A) B) C)

199) _____ D)

200) _____ D)

201) _____ D)

202) _____ D)

Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian. 203) 4x + 3y + 10z = -7 and 9x + 10y + 2z = -10 A) 1.459 rad B) 0.601 rad C) 1.479 rad D) 0.970 rad 204) 9x - 3y + 9z = -10 and -3x - 7y + 10z = -1 A) 1.479 rad B) 1.034 rad 205) 2x - 6y - 6z = -3 and- 5x + 3y - 10z = 2

194) _____

195) _____

196) The distance from the point S(-3, -5, -5) to the line x = -5 + 2t, y = 6 + 11t, A)

193) _____

203) _____

204) _____ C) 0.536 rad

D) 0.575 rad 205) _____

A) 1.030 rad

B) 1.472 rad

C) 1.248 rad

D) 0.323 rad

206) -5x - 4y - 9z = -6 and 6x + 10y - 9z = -3 A) 1.120 rad B) 1.460 rad

C) 1.503 rad

D) 0.068 rad

Find the intersection. 207) x = -5 + 5t, y = 8 + 10t, z = 9 + 7t ; 4x + 5y + 4z = 8 A) C) (-10, -2, 2)

206) _____

207) _____ B) D) (0, 18, 16)

208) x = -9 + 8t, y = -4 - 9t, z = 2 + 8t ; 9x + 5y - 10z = 9 A) (-17, 5, -6) C)

209) x = -8 - 5t, y = -6 + 3t, z = 2 + 3t ; -8x - 9y - 7z = -9 A) C) (-13, -3, 5)

208) _____ B) (-1, -13, 10) D)

209) _____ B) (-3, -9, -1) D)

210) 9x - 4y - 3z = 4, 8x - 5y - 7z = 8 A) x = 13t - 12, y = 39t + 40, z = 13t C) x = -13t + 12, y = -39t + 40, z = 13t

210) _____ B) x = -169t - 12, y = -507t - 40, z = 13t D) x = 13t -

211) 9x + 7y = 8, 4y - 7z = -6 A) x = -49t + 74, y = 63t - 6, z = 36t C)

, y = 39t -

, z = -13t 211) _____

B) x = -49t -

, y = 63t +

, z = -36t

x = -49t +

, y = 63t -

, z = 36t

D) x = -49t +

, y = 63t -

, z = 36

212) x + y + z = 8, x + y = 10 A) x = -t, y = 10 + t, z = 2 C) x = t, y = 10 - t, z = -2 Match the equation with the surface it defines. 213) + = 16

212) _____ B) x = -1, y = 1 + 10t, z = -2t D) x = -t, y = 10 + t, z = -2

____ _

213)

A) Figure 1

B) Figure 3

C) Figure 2

D) Figure 4 214) _____

214) +

A) Figure 3

B) Figure 1

215) +

C) Figure 4

D) Figure 2

____ _

215)

A) Figure 4

B) Figure 1

C) Figure 2

D) Figure 3 216) _____

216) +

A) Figure 3 217) +

B) Figure 2

C) Figure 1

D) Figure 4

____ _

217)

A) Figure 2

B) Figure 4

C) Figure 3

D) Figure 1 218) _____

218) -

A) Figure 4

B) Figure 2

219) -

C) Figure 3

D) Figure 1

____ _

219)

A) Figure 2

B) Figure 3

C) Figure 1

D) Figure 4 220) _____

220) -

A) Figure 2

B) Figure 3

C) Figure 1

D) Figure 4

Identify the type of surface represented by the given equation. 221) + =9 A) Parabolic cylinder B) Ellipsoid C) Cylinder D) Paraboloid 222)

223)

x = -2 , no limit on y A) Cylinder C) Parabolic cylinder

221) _____

222) _____ B) Hyperboloid of two sheets D) Sphere

____ _

223) + =2 A) Parabolic cylinder C) Ellipsoid

B) Elliptical cylinder D) Paraboloid 224) _____

224) + + =1 A) Ellipsoid

B) Paraboloid

C) Sphere

D) Elliptical cone 225) _____

225) + = A) Elliptical cone

B) Ellipsoid

C) Elliptical paraboloid

D) Hyperbolic paraboloid 226) _____

226) + = A) Ellipsoid

B) Elliptical cone

C) Elliptical paraboloid

D) Hyperbolic paraboloid 227) _____

227) + = A) Ellipsoid C) Paraboloid

B) Elliptical cone D) Hyperbolic paraboloid 228) _____

228) + =1 A) Hyperboloid of two sheets C) Ellipsoid

B) Elliptical cone D) Hyperboloid of one sheet 229) _____

229) =1 A) Elliptical cone C) Ellipsoid

B) Hyperboloid of two sheets D) Hyperboloid of one sheet 230) _____

230) = A) Parabolic cylinder C) Paraboloid

B) Hyperbolic paraboloid D) Ellipsoid

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Sketch the given surface. 231) + =9

____ ____ ____

231)

232)

233)

234)

232) ____________

= 16

233) ____________

____ ____ ____

234)

235)

236)

x=1-

235) ____________

236) ____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 237) Find the volume of the solid bounded by the paraboloid and A)

(The area of an ellipse with semiaxes a and b is πab.) 48 π

and the planes

237) _____

24 π

238) _____

238) Find the volume of the solid bounded by the elliptical cone and A)

(The area of an ellipse with semiaxes a and b is πab.) B) C) π

and the planes D)

π 239) _____

239) Find the volume of the solid bounded by the ellipsoid and A)

(The area of an ellipse with semiaxes a and b is πab.) B) C) 34.67π 29.33π 26.67π

= 1 and the planes D)

469.33π 240) _____

240) Find the volume of the solid bounded by the hyperboloid of one sheet planes and A) 672.00π

(The area of an ellipse with semiaxes a and b is πab.) B) C) D) 62.40π 52.80π 67.20π

and the

1) C 2) C 3) B 4) B 5) C 6) D 7) C 8) B 9) B 10) B 11) C 12) C 13) D 14) D 15) C 16) C 17) A 18) D 19) D 20) B 21) B 22) B 23) B 24) C 25) A 26) B 27) D 28) C 29) C 30) D 31) C 32) C 33) C 34) B 35) D 36) C 37) A 38) C 39) A 40) A 41) C 42) A 43) D 44) B 45) D 46) D 47) C 48) D 49) C 50) A 51) A

52) C 53) C 54) D 55) C 56) D 57) C 58) B 59) D 60) D 61) A 62) C 63) C 64) B 65) C 66) D 67) B 68) D 69) A 70) D 71) B 72) D 73) D 74) B 75) D 76) C 77) B 78) C 79) A 80) D 81) C 82) D 83) B 84) A 85) D 86) C 87) B 88) D 89) D 90) B 91) C 92) C 93) A 94) B 95) C 96) C 97) B 98) C 99) D 100) A 101) A 102) B 103) C

104) B 105) A 106) D 107) C 108) B 109) B 110) D 111) B 112) D 113) D 114) B 115) D 116) C 117) D 118) B 119) A 120) D 121) D 122) C 123) A 124) D 125) A 126) D 127) A 128) A 129) D 130) B 131) D 132) A 133) A 134) B 135) B 136) A 137) u =

a=u+v=(

j and v =

j , so

)j and b = u - v = (

)i+(

)(

)i+(

Take the dot product a ∙ b: a ∙ b = (u + v) ∙ (u - v) = ( =

)+(

)-(

) +

)

= =1-1=0 Since the dot product of the two non-zero vectors is zero they are orthogonol. 138) Take the dot product: (

a) ∙ ( =(

b) ∙ (

b∙b +

a) = (

b) + (

a) ∙ (

a∙b-

a) ∙

b-(

b) - (

b) ∙ (

a∙b-

a∙a

a) ∙

a) - (

a) ∙ (

= =0 Since the dot product is zero and since neither vector is identically zero, then the vectors are orthogonal. 139)

Verif . y that Cancel the 2's and square both sides:

140)

= 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.

A∙B = (au + bv)∙(bu - av) = abu∙u u∙v + 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. 141) D 142) B 143) D 144) C 145) B 146) D 147) B 148)

149)

150)

151)

152) B 153) B 154) C 155) D 156) D 157) C 158) D 159) A 160) A 161) A 162) A 163) A 164) Always true by definition 165) Always true by definition of 0 166) Always true by distributive property 167) Not always true; (u × v) ∙ w = u ∙ (v × w), but v × w = -(w × v) from which it follows that the original equation false if 168) Always true by definition of the cross product 169) Always true because u × v and v are orthogonal 170) Not always true; The statement is false if u ≠ v. 171) Not always true; The statement if false if c ≠ 0,1. 172) Not always true; The statement if false if c ≠ 0,1. 173) B 174) D 175) A

176) A 177) C 178) C 179) D 180) D 181) C 182) A 183) A 184) C 185) D 186) A 187) C 188) C 189) C 190) B 191) C 192) D 193) D 194) D 195) B 196) C 197) A 198) D 199) D 200) A 201) D 202) B 203) D 204) B 205) C 206) C 207) A 208) C 209) A 210) D 211) D 212) D 213) C 214) A 215) B 216) D 217) D 218) C 219) D 220) B 221) C 222) C 223) B 224) A 225) C 226) C 227) B

228) D 229) B 230) B 231)

232)

233)

234)

235)

236)

237) D 238) C 239) B 240) D

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. If r(t) is the position vector of a particle in the plane at time t, find the indicated vector. 1) Find the velocity vector. r(t) = (6 A) v=

+ 14)i +

j B)

i + (12t)j

v = (12t)i +

v = (12t)i -

D) v = (12)i +

2) Find the acceleration vector. r(t) = (9 cos t)i + (4 sin t)j A) a = (-9 sin t)i + (-4 cos t)j C) a = (9 sin t)i + (4 cos t)j 3) Find the acceleration vector. r(t) = (cos 3t)i + (5 sin t)j A) a = (-3 cos 3t)i + (5 sin t)j C) a = (-9 cos 3t)i + (-5 sin t)j 4) Find the velocity vector. r(t) = (cot t)i + (csc t)j A) v=( t)i + (cot t csc t)j C) v = (t)i - (tan t sec t)j

2) _______ B) a = (9 cos t)i + (4 sin t)j D) a = (-9 cos t)i + (-4 sin t)j 3) _______ B) a = (9 cos 3t)i + (-5 sin t)j D) a = (-9 cos 3t)i + (-25 sin t)j 4) _______ B) D)

v=(

t)i + (tan t sec t)j

v = (-

t)i - (cot t csc t)j 5) _______

5) Find the acceleration vector. r(t) = (2 ln(4t))i + (5 )j A) a=

i + 15tj

a=-

i + 30tj

D) a=

a=-

i + 30tj

The position vector of a particle is r(t). Find the requested vector. 6) The velocity at t = 3 for r(t) = (6 + 3t + 6)i - 5 j + (10 - )k A) v(3) = 33i - 135j - 6k B) v(3) = 39i + 135j + 6k C) v(3) = 21i - 45j - 3k D) v(3) = 39i - 135j - 6k

6) _______

7) _______

7) The velocity at t = A) v

for r(t) = 3

(t)i - 4tan(t)j + 6 k B)

= -8j + 3πk

= -8j - 3πk

= 12i - 8j + 3πk

D) v

1) _______

= 12i + 8j - 3πk

Th velocity e at t = 0

__ __ __ _

for r(t) = 8) cos(3t)i + 8ln(t 6)j -

k A)

B) v(0) =

10)

C) v(0) = 3i -

v(0) = - j

The velocity at t = 2 for r(t) = (2 - 2 )i + (2t + 3)j A) v(2) = 8i + 2j + 2 k C) v(2) = -8i + 2j + 2 k The velocity at t = 0 for r(t) = ln( A) v(0) =

-3

+ 1)i -

k B) D)

v(0) = -3i -

j 9) _______

v(2) = -8i + 2j - 2

v(2) = -4i +2j + 2

k 10) ______

j - 6cos(t)k B) v(0) = 1i -

D) v(0) = 1i -

j + 6k

v(0) = -

j 11) ______

11) The acceleration at t = A) a

for r(t) = (4 sin 4t)i - (10 cos 4t)j + (7 csc 4t)k B)

= -64i - 112k

C) a 12)

= 64i + 112k

= -64i + 112k

D) = 160j + 112k

12) ______

The acceleration at t = 3 for r(t) = (10t - 3 )i + (9 - t)j + (7 - 4t)k A) a(3) = 324i + 14k B) a(3) = -81i + 14k C) a(3) = -324i - j + 14k D) a(3) = -324i + 14k

13) ______

13) The acceleration at t = 1 for r(t) = A)

i + 4ln

a(1) = 20i + j + 6k C) a(1) = 20i - 4j - 6k

k B) a(1) = 20i + j - 6k D) a(1) = 20i + 4j + 6k 14) ______

14) The acceleration at t = A) a

for r(t) =

= - 3πi + 32j + 16

)i + 2tan(2t)j + B)

= - 3πi - 32j + 16

= 3πi + 32j + 16

D) a

15)

= - 3πi - 32j + 16k

The acceleration at t = 0 for r(t) = A) B) a(0) = 2i -

i + (4

a(0) = 2i -

- 4)j +

15) ______

k C)

D) a(0) = 2i -

a(0) = 2i +

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) 16) ______ r(t) = (3 + 1)i + (4 - 4t)k A) B) C) 0 D) π

17) ______

17) r(t) = ti + ( A) 0

18)

19)

r(t) = A) 0

r(t) = A)

)k B) π

(4t)i + ln(5

i + (2 +

+ 1)j + B)

18) ______

k C)

19) ______

)j + (3 cos 5t)k B)

C) 0

For the smooth curve r(t), find the parametric equations for the line that is tangent to r at the given parameter value 20) 20) ______ r(t) = (2 - 6t)i + (t + 7)j + k ; =2 A) x = -4 + 2t, y = 9 + t, z = 0 B) x = 2t, y = t, z = t C) x = -4 + t, y = t, z = t D) x = -4 + 2t, y = 9 + t, z = 1 21) ______

21) r(t) = (3 tan t)i - (3 sin t)j + (2 A) x = 3 + 6t, y = -

t)k ;

B) t, z = 1 + 2t

x = 6 + 3t, y = -

t, z= -2 + 1t

D) x = 3 + 6t, y = -

22)

t, z = 2 - 2t

r(t) = (10 sin t)i - (6 cos 6t)j + A) x = 10t, y = 6, z = 1 + t C) x = 10, y = -6t, z = -2 + t

x = 3 + 6t, y = -

t, z = 1 - 2t 22) ______

=0 B) x = 10t, y = -6, z = 1 - 2t D) x = 10t, y = -6, z = 1 - t

Provide an appropriate response. 23) 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?

23) ______

(1) r(t) = i + j (2) r(t) = cos (3t)i + sin (2t)j (3) r(t) = ti + j (4) r(t) = cos (-10t)i + sin (-10t)j A) Path (3) C) Path (2) and Path (4)

B) Path (4) D) Path (1)

24) The following equations each describe the motion of a particle. For which path is the particle's speed constant?

(1) = r(t)

j (2) r(t) = cos (7t)i + sin (7t)j (3) r(t) = ti + tj (4) r(t) = cos

___ ___

24)

(-2 )i + sin (-2 )j A) Path (1) C) Path (3)

B) Path (2) and Path (3) D) Path (4) and Path (2) 25) ______

25) The position of a particle is given by r(t) = sin 3t i + cos 3t j. Find the velocity vector for the particle. A) 3 cos 3t i - 3 sin 3t j B) 3 sin 3t i + 3 cos 3t j C) 3 i + 3 j D) cos 3t i - sin 3t j

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 26) Show that if a particle's velocity vector is always orthogonal to its acceleration vector 26) _____________ then the particle's speed is constant. 27) Prove that if u is a differentiable function of t and f is a differentiable scalar function of t, then

27) _____________

. 28) _____________

28) Prove that if v and u are differentiable functions of t, then

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the integral. 29)

B) 4i - 27j + 9k 8i - 27j +

C) 8i - 27j + 9k

29) ______

D) 8i + 27j +

k 30) ______

30)

B) 8i +

j + 7(1 -

8i +

j + 7(1 +

8i +

j + 7(1 -

D) 8i +

j + 7(1 +

31) ______

31)

B) 8i + 4 ln 2j + (1 8i + 4 ln 2j +

C) 8i + 4 ln 2j + (1 +

D) 2i + 4 ln 2j + (1 -

)k 32) ______

32)

B) +4i +

j + 72k

C) -4i -

j - 72k

D) -4i +

j + 72k

-4i -

j + 72k 33) ______

33)

B) j+

πi +

D) i+

k 34) ______

34)

B) 10i - 6j + 3k 5i - 3j -

C) 10i + 6j - 3k

D) 5i + 3j -

k 35) ______

35)

A) 2

i+3

B) 2

C) 0

D) 3

i 36) ______

36)

B) -

- 1) j

- 1)

D) i+

- 1) j

Solve the initial value problem. 37)

37) ______

Differential Equation: = (3t2 + 2t)i + (4 - 8)j Initial Condition: r(0) = -8j A) r(t) = (t3 + t2)i + ( )j C) r(t) = (-t3 - t2)i + (12 - 8)j

r(t) = (t3 + t2)i + ( - 8t - 8)j

r(t) = (t3 + t2)i + (

- 8)j 38) ______

38) Differential Equation: = Initial Condition: r(0) = 0 A) r(t) = [ ]i + ( C) r(t) = i+ j

i+ - 1)j

j B) D)

r(t) = [

]i + (

- 1)j

r(t) = [

]i + (

+ 1)j

39) ______

39) Differential Equation: = ( + 4 )i + 4tj Initial Condition: r(0) = -5i + 6j A) r(t) =

i+2 j

r(t) =

i+(

+ 6)j

r(t) =

i + (2

- 5)j

D) r(t) =

i + (2

+ 6)j

40) ______

40) Differential Equation: = -3ti + 4tj + 5tk Initial Condition: r(0) = -6i + 5k A) r(t) =

i+2 j +

r(t) = (-3

- 12)i + 4 j + (5

+ 10)k

i+2 j -

D) r(t) =

i+2 j +

r(t) =

41) ______

41) Differential Equation: = i + (9 - 2t)j + Initial Condition: r(0) = 2j + (ln 2)k A) r(t) = ti +

k B)

j + ln(t + 2)k

r(t) = ti +

j + (ln t)k

r(t) = i +

j + (ln 2)k

D) r(t) = ti +

j + (ln 2)k

42) ______

42) Differential Equation: = (9 - 4)i - j + k Initial Condition: r(0) = -8i + 6j + 4k A) r(t) = (9 - 4t - 8)i + (6 - t)j + ( + 2)k B) r(t) = (3 + 4t + 8)i + (6 + t)j + ( + 2)k C) r(t) = (3 - 4t - 8)i + (6 - t)j + (2 + 2)k D) r(t) = (3 - 4t - 8)i + (6 - t)j + ( - 2)k

43) ______

43) Differential Equation:

= (4 sec 4t tan 4t) i +

Initial Condition: r(0) = 6i + A)

j - 3k

r(t) = (csc 4t + 5)i +

B) r(t) = (sec 4t + 5)i -

C) r(t) = (csc 4t + 5)i + D)

j+(

- 3)k

r(t) = (sec 4t + 5)i +

k 44) ______

44) Differential Equation:

= (2t - 8)i

Initial Conditions:

= -k, r(0) = 3i + 5j + 8k

B) r(t) =

i + 5j + (8 - t)k

r(t) =

i - 5j + (8 - t)k

r(t) =

i - 5j + (t - 8)k

D) r(t) =

i + 5j + (8 - t)k

45) ______

45) Differential Equation:

= 2 i - tj

Initial Conditions: A) r(t) =

= -3i, r(0) = 5i - 9k B) i-

j - 9k

r(t) =

j - 9k

r(t) =

j - 9k

D) r(t) =

Solve the problem. 46) At time

j - 9k

a particle is located at the point (2, -3, 1). It travels in a straight line to the point

has speed 2 at (2, -3, 1) and constant acceleration position vector r(t) of the particle at time t. A)

46) ______

Find an equation for the

r(t) =

B) j-

C) D)

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 47) An ideal projectile is launched from the origin at an angle of α radians to the horizontal and an 47) ______ initial speed of Find the position function r(t) for this projectile. A) B) r(t) = (125t cos α - 32 )i + (125t sin α)j r(t) = (125t sin α - 16 )i + (125t cos α)j C) D) r(t) = (125t cos α)i + (125t sin α - 16 )j r(t) = (125t sin α)i + (125t cos α - 16 )j 48) An ideal projectile is launched from level ground at a launch angle of 26° and an initial speed of 48 m/sec. How far away from the launch point does the projectile hit the ground? A) ≈ 60 m B) ≈ 290 m C) ≈ 230 m D) ≈ 185 m

48) ______

49) A projectile is fired at a speed of 880 m/sec at an angle of 31°. How long will it take to get 22 km downrange? Round your answer to the nearest whole number. A) It will never get that far downrange. B) 27 sec C) 31 sec D) 29 sec

49) ______

50) A projectile is fired with an initial speed of 500 m/sec at an angle of 45°. What is the greatest height reached by the projectile? Round your answer to the nearest tenth. A) 62,500.0 m B) 25,510.2 m C) 6377.6 m D) 72.2 m

50) ______

51) A spring gun at ground level fires a tennis ball at an angle of 48°. The ball lands 12 m away. What was the ball's initial speed? Round your answer to the nearest tenth. A) 3.5 m/sec B) 10.9 m/sec C) 118.2 m/sec D) 12.6 m/sec

51) ______

52) Find the muzzle speed of a gun whose maximum range is 19.1 km. Round your answer to the nearest tenth. A) 187.7 km/sec B) 14.2 km/sec C) 13.7 km/sec D) 187.2 km/sec

52) ______

53) A fan in the bleachers at Wrigley Field throws an opposing player's home run baseball back onto the playing field. Assume that the fan is 30 feet above the field and that the ball is launched at an angle of 37°. When will the ball hit the ground if its initial speed is 36 ft/sec? Round your answer to the nearest tenth. A) 0.9 sec B) 2.4 sec C) 2.2 sec D) 5.5 sec

53) ______

54) An athlete puts a 16-lb shot at an angle of 36° to the horizontal from 5.6 ft above the ground at an initial speed of 47 ft/sec. How far forward does the shot travel before it hits the ground? Round your answer to the nearest tenth. A) 72.6 ft B) 1.9 ft C) 7 ft D) 221.8 ft

54) ______

55) A baseball is hit when it is 2.9 ft above the ground. It leaves the bat with an initial velocity of

55) ______

at a launch angle of 21°. At the instant the ball is hit, an instantaneous gust of wind blows against the ball, adding a component of -12.6i (ft/sec) to the ball's initial velocity. Find a vector equation for the path of the baseball. A) r = (137 cos 21° - 12.6)ti + (2.9 + (137 sin 21°)t - 16 )j B) r = (137 sin 21° - 12.6)ti + (2.9 + (137 cos 21°)t - 16 )j C) r = ((137 cos 21°)t - 12.6)i + (2.9 + (137 sin 21°)t - 16 )j D) r = (137 cos 21° - 12.6)ti + (2.9 + (137 sin 21°)t + 16 j 56) A baseball is hit when it is 2.7 ft above the ground. It leaves the bat with an initial velocity of

56) ______

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 -9.6i (ft/sec) to the ball's initial velocity. How high does the baseball go? Round your answer to the nearest tenth. A) 68.6 ft B) 65.9 ft C) 65.0 ft D) 70.6 ft SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 57) A human cannonball is to be fired with an initial speed of 70 ft/sec. The circus performer If the proper hopes to land on a cushion located 140 feet downrange at the same height as the muzzle so, firing angle? (g of the cannon. The circus is being held in a large room with a flat ceiling 30 feet higher wh = 32 ft/ ) than the muzzle. Can the performer be fired to the cushion without striking the ceiling? at is

57)

_____________ 58) A golf ball leaves the ground at a 29° angle at a speed of 91 ft/sec. Will it clear the top of a 30 ft tree that is in the way, 136 ft down the fairway? Explain.

58) _____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 59) What two angles of elevation will enable a projectile to reach a target downrange on the

59) ______

same level as the gun if the projectile's initial speed is Assume there is no wind resistance. A) 24.7° and 65.3° B) 49.4° and 40.6° C) 24.7° D) 0.02° and 89.98° 60) ______

60) Increasing the initial speed of a projectile by a factor of 3 increases its range by what factor? Assume the elevation is the same. A) Factor of 1.73 B) The elevation must be specified before an answer can be found. C) Factor of 9 D) Factor of 3

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 61) A baseball is hit when it is 3.1 feet above the ground. It leaves the bat with an initial 61) _____________ velocity of 145 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 5i ft/sec to the ball's initial velocity. Find the range and flight time of the ball, assuming that the ball is not caught. 62) _____________

62) Derive the equations x=

cos α)t,

y= +( sin α)t - g by solving the following initial value problem for a vector r in the plane.

Differential equation: Initial conditions:

= -gj r(0) =

j (0) =

cos α)i + (

sin

α)j 63) Derive the equations x=

( +

(1 -

)cos α,

y= + (1 )sin α + (1 - kt ) by solving the following initial value problem for a vector r in the plane.

Differential equation: Initial conditions:

= -gj - kv = -gj - k r(0) =

cos α)i +

( sin α)j The drag coefficient k is a positive constant representing resistance due ( to air density, 0) = and α are the projectile's =

___ ___ ___ ___ _

initial 63) speed and launch angle, and g is the accelerati on of gravity.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 64) A baseball is hit when it is 3.1 feet above the ground. It leaves the bat with an initial speed of 152 ft/sec, making an angle of 22° with the horizontal. Assuming a drag coefficient high does the ball go, and when does it reach maximum height?

64) ______

how

For projectiles with linear drag: x=

(1 -

) cos α,

(1 -

) sin α +

where k is the drag coefficient,

(1 - kt -

)

and α are the projectile's initial speed and launch angle, and g

is the acceleration of gravity (32 ft/ ). A) Maximum height = 3.43 feet; time to maximum height = 0.06 sec C) Maximum height = 3.47 feet; time to maximum height = 0.02 sec

B) Maximum height = 8.67 feet; time to maximum height = 0.74 sec D) Maximum height = 0.17 feet; time to maximum height = -0.21 sec

65) A baseball is hit when it is 3.1 feet above the ground. It leaves the bat with an initial speed of 150 ft/sec, making an angle of 24° with the horizontal. Assuming a drag coefficient range and the flight time of the ball.

65) ______

find the

For projectiles with linear drag: x=

(1 -

)cos α,

(1 -

)sin α +

where k is the drag coefficient,

(1 - kt -

)

and α are the projectile's initial speed and launch angle, and g

is the acceleration of gravity (32 ft/ ). A) Flight time = 2.82 sec; range = 393.5 feet C) Flight time = 3.62 sec; range = 402.2 feet

B) Flight time = 3.97 sec; range = 424.4 feet D) Flight time = 3.74 sec; range = 413.5 feet

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 66) 66) _____________ Prove that 67)

dt = k

dt for any scalar constant k. Pro that ve

___ ___ ___ ___ _

67)

dt =

dt +

dt 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). 68) r(t) = 2 i + 11 j + 10 k, 1 ≤ t ≤ 3 A) 130 B) 1950 C) 390 D) 5850

69) ______

69) r(t) = (5 - 6t)i + (1 + 9t)j + (2t - 1)k, -4 ≤ t ≤ 1 A) 55 B) -33 70)

r(t) = (7 + 2 )i + (2 A) 189

68) ______

C) -363

- 4)j + (1 - )k, -1 ≤ t ≤ 4 B) 195

D) 605 70) ______

C) 585

D) 567 71) ______

71) r(t) = 3ti + A) 15

k; -9≤ t ≤ -6 B) -375

C) -75

D) 75 72) ______

72) r(t) = (2 A) 1

3t)j + (2

3t)k; B) 3

π≤t≤

π C) 6

D) 0 73) ______

73) r(t) = A) 120

i + (4t sin t)j + (4t cos t)k ; -4 ≤ t ≤ 6 B) 80

C) 216

D) 112 74) ______

74) r(t) = (3t sin t + 3 cos t)i + (3 t cos t - 3 sin t)j ; -4 ≤ t ≤ 5 A) 123 B) 27 C) 61.5

D) 13.5

75) Following the curve r(t) = 5ti + (12 cos t)j + (12 sin t)k in the direction of increasing of arc length, find the point that lies 52π units away from the point where A) (20, 12, 0) B) (20π, 0, 12) C) (20π, 0, 0) Find the unit tangent vector of the given curve. 76) r(t) = 4 i - 3 j + 12 k A) T=

D) (20π, 12, 0)

76) ______ B)

D) T=

75) ______

77) ______

77) r(t) = (2 - 2t)i + (2t - 8)j + (4 + t)k A) T=-

C) T= 78)

T=-

D) i-

k 78) ______

r(t) = (9 + 2 )i + (2 + 10 )j + (6 + 11 )k A) T= i+ j+ C) T = 2i + 10j + 11k

i + 2j +

D) j+

k 79) ______

79) r(t) = A)

i +

j - 12tk

i +

j -

i +

j -

i +

j -

i +

j -

B) k

C) D)

80)

80) ______

r(t) = (6 9t)i + (6 9t)j A) T = (6 cos 9t)i - (6 sin 9t)j C) T = (sin 9t)i - (cos 9t)j

B) T = (6 sin 9t)i - (6 cos 9t)j D) T = (162 sin 9t)i -(162 cos 9t)j

81) r(t) = (9t cos t - 9 sin t)j + (9t sin t + 9 cos t)k A) T = - (sin t)j + (cos t)k C) T = (-sin t)j + (cos t)k

81) ______ B) T = (-9 sin t)j + (9 cos t)k D) T = (9 cos t)j - (9 sin t)k

Find the arc length parameter along the curve from the point where t = 0 by evaluating 82) r(t) = (5cos t)i + (5sin t)j + 2tk A) B) C) t t

82) ______ D) 3

t 83)

r(t) = ( cos t)i + ( sin t)j + k A) B) -3 3

84) r(t) = (1 + 4t)i + (1 + 7t)j + (5 - 5t)k A) B) t t 85) r(t) = (3 + 2t)i + (2 + 3t)j + (2 - 6t)k

83) ______ -3

84) ______ C) 3

t 85) ______

A) 5t

B) 9t

C) 8t

Find the length of the indicated portion of the curve. 86) r(t) = (4cos t)i + (4sin t)j + 5tk, 0 ≤ t ≤ π/2 A) B) π 87)

88) r(t) = (1 + 5t)i + (1 + 6t)j + (2 - 2t)k, -1 ≤ t ≤ 0 A) B) 89) r(t) = (3 + 2t)i + (8 + 3t)j + (2 - 6t)k, -1 ≤ t ≤ 0 A) 9 B) 5 Find the principal unit normal vector N for the curve r(t). 90) r(t) = (5 + t)i + (1 + ln(cos t))k, -π/2 < t < π/2 A) N = (-cos t)i - (tan t)k C) N = (-cos t)i - (ln(cos t))k 91)

r(t) = ( A)

86) ______ C)

r(t) = ( cos t)i + ( sin t)j + k, -ln 2 ≤ t ≤ 0 A) B)

D) 7t

π 87) ______

88) ______ C) 2

D) 89) ______

C) 8

D) 7

90) ______ B) N = (-sin t)i - (cos t)k D) N = (cos t)i + (sin t)k 91) ______

+ 4)j + (2t - 5)k

N=-

B) j+

N=-

D) N=

92) r(t) = (5t sin t + 5 cos t)i + (5t cos t - 5 sin t))k A) N=(cos t)i (sin t)k C) N = (-sin t)i - (cos t)k

92) ______ B) N = (cos t)i - (sin t)k D) N = (sin t)i + (cos t)k 93) ______

93) r(t) = (9 sin t)i + (9 cos t)j + 6k A) N = (-sin t)i - (cos t)j C) N = (sin t)i + (cos t)j

B) N = (-sin t)i + (cos t)j D) N = (-cos t)i - (sin t)j

Find the curvature of the curve r(t). 94) r(t) = (2 + ln(sec t))i + (6 + t)k, -π/2 < t < π/2 A) κ = 1 - cos t B) κ = -cos t 95) r(t) = (8 + 8 cos 5t) i - (4 + 8 sin 5t)j + 8k A) B) κ=

94) ______ C) κ = sin t

D) κ = cos t 95) ______

C) κ = 8

D) κ=

κ=

96) r(t) = (9 + cos 6t - sin 6t)i + (9 + sin 6t + cos 6t)j + 4k A) κ = 3 B) κ = 3

96) ______ C) κ =

D) κ=

97)

97) ______

r(t) = (6t + 9)i - 6j + (9 - 3 )k A) B) κ=6 κ=6

D) κ = κ=

Find the curvature of the space curve. 98) r(t) = 12ti + A)

98) ______

k B)

κ=

C) κ=

D) κ=

κ= 99) ______

99) r(t) = (t + 10)i + 6j + (ln(sec t) + 4)k A) κ = sec t B) κ = sin t

C) κ = cos t

D) κ = csc t

100) r(t) = -10i + (t + 1)j +(ln(cos t) + 6)k A) κ = cos t B) κ = sec t

C) κ = sin t

D) κ = csc t

101)

r(t) = -6i + (10 + 2t)j + ( A)

100) _____

101) _____

+ 10)k B)

κ=

D) κ=

κ=-

102) r(t) = (8 t sin t + 8 cos t)i + 8j + (8t cos t - 8 sin t)k A) B) κ = 8t

102) _____ C)

κ=-

D) κ=

κ=

103) _____

103) r(t) = ti + (sinh t)j + (cosh t)k A) B) κ= t

κ=

κ= 104) _____

104) r(t) = A)

B) κ=

C) κ=

κ=

For the curve r(t), write the acceleration in the form 105) r(t) = i+ A) a = T + 144N C) a = 16T

T+

κ=

N. 105) _____

j + 5tk

106) r(t) = (t - 4)i + (ln(sec t) - 4)j + 9k, -π/2 < t < π/2 A) a=( t)T + (cos t)N C) a = (cos t)T + (cos t)N

B) a = 144T + 144N D) a = 16N 106) _____ B) a = (csc t)T + (sec t)N D) a = (sec t tan t)T + (sec t)N

107) _____

107) r(t) = (t + 7)i + (ln(cos t) - 6)j + 9k A) a = (-sec t tan t)T + (sec t)N C) a = (-sec t tan t)T - (sec t)N 108)

r(t) = ( A)

B) a = (sec t tan t)T - (sec t)N D) a = (sec t tan t)T + (sec t)N 108) _____

- 7)i + ( 2t - 10)j + 10k B)

T+

D) a=

T+

109) r(t) = (2t sin t + 2 cos t)i + (2t cos t - 2 sin t)j + 5k A) a = 2tN B) a = 2T + 2tN

109) _____ C)

D) a = 2T +

N 110) _____

110) r(t) = (cosh t)i + (sinh t)j + tk A) a = (sinh t)T + N C) a = (

B) a = (-

sinh t)T + N D) a = (-sinh t)T + N

sinh t)T + N

111) _____

111) r(t) = A)

tk B)

D) a=T+

Find T, N, and B for the given space curve. 112) r(t) = i + A)

112) _____

j + 3tk

(cos 4t)i -

(sin 4t)j +

k ; N = (-sin 4t)i - (cos 4t)j ; B =

(cos 4t)i -

(sin 4t)j -

(cos 4t)i -

(sin 4t)j +

k ; N = (-sin 4t)i - (cos 4t)j ; B = -

(sin 4t)i -

(cos 4t)j ; N = (-sin 4t)i - (cos 4t)j ; B =

(cos 4t)i -

(sin 4t)j -

(cos 4t)i -

(sin 4t)j ; N = (-sin 4t)i - (cos 4t)j ; B =

(cos 4t)i -

(sin 4t)j -

B) C) D)

113) r(t) = (1 + t)i + (7 + ln(sec t))j - 9k, -π/2 < t < π/2 A) T = (cos t)i + (sin t)j; N = (- sin t)i + (cos t)j ; B = k 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

113) _____

114) r(t) = (ln(cos t) + 4)i + 10j + (9 + t )k, -π/2 < t < π/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

114) _____

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 115)

r(t) = ( A)

- 5)i

115) _____

+ (2t - 7)j + 2k

i +

j; N =

i -

j; B = -k

i -

j; N = -

i -

j; B = -k

i +

j; N =

i -

j; B = k

B) C) D) T=

i +

j; N =

i -

j; B = -k

116) r(t) = (3t sin t + 3cos t)i + (3t cos t - 3 sin t)j - 2k A) T = (cos t)i + (sin t)j; N = (-sin t)i + (cos t)j; B = k B) T = (-cos t)i - (sin t)j; N = (sin t)i - (cos t)j; B = -2k 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 = (-sin t)i - (cos t)j; B = -k

116) _____

117) r(t) = (cosh t)i + (sinh t)j + tk A)

117) _____

(tanh t)i +

j +

(tanh t sech t)i +

T=-

(tanh t)i +

T=-

(tanh t sech t)i +

(sech t)k; N = (sech t)i - (tanh t)k;

B) j +

(sech t)k; N = (

t)i - (sinh t)k;

C) j +

(sech t)k; N = (-sech t)i - (sinh t)k ;

118)

r(t) = 4 A)

j +

(sech t)k; N = (-sech t)i - (tanh t)k;

118) _____

j + 3tk

T=6

i-6

j + 3k; N =

j; B = -

j+6

k C)

k; N =

i +

j; B =

k; N =

j; B = -

k; N =

j;B=-

D) T=

k For the curve r(t), find an equation for the indicated plane at the given value of t. 119) r(t) = (8 sin A)

119) _____

t + 9)i + (8 cos 20t) - 6)j + 12tk; osculating plane at t = 2.5π. B)

(x - 9) +

(y - 6) +

(y + 6) +

D) (y + 6) +

120) _____

120) r(t) = (t - 4)i + (ln(sec t) - 3)j + 9k, -π/2 < t < π/2; normal plane at t = 4π. A) x = -4π - 4 B) x = 4π + 4 C) x = 4π - 4

D) x = 4 - 4π 121) _____

121) r(t) = (t + 10)i + (ln(cos t) - 6)j + 9k, -π/2 < t < π.2; rectifying plane at t = A) +

B) (x - (1 + 10)) = 0 C) -

D) 122)

=0 122) _____

r(t) = ( - 7)i + (2t - 3)j + 5k; osculating plane at t = 5. A) z = 5 B) x + y + (z + 5) = 0 C) z = -5 D) x + y + (z - 5) = 0

123) _____

123) r(t) = (8t sin t + 8 cos t)i + (8t cos t - 8 sin t)j + 8k; normal plane at t = 3.5π. A) x + y + z = -8 B) x - y + z = -8 C) y = -8

D) y = 8

124) r(t) = (cosh t)i + (sinh t)j + tk; rectifying plane at t = 0. A) x - y = -1 B) x = -1 C) x = 1

D) x - y = 1

124) _____

125) _____

125) r(t) = A)

j + B)

x-y=-

tk; osculating plane at t = 0. C)

x-y=

Find the torsion of the space curve. 126) r(t) =

i + 3tj +

D) x - y = 0

126) _____ k

B) τ=-

C) τ=

τ=

127) r(t) = (t - 10)i + (ln(sec t) + 5)j - 4k, -π/2 < t < π/2 A) τ = 1 B) τ = 0 128) r(t) = (t - 6)i + (ln(cos t) - 4)j + 10k, -π/2 < t < π/2 A) τ = -1 B) τ = 1 129)

r(t) = (2t - 9)i + ( A) τ = 1

D) τ= 127) _____

C) τ = -1

D) Undefined 128) _____

C) Undefined

D) τ = 0 129) _____

- 3)j + 6k B) Undefined

130) r(t) = (3t sin t + 3 cos t)i + (3t cos t) - 3 sin t)j - 7k A) τ = 0 B) τ = -1

C) τ = 0

D) τ = -1 130) _____

C) Undefined

D) τ = 1 131) _____

131) r(t) = (cosh t)i + tj + (sinh t)k A) B) τ = 0

τ=-

D) τ=-

Find the velocity vector in terms of 132) r = a(8 - cos θ) and A) v = (5a cos θ) C) v = (8a sin θ)

and

τ=

. 132) _____

=5 + 5a(8 - sin θ)

B) v = 5a(8 - cos θ)

+ (5a sin θ)

+ 8a(5 - cos θ)

D) v = (5a sin θ)

+ 5a(8 - cos θ) 133) _____

133) r= A) C)

and

v = 3a

v=3

B) D)

+ 8cos 3t

Find the acceleration vector in terms of 135)

+ 3a

v = 3a

+ 3a 134) _____

134) r = 2 cos 3t and θ = 4t A) v = (6 sin 3t) + 8cos 3t C) v = (-6 sin 3t)

v=3

and

B) v = (-3 sin 3t)

+ 8cos 3t

D) v = (-8 sin 4t)

+ 6cos 4t

. 135) _____

r = a(6 - cos θ) and =4 A) a = 36a( cos θ - 4a + a cos θ)

+ (72a sin θ)

B) a = 16a( cos θ + 6a - a cos θ)

- (32a sin θ)

C) a = 16a(2 cos θ - 6)

+ (32a sin θ) D) a = 16a( cos θ - a + a cos θ) + (32a sin θ) 136) r = 2 cos 4t and θ = 3t A) a = (-50 sin 4t) - 48 cos 4t C) a = (-50 cos 4t) - 48 sin 4t

136) _____ B) a = (-50 cos 3t)

- 48 sin 3t

D) a = (-50 cos 4t)

+ 48 sin 4t

Solve the problem. 137) The orbit of a satellite had a semimajor axis of a = 6955 km. Calculate the period of the satellite. and B) 9250 hr

A) 0.18 sec

C) 96.2 min

D) 124 min

138) The orbit of a satellite had a semimajor axis of a = 8872 km. Calculate the period of the satellite. and B) 12 hr

A) 1.98 hr

C) 138.6 min

139) _____

and B)

4.217 × km

C) 2751 km

2.751 × km

140) The orbit of a satellite had a period of T = 93.11 minutes. Calculate the semimajor axis of the orbit. A) 444 km

140) _____

and B)

6.805 × km

4.440 × km

D) 6805 km

141) A space shuttle is in a circular orbit 250 km above the Earth's surface. Use Kepler's third law( with k

138) _____

D) 78.2 min

139) The orbit of a satellite has a period of T = 1436.8 minutes. Calculate the semimajor axis of the satellite. A) 42,170 km

137) _____

) to find the orbital period of the satellite.(G = 6.67 ×

141) _____

, and Earth's radius is A) T = 9 min C) T = 1 hr

). B) T = 2 hr D) T = 90 min 142) _____

142) Find the values of

in the equation e =

- 1 that make the orbit described by

parabolic. A)

B) =

= 2GM

D) =

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 143) Show that a planet in a circular orbit moves with constant speed. 143) ____________

1) B 2) D 3) C 4) D 5) B 6) D 7) D 8) C 9) C 10) D 11) D 12) D 13) D 14) A 15) C 16) B 17) D 18) B 19) D 20) D 21) D 22) B 23) B 24) C 25) A 26) Given: v and a are orthogonal. Thus v∙a = v∙(dv/dt) = So, v∙v =

= constant and thus

d(v∙v)/dt = 0.

= constant.

27) (f u) =

i +

j) + f

u + f

= 28)

29) C 30) D 31) D 32) D 33) C 34) D 35) B 36) A 37) B 38) B

(

i +

39) C 40) A 41) A 42) C 43) D 44) A 45) D 46) A 47) C 48) D 49) D 50) C 51) B 52) C 53) C 54) A 55) A 56) A 57) Yes: firing angle = 33.1° under these conditions. 58) Flight time to tree: t =

= 1.709 seconds;

height at 1.709 seconds: y = (

sinθ)t -

= 28.66 ft. Tree is 30 ft high

59) A 60) C 61) Immediately after the wind gust, the velocity is v = 135.33i + 63.56j. Flight time ≈ 3.97 sec Range ≈ 517.75 ft 62) x-coordinate:

= 0, so Then, x =

= constant =

dt =

c os α cos α t +

y-coordinate:

= -g , so

y= dt = 63) Problem is separable. x-coordinate:

= -k

dt =

dt = -gt +

dt =

sin α t -

dt =

dt = -

sin α - gt

= -k dt. Integrate to get ln

= Next, x =

= -kt or

α, so At t = 0, x(0) =

(1 -

)cos α.

= y-coordinate: -

+C or C=

= -g-k

= - kw Replace w:

. Let w =

, then

= -k dt. Integrate to get: ln

. Make substitutions to get: = -kt or w =

+ =

Thus, x=

Finally, y =

to get

dt =

+ At t = 0, y = ( y=

. Solve for

(g + k

)+C

or C =

(g + k

)

t +

(g + k

)

dt = -

(g + k

)

t + C.

(g + k

). Thus,

)

1). Also,

= Also,

)(1 -

sin α. Substituting and rearranging:

= cos

y= 64) C 65) C 66)

(1 -

)sin α +

(1 - kt -

dt =

)

dt =

dt + k

dt = k

= k

67) dt =

= 68) C 69) A 70) B

dt +

71) A 72) B 73) B 74) D 75) D 76) B 77) D 78) D 79) D 80) C 81) C 82) B 83) B 84) C 85) D 86) C 87) B 88) B 89) D 90) B 91) C 92) C 93) A 94) D 95) A 96) D 97) D 98) A 99) C 100) A 101) B 102) D 103) D 104) B 105) D 106) D 107) D 108) A 109) B 110) C 111) B 112) A 113) A 114) B 115) D 116) D 117) A 118) D 119) D 120) C 121) A 122) A

123) C 124) C 125) D 126) B 127) B 128) D 129) C 130) A 131) C 132) D 133) A 134) C 135) C 136) C 137) C 138) C 139) A 140) D 141) D 142) A 143) Kepler's first law says that the eccentricity of a planet's elliptical orbit e =

and solving for the speed terms for computing

with e = 0 gives are constant, so

. Since

is also constant.

-1. The eccentricity of a circle is 0,

is a fixed radius for a circular orbit, all three

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the specific function value. 1) Find f(7, 2) when f(x, y) = 5x + 3y - 2. A) 41 B) 34 C) 36 D) 39

2) _______

2) Find f(-3, 5) when f(x, y) = 8y2 - 6xy. A) 167 B) 290 3)

Find f(3, 6) when f(x, y) = A) 3

C) 293

D) 162 3) _______

. B) 9

C) 10

D) 5

C) 6

D) 30

4) _______

4) Find f(100, 3) when f(x, y) = y log x. A) 3 B) 60

5) _______

5) Find f(3, 4) when f(x, y) = A) -

. B)

C) -

D) -

6) _______

6) Find f(9, 3) when f(x, y) = (x + y)3. A) 1728 7)

1) _______

B) 144

Find f(3, 0, 8) when f(x, y, z) = 3 +3 A) 91 B) -55

C) 36 -

Find f(0, 1, -1) when f(x, y, z) = - 9yz + 6x. -8 A) B) 10

D) 756 7) _______

. C) -31

D) -37 8) _______

C) 9

D) -9

Find the domain and range and describe the level curves for the function f(x,y). 9) f(x, y) = 9x + 2y A) Domain: all points in the xy-plane; range: all real numbers; level curves: lines

9) _______ c

>0 B) Domain: all points in the xy-plane; range: real numbers z ≥ 0; level curves: lines c≤0 C) Domain: all points in the xy-plane; range: real numbers z ≥ 0 ; level curves: lines c≥0 D) Domain: all points in the xy-plane; range: all real numbers; level curves: lines 10)

10) ______

f(x, y) = A) Domain: all points in the xy-plane; range: real numbers z ≤ 0; level curves: lines c≤0 B) Domain: all points in the xy-plane; range: all real numbers; level curves: lines ≥0 C) Domain: all points in the xy-plane; range: all real numbers; level curves: lines D) Domain: all points in the xy-plane; range: real numbers z ≥ 0; level curves: lines

11) ______

11) f(x, y) = A) Domain: all points in the xy-plane except (0, 0); range: all real numbers; level curves: ellipses B) Domain: all points in the xy-plane; range: real numbers > 0; level curves: ellipses C) Domain: all points in the xy-plane; range: all real numbers; level curves: ellipses D) Domain: all points in the xy-plane except (0, 0); range: real numbers > 0; level curves: ellipses

12) ______

12) f(x, y) = A) Domain: all points in the xy-plane excluding x = 0; range: real numbers z ≥ 0; level curves: parabolas B) Domain: all points in the xy-plane; range: real numbers z ≥ 0; level curves: parabolas C) Domain: all points in the xy-plane excluding x = 0; range: all real numbers; level curves: parabolas D) Domain: all points in the xy-plane; range: all real numbers; level curves: parabolas

13) f(x, y) = ln (4x + 2y) A) Domain: all points in the xy-plane satisfying 4x + 2y > 0; range: all real numbers; level

13) ______

curves: lines B) Domain: all points in the xy-plane satisfying 4x + 2y > 0; range: real numbers z ≥ 0; level curves: lines C) Domain: all points in the xy-plane; range: all real numbers; level curves: lines D) Domain: all points in the xy-plane satisfying 4x + 2y ≥ 0; range: all real numbers; level curves: lines 14)

14) ______

f(x, y) = ( + ) A) Domain: all points in the xy-plane; range: all real numbers; level curves: circles with centers at (0, 0) B) Domain: all points in the xy-plane; range: real numbers with centers at (0, 0)

≤z≤

; level curves: circles

C) Domain: all points in the xy-plane satisfying

≤ 1; range: real numbers -

≤z≤

≤ 1; range: real numbers -

≤z≤

; level curves: circles with centers at (0, 0) D) Domain: all points in the xy-plane satisfying

; level curves: circles with centers at (0,0) and radii r, 0 < r ≤ 1

15)

16)

f(x, y) = A) Domain: all points in the xy-plane; range: real numbers z > 0; level curves: lines x + y = c B) Domain: all points in the first quadrant of the xy-plane; range: all real numbers; level curves: lines x + y = c C) Domain: all points in the first quadrant of the xy-plane; range: real numbers z > 0; level curves: lines x + y = c D) Domain: all points in the xy-plane; range: all real numbers; level curves: lines x + y = c f(x, y) = A) Domain: all points in the xy-plane; range: all real numbers; level curves: circles with

15) ______

16) ______

centers at B)

Domain: all points in the xy-plane satisfying + ≤ 25; range: real numbers 0 ≤ z ≤ 5; level curves: circles with centers at (0, 0) and radii r, 0 < r ≤ 5 C) Domain: all points in the xy-plane; range: real numbers 0 ≤ z ≤ 5; level curves: circles with centers at (0, 0) and radii r, 0 < r ≤ 5 D) Domain: all points in the xy-plane satisfying + = 25; range: real numbers 0 ≤ z ≤ 5; level curves: circles with centers at (0, 0) and radii r, 0 < r ≤ 5 17) ______

17) f(x, y) = A) Domain: all points in the xy-plane except y = 0; range: all real numbers; level curves: parabolas B)

Domain: all points in the xy-plane; range: all real numbers; level curves: parabolas C) Domain: all points in the xy-plane except y = 0; range: real numbers z ≥ 0 ; level curves: parabolas D) Domain: all points in the xy-plane; range: real numbers z ≥ 0; level curves: parabolas

18)

f(x, y) = 10 -8 A) Domain: all points in the xy-plane; range: all real numbers; level curves: hyperbolas B) Domain: all points in the xy-plane; range: all real numbers; level curves: ellipses C) Domain: all points in the first quadrant of the xy-plane; range: all real numbers; level curves: hyperbolas D) Domain: all points in the xy-plane; range: real numbers z ≥ 0; level curves: ellipses

Match the surface show below to the graph of its level curves. 19)

18) ______

___ ___

19)

20)

___ ___

20)

21)

______ A)

22) ______

22)

23) ______

23)

24) ______

24)

25) ______

25)

26) ______

26)

Sketch the surface z = f(x,y). 27) f(x, y) = A)

28)

f(x, y) = A)

27) ______

28) ______

29)

30)

29) ______

f(x, y) = 3 A)

f(x, y) = A)

30) ______

31)

31) ______

f(x, y) = A)

32) ______

32) f(x, y) =

33)

f(x, y) = 4 A)

34)

33) ______

+2 B)

f(x, y) = 1 -

___ ___

34)

35)

f(x, y) = 2 A)

35) ______

36) ______

36) f(x, y) = 1 - x - 2y A)

Solve the problem. 37) Find an equation for the level curve of the function f(x, y) = 4 point A) 38)

= 12

Find an equation for the level curve of the function f(x, y) = B) y = -81

C) y = 9

= 25

37) ______

=-8

that passes through the

that passes through the +

38) ______

D) y = ± 9

Find an equation for the level curve of the function f(x, y) = point . A) x + y = 5

40)

that passes through the

point . A) y = 81 39)

39) ______

=7 Fin equatio d ann for

___ ___

the level 40) curve of the function f(x, y) =

dt that passes through the point . A)

= 181

= 19

= -19 41) ______

41) Find an equation for the level curve of the function f(x, y) = point A)

. B)

C) x = 4y

≤ 42)

Find an equation for the level surface of the function f(x, y, z) = x + point A)

that passes through the

B) D)

= 36

= 19

Find an equation for the level surface of the function f(x, y, z) = through the point A) + + = 13 C) x + y + z = ± 13

43) ______

that passes

. B) x + y + z = 13 D)

= 169 44) ______

44) Find an equation for the level surface of the function f(x, y, z) = ln point A)

that passes through the

. B) ln

C) =

D) =

=9 45) ______

45) Find an equation for the level surface of the function through the point (π, π, 1) . A) B) C) =1 46)

42) ______

x+ = 30 C) ln(x) + y + z = 36

43)

that passes through the

cos

that passes D) = 2π

cos

= 2π Fin equatio d ann for

___ ___

the level 46) surface of the function f(x, y, z) =

d θ -

d t that passes through the point (0, ln 10, ln 6) . A)

B) +

= -4

C) +

Sketch a typical level surface for the function. 47) f(x, y, z) = A)

48)

= -4

D) -

47) ______ B)

f(x, y, z) =

___ ___

48)

49)

f(x, y, z) = 2 A)

49) ______ B)

50)

f(x, y, z) = x A)

51)

f(x, y, z) = A)

50) ______

51) ______ B)

52)

f(x, y, z) = cos ( A)

53)

f(x, y, z) = A)

52) ______

) B)

53) ______ B)

Find the equation for the level surface of the function through the given point. 54) f(x, y, z) = A)

, (6, 4, 8) B)

= C)

54) ______

= 144

= 55)

f(x, y, z) = , (1, 5, 8) A) ln( + - z) = 18 C) + -z=

55) ______ B)

- z = ln(18)

= ln(18) 56) ______

56) f(x, y, z) = A)

, (5, 4, 4) B) The level surfaces cannot be determined. =

D) =

57)

f(x, = y, z)

___ ___

57)

, (3,

) A) -3

-6

+1=

-4

-5

+1=

-3

-6

+ 1 = ln θ + 1 + t

-3

-6

-8=

C) D)

Provide an appropriate response. 58) Find the extrema of f(x, y, z) = x + yz on the line defined by x = 3(8 + t), y = t - 3, and z = t + 8. Classify each extremum as a minimum or maximum. (Hint: w = f(x, y, z) is a differentiable function of t.) A) (36, 1, 12), minimum; (0, -3, 0), minimum B) (12, -7, 4), minimum; (36, 1, 12), maximum C) (36, 1, 12), minimum D) (12, -7, 4), minimum 59)

Find the value(s) of t corresponding to the extrema of f(x, y, z) = sin(

)cos(z) subject to the

58) ______

59) ______

constraints + = 6t, and z = . Classify each extremum as a minimum or maximum. (Hint: w = f(x, y, z) is a differentiable function of t.) A) B) t=

, minimum; t = 0, maximum

, minimum; t = -

, minimum

, maximum

D) t=-

, maximum

60) ______

60) 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 ≤ ii. The xy-plane iii. The half-space x > 0 iv. Space itself A) ii only B) ii, iii, and iv

C) iv only

D) ii and iv

1) D 2) B 3) A 4) C 5) D 6) A 7) D 8) B 9) D 10) C 11) D 12) C 13) A 14) D 15) A 16) B 17) A 18) A 19) D 20) D 21) D 22) D 23) D 24) B 25) D 26) A 27) C 28) C 29) D 30) D 31) B 32) D 33) A 34) D 35) D 36) B 37) A 38) C 39) B 40) C 41) B 42) B 43) D 44) C 45) C 46) C 47) A 48) A 49) A 50) B 51) A

52) B 53) A 54) B 55) B 56) D 57) A 58) D 59) D 60) D

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit. 1) A) -1

C) 1

1) _______

D) No limit

2) _______

2) A) 143

D) No limit

3) _______

3) ln A) 0

B) -ln 2

C) ln 2

D) No limit 4) _______

4) A) 1

B) ∞

C) 0

D) No limit 5) _______

5) sin A) 1

C) 0

D) No limit

6) _______

B) 2

C) 0

7) _______

A) -

-1

B) 2

C) -2

+1 8) _______

8) A) -7

D) No limit

9) _______

9) x ln y A) ln (3) - 1 10)

B) 3

C) ln 3

D) No limit 10) ______

A) 1

C) 0

D) 11) ______

11)

A) 0

C) 1

D) No limit

12) ______

12)

A) 1

B) 8

C) -4

D) 0 13) ______

13)

A) 0

B) 8

C) 4

D) No limit 14) ______

14)

A) 7

B) 25

C) 0

15) ______

15)

A) 0

B) 4

C) 2

D) No limit 16) ______

16)

B) -4

A) 0

C) ∞

D) 10 17) ______

17) x-

y+z B) 2

A) 5

C) 4

D) 1 18) ______

18) A) 8

B) -2

C) 2

D) - 4 19) ______

19) ln (z A) ln

) B) ln 25

C) 0

D) ln 5 20) ______

20) A) π

C) 0

D) 1

At what points is the given function continuous? 21) f(x, y) = A) All (x, y) C) All (x, y) ≠ (0, 0)

21) ______ B) All (x, y) satisfying x + y ≥ 0 D) All (x, y) in the first quadrant 22) ______

22) f(x, y) = A) All (x, y) ≠ (0, 0) C) All (x, y) such that x ≠ y

B) All (x, y) such that x ≠ - y D) All (x, y) 23) ______

23) f(x, y) = tan (x + y) A) All (x, y) ≠ (0, 0) B) All (x, y) C) All (x, y) ≠ D) All (x, y) such that x + y ≠

, where n is an integer 24) ______

24) f(x, y) = A) All (x, y) satisfying x - y ≠ 0 C) All (x, y) such that x ≠

B) All (x, y) such that x ≠ 0 D) All (x, y)

and x ≠ -2 25) ______

25) f(x, y) = A) All (x, y) such that 10x + 8y ≠ 0 C) All (x, y) such that x + y ≥ 0

B) All (x, y) such that 10x + 8y ≥ 0 D) All (x, y)

26) f(x, y, z) = ln(x + y + z - 8) A) All (x, y, z) such that x + y + z > 8 C) All (x, y, z)

26) ______ B) All (x, y, z) such that x + y + z ≥ 8 D) All (x, y, z) in the first octant 27) ______

27) f(x, y, z) = A) All (x, y, z) ≠ ± (5, -6, 2) C) All (x, y, z)

B) All (x, y, z) ≠ (0, 0, 0) D) All (x, y, z) ≠ (5, -6, 2) 28) ______

28) f(x, y, z) = A) All (x, y, z) such that C) All (x, y, z)

≠4

B) D)

All (x, y, z) such that

≠0

All (x, y, z) such that

≠2 29) ______

29) f(x, y, z) = yz cos A) All (x, y, z) such that 0 ≤ C) All (x, y, z)

B) All (x, y, z) such that x ≠ 0 ≤ 2π D)

All (x, y, z) such that x ≠ 30) ______

30) f(x, y, z) = A) All (x, y, z) C) All (x, y, z) such that x ≥ 0 and y + z ≥ 0

B) All (x, y, z) such that x - y - z ≥ 0 D) All (x, y, z) in the first octant

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). 31) 31) _____________ f(x, y) = 32) _____________

32) f(x, y) =

33) _____________

33) f(x, y) =

34) _____________

34) f(x, y) =

35) _____________

35) f(x, y) =

36) _____________

36) f(x, y) =

37) _____________

37) f(x, y) =

38) _____________

38) f(x, y) = Provide an appropriate response. 39) We say that a function f(x, y, z) approaches the limit L as (x, y, z) approaches (

) and write f(x, y, z) = L if for every number ε > 0, there exists a corresponding number δ > 0 such that for all in the domain of f, Show that the

requirement in this definition is equivalent to

39) _____________

40) 41)

Show that f(x, y, z) =

is continuous at every point (

Show that f(x, y, z) =

40) _____________

41) _____________

is continuous at the origin.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 42)

42) ______

Define f(0,0) in a way that extends f(x, y) = to be continuous at the origin. 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. 43) ______

43) Define f(0, 0) in a way that extends f(x, y) = A) f(0, 0) = 2 B) f(0, 0) = 0

to be continuous at the origin. C) f(0, 0) = 16 D) f(0, 0) = 8

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 44) If f( , 44) _____________ ) = -2 and the limit of f(x, y) exists as (x,y) approaches ( , ), what can you say about the continuity of

at the point

Give reasons for your answer. 45) _____________

45) Does knowing that 5 -

< 5 tell you anything about

Give reasons for your answer. 46) _____________

46) Does knowing that ≤ 1 tell you anything about Give reasons for your answer.

47) _____________

47) Does knowing that ≤ 1 tell you anything about Give reasons for your answer.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0). 48) f(x, y) = A) 2

B) 1

C) 0

48) ______

D) No limit 49) ______

49) f(x, y) = A) 0

B) 1

C) 2

D) No limit 50) ______

50) f(x, y) = cos A) 0

B) 1

N o li mi t 51) ______

51) f(x, y) = A) 0

B) -1

C) 1

D) π 52) ______

52) f(x, y) = A) π

B) 1

D) No limit

53) ______

53) f(x, y) = A) 0

B) π

C) 1

D) No limit

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Give an appropriate answer. 54) Given the function f(x, y) and the positive number ε as in the formal definition of a limit, 54) _____________ find a positive number δ as in the definition that insures

f(x, y) =

< ε.

; ε = 0.02

55) Given the function f(x, y) and the positive number ε as in the formal definition of a limit, find a positive number δ as in the definition that insures

55) _____________

< ε.

f(x, y) = (1 + cos x)(x + y); ε = 0.16 56) Given the function f(x, y) and the positive number ε as in the formal definition of a limit, find a positive number δ as in the definition that insures

56) _____________

< ε.

f(x,y) = x + y ; ε = 0.02 57) Given the function f(x, y) and the positive number ε as in the formal definition of a limit, find a positive number δ as in the definition that insures

f(x, y) =

57) _____________

< ε.

; ε = 0.06

58) Given the function f(x, y, z) and the positive number ε as in the formal definition of a limit, find a positive number δ as in the definition that insures

58) _____________

< ε.

f(x, y, z) = x + y + z ; ε = 0.06 59) Given the function f(x, y, z) and the positive number ε as in the formal definition of a

lim it, find a

___ ___ ___ ___ _

positive 59) number δ as in the definitio n that insures < ε. f(x, y, z) =

x+ y+

z;ε = 0.03 60) Given the function f(x, y, z) and the positive number ε as in the formal definition of a limit, find a positive number δ as in the definition that insures

f(x, y, z) =

60) _____________

< ε.

; ε = 0.04

61) Given the function f(x, y, z) and the positive number ε as in the formal definition of a limit, find a positive number δ as in the definition that insures f(x, y, z) = x + y - z ; ε = 0.09

< ε.

61) _____________

1) B 2) B 3) C 4) A 5) C 6) D 7) B 8) B 9) C 10) D 11) A 12) B 13) B 14) D 15) C 16) B 17) A 18) D 19) D 20) B 21) A 22) B 23) D 24) C 25) B 26) A 27) D 28) D 29) B 30) A 31) Answers will vary. One possibility is Path 1: x = t, y = t ; Path 2: x = 0, y = t 32) Answers will vary. One possibility is Path 1: x = t, y = 0 ; Path 2: x = 0, y = t 33) Answers will vary. One possibility is Path 1: x = t, y = t ; Path 2: x = t, y = 34) Answers will vary. One possibility is Path 1: x = t, y = t ; Path 2: x = t, y = -t 35) Answers will vary. One possibility is Path 1: x = t, y = t ; Path 2: x = t, y = 36) Answers will vary. One possibility is Path 1: x = t, y = t ; Path 2: x = 0, y = t 37) Answers will vary. One possibility is Path 1: x = 0, y = t ; Path 2: x = - , y = t 38) Answers will vary. One possibility is Path 1: x = t, y = t ; Path 2: x = t, y = 2t 39) The sphere centered at ( , , ) of radius δ circumscribes the cube centered at Therefore,

and

imply

Likewise, the cube centered at sphere with radius δ. Thus,

and with sides

implies

with sides 2δ circumscribes the and

The requirements are equivalent. 40) f(x,y,z) = assertion.

which proves the

41) f(x, y, z) =

which proves the

assertion. 42) C 43) D 44) The function is not necessarily continuous at (

). It is continuous only if

45) 5-

= 5. Hence. by the Sandwich Theorem,

46) ≤ 1 implies -1 ≤ sin

≤ 1, which allows us to obtain

and

for

the Sandwich Theorem implies

47) ≤ 1 implies -1 ≤ cos

≤ 1, which allows us to obtain

and

the Sandwich Theorem implies

48) C 49) D 50) D 51) A 52) C 53) A 54) Let δ = 0.01. Then if

< δ and

< δ,

55) Let δ = 0.04. Then if

< δ and

< δ,

56) Let δ = 0.01. Then if

< δ and

< δ,

Let δ = 0.02. Then if

< δ and

< δ,

58) Let δ = 0.02. Then if

< δ and

≤

57)

59)

Let δ =

. Then if

< δ and

≤

< δ, < δ and

= < δ,

60) Let δ = <a>. Then if 61) Let δ = 0.03. Then if ≤

< δ, < δ,

< δ, and

< δ,

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find

and . 1) f(x, y) = 3x - 4 A)

1) _______

-4 B)

= -1;

= -8y - 4

= -8y

= 3x;

= -8y

D) = -8y;

= 3;

2) _______

f(x, y) = A) = 2(2

= 2(2

- 8);

- 8)

B) =6

;

C) = 12

- 8);

= 12

- 8)

= 12

- 8);

= 12

- 8)

3) _______

3) f(x, y) = A) =-

;

= -

B) =

;

C) =-

;

= -

D) =-

;

= 4) _______

4) f(x, y) = ln A)

;

f(x, y) = A)

;

(10x = 20

D) = -ln

;

= -ln

= ln

;

= ln 5) _______

- y)

sin (10x

- y) cos (10x

- y);

= 2sin (10x

- y) cos (10x

- y)

= 2sin (10x

(10x

- y);

= 2sin(10x

- y) cos(10x

- y)

- y) cos C) = 2sin (10x

- y) cos (10x

- y);

= (40x - 2) sin (10x

- y) cos (10x

- y)

D) = 20 6)

sin (10x

- y) cos (10x

- y);

= (40xy - 2) sin (10x

- y) cos (10x

- y) 6) _______

f(x, y) = ln A)

= 0;

= -

= ln y;

= - xln y

= xln y;

D) = ln y;

7) _______

7) f(x, y) = A) =-

;

= -

B) =-

;

= -

C) =-

;

= -

D) =

;

= 8) _______

8) f(x, y) = A)

B) =-

;

= -

C) ;

= -

;

= -

;

= - xy

;

= x

;

= x

(1 - y)

D) =y -9

;

= x

(y - 1)

10) ______

y - 2x B)

=3 C)

f(x, y) = A)

9) _______

f(x, y) = xy A) C)

10)

;

D) =

+ 2xy - 2

;

= -9

+ 3x

;

= -9

- 6x

D) =3

18xy - 2 = -9

;

9x y2

- 6x

;

= -9 2x

Find

, , and . 11) f(x, y, z) = y+ z+x A) = 2xy + ; = + 2yz; 2xz C) = 2xy; = + 2yz; =

11) ______ =

= 2y +

+ 2xz

;

= 2xy +

= ;

+ 2z; =

+ yz;

+ 2x =

xz 12) ______

12) f(x, y, z) = xz A) =z

;

B) ;

C) =z

;

13)

;

=x 13) ______

f(x, y, z) = ln A)

= z ln

;

= z ln

;

= ln (xy)

C) = 14)

;

= z ln

f(x ,y, z) = sin (xy) cos (y ) A) = y cos (xy) cos (y ); sin (y B)

;

= ln (xy)

D) =

;

= ln (xy) 14) ______

= x cos (xy) cos (y

sin (xy) sin (y

);

= -2yz sin (xy)

= x cos (xy) cos (y

sin (xy) sin (y

);

= 2yz sin (xy)

)

= y cos (xy) cos (y

);

sin (y

) C) D)

= y cos (xy) cos (y

);

= x cos (xy) cos (y

= y cos (xy) cos (y

);

sin (y 15)

16)

);

sin (xy) sin (y

= -2yz sin (xy) sin (y )- x cos (xy) cos (y

);

) = 2yz sin (xy)

) 15) ______

f(x, y, z) = z A) =z ; C) = zx ;

f(x,y,z) = x A) =2 B) = (1 + 2 C) = (1 + 2 D) = (1 + 2

;

= zy

= ;

= zy

;

= zx

;

= zy

;

= zx

;

= 16) ______

;

= xy

;

= 2xz

)

;

)

;

= 2xy

;

= 2xz

)

;

=x 17) ______

17) f(x, y, z) = A) =-

;

B) =

;

C) ;

D) =18)

;

f(x, y, z) = cos x A) = sin x B) = -sin x C) = sin x D) = -sin x

;

=18) ______

(yz) (yz);

= - cos x sin (yz) cos (yz);

= -cos x sin (yz) cos (yz)

(yz);

= z cos x sin (yz) cos (yz);

= y cos x sin (yz) cos (yz)

(yz);

= -2z cos x sin (yz) cos (yz);

= - 2y cos x sin (yz) cos (yz)

(yz);

= 2z cos x sin (yz) cos (yz);

= 2y cos x sin (yz) cos (yz) 19) ______

19) f(x, y, z) = A) =-

B) ;

;

C) =

;

D) ;

= ;

; =-

20)

20) ______

f(x, y, z) = A) = (cos x + yz) B) = ; C) = (cos x) D) = (cos x)

;

; ;

;

=y ; ;

=y =

Find all the second order partial derivatives of the given function. 21) f(x, y) = +yA) =2+

;

=2-

;

21) ______

B) ;

C) =2-

;

D) =122)

f(x, y) = cos (x A)

;

=22) ______

)

sin x

;

= 2y

;

2 B)

sin (x

;

= 2[2

cos (x

) - sin (x

)] ;

= 2y[

cos (x

)]

C) =-

sin x

;

= 2[ sin (x

)- 2

cos (x

)] ;

= 2y [sin (x

cos (x

)] D) =- 2y[x

cos x

;

cos (x

= - 2x[2x

) + sin(x

cos (x

) + sin(x

)];

)]; 23) ______

23) f(x, y) = x ln (y - x) A) =

;

= -

;

= -

;

= -

;

= -

24)

f(x, y) = x A)

24) ______

= 2y

(1 + 2

);

= 2y

;

= 2x;

= 2x

= 2y

;

= 2x;

= 2y + 2x

(1 + 2

);

= 2x;

= 2y + 2x

D) = x;

= y+x 25) ______

25) f(x, y) = A) =-

;

B) =

;

C) =-

;

26)

f(x, y) = ln ( A) =

26) ______

y - x)

;

= -

B) =

;

= -

C) =

;

= -

27)

;

= 27) ______

f(x, y) = xy A) =y

;

= 2xy

- 6);

(1 - 2

)

B) =0;

= 2xy

- 3);

(1 -

)

C) =y

;

= 2xy

(

- 1);

(1 -

)

D) = 0; 28)

= 2xy

- 3);

(1 - 2

) 28) ______

f(x, y) = A) =

;

C) ;

D) ;

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the limit definition of the partial derivative to compute the indicated partial derivative of the function at the specified point. 29) 29) _____________ Find

at the point (-10, -4): f(x, y) = 2

+ 2xy + 2 30) _____________

30) Find

at the point (8, -3): f(x ,y) = 5

+ 10xy + 6 31) _____________

31) Find

at the point (6, -5): f(x, y) = 7 - 8xy + 10x 32) _____________

32) Find

at the point (-9, 7): f(x, y) = 3 - 5xy + 10x

33) _____________

33) Find

at the point (10, -1, 4): f(x, y, z) = 7

y+8

+ 9z 34) _____________

34) Find

at the point (-9, -9, -10): f(x, y, z) = 2

y+2

+ 10z 35) _____________

35) Find

at the point (-3, -3, -3): f(x, y, z) = xyz - 10

- 2z 36) _____________

36) Find

at the point (1, 1, 1): f(x, y, z) = xyz - 8

- 4z

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the given function satisfies a Laplace equation. 37) f(x, y) = A) No

B) Yes 38) ______

38) f(x, y) = cos(x) sin(-y) A) No 39)

40)

41)

42)

f(x, y) = A) No

B) Yes 39) ______

sin 3x B) Yes

40) ______

f(x, y, z) = 4x + 7 A) No f(x, y, z) = 10 A) No

-4

B) Yes 41) ______

-6

f(x, y, z) = cos (4x) sin (-6y) A) Yes

Show that the function is a solution of the wave equation. 43) w(x, t) = sin (9x + 9ct) A) No

B) Yes 42) ______ B) No

43) ______ B) Yes 44) ______

44) w(x, t) = ln -5cxt A) No

B) Yes

45) w(x, t) = cos (-4ct) sin (-4x) A) Yes

B) No

46)

37) ______

w(x, t) = A) No

47) w(x, t) = cos (10t - 10cx), c ≠ 1 A) No

45) ______

46) ______ B) Yes 47) ______ B) Yes

48)

w(x, t) = A) Yes

48) ______

cos 5x B) No

Solve the problem. 49) The Van der Waals equation provides an approximate model for the behavior of real gases. The

49) ______

equation is P(V, T) = , where P is pressure, V is volume, T is Kelvin temperature, and a,b , and R are constants. Find the partial derivative of the function with respect to each variable. A) B) =-

;

D) =

;

50) The Redlich-Kwong equation provides an approximate model for the behavior of real gases. The

50) ______

equation is P(V, T) = , where P is pressure, V is volume, T is Kelvin temperature, and a,b , and R are constants. Find the partial derivative of the function with respect to each variable. A) =

;

B) =

;

C) D)

Provide an appropriate response. 51) Which order of differentiation will calculate

faster, x first or y first?

f(x, y) = y ln(x) - 3 sin(x) A) y first 52) Which order of differentiation will calculate

B) x first faster, x first or y first?

f(x, y) = , g(y) ≠ 0 A) y first 53) Which order of differentiation will calculate

52) ______

B) x first faster, x first or y first?

f(x, y) = y+ A) x first 54) For which of the following functions do both

51) ______

53) ______

B) y first and

exis t?

___ ___

54)

A) I. and II.

B) IV. only

C) II., III., and IV.

D) II. and IV.

1) B 2) D 3) D 4) A 5) D 6) C 7) B 8) C 9) B 10) C 11) A 12) D 13) D 14) A 15) D 16) C 17) A 18) D 19) D 20) C 21) B 22) D 23) B 24) A 25) C 26) C 27) D 28) A 29) = = =

-48 + 2h = -48

44 + 6h = 44

30) = = 31) = =

-648 + 60h = -648

32) = = 33)

455 = 455

= =

684 + 8h = 684

324 - 18h = 324

69 - 10h = 69

34) = = 35) = = 36) = = 37) A 38) A 39) B 40) A 41) B 42) B 43) B 44) A 45) A 46) B 47) A 48) B 49) D 50) D 51) A 52) B 53) A 54) D

-3 = -3

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Evaluate A) 4

at t =

π for the function w = B) -2

1) _______

- 4x; x = cost, y = sin t. C) -8 D) -6 2) _______

2) Evaluate A)

at t = 4 for the function w = B) 2

- ln x; x = C)

, y = ln t. D) 3) _______

3) Evaluate A) -

at t =

π for the function w = B)

; x = sin t, y = cost, z = C)

. D) -

4) _______

4) Evaluate A)

at t = 4 for the function w = B)

; x = t, y = t, z = C) 0

. D)

e 5) _______

5) Evaluate A) -1

at (u, v) = (1, 2) for the function z = xy ; x = u - v, y = uv. B) -8 C) 3 D) 1 6) _______

6) Evaluate A) 27

at (u, v) = (1, 3) for the function z = x - ln x; x = B) C) -1 12 - 1 15 - 1

, y = uv. D) -1 7) _______

7) Evaluate A) 40

at (u, v) = (4, 3) for the function w = xz + yz B) 0 C) 48

; x = uv, y = uv, z = u. D) -8 8) _______

8) Evaluate A) 11

at (x, y, z) = (2, 2, 3) for the function u = B) 5 C) 13

- r; p = xy, q =

, r = xz. D) 19 9) _______

9) Evaluate A) 160

at (x, y, z) = (4, 2, 4) for the function u = - r; p = y - z, B) -192 C) 320

D) 640 10) ______

10) Evaluate A) 0

at (x, y, z) = (4, 3, 0) for the function u = cos(r); p = B) C) 324

, D) 108

Write a chain rule formula for the following derivative. 11) for z = f(r, s); r = g(t), s = h(t) A) =

11) ______ B)

D) =

+ 12) ______

12) for w = f(p, q, r); p = g(t), q = h(t), r = k(t) A) =

D) =

+ 13) ______

13) for u = f(v); v = h(s, t) A) =

C) =

D) =

= 14) ______

14) for w = f(p, q); p = g(x, y), q = h(x, y) A)

D) =

+ 15) ______

15) for w = f(x, y, z); x = g(s, t), y = h(s, t), z = k(s) A)

D) =

+ 16) ______

16) for u = f(x); x = g(p, q, r) A)

= C)

D) =

= 17) ______

17) for u = f(p, q); p = g(x, y, z), q = h(x, y, z) A) =

B) =

D) =

+ 18) ______

18) for w = f(x, y, z); x = g(r, s, t), y = h(r, s, t), z = k(r, s, t) A) B) =

D) =

+ 19) ______

19) for w = f(x, y, z); x = g(r, s), y = h(t), z = k(r, s, t) A) =

D) =

= 20) ______

20) for u = f(r, s, t); r = g(y), s = h(z), t = k(x, z) A) B) =

Use implicit differentiation to find the specified derivative at the given point. 21) Find A)

at the point (1, 1) for 5 B) -

21) ______

+ 4xy = 0. C)

D) 22) ______

22) Find A)

at the point (-1, 1) for 2x B)

= 0. C)

D) -

23) ______

23) Find A) 1

at the point (2, 1) for ln x + x B)

+ ln y = 0. C) -1

D) 24) ______

24) Find A) 1

at the point (1, 0) for cos xy + y B) 0

= 0. C) e

25) ______

25) Find A)

at the point (1, -1) for -4x -

y + 6x = 0. B)

D) - 1 26) ______

26) Find A)

at the point (4, 6, 6) for -2 B)

+ 4 ln xz - 7y

-4 C)

= 0. D)

27) ______

27) Find A)

at the point (10, 1, -1) for ln B)

= 0. C)

28) ______

28) Find A)

at the point

for

cos yz = 0.

D) -

29) ______

29) Find A)

at the point (5, 1, 2) for B) -

= 0. C)

30) ______

30) Find at the point (1, 2, ) for ln A) - 27 B)

= 0. C)

D) 27

Provide an appropriate answer. 31) Find A)

31) ______

when r = 4 and s = 3 if w(x, y, z) = xz + y^2, x = 5r + 7, y = r + s, and z = r - s. B) C) D) = 46

= 30

= 26 32) ______

32) Find A)

when u = 3 and v = -2 if w(x, y, z) = B) =

, x = , y = u + v, and z = C)

=33) ______

33) Find A)

when u = 0 and v = B)

if z(x, y) = sin x + cos y, x = u ∙ v, and y = u + v. C) D)

= -1

=2 34) ______

34) Find A)

when u = 5 and v = 3 if z(x) =

and x = u ∙ v.

=0 =

Solve the problem. 35) A simple electrical circuit consists of a resistor connected between the terminals of a battery. The 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 given by P =

35) ______

. Use the equation

= + to find how much the power is changing at the instant when R = 5 ohms, V = 2 volts, A) -0.63 watts

and dV/dt = -0.01 volts/sec. B) 0.65 watts

C) 0.63 watts

D) -0.65 watts 36) ______

36) The radius r and height h of a cylinder are changing with time. At the instant in question, and volume changing at that instant? A) B) 2.73 /sec 0.66

/sec

At what rate is the cylinder's C)

1.04

/sec

0.09

/sec

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 37) If f(x, y, z) is differentiable, x = r - s, y = s - t, and z = t - r, show that 37) _____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 38) For the space curve x = t, y = , z = t , find the points at which the function f(x, y) takes on extreme values if = 36, A) t = 0 C) t = -6 and t = -4

, and

= 48t. B) t = -12 and t = -8 D) t = - 3 and t = - 2 39) ______

39) Find at t = 0 for the function w = sin(x) cos(y) ln(z) where x = 6 -20 A) B) 1 C) -10 40)

38) ______

Let T(x, y) = 4

- 2xy + 16

;

= 56;

, y = t, z = D) -10

and = 16

= = 72

= 120;

= 136

40) ______

be the temperature at the point (x, y) on the ellipse

Find the minimum and maximum temperatures, respectively, on the ellipse. A) B) = 0; =

+ 7t +

41) ______

41) Find A)

(x) if F(x) =

dt. B)

D) x

42) ______

42) Find A)

(x) if F(x) =

dt. B)

dt C)

D) dt - x

dt -

1) A 2) A 3) B 4) C 5) B 6) C 7) A 8) C 9) C 10) D 11) C 12) D 13) D 14) C 15) C 16) D 17) B 18) D 19) A 20) C 21) B 22) A 23) D 24) B 25) B 26) C 27) A 28) B 29) B 30) B 31) A 32) C 33) A 34) C 35) D 36) C 37) This follows from applying the chain rule. 38) C 39) C 40) C 41) B 42) D

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Compute the gradient of the function at the given point. 1) A) ∇f = 56i + 12j B) ∇f = 7i - 2j C) ∇f = 8i + 6j D) ∇f = 56i - 12j

1) _______

2) _______

2) A) ∇f = 48i + 4j

B) ∇f = 384i + 28j

C) ∇f = 192i + 28j

D) ∇f = 48i + 28j 3) _______

3) A)

B) ∇f = -

∇f = -

D) ∇f =

j 4) _______

4) A)

B) ∇f =

∇f =

D) ∇f =

j 5) _______

5) A) ∇f = -5i - 9j + 10k C) ∇f = -15i + 36j - 20k

B) ∇f = -15i - 36j - 20k D) ∇f = 3i + 4j - 2k 6) _______

6) A) ∇f = 8192i + 8192j + 6144k C) ∇f = 8192i + 12,288j + 4096k

B) ∇f = 16,384i + 8192j + 24,576k D) ∇f = 16,384i + 12,288j + 16,384k 7) _______

7) A)

B) ∇f = -

C) ∇f =

D) ∇f =

∇f =

k 8) _______

8) A)

B) ∇f =

∇f =

D) ∇f =

Find the derivative of the function at in the direction of u. 9) f(x, y) = 5x - 8y, (-2, -6), u = 4i - 3j A) 12 B)

9) _______ C)

D) - 4 10)

f(x, y) = 2 A)

(-4, -2), B)

- 4y,

10) ______

u = 3i - 4j C) - 16

D) -

11) f(x, y) = ln(6x - 3y),

(-7, -4), B)

11) ______

u = 6i + 8j D) 0

12) ______

12) f(x, y) = A)

(10, -5), B)

u = 12i - 5j C)

13) f(x, y, z) = 3x - 9y - 6z,

(6, -3, -8), B)

14)

f(x, y, z) = -3x A) - 354,294

13) ______

u = 3i - 6j - 2k C)

14) ______

(-3, -27, 9), u = -2i + j - 2k B) - 531,441 C) - 885,735

D) - 708,588 15) ______

15) f(x, y, z) = A)

16)

(7, 0, 0), B)

f(x, y, z) = ln( A)

-6

-9

), B)

u = 12i - 3j + 4k C)

(-6, -6, -6),

16) ______

u = 3i + 4j C)

Provide an appropriate response. 17) Find the direction in which the function is increasing most rapidly at the point f(x, y) = x A)

-y

17) ______

18) ______

(1, -2) B)

D) i+

18) Find the direction in which the function is increasing most rapidly at the point f(x, y) = x A)

- ln(x),

(4, 0) B)

i+ C)

D) i

19) Find the direction in which the function is increasing most rapidly at the point f(x, y, z) = xy - ln(z), A)

20) ______

(-4i + 4j - k) D)

(-4i - 4j + k)

(-4i + 4j - k)

20) Find the direction in which the function is increasing most rapidly at the point f(x, y, z) = x A)

(1, 1, 2) B)

Solve the problem. 21) Find the derivative of the function which the function increases most rapidly. A) 3 B)

24)

23)

19) ______

(2, -2, 2)

(-4i + 4j - k)

22)

Find the derivative of the function which the function decreases most rapidly. A) -3 B) -

at the point

in the direction in

21) ______

at the point

in the direction in

C) -

Find the derivative of the function the function increases most rapidly. A) 10 B) 20

at the point

Find the derivative of the function the function decreases most rapidly. A) -7 B) -8

at the point

22) ______

D) in the direction in which

C) 11

23) ______

D) 30 in the direction in which

C) -16

24) ______

D) -24 25) ______

25) Find the derivative of the function which the function increases most rapidly. A) B)

at the point C)

in the direction in D)

26) ______

26) Find the derivative of the function which the function decreases most rapidly. A) B) -

at the point C)

27) Find the derivative of the function direction in which the function increases most rapidly.

in the direction in D)

at the point

in the

27) ______

28) Find the derivative of the function

at the point

the direction in which the function decreases most rapidly. A) B) C) -

in D)

29) ______

29) Find the derivative of the function direction in which the function increases most rapidly. A) B) C)

at the point

in the D)

30) ______

30) Find the derivative of the function direction in which the function decreases most rapidly. A) B) C) -

at the point

in the D)

31) ______

31) Write an equation for the tangent line to the curve A) B) +

32) Write an equation for the tangent line to the curve A) 3x + 5y = 30 33)

34)

28) ______

B) 5x + 3y = 15

Write an equation for the tangent line to the curve A) x + y = 1 B) x - y + 2 = 0 Write an equation for the tangent line to the curve A) x - 2 y + 21 = 0 C) x - 2y + 1 = 0

at the point C)

D) +

at the point C) 3x + 5y = 15

C) y = x - 2

32) ______ D) 5x + 3y = 30

at the point D) y = x + 1 at the point

B) x - 2

y+1=0 D) x - 2y + 21 = 0

33) ______

34) ______

1) B 2) A 3) D 4) C 5) A 6) D 7) A 8) D 9) B 10) D 11) C 12) A 13) B 14) B 15) B 16) B 17) B 18) D 19) D 20) C 21) C 22) C 23) A 24) B 25) B 26) B 27) C 28) B 29) A 30) D 31) C 32) A 33) B 34) A

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Find the equation for the tangent plane to the surface at the point A) 5x - 3y + 10z = 12 B) 5x + 3y + 5z = 12 C) 5x + 3y + 5z = 13 D) 5x - 3y + 10z = 13 2) Find parametric equations for the normal line to the surface A) C) 3)

Find parametric equations for the normal line to the surface

Find the equation for the tangent plane to the surface

10)

B) x - y = 0

C) z = 0

Find parametric equations for the normal line to the surface

5) _______

at the point

6) _______

at the point

7) _______

D) x + y + z = 0 at the point

8) _______

B) D)

Find the equation for the tangent plane to the surface A) z = -1 B) z = 1 C) z = 2

Find parametric equations for the normal line to the surface A) C)

at the point

B) D)

Find the equation for the tangent plane to the surface

A) C)

4) _______

B) x + y + z = 1 D) -7x - 7y + 9z = 5

Find parametric equations for the normal line to the surface

A) x + y = 0 8)

at the point

B) D)

A) C) 7)

3) _______

Find the equation for the tangent plane to the surface at the point A) 32x - 16y - z = 14 B) 2x + y + 24z = 27 C) 2x + y + 24z = 1 D) 32x - 16y - z = 24

A) x + y + z = 5 C) -7x - 7y + 9z = 1 6)

2) _______

B) D)

A) C) 5)

at the point

1) _______

B) D)

at the point D) z = 0

at the point

9) _______

10) ______

11) ______

11) Write parametric equations for the tangent line to the curve of intersection of the surfaces and

at the point

A) C)

B) D) 12) ______

12) Write parametric equations for the tangent line to the curve of intersection of the surfaces and

at the point

A) C)

B) D) 13) ______

13) Write parametric equations for the tangent line to the curve of intersection of the surfaces and

at the point

A) C)

B) D) 14) ______

14) Write parametric equations for the tangent line to the curve of intersection of the surfaces and

at the point

A) C)

B) D)

15) About how much will

change if the point

a distance of A)

16)

About how much will distance of A) 2400

unit in the direction C)

17) About how much will

C) 1600

moves from

unit in the direction of C)

16) ______

change if the point

a distance of A)

change if the point unit in the direction of B)

15) ______

moves from

17) ______

18) ______

18) About how much will a distance of A)

change if the point unit in the direction of B)

Find the linearization of the function at the given point. 19) at

moves from

D) -

19) ______

A) C)

B) D)

20)

20) ______

at A) C)

B) D)

21)

21) ______

at A) C)

B) D)

22)

22) ______

at A) C)

B) D) 23) ______

23) at A)

24)

24) ______

at A) C)

B) D)

25)

25) ______

at A) C)

B) D) 26) ______

26) at A)

L(x, y, z) = - 3x + 5y +

27)

27) ______

at A) L(x, y, z) =

L(x, y, z) =

z + ln 3 - 1

27 -

B) y+

C) y+

D) y+

27 -

28)

28) ______

at A)

L(x, y, z) =

y - 3z

L(x, y, z) =

y - 3z + 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. 29) 29) ______ at ; R: |x + 4| ≤ 0.1, |y + 6| ≤ 0.1 A) |E| ≤ 0 B) |E| ≤ 0.02 C) |E| ≤ 0.04 D) |E| ≤ 0.01 30)

; R: |x + 9| ≤ 0.1, |y - 9| ≤ 0.1 B) |E| ≤ 0.16 C) |E| ≤ 0.32

A) |E| ≤ 0.08 31)

32)

at A) |E| ≤ 12.16512

; R: |x - 2| ≤ 0.2, |y - 1| ≤ 0.2 B) |E| ≤ 17.47104 C) |E| ≤ 17.23392

at A) |E| ≤ 9.7335

; R: |x| ≤ 0.1, |y| ≤ 0.1 B) |E| ≤ 8.5168

33)

D) |E| ≤ 0.24 31) ______ D) |E| ≤ 22.30272 32) ______

C) |E| ≤ 4.8667

D) |E| ≤ 17.0336 33) ______

A) |E| ≤ 0.0089

; R: |x - 1| ≤ 0.1, |y - 1| ≤ 0.1 B) |E| ≤ 0.0139 C) |E| ≤ 0.0073

A) |E| ≤ 0.35

at ; R: |x - 1| ≤ 0.1, |y - 1| ≤ 0.1, |z - 1| ≤ 0.1 B) |E| ≤ 0.375 C) |E| ≤ 0.42 D) |E| ≤ 0.45

A) |E| ≤ 0.8

at ; R: |x - 1| ≤ 0.1, |y + 2| ≤ 0.1, |z - 3| ≤ 0.1 B) |E| ≤ 0.6 C) |E| ≤ 1.2 D) |E| ≤ 0.9

A) |E| ≤ 0.0148

at (1, 1, 1); R: |x - 1| ≤ 0.1, |y - 1| ≤ 0.1, |z - 1| ≤ 0.1 B) |E| ≤ 0.0138 C) |E| ≤ 0.0185 D) |E| ≤ 0.0123

34)

35)

36)

37)

30) ______

at A) |E| ≤ 0.00000985 C) |E| ≤ 0.00001847

38)

; R: |x| ≤ 0.1, |y| ≤ 0.1, |z| ≤ 0.1 B) |E| ≤ 6.8328 C) |E| ≤ 7.3209

34) ______

35) ______

36) ______

37) ______

; R: |x - 3| ≤ 0.2, |y - 3| ≤ 0.2, |z - 3| ≤ 0.2 B) |E| ≤ 0.00001477 D) |E| ≤ 0.00001108 at

A) |E| ≤ 8.4597

D) |E| ≤ 0.011

38) ______ D) |E| ≤ 7.8089

Solve the problem. 39) If the length, width, and height of a rectangular solid are measured to be 9, 6, and 3 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) 1237.50 B) 990.00 C) 825.00 D) 1100.00 40) If the length, width, and height of a rectangular solid are measured to be 7, 2, and 7 inches respectively and each measurement is accurate to within 0.1 inch, estimate the maximum percentage error in computing the volume of the solid. A) 7.07% B) 6.29% C) 9.43% D) 7.86%

39) ______

40) ______

41) The resistance R produced by wiring resistors of

and

ohms in parallel can be

41) ______

calculated from the formula

If and are measured to be 10 ohms and 8 ohms respectively and if these measurements are accurate to within 0.05 ohms, estimate the maximum possible error in computing R. A) 0.015 B) 0.030 C) 0.020 D) 0.025 42) The resistance R produced by wiring resistors of

and

ohms in parallel can be

42) ______

calculated from the formula

If and are measured to be 9 ohms and 4 ohms respectively and if these measurements are accurate to within 0.05 ohms, estimate the maximum percentage error in computing R. A) 1.66% B) 1.24% C) 2.07% D) 1.04% 43) The resistance R produced by wiring resistors of

, and

ohms in parallel can be

43) ______

calculated from the formula

If , , and are measured to be 4 ohms, 7 ohms, and 8 ohms respectively, and if these measurements are accurate to within 0.05 ohms, estimate the maximum possible error in computing R. A) 0.018 B) 0.011 C) 0.015 D) 0.022 44) The resistance R produced by wiring resistors of

, and

ohms in parallel can be

44) ______

calculated from the formula

If , , and are measured to be 7 ohms, 6 ohms, and 9 ohms respectively, and if these measurements are accurate to within 0.05 ohms, estimate the maximum percentage error in computing R. A) 0.86% B) 0.72% C) 0.58% D) 0.43% 45) ______

45) The surface area of a hollow cylinder (tube) is given by where h is the length of the cylinder and

and

are the outer and inner radii. If h,

, and are measured to be 7 inches, 3 inches, and 8 inches respectively, and if these measurements are accurate to within 0.1 inches, estimate the maximum possible error in computing S. A) 13.19 B) 15.08 C) 11.31 D) 16.96 46) The surface area of a hollow cylinder (tube) is given by

whe is the re length h of the

___ ___

cylinder 46) and and are the outer and inner radii. If h,

and are measure d to be 5 inches, 2 inches, and 9 inches respectiv ely, and if these measure ments are accurate to within 0.1 inches, estimate the maximu m percenta ge error in computi ng S. A) -0.100%

B) -0.118%

C) -0.091%

D) -0.082%

1) B 2) C 3) D 4) C 5) D 6) C 7) C 8) B 9) B 10) B 11) A 12) C 13) A 14) C 15) C 16) D 17) A 18) D 19) B 20) A 21) D 22) C 23) A 24) C 25) C 26) D 27) D 28) C 29) A 30) C 31) D 32) C 33) D 34) D 35) D 36) D 37) B 38) C 39) B 40) D 41) D 42) D 43) A 44) B 45) B 46) C

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find all the local maxima, local minima, and saddle points of the function. 1) A) f(-10, 8) = 102, saddle point C) f(10, -8) = 246, saddle point

1) _______

B) f(10, 8) = 502, local maximum D) f(-10, -8) = -154, local minimum 2) _______

2) A) f

= - 10, saddle point; f

= - 10, local minimum; f

= 30, local maximum

= - 10, saddle point

B) = - 10, local minimum

C) D)

3) _______

3) A) f(2, 0) = 7, local minimum; f(0, 2) = 7, local minimum B) f(2, 2) = 39, local maximum C) f(0, 0) = 7, test is inconclusive D) f(0, 0) = 7, saddle point; f(2, 2) = 39, local maximum

4) _______

4) A) f(0, 0) = 0, saddle point; f(8, 8) = 640, local maximim B) f(0, 0) = 0, saddle point C) f(8, 0) = 64, local minimum; f(0, 8) = 64, local minimum D) f(8, 8) = 640, local maximim

5) _______

5) A) f(-8, -5) = 1276, local maximum B) f(8, 5) = -1272, local minimum; f(8, -5) = -772, saddle point; f(-8, 5) = 776, saddle point; local maximum C) f(8, -5) = -772, saddle point; f(-8, 5) = 776, saddle point D) f(-8, -5) = 1276, local maximum; f(8, 5) = -1272, local minimum

6) _______

6) A) f(0, 0) = 2, local maximum; f(2, 2) = -254, local minimum B) f(2, 2) = -254, local minimum C) f(2, 0) = 2, saddle point; f(0, 2) = 2, saddle point D) f(0, 0) = 2, local maximum

7) _______

f = , local minimum C) f(0, 0) = 0, saddle point

f = , local minimum D) f(32, 32) = 393,216, local maximum 8) _______

A) f(54, 54) = 419,904, local maximum; f(0, 0) = 0, local minimum B) f(6, 9) = 10,773, saddle point; f(9, 6) = 8748, saddle point C) f(54, 54) = 419,904, local maximum D) f(0, 0) = 0, local minimum 9) _______

9) A) f(0, 0) = 97, local maximum; f(0, 2) = 81, saddle point; f(3, 0) = 16, saddle point; f(3, 2) = 0, local minimum; f(-3, -2) = 0, local minimum B) f(0, 0) = 97, local maximum; f(0, 2) = 81, saddle point; f(0, -2) = 81, saddle point; f(3, 0) = 97, saddle point; f(3, 2) = 0, local minimum; f(3, -2) = 0, local minimum; f(-3, 0) = 16, saddle point; f(-3, 2) = 0, local minimum; f(-3, -2) = 0, local minimum C) f(0, 0) = 97, local maximum; f(3, 2) = 0, local minimum; f(3, -2) = 0, local minimum; local minimum; f(-3, -2) = 0, local minimum D) f(0, 0) = 97, local maximum; f(-3, -2) = 0, local minimum

10) ______

10) A) f(0, 4) = 1296, local maximum; f(0, -4) = 1296, local maximum; f(6, 0) = -256, local minimum; f(-6, 0) = -256, local minimum B) f(0, 0) = 1040, saddle point; f(0, 4) = 1296, local maximum; f(0, -4) = 1296, local maximum; f(6, 0) = -256, local minimum; f(6, 4) = 0, saddle point; f(6, -4) = 0, saddle point; f(-6, 0) = -256, local minimum; f(-6, 4) = 0, saddle point; f(-6, -4) = 0, saddle point C) f(0, 0) = 1040, saddle point; f(0, 4) = 1296, local maximum; f(6, 0) = -256, local minimum D) f(0, 0) = 1040, saddle point; f(6, 4) = 0, saddle point; f(6, -4) = 0, saddle point; f(-6, 4) = 0, saddle point; f(-6, -4) = 0, saddle point Find the absolute maxima and minima of the function on the given domain. 11) on the closed triangular region with vertices A) Absolute maximum: 5 at absolute minimum: 0 at

12)

13)

14)

B) Absolute maximum: 15 at C) Absolute maximum: 10 at

absolute minimum: 5 at

D) Absolute maximum: 10 at

absolute minimum: 0 at

11) ______

and

absolute minimum: 5 at

on the closed triangular region bounded by the lines A) Absolute maximum: 132 at B) Absolute maximum: 132 at

absolute minimum: 108 at

C) Absolute maximum: 108 at D) Absolute maximum: 108 at

absolute minimum: 0 at

and

absolute minimum: 0 at absolute minimum: 48 at 13) ______

on the disk bounded by the circle A) Absolute maximum: 90 at

and

absolute minimum: 36 at

B) Absolute maximum: 90 at C) Absolute maximum: 126 at

and

absolute minimum: 0 at

D) Absolute maximum: 36 at

and

on the square

12) ______

and

absolute minimum: 0 at absolute minimum: 0 at 14) ______

A) Absolute maximum: 300 at

and

absolute minimum: 100 at

and B) Absolute maximum: 100 at C) Absolute maximum: 300 at

and

absolute minimum: 0 at

and

absolute minimum: 0 at

D) Absolute maximum: 100 at

and

absolute minimum: 75 at

and 15)

on the trapezoidal region with vertices A) Absolute maximum: 9 at B) Absolute maximum: 9 at

absolute minimum: 5 at

C) Absolute maximum: 8 at D) Absolute maximum: 10 at

absolute minimum: 0 at

16)

15) ______

and

absolute minimum: 0 at absolute minimum: 5 at

on the trapezoidal region with vertices

and

A) Absolute maximum: 18 at B) Absolute maximum: 19 at

absolute minimum: 9 at

C) Absolute maximum: 19 at D) Absolute maximum: 20 at

absolute minimum: 0 at

17)

absolute minimum: 9 at absolute minimum: 0 at 17) ______

on the rectangular region A) Absolute maximum:

absolute minimum:

B) Absolute maximum:

absolute minimum:

C) Absolute maximum: D) Absolute maximum:

absolute minimum:

18)

18) ______

on the diamond-shaped region A) Absolute maximum:

and

minimum: 0 at B) Absolute maximum: 50 at

absolute

and

minimum: 0 at C) Absolute maximum:

and D) Absolute maximum:

minimum: 50 at

and

absolute

absolute minimum: 0 at and

absolute

and

Find the absolute maximum and minimum values of the function on the given curve. 19) Function: curve: , (Use the parametric equations A) Absolute maximum: 5

absolute minimum: -5 at

B) Absolute maximum: 5 at

absolute minimum: -5 at

C) Absolute maximum: 5 D)

16) ______

absolute minimum: -5

19) ______

Absolute maximum: 5 at 20)

Function:

absolute minimum: -5 curve:

at (Use the parametric equations

20) ______

A) Absolute maximum:

absolute minimum: 0 at

and

Absolute maximum:

absolute minimum: 0 at

and

Absolute maximum:

absolute minimum: 0 at

and

Absolute maximum:

absolute minimum: 0 at

and

B) C) D)

21) ______

21) Function:

curve:

(Use the parametric equations

A) Absolute maximum: 15 at

absolute minimum: - 15 at

Absolute maximum:

absolute minimum: - 15 at

B) at

C) Absolute maximum: 15 at

absolute minimum: -

Absolute maximum:

absolute minimum: -

D) at

22) ______

22) Function:

curve:

(Use the parametric

equations A) Absolute maximum: 100 at

absolute minimum: 25 at

Absolute maximum: 100 at

absolute minimum: 50 at

Absolute maximum: 200 at

absolute minimum: 50 at

Absolute maximum: 200 at

absolute minimum: 25 at

B) C) D)

23)

Function: curve: A) Absolute maximum: 201 at B) Absolute maximum: 202 at C) Absolute maximum: 202 at D) Absolute maximum: 201 at

Solve the problem.

23) ______ absolute minimum: 11 at absolute minimum: 2 at absolute minimum: 11 at absolute minimum: 2 at

24) Find the least squares line through the points A) y = 6x - 14

B) y = -10x - 18

25) Find the least squares line through the points A) y = 39x + 50

B) y = 39x + 82

24) ______

and C) y = 6x - 18

D) y = -10x - 14 25) ______

and C) y = -15x + 82

D) y = -15x + 50

26) 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.

A) y = 8.13x - 12.79 C) y = 7.97x - 13.54

26) ______

B) y = 7.87x - 13.08 D) y = 8.13x

27) Amarillo Motors manufactures an economy car called the Citrus, which is notorious for its 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..

A) ln(V) = -0.303a + 2.953 C) ln(V) = 0.315a

27) ______

B) ln(V) = 0.299a - 2.951 D) ln(V) = 0.314a - 2.986

Provide an appropriate response. 28) Find any local extrema (maxima, minima, or saddle points) of

given that

28) ______

and A)

B) Local maximum at

C) Saddle point at

Local minimum at D) Saddle point at

29) Find any local extrema (maxima, minima, or saddle points) of and A)

Lo cal

given that

29) ______

maximum at

C) Saddle point at

Lo cal mi ni m u m at

D) Local minimum at

30) Find any local extrema (maxima, minima, or saddle points) of

given that

30) ______

and A)

B) Saddle point at Local minimum at

C) Local maximum at

D) Local maximum at

31) Find any local extrema (maxima, minima, or saddle points) of

given that

and A) Local maximum at

B) Local minimum at

C) Local maximum at

D) Saddle point at

32) Find any local extrema (maxima, minima, or saddle points) of

given that

31) ______

32) ______

and A) Local maximum at

saddle point at

B) Local minimum at

saddle point at

C) Local minimum at

saddle point at

D) Local maximum at

saddle point at 33) ______

33) Determine whether the function has a maximum, a minimum, or neither at the origin. A) Maximum B) Minimum 34) Determine the point on the plane A) (2, -1, 3) 35)

B) (-2, 1, -3)

Determine the point on the paraboloid A) (2, 3, 80)

B) (-2, 3, 80)

C) Neither that is closest to the point

C) (-2, -1, -3)

D) (2, 1, -3)

that is closest to the point C) (3, -2, 50)

34) ______

D) (3, 2, 50)

35) ______

1) D 2) D 3) C 4) B 5) B 6) D 7) C 8) D 9) B 10) B 11) D 12) B 13) B 14) C 15) B 16) C 17) C 18) A 19) A 20) C 21) A 22) D 23) B 24) A 25) D 26) C 27) A 28) D 29) D 30) C 31) B 32) C 33) B 34) A 35) C

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the extreme values of the function subject to the given constraint. 1) A) Minimum: B) Minimum: C) Minimum: D) Minimum:

6 at (±1, 0); maximum: 6 at (0, ±1); maximum: 6 at (±1, 0); maximum: 6 at (0, ±1); maximum:

1) _______

0 at (0, 0) 7 at (±1, 0) 7 at (0, ±1) 0 at (0, 0) 2) _______

2) A) Maximum: 100 at

and

B) Maximum: 100 at C) Maximum: 100 at

minimum: 0 at

D) Maximum: 100 at

minimum: -100 at

and

minimum: -100 at

and

minimum: -100 at

and

3) _______

3) A) Maximum: 192 at minimum: -192 at B) Maximum: none; minimum: 192 at C) Maximum: 192 at minimum: 0 at D) Maximum: none; minimum: 0 at

4) _______

4) A) Maximum: 36 at B) Maximum: 36 at C) Maximum: 72 at D) Maximum: 72 at

minimum: -72 at minimum: -36 at minimum: -72 at minimum: -36 at 5) _______

5) A) Maximum: 36 at B) Maximum: 26 at

minimum: 0 at

C) Maximum: 26 at D) Maximum: 36 at

minimum: 0 at

minimum: -26 at minimum: -36 at 6) _______

6) A) Maximum: 18 at B) Maximum: 18 at

minimum: -18 at

C) Maximum: 4 at D) Maximum: 4 at

minimum: -4 at

minimum: -18 at minimum: -4 at 7) _______

7) A) Maximum: 7 at B)

minimum: -2 at

Maximum: 4 at

minimum: -2 at

C) Maximum: 7 at

minimum: -5 at

D) Maximum: 4 at

minimum: -5 at 8) _______

8) A) Maximum: 8 at B) Maximum: 4 at

minimum: -31 at

C) Maximum: 4 at D) Maximum: 8 at

minimum: -31 at

minimum: -4 at minimum: -4 at 9) _______

9) A) Maximum: 3 at

minimum: -3 at

B) Maximum: 3 at

minimum: -3 at

C) Maximum: 3 at

minimum: -3 at

Maximum: 3 at

and

D) minimum: -3 at

and

10) ______

10) A) Maximum: none; minimum: 30 at B) Maximum: 30 at minimum: 24 at C) Maximum: none; minimum: 24 at D) Maximum: 51 at

minimum: 30 at 11) ______

11) A) Maximum: 8 at

minimum: -8 at

B) Maximum: 8 at C) Maximum: 8 at

minimum: 0 at minimum: -8 at

D) Maximum: 8 at

minimum: 0 at 12) ______

12) A) Maximum: 1 at B) Maximum: 9 at C) Maximum: 8 at D) Maximum: 1 at

minimum: -1 at minimum: -9 at minimum: -8 at minimum: -1 at 13) ______

13) A)

Maximum: none;

minimum:

B) Maximum: none; minimum:

C) Maximum: none; minimum: 2 at D) Maximum: none; minimum:

at 14) ______

14) A) Maximum: 49 at B) Maximum: 77 at

minimum: 41 at

C) Maximum: 81 at D) Maximum: 77 at

minimum: 9 at

minimum: 13 at minimum: 13 at 15) ______

15) 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 B) Maximum: 9 at C) Maximum: 6 at

minimum: 3 at

D) Maximum: 6 at

minimum: 3 at

minimum: -3 at

16) ______

16) A) Maximum: none; minimum:

Maximum: none; minimum:

B) C) D)

17) ______

17) A) Maximum: none; minimum:

Maximum: none; minimum:

B) C) D)

18) ______

18) A) Maximum: none; minimum:

B) Maximum: none; minimum:

C) Maximum: none; minimum:

Maximum: none; minimum:

Solve the problem. 19) Find the points on the curve A) (4, ±4 ) B) (4, 4 20) Find the point on the line A)

that is closest to the origin. C)

that is closest to the point C)

21) Find the point on the line A)

22)

)

19) ______

that are closest to the origin. C) (64, ± )

A rectangular box with square base and no top is to have a volume of 32 amount of material required? A) B) C) 48 36 40

D) (64,

) 20) ______

21) ______ D)

. What is the least D)

23) A rectangle with sides parallel to the axes is inscribed in the region bounded by the axes and the line A)

Find the maximum area of this rectangle. B) C)

24) Find the point on the plane A) (2, 4, -2)

B) (4, 4, 0)

25) Find the distance from the point A)

26)

Find the point on the sphere A) C)

27)

Find the point on the sphere A) C)

that is nearest the origin. C) (-2, 8, 2) to the plane C)

that is closest to the point B)

22) ______

23) ______

24) ______ D) (2, 4, 0) 25) ______ D)

26) ______

that is farthest from the point B) D)

27) ______

28)

29)

A rectangular box is to be inscribed inside the ellipsoid possible volume for the box. A) 16 B) 18 C) 15

Find the largest D) 12

Find the extreme values of

and

subject to

28) ______

29) ______

A) Maximum: none; minimum:

Maximum: none; minimum:

B) C) D)

30)

31)

Find the extreme values of A) Maximum: at

subject to minimum: -

B) Maximum:

minimum: -

C) Maximum:

minimum: -

D) Maximum:

minimum: -

Find the extreme values of A)

30) ______

and

subject to

Maximum:

minimum: -

Maximum: 4

minimum: - 4

Maximum: 2

Maximum: 3

31) ______

and at

B) at

C) minimum: - 2

D) minimum: - 3

32) Find the point on the line of intersection of the planes is closest to the origin. A)

33)

34)

Find the maximum value of A) 3 + B) 3 + Find the extreme values of A) Maximum: at B) Maximum: at C) Maximum:

D) Maximum:

at and

that

subject to C) 3 -

and D) 3 -

subject to minimum: minimum: minimum: minimum:

and at at at at

32) ______

33) ______

34) ______

35)

Find the point on the curve of intersection of the paraboloid A) (1, -1, 1)

36)

that is closest to the origin. B) (1, 1, 1)

C) (1, 1, -1)

Find the point on the curve of intersection of the paraboloid A) (2, -2, 2)

that is farthest from the origin. B) (-2, -2, 2)

C) (2, -2, -2)

and the plane

35) ______

D) (-1, 1, 1) and the plane

36) ______

D) (-2, 2, 2)

Answer the question. 37) The graph below shows the level curves of a differentiable function f(x, y) (thin curves) as well

37) ______

as the constraint g(x, y) = = 0 (thick circle). Using the concepts of the orthogonal 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) (1.3, 0.7), (-1.3, 0.7), (-1.3,-0.7), (1.3,-0.7) C) (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) D) (1.5, 0), (0, 1.5), (-1.5, 0), (0, -1.5) 38) 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 perpendicular to the mountain's contour line which passes through that point.

38) ______

C) At the bottom of the dip, the trail is headed in the direction of the mountain's steepest ascent. D) The mountainside rises in all directions relative to the dip. 39) 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, 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. 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, 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. 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.

39) ______

40) Consider a function f(x, y, z), where the independent variables are constrained to lie on the

40) ______

curve (t) = x(t)i + y(t)j + z(t)k. What mathematical fact forms the basis for the method of Lagrange multipliers? A) f approaches a local extremum as λ → 0. B) ∇g = 0 C) D) ∇f∙∇g = 0 ∇f∙ = 0

1) B 2) A 3) B 4) B 5) B 6) D 7) C 8) B 9) D 10) C 11) A 12) B 13) A 14) C 15) A 16) A 17) D 18) D 19) A 20) D 21) B 22) A 23) C 24) A 25) C 26) B 27) A 28) A 29) B 30) D 31) D 32) C 33) A 34) C 35) D 36) C 37) B 38) A 39) D 40) C

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use Taylor's formula to find the requested approximation of near the origin. 1) Quadratic approximation to A)

8x + y + 4

8x + y + 4 D) 8x + y

C) 8x + y + 4

xy + 2) _______

2) Cubic approximation to A)

B) 6x + y - 36

- 18

y - 3x

C) -

6x + y - 72

- 18

x + 9y +

y - 3x

3) _______

Quadratic approximation to A)

1 + x + 9y +

+ 9xy +

1 + x + 9y +

D) x + 9y +

6x + y - 72 D)

6x + y - 36 3)

1) _______

xy +

4) _______

Cubic approximation to A) 1 + x + 9y +

xy +

1 + x + 9y +

+ 9xy +

1 + x + 9y +

+ 9xy +

y + 27x

B) +

C) xy +

D) +

+ 5) _______

5) Quadratic approximation to A)

B) 7x + y -

- 7xy -

1 + 7x + y -

xy -

D) 1 + 7x + y -

- 7xy -

7x + y -

A) 1 + 2x + y - 2

- 2xy -

y + 2x

1 + 2x + y - 2

- 2xy -

B) x

C) - 2xy -

xy 6) _______

6) Cubic approximation to

2x + y - 2

D) 2x + y - 2

- 2xy -

y + 2x

+ 7) _______

7) Quadratic approximation to A) 1 + 7x + y +49 + 7xy + C) 1 - 7x - y + 49 + 14xy +

B) D)

1 + 7x + y + 49

+ 14xy +

1 - 7x - y + 49

+ 7xy + 8) _______

8) Cubic approximation to A) 1 - 4x - y + 16 + 8xy + B) 1 - 4x - y + 16 + 8xy + C) 1 - 4x - y + 16 + 4xy + D) 1 - 4x - y + 16 + 4xy +

- 64

- 48

y - 12x

- 64

+ 48

y - 12x

- 64

+ 48

y - 12x

- 64

- 48

y - 12x

9) _______

9) Quadratic approximation to A) 1 - 2x - 20y + 3 + 60xy + 300 C) 1 + 2x + 20y + 3 + 60xy + 300

B) D)

1 + 2x + 20y + 3

- 60xy + 300

1 - 2x - 20y + 3

- 60xy + 300 10) ______

10) Cubic approximation to A) 1 - 2x - 18y + 3 + 72xy + 243 B) 1 + 2x + 18y - 3 - 54xy - 243 C) 1 + 2x + 18y - 3 - 72xy - 243 D) 1 - 2x - 18y + 3 + 54xy + 243

-4

- 108

y - 972x

- 2916

+ 108

y + 972x

+ 2916

+ 108

y + 972x

+ 2916

-4

- 108

y - 972x

- 2916

Estimate the error in the quadratic approximation of the given function at the origin over the given region. 11) A) |E(x, y)| ≤ 0.864 B) |E(x, y)| ≤ 0.432 C) |E(x, y)| ≤ 0.216 D) |E(x, y)| ≤ 0.288

12) ______

12) A) |E(x, y)| ≤ 0.032 C) |E(x, y)| ≤ 0.216

B) |E(x, y)| ≤ 0.288 D) |E(x, y)| ≤ 0.864 13) ______

13) A) |E(x, y)| ≤ 0.1273 C) |E(x, y)| ≤ 0.3819

B) |E(x, y)| ≤ 0.0955 D) |E(x, y)| ≤ 0.191 14) ______

14) A) |E(x, y)| ≤ 0.191 C) |E(x, y)| ≤ 0.0955 15)

11) ______

B) |E(x, y)| ≤ 0.1273 D) |E(x, y)| ≤ 0.3819 15) ______

A) |E(x, y)| ≤ 0.1422 C) |E(x, y)| ≤ 0.1707

B) |E(x, y)| ≤ 0.2327 D) |E(x, y)| ≤ 0.0053

1) D 2) A 3) B 4) D 5) A 6) D 7) C 8) A 9) A 10) D 11) D 12) B 13) A 14) B 15) C

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the requested partial derivative. 1) if and A) B) 3x(3x + 2 ) + 18y(2 + z) 3x(x + 2 ) + 18y(2 + 3z) C) D) 3x(3x + 2 ) + 18y(2 + 3z) 3x(x + 2 ) + 18y(2 + z) 2)

2) _______

if A)

+ 6xyz)

+ 3yz)

+ 3xyz) 3) _______

and D)

4) _______ C) 30

D) 36 5) _______

if B)

6) _______

and

A) 9

B) 1

+ 6yz)

C) 44

= B) 54

A) 42

if B) 88

1) _______

C) 15

A) 8

D) 6 7) _______

and

B) 2

C) 1

D) 13 8) _______

8) if A)

and B)

9) _______

9) at A)

if B)

and C)

10) ______

10) if A) 3(

and +

) Provide an appropriate answer. 11) Suppose that w = +

. +

)

+ 15z + t and x + 3z + t = -1. Assuming that the independent variables

are x, y,

and z, find

___ ___

11) . A)

= 2x - 5

= 2x + 1

= 2x - 1

= 2x + 5 12) ______

12) Suppose that A)

and x = r cos θ, as in polar coordinates. Find B)

= r(cos θ - sin θ)

= -r sin θ

D) =0

= r cos θ

Solve the problem. 13) Find the maximum value of A) 4 B) 10

subject to C) 1

14) Find the least squares line for the points A) y = -2 + 3x

y=-

D) 2 14) ______

, C) y = -3 + 4x

13) ______

D) y = -5 + 5x

+ 4x 15) ______

15) Find the least squares line for the points A)

B) y=

C) y=

D) y=

16) ______

16) Find the least squares line for the points A) y = 32 - 9x 17)

18)

Maximize A)

B) y = 32 + 3x

Maximize A) 0

C) y = 38 + 3x

subject to B)

D) y = 38 - 9x 17) ______

and C)

subject to B) 17

, C) 1

and

18) ______

D) e 19) ______

19) Find the distance between the skew lines and A) 2 20)

21)

B) 2

C) 3

D) 3

Find the point on the paraboloid A) B)

that is closest to the point C) D)

What points of the surface

are closest to the origin?

20) ______

21) ______

A) (-2, -4, ±2) 22)

B) (-2, 4, ±2)

What is the distance from the surface A) 2 B) 6

C) (2, 4, ±2)

D) (2, -4, ±2) 22) ______

to the origin? C) 2

D) 6

1) D 2) B 3) A 4) A 5) B 6) A 7) A 8) B 9) D 10) D 11) C 12) D 13) D 14) D 15) D 16) D 17) B 18) C 19) C 20) C 21) B 22) A

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the integral. 1)

A) 22

B) 1

C) -26

D) 20 2) _______

C) -

D) 3) _______

A) 480

B) - 600

C) - 1080

D) - 1440 4) _______

A) 6

B) 24

C) 192

D) 768 5) _______

A) 7

B) - 3

C) 58

D) - 7 6) _______

A) - 40

B) - 20

C) - 180

D) - 360 7) _______

A) - 350

B) - 175

C) - 35

D) - 1750 8) _______

A) 324

B) 972

C) 54

D) 162 9) _______

C) -

10)

1) _______

D) -

___ ___

10)

A) 6π

B) 13π

C) 12π

D) 7π 11) ______

11)

B) 50

D) 60

Evaluate the double integral over the given region. 12) R: 4 ≤ x ≤ 8, 4 ≤ y ≤ 8 B)

12) ______

ln 13) ______

13) R: 8 ≤ x ≤ 9, 5 ≤ y ≤ 9 B)

C) 476

D) 238

14) ______

14) R: 0 ≤ x ≤ B)

,0≤y≤π C)

D) π

15) ______

15) R: 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 A)

B) (

- 1)

(

+ 1)

(

+ 1)

D) (

- 1)

16) ______

16)

R: 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 B)

17) ______

17)

A) 8π

R: 0 ≤ x ≤ π, 0 ≤ y ≤ 1 B) π

C) 8π - 8

18) ______

18) R: 0 ≤ x ≤ 1, 0 ≤ y ≤ 5 B)

A) ln 2 ln 6

C) 6 ln 2

D) ln 12

ln 2 19) ______

19) R: 0 ≤ x ≤ 1, 0 ≤ y ≤ 2 B)

Integrate the function f over the given region. 20) f(x, y) = A)

20) ______

over the square 6 ≤ x ≤ 10, 6 ≤ y ≤ 10 B)

D) ln

21) f(x, y) = xy over the rectangle 3 ≤ x ≤ 9, 5 ≤ y ≤ 6 A) 132 B) 264

21) ______ C) 198

D) 396 22) ______

22) f(x, y) = sin 15x over the rectangle 0 ≤ x ≤ A) B) π

,0≤y≤π C)

23) ______

23) f(x, y) = e2x + 3y over the rectangle 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 A) B) (e5 - e3 - e2 - 1) C)

(e5 - e3 - e2 + 1) D)

(e5 - e3 - e2 + 1) 24) f(x, y) = A)

(e5 - e3 - e2 - 1)

over the rectangle 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 B) C)

25) f(x, y) = 5x sin xy over the rectangle 0 ≤ x ≤ π, 0 ≤ y ≤ 1 A) 5π - 5 B) 5π C) π

24) ______ D)

25) ______ D)

Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries. 26) z = 6 y; 0 ≤ x ≤ 4, 0 ≤ y ≤ 3 A) 1256 B) 2256 C) 576 D) 676 27)

z= A)

; 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 B)

26) ______

27) ______ C)

28) z = 8x + 4y + 7; 0 ≤ x ≤ 1, 1 ≤ y ≤ 3 A) 38 B) 36 29)

z=4 A)

; 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 B)

28) ______ C) 28

D) 26 29) ______

30) ______

30) z = ; 0 ≤ x ≤ 1, 1 ≤ y ≤ e A)

1) D 2) B 3) A 4) C 5) B 6) D 7) D 8) B 9) D 10) C 11) D 12) C 13) D 14) B 15) B 16) C 17) A 18) A 19) A 20) B 21) C 22) D 23) B 24) A 25) B 26) C 27) B 28) A 29) B 30) C

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the integral. 1)

A) - 1

B) 3

1) _______

D) 2) _______

3) _______

4) _______

5) _______

6) _______

A) 9

B) 4

C) 8

D) 18 7) _______

8) _______

__ __ __ _

B) (e - 1)2

C) (e2 - e)2

D) (e2 - e)2

Integrate the function f over the given region. 10) f(x, y) = xy over the triangular region with vertices (0, 0), (10, 0), and (0, 8) A) B) C)

(e - 1)2

10) ______ D)

11) ______

11) f(x, y) = A) 20

over the trapezoidal region bounded by the x-axis, y-axis, line B)

and line

D) 12

12) ______

12) f(x, y) = A) 1 13)

f(x, y) = 1, y = 0 A) e

over the region bounded by the x-axis, line B) 11 C) 9

D) 10

over the triangular region in the first quadrant bounded by the lines x = y/3, x = B)

C) 9(e - 1)

13) ______

D) 27(e + 1)

(e - 1)

Write an equivalent double integral with the order of integration reversed. 14)

14) ______

15) ______

15)

16)

and curve

___ ___

16)

17) ______

17)

18) ______

18)

19) ______

19)

20) ______

20)

21)

21) ______

22) ______

22)

23) ______

23)

Reverse the order of integration and then evaluate the integral. 24)

A) 1 + cos 8

B) 1 - cos 8

24) ______

C) - cos 8

D) cos 8 25) ______

25)

A) 9 sin 10

B) 10 cos 9

C) 9 cos 10

D) 10 sin 9 26) ______

26)

- 1)

C) (

- 1) 27) ______

27)

A) 22 28)

B) 18

C) 25

D) 15 28) ______

(e - 1)

D) (2e - 1)

(2e - 1)

(e - 1) 29) ______

29)

D) 5(1 - cos 16) 5

(1 - cos 16)

30) ______

30)

A) ln 5

B) ln 5 - 1

31) ______

31)

A) C)

B) 405

81 -

ln 6562

405

81 - 5 ln 6562

81 -

ln 6562

D) 81 - 5 ln 6562 32) ______

32)

- 68

-1

33) ______

33)

A) cos 3 - 1

C) 2(1 - cos 3)

(1 - cos 3) Find the volume of the indicated region. 34) the region that lies under the paraboloid

(1 - cos 3)

and above the triangle enclosed by the lines

34) ______

, and A)

C) 13,122

35) ______

35) the tetrahedron bounded by the coordinate planes and the plane A) 40 B) 60 C) 30

D) 20 36) ______

36) the region that lies under the plane A) B)

and above the square C)

37)

38)

the solid cut from the first octant by the surface A) B)

the region under the surface cylinder A)

39)

37) ______ C)

, and bounded by the planes

the region bounded by the paraboloid A) B) 1024π

and the xy-plane C)

D) 2048π

the region bounded by the paraboloid A) 64π B)

the region bounded by the paraboloid A) B) 27π

and the plane C) 128π

40) ______ D) π

, the cylinder C)

, and A)

43) the region that lies under the plane

41) ______

, and the D)

42) the region that lies under the plane

, A) 105

π ,

42) ______

and over the triangle bounded by the lines

43) ______

and over the triangle with vertices at C)

, and B) 85

C) 75

D) 95

Evaluate the improper integral. 44)

A) 126

38) ______

39) ______

π 41)

and the

π 40)

and

44) ______

B) 28

C) 12

D) 84 45) ______

45)

C) 12(ln 12 - 2)

D) 12(ln 12 - 1)

46) ______

46)

47) ______

47)

48) ______

48)

49) ______

49)

50) ______

50)

A) 48(1 - ln 2)

B) 288(1 - ln 2)

C) 96(1 - ln 2)

D) 576(1 - ln 2) 51) ______

51)

A) 60

B) 36

C) 72

D) 48 52) ______

52)

A) 75

B) 90

C) 45

D) 60 53) ______

53)

Solve the problem. 54) What region in the xy-plane maximizes the value of

54) ______

? R

B) The ellipse

The ellipse 64

=1 + 49

The ellipse

= 3136

The ellipse 49

=1 + 64

55) What region in the xy-plane minimizes the value of

55) R

______ A) C)

B) The ellipse

The ellipse 4

+ 100

The ellipse D)

The ellipse 100

=1 +4

= 400 56) ______

56) Write

+ as a single iterated integral. A)

57) ______

57) Write

+ as a single iterated integral. A)

58) ______

58) Write

+ as a single iterated integral. A)

+ B)

59) ______

59) Write

as a single iterated integral with the same order of integration. A) B)

60) ______

60) Write

as a single iterated integral with the order of integration reversed. A) B)

61) ______

61) Evaluate

by writing the integrand as an integral. A) B)

D) ln 62) ______

62) Evaluate

by writing the integrand as an integral. A) B) ln

D) ln

ln 63) ______

63) Evaluate

by writing the integrand as an integral. A) B) ln

D) ln

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 64) Write an "area" word problem for which finding the solution would involve evaluating 64) _____________ the double integral

65) Write an "area" word problem for which finding the solution would involve evaluating the double integral

65) _____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 66) True or false? Consider the double integral

66) ______

The first step in calculating this integral involves integrating with respect to x. A) False B) True 67) ______

67) True or false? Consider the double integral

The first step in calculating this integral involves integrating with respect to y. A) False B) True 68) ______

68) True or false? Consider the double integral

The first step in calculating this integral involves holding y constant. A) False B) True 69) ______

69) True or false? Consider the double integral

The first step in calculating this integral involves holding x constant. A) False B) True SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 70) Write a "volume" word problem for which finding the solution would involve 70) _____________ evaluating the double integral

71) Write a "volume" word problem for which finding the solution would involve evaluating the double integral

71)

________ _____

1) A 2) C 3) D 4) C 5) A 6) C 7) D 8) D 9) D 10) B 11) C 12) C 13) B 14) B 15) B 16) B 17) D 18) B 19) C 20) B 21) A 22) D 23) C 24) B 25) A 26) A 27) B 28) A 29) D 30) C 31) A 32) C 33) D 34) A 35) D 36) A 37) D 38) B 39) D 40) C 41) D 42) C 43) B 44) D 45) C 46) C 47) A 48) C 49) C 50) D 51) D

52) B 53) B 54) A 55) B 56) C 57) B 58) D 59) A 60) C 61) C 62) D 63) C 64) Find the area of the region bounded by the curve y = x 2 and the lines

and x = 5.

65) Find the area of the region between the curves y = x2 and

for x between 0 and 2. 66) B 67) A 68) B 69) A 70) Find the volume bounded above by f(x,y) = y + x which lies over the region for which

and

71) Find the volume bounded above by f(x,y) = y + x which lies over the region for which

and

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. 1) The coordinate axes and the line A) B)

2) _______

2) The coordinate axes and the line A)

3) The lines

1) _______

3) _______

, and

The parabola A)

4) _______

and the line B)

5) The curve

and the lines

5) _______

and

6) _______

6) The parabola A)

and the line B)

The curve A)

and the lines

7) _______

and B)

8) _______

8) The lines A)

, and B)

9) _______

9) The lines A)

, and B)

10) The curves

10) ______

and

Find the area of the region specified by the integral(s). 11)

A) 8

B) 4

11) ______

12) ______

12)

A) 3

C) 15

D) 20

13) ______

13)

14) ______

14)

A) 3(ln 3 - 1)

B) 3 ln 3

C) ln 3 + 1

D) ln 3 15) ______

15)

-7

-4

16) ______

16)

A) 504

B) 180

C) 360

D) 720 17) ______

17)

A) 6

B) 12

C) 6

D) 1 18) ______

18) + A) 5

B) 100

C) 10

D) 200 19) ______

19) + A)

B) 49

C) 7

D) 98

20) ______

20) + A)

B) 8

Find the average value of the function f over the given region. 21) f(x, y) = 7x + 9y over the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. A) B) 8

21) ______ C) 16

22) f(x, y) = 10x + 5y over the rectangle 0 ≤ x ≤ 5, 0 ≤ y ≤ 3. A)

22) ______

23) f(x, y) = 8x + 3y over the triangle with vertices A)

23) ______

, and C)

24) f(x, y) = 9x + 7y over the region bounded by the coordinate axes and the lines

24) ______

and

. A)

25) ______

25) f(x, y) = sin A) 4

(x + y) over the square 0 ≤ x ≤ B)

,0≤y≤ . C)

26) ______

26) f(x, y) = A) e - 1

over the square 0 ≤ x ≤ B)

,0≤y≤

. C)

D) 2e - 1

27) ______

27) f(x, y) = A)

over the square 1 ≤ x ≤ 9, 1 ≤ y ≤ 9. B)

28) ______

28) f(x, y) = A)

over the square 1 ≤ x ≤ 7, 1 ≤ y ≤ 7. B)

29) ______

29) f(x, y) = over the region bounded by A) 42 ln 2 B)

, C)

, and

D) ln 2

ln 2 30)

f(x, y) = over the region bounded by A) 2e - 1 B)

ln 2

, C)

, and

30) ______

. D) e - 1

Solve the problem. 31) The northern third of Indiana is a rectangle measuring 96 miles by 132 miles. Thus, let Assuming that the total annual snowfall (in inches), S(x,y), at given by the function S(x,y) = 60 A) 51.14 inches B) 51.78 inches

31) ______ is

with (x,y) ∈ D, find the average snowfall on D. C) 52.06 inches D) 52.44 inches

32)

If f(x, y) = (3000 )/(1 + /2) represents the population density of a planar region on Earth, where x and y are measured in miles, find the number of people within the rectangle -7 ≤ x ≤ 7 and -3 ≤ y ≤ 0. A) B) 6000(1 ) ln 4 ≈ 12,527 12,000(1 ) ln ≈ 17,150 C) D) 6000(1 -

) ln

≈ 8575

3000(1 -

) ln

≈ 4288

32) ______

1) D 2) B 3) C 4) D 5) C 6) D 7) D 8) D 9) D 10) A 11) A 12) C 13) B 14) A 15) D 16) B 17) A 18) B 19) B 20) A 21) B 22) A 23) C 24) B 25) C 26) A 27) A 28) B 29) D 30) D 31) A 32) B

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Change the Cartesian integral to an equivalent polar integral, and then evaluate. 1)

1) _______

2) _______

3) _______

A) 200π

B) 400π

C) 10π

D) 100π 4) _______

5) _______

6) _______

7) _______

8) _______

A) π(9 ln 10 + 9)

B) π(10 ln 10 + 9)

C) π(9 ln 10 - 9)

D) π(10 ln 10 - 9)

9) _______

B) π

C) π

D) π

π 10) ______

10)

Find the area of the region specified in polar coordinates. 11) the region enclosed by the curve r = 10 cos θ A) 50π B) 100π

11) ______ C)

D) 25π π

12) the region enclosed by the curve r = 5 + cos θ A) 17π B) 13π

12) ______ C)

D) 26π π 13) ______

13) the smaller loop of the curve r = 7 + 14 sin θ A) B) (2π - 3

)

(2π + 3

C) )

14) the region enclosed by the curve r = 10 - 6 cos θ A) 118π B) 86π 15) the region enclosed by the curve r = 10 cos 3θ A) B) 25π

D) (2π + 3

)

(2π - 3

) 14) ______

C) 68π

D) 136π 15) ______

D) 50π π

16) one petal of the rose curve r = 7 cos 3θ A) B) π

16) ______ C)

17) the region enclosed by the curve r = 7 sin 2θ A) B) π

π 17) ______

18) one petal of the rose curve r = 8 sin 2θ A) B) 8π

D) π

π 18) ______

C) 32π

D) 16π

π 19) the region inside both A)

B) (π - 1)

19) ______

and C) (π - 2)

D) (π - 1)

(π - 2)

20) the region inside A)

20) ______

and outside B)

D) 9

Solve the problem. 21) Find the average height of the paraboloid

above the disk

21) ______

in the

. A)

22)

Find the average height of the paraboloid

above the disk

22) ______

in the

. A)

23)

24)

27)

28)

D) 17

above the annular region

in the A) 20

C) 22

B) 38

D) 56 24) ______

above the annular region

. B) 146

Find the average height of the part of the paraboloid B) 50

D) 150

in the first quadrant of the disk

Find the average distance from a point to the origin. A) B)

in the first two quadrants of the disk

Find the average distance from a point origin. A) B)

in the annular region

29) Find the average distance from a point

25) ______

that lies above the

Find the average distance from a point origin. A) B)

origin.

23) ______

Find the average height of the paraboloid

. A) 250

26)

C) 11

Find the average height of the paraboloid

in the A)

25)

B) 16

C) 1

in the region bounded by

to the

26) ______

D) 1

27) ______

to the

28) ______

to the

29) ______

30) Find the average distance from a point origin. A)

in the region bounded by

to the D)

31) ______

31) Integrate A) 36π(2 ln 9 - 1)

over the region 0 ≤ B) 36π(ln 9 - 1)

+ ≤ 81. C) 18π(2 ln 9 - 1)

D) 18π(ln 9 - 1) 32) ______

32) Integrate A) 2π 33)

34)

30) ______

over the region 1 ≤ B) π ln 10

Integrate A) 2π(1 - cos 81)

over the region 0 ≤ B)

Integrate A) π(sin 4 - cos 4) C) 2π(sin 4 - cos 4)

2π(1 -

over the region 0 ≤

+ C)

≤ 100. π

+ ≤ 81. C) π(1 - cos 81)

D) 2π ln 10 33) ______ D)

π(1 -

9) 34) ______

+ ≤ 16. B) π(sin 4 - 4 cos 4) D) 2π(sin 4 - 4 cos 4)

35) ______

35) Given that

evaluate

36) ______

36) Given that

evaluate

. A)

37) Evaluate

___ ___

37) A)

38) ______

38) Evaluate

39) ______

39) Evaluate

40) ______

40) Evaluate

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Write an iterated triple integral in the order for the volume of the rectangular solid in the first octant bounded by the planes A)

2) Write an iterated triple integral in the order the first octant bounded by the planes A)

for the volume of the rectangular solid in ,

, and B)

2) _______

3) Write an iterated triple integral in the order the first octant by the plane A)

, and B)

1) _______

for the volume of the tetrahedron cut from

3) _______

for the volume of the tetrahedron cut from

4) _______

4) Write an iterated triple integral in the order the first octant by the plane A)

5) Write an iterated triple integral in the order octant enclosed by the cylinder A)

for the volume of the region in the first and the plane B)

5) _______

6) Write an iterated triple integral in the order octant enclosed by the cylinder A)

for the volume of the region in the first and the plane B)

6) _______

7) Write an iterated triple integral in the order the paraboloids A)

and

for the volume of the region enclosed by

7) _______

8) _______

8) Rewrite the integral

in the order A)

9) _______

9) Rewrite the integral

in the order A)

10) ______

10) Rewrite the integral

in the order A)

Evaluate the integral. 11)

11) ______

D) 35

12) ______

12)

A) 125

B) 25

C) 3125

D) 625

13) ______

13)

A) 5832

B) 7776

C) 3888

D) 11,664 14) ______

14)

D) 15) ______

15)

16) ______

16)

A) 120

B) 360

C) 40

D) 240 17) ______

17)

A) 2

B) 4

D) 6

18) ______

18)

B) 0

A) 16

C) 23.2

D) 24 19) ______

19)

B) cos 2

D) π cos 2

π cos 2

cos 2 20) ______

20)

Find the volume of the indicated region. 21) the tetrahedron cut off from the first octant by the plane A) 10 B) 20 C) 15 22)

21) ______ D) 30 the region

___ ___

bounded 22) by the paraboloi d and the xy-plane A) 72π

B) 432π

C) 108π

D) 648π 23) ______

23) the region bounded by the paraboloid A) B) 196π

and the xy-plane C) 28π

D) 224π

π 24)

25)

the region bounded by the cylinder A) 160π B) 40π

and the planes C) 400π

24) ______

and D) 1600π

26) the region in the first octant bounded by the coordinate planes and the planes A) 12

27)

B) 6

the region common to the interiors of the cylinders A) B) C)

30)

B) 270

D) 1372

28) ______

, D)

C) 90

the region bounded by the paraboloid A) B)

, and the plane

30) ______ D)

Find the average value of over the given region. 31) over the cube in the first octant bounded by the coordinate planes and the planes , , A) 2592

B) 36

C) 288

29) ______

and the cylinder C)

26) ______

27) ______

and

the region bounded by the coordinate planes, the parabolic cylinder A)

, D) 24

28) the region bounded by the coordinate planes and the planes A) B) 2 C) 4

29)

25) ______

the region in the first octant bounded by the coordinate planes and the surface A) B) C) D) 8192

D) 324

31) ______

32)

over the rectangular solid in the first octant bounded by the coordinate planes and the planes A)

, B)

33) ,

B) 126

over the rectangular solid in the first octant bounded by the coordinate planes and the planes A) 61

35)

, B)

35) ______

, B)

36)

over the rectangular solid in the first octant bounded by the coordinate planes and the planes A) 2

+ the planes A) 72

39)

C) 108

D) 6

over the rectangular solid in the first octant bounded by the coordinate

planes and the planes A) 23

, B)

38) ______

, C)

over the cube in the first octant bounded by the coordinate planes and the ,

37) ______

, B) 36

38)

D) 3

over the cube in the first octant bounded by the coordinate planes and

36) ______

, B)

37)

planes A)

34) ______

over the cube in the first octant bounded by the coordinate planes and the planes , A)

33) ______

34)

40)

over the cube in the first octant bounded by the coordinate planes and the planes A) 81

32) ______

39) ______

, B)

over the rectangular solid in the first octant bounded by the coordinate planes and the planes A)

, B)

, C)

Evaluate the integral by changing the order of integration in an appropriate way. 41)

40) ______

___ ___

41)

B) 1 - sin 5

C) 1 + cos 5

42) ______

42)

A) 2 sin 4

B) 2(1 - cos 4)

C) 2(1 + sin 4)

D) 2 cos 4 43) ______

43)

A) - 4 ln 2 ln cos 7

B) 4 ln 4 ln cos 7

C) - 4 ln 4 ln cos 7

D) 4 ln 2 ln cos 7 44) ______

44)

A) - 6 ln 2 ln cos 5

B) 6 ln 2 ln cos 5

C) - 3 ln 2 ln cos 5

D) 3 ln 2 ln cos 5 45) ______

45)

2 46) ______

46)

47) ______

47)

A) 4 ln 217

B) 2 ln 217

C) 2 ln 1297

D) 4 ln 1297 48) ______

48)

49) ______

49)

B) ln 3 (e - 1)

50)

C) 4 ln 3 (e - 1) ln 2 (e - 1)

D) 4 ln 2 (e - 1)

___ ___

50)

B) ln 8 (1 - sin 9)

ln 8 (1 - sin 9)

D) ln 8 (1 - cos 9)

ln 8 (1 - cos 9)

Solve the problem. 51) Solve for a:

51) ______

B) a=

C) a = 2

a= 52) ______

52) Solve for a:

C) a=

D) a=

53) ______

53) Solve for a:

B) a=

C) a = 3

D) a=

54) For what value of a is the volume of the tetrahedron formed by the coordinate planes and the plane A)

equal to 10? B) a=

D) a=

54) ______

a= 55) ______

55) For what value of b is the volume of the ellipsoid A) B) b= 56) Let

equal to 8π? D)

C) b=

be the volume of the tetrahedron bounded by the coordinate planes and the plane , and let

be the volume of the ellipsoid

. Find the quotient

56) ______

A) 8π

B) 4π

57) ______

57) What domain D in space maximizes the value of the integral

D A)

D= B) No such minimum domain exists. C) D = the boundary of the ellipsoid

D) D = the boundary and interior of the ellipsoid

. 58) ______

58) What domain D in space minimizes the value of the integral

D A) B)

D = the boundary and interior of the sphere

D = the boundary of the sphere D) No such maximum domain exists.

Use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. 59) over the rectangular solid , , A) - 21,924 B) - 65,772 C) - 14,616 D) - 10,962 60)

over the rectangular solid B) - 112,896 C)

A) - 127,008

59) ______

60) ______

, D) - 84,672

61)

over the cylinder bounded by A)

and the planes π

D) π

A) 92,672

π over the rectangular cube B) 98,816 C) 94,720

62) ______ D) 96,768 63) ______

63)

64)

61) ______

B) π

62)

over the rectangular cube C)

___ ___

64) over the rectangul ar cube

65)

over the tetrahedron bounded by the coordinate planes and the plane A)

66)

over the tetrahedron bounded by the coordinate planes and the plane

A) 28

67)

B) 42

C) 21

68)

67) ______

D) 101

over the tetrahedron bounded by the coordinate planes and the plane

66) ______

over the tetrahedron bounded by the coordinate planes and the plane

65) ______

D) 710

68) ______

1) C 2) A 3) B 4) A 5) C 6) B 7) D 8) B 9) C 10) C 11) B 12) D 13) A 14) D 15) C 16) A 17) D 18) D 19) D 20) D 21) A 22) D 23) C 24) A 25) B 26) A 27) C 28) A 29) C 30) A 31) B 32) A 33) A 34) D 35) D 36) D 37) B 38) A 39) D 40) C 41) A 42) B 43) A 44) C 45) B 46) A 47) C 48) A 49) B 50) C 51) C

52) B 53) D 54) D 55) D 56) A 57) A 58) B 59) A 60) D 61) B 62) C 63) D 64) D 65) A 66) D 67) C 68) A

2) _______

2) Find the centroid of the region cut from the first quadrant by the line A) B) C) = 3,

. D)

3) Find the center of mass of the thin semicircular region of constant density and the curve A) =

= 0,

bounded by the

3) _______

D) = 0,

= 0,

Find the centroid of the region cut from the fourth quadrant by the circle A) B) =

. B)

= 0, 4)

1) _______

= 4) _______

= 100.

D) =

5) Find the centroid of the region in the first quadrant bounded by the curve y = 9 cos x and the axes. A) B) =

5) _______

D) =

6) Find the centroid of the infinite region in the first quadrant bounded by the coordinate axes and the curve A) =

6) _______

. B) ,

C) =

D) =

7) Find the moment of inertia about the origin of a thin plane of constant density the coordinate axes and the line A) B)

bounded by

7) _______

bounded by

8) _______

. C)

8) Find the moment of inertia about the x-axis of a thin plane of constant density the coordinate axes and the line

9) _______

9) Find the moment of inertia about the y-axis of the thin semicircular region of constant density bounded by the x-axis and the curve A) 9216π B) 12,288π

. C) 6144π

D) 4608π 10) ______

10) Find the moment of inertia about the y-axis of the thin infinite region of constant density in the first quadrant bounded by the coordinate axes and the curve A) B) C)

. D)

11) Find the mass of a thin triangular plate bounded by the coordinate axes and the line

11) ______

. A)

12) Find the center of mass of a thin triangular plate bounded by the coordinate axes and the line if A)

= 6,

= 3,

D) =

13) Find the moment of inertia about the x-axis of a thin triangular plate bounded by the coordinate axes and the line A) 6561

if B)

the curve A)

15) Find the center of mass of a thin infinite region in the first quadrant bounded by the coordinate

if B)

15) ______

. C) =

16) 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 A) B)

17)

14) ______

. B)

axes and the curve A)

13) ______

14) Find the mass of a thin infinite region in the first quadrant bounded by the coordinate axes and

12) ______

Find the mass of a thin circular plate bounded by A) 32π B) 2π

16) ______

if C) 128π

. D) 8π

17) ______

18)

Find the mass of a thin plate bounded by A) B)

if C)

19) Find the centroid of the rectangular solid defined by A)

= 4,

= 7,

18) ______

19) ______

defined by

20) ______

= 2, = C)

D) = 1,

20) Find the moment of inertia ,

of the rectangular solid of density

, where L is the line through the points B) C) 8288

. D)

21) Find the center of mass of a tetrahedron of constant density bounded by the coordinate planes and the plane A) = 1, = 4,

21) ______

. B)

D) =

= 2,

22) Find the moment of inertia

of a tetrahedron of constant density

coordinate planes and the plane A) 1365 B)

bounded by the

22) ______

. C) 2415

23) ______

23) Find the center of mass of the region of constant density bounded by the paraboloid and the xy-plane. A) C)

= 0,

= 18

= 0,

= 12

= 0,

24) Find the moment of inertia paraboloid A)

and the xy-plane. B) π

25)

of the region of constant density

bounded by the

D) π

Find the center of mass of the hemisphere of constant density bounded the A)

and

B) = 0,

= 0,

D) = 0,

= 0,

24) ______

25) ______

26) Find the mass of the rectangular solid of density . A) 2160

B) 4050

defined by C) 5400

26) ______

D) 2700

27) Find the center of mass of the rectangular solid of density ,

defined by

27) ______

B) = 2,

= 4,

D) =

= 1,

28) Find the moment of inertia ,

= 2,

of the rectangular solid of density

defined by

29) Find the mass of a tetrahedron of density and the plane A) 700

bounded by the coordinate planes

. B)

coordinate planes and the plane A) =

29) ______

30) Find the center of mass of a tetrahedron of density

28) ______

30) ______

bounded by the

. B)

D) =

31) Find the moment of inertia

of a tetrahedron of constant density

by the coordinate planes and the plane A) B)

. C)

bounded

31) ______

32) ______

32) Find the mass of the region of density A) 2401π

and the xy-plane. B) 343π

bounded by the paraboloid C) 7π

D) 49π 33) ______

33) Find the center of mass of the region of density paraboloid A)

and the xy-plane.

bounded by the

= 0, 8

= 0,

= 0, = 0, =

= 0,

D) = 0,

= 0,

= 34) ______

34) Find the mass of the region of density A) 8π

and the xy-plane. B) 16π

bounded by the hemisphere C) 4π

D) 32π

1) B 2) A 3) B 4) C 5) C 6) D 7) B 8) D 9) D 10) B 11) D 12) D 13) B 14) B 15) D 16) C 17) C 18) C 19) B 20) B 21) C 22) B 23) C 24) B 25) A 26) B 27) D 28) C 29) C 30) D 31) D 32) D 33) B 34) A

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the cylindrical coordinate integral. 1)

A) 648π

B) 5832π

C) 52,488π

1) _______

D) 61,236π 2) _______

A) 12π

B) 96π

C) 48π

D) 6π 3) _______

B) π

C) π

D) 20π π 4) _______

A) 20,634

B) 185,706

C) 340,461

D) 30,951 5) _______

A) 120

B) 60π

C) 180π

D) 180 6) _______

A) 5 - ln 2

B) 10π + ln 2

C) 5π + ln 2

D) 10 - ln 2 7) _______

8) _______

9) _______

A) 86,184π

C) π

D) π

11 4,9 12 π 10) ______

10)

B) 35

C) 70

Evaluate the integral. 11)

11) ______

B) 1183

12) ______

12)

B) 513

D) 1026

13) ______

13)

A) 18,879

B) 13,020

C) 1302

D) 7161 14) ______

14)

679

15) ______

15)

17,248

16) ______

16)

17) ______

17)

B) π

C) π

D) π

18) ______

18)

A) 96π

B) 64π

C) 32π

D) 48π 19) ______

19)

20) ______

20)

D) π

Solve the problem. 21) Let D be the region bounded below by the xy-plane, above by the sphere on the sides by the cylinder

22)

and

. Set up the triple integral in cylindrical coordinates that

gives the volume of D using the order of integration A) B)

22) ______

Let D be the region bounded below by the xy-plane, above by the sphere on the sides by the cylinder

and

. Set up the triple integral in cylindrical coordinates that

gives the volume of D using the order of integration A) B)

23)

Let D be the region bounded below by the xy-plane, above by the sphere on the sides by the cylinder

21) ______

. Set up the triple integral in cylindrical coordinates that

gives the volume of D using the order of integration A) B)

and

23) ______

24)

Let D be the region bounded below by the xy-plane, above by the sphere on the sides by the cylinder

and

24) ______

. Set up the triple integral in cylindrical coordinates that

gives the volume of D using the order of integration A)

+ B) + C) + D) + 25)

Let D be the region bounded below by the cone

25) ______

and above by the sphere

. Set up the triple integral in cylindrical coordinates that gives the volume of using the order of integration A)

. B)

26)

Let D be the region bounded below by the cone

26) ______

and above by the sphere

. Set up the triple integral in cylindrical coordinates that gives the volume of D using the order of integration A)

+ B) + C) + D) + 27) Let D be the region bounded below by the xy-plane, on the side by the cylinder on top by the paraboloid

and

. Set up the triple integral in cylindrical coordinates that gives

the e of D vol using um the

___ ___

order of 27) integrati on . A)

28) Let D be the region bounded below by the xy-plane, on the side by the cylinder on top by the paraboloid

and

28) ______

. Set up the triple integral in cylindrical coordinates that gives

the volume of D using the order of integration A)

. B)

Set up the iterated integral for evaluating

over the given region D. D

29) D is the right circular cylinder whose base is the circle lies in the plane A)

in the xy-plane and whose top

29) ______

in the xy-plane and whose top

30) ______

30) D is the right circular cylinder whose base is the circle lies in the plane A)

in the xy-plane and whose top

31) ______

32) D is the solid right cylinder whose base is the region in the xy-plane that lies inside the cardioid

32) ______

31) D is the right circular cylinder whose base is the circle lies in the plane A)

and outside the circle

and whose top lies in the plane

33) D is the solid right cylinder whose base is the region in the xy-plane that lies inside the cardioid and outside the circle

and whose top lies in the plane

33) ______

34) D is the solid right cylinder whose base is the region between the circles and whose top lies in the plane A)

and

34) ______

35) D is the solid right cylinder whose base is the region between the circles and whose top lies in the plane

35) ______

and

36) D is the rectangular solid whose base is the triangle with vertices at whose top lies in the plane A)

and

, and

36) ______

and

37) ______

37) D is the rectangular solid whose base is the triangle with vertices at and whose top lies in the plane A)

Evaluate the spherical coordinate integral. 38)

38) ______

39) ______

39)

π 40) ______

40)

π 41) ______

41)

C) π

D) π

42) ______

42)

B) π(15π - 8)

C) π(15π - 16)

D) π(15π - 16)

π(15π - 8) 43) ______

43)

D) π 44) ______

44)

D) π

π 45) ______

45)

B) π

46)

C) π

D) π

___ ___

46)

A) 400π

B) 480π

C) 240π

D) 800π 47) ______

47)

Evaluate the integral. 48)

48) ______

49) ______

49)

D) π 50) ______

50)

B) 1944π

π 51) ______

51)

A) 5π

C) π

D) π

π 52) ______

52)

B) π(15π - 16)

C) π(15π - 8)

D) π(15π - 8)

π(15π - 16) 53) ______

53)

B) π

54)

D) π 54) ______

B) π

π 55) ______

55)

B) π

D) π

π 56) ______

56)

π 57) ______

57)

B) 726π

D) 726π

Use a spherical coordinate integral to find the volume of the given solid. 58) the solid between the sphere and the hemisphere A) B) C) π

59) the solid between the spheres A)

59) ______ C)

60) ______ C)

and

A) 852π

D) π

60) the solid between the spheres

61) the solid between the sphere

and π

A) 109π

D) π

B) π

58) ______

D) π

and the cardioid of revolution C) 426π

π 61) ______ D) 1065π

π 62) the solid bounded below by the hemisphere A)

B) 148π π

and above by the cardioid of revolution C)

D) 185π π 63) ______

63) the solid enclosed by the cardioid of revolution A) 435π

B) 3480π

64) the solid enclosed by the cardioid of revolution A)

62) ______

C) 1740π

D) 870π 64) ______

π π 65)

the solid bounded below by the sphere A) B) π

and above by the cone C)

B) 128π

66) the solid bounded below by the xy-plane, on the sides by the sphere cone A) 256π

65) ______

D) π

π Solve the problem. 67) Set up the triple integral for the volume of the sphere A) B)

66) ______

and above by the

in spherical coordinates.

67) ______

in cylindrical coordinates.

68) ______

in rectangular coordinates.

69) ______

and above by the sphere

up triple the integral

68) Set up the triple integral for the volume of the sphere A)

69) Set up the triple integral for the volume of the sphere A) 8 B)

8 C)

70) Let D be the region that is bounded below by the cone

Set

___ ___

for the 70) volume of D in spherical coordinat es. A)

71) ______

71) Let D be the region that is bounded below by the cone and above by the sphere up the triple integral for the volume of D in cylindrical coordinates. A) B)

Set

72) Let D be the smaller cap cut from a solid ball of radius 7 units by a plane 6 units from the center of the sphere. Set up the triple integral for the volume of D in spherical coordinates. A)

72) ______

73) Let D be the smaller cap cut from a solid ball of radius 10 units by a plane 9 units from the center of the sphere. Set up the triple integral for the volume of D in cylindrical coordinates. A) B)

73) ______

74) Let D be the smaller cap cut from a solid ball of radius 9 units by a plane 8 units from the center of the sphere. Set up the triple integral for the volume of D in rectangular coordinates. A)

74) ______

Find the volume of the indicated region. 75) the region bounded above by the sphere A) 54π(2 B) 72π(2 ) ) 76)

77)

78)

and below by the cone C) 54π(2 D) 72π(2 )

the region enclosed by the paraboloids A) 768π B) 512π

and C) 1024π

the region enclosed by the sphere A) 1152(3π - 4) B) 384(3π - 4)

and the cylinder C) 360(3π - 4)

the region enclosed by the cone A) B)

between the planes C)

76) ______ D) 256π 77) ______ D) 288(3π - 4) 78) ______

and D)

79) the region bounded below by the xy-plane, laterally by the cylinder B) 8π

and above by

C) 4π

B) π

81)

the region that lies inside the sphere A) C)

82)

83)

and above by the

C) π

the region enclosed by the cylinder A) 64π B) 192π

79) ______

D) 2π

80) the region bounded below by the xy-plane, laterally by the cylinder plane A)

)

the plane A) 16π

75) ______

80) ______

D) π

π and outside the cylinder

81) ______

B) D)

and the planes C) 256π

82) ______

and D) 128π

the region

___ ___

inside 83) the solid sphere that lies between the cones

and

A) 243π

C) π

84) the region bounded by the cylinders A) 48π

and the planes C) 192π

B) 144π

84) ______ D) 96π

Find the average value of the function over the region. 85) f(r, θ, z) = r over the region bounded by the cylinder r = 3 between the planes z = -7 and z = 7 A) 252π B) 2 C) D) 3

86)

f(r, θ, z) = A)

over the region bounded by the cylinder r = 8 between the planes z = -9 and z = 9 B) C) 32 D) 18,432π

85) ______

86) ______

87) ______

87) f(ρ, φ, θ) = ρ over the solid ball ρ ≤ 4 A) B) 4

C) 4π

D) 3

88) ______

88) f(ρ, φ, θ) = ρ over the "ice cream cone" cut from the solid sphere p ≤ 5 by the cone φ = π/3 A) B) C) D) π 89) f(ρ, φ, θ) = ρ cos φ over the solid lower ball ρ ≤ 5, π/2 ≤ φ ≤ π A) B) C)

Solve the problem. 90) A solid of constant density is bounded below by the plane

(0, 0,

)

D) -

on the sides by the cylinder A) B)

89) ______

Find the center of mass. C) (0, 0, 2) (0, 0,

above by the cone

and

90) ______

D) (0, 0, 1)

)

91) Find the center of mass of the region of constant density bounded from above by the sphere and from below by the cone A) =

91) ______

= 0, (2 +

= 0,

)

= 0, = ( 2+ ) D)

C) = 0, 92)

= 0,

(2 +

)

= 0,

Find the centroid of the region bounded above by the sphere the plane A) = 0,

(2 +

) and below by

92) ______

if the density is constant. B) = 0,

= 0,

D) = 0,

= 0,

93) Find the centroid of a hemispherical shell having outer radius 10 and inner radius 4 if the density is constant. A) B) = 0,

= 0,

93) ______

D) = 0,

= 0,

94) Find the centroid of the first-octant portion of a solid ball of radius 3 centered at the origin. A) B) ( , , )=

94) ______

( , , )=

D) ( , , )=

( , , )= 95) ______

95) Find the centroid of the region enclosed between the cone with equation plane with equation A)

and the

( , , )=

D) ( , , )=

( , , )=

96) Find the moment of inertia about the z-axis of a region of constant density δ enclosed by the paraboloid A)

and the xy-plane.

96) ______

δπ δπ

δπ

δπ 97) ______

97) Find the moment of inertia of a sphere of radius 6 and constant density δ about a diameter. A) 4536δπ B) C) 5184δπ D) δπ

δπ

98) Find the moment of inertia about the z-axis of a thick-walled right circular cylinder bounded on the inside by the cylinder r = 1, on the outside by the cylinder r = 6, and on the top and bottom by the planes A) 2592π

B) 2590π

99) Find the mass of a sphere of radius 5 if A) 25kπ 100)

B) 1250kπ

C) 1295π

98) ______

D) 5180π 99) ______

k a constant. C) 625kπ

D) 125kπ

Find the mass of the solid in the first octant between the spheres

100) _____

and

if the density at any point is inversely proportional to its distance from the origin. A)

B) kπ

101)

C) 154kπ

kπ

Find the center of mass of the hemisphere distance from the center. A) B)

kπ if the density is proportional to the C)

102) _____

102) Find the centroid of a hemispherical shell having outer radius 10 and inner radius 2 if the density varies as the square of the distance from the base. A) B) = 0,

= 0,

101) _____

= 0,

D) = 0,

= 0,

103) Find the center of mass of the upper half of a solid ball of radius 4 centered at the origin if the

103) _____

density at any point is proportional to the distance from that point to the A) B) ( , , )= C)

( , , )= D)

( , , )= 104)

( , , )=

Find the center of mass of the solid enclosed between the cone with equation the plane with equation point to the axis of the cone. A) ( , , ) =

and

if the density at any point is proportional to the distance from that B) ( , , )=

104) _____

D) ( , , ) = ( , , )= 105) _____

105) Find the center of mass of the solid enclosed between the cone with equation the plane with equation point to the plane A)

and

if the density at any point is proportional to the distance from that

. B)

( , , )=

D) ( , , ) = ( , , )=

106) A solid right circular cylinder is bounded by the cylinder r = 2 and the planes z = 0 and z = 5. Find the moment of inertia about the z-axis if the density is δ = 3z. A) 600π B) 100π C) 200π D) 300π Provide an appropriate response. 107) What form do planes perpendicular to the z-axis have in spherical coordinates? A) ρ = a sin φ B) ρ = a csc φ C) ρ = a sec φ D) ρ = a cos φ

106) _____

107) _____

108) What form do planes perpendicular to the y-axis have in cylindrical coordinates? A) r = a sec θ B) r = a cos θ C) r = a csc θ D) r = a sin θ

108) _____

109) What form do planes perpendicular to the y-axis have in spherical coordinates? A) ρ = a sin φ sin θ B) ρ = a csc φ C) ρ = a csc φ csc θ D) ρ = a csc θ

109) _____

110)

110) _____ What does the graph of the equation A) A line B) A cylinder

look like? C) A plane

D) A cone 111) _____

111) What does the graph of the equation A) The xz-plane B) The z-axis 112) What does the graph of the equation A) The yz-plane

B) The y-axis

113) What does the graph of the equation A) The xz-plane

B) The x-axis

114) What does the graph of the equation A) A line

B) A plane

look like? C) The y-axis look like? C) The z-axis look like? C) The yz-plane look like? C) A sphere

D) The xy-plane 112) _____ D) The xy-plane 113) _____ D) The y-axis 114) _____ D) A cylinder

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 115) Give a geometric interpretation of the triple integral 115) ____________

116) Give a geometric interpretation of the triple integral

Describe the three-dimensional region associated with this integral in detail.

116) ____________

1) B 2) D 3) A 4) D 5) A 6) A 7) C 8) B 9) B 10) C 11) D 12) D 13) C 14) C 15) A 16) B 17) B 18) A 19) A 20) A 21) C 22) A 23) A 24) D 25) A 26) D 27) A 28) C 29) C 30) A 31) B 32) C 33) A 34) D 35) B 36) D 37) D 38) D 39) D 40) D 41) A 42) C 43) C 44) A 45) A 46) C 47) C 48) D 49) C 50) D 51) C

52) D 53) C 54) C 55) C 56) D 57) C 58) A 59) D 60) D 61) A 62) C 63) C 64) C 65) B 66) C 67) D 68) A 69) D 70) C 71) C 72) D 73) B 74) A 75) D 76) D 77) B 78) C 79) C 80) D 81) C 82) D 83) A 84) A 85) B 86) C 87) D 88) C 89) B 90) B 91) A 92) B 93) A 94) C 95) A 96) D 97) B 98) B 99) C 100) A 101) D 102) D 103) D

104) B 105) A 106) D 107) C 108) C 109) C 110) D 111) D 112) C 113) A 114) B 115) This integral represents the volume of the region enclosed between the paraboloid with equation and the xy-plane. 116) This integral represents the volume of a wedge cut from a sphere of radius 1 by two half-planes whose edges meet at a 45° angle along a diameter of the sphere. The region resembles a section of an orange and is sphere.

of the entire

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the Jacobian 1) A) 21

(as appropriate) using the given equations. 1) _______ B) -9

C) -21

D) 9 2) _______

2) A) 5

B) -5

D) 3) _______

3) A) 60

B) -48

C) 2880

D) 48

A) -60

D) -20

4) _______

4) 5) x = 10u cos 8v, y = 10u sin 8v A) 640v B) 800u

5) _______ C) 640u

D) 800v

C) 810u

D) 810v

6) _______

6) A) 2700v

B) 2700u

7) _______

7) A)

8) x = 7u cosh 4v, y = 7u sinh 4v A) 196u B) 112u

56 8) _______

C) 196v

D) 112v

C) 2016u

D) 2016v

9) _______

9) A) 2688v

B) 2688u

Sketch D and g(D) from the description of D and change of variables 10) x = 3u, y =

v where D is the rectangle 0 ≤ u ≤ 2, 1 ≤ v ≤ 4

A) None of these.

10) ______

11) ______

11) u = xy, v =

where D is the rectangle 1 ≤ x ≤ 4,

A) None of these. B)

≤y≤2

12) x = u + v, y = 2u - v where D is the rectangle 0 ≤ u ≤ 1, -1 ≤ v ≤ 1

12) ______

D) None of these. 13)

, y = 2uv where D is the rectangle 0 ≤ u ≤ 1, 0 ≤ v ≤ 1

A) None of these. B)

13) ______

14) ______

14) x = u cos v, y = u sin v where D is the rectangle 1 ≤ u ≤ 2, 0 ≤ v ≤

B) None of these. C)

Use the given transformation to evaluate the integral. 15)

15) ______

R where R is the parallelogram bounded by the lines A) 2 B) 4

C) -4

D) -2 16) ______

16)

R where R is the parallelogram bounded by the lines A) B) 17

D) 13

17) ______

17)

R where R is the parallelogram bounded by the lines A) B)

18) ______

18)

R where R is the parallelogram bounded by the lines A) B) 1092

C) 2184

19) ______

19)

R where R is the parallelogram bounded by the lines A) B) 780

C) 390

20) ______

20)

R where R is the parallelogram bounded by the lines A)

21) ______

21)

R where R is the parallelogram bounded by the lines A) B) (1 + cos 5)

(1 + cos 5)

D) (1 - cos 5)

(1 - cos 5) 22) ______

22)

R where R is the trapezoid with vertices at A) 15 sin 1 B) 30 sin 2

C) 15 sin 2

D) 30 sin 1 23) ______

23)

R where R is the trapezoid with vertices at A) B)

24)

___ ___

R 24) where R is the trapezoid with vertices at

25) ______

25)

R where R is the interior of the ellipsoid A) 18π B) 36π

D) π

π 26) ______

26)

R where R is the interior of the ellipsoid A) 12π B)

C) 9π

π 27) ______

27)

R where R is the interior of the ellipsoid A) B) π

D) π

π 28) ______

28)

R where R is the interior of the ellipsoid A) B)

29) ______

29)

R where R is the interior of the ellipsoid A) B)

D) π

π 30) ______

30)

R where R is the interior of the ellipsoid A) 360π B) 180π

C) 270π

D) 300π 31) ______

31)

R where R is the parallelepiped bounded by the planes A) 96

D) 192

32) ______

32)

R where R is the parallelepiped bounded by the planes A) 45

B) 270

C) 30

D) 120 33) ______

33)

R where R is the parallelepiped bounded by the planes A) 1875

B) 3750

C) 33,750

D) 16,875 34) ______

34)

R where R is the parallelepiped bounded by the planes A)

Solve the problem. 35) Evaluate

35) ______

by integrating

36) ______

36) Evaluate

,a>0 by integrating

37) ______

37) Evaluate

R where R is the interior of the ellipsoid then convert to spherical coordinates. A) 54π B) 72π

. Hint: Let C)

and D)

54 38) ______

38) Evaluate

by transforming to cylindrical or spherical coordinates. A) B) C)

39) ______

39) Evaluate

by transforming to cylindrical or spherical coordinates. A) B) C) 16,875π π

D) π

40) ______

40) Evaluate

by transforming to cylindrical or spherical coordinates. A) B) C) π

D) 9π π 41) ______

41) Find the volume of the parallelepiped enclosed by the planes A)

42) ______

42) Evaluate

R where R is the parallelepiped enclosed by the planes A)

43) ______

43) Evaluate

R where R is the parallelepiped enclosed by the planes A)

44) ______

44) Evaluate

R where R is the parallelepiped enclosed by the planes A)

1) D 2) C 3) A 4) B 5) B 6) C 7) B 8) A 9) B 10) B 11) D 12) B 13) C 14) C 15) D 16) B 17) D 18) A 19) D 20) B 21) C 22) D 23) A 24) D 25) B 26) D 27) D 28) A 29) A 30) A 31) C 32) C 33) A 34) C 35) D 36) A 37) D 38) D 39) D 40) D 41) A 42) D 43) A 44) A

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Match the vector equation with the correct graph.

1) r(t) = 3i - 2j - tk; -1 ≤ t ≤ 1 1) _______ A) Figure 6

B) Figure 3 C) Figure 5 D) Figure 8

2) r(t) = 2 cos ti + sin tk; 0 ≤ t ≤ π 2) _______ A) Figure 4 B) Figure 6 C) Figure 2 D) Figure 7

r(t) = (3 - 2t)i + tj; 0 ≤ t ≤ 3) _______ A)

Figure 1 B) Figure 5 C) Figure 3 D) Figure 8

4) r(t) = (1 -

)j + 3tk; -1 ≤ t ≤ 1

4) _______ A) Figure 2 B) Figure 8 C) Figure 7 D) Figure 4

5) r(t) = tj; -2 ≤ t ≤ 2 5) _______ A)

Figure 1 B) Figure 5 C) Figure 6 D) Figure 3

r(t) = sin tj - cos tk; 0 ≤ t ≤ 6) _______ A) Figure 4 B) Figure 7 C) Figure 1 D) Figure 2

r(t) =

ti +(2 - t)k; 0 ≤ t ≤ 2

_______ A) Figure 3 B) Figure 8 C) Figure 1 D) Figure 5

8) r(t) = -3ti + 2tj + 2tk; 0 ≤ t ≤ 1 8) _______ A) Figure 8 B) Figure 5 C) Figure 1 D) Figure 3

Evaluate the line integral along the curve C. 9)

ds , C is the straight-line segment x = 0, y = 4 - t, z = t from (0, 4, 0) to 9) _______ A) 0 B) 8 C) 16 D) 16

10)

ds , C is the curve r(t) = 4ti + (5 cos

t)j + (5 sin

t)k , 0 ≤ t ≤

10) ______ A)

+ B)

+ C)

11)

ds, C is the curve r(t) = (5 - 2t)i + tj - 2tk , 0 ≤ t ≤ 1 11) ______ A) 20 B) - 10 C)

12)

ds , C is the curve r(t) = (4

cos t)i + (4

sin t)j + 4 k , 0 ≤ t ≤ 7

12) ______ A)

(4 +

)

(4 -

) C)

(4 -

) D)

13)

ds , C is the path from (0, 0, 0) to (4, -4, 1) given by r(t) = 4 i - 4tj, 0 ≤ t ≤ 1 r(t) = 4i - 4j + (t-1)k, 1 ≤ t ≤ 2 13) ______ A)

+ B)

14)

ds , C is the path from (0, 0, 0) to (1, 1, 1) given by r(t) = ti, 0 ≤ t ≤ 1 r(t) = i + tj, 0 ≤ t ≤ 1 r(t) = i + j + tk, 0 ≤ t ≤ 1 14) ______ A) 11 + B) 27 + 6

+ ln 2 C)

8+ D) 22 + 5

+ ln 2

15)

ds, C is the curve r(t) = (10 - t)i - j + (10 - t) k, 0 ≤ t ≤ 1 15) ______ A)

16)

ds, C is the curve r(t) = (5 sin

t)i + (5 cos

t)j + 5tk, 2 ≤ t ≤ 4

16) ______ A)

17)

ds , C is the path from (1, 1, 0) to ( , r(t) =

j, 0 ≤ t ≤ 1

, 1) given by

r(t) =

j + 3tk, 0 ≤ t ≤ 1

17) ______ A)

(

- 1) B)

(

- 1) + 3

+9 C)

(

- 1) +

+ D)

(

- 1) + 3

18)

ds , C is the path given by r(t) = (3 cos t)i + (3 sin t)j from (3, 0, 0) to (0, 3, 0) r(t) = (3 sin t)j + (3 cos t)k from (0, 3, 0) to (0, 0, 3) r(t) = (3 sin t)i + (3 cos t)k from (0, 0, 3) to (3, 0, 0) 18) ______ A)

π C)

π D)

Evaluate the line integral of f(x,y) along the curve C. 19)

,Cy=

f(x, y) =

, 0≤x≤2

19) ______ A) 32 B)

C) 8 D) 0

20) f(x, y) =

, C y = 3x + 2, 0 ≤ x ≤ 3

20)

______ A) 156 B) 336 C) 156 D) 52

21) f(x, y) = y + x, C

= 16 in the first quadrant from (4, 0) to (0, 4)

21) ______ A) 32 B) 16 C) 0 D) 64

22) f(x, y) =

, C the perimeter of the circle

22)

______ A) 54π B) 18π C) 6π D) 27π

23)

f(x, y) = x, C y =

, 0≤x≤

23) ______ A)

B) 63 C)

D) 21

24)

f(x, y) = cos x + sin y, C y = x, 0 ≤ x ≤ 24) ______ A) 2 B) 2 C)

D) 0

25) f(x,y) = 3

,Cy=

, 0≤x≤5

25) ______ A) 1 B) 1 C) 1[2

] D)

]

Find the mass of the wire that lies along the curve r and has density δ. 26)

r(t) =

i + 6tj, 0 ≤ t ≤ 1; δ = 4t

26) ______ A)

units B) 2 units C)

units D)

units

27) r(t) = 10i + (5 - 4t)j + 3tk, 0 ≤ t ≤ 2π ; δ = 8(1 + sin 4t) 27) ______ A) 16π units B) 20 + 80π units

C) 40 + 80π units D) 80π units

28) r(t) = (4 cos t)i + (4 sin t)j + 4tk, 0 ≤ t ≤ 2π; δ = 4 28) ______ A) 8π

units B)

32π

units C)

8π units D) 128π

units

29)

r(t) = (3 cos t)i + (3 sin t)j, 0 ≤ t ≤

;

r(t) = 3j + tk, 0 ≤ t ≤ 1; δ = 5 29) ______ A)

units B)

units C)

units D)

units

Find the center of mass of the wire that lies along the curve r and has density δ. 30) r(t) = (4 + 3t)i + j + 4tk, 0 ≤ t ≤ 1; δ(x, y, z) = x + 30) ______ A)

31) r(t) = (8 cos 3t)i +(8 sin 3t) j + 24tk, 0 ≤ t ≤ 2π; δ(x, y, z) = 4(1 + sin 3t cos 3t) 31) ______ A)

32) r(t) = (2

- 4)i + 6tk, -1 ≤ t ≤ 1;

32) ______ A)

33)

r(t) = 4ti + 7tj + 4 k, 0 ≤ t ≤ 1; δ(x, y, z) = 33) ______ A)

Find the required quantity given the wire that lies along the curve r and has density δ. 34) Moment of inertia

about the z-axis, where r(t) = (4 sin 3t)i + (4 cos 3t)j +

34) ______

k, 0 ≤ t ≤ 1;

= B)

= C)

(

- 1) D)

[

]

35)

Moment of inertia

about the z-axis, where r(t) = (5 cos t)i + (5 sin t)j, 0 ≤ t ≤

;

35) ______ A)

= 375 B) =0 C)

= 15 D)

= 375

36) Moment of inertia

about the y-axis, where r(t) =

i + 4tj, 0 ≤ t ≤ 1;

36) ______ A) = 3225 B) = - 3225 C) = - 645 D) = 645

37)

Moment of inertia

about the x-axis, where r(t) =

i + (5t cos t)j + (5t sin t)k,

37) ______ A) = 100 B) = 75 C)

= 50 D) =0

Find the gradient field of the function. 38)

f(x, y, z) =

38) ______ A)

i+3

∇f = 5

k B)

∇f = (5

)i + 3

k C)

∇f = (5

)i + 3

k D)

∇f =

i+3

39) f(x, y, z) =

39) ______

A) ∇f = (1 + x)

k B)

∇f = (4 + 5x)

i+6

j+5

k C)

∇f = (4 + 5x)

i+6

j+5

k D)

∇f = (4 + 5x)

i+(

)j + (

40)

f(x, y, z) = 40) ______ A)

∇f =

k B)

∇f =

(

)i +

k C)

∇f =

i + 2yj + 2zk D)

∇f =

41) f(x, y, z) = ln (

)

41) ______ A)

∇f =

k B)

∇f =

k C)

∇f =

k D)

∇f =

ln (

)i +

ln (

)j +

ln (

42) f(x, y, z) = 42) ______ A) ∇f = 3

i + 10

j+9

k B)

∇f =

k C)

∇f =

k D)

i + 10

∇f = 3

j+9

43) f(x, y, z) = z sin (x + y + z) 43) ______ A) ∇f = -cos xi - cos yj + (sin z - z cos z)k B) ∇f = cos xi + cos yj + (sin z + z cos z)k C) ∇f = -z cos(x + y + z)i - z cos(x + y +z)j + (sin(x + y + z) - zcos(x + y + z))k D) ∇f = z cos(x + y + z)i + z cos(x + y +z)j + (sin(x + y + z) + zcos(x + y + z))k

44)

f(x, y, z) = 44) ______ A)

∇f = -

k B)

∇f =

k C)

∇f =

k D)

∇f = -

45) f(x, y, z) = (

)

45) ______ A) ∇f = 10

i + (10

- 2y)j + 2

k B)

∇f = 10

i + (2

y - 10

)

[

z(2

- 3) - 2

]k C)

∇f = 10

i + (2

y - 10

)

j + (3 - 2

)

k D)

∇f = 10

i + (10

-2

y)j + (2

- 3)

46)

f(x, y, z) = ln

46)

______ A)

∇f =

k B)

∇f =

k C)

∇f =

k D)

∇f =

47)

f(x, y, z) = 47) ______ A)

∇f =

j-6

k B)

∇f = 6

j-6

k C)

∇f = 6

j+6

k D)

∇f = 6

j+6

Find the work done by F over the curve in the direction of increasing t. 48) F = -10zi + 3xj + 9yk; C r(t) = ti + tj + tk, 0 ≤ t ≤ 1 48) ______ A) W=2 B) W=4 C)

W= D) W=1

49)

k; C r(t) =

k, 0 ≤ t ≤ 1

49) ______ A) W = 17 B) W=0

C) W=1 D)

50) F = -7yi + 7xj + 4

k; C r(t) = cos ti + sin tj, 0 ≤ t ≤ 3

50) ______ A)

W= B) W=0 C) W = 42 D) W = 21

51)

F=-

k; C r(t) =

j + 7tk, 0 ≤ t ≤

51) ______ A)

+ B)

+ C)

+ D)

52)

i + j + 6k; C r(t) =

j + tk, 0 ≤ t ≤ 1

52) ______ A)

W = ln

+8 B)

W = ln

+ C)

W = ln D)

W = ln

53) F=x

k ; the path is

and

is the straight line from (1, 1, 0) to (1, 1, 1)

∪

where

is the straight line from (0, 0, 0) to

53) ______ A)

+ B)

+ C)

+ D)

54)

F = xyi + 2j + 4xk; C r(t) = cos 3ti + sin 3tj + tk, 0 ≤ t ≤ 54) ______ A) W=0 B)

W= D) W=3

55) j + (2x + 6z)k; C r(t) = ti +

F = 6yi +

j + tk, 0 ≤ t ≤ 2

55) ______ A)

W = 32 + B) W = 32 + 20 C) W=0 D) W = 56 + 20

56)

F = ti +

j + k ; C r(t) =

j + (-5

+ t)k, -1 ≤ t ≤ 1

56) ______ A) W = -8

B) W=2 C) W=

+2 D)

-8

57) F = -5xi + 3

j + (-4z - 9

)k ; the path is

where

is the straight line from

to is the straight line from (1, 0, 0) to (1, 1, 0), and from (1, 1, 0) to (1, 1, 1)

is the straight line

57) ______ A)

W=B) W = - 12 C) W = - 13 D) W=-2

Calculate the circulation of the field F around the closed curve C. 58)

F=-

yi -

j; curve C is r(t) = 3 cos ti + 3 sin tj, 0 ≤ t ≤ 2π

58) ______ A) -9 B) -3 C)

D) 0

59) F = xyi + 2j , curve C is r(t) = 3 cos ti + 3 sin tj, 0 ≤ t ≤ 2π 59) ______ A) 4 B) -2 C) 10 D) 0

60) F=

j; curve C is the counterclockwise path around the rectangle with vertices at

, and 60) ______ A)

C) 512 D) 0

61) F = x i + yj; curve C is the counterclockwise path around 0≤t≤π r(t) = ti , -7 ≤ t ≤ 7

∪

61) ______ A) 49 B) 7 C) 98 D) 0

62) F=

i + j; curve C is the counterclockwise path around the triangle with vertices at , and

62) ______ A)

D) 0

63) F = (-x - y)i + (x + y)j , curve C is the counterclockwise path around the circle with radius 6 centered at 63) ______ A) 144π B) 72(1 + π) C)

72(1 + π) + 120 D) 72π

Calculate the flux of the field F across the closed plane curve C. 64) F = xi + yj; the curve C is the counterclockwise path around the circle

64) ______ A) 36π B) 0 C) 6π D) 18π

65) F=

j; the curve C is the closed counterclockwise path formed from the semicircle 0 ≤ t ≤ π, and the straight line segment from (-3, 0) to (3, 0)

65) ______ A) -6 B)

0 C) 12 D) 6

66) F= i+ vertices at

j; the curve C is the closed counterclockwise path around the triangle with and

66) ______ A)

(1 -

(

- 1) - 8 B)

(1 -

(

- 1) + 8 C)

0 D)

(

- 1) +

(1 -

)-8

67) F= at

j; the curve C is the closed counterclockwise path around the triangle with vertices and

67) ______

A) 0 B) 84 C) 28 D) 4

68) F = xi + yj; the curve C is the closed counterclockwise path around the rectangle with vertices at and 68) ______ A) 5 B) 12 C) 0 D) 13

69) F = (x+y)i + xyj; the curve C is the closed counterclockwise path around the rectangle with vertices at and

69) ______ A) 20 B) -3 C) 12 D) 13

70) F = yi - xj; the curve C is the circle

= 49

70) ______ A) -112 B) 0 C) 224 D) 112

71) F = xi + yj; the curve C is the circle

= 144

71) ______ A) 288π + 12 B) 288π C) 2π D) 0

Calculate the flow in the field F along the path C. 72)

F = -2i + 6j + 10k; C is the curve r(t) = 2 cos 5ti + 2 sin 5tj + 4tk , 0 ≤ t ≤

72) ______ A) -8 + 80π B) 16 + 80π C) 32 + 80π D) 16π

73) i + zj + xk; C is the curve r(t) = (-3 + 4t)i + 3tj - 4tk , 0 ≤ t ≤ 1

F= 73)

______ A) - 14 B) 52 C) 10 D) 20

74)

j + 4k , C is the curve r(t) = 3 i + 6tj + (3 - 3t)k , 1 ≤ t ≤ 6

74) ______ A) 3(

) + ln 6 - 60 B)

(

) + ln 6 + 60 C)

(

) + ln 6 - 60 D)

(

) + ln 6 - 60

75) F = (x - y)i - (

)j; C is curve from (3, 0) to (-3, 0) on the upper half of the circle

75) ______ A)

π D) 9π

76)

F=-

j +

k , C is the curve

0≤t≤1 76) ______ A)

C) 0 D)

77) F = ∇(x

); C is the line segment from (6, 1, 1) to (7, 1, -1)

77) ______ A) -13 B) -8 C) 14 D) 23

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Sketch the vector field in the plane along with its horizontal and vertical components at a representative assortment of points on the circle

= 4. 78)

i -

78) _____________

79)

i +

79) _____________

80)

F=-

i -

80) _____________

81) F = -xi - yj 81) _____________

82) F = -xi + yj 82) _____________

83) F = xi - yj 83) _____________

Solve the problem. 84) a). For the field F(x,y) = Mi + Nj and the closed counterclockwise plane curve C in the xy-

plane, show that ds = b). How would the equality change if the closed path is followed clockwise? 84) _____________

85) The velocity field F of a fluid has a constant magnitude k and always points towards the origin. Following the smooth curve y = f(x) from (a, f(a)) to (b, f(b)), show that the flow along the curve is

ds = k[

]

85) _____________

86)

The velocity field F of a fluid is the spin field F = -

smooth curve

j. Following the

from (a, f(a)) to (b, f(b)), show that the flux across the curve is

86) _____________

87) Imagine a force field in which the force is always perpendicular to dr. What is special about the work done in moving a particle in such a field? 87) _____________

88) Imagine a force field in which the force is always parallel to dr. What is special about the work done in moving a particle in such a field? 88) _____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 89) Find a field G = P(x, y)i + Q(x, y)j in the xy-plane with the property that at any point (a, b) ≠ (0, 0), G is a vector of magnitude in the clockwise direction.

tangent to the circle

and pointing

89) ______ A) yi - xj B) xi + yj C) xi - yj

D) -yi + xj

90) Find a field G = P(x, y)i + Q(x, y)j in the xy-plane with the property that at any point (a, b) ≠ (0, 0), G is a unit vector pointing away from the origin. 90) ______ A) xi + yj B)

91) The radial flow field of an incompressible fluid is shown below. Which of the closed paths would exhibit a non-zero flux?

91) ______ A) A and B B) C C) none of these D) A, B, and C

92) The radial flow field of an incompressible fluid is shown below. For which of the closed paths is the circulation not necessarily zero?

92) ______ A) none of these B) C only C) A and B only D) A, B, and C

Test the vector field F to determine if it is conservative. 93) F=5

i+5

j+5

93)

______ A) Conservative B) Not conservative

94) F=

94) ______ A) Not conservative B) Conservative

95)

F=3

95) ______ A) Not conservative B) Conservative

96)

F = xyi + yj + zk 96) ______ A) Not conservative B) Conservative

97) F = -cos x cos yi + sin x sin yj -

97) ______ A) Conservative B) Not conservative

98) F=-

x csc yi - cot x cot y csc yj - cos xk

98) ______ A) Not conservative B) Conservative

99)

i+z

99) ______ A) Conservative B) Not conservative

100)

100) _____ A) Not conservative B) Conservative

101)

101) _____ A)

Conservative B) Not conservative

102)

102) _____ A) Conservative B) Not conservative

Find the potential function f for the field F. 103) F=5

i + 10

j + 10

103) _____ A) f(x, y, z) =

+C B)

f(x, y, z) =

+ 10

+C C)

f(x, y, z) =

D) f(x, y, z) =

104)

F=-

104) _____ A)

f(x, y, z) =

+C B)

f(x, y, z) =

+C C)

f(x, y, z) = -

f(x, y, z) =

105)

i - 6j -

105) _____ A)

f(x, y, z) =

-6+C B)

f(x, y, z) =

- 6y + C C)

f(x, y, z) =

+C D)

f(x, y, z) =

- 6y + C

106) F=

yi + 2 tan x sin y cos yj + 8

106) _____ A) f(x, y, z) = tan x

+C B)

f(x, y, z) = -sec x

+C C)

f(x, y, z) = sec x

+C D)

f(x, y, z) = sec x

y+z+C

107) F = 2x

i + 2y

107) _____ A)

f(x, y, z) =

B) f(x, y, z) = 2

+C C)

f(x, y, z) =

+C D)

f(x, y, z) =

108)

F=-

108) _____ A)

f(x, y, z) =

+C B)

f(x, y, z) = ln (y - x - z) + C C)

f(x, y, z) =

+C D)

f(x, y, z)= ln

109)

F=-

109) _____ A)

f(x, y, z) = -

+C B)

f(x, y, z) =

+C D)

f(x, y, z) =

110)

110) _____ A) f(x, y, z) =

+ ln z + C

f(x, y, z) =

+C C)

f(x, y, z) =

+C D)

f(x, y, z) =

111) F=(

- sin x)i + 2xy

j+k

111) _____ A) f(x, y, z) = cos x +

+z +C B)

f(x, y, z) = cos x + xz

+C C)

f(x, y, z) = cos x + x

+z +C D)

f(x, y, z) = cos x +

(x + z) + C

112) F = (y - z)i + (x + 2y - z)j - (x + y)k

112) _____ A) f(x, y, z) = xy +

-x-y+C B)

f(x, y, z) = x(y +

) - xz - yz + C C)

f(x, y, z) = x +

- xz - yz + C D)

f(x, y, z) = xy +

- xz - yz + C

Evaluate. The differential is exact. 113)

113) _____ A)

B) 3 C) 1 D) 0

114)

114) _____ A) 2 B) -2 C) 1 D) 0

115)

115) _____ A) 640 B) 320 C)

832 D) 0

116)

116) _____ A) 0 B) -3 C) -1 D) +

-1

117)

117) _____ A) 0 B)

+ ln 3 C)

+ ln 3 D)

+ ln 3

118)

118) _____ A) 0 B) 4 C) 1 D) 2

119)

119) _____ A) ln 100 B) 0 C)

ln D) ln 14

120)

120) _____ A) 5 B) 7 C) 8 D) 1

121)

121) _____ A) 19 B) 17 C) 0 D) 18

122)

122) _____ A) 0 B) 2

+9 C)

+ 19 D)

+ 18

Evaluate the work done between point 1 and point 2 for the conservative field F. 123) F = (y + z)i + xj + xk;

(0, 0, 0),

(1, 1, 8)

123) _____ A) W=0 B) W = -7 C) W=1 D) W=9

124) F = 6xi + 6yj + 6zk;

(1, 5, 5) ,

(4, 6, 7)

124) _____ A) W = - 150 B) W = 150 C)

W=0 D) W = 456

125)

F = 4 sin 4x cos 9y cos 2zi + 9 cos 4x sin 9y cos 2zj + 2 cos 4x cos 9y sin 2zk ;

(0, 0, 0) ,

125) _____ A) W=2 B) W=0 C) W=1 D) W = -2

126) F = 16x

i - 8y

j - 12z

(0, 0, 0) ,

(1, 1, 1)

126) _____ A) W= B)

-1 C)

W=0 D) W=

-1

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 127) Consider a small region inside an elastic material such as gelatin. As the material "jiggles", this small region oscillates about its equilibrium position ( , , ). The force that tends to restore the small region to its equilibrium position can be approximated as F = - k(x - )i k(y -

)j - k(z -

)k Find a potential function f for this force field.

127) ____________

128)

Find the values of b and c that make F =

k a gradient field.

128) ____________

129) Find the values of a, b and c that make a gradient field. 129) ____________

130) For some inexact differential forms df, a function g(x, y, z) can be found such that dh = g(x, y, z) df is exact. When it exists, the function g(x,y,z) is called an "integrating factor". Show that g(x, y, z) =

is an integrating factor for the inexact differential df = -

dx +

dy +

dz.

130) ____________

131) In thermodynamics, the differential form of the internal energy of a system is dU = T dS - P dV, where U is the internal energy, T is the temperature, S is the entropy, P is the pressure, and V is the volume of the system. The First Law of Thermodynamics asserts that dU is an exact differential. Using this information, justify the thermodynamic relation

131) ____________

132)

Show that df = -

dx +

dy -

dz is exact.

132) ____________

133) Show that the value of the integral does not depend on the path taken from A to B.

+ 2y dy + 7xz dz 133) ____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. 134) F=(

)i + (x - y)j; C is the rectangle with vertices at (0, 0), (1, 0), (1, 3), and (0, 3)

134) _____ A) 12 B) 6 C) 0 D) -6

135)

F = sin 3yi + cos 8xj; C is the rectangle with vertices at (0, 0),

, and

135) _____ A)

π B)

C) 0 D)

136) F = (x - y)i + (x + y)j; C is the triangle with vertices at (0, 0), (6, 0), and (0, 4) 136) _____ A) 24 B) 96 C) 48 D) 0

137) F = xyi + xj; C is the triangle with vertices at (0, 0), (2, 0), and (0, 10) 137) _____ A)

C) 0 D)

138) F = (x - cos y)i + (x + quadrant.

sin y)j; C is the lobe of the lemniscate

= sin 2θ that lies in the first

138) _____ A) 1 B)

C) 0 D)

139)

F = (-y - cos x)i + (y the first quadrant.

sin x)j; C is the right lobe of the lemniscate

= cos 2θ that lies in

139) _____ A) 0 B) 1 C)

140)

F = ln ( + ≤ 5 and

)i +

j; C is the region defined by the polar coordinate inequalities 1 ≤ r

140) _____ A) 48 B) -8 C) 0 D)

141) F=6 and

j; C is the region defined by the polar coordinate inequalities 1 ≤ r ≤

141) _____ A) 30 B) 0 C) 37 D) 70

142)

F=-

i; C is the region defined by the polar coordinate inequalities 1 ≤ r ≤ 2 and

142) _____ A) 0 B)

143) F = (-2x + 10y)i + (6x - 8y)j; C is the region bounded above by y = -3

+ 7 and below by

in the first quadrant 143) _____ A) 24 B)

Using Green's Theorem, find the outward flux of F across the closed curve C. 144) F =(

)i + (x - y)j ; C is the rectangle with vertices at (0, 0), (1, 0), (1, 4), and

144) _____

A) -12 B) 0 C) 8 D) 20

145)

F = sin 3yi + cos 2xj ; C is the rectangle with vertices at (0, 0),

, and

145) _____ A) 0 B)

π C)

π D)

146)

F = (x - y)i + (x + y)j; C is the triangle with vertices at (0, 0), (3, 0), and (0, 10) 146) _____ A) 300 B) 0 C) 30 D) 60

147) F = xyi + xj; C is the triangle with vertices at (0, 0), (10, 0), and (0, 10) 147) _____ A)

B) 0 C)

148) F = (x - cos y)i + (x + quadrant

sin y)j; C is the lobe of the lemniscate

= sin 2θ that lies in the first

148) _____ A) 0 B)

C) 1 D)

149) F = (-y - cos x)i + (y the first quadrant.

sin x)j ; C is the right lobe of the lemniscate

= cos 2θ that lies in

149) _____ A)

C) 1 D) 0

150)

F = ln (

+ )i+ and

j; C is the region defined by the polar coordinate inequalities

150) _____ A) 72 B) 90 C) 144 D) 0

151) F=-

j ; C is the region defined by the polar coordinate inequalities

and 151) _____ A) 0

B) 42 C) 29 D) 21

152)

F=-

i ; C is the region defined by the polar coordinate inequalities

and

152) _____ A)

D) 0

153) F = (9x + 7y)i + (7x - 4y)j; C is the region bounded above by y = -5 in the first quadrant

+ 10 and below by

153) _____ A) -45 B) - 35 C) - 98 D)

Apply Green's Theorem to evaluate the integral. 154)

(7y + x) dx + (y + 4x) dy C The circle

154) _____ A) -12π B) -72 C) -12 D) 12π

155)

( + 6) dx + ( + 4) dy C The triangle bounded by x = 0, x + y = 4, y = 0 155) _____ A)

B) 0 C) 256 D) -192

156)

(5y dx + 3y dy) C The boundary of 0 ≤ x ≤ π, 0 ≤ y ≤ sin x 156) _____ A) 0 B)

-2 C) 4 D) -4

157)

(6x + ) dx + (3x + 5y) dy C Any simple closed curve in the plane for which Green's Theorem holds 157) _____ A) 2 B) There is not enough information to evaluate the integral. C) -2 D) 0

Using Green's Theorem, calculate the area of the indicated region. 158) The area bounded above by y = 3x and below by y = 4 158) _____ A)

159)

The area bounded above by y = 8 and below by y = 159) _____ A)

C) 0 D)

160) The area bounded above by y = 2

and below by y = 2

160) _____ A)

161) The circle r(t) = (8 cos t)i + (8 sin t)j, 0 ≤ t ≤ 2 π 161) _____ A) 16π B) 64π C) 2π

D) 8π

162) The astroid r(t) = (3

t)i + (3

t)j, 0 ≤ t ≤ 2π

162) _____ A) 2π B)

π C)

π D)

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 163)

Consider the counterclockwise integral where C is a closed path in a region where Green's Theorem applies. To evaluate the integral, should one use the fluxdivergence form or the circulation-flow form of Green's theorem? Explain. 163) ____________

164)

Assuming C is a closed path , what is special about the integral Give reasons for your answer.

164) ____________

165)

Assuming C is a simple closed path , what is special about the integral

? Give reasons for your answer. 165) ____________

166)

Assuming C is a simple closed path , what is special about the integral

? Give reasons for your answer. 166) ____________

167)

Assuming all the necessary derivatives exist, show that if

= 0 for all closed

curves C to which Green's Theorem applies, then f satisfies the Laplace equation = 0 for all regions bounded by closed curves C to which Green's Theorem applies.

167) ____________

168)

Let M = and N = . Show that where R is the region bounded by the unit circle C centered at the origin. Why is Green's Theorem failing in this case? 168) ____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 169) Let f(x,y) = ln ( + ). Which one of the following curves is a simple closed path in the domain of the function f? 169) _____ A) The rectangle with vertices at (-3, 2), (3, 2), (3, -2), and (-3, -2) B) The unit circle

=1 C)

The ellipse

=1 D)

The unit circle

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Parametrize the surface S. 170) S is the portion of the plane 8x + 2y - 7z = 5 that lies within the cylinder

170) ____________

171)

S is the portion of the cone

that lies between z = 6 and z = 8.

171) ____________

172) S is the portion of the cylinder

= 36 that lies between z = 3 and z = 8.

172) ____________

173)

S is the cap cut from the paraboloid z =

-9

173) ____________

by the cone z =

174) S is the portion of the paraboloid z = 3

that lies between z = 2 and z = 7.

174) ____________

175) S is the lower portion of the sphere

= 9 cut by the cone z =

175) ____________

176) S is the portion of the sphere

= 64 between z = - 4

and z = 4

176) ____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Calculate the area of the surface S. 177) S is the portion of the plane 5x + 2y + 7z = 5 that lies within the cylinder

177) _____ A)

π B)

π C)

π D)

178)

S is the portion of the cone

that lies between z = 2 and z = 5.

178) _____ A)

π B)

π C)

π D)

179) S is the portion of the cylinder

= 4 that lies between z = 1 and z = 2.

179) _____ A) 6π B) 2π C) 12π D) 4π

180)

S is the cap cut from the paraboloid z =

-9

by the cone z =

180) _____ A)

181) S is the portion of the paraboloid z = 2

that lies between z = 4 and z = 5.

181) _____ A)

182) S is the lower portion of the sphere

= 9 cut by the cone z =

182) _____ A)

π B)

9 C)

π D)

183)

S is the portion of the sphere

= 25 between z = -

and z =

183) _____ A) 50π B) 5

π C)

π D)

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the equation for the plane tangent to the parametrized surface S at the point P. 184)

S is the cone r(r, θ) = r cos θi + r sin θj + 4rk; P =

184) ____________

185) S is the sphere r(θ, φ) = 5 cos θ sin φi + 5 sin θ sin φj + 5 cos φk; P is the point corresponding to

185) ____________

186)

S is the paraboloid r(θ, r) = r cos θi + r sin θj + 5 k ; P is the point corresponding to . 186) ____________

187)

S is the cylinder r(θ, z) = 2 cos θi + 2 sin θj + zk ; P is the point corresponding to . 187) ____________

188) S is the parabolic cylinder r(x, z) = xi + 3 .

j + zk; P is the point corresponding to

188) ____________

189)

S is the cylinder r(θ, z) = 14

θi + 7 sin 2θj + zk; P is the point corresponding to

. 189) ____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the surface area of the surface S. 190)

S is the paraboloid

- z = 0 below the plane z =

190) _____ A)

π B)

π C)

π D)

191)

S is the paraboloid

- z = 0 between the planes z =

and z =

191) _____ A) 14π B)

π C)

π D)

192) S is the intersection of the plane 3x + 4y + 12z = 7 and the cylinder with sides y = 4 8-4

and y =

192) _____ A)

193)

S is the surface = x.

+ 11z = 0 that lies above the region bounded by the x-axis,

and y

193) _____ A)

B) 7 C)

194) S is the upper cap cut from the sphere

= 169 by the cylinder

194) _____ A) 3744π B) 208π C) -208π D) 104

195) S is the cap cut from the sphere

= 25 by the cone z = 4

195) _____ A)

π B)

25 C)

π D)

100

196) S is the area cut from the plane z = 9y by the cylinder

= 81.

196) _____ A) π

B) 81

π C)

81 D) 9

197) S is the portion of the paraboloid z = 25 the

that lies above the ring 4 ≤

≤ 16 in

197) _____ A)

[65

- 17

] B)

[65

- 17

] C)

101

[65

- 17

] D)

[65

- 17

]

198) S is the portion of the surface and in the x-y plane.

that lies above the rectangle

198) _____ A)

[40 + ln 9] B) 144[40 + ln 9] C) 16[40 + ln 9] D) 32[40 + ln 9]

199) S is the portion of the surface 5x + 12z = 5 that lies above the rectangle 2 ≤ x ≤ 7 and in the 199) _____ A) 234

102

B) 260 C)

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 200) The curve x = is revolved about the x-axis. Find a parametrization of the surface of revolution and then find the equation of the tangent plane at the point 200) ____________

201)

Find a parametrization for the ellipsoid of an ellipse

= 1. (Recall that the parametrization

= 1 is x = 10 cos θ, y = 2 sin θ, 0 ≤ θ < 2π).

201) ____________

202)

For a surface parametrized in the parameters u and v and a force F, show that dudv.

103

dσ =

202) ____________

203) Suppose that the parametrized plane curve C (f(u), g(u)) is revolved about the x-axis, where

g(u) > 0 and a ≤ u ≤ b. Show that the surface area of the surface of revolution is 2π

du 203) ____________

204)

For the surface z = f(x,y), show that the surface integral

dσ =

dxdy. 204) ____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the surface integral of G over the surface S. 205) S is the parabolic cylinder y = 2

, 0 ≤ x ≤ 5 and 0 ≤ z ≤ 6; G(x, y, z) = 3x

205) _____ A)

104

206) S is the cylinder

= 144, z ≥ 0 and 3 ≤ x ≤ 9; G(x, y, z) = z

206) _____ A) 1728 B) 144 C) 864 D) 3456

207) S is the hemisphere

= 5, z ≥ 0; G(x,y,z) =

105

207) _____ A)

π B) 100π C)

π D)

208) S is the plane x + y + z = 2 above the rectangle 0 ≤ x≤ 4 and 0 ≤ y ≤ 4; G(x,y,z) = 8z 208) _____ A) - 256 B) 768 C) 256 D) -256

106

209) S is the portion of the cone z = 5

, 0 ≤ z ≤ 2; G(x, y, z) = z - y

209) _____ A)

π C)

π D)

210)

S is the dome z = 4 - 8

-8

, z ≥ 0; G(x, y, z) =

210) _____ A) 32π B) 4π C)

107

π D) 32

211)

S is the cap cut from the sphere

= 16 by the cone z =

;

211) _____ A) 4π B) 0 C) 16π D) 8π

Evaluate the surface integral of the function g over the surface S. 212) G(x,y,z) = planes

+ ; S is the surface of the cube formed from the coordinate planes and the and

212) _____ A)

108

C) 80 D) 112

213) G(x, y, z) = x + z; S is the surface of the wedge formed from the coordinate planes and the planes and y = 2 213) _____ A) 16 + 8 B)

+8 C)

+4 D)

214)

109

G(x, y, z) = ; S is the surface of the rectangular prism formed from the coordinate planes and the planes x = 2, y = 3, and z = 3 214) _____ A)

C) 30618 D)

215) G(x, y, z) = = ± 2, and

; S is the surface of the rectangular prism formed from the planes x = ± 1, y

215) _____ A) 88 B) 4 C) 44 D)

110

132

216)

G(x, y, z) = ; S is the surface of the parabolic cylinder 32 x = 1, y = 0, and z = 0 planes

+ 8z = 16 bounded by the

216) _____ A) 2 B)

Find the flux of the vector field F across the surface S in the indicated direction. 217) F= i + yj - zk; S is the portion of the parabolic cylinder z = 9 ≤ 10; direction is outward (away from the x-y plane)

for which

and 8 ≤ x

217) _____ A) 54 B)

111

108 C) -108 D) 0

218) F = 2 j - k; S is the portion of the parabolic cylinder y = 2 ≤ 2; direction is outward (away from the y-z plane)

for which 0 ≤ z ≤ 4 and -2 ≤ x

218) _____ A)

B) -128 C)

D) 128

219)

k ; S is the upper hemisphere of

219) _____

112

= 4; direction is outward

π B)

220) F = 2xi + 2yj + zk; S is portion of the plane x + y + z = 7 for which 0 ≤ x ≤ 2 and direction is outward (away from origin) 220) _____ A) 11 B) 17 C) 10 D) 34

221)

113

F = 4xi + 4yj + outward

k; S is portion of the cylinder

= 16 between z = 0 and z = 4; direction is

221) _____ A) 512 B) -512π C) 512π D) 256

222) F = yi - zk; S is portion of the cone z = 3 outward

between z = 0 and z = 3; direction is

222) _____ A) - 2π B) 2π C) - 6π D) -1

114

223) F = xi + yj + outward

k; S is portion of the cone z = 2

between z = 2 and z = 4; direction is

223) _____ A)

π B)

224) F = 8xi + 8yj + 6k; S is "nose" of the paraboloid z = 6 outward

cut by the plane z = 2; direction is

224) _____ A)

π B)

115

225) F(x, y, z) = 5i + 7j + 2k , S is the rectangular surface z = 0, 0 ≤ x ≤ 9, and 0 ≤ y ≤ 5, direction k 225) _____ A) 315 B) 225 C) 90 D) 0

226) F(x, y, z) = 6i + direction i

k , S is the rectangular surface x = 0, -9 ≤ y ≤ 9, and -3 ≤ z ≤ 3,

226) _____ A) 108

116

B) 0 C) 648 D) 324

227) F(x, y, z) = zk , S is the surface of the sphere away from the origin

= 16 in the first octant , direction

227) _____ A) 16π B)

π C) 0 D)

228) F(x, y, z) = 3xi + 3yj + 3zk , S is the surface of the sphere direction away from the origin 228)

117

= 4 in the first octant,

_____ A) 24π B) 8π C) 12π D) 0

229) F(x, y, z) = 10xi + 10yj + 2k , S is the surface cut from the bottom of the paraboloid z = by the plane direction is outward

229) _____ A) -1584π B) 792π C) 828π D) 84π

230) F(x, y, z) = xyi + yzj + xzk , S is the surface of the rectangular prism formed from the

118

coordinate planes and the planes x = 3, y = 4, and z = 4, direction is outward 230) _____ A) 132 B) 72 C) 528 D) 264

231) F(x, y, z) = xzi + yzj + k , S is the cap cut from the sphere direction is outward

= 16 by the plane

231) _____ A) 60π B) 168π C) 84π D) - 84π

119

Solve the problem. 232) The shape and density of a thin shell are indicated below. Find the coordinates of the center of mass. Shell portion of the sphere Density constant

= 1 that lies in the first octant

232) _____ A)

233) The shape and density of a thin shell are indicated below. Find the coordinates of the center of mass. Shell cylinder + Density constant

= 4 bounded by y = 0 and y = 9

233) _____ A)

120

234) The shape and density of a thin shell are indicated below. Find the coordinates of the center of mass. + Shell cone Density constant

= 0 between z = 2 and z = 3

234) _____ A)

235)

121

The shape and density of a thin shell are indicated below. Find the coordinates of the center of mass. Shell "nose" of the paraboloid

= 2z cut by the plane z = 4

Density δ = 235) _____ A)

236) The shape and density of a thin shell are indicated below. Find the moment of inertia about the z-axis. Shell portion of the cone Density δ = 3

= 0 between z = 3 and z = 4

236) _____ A)

π B)

122

= 525π C) = 525 D)

237) The shape and density of a thin shell are indicated below. Find the moment of inertia about the z-axis. Shell "nose" of the paraboloid

= 2z cut by the plane z = 1

Density δ = 237) _____ A)

π B)

= 16π C) = 2π D)

238) The shape and density of a thin shell are indicated below. Find the moment of inertia about the z-axis.

123

Shell upper hemisphere of Density δ = 4

= 9 cut by the plane z = 0

238) _____ A) = 36π B) = 432π C) = 54π D) = 216π

Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction. 239)

F = 2yi + yj + zk; C the counterclockwise path around the boundary of the ellipse

239) _____ A) -16π B) 8π C) -16 D)

124

16π

240) F = 5yi + 6xj + k; C the counterclockwise path around the perimeter of the triangle in the x-y plane formed from the x-axis, y-axis , and the line y = 5 - 4x 240) _____ A)

241) F = i + xyj + yk; C the counterclockwise path around the perimeter of the rectangle in the xy plane formed from the x-axis, y-axis , x = 2 and y = 4 241) _____ A) 32 B) 16

125

C) -32 D) - 16

242) F = 6yi + 3xj - 4

k ; C the portion of the plane 7x + 6y + 8z = 9 in the first quadrant

242) _____ A)

C) 0 D) -3

243) F = 3xi + 2xj + 7zk; C the cap cut from the upper hemisphere

= 16

by the

cylinder 243) _____ A) 4π

126

B) 2π C) 8π D) 3π

244) F = -3

i+3

j+2

k ; C the portion of the paraboloid

= z cut by the cylinder

244) _____ A) - 72π B) -144π C) 72π D) 144π

Find the flux of the curl of field F through the shell S. 245) F=

j + 2xyk; S is the portion of the paraboloid 1 -

245)

127

= z that lies above the

_____ A) -2π B) 0 C) -2 D) 2

246) F = -3

yi + 3x

k; S is the portion of the paraboloid 3 -

= z that lies above the

246) _____ A) 27π B)

π C) 162 D) 162π

247) F = (x-y)i + (x-z)j + (y-z)k; S is the portion of the cone z = 4

128

below the plane z = 3

247) _____ A)

π B)

π C)

π D)

248) F = -6zi + 4xj + 4yk; S is the portion of the cone z = 3

below the plane z = 1

248) _____ A)

π B)

π C)

129

249) F = (4 - y)i + (8 + x)j +

k; S is the upper hemisphere of

249) _____ A) -4π B) 2π C) 18π D) 36π

250) F=

i + 9xj + 9k; S is the upper hemisphere of

= 16

250) _____ A) 144π B) 16 C)

130

16π D) π

251) F = 5yi + 8zj + 9xk; S r(r, θ) = r cos θi + r sin θj + (36 -

)k, 0 ≤ r ≤ 6 and 0 ≤ θ ≤ 2π

251) _____ A) -30π B) -180 C) -180π D) 180π

252) F = (z - y)i + (x - z)j + (z - x)k; S r(r,θ ) = r cosθi + r sinθj + (9 -

)k, 0 ≤ r ≤ 3 and

252) _____ A) -36π B) 36π C)

131

-18π D) 18π

253) F=1

yi - 1x

j + ln zk ; S r(r, θ) = r cos θi + r sin θj + 3rk, 0 ≤ r ≤ 3 and 0 ≤ θ ≤ 2π

253) _____ A) - 18π B) - 18 C)

π D) 18

254) F = -5zi - 2xj + 5yk ; S r(r, θ) = r cos θi + r sin θj + 4rk, 0 ≤ r ≤ 4 and 0 ≤ θ ≤ 2π 254) _____ A) 0 B) -32π

132

C) 32π D) -32

255)

F = (3y + 3)i - 4xj + (

- 2)k; S r(r, θ) = 2 sin φ cos θi + 2 sin φ sin θj + 2 cos φk, 0 ≤ θ ≤ 2π and

0≤φ≤ 255) _____ A) 56 B) -56π C) -28π D) 28π

256) F = 5yi + 4xj + cos(z)k ; S r(r,θ) = 5 sin φ cos θi + 5 sin φ sin θj + 5 cos φk, 0 ≤ θ ≤ 2π and 0 ≤ φ≤ 256) _____ A)

133

-25 B) -50π C) -25π D) 50π

Solve the problem. 257) The flow F of a fluid in a plane is illustrated below. At which point or points would point out from the page?

257) _____ A) B B) A C) A and C D)

134

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 258) Assume the curl of a vector field F is zero. Can one automatically conclude that the

circulation

for all closed paths C? Explain or justify your answer.

258) ____________

259) Consider a fluid with a flow field F = i + 7 j + zk. A miniature paddlewheel (idealized) Find a vector describing the orientation of is to be inserted into the flow at the point the paddlewheel axis which produces the maximum rotational speed. 259) ____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 260) Find values for a, b, and c so that ∇ × F = 0 for F = axi + byj + czk. 260) _____ A) a = 2, b = -1, c = -1 B) a = 1, b = 1, c = 1 C)

135

∇ × F = 0 for any a, b, and c. D) There is no possible way to make ∇ × F = 0.

Find the divergence of the field F. 261) F = -40x

i + 12yj + 5

261) _____ A) -40

+ 12 B)

12 C) -80 D) -80

+ 12

262) F = -4

i + 7xyj - 2xzk

262) _____ A) -20

+ 5x B)

136

-20

+ 7y - 2z C)

-20

+ 5x - 15 D)

-15

263) F = -7

i+2

j-3

263) _____ A) -7

-3 B)

0 C) -72 D) -63

+ 18

- 27

264)

F= 264) _____ A) 0

137

265)

F= 265) _____ A)

C) 0 D)

138

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. 266) F = zi + xyj + zyk; D the solid cube cut by the coordinate planes and the planes z=1

y = 1, and

266) _____ A)

C) 2 D) 1

267) F= i+ z=1

j + zk; D the solid cube cut by the coordinate planes and the planes x = 1, y = 1, and

267) _____ A) 3 B) 4 C)

139

1 D) 2

268) F = (y-x)i + (z-y)j + (z-x)k ; D the region cut from the solid cylinder z = 0 and

≤ 36 by the planes

268) _____ A) 288π B) 0 C) -288 D) -288π

269) F = 4x i + 4 = 0 and

yj + 6xyk ; D the region cut from the solid cylinder

≤ 16 by the planes z

269) _____ A) 6144π B)

140

3072π C) 1536π D) 12,288π

270) F=

i + 5yj + 3

k ; D the solid sphere

≤9

270) _____ A) 423π B) 36π C) 540π D) 180π

271) F=5

i+5

j+5

k ; D the thick sphere 4 ≤

≤9

271) _____ A) 12,660π B)

141

285π C) 2532π D)

272) F = sin yi + xzj + 2zk ; D the thick sphere 4 ≤

≤9

272) _____ A)

π B) 54π C) 152π D)

273) F=x

i+y

j+z

k ; D the thick cylinder 2 ≤

142

≤3,