TEST BANK FOR BRIEF CALCULUS AN APPLIED APPROACH 9TH EDITION by QuentinAxel

Chapter 1: Functions, Graphs, and Limits 1. Plot the points (–4, 2), (3, –3), (–5, –2), (4, 0), (3, –4) in the Cartesian plane. A)

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Ans: C 2. Find the distance between the points (3, 4) and (7, 7) . Round your answer to the nearest hundredth. A) 25.00 B) 2.65 C) 5.00 D) 14.87 E) 4.58 Ans: C 3. Find the midpoint of the line segment joining the points (5,1) and (7, 7) . Round your answer to the nearest hundredth. A) (12, 28) B) (6, 4) C) (–1, –3) D) (3, 7) E) none of these choices Ans: B

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4. Find the length of each side of the right triangle from the following figure.

A) a = 15, b = 8, c = 17 B) a = 8, b = 15, c = 17 C) a = 17, b = –8, c = 15 D) a = 15, b = 8, c = –17 E) a = 15, b = 15, c = 17 Ans: A 5. Find x such that the distance between the points (5, 2) and ( x,8) is 10. A) x = 13 or x = –13 B) x = –3 or x = 3 C) x = 13 or x = –3 D) x = –11 or x = 11 E) x = 15 or x = –15 Ans: C

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6. Assume that the number (in millions) of basic cable television subscribers in the United States from 1996 through 2005 is given in the following table. Use a graphing utility to graph a scatter plot of the given data. Describe any trends that appear within the last four years. Year 1996 1997 1998 1999 2000 2001 2002 2003 2004 2 Subscribers 62.3 63.6 64.7 65.5 66.3 66.7 66.5 66.0 65.7 6 A)

The number of subscribers appears to be increasing. B)

The number of subscribers appears to be decreasing. C)

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The number of subscribers appears to be linearly decreasing. D)

The number of subscribers appears to be decreasing. E)

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The number of subscribers appears to be linearly increasing. Ans: B

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7. Assume that the number (in millions) of cellular telephone subscribers in the United States from 1996 through 2005 is given in the following table. Use a graphing utility to graph a line plot of the given data. Describe any trends that appear within the last four years. Year 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Subscribers 200.5 195.1 192.9 185.3 146.6 125.6 102.5 76.7 65.9 57.3 A)

The number of subscribers appears to be decreasing. B)

The number of subscribers appears to be increasing. C)

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The number of subscribers appears to be constant. D)

The number of subscribers appears to be decreasing. E)

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The number of subscribers appears to be increasing linearly. Ans: A

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8. Assume that the median sales prices of existing one family homes sold (in thousands of dollars) in the United States from 1990 through 2005 are as given in the following figure. Use the following figure to estimate the percent increase in the value of existing one-family homes from 1997 to 1998.

A) 0.021% B) 0.058% C) 2.06% D) 55.5% E) 5.8% Ans: E 9. Use the Midpoint Formula repeatedly to find the three points that divide the segment joining x1, y1 and x2, y2 into four equal parts.

(

)

(

)

 3 x1 + x2 3 y1 + y2   x1 + x2 y1 + y2   x1 + 3 x2 y1 + 3 y2  , , ,  , ,  2   4 4   4 4   2 B)  3 x1 + x2 3 y1 + y2   x1 + x2 y1 + y2   x1 + 3 x2 y1 + 3 y2  , , ,  , ,  4   2 2   4 4   4 C)  3 x1 + x2 y1 + y2   x1 + 3 x2 y1 + 3 y2   x1 + 3 x2 y1 + 3 y2  , , ,  , ,  2   4 4   4 4   2  3 x1 + x2 y1 + y2   x1 + 3 x2 y1 + 3 y2   x1 + x2 y1 + y2  , , ,  , ,  2   4 4   4 4   2 D)  3 x1 + x2 y1 + 4 y2   x1 + 4 x2 y1 + 4 y2   x1 + 4 x2 y1 + 4 y2  , , ,  , ,  2 4 4 4 4  2      E)  x1 + x2 y1 + y2   x1 + x2 y1 + y2   x1 + x2 y1 + y2  , , ,  , ,  2   4 2   2 4   4 Ans: B

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10. The red figure is translated to a new position in the plane to form the blue figure. Find the vertices of the transformed figure from the following graph. (In case your exam is printed in black and white - the red figure has one vertex at (0,0)).

A) (–2, 0), (–3, 2), (1, 0) B) (–2, 0), (–3, 2), (0,1) C) (–2, 0), (–3,1), (0,1) D) (–2, –3), (–3, 2), (0, 0) E) (–2, 0), (–3, 0), (0,1) Ans: B

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11. Which of the following is the correct graph of y= 4 − x ? A)

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Ans: C

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12. Which of the following is the correct graph of y = 2x – x2? A)

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Ans: A

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13. Which of the following is the correct graph of the given equation? y= − 1 − x2 A)

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Ans: D

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14. Sketch the graph of the equation. y= x + 4 A)

E) None of the above Ans: D

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15. Which of the following is the correct graph of y = x – x3? A)

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Ans: B 16. Find the x- and y- intercepts of the graph of the equation= y

16 − x 2 .

A) x-intercepts: (4, 0) , (−4, 0) ; y-intercepts: (0, 4) B) x-intercepts: (0, 4) , (0, −4) ; y-intercepts: (4, 0) C) x-intercepts: (0,16) , (0, −16) ; y-intercepts: (16, 0) , (−16, 0) D) x-intercepts: (0,16) , (0, 4) ; y-intercepts: (16, 0) , (4, 0) E) x-intercept: (4, 0) ; y-intercept: (0, 4) Ans: A 17.

Find the x- and y- intercepts of the graph of the equation y =

x 2 − 36 . x+6

A) x-intercept: (–6, 0) ; y-intercept: (0, 6) B) x-intercepts: (6, 0) , (–6, 0) ; y-intercepts: (0, –6) , (0, 6) C) x-intercept: (6, 0) ; y-intercept: (0, –6) D) x-intercepts: (0, –6) , (0, 6) ; y-intercepts: (6, 0) , (–6, 0) E) x-intercept: (36, 0) ; y-intercept: (0,36) Ans: C

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18. Sketch the graph of the function y = x –1. A)

Ans: A

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19. Sketch the graph of the equation: x = 3 – y2. A)

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Ans: C 20. Write the general form of the equation of the circle with center (4, 6) and solution point (0, 0) . A) x 2 − y 2 – 8 y –12 x = 0 2 2 B) x + y – 8 y +12 x = 0 2 2 C) x − y – 8 x –12 y = 0 2 2 D) x + y + 8 x –12 y = 0 2 2 E) x + y – 8 x –12 y = 0 Ans: E 21. Write the general form of the equation of the circle with endpoints of a diameter at (0, 0) and (–14,18) . A) x 2 − y 2 +14 y –18 x = 0 2 2 B) x + y +14 y +18 x = 0 2 2 C) x − y +14 x –18 y = 0 2 2 D) x + y –14 x –18 y = 0 2 2 E) x + y +14 x –18 y = 0 Ans: E 22. Find the points of intersection (if any) of the graphs of the equations 2 x + y = –20 and 6x − 4 y = –32 . A) (–8, –4) B) (–4, –8) C) (8, –4) D) (–4,8) E) (8, 4) Ans: A

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23. A manufacturer of DVD players has monthly fixed costs of $8600 and variable costs of $75 per unit for one particular model. For this model DVD player, find the function C ( x ) for monthly total costs where x denotes the number of units produced and sold.. A) C (= x ) 75 x − 8600 B)

C (= x ) 75 x + 8600

C= ( x ) 100 x + 8600

C ( x ) = 100 x

C (= x ) 25 x − 8600 Ans: B E)

24. A small business recaps and sells tires. The business has a revenue function R ( x) = 115 x and a cost function C= ( x) 3500 + 80 x , where x represents the number of sets of four tires recapped and sold. Find the number of sets of recaps that must be sold to break even. A) 100 B) 500 C) 35 D) 700 E) 80 Ans: A 25. Find the market equilibrium point for the following demand and supply functions below, where p is price per unit and q is the number of units produced and sold. Demand: p = −2q + 320 Supply: = p 6q + 2 A) = q 39.75, = p 240.50 B) = q 79.50, = p 161.00 C) = q 40.25, = p 239.50 D) = q 80.50, = p 159.00 E) = q 40.00, = p 240.00 Ans: A 26. Find the equilibrium point for the following supply and demand functions below, where p is price per unit and q is the number of units produced and sold. Demand: = p 520 − 3q Supply: = p 17 q + 80 A) = q 30, = p $430 B) = q 60, = p $340 C) = q 44, = p $388 D) = q 22, = p $454 E) = q 26, = p $442 Ans: D

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27. Estimate the slope of the line from the graph.

A) B) C) D)

1 5 2 5

1 5 E) 1 − 2 Ans: C −

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28. Estimate the slope of the line from the graph.

−

1 6

B) C)

6 1 6 D) –6 E) None of the above Ans: D 29. Find the slope of the line passing through the pair of points. ( –8,5) , ( 6, –12 ) A) 17 14 B) 17 14 C) 14 17 D) 14 17 E) None of the above Ans: B

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30. Find the slope of the line passing through the given pair of points.

( 7, 4 ) and ( 7, –4 ) A) B)

–8 1 – 7 C) –7 D) 0 E) The slope is undefined. Ans: E 31. Find the slope of the line passing through the given pair of points.

(17, –43) and ( 23, –25) A) B)

–18 1 3 C) 3 D) 1 – 3 E) The slope is undefined. Ans: C 32. Find the slope of the line passing through the given pair of points.

( –3,10 ) and ( –14, –1) A)

–

1 11

B) 1 C) –11 D) –17 E) The slope is undefined. Ans: B 33. Use the point (3,5) on a line having slope m = –4 to find two additional points through which the line passes. A) (–4,33), (2,9) B) (–4, −33), (2,9) C) (–4,33), (2, −9) D) (–4, −33), (2, −9) E) (–4,33), (−2, −9) Ans: A

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34. Find the slope m and y-intercept b of the line whose equation is given below. = y

3 1 x− 2 5

1 3 m= − ,b= 5 2 B) 2 1 m= , b= − 5 3 C) 3 1 m= , b= − 5 2 D) 3 1 m= , b= 5 2 E) 3 1 m= − ,b= 2 5 Ans: C

35. Find the slope m and y-intercept b of the line whose equation is given below. 2x + 5 y = 10 A) 2 m = − , b = –5 5 B) 2 m= − ,b=2 5 C) 5 m= , b=5 2 D) 2 m= , b=2 5 E) 5 m = − , b = –2 2 Ans: B

36. Find the slope m and y-intercept b of the line whose equation is given below. x= −

1 3

1 m= − , b=0 3 B) m = 0, b = 0 C) 1 1 m= − , b= 3 3 D) 1 m = 0, b = − 3 E) Both m and b are undefined. Ans: E

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37. Find the slope m and y-intercept b of the line whose equation is given below. y = –5 A) m = –5, b = 0 B) m = 0, b = 0 C) m = –5, b = 5 D) m = 0, b = –5 E) Both m and b are undefined. Ans: D

38. Write the equation of the line passing through the given pair of points.

( –6,5) and ( 5, 6 ) y= x − 1 1 61 = y x+ 11 11 C) 1 y= − x + 61 11 D) y =− x + 11 E) 1 11 = y x+ 11 61 Ans: B

A) B)

39. Write the equation of the line passing through the given pair of points.

( 3,10 ) and ( 9, 4 ) A) y =− x + 13 B) = y –5 x + 13 C) = y –13 x + 13 D) y = −x – 5 E) y = x–5 Ans: A

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40.

 9 5  10 16  Find an equation of the line that passes through the points  – ,  and  – , –  .  2 4  11 3  A) 11 y=– x 6 B) 11 y=– x–7 6 C) 11 y = – x – 14 6 D) 11 y = – x+7 6 E) 11 y = – x + 14 6 Ans: B

41. Find an equation of the line that passes through the point (–6,12) and has the slope m that is undefined. A) y = –6 B) x = –6 C) y = 12 D) x = 12 E) y = –6x Ans: B 42. Write the equation and graph the line that passes through the given point and has the slope indicated.

( –4, –4 ) with 0 slope A) y= x + 4 B) y = –4 y=x C) D) x=4 E) y= x − 4 Ans: B

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43. Write the equation of the line that passes through the given point and has the slope indicated.

( –4, –1) with slope −

3 5

3 y = − x –17 5 B) 3 17 y= − x– 5 5 C) 3 y = − x +5 5 D) 3 3 y= − x– 5 5 E) 3 17 y = − x+ 5 5 Ans: B

44. True or False: These three points are collinear. (1, 3), (0, 2), (–2, 1) A) true B) false Ans: B 45. Write the equation of the line through ( –7, –3) that is parallel to 4 x − 5 y = 6. A)

5 23 y = − x+ 4 5 B) 4 13 y = x+ 5 5 C) 4 23 y= x– 5 5 D) 4 y = x + 23 5 E) 4 13 y= − x– 5 5 Ans: B

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46. Write the equation of the line through ( –4, –7 ) that is perpendicular to = x 4 y + 8. A) 1 y = x – 23 4 B) y = 4x + 9 C) y = –4 x + 23 D) 1 y = x–9 4 E) y = –4 x – 23 Ans: E 47. Write an equation of the line that passes through the point (i) parallel to the given line, and (ii) perpendicular to the given line. Point Line

( –5, 4 )

–3 x – 9 y = –12

(i) parallel: –3 x – 9 y = –57 (ii) perpendicular: 9 x – 3 y = –21 B) (i) parallel: –3 x – 9 y = –21 (ii) perpendicular: 9 x – 3 y = –57 C) (i) parallel: 3 x – 9 y = –21 (ii) perpendicular: –9 x – 3 y = –57 D) (i) parallel: –3 x + 9 y = 51 (ii) perpendicular: 9 x – 3 y = –57 E) (i) parallel: 3 x – 9 y = –57 (ii) perpendicular: –3 x – 9 y = –21 Ans: B A)

48. Write an equation of the line that passes through the point (i) parallel to the given line, and (ii) perpendicular to the given line. Point Line ( 3, –6 ) x = –8 A) (i) parallel: x = 3 (ii) perpendicular: y = –6 B) (i) parallel: y = –6 (ii) perpendicular: x = 3 C) (i) parallel: x = –6 (ii) perpendicular: y = 3 D) (i) parallel: x = –3 (ii) perpendicular: y = 6 E) (i) parallel: y = 6 (ii) perpendicular: x = 3 Ans: A

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49. Find a linear equation that expresses the relationship between the temperature in degrees Celsius and degrees Fahrenheit. Use the fact that water freezes at 0°C ( 32° F ) and boils at 100°C ( 212° F ). Use the equation to convert 76° F to Celsius. A) 24°C B) 10°C C) 60°C D) 79°C E) 105°C Ans: A 50. Suppose the resident population of South Carolina (in thousands) was 4020 in 2000 and 4257 in 2007. Assume that the relationship between the population y and the year t is linear. Let t = 0 represent 2000. Estimate the population in 2004 by using linear model for the given data. Round your answer to the nearest thousand residents. A) 4336 thousand residents B) 3885 thousand residents C) 4122 thousand residents D) 4155 thousand residents E) 4392 thousand residents Ans: D 51. In 2004, a product has a value of $2875. Over the next five years, its value will increase by $150 per year. Write a linear equation that gives the dollar value V in terms of the year t. (Let t = 0 represent 2000.) A) V = 150t + 2875 B) V = 150t − 2875 C) V = 150t + 2275 D) V = 150t + 3475 E) V = 150t − 2275 Ans: C 52. A small business purchases a piece of equipment for $1030. After 10 years, the equipment will be outdated, having no value. Write a linear equation giving the value V of the equipment in terms of time t in years, 0 ≤ t ≤ 10. A) V = –103t − 1030 B) V = 103t + 1030 C) V = 103t − 1030 D) V = –103t + 1030 E) V = –103t + 103 Ans: D 53. If y 2 = 8 x 2 , is y a function of x? A) Yes B) No Ans: B

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54. If y 2 = 3 x, is y a function of x? A) Yes B) No Ans: B 55. Determine whether y is a function of x. y − 8x2 = 5 A) Yes B) No Ans: A 56. Determine whether y is a function of x. xy − x 2 = 9 y + x A) No B) Yes Ans: B 57. Determine the range of the function f ( x) = 5 x 2 − 10 x + 9 . A) [ 4, ∞) B) (6, ∞) C) (−∞,1] D) (−∞, –1) E) ( −∞, ∞ ) Ans: A

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58.

Determine the range of the function f ( x) =

4x . x

A) {−4, 4} B) [ −4,5 ] C) (−5,5) D) (−∞, 4 ] E) (−4, 4) Ans: A 59. Evaluate (if possible) the function at the given value of the independent variable. Simplify the results. f ( x ) = –9 x + 6, f ( –1) A) 15 B) 3 C) 5 D) –7 E) undefined Ans: A

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60. If C ( = x) A)

( x − 1) / x, find C ( ) . 2

1 5

24 25 B) 24 5 C) –5 D) 24 − 5 E) –24 Ans: D −

61. Simplify the expression using the given function definition. f ( x ) − f ( –7 ) f ( x ) = –13 x – 14, x+7 A) –8 B) –12 C) –13 D) –16 E) undefined Ans: C

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62. Use the Vertical Line Test to determine which of the following graphs shows y as a function of x. A)

E) Larson, Calculus: An Applied Approach, 9e

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Ans: E 63. Given f ( x) = x and g ( x= ) x 2 − 25 , find f ( g ( x)). A) f (= g ( x)) x ( x 2 − 25) B)

f ( g (= x))

x 2 − 25

f ( g ( x= ))

x −5

f ( g ( x= )) x − 25 E) f ( g ( x))= x − 25 Ans: B

64. Given f ( x= ) x 2 + 1 and g ( x)= x − 9 , evaluate f ( g (3)). A) 1 B) 4 C) –60 D) 37 E) 16 Ans: D

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65. Use the Horizontal Line Test to determine whether the functions are one-to-one.

A) f(x) and g(x) both are one-to-one. B) f(x) is not one-to-one and g(x) is one-to-one. C) f(x) and g(x) both are not one-to-one. D) f(x) is one-to-one and g(x) is not one-to-one. Ans: B

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66. Use the graph of f ( x) = x to sketch y = A)

x – 3.

E) Larson, Calculus: An Applied Approach, 9e

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Ans: B

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67. Use the graph of f ( x) = x below to sketch the graph of the following function:

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Ans: B

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68. The inventor of a new game believes that the variable cost for producing the game is $0.97 per unit. The fixed cost is $5000. Find a formula for the average cost per unit C = C / x . A) 0.97 x + 5000 B) 0.97 x − 5000 C) 5000 0.97 + x D) 5000 0.97x − x E) 0.97 − 5000 x Ans: C 69. A manufacturer charges $70 per unit for units that cost $60 to produce. To encourage large orders from distributors, the manufacturer will reduce the price by $0.02 per unit for each unit in excess of 100 units. (For example, an order of 101 units would have a price of $69.98 per unit, and an order of 102 units would have a price of $69.96 per unit.) This price reduction is discontinued when the price per unit drops to $64. Express the price per unit as a function of the order size. A) 0 ≤ x ≤ 100 70  = p 71 + 0.02 x 100 < x ≤ 400 64 x > 400  B) 0 ≤ x ≤ 100 70  p= 70 − 0.02( x − 100) 100 ≤ x ≤ 400 64 x > 400  C) 0 ≤ x ≤ 100 70  p= 70 − 0.02( x − 100) 100 < x ≤ 400 64 x > 400  D) 0 ≤ x ≤ 100 70  = p 70 + 0.02 x 100 < x ≤ 400 64 x ≥ 400  E) 0 ≤ x < 100 70  = p 69 − 0.02 x 100 < x ≤ 400 64 x ≥ 400  Ans: C

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70. Complete the table and use the result to estimate the limit. x–3 lim 2 x →3 x + x – 12 x 2.9 f(x) A) 0.142857 B) 0.642857 C) 0.517857 D) 0.767857 E) –0.232143 Ans: A

2.99

2.999

3.001

3.01

3.1

–1.99

–1.9

–7.99

–7.9

71. Complete the table and use the result to estimate the limit. –6 x – 10 − 2 lim x → –2 x+2 x –2.1 f(x) A) 2.12132 B) –1.99632 C) –2.12132 D) 1.954654 E) 1.87132 Ans: C

–2.01

–2.001

–1.999

72. Complete the table and use the result to estimate the limit. 1 1 + lim x – 3 11 x → –8 x +8 x –8.1 f(x) A) 0.121736 B) 0.101736 C) –0.138264 D) –0.008264 E) –0.118264 Ans: D

–8.01

–8.001

–7.999

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73. Suppose that lim f ( x) = 8 and lim g ( x) = –11 . Find the following limit: x →c

lim [ f ( x) + g ( x) ]

x →c

A) –88 B) 19 C) 0 D) –3 E) –11 Ans: D 74. Suppose that lim f ( x) = –12 and lim g ( x) = –11 . Find the following limit: x →c

lim [ f ( x) g ( x) ]

x →c

A) –12 B) –23 C) –1 D) 132 E) 11 Ans: D

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75. Let

 x 2 + 4, x ≠ 1 . f ( x) =  x =1  1, Determine the following limit. (Hint: Use the graph of the function.) lim f ( x) x →1

A) 5 B) 1 C) 4 D) 16 E) does not exist. Ans: A

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76. A graph of y = f ( x) is shown and a c-value is given. For this problem, use the graph to find lim f ( x) . x →c

c = −2

A) 0 B) 2 C) –6 D) –4 E) does not exist Ans: A 77. Use the graph of y = f ( x) and the given c-value to find lim+ f ( x) . x →c

c = −4.5

A) −6 B) –5 C) –7 D) 3 E) does not exist Ans: A 78. Find the limit (if it exists):

( x + ∆x ) – 11( x + ∆x ) + 2 − ( x 2 – 11x + 2 ) 2

lim

∆x

∆x → 0

1 3 11 2 x – x + 2x 3 2 3 B) x – 11x 2 + 2 x C) 0 D) 2 x – 11 E) x 2 – 11x + 2 Ans: D

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79.

1 . x→ – 4 x + 4 4 0

Find lim – A) B) C) D) –4 E) inf Ans: C

80.

x+7 . x →13 x – 13

Find the limit: lim+ A) –∞ B) ∞ C) 0 D) –1 E) 1 Ans: B

81.

Find lim+ x→ 3

–1

( x – 3)

A) 3 B) inf C) 0 D) –3 E) Ans: E

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82. Determine the following limit. (Hint: Use the graph of the function.) lim x →1

1 x −1

A) 0 B) does not exist C) 1 D) –1 E) –2 Ans: B 83. Graph the function with a graphing utility and use it to predict the limit. Check your work either by using the table feature of the graphing utility or by finding the limit algebraically. x3 − 2 x 2 − 24 x lim 2 x →3 x − 9 x + 18 A) 9 7 B) 21 C) 7 9 D) 0 E) does not exist Ans: E

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84. The cost (in dollars) of removing p % of the pollutants from the water in a small lake is 26, 000 p given by C , 0 ≤ p < 500 . Evaluate lim C . = p →500− 500 − p A) ∞ B) 26, 000 C) 0 D) −∞ E) −26, 000 Ans: A 85. Consider a certificate of deposit that pays 14% (annual percentage rate) on an initial deposit of $4000. The balance after 14 years= is A 4000(1 + 0.14 x)14 / x . Estimate lim+ A , where x is the length of the compounding period (in years). Round your answer x →0

to the nearest hundredth. A) 28,397.31 B) 1471.52 C) 4000.00 D) 56,000.00 E) 4560.00 Ans: A 86. Determine whether the given function is continuous. If it is not, identify where it is discontinuous. y = 5x2 − 6 x + 8 A) discontinuous at x = 9 B) discontinuous at x = 0 C) discontinuous at x = −9 D) discontinuous at x = 18 E) continuous everywhere Ans: E 87. Find the x-values (if any) at which the function f ( x) = – x 2 – 7 x + 3 is not continuous. Which of the discontinuities are removable? A) continuous everywhere B) x = 3 , removable C) 7 x = – , removable 2 D) 7 x = – , not removable 2 E) both B and C Ans: A

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88.

Describe the interval ( s ) on which the function f ( x) = A) B) C) D) E)

(−∞, −11], (−11,11] & (11, ∞) (−∞,11), (11,11) & (11, ∞) (−∞, −11), (−11,11) & (11, ∞) (−∞, −11], (−11,11) & (11, ∞)

x − 11 is continuous. x 2 − 121

(−∞, −11], [ −11,11] & [11, ∞) Ans: C

89. Determine whether the given function is continuous. If it is not, identify where it is discontinuous and which condition fails to hold. You can verify your conclusions by graphing the function with a graphing utility, if one is available. 4x − 7 y= 2 x +9 A) discontinuous at x = −9 B) discontinuous at x = 3 C) discontinuous at x = −3 D) discontinuous at x = 9 E) continuous everywhere Ans: E 90. Find the x-values (if any) at which f(x) is not continuous and identify whether they are removable or nonremovable. −14 x + 15, x < 1 f ( x) =  2 x ≥1 x , A) x = 1 is a removable discontinuity B) x = 1 is a nonremovable discontinuity C) x = -1 is a removable discontinuity D) x = -1 is a nonremovable discontinuity E) f(x) has no discontinuities Ans: E 91.

Find the x-values (if any) at which the function f ( x) =

x is not continuous. Which x +4 2

of the discontinuities are removable? A) 2 and -2, not removable B) continuous everywhere C) 2 and -2, removable D) discontinuous everywhere E) none of the above Ans: B

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92.

Find the x-values (if any) at which the function f ( x) =

x–9 is not continuous. x – 6 x – 27 2

Which of the discontinuities are removable? A) no points of discontinuity B) x = 9 (not removable), x = –3 (removable) C) x = 9 (removable), x = –3 (not removable) D) no points of continuity E) x = 9 (not removable), x = –3 (not removable) Ans: C

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93.

Sketch the graph of the function f ( x) =

x 2 − 81 and describe the interval(s) on which x−9

the function is continuous. A) (−∞,9] and [9, ∞)

(−∞, −9] and [9, ∞)

(−∞,9] and [ −9, ∞)

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(−∞,9) and (9, ∞)

E) none of these choices Ans: D 94.

 x 2 − 16, Describe the interval(s) on which the function f ( x) =  4 x + 16, A) (−∞, 0] and (0, ∞) B) (−∞, 0) and [0, ∞) C) (−∞, 0) and (0, ∞) D) (−∞, ∞) E) none of these choices Ans: C Larson, Calculus: An Applied Approach, 9e

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x≤0 is continuous. x>0

95. Find constants a and b such that the function

x ≤ –9 24,  f ( x)= ax + b, –9 < x < 7  –24, x≥7  is continuous on the entire real line. A) a = 3 , b = 0 B) a = 3 , b = –3 C) a = 3 , b = 3 D) a = –3 , b = 3 E) a = –3 , b = –3 Ans: E 96. A deposit of $7500 is made in an account that pays 6% compounded every 5 months. 12   t

The amount A in the account after t years= is A 7500(1 + 0.025)  5  , t ≥ 0 . What are the 12   t

points of discontinuity of graph = of A 7500(1 + 0.025)  5  ? (Here, the brackets indicate the greatest integer function.) A) 1 2 3 0, , , ,... 5 5 5 B) 0,1, 2,... C) 5,10,15,... D) 1, 2,3,... E) 5 5 5 , , ,... 12 6 4 Ans: E

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Chapter 2: Differentiation 1. Find the slope of the tangent line to the graph of the function below at the given point.

f ( x)  2 x –10, (3, –4) A) 2 B) –2 –10 C) D) 12 E) none of the above Ans: A 2. Find the slope of the tangent line to the graph of the function at the given point. f ( x)  –5 x 2 +10, (–2, –10) A) 20 –5 B) –10 C) –20 D) E) none of the above Ans: A

3. Find the slope of the tangent line to the graph of the function at the given point. f ( x)  2 x 2 + 6, (3, 24) A) 12 B) 2 –6 C) D) 18 E) none of the above Ans: A

4. Use the limit definition to find the slope of the tangent line to the graph of f ( x)  4 x  29 at the point (5, 7) . A) 2 7 B) 2  7 C) 1 7 D) 1  7 E) 1 5 Ans: A

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5. Find the derivative of the following function using the limiting process.

f ( x)  –2 x 2 – 9 x A) –2 –4 x – 9 B) –4 x + 9 C) –4x D) E) none of the above Ans: B 6. Find the derivative of the following function using the limiting process. f ( x)  9 x – 6 A) 9 f ( x)  2 9x – 6 B) 9 f ( x)   2 9x – 6 C) 9 1/ 2 f ( x)   9 x – 6  2 D) 9 f ( x)   9x – 6 E) either B or D Ans: A 7. Find the derivative of the following function using the limiting process. f ( x) 

A) B)

2 x–9

f ( x) 

 x – 9

f ( x)  

 x + 9

f ( x)  

2  x + 9

f ( x)  

 x – 9

E) none of the above Ans: D

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8. Find an equation of the line that is tangent to the graph of f and parallel to the given line. f ( x)  5 x 2 , 20 x  y  2  0 A) y  20 x – 20 y  20 x + 20 B) C) y  –20 x + 20 y  –20 x – 20 D) E) none of the above Ans: A 9. Find an equation of the a line that is tangent to the graph of f and parallel to the given line. f ( x)  5 x 3 , 135 x  y  6  0 A) y  –135 x – 270 B) y  135 x + 270 y  –135 x + 270 C) D) y  135 x – 270 E) both B and D Ans: E

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10. Identify a function f ( x) that has the given characteristics and then sketch the function. f (0)  3; f '( x)  4,   x   f ( x)  4 x  3 A)

f ( x)  –4 x  3

f ( x)  4 x  3

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f ( x)  –4 x  3

f ( x)  3x + 4

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Ans: A 11. Find the derivative of the function. f ( x)  x 7 A) f ( x)  7 x 7 B) f ( x)  7 x 6 C) f ( x)  6 x 6 D) f ( x)  6 x8 E) none of the above Ans: B 12. Find the derivative of the function. f ( x)  2 x 3 – 3x 2 +1 A) f ( x)  6 x 2 – 6 x B) f ( x)  4 x 2 – 3 x C) f ( x)  4 x – 3 x 2 D) f ( x)  6 x 2 – 6 x +1 E) none of the above Ans: A

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13. For the function given, find f '( x). f ( x)  x 3  15 x  6 A) x 2  15 B) 3x 2  6 C) 3x 2  15 D) 3x 3  15 x E) x3  15 x  6 Ans: C 14. Find the derivative of the function. h( x)  15 x 23  11x13  4 x10  3 x  7 A) 330 x 22  132 x12  36 x 9  3 B) 345 x 23  143x13  40 x10  3x C) 15 x 22  11x12  4 x9  3 D) 345 x 22  143x12  40 x 9  3 E) 330 x 23  132 x13  36 x10  3x Ans: D 15. Find the derivative of the function h( x)  x5/ 3 . A) 5 h '( x)  x8/ 3 3 B) 5 h '( x)   x 2 / 3 3 C) 5 2/3 h '( x)  x 3 D) 5 h '( x)   x8 / 3 3 E) 5 h '( x)  x 2 / 3 3 Ans: C 16. Find the derivative of the function s (t )  2 x 2  8 . A) 4 s '(t )  3 x B) 4 s '(t )   3 x C) 4 s '(t )   3  8 x D) 4 s '(t )  3  8 x E) s '(t )  2 x 3 Ans: B

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17. Find the derivative of the function. f ( x)  A)

1 x3

2 x4 B) 3 f ( x)   2 x C) 3 f ( x)   4 x D) 4 f ( x)   4 x E) none of the above Ans: C f ( x)  

18. Differentiate the given function. 3 y 4 4x A) 12  5 x B) 3  4 x C) 12  4 x D) 3  5 x E) 4  5 x Ans: D 19. Differentiate the given function. 5 y (4 x) 4 A) 80 (4 x)5 B) 20  (4 x)5 C) 80  (4 x)5 D) 20 (4 x)5 E) 20  (4 x)3 Ans: C Larson, Calculus: An Applied Approach (+Brief), 9e

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20. Determine the point(s), (if any), at which the graph of the function has a horizontal tangent. y ( x)  x 4  32 x  1 0 A) B) 0 and 2 C) 0 and –2 D) 2 E) There are no points at which the graph has a horizontal tangent. Ans: D 21. The graph shows the number of visitors V to a national park in hundreds of thousands during a one-year period, where t = 1 represents January. Estimate the rate of change of V over the interval 5,8 . Round your answer to the nearest hundred thousand visitors per year.

A) 176.92 hundred thousand visitors per year B) 328.57 hundred thousand visitors per year C) 166.67 hundred thousand visitors per year D) 383.33 hundred thousand visitors per year E) 766.67 hundred thousand visitors per year Ans: C 22. Find the marginal cost for producing x units. (The cost is measured in dollars.) C  205, 000  9800 x $9800 A) $9850 B) $8800 C) $8850 D) $9750 E) Ans: A Larson, Calculus: An Applied Approach (+Brief), 9e

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23. Find the marginal revenue for producing x units. (The revenue is measured in dollars.) R  50 x  0.5 x 2 50  x dollars A) 50  x dollars B) 50 dollars C) 50  0.5 x dollars D) 50  0.5 x dollars E) Ans: A 24. Find the marginal profit for producing x units. (The profit is measured in dollars.) P  2 x 2  72 x  145 4 x  72 dollars A) 4 x  72 dollars B)  4 x  72 dollars C) 4 x  72 dollars D) 4  72 x dollars E) Ans: A 25. The cost C (in dollars) of producing x units of a product is given by C  3.6 x  500 . Find the additional cost when the production increases from 9 t o10. $0.58 A) $0.36 B) $0.62 C) $0.12 D) $0.64 E) Ans: A 26. The profit (in dollars) from selling x units of calculus textbooks is given by p  0.05 x 2  20 x  3000 . Find the additional profit when the sales increase from 145 to 146 units. Round your answer to two decimal places. A) $5.45 B) $20.00 C) $5.55 D) $11.00 E) $10.80 Ans: A 27. The profit (in dollars) from selling x units of calculus textbooks is given by p  0.05 x 2  20 x  1000 . Find the marginal profit when x  148 . Round your answer to two decimal places. A) $34.80 B) $864.80 C) $5.20 D) $20.00 E) $859.55 Ans: C Larson, Calculus: An Applied Approach (+Brief), 9e

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28. The population P ( in thousands) of Japan from 1980 through 2010 can be modeled by P  15.56t 2  802.1t  117, 001 where t is the year, with t =0 corresponding to 1980. Determine the population growth rate, dP dt . A) dP dt  31.12t  802.1 B) dP dt  31.12t  802.1 C) dP dt  31.12t  802.1 D) dP dt  31.12t  802.1 E) dP dt  31.12  802.1t Ans: A 29. When the price of a glass of lemonade at a lemonade stand was $1.75, 400 glasses were sold. When the price was lowered to $1.50, 500 glasses were sold. Assume that the demand function is linear and that the marginal and fixed costs are $0.10 and $ 25, respectively. Find the profit P as a function of x, the number of glasses of lemonade sold. A) P  0.0025 x 2  2.65 x  25 B) P  0.0025 x 2  2.65 x  25 C) P  0.0025 x 2  2.65 x  25 D) P  0.0025 x 2  2.65 x  25 E) P  0.0025 x 2  2.65 x  25 Ans: A 30. When the price of a glass of lemonade at a lemonade stand was $1.75, 400 glasses were sold. When the price was lowered to $1.50, 500 glasses were sold. Assume that the demand function is linear and that the marginal and fixed costs are $0.10 and $ 25, respectively. Find the marginal profit when 300 glasses of lemonade are sold and when 700 glasses of lemonade are sold. A) P  300   1.15, P  700   0.85 B) P  300   0.85, P  700   1.15 C)

P  300   1.15, P  700   0.85

P  300   0.85, P  700   1.15

P  300   1.15, P  700   0.85

Ans: A 31. Use the product Rule to find the derivative of the function f  x   x  x 2  3 . A)

f   x   3x 2  3

f   x   3x 2  1

f   x   x2  3

f   x   3x 2  3

f   x   3x 2  1

Ans: A Larson, Calculus: An Applied Approach (+Brief), 9e

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32.

Find the derivative of the function f  x   A)

f   x   x2  2

f   x   x2  6

f   x   x2  2x

f   x   x2  x

f   x   x2  2x

x3  6 x . 3

Ans: A 33.

x 2  x  20 . State which differentiation Find the derivative of the function f  x   x4 rule(s) you used to find the derivative. A) 1, Product Rule. B) 1, Quotient Rule C) 5, Product Rule. D) 5, Quotient Rule E) x+3, Product Rule. Ans: A

34. Find the point(s), if any, at which the graph of f has a horizontal tangent line. x2 f  x  x 1 A)  0, 0  ,  2, 4  B) C) D) E)

 0, 2  ,  0, 4   4, 0  ,  2, 0   0, 4  ,  2, 0   0, 0  ,  4, 2 

Ans: A 35. A population of bacteria is introduced into a culture. The number of bacteria P can be 4t   where t is the time (in hours). Find the rate of change modeled by P  500 1  2   50  t  of the population when t = 2. 31.55 bacteria/hr A) 29.15 bacteria/hr B) 33.65 bacteria/hr C) 32.75 bacteria/hr D) 30.25 bacteria/hr E) Ans: A

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36. Use the given information to find f   2  of the function f  x   g  x  h  x  . g  2   3 and g   2   2, h  2   1 and h  2   4

f   2   14

f   2   11

f   2   17

f   2   9

f   2   12

Ans: A 37. Find an equation of the tangent line to the graph of f at the given point. f ( s )  ( s  4)( s 2  3), at 1, 6 

y  8s  14 A) B) y  2s – 8 C) y  –8s  2 y  –8s + 14 D) E) y  –8 + 14 s Ans: D 38. Find an equation of the tangent line to the graph of f at the given point. f ( s )  ( s  5)( s 2  6), at  3, –6  A) y  9 s  21 B) y  33s – 9 y  –9s  33 C) D) y  –9s + 21 E) y  –9 + 21s Ans: D 39.

 4p  Use the demand function x  325 1   to find the rate of change in the demand  5p  4  x for the given price p  $2.00 . Round your answer to two decimal places. A) 26.53 units per dollar B) –6.63 units per dollar C) 6.63 units per dollar D) 36.11 units per dollar E) –26.53 units per dollar Ans: E

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40. A population of bacteria is introduced into a culture. The number of bacteria P can be 4t   where t is the time (in hours). Find the rate of change modeled by P  225  1  2   45  t  of the population when t  4.00 . A) 31.03 units per dollar B) 1.75 units per dollar C) 7.01 units per dollar D) 3.63 units per dollar E) 7.76 units per dollar Ans: C 41. Find dy du , du dx , and dy dx of the functions y  u 2 , u  4 x  7 . A) dy du  2u , du dx  4, and dy dx  32 x  56 B) dy du  2u , du dx  2, and dy dx  16 x  49 C) dy du  4u , du dx  4, and dy dx  32 x  56 D) dy du  4u , du dx  2, and dy dx  32 x  56 E) dy du  2u , du dx  4, and dy dx  16 x  49 Ans: A 42.

Find A)

dy of y  u , u  7 – x 2 . dx x

7 – x2 1

2 7 – x2 –x

7 – x2 1



2 7 – x2 E) none of these choices Ans: C

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43. Find the derivative of the function. 4

f (t )  (1  3t ) 7 –3 A) 1 f (t )  (1  3t ) 7 7 –3 B) 12 f (t )  (1  3t ) 7 4 –3 C) 12 f (t )  (1  3t ) 4 7 –3 D) 3 f (t )  (1  3t ) 7 7 –3 E) 12 f (t )  (1  3t ) 7 7 Ans: E

44. Differentiate the given function. y  5 x9  9 x 1/ 2 A) 1 45 x8  9   2 1/ 2 B) 1 5 x9  9 x   2 1/ 2 C) 1 45 x 9  9 x   5 x9  9   2 1/ 2 D) 1 5 x9  9 x   45 x8  9   2 3/ 2 E) 1   5 x9  9 x   45 x8  9  2 Ans: D 45. Find the derivative of the function. f ( x)  x8 (7  6 x) 4 A) f ( x)  x 3 (7  6 x)7  56  72 x  B)

f ( x)  6 x8 (7  6 x)3  56  72 x 

f ( x)  x 7 (7  6 x) 4  56  72 x 

f ( x)  x 7 (7  6 x)3  56  72 x 

f ( x)  x 7 (7  6 x)3  56  6 x 

Ans: D

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46. Find the derivative of the given function. Simplify and express the answer using positive exponents only. c( x)  3 x x 7  5 A) 3  9 x 7  10 

  3  7 x  10  2  x  5 3  7 x  10   x  5 3  9 x  10  2  x  5 3  7 x  10   x  5 2 x7  5

Ans: D 47. Find the derivative of the function. f ( x)  x8 4  2 x A) x 7  64  34 x  f ( x)  2 4  2x 7 B) x  64  34 x  f ( x)  2 4  2x 7 C) x  4  34 x  f ( x)  2 4  2x D) x 7  64  2 x  f ( x)  2 4  2x E) x7  4  2 x  f ( x)  2 4  2x Ans: A

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48. Find the derivative of the function.  x5  g ( x)   2   x 5 5 A) 5  5  10 x  x 2    5  x     g ( x)   5  x   5  x 2    5  x 2   4 B) 5  5  10 x  x 2   5  x  g ( x)  6 5  x2  5

g ( x) 

5  5  10 x  x 2   5  x 

5  x 

2 4

g ( x)  

5  5  10 x  x 2   5  x 

5  x  5  5  10 x  x   5  x  g ( x)  5  x 

2 6

Ans: E 49. You deposit $ 4000 in an account with an annual interest rate of change r (in decimal 48

r   form) compounded monthly. At the end of 4 years, the balance is A  4000 1   .  12  Find the rates of change of A with respect to r when r  0.13 . A) 6709.32 B) 318,595.99 C) 559.11 D) 26549.67 E) 26,265.13 Ans: D 50. The value V of a machine t years after it is purchased is inversely proportional to the square root of t  5 . The initial value of the machine is $ 10,000. Find the rate of depreciation when t  2 . Round your answer to two decimal places. A) –603.68 per year B) –1889.82 per year C) 1767.77 per year D) 447.21 per year E) –1207.36 per year Ans: A

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51. Find the second derivative of the function. 3

f ( x)  5 x 13 A) –150 10 f ( x)  x 13 169 B) 3 –23 f ( x)  x 13 169 C) 845 –23 f ( x)  x 13 169 D) –150 –23 f ( x)  x 13 169 E) None of the above Ans: D

52. Find the third derivative of the function f  x   x 5  3 x 4 . A) 60 x 2  72 x B) 30 x 2  36 x C) 60 x 2  72 x 2 D) 60 x 2  36 x E) 30 x 2  36 x Ans: A 53. Find the f  6  x  of f  4  x   x 2  1 2 .   A) 12 x 2  4 B) 12 x 2  2 C) 6x2  4 D) 6x2  2 E) 12 x 2  1 Ans: A 54. Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If y  f  x  g  x  , then y  f   x  g   x  A) True B) False. The product rule is  f  x  g  x    f  x  g   x   g  x  f   x  Ans: B

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55. Find the third derivative. 2 y 5 x A) –420 x7 B) 420 x8 0 C) D) 84 x7 E) –420 x8 Ans: E 56. Find the value g (4) for the function g (t )  3t 8  6t 6  1 . A) 734,208 B) 430,080 C) 221,185 D) 430,081 E) 3,403,776 Ans: A 57. Find the indicated derivative. Find y (4) if y  x8  4 x3 . A) 336x 5 B) 336 x 4 C) 336 x 4  24 x D) 1680 x 5  24 x E) 1680 x 4 Ans: E 58. Find the second derivative for the function f ( x)  4 x3 +12 x 2  20 x  18 and solve the equation f ( x)  0 . A) –1 B) 4 C) 0 D) 18 E) 20 Ans: A

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59.

Find the second derivative for the function f ( x)  f ''( x)  0 . A) 0 B) 7 C) no solution D) –7 E) 1  7 Ans: C

5x and solve the equation 5x + 7

60. A brick becomes dislodged from the Empire State Building (at a height of 1025 feet) and falls to the sidewalk below. Write the position s(t), velocity v(t), and acceleration a(t) as functions of time. A) s (t )  16t 2  1025 ; v(t )  32t ; a (t )  32 B) s (t )  16t 2  1025 ; v(t )  32t ; a(t )  32 C) s (t )  16t 2  1025 ; v(t )  32t ; a(t )  32 D) s (t )  16t 2  1025 ; v(t )  32t ; a(t )  32 E) s (t )  16t 2  1025 ; v(t )  32 ; a(t )  32t Ans: C 61. Find y implicitly for 6 x9  y 9  3. A) 6x9 y  9 y B) y9  y  9 6x C) 6x8 y  8 y D) y8  y  8 6x E) x8 y  8 6y Ans: C

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62.

dy 9x  7 y  4. for the equation 5x  6 y dx A) dy 11  dx 31 B) dy 29  dx 31 C) dy 11  dx 31 D) dy 29  dx 31 E) dy 4 dx Ans: C Find

63. Find the slope of the graph at the given point.

A) 0 B) 3 C) 5 D) 4 E) 7 Ans: A

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64. Find the slope of the graph at the given point.

A) 2 B) 0 C) 1 D) 3 E) 5 Ans: A 65. Find the rate of change of x with respect to p. 2 p x0 0.00001x3  0.1x A) 2  2 p  0.00003x 2  0.1 B)



2 p  0.00003x 2  0.1



2 p x  0.00003x 2  0.1



2 px  0.00003x 2  0.1



2x p  0.00003x 2  0.1

Ans: A

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66. Find the rate of change of x with respect to p. 200  x , 0  x  200 p 2x A) 4 xp  2 2 p 1 B) 4 xp 2 p2  1 C) 4x  2 2 p 1 D) 4x 2 p2  1 E) 4 xp  2 p 1 Ans: A 67. Find dy dx implicitly and explicitly(the explicit functions are shown on the graph) and show that the results are equivalent. Use the graph to estimate the slope of the tangent line at the labeled point. Then verify your result analytically by evaluating dy dx at the point.

1 1 , 2y 2 B) 1 1  , 2y 2 C) 1 1  , 2y 2 D) 1 1 , 2y 2 E) 1 1 , 2 2 Ans: A

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68. Let x represent the units of labor and y the capital invested in a manufacturing process. When 135,540 units are produced, the relationship between labor and capital can be modeled by 100 x 0.75 y 0.25  135,540 . Find the rate of change of y with respect to x when x  1500 and y  135,540 . A) -2 B) 0 C) 3 D) -7 E) 5 Ans: A 69. Find dy/dx for the following equation: 2 x  y 2  5 y  9  0. A) dy 5  dx 2  2 y B) dy 2  dx 5  2 y C) dy 1  dx 5  y D) dy 5  dx 5  y E) dy 1  dx 2  y Ans: B 70.

dy for the equation xy  x  20 y by implicit differentiation and evaluate the dx derivative at the point (50, 2) . A) 1  25 B) 1 25 C) 3 25 D) 3  25 E) 0 Ans: B Find

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71. Assume that x and y are differentiable functions of t. Find dy/dt using the given values. y  4 x 3  6 x 2  x for x  3, dx / dt  2. A) 288 B) 159 C) 318 D) 286 E) 143 Ans: D 72.

Given xy  10, find

dx dy  3. when x = –9 and dt dt

dy 260  dt 27 B) dy 10  dt 27 C) dy 10 – dt 27 D) dy 27 – dt 10 E) dy 27 – dt 260 Ans: C 73. Assume that x and y are differentiable functions of t. Find dx/dt given that x  2 , y  8 , and dy / dt  3. y 2  x 2  60 A) 1.50 B) 5.33 C) 0.75 D) 24.00 E) 12.00 Ans: E

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74. Area. The radius, r, of a circle is increasing at a rate of 5 centimeters per minute. Find the rate of change of area, A, when the radius is 4 . A) dA  20 dt B) dA  160 dt C) dA  –160 dt D) dA  40 dt E) dA  –40 dt Ans: D 75. Volume and radius. Suppose that air is being pumped into a spherical balloon at a rate of 4 in.3 / min . At what rate is the radius of the balloon increasing when the radius is 7 in.? A) dr 4  dt 49 B) dr 1  dt 7 C) dr 49  dt 4 D) dr 7  dt 4 E) dr 1  dt 49 Ans: E 76. The radius r of a sphere is increasing at a rate of 3 inches per minute. Find the rate of change of volume when r = 8 inches. Round your answer to one decimal place. A) 804.2 cubic inches per minute B) 2144.7 cubic inches per minute C) 6434.0 cubic inches per minute D) 2412.7 cubic inches per minute E) 7238.2 cubic inches per minute Ans: D

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77. Profit. Suppose that the monthly revenue and cost (in dollars) for x units of a product x2 and C  4000  30 x. At what rate per month is the profit changing if are R  900 x  50 the number of units produced and sold is 100 and is increasing at a rate of 10 units per month? A) $86,960 per month B) $8660 per month C) $8960 per month D) $260 per month E) $89,960 per month Ans: B 78. The lengths of the edges of a cube are increasing at a rate of 8 ft/min. At what rate is the surface area changing when the edges are 15 ft long? A) 384 ft2/min B) 1440 ft2/min C) 720 ft2/min D) 5760 ft2/min E) 120 ft2/min Ans: B 79.

A point is moving along the graph of the function y  9 x 2  2 such that centimeters per second. Find dy/dt for the given values of x. (a) x  4 A)

(b) x  8

dy 4 dt dy  216 dt

dy  432 dt dy  432 dt

dy  432 dt

dy  216 dt

dy 8 dt dy 8 dt

dy  –216 dt dy  432 dt

Ans: B

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dx 3 dt

80.

A point is moving along the graph of the function y 

1 dx such that 5 7x  2 dt 2

centimeters per second. Find dy/dt when x  3 . A) dy 42 – dt 4225 B) dy 42 – dt 845 C) dy 42  dt 845 D) dy 42  dt 4225 E) dy 42 – dt 13 Ans: B 81. Boat docking. Suppose that a boat is being pulled toward a dock by a winch that is 21 ft above the level of the boat deck. If the winch is pulling the cable at a rate of 23 ft/min, at what rate is the boat approaching the dock when it is 28 ft from the dock? Use the figure below.

28.75 ft/min A) 23.00 ft/min B) 38.33 ft/min C) D) 17.25 ft/min 13.80 ft/min E) Ans: A

82. An airplane flying at an altitude of 5 miles passes directly over a radar antenna. When the airplane is 25 miles away (s = 25), the radar detects that the distance s is changing at a rate of 250 miles per hour. What is the speed of the airplane? Round your answer to the nearest integer. A) 255 mi/hr B) 236 mi/hr C) 510 mi/hr D) 128 mi/hr E) 118 mi/hr Ans: A

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83. A baseball diamond has the shape of a square with sides 90 feet long (see figure). A player running from second base to third base at a speed of 30 feet per second is 80 feet from third base. At what rate is the player’s distance s from home plate changing? Round your answer to one decimal place.

A) –58.2 feet/second B) –0.2 feet/second C) –0.7 feet/second D) –19.9 feet/second E) –1.9 feet/second Ans: D 84. A retail sporting goods store estimates that weekly sales and weekly advertising costs are related by the equation S  2270  60 x  0.35 x 2 . The current weekly advertising costs are $1700, and these costs are increasing at a rate of $130 per week. Find the current rate of change of weekly sales. A) 162,500 dollars per week B) 164,770 dollars per week C) 87,420 dollars per week D) 85,150 dollars per week E) 1,021,570 dollars per week Ans: A

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Chapter 3: Applications of the Derivative 1. Use the graph of y = f ( x) to identify at which of the indicated points the derivative f '( x) changes from positive to negative.

A) (5,6) B) (-1,2), (5,6) C) (2,4) D) (2,4), (5,6) E) (-1,2) Ans: A 2. Use the graph of y = f ( x) to identify at which of the indicated points the derivative f '( x) changes from negative to positive.

A) (2,4) B) (-1,2) C) (-1,2), (5,6) D) (5,6) E) (2,4), (5,6) Ans: B 3. Identify the open intervals where the function f ( x) = 4 x 2 – 3x + 2 is increasing or decreasing. A) 3   3  decreasing:  −∞,  ; increasing:  , ∞  8   8  B) 3   3  increasing:  −∞,  ; decreasing:  , ∞  8   8  C) increasing on ( −∞, ∞ ) D) decreasing on ( −∞, ∞ ) E) none of the above Ans: A

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4. Both a function and its derivative are given. Use them to find all critical numbers. x1/ 3 − 6 f ( x) = x − 9x2 / 3 + 6 f ′( x) = 1/ 3 x A) x=0 B) x = 216 C) = x 0,= x –102 D) = x 0,= x 216 E)= x –102, = x 216 Ans: D 5. Identify the open intervals where the function f ( x) = –5 x 2 + 2 x – 4 is increasing or decreasing. A) 1  1  increasing:  −∞,  ; decreasing:  , ∞  5  5  B) 1  1  decreasing:  −∞,  ; increasing:  , ∞  5  5  C) increasing on ( −∞, ∞ ) D) decreasing on ( −∞, ∞ ) E) none of the above Ans: A 6. For the given function, find all critical numbers. y =x 3 − 9 x 2 − 48 x + 4 A) x=0 B) x = –8 and x = –2 C) x = –8 and x = 2 D) x = –2 and x = 8 E) x = 2 and x = 8 Ans: D 7. Find any critical numbers of the function g= (t ) t 5 − t , t < 5. A) 0 B) 10 − 3 C) 10 3 D) both A and B E) both A and C Ans: C

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8. Identify the open intervals where the function f= ( x) x 14 − x 2 is increasing or decreasing. A) decreasing: −∞, 7 ; increasing: 7, ∞ B) C) D)

( ) ( ) increasing: ( − 7, 7 ) ; decreasing: ( − 14, − 7 ) ∪ ( 7, 14 ) increasing: ( −∞, 14 ) ; decreasing: ( 14, ∞ ) increasing: ( − 14, − 7 ) ∪ ( 7, 14 ) ; decreasing: ( − 7, 7 )

E) decreasing for all x Ans: B 9. For the given function, find the critical numbers. x 4 x3 y= − −6 4 3 A) = x 0= and x 1 B)= x 0= and x 6 C)= x 0= and x –6 D) x = 0 and x = −1 E) x= −1 and x = 1 Ans: A 10.

Find the open intervals on which the function f ( x) =

x is increasing or x +9 2

decreasing. A) The function is increasing on the interval −3 < x < 3 , and decreasing on the intervals −∞ < x < −3 and 3 < x < ∞ . B) The function is increasing on the interval −∞ < x < −3 , and decreasing on the intervals −3 < x < 3 and 3 < x < ∞ . C) The function is increasing on the interval 3 < x < ∞ , and decreasing on the intervals −∞ < x < −3 and −3 < x < 3 . D) The function is decreasing on the interval −3 < x < 3 , and increasing on the intervals −∞ < x < −3 and 3 < x < ∞ . E) The function is decreasing on the interval −∞ < x < −3 , and increasing on the intervals −3 < x < 3 and 3 < x < ∞ . Ans: A

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11.

3 x + 2, x ≤ 2 Find the open intervals on which the function y =  is increasing or 2 5 − x , x > 2 decreasing. A) The function is increasing on the interval −∞ < x < 0 and decreasing on the interval 0 < x < ∞ . B) The function is increasing on the interval 2 < x < ∞ and decreasing on the interval −∞ < x < 2 . C) The function is increasing on the interval −∞ < x ≤ 2 and decreasing on the interval 2 < x < ∞ . D) The function is increasing on the interval 0 < x < ∞ and decreasing on the interval −∞ < x < 0 . E) The function is increasing on the interval −∞ < x < 2 and decreasing on the interval 2 < x < ∞ . Ans: E

12. Suppose the number y of medical degrees conferred in the United States can be modeled by y = 0.813t 3 − 55.70t 2 + 1185.2t + 7752, for 0 ≤ t ≤ 32 , where t is the time in years, with t = 0 corresponding to 1975. Use the test for increasing and decreasing functions to estimate the years during which the number of medical degrees is increasing and the years during which it is decreasing. A) The number of medical degrees is increasing from 1975 to 1992 and 2000 to 2005, and decreasing during 1992 to 2000. B) The number of medical degrees is increasing from 1975 to 1991 and 1999 to 2005, and decreasing during 1991 to 1999. C) The number of medical degrees is increasing from 1975 to 1992 and 1999 to 2005, and decreasing during 1992 to 1999. D) The number of medical degrees is increasing from 1975 to 1993 and 1999 to 2005, and decreasing during 1993 to 1999. E) The number of medical degrees is increasing from 1975 to 1992 and 1998 to 2005, and decreasing during 1992 to 1998. Ans: C

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13. A fast-food restaurant determines the cost model, C= 0.3 x + 4500, 0 ≤ x ≤ 30000 and 1 revenue model, R = (45000 x − x 2 ) for 0 ≤ x ≤ 30000 where x is the number of 20000 hamburgers sold. Determine the intervals on which the profit function is increasing and on which it is decreasing. A) The profit function is increasing on the interval (19500,30000) and decreasing on the interval (0,19500) . B) The profit function is increasing on the interval (0,12500) and decreasing on the interval (12500,30000) . C) The profit function is increasing on the interval (0,19500) and decreasing on the interval (19500,30000) . D) The profit function is increasing on the interval (12500,30000) and decreasing on the interval (0,12500) . E) The profit function is increasing on the interval (0, 4500) and decreasing on the interval (4500,30000) . Ans: C 14. For the given function, find the relative minima. y =x 3 − 3 x 2 − 72 x + 16 A) ( –4,192 ) B) ( 6, –308) C) D)

( –4, –256 ) ( –6,124 )

E) no relative minima Ans: B 15. Find the x-values of all relative maxima of the given function. y = 13 x 3 − 4 x 2 + 12 x + 8 A) x=0 B) x=6 C) x=4 D) x=2 E) no relative maxima Ans: D

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16. For the function f ( x) =4 x 3 − 36 x 2 + 1 : (a) Find the critical numbers of f (if any); (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. Then use a graphing utility to confirm your results. A) (a) x = 0 , 6 (b) increasing: ( −∞, 0 ) ∪ ( 6, ∞ ) ; decreasing: ( 0, 6 ) B)

(c) relative max: f (0) = 1 ; relative min: f (6) = –431 (a) x = 0 , 6 (b) decreasing: ( −∞, 0 ) ∪ ( 6, ∞ ) ; increasing: ( 0, 6 )

(c) relative min: f (0) = 1 ; relative max: f (6) = –431 (a) x = 0 , 2 (b) increasing: ( −∞, 0 ) ∪ ( 2, ∞ ) ; decreasing: ( 0, 2 )

(c) relative max: f (0) = 1 ; relative min: f (2) = –111 (a) x = 0 , 2 (b) decreasing: ( −∞, 0 ) ∪ ( 2, ∞ ) ; increasing: ( 0, 2 )

(c) relative min: f (0) = 1 ; relative max: f (2) = –111 (a) x = 0 , 2 (b) increasing: ( −∞, 0 ) ∪ ( 2, ∞ ) ; decreasing: ( 0, 2 )

(c) relative max: f (0) = 1 ; no relative min. Ans: A 17. Find all relative maxima of the given function. y =x 4 − 8 x 3 + 16 x 2 + 3 A) ( 0,3) B) C) D)

( 2,19 ) ( 4,3) ( 0,3) , ( 4,3)

E) no relative maxima Ans: B 18. Find all relative minima of the given function. y =x 4 − 8 x 3 + 16 x 2 + 4 A) ( 0, 4 ) B) C) D)

( 2, 20 ) ( 4, 4 ) ( 0, 4 ) , ( 4, 4 )

E) no relative minima Ans: D

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19. Locate the absolute extrema of the function f ( x) = –3 x 2 – 6 x + 2 on the closed interval

[ –2, 2] .

A) no absolute max; absolute min: f(–1) = 5 B) absolute max: f(2) = –22 ; absolute min: f(–1) = 5 C) absolute max: f(–1) = 5 ; no absolute min D) absolute max: f(–1) = 5 ; absolute min: f(2) = –22 E) no absolute max or min Ans: D 20. Locate the absolute extrema of the function f ( x= ) x 3 − 12 x on the closed interval [0,5]. A) absolute max: f(5) = 65 ; absolute min: f(2) = –16 B) absolute max: f(2) = –16 ; absolute min: f(5) = 65 C) absolute max: f(5) = 65 ; no absolute min D) no absolute max; absolute min: f(5) = 65 E) no absolute max or min Ans: A 21. Find the x-value at which the absolute minimum of f (x) occurs on the interval [a, b]. f ( x) =x 3 − 27 x + 6, [ –9, 4] A) x = –6 B) x = –3 C) x=0 D) x=3 E) x=4 Ans: A 22. Locate the absolute extrema of the given function on the closed interval [–36,36]. 36 x x + 36 A) absolute max: f(6) = 3 B) absolute min: f(-6) = –3 C) no absolute max D) no absolute min E) both A and D F) both A and B Ans: F f ( x) =

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23. Find the absolute extrema of the function h(t )= (t − 1) 2 / 3 on the closed interval [ –4, 4] . Round your answer to two decimal places. A) The maximum of the function is 1 and the minimum of the function is 0. B) The maximum of the function is 2.92 and the minimum of the function is 1. C) The maximum of the function is 2.92 and the minimum of the function is 0. D) The maximum of the function is1 and the minimum of the function is 2.08. E) The maximum of the function is 0 and the minimum of the function is 2.08. Ans: C 24. Approximate the critical numbers of the function shown in the graph and determine whether the function has a relative maximum, a relative minimum, an absolute maximum, an absolute minimum, or none of these at each critical number on the interval shown.

The critical number x = 0 yields an absolute maximum and the critical number x = 1 yields an absolute minimum.. B) Both the critical numbers x = 0 & x = 1 yield an absolute maximum. C) The critical number x = 0 yields an absolute minimum and the critical number x = 1 yields an absolute maximum. D) Both the critical numbers x = 0 and x = 1 yield an absolute minimum. E) The critical number x = 0 yields a relative minimum and the critical number x = 1 yields a relative maximum. Ans: C A)

25.

20 x on the interval [0, ∞) . x2 + 1 A) The maximum of the function is 1 and the minimum of the function is 0. B) The maximum of the function is 0 and the minimum of the function is –10. C) The maximum of the function is –10 and the minimum of the function is 0. D) The maximum of the function is 10 and the minimum of the function is 0. E) The maximum of the function is 0 and the minimum of the function is 10. Ans: D

Find the absolute extrema of the function f ( x) =

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26. Graph a function on the interval [ –1,3] having the following characteristics. Absolute maximum at x = 3 Absolute minimum at x = –1 Relative maximum at x = 0.2 Relative minimum at x = 2 A)

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Ans: A 27. Medication. The number of milligrams x of a medication in the bloodstream t hours 4000t after a dose is taken can be modeled by x(t ) = 2 t > 0 . Find the t-value at which t +5 x is maximum. Round your answer to two decimal places. A) 0 hours B) 2.24 hours C) 894.43 hours D) 4.24 hours E) 5.46 hours Ans: B 28. Medication. The number of milligrams x of a medication in the bloodstream t hours 4000t after a dose is taken can be modeled by x(t ) = 2 t > 0 . Find the maximum value t +7 of x. Round your answer to two decimal places. A) 2.65 mg B) 755.93 mg C) 1663.04 mg D) 8.20 mg E) 1500.40 mg Ans: B

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29. Suppose the resident population P(in millions) of the United States can be modeled by = P 0.00000583t 3 + 0.005003t 2 + 0.13776t + 4.658, −3 ≤ t ≤ 198 , where t = 0 corresponds to 1800. Analytically find the minimum and maximum populations in the U.S. for −3 ≤ t ≤ 198 . A) The population is minimum at t = −3 and maximum at t = 0 . B) The population is minimum at t = 0 and maximum at t = 198 . C) The population is minimum at t = 198 and maximum at t = −3 . D) The population is minimum at t = −3 and maximum at t = 198 . E) The population is minimum at t = 0 and maximum at t = −3 . Ans: D 30. Determine the open intervals on which the graph of f ( x) = 8 x 2 – 7 x + 8 is concave downward or concave upward. A) concave upward on ( −∞, 0 ) ; concave downward on ( 0, ∞ ) B) concave downward on ( −∞, ∞ ) C)

concave upward on ( −∞, ∞ )

concave downward on ( −∞, 0 ) ; concave upward on ( 0, ∞ )

concave upward on ( −∞,1) ; concave downward on (1, ∞ ) Ans: C E)

31. Determine the open intervals on which the graph of y = –8 x 3 + 6 x 2 + 6 x – 5 is concave downward or concave upward. A) concave downward on (−∞, ∞) B) 1  1  concave downward on  −∞,  ; concave upward on  , ∞  4  4  C) 1  1   concave upward on  −∞, –  ; concave downward on  – , ∞  4  4   D) 1  1   concave downward on  −∞, –  ; concave upward on  – , ∞  4  4   E) 1 1   concave upward on  −∞,  ; concave downward on  , ∞  4 4   Ans: E 32. Find all relative extrema of the function f ( x) = 3 x 2 + 24 x + 49 . Use the Second Derivative Test where applicable. A) relative min: f (–4) = 1 B) relative max: f (0) = 49 C) no relative max D) no relative min E) both A and C F) both B and D Ans: E

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33. Find all relative extrema of the function f ( x) = 2 x 4 – 16 x 3 + 3 Use the Second Derivative Test where applicable. A) relative max: f (12) = 13,827 ; no relative min B) relative max: f (6) = 861 ; no relative min C) no relative max or min D) relative min: f (12) = 13,827 ; no relative max E) relative min: f (6) = –861 ; no relative max Ans: E 34. Find all relative extrema of the function f ( x) = x 2 / 9 – 7 . Use the Second Derivative Test where applicable. A) relative max: f(1) = –6 B) relative min: f(0) = –7 C) no relative max or min D) both A and B E) none of the above Ans: B 35. Find all relative extrema of the function f = ( x) 36 − x 2 . Use the Second-Derivative Test when applicable. A) The relative minimum is (0, 6) and the relative maximum is (−6, 0) . B) The relative maximum is (0, 6) . C) The relative minimum is (0, 6) . D) The relative maximum is (0, 6) and the relative minima are (6, 0) and (−6, 0) . E) The relative minimum is (0, 6) and the relative maximum is (6, 0) . Ans: B 36.

Find all relative extrema of the function f ( x) =

10 . Use the Second-Derivative Test x2 + 1

when applicable. A) The relative maximum is (1, 0) . B) The relative minimum is (0,1) . C) The relative maximum is (0,10) . D) The relative minimum is (0,10) . E) The relative maximum is (10, 0) . Ans: C

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37. State the signs of f '( x) and f ''( x) on the interval (0, 2).

f '= 0 f "> 0 B) f '< 0 f "< 0 C) f '> 0 f "> 0 D) f '< 0 f "> 0 E) f '> 0 f "< 0 Ans: B 38. Find the x-value at which the given function has a point of inflection. y = 13 x 3 − 4 x 2 + 12 x + 5 A) x=0 B) x=6 C) x=4 D) x=2 E) no point of inflection Ans: C

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39. Find the points of inflection and discuss the concavity of the function. f ( x) = –5 x3 + 4 x 2 – 3 x – 3 A) 4 4  inflection point at x = ; concave upward on  −∞,  ; concave downward on 15 15    4   ,∞  15  B) 4 4  inflection point at x = ; concave downward on  −∞,  ; concave upward on 15 15    4   ,∞  15  C) 4 4  inflection point at x = – ; concave upward on  −∞, –  ; concave 15 15    4  downward on  – , ∞   15  D) 4 4  inflection point at x = – ; concave downward on  −∞, –  ; concave 15 15    4  upward on  – , ∞   15  E) none of the above Ans: A 40. A function and its graph are given. Use the second derivative to locate all x-values of points of inflection on the graph of y = f ( x) . Check these results against the graph shown. y = 18 x 4 − 54 x 2 + 36

A) B) C)

x= − x=0

2 2

2 2 D) 2 2 , x= − x= 2 2 E) 2 2 , x =0, x = − x= − 2 2 Ans: D x=

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41. Sketch a graph of a function f having the following characteristics.

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Ans: C

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42. The graph of f is shown in the figure. Sketch a graph of the derivative of f.

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Ans: E

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43. The graph of f is shown in the figure. Sketch a graph of the derivative of f.

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Ans: C

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44. The graph of f is shown in the figure. Sketch a graph of the derivative of f.

B) C)

The derivative of f does not exist.

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Ans: D

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45. The graph of f is shown. Graph f, f' and f'' on the same set of coordinate axes.

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E) none of the above Ans: C

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46. The graph of f is shown in the figure. Sketch a graph of the derivative of f.

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Ans: D 47. Production. Suppose that the total number of units produced by a worker in t hours of an 8-hour shift can be modeled by the production function 6*a t + 3*(a–1)t 2 − 2t 3 . Find the number of hours before the rate of P (t ) : P(t ) = production is maximized. That is, find the point of diminishing returns. A) t =0 B) t = inf C) t =5 D) t =8 t =a E) Ans: B

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48. The profit P (in thousands of dollars) for a company spending an amount s (in thousands 1 of dollars) on advertising is P = − s 3 + 24 s 2 + 100. The point of diminishing returns 10 is the point at which the rate of growth of the profit function begins to decline. Find the point of diminishing returns. Round your answer to the nearest thousand dollars. A) 40 thousand dollars B) 120 thousand dollars C) 96 thousand dollars D) 160 thousand dollars E) 80 thousand dollars Ans: E 49. The number of people who donated to a certain organization between 1975 and 1992 can be modeled by the equation D(t ) = –10.46t 3 + 208.808t 2 –168.202t + 9775.234 donors, where t is the number of years after 1975. Find the inflection point(s) from t = 0 through t = 17 , if any exist. A) There are no inflection points from t = 0 through t = 17 . B) There is one inflection point at t = 6.65 . C) There are inflection points at t = 0 and t = 17 . D) There is one inflection point at t = 0.15 . E) There are inflection points at t = 0 , t = 0.15 , and t = 17 . Ans: B 50. Find the length and width of a rectangle that has perimeter 8 meters and a maximum area. A) 1, 3 B) 1, 3 C) 2, 2 D) 3, 1 E) 6, –2 Ans: C

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51. A rancher has 520 feet of fencing to enclose two adjacent rectangular corrals (see figure). What dimensions should be used so that the enclosed area will be a maximum?

A) x = 65.00 and y = 86.67 B) x = 13.00 and y = 156.00 C) x = 26.00 and y = 173.33 D) x = 86.67 and y = 65.00 E) x = 39.00 and y = 104.00 Ans: A 52. Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 361 square meters. A) 19 6 19 6 square base side ; height 3 3 B) 19 6 19 6 square base side ; height 3 6 C) 19 6 19 6 square base side ; height 6 3 D) 19 6 19 6 square base side ; height 6 6 E) 19 6 361 6 square base side ; height 6 6 Ans: D

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53. A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). Find the dimensions of a Norman window of maximum area if the total perimeter is 24 feet. Round yours answers to two decimal places.

A) x = 6.72 feet and y = 3.36 feet B) x = 3.36 feet and y = 7.68 feet C) x = 2.24 feet and y = 9.12 feet D) x = 5.72 feet and y = 4.65 feet E) x = 7.72 feet and y = 2.08 feet Ans: A 54. Volume. A rectangular box with a square base is to be formed from a square piece of metal with 36-inch sides. If a square piece with side x is cut from each corner of the metal and the sides are folded up to form an open box, the volume of the box is = V (36 − 2 x) 2 x. What value of x will maximize the volume of the box?

A) 18 B) 1 C) 6 D) 15 E) 9 Ans: C 55. A rectangular page is to contain 225 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used. A) 17, 17 B) 15, 15 C) 13, 13 D) 16, 16 E) 14, 14 Ans: A

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56. Find the dimensions of the rectangle of maximum area bounded by the x-axis and y-axis 2− x and the graph of y = . 2

A) length 0.75; width 0.625 B) length 1; width 0.5 C) length 0.25; width 0.875 D) length 0.5; width 0.75 E) none of the above Ans: B 57. Find the point on the graph of f ( x) = x 2 that is closest to the point (3, 0.5). Round your answer to two decimal places. A) (1.14, 1.30) B) (1.44, 2.07) C) (1.82, 3.31) D) (1.00, 1.00) E) (0.91, 0.83) Ans: A

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58. Minimum cost. From a tract of land, a developer plans to fence a rectangular region and then divide it into two identical rectangular lots by putting a fence down the middle. Suppose that the fence for the outside boundary costs $8 per foot and the fence for the middle costs $2 per foot. If each lot contains 4100 square feet, find the dimensions of each lot that yield the minimum cost for the fence. A) Dimensions are 48.07 ft for the side parallel to the divider and 85.29 ft for the other side. B) Dimensions are 85.29 ft for the side parallel to the divider and 48.07 ft for the other side. C) Dimensions are 64.03 ft for the side parallel to the divider and 64.03 ft for the other side. D) Dimensions are 60.37 ft for the side parallel to the divider and 67.91 ft for the other side. E) Dimensions are 67.91 ft for the side parallel to the divider and 60.37 ft for the other side. Ans: D 59. You are in a boat 2 miles from the nearest point on the coast. You are to go to point Q located 3 miles down the coast and 1 mile inland (see figure). You can row at a rate of 1 miles per hour and you can walk at a rate of 2 miles per hour. Toward what point on the coast should you row in order to reach point Q in the least time?

A) 3 miles B) 8 miles C) 2 miles D) 1 mile E) 5 miles Ans: D

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60. A wooden beam has a rectangular cross section of height h and width w (see figure). The strength S of the beam is directly proportional to the width and the square of the height. What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 23 inches? Round your answers to two decimal places. [Hint: S = kwh 2 , where k > 0 is the proportionality constant.]

A) w = 13.28 inches and h = 18.78 inches B) w = 7.67 inches and h = 21.68 inches C) w = 19.92 inches and h = 11.50 inches D) w = 16.26 inches and h = 16.27 inches E) w = 18.78 inches and h = 13.28 inches Ans: A 61. If the total revenue function for a blender is R (= x) 50 x − 0.25 x 2 , determine how many units x must be sold to provide the maximum total revenue in dollars. A) 2500 B) 1875 C) 50 D) 150 E) 100 Ans: E 62. If the total revenue function for a blender is R (= x) 45 x − 0.05 x 2 , find the maximum revenue. A) $450 B) $8125 C) $45 D) $250 E) $10,125 Ans: E 63. A firm has total revenue given by R ( x) =600 x − 95.5 x 2 − x 3 dollars for x units of a product. Find the maximum revenue from sales of that product. A) $1200 B) $914 C) $303 D) $2200 E) $631 Ans: B

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64. If the total cost function for a product is C ( x) = 500 + 9 x + 0.03 x 2 dollars, determine how many units x should be produced to minimize the average cost per unit? A) 149 units B) 500 units C) 91 units D) 129 units E) 81 units Ans: D 65. If the total cost function for a product is C ( x) = 200 + 5 x + 0.07 x 2 dollars. Find the minimum average cost. A) $13.10 B) $20.00 C) $12.92 D) $12.48 E) $19.00 Ans: D 66. Average costs. Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by 1125 C ( x) = 5x + , x > 0 , where x is the number of machines used. How many x machines give minimum average costs? A) Using 15 machines gives the minimum average costs. B) Using zero machines gives the minimum average costs. C) Using 25 machines gives the minimum average costs. D) Using 30 machines gives the minimum average costs. E) Using 35 machines gives the minimum average costs. Ans: A 67. Average costs. Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by 500 C ( x) = 5x + , x > 0 , where x is the number of machines used. What is the x minimum average cost? A) $0 B) $10 C) $100 D) $50 E) $505 Ans: B

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68. A firm can produce 100 units per week. If its total cost function is = C 500 + 1800 x 2 dollars, and its total revenue function = is R 1900 x − x dollars, how many units x should it produce to maximize its profit? A) 1850 units B) 950 units C) 94 units D) 50 units E) 100 units Ans: D 69. A firm can produce 100 units per week. If its total cost function is = C 600 + 1200 x 2 dollars, and its total revenue function = is R 1300 x − x dollars, find the maximum profit. A) $6319 B) $1900 C) $8236 D) $6921 E) $2806 Ans: B 70. A travel agency will plan a tour for groups of size 28 or larger. If the group contains exactly 28 people, the price is $900 per person. However, each person’s price is reduced by $20 for each additional person above the 28 . If the travel agency incurs a price of $100 per person for the tour, what size group will give the agency the maximum profit? A) 6 B) 38 C) 34 D) 52 E) 20 Ans: C 71. A power station is on one side of a river that is 0.5 mile wide, and a factory is 6.00 miles downstream on the other side of the river. It costs $ 18 per foot to run overland power lines and $ 21 per foot to run underwater power lines. Estimate the value of x that minimizes the cost. A) 0.51 B) 0.83 C) 0.87 D) 1.86 E) 0.52 Ans: B

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72. Find the speed v, in miles per hour, that will minimize costs on a 105-mile delivery trip. v2 The cost per hour for fuel is C = dollars, and the driver is paid W = $7 dollars 700 per hour. (Assume there are no costs other than wages and fuel.) A) 595 mph B) 105 mph C) 700 mph D) 70 mph E) 35 mph Ans: D 73. Suppose the sales S (in billions of dollars per year) for Proctor & Gamble for the years 1999 through 2004 can be modeled = by S 2.0931t 2 − 1.9682t + 40.831, 1999 ≤ t ≤ 2004 where t represents the year. During which year were the sales increasing at the lowest rate? A) Sales are increasing at the lowest rate in the year 2004. B) Sales are increasing at the lowest rate in the year 1999. C) Sales are increasing at the lowest rate in the year 2000. D) Sales are increasing at the lowest rate in the year 2002. E) Sales are increasing at the lowest rate in the year 2001. Ans: C 74. p is in dollars and q is the number of units. Find the elasticity of the demand function 9 p + 3q = 160 at the price p = $15 . A) –5.40 B) 1.00 C) 5.40 D) –3.00 E) 3.00 Ans: C

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75. A function and its graph are given. Use the graph to find the vertical asymptotes, if they exist, where A = 66. Confirm your results analytically. 22 x 2 f ( x) = ( x − 2) 2

A) x=0 B) x=2 C) x=5 D) x =1 E) no vertical asymptotes Ans: B 76. A function and its graph are given. Use the graph to find the horizontal asymptotes, if they exist, where A = 45. Confirm your results analytically. 15 x 2 f ( x) = ( x − 2) 2

A) y =1 B) y = 15 C) y=2 D) y=5 E) no horizontal asymptotes Ans: B

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77. A function and its graph are given. Use the graph to find the vertical asymptotes, if they exist. Confirm your results analytically. 8 f ( x) = x+2

A) x=6 B) x=2 C) x=0 D) x = –2 E) no vertical asymptotes Ans: D 78. A function and its graph are given. Use the graph to find the horizontal asymptotes, if they exist. Confirm your results analytically. 40 f ( x) = x+2

A) y =8 B) y=6 C) y=0 D) y=4 E) no horizontal asymptotes Ans: C 79. Analytically determine the location(s) of any vertical asymptote(s). 500 x + 8000 f ( x) = 2 x − 50 x − 1665 A) x = 72.85 B) x = –22.85 C) x = 72.85 , x = –22.85 D) x=0 E) no vertical asymptotes Ans: C

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80. Analytically determine the location(s) of any horizontal asymptote(s). 700 x + 5000 f ( x) = 2 x − 27 x − 2194 A) y = 62.25 B) y = –35.25 C) y = 62.25 , y = –35.25 D) y=0 E) no horizontal asymptotes Ans: D 81. This problem contains a function and its graph, where A = 15. Use the graph to determine, as well as you can, the vertical asymptote. Check your conclusion by using the function to determine the vertical asymptote analytically. 3 ( x − 3) f ( x) = x−2

A) x = –2 B) x=0 C) x = −1 D) x=2 E) no vertical asymptote Ans: D

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82. This problem contains a function and its graph, where A = 20. Use the graph to determine, as well as you can, the horizontal asymptote. Check your conclusion by using the function to determine the horizontal asymptote analytically. 4 ( x − 3) f ( x) = x−2

A) y = –2 B) y=0 C) y=4 D) y = −1 E) y = –20 Ans: C 83. Find any horizontal asymptotes for the given function. 3x − 3 y= x+9 A) y =1 B) y=9 C) y=0 D) y=3 E) no horizontal asymptotes Ans: D 84. Find any vertical asymptotes for the given function. 2x − 2 y= x+9 A) x =1 B) x=9 C) x = –9 D) x=0 E) no vertical asymptotes Ans: C

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85. Find any horizontal asymptotes for the given function. 8x y= 4 − x2 A) y=2 B) y =8 C) y=4 D) y=0 E) no horizontal asymptotes Ans: D 86. Find all vertical asymptotes for the given function. 25 x y= 9 − x2 A) x=0 B) x = ±3 C) x=3 D) x = −3 E) no vertical asymptotes Ans: B 87. Analytically determine the location of any vertical asymptotes. x − 90 f ( x) = 2 x + 2000 A) x = 9.486833 B) x = 0.045 C) x = 44.72136 D) x = –44.72136 E) no vertical asymptotes Ans: E 88. Analytically determine the location(s) of any horizontal asymptote(s). x − 80 f ( x) = 2 x + 1200 A) y=0 B) y = 0.066667 C) y = 34.641016 D) y = –34.641016 E) no horizontal asymptotes Ans: A

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89. Find any horizontal asymptotes for the given function. 9 x3 y= 2 9x + 2 A) y =1 B) y=9 C) y=2 D) y=0 E) no horizontal asymptotes Ans: E

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90.

Match the function f ( x) =

3x 2 with one of the following graphs. x2 + 2

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Ans: D

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91.

Match the function f ( x) =

3x with one of the following graphs. x +2 2

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Ans: B 92. Find the limit: x – 15 – x + 11 A) –∞ B) ∞ C) 0 D) –1 E) 1 Ans: B lim

x →11+

93. Find the limit: 1  lim  x 6 –  x  A) 0 B) –∞ C) 1 D) –1 E) ∞ Ans: E x → 0−

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94.

For the function f ( x) =

, use a graphing utility to complete the table and 64 x 2 – 6 estimate the limit as x approaches infinity. x 100 f(x) A) –0.525 B) 1.375 C) 0.125 D) 2.555 E) does not exist Ans: C

101

102

103

104

105

95. Use a table utility with x-values larger than 10,000 to investigate lim f ( x) .What does x →+∞

the table indicate about lim f ( x) ? x →+∞

16 x − 3 x 5 − 8 x3 A) −2 B) 2 C) 16 D) 0 E) does not exist Ans: A f ( x) =

96.

5x – 4 , use a graphing utility to complete the table and 3x + 4 estimate the limit as x approaches infinity.

For the function f ( x) =

x 100 f(x) A) 0.6 B) 1.666667 C) 2.666667 D) 1.6 E) –0.4 Ans: B

101

102

103

104

97. Use analytic methods to find the limit as x → +∞ for the given function. 3000 x f ( x) = 3600 − 10 x A) 300 B) 360 C) −360 D) −300 E) does not exist Ans: D

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105

98. Use analytic methods to find the limit as x → −∞ for the given function. 9000 x f ( x) = 4650 − 5 x A) −1800 B) 930 C) −930 D) 1800 E) does not exist Ans: A 99. Find the limit. 7x – 1 lim x →∞ 4 x – 2 A) 1 2 B) 7 4 C) 1 D) 0 E) does not exist Ans: B 100. Find the limit. –2 x + 3 lim 2 x →∞ 3 x – 4 A) ∞ B) 1 C) 0 D) 2 – 3 E) 3 – 4 Ans: C 101. Find the limit. 5 x 2 − 3 x − 14 lim 2 x→∞ 2 − 5 x − 8 x A) 5 − 8 B) 7 C) –7 D) 5 8 E) 3 5 Ans: A Larson, Calculus: An Applied Approach (+Brief), 9e

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102. Find the limit. 5x2 lim x→−∞ x + 6 A) 0 B) 5 − 6 C) 5 6 D) ∞ E) −∞ Ans: E

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103.

Sketch the graph of the function f ( x) =

2+ x using any extrema, intercepts, symmetry, 2− x

and asymptotes. A)

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Ans: E

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104.

Sketch the graph of the function f ( x) =

x2 using any extrema, intercepts, x2 − 4

symmetry, and asymptotes. A)

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Ans: A

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105. Sketch the graph of the relation xy 2 = 4 using any extrema, intercepts, symmetry, and asymptotes. A)

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Ans: B

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106. Sketch the graph of the relation x 2 y = 3 using any extrema, intercepts, symmetry, and asymptotes. A)

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Ans: E

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107. Sketch the graph of the equation given below. Use intercepts, extrema, and asymptotes as sketching aids.

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Ans: A 108. A business has a cost (in dollars) of= C 0.9 x + 400 for producing x units. What is the limit of C as x approaches infinity? A) ∞ B) $0.90 C) $400.90 D) $400.00 E) $399.10 Ans: B 109. The cost C (in millions of dollars) for the federal government to seize p% of a type of illegal drug as it enters the country is modeled by C 580 /(100 − p ) , for 0 ≤ p ≤ 100 . = − Find the limit of C as p → 100 . A) 580 B) 100 C) −100 D) ∞ E) 0 Ans: D

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110. Analyze and sketch a graph of the function = y x3 – 9 x 2 + 3 . A)

C) Larson, Calculus: An Applied Approach (+Brief), 9e

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Ans: E

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111. Analyze and sketch a graph of the function= y 2 – x − x3 . A)

C) Larson, Calculus: An Applied Approach (+Brief), 9e

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Ans: C

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112. Sketch the graph of the function below. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph.

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Ans: A

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113. Sketch the graph of the function given below. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph.

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Ans: B

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114.

Analyze and sketch a graph of the function f ( x) =

x . x +1 4

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Ans: C

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115.

Analyze and sketch a graph of the function f ( x) =

x 2 – 3 x + 10 . x +1

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E) none of the above Ans: E Larson, Calculus: An Applied Approach (+Brief), 9e

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116. Analyze and sketch a graph of the function= y x 1− x . A)

C) Larson, Calculus: An Applied Approach (+Brief), 9e

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Ans: A

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117. Analyze and sketch a graph of the function= y x 1 − x2 . A)

C) Larson, Calculus: An Applied Approach (+Brief), 9e

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Ans: D

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118.

Analyze and sketch a graph of the function f ( x) =

x4 . x4 + 1

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Ans: B

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119. Use the graph f ' to sketch the graph of f .

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Ans: C

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120. Use the graph f ′′ to sketch the graph of f .

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Ans: D 121. An employee of a delivery company earns $ 22.50 per hour driving a delivery van in an area where gasoline costs $ 2.50 per gallon. When the van is driven at a constant speed 225 s (in miles per hour, with 45 ≤ s ≤ 60 ), the van gets miles per gallon. Determine s the most economical speed s for a 100-mile trip on an interstate highway. A) The most economical speed is 47.0 mph. B) The most economical speed is 43.0 mph. C) The most economical speed is 22.5 mph. D) The most economical speed is 45.0 mph. E) The most economical speed is 48.0 mph. Ans: D

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122. Find the differential dy of the function y = 4 x 2 – x – 3 . A) (8 x 2 – x – 3)dx B) (4 x – 1)dx C) 4 1 ( x3 – x 2 – 3x)dx 3 2 3 D) (4 x – x 2 – 3 x)dx E) (8 x – 1)dx Ans: E 123. Find the differential dy of the function y = 3 x 3/ 8 . A) x 3/ 8 ln(3) B) 3 −5/ 8 x dx 8 C) 3x −5/ 8 dx D) 9 −5/ 8 x dx 8 E) 3 – x −5/ 8 dx 8 Ans: D 124. Compare dy and ∆y for y = 3 x 2 – 2 at x = 1 with dx = 0.05. Give your answers to four decimal places. A)= dy 0.2900; = ∆y 0.3073 B)= dy 0.3000; = ∆y 0.3075 C)= dy 0.2700; = ∆y 0.3074 D)= dy 0.3200; = ∆y 0.3072 E)= dy 0.3200; = ∆y 0.3074 Ans: B 125. Compare dy and ∆y for y = –2 x 2 – 4 at x = –1 with ∆x = dx = 0.07. Give your answers to four decimal places. A)= dy 0.2700; = ∆y 0.2700 B)= dy 0.2800; = ∆y 0.2702 C)= dy 0.2500; = ∆y 0.2701 D)= dy 0.3000; = ∆y 0.2699 E)= dy 0.3000; = ∆y 0.2701 Ans: B

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126. Compare dy and ∆y for y = 2 x 4 + 2 at x = 0 with dx = 0.09. Give your answers to four decimal places. A) dy = 0.0200 ; ∆y = –0.0002 B) dy = –0.0100 ; ∆y = –0.0001 C) dy = –0.0300 ; ∆y = 0.0000 D) dy = 0.0000 ; ∆y = 0.0001 E) dy = 0.0200 ; ∆y = 0.0000 Ans: D 127.

1 . Let x = 4. x ∆y − dy dy / ∆y

Complete the table for the function y = dx = ∆x dy 4.00000 2.00000 0.40000 A) dx = ∆x 4.00000 2.00000 0.40000 B) dx = ∆x 4.00000 2.00000 0.40000 C) dx = ∆x 4.00000 2.00000 0.40000 D) dx = ∆x 4.00000 2.00000 0.40000 E) dx = ∆x 4.00000 2.00000 0.40000 Ans: C

∆y

dy –0.25 –0.125 –0.0561 dy –0.25 –0.125 –0.025 dy –0.25 –0.125 –0.025 dy –0.1989 –0.125 –0.0561 dy –0.1989 –0.125 –0.0561

∆y –0.125 –0.08333 –0.02273 ∆y –0.125 0.22667 –0.02273 ∆y –0.125 –0.08333 –0.02273 ∆y –0.125 –0.08333 –0.02273 ∆y –0.125 0.22667 –0.02273

∆y − dy 0.125 0.04167 0.00227 ∆y − dy 0.125 0.04167 0.00227 ∆y − dy 0.125 0.04167 0.00227 ∆y − dy 0.125 0.04167 0.00227 ∆y − dy 0.125 0.04167 0.00227

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dy / ∆y 2 1.50006 1.10208 dy / ∆y 2 1.50427 1.09987 dy / ∆y 2 1.50006 1.09987 dy / ∆y 2 1.50006 1.10208 dy / ∆y 2 1.50427 1.10208

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128. Complete the table for the function y = x . Let x = 3. dx = ∆x dy ∆y ∆y − dy dy / ∆y 3.00000 1.50000 0.30000 A) dx = ∆x dy ∆y ∆y − dy 3.00000 0.86603 0.71744 –0.14859 1.50000 0.43301 0.38927 –0.04374 0.30000 0.0555 0.08454 –0.00206 B) dx = ∆x dy ∆y ∆y − dy 3.00000 0.86603 0.71744 –0.14859 1.50000 0.43301 0.69927 –0.04374 0.30000 0.0866 0.08454 –0.00206 C) dx = ∆x dy ∆y ∆y − dy 3.00000 0.91713 0.71744 –0.14859 1.50000 0.43301 0.69927 –0.04374 0.30000 0.0555 0.08454 –0.00206 D) dx = ∆x ∆y − dy dy ∆y 3.00000 0.91713 0.71744 –0.14859 1.50000 0.43301 0.38927 –0.04374 0.30000 0.0555 0.08454 –0.00206 E) dx = ∆x ∆y − dy dy ∆y 3.00000 0.86603 0.71744 –0.14859 1.50000 0.43301 0.38927 –0.04374 0.30000 0.0866 0.08454 –0.00206 Ans: E

dy / ∆y 1.20711 1.11236 1.02658 dy / ∆y 1.20711 1.11657 1.02437 dy / ∆y 1.20711 1.11657 1.02658 dy / ∆y 1.20711 1.11236 1.02658 dy / ∆y 1.20711 1.11236 1.02437

129. The revenue R for a company selling x units is= R 500 x − 0.1x 2 . Use differentials to approximate the change in revenue if sales increase from x = 1000 to x = 1100 units. A) 28,000 dollars B) 30,000 dollars C) 25,000 dollars D) 33,000 dollars E) 40,000 dollars Ans: B 130. The variable cost for the production of a calculator is $ 16.25 and the initial investment is $ 530,000. Use differentials to approximate the change in the cost C for a one-unit increase in production when x = 80, 000 , where x is the number of units produced. A) 1300000.00 dollars B) 17.25 dollars C) 1301000.00 dollars D) 16.25 dollars E) 26.25 dollars Ans: D

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131. The measurement of the circumference of a circle is found to be 43 centimeters, with a possible error of 0.9 centimeters. Approximate the percent error in computing the area of the circle. A) 4.65 % B) 2.09 % C) 4.19 % D) 8.37 % E) 2.33 % Ans: C 132. The measurement of the edge of a cube is found to be 11 inches, with a possible error of 0.01 inch. Use differentials to estimate the propagated error in computing (a) the volume of the cube and (b) the surface area of the cube. Give your answers to two decimal places. A) 4.36, 1.32 B) 2.90, 1.19 C) 3.27, 1.19 D) 2.90, 1.06 E) 3.63, 1.32 Ans: E

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Chapter 4: Exponential and Logarithmic Functions 1.

Evaluate the expression 243 5 . A) 243 B) 3 C) 9 D) 11 E) 13 Ans: C 6

 −1 62  Use the properties of exponents to simplify the expression (5 )(5 )  .   A) 1 25 B) 25 C) 1 15625 D) 1 625 E) 625 Ans: D

3. After t years, the remaining mass y(in grams) of 20 grams of a radioactive element t / 35

1 whose half-life is 35 years is given by y = 20   , for t ≥ 0 . How much of the 2 initial mass remains after 140 years? Round your answer to two decimal places. A) 2.50 grams B) 2.45 grams C) 3.55 grams D) 3.40 grams E) 1.25 grams Ans: E

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4. Sketch the graph of the function f ( x) = 2 x . A)

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Ans: B 5. With an annual rate of inflation of 4% over the next 10 years, the approximate cost of goods or services during any year in the decade is given by = C (t ) P (1.04)t , 0 ≤ t ≤ 10 where is the time (in years) and is the present cost. The price of an oil change for a car is presently $24.95.Estimate the price 10 years from now. A) $37.09 B) $36.93 C) $89.00 D) $63.90 Ans: B

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1 −x Use a graphing utility to graph the function = f ( x) =  13 . 13   A)

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Ans: C

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7. Use a graphing utility to graph the function f ( x) = 3− x 2 . A)

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Ans: B 8. Assume the population P (in millions) of the United States from 1992 through 2005 can be modeled by the exponential function P(t ) = 254.82(1.606)t , where t is the time in years, with t = 2 corresponding to1992. Use the model to estimate the population in the year 2007. Round your answer to the nearest million. A) 7022 million B) 4372 million C) 11,277 million D) 657 million E) 7021 million Ans: A 9. After t years, the value of a car that originally cost $ 19,000 depreciates so that each 3 year it is worth of its value for the previous year. Find a model for V(t), the value of 4 the car after t years. A)  3t  V(t) = 19,000   4 B) 3 V(t) = 19,000t   4 t C) 3 V(t) = 19,000   4 D) 3 V(t) = 19,000  t  4  t E) 3 V(t) = 19,000t   4 Ans: C

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10. Suppose that the annual rate of inflation averages 4% over the next 10 years. With this rate of inflation, the approximate cost C of goods or services during any year in that decade will be given by C(t) = P(1.04)t, 0 ≤ t ≤ 10 where t is time in years and P is the present cost. If the price of an oil change for your car is presently $ 22.95, estimate the price 9 years from now. Round your answer to two decimal places. A) $ 33.97 B) $ 34.67 C) $ 35.97 D) $ 37.67 E) $ 32.67 Ans: E 11. Use the properties of exponents to simplify the expression (e −6 )(e −7 / 2 ) . A) e −21 B) e6.5 C) e 21 D) e −6.5 E) e 23 Ans: C

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12. Sketch the graph of the function f ( x) = e3 x + 2 . A)

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Ans: B

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13. Sketch the graph of the function f ( x) = e 2 x . A)

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Ans: C

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14.

Use a graphing utility to graph the function g ( x) =

11 . Be sure to choose an 1 + e− x

appropriate viewing window. A)

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Ans: A 15. Determine whether the function below has any horizontal asymptotes.

A) horizontal asymptotes: y = 1 B) no horizontal asymptotes C) horizontal asymptotes: y = 0 and y = 2 D) horizontal asymptotes: y = 3 E) horizontal asymptotes: y = 1 and y = 3 Ans: B

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16. Determine the continuity of the function below.

A) discontinuous at x = 0 B) continuous on the entire real number line C) discontinuous at x = 1 D) discontinuous at x = 2 E) discontinuous at x = 4 Ans: B 17. What is the resulting balance if $6800 is invested for 5 years at an annual rate of 12% compounded monthly? A) $ 7146.87 B) $ 7662.41 C) $ 10,880.00 D) $ 12,353.54 E) $ 40,062.90 Ans: D 18. How much more interest will be earned if $4000 is invested for 7 years at an annual rate of 12% compounded continuously, instead of at 12% compounded quarterly? A) $38.58 B) $75.18 C) $113.76 D) $1791.71 E) $1866.89 Ans: C 19. What lump sum should be deposited in an account that will earn at an annual rate of 10%, compounded quarterly, to grow to $180,000 for retirement in 35 years? A) $177,957.47 B) $5514.93 C) $12,000.00 D) $40,000.00 E) $5674.55 Ans: E 20. To help their son buy a car on his 18th birthday, a boy’s parents invest $1600 on his 12th birthday. If the investment pays an annual rate of 11% compounded continuously, how much is available on his 18th birthday? A) $3068.20 B) $3095.67 C) $2992.66 D) $2656.00 E) $22,421.13 Ans: B

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21. What is the annual percentage yield (or effective annual rate) for a nominal rate of 8.1% compounded quarterly? A) 8.10% B) 8.41% C) 8.44% D) 8.35% E) 8.26% Ans: D 22. Find the future value if $2900 is invested for 4 years at an annual rate of 10% compounded quarterly. A) $ 4640.00 B) $ 4284.62 C) $ 3201.06 D) $ 4245.89 E) $ 4305.07 Ans: E 23.

3   The demand function for a product is modeled= by p 3000 1 − . Find the −0.0002 x   3+ e  price of the product if the quantity demanded is x = 200. Round your answer to two decimal places where applicable. A) $ 547.90 B) $ 2282.47 C) $ 727.73 D) $ 738.03 E) $ 2272.27 Ans: C

24.

3   The demand function for a product is modeled= by p 5000 1 − . What is −0.0003 x   3+ e  the limit of the price as x increases without bound? Round your answer to two decimal places where applicable. A) The limit of the price as x increases without bound is -1. B) The limit of the price as x increases without bound is 1. C) The limit of the price as x increases without bound is 0. D) The limit of the price as x increases without bound is 5000 . E) The limit of the price as x increases without bound is 3 . Ans: C

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25. The average time between incoming calls at a switchboard is 3 minutes. If a call has just come in, the probability that the next call will come within the next t minutes is 1 P(t ) = 1 − e − t / 3 . Find the probability that the next call will come within the next 8 minute. Round your answer to two decimal places. A) 4.08% B) 0.41% C) 195.92% D) 6.31% E) 3.95% Ans: A 26. Find the derivative of f ( x) = x −5 – 3e x . A) f ′( x) = 5 x 4 – 3e x B) 5 f ′( x) = − 6 – 3 xe x −1 x C) 5 f ′( x) = − 6 – 3e x x D) f ′( x) = 5 x 4 – 3 xe x −1 E) 5 f ′( x) = − 6 – 3 xe x x Ans: C 27. Find the derivative of the following function. = y 5 x 4 − 4e x A) = y′ 20 x3 − 4 xe x −1 B) = y′ 5 x3 − 4 xe x −1 C) = y ′ 5 x 3 − 4e x D) = y′ 20 x3 − e x E) = y′ 20 x3 − 4e x Ans: E 28. Find the derivative of the following function. 7

y= 2 − 5e − x 7 A) y′ = 35 x 6 e − x B)

y′ = −35 x 6 e − x

y′ = 5e − x

y′ = 5 x 7 e− x

7 7

y′ = −5 x 7 e − x Ans: A

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29. Find the derivative of the following function. 2

y = 2e6 x − 4 2 A) y′ = 12 xe6 x − 4 B)

y′ = 24 xe6 x − 4

y′ = 24e6 x − 4

y′ = 12e6 x − 4

y′ = 4 xe6 x − 4 Ans: B 30.

10 dy if y = e6 x . dx A) dy = 10e6 x dx 10 B) dy = e6 x dx 10 C) dy = 6 x10 e6 x −1 dx D) dy = 60 x 9 ln ( 610 ) dx 10 E) dy = 60 x 9 e6 x dx Ans: E

Find

31. Find the derivative of the following function.

= y 5e8 x + 3 A) 20e8 x y′ = x 8 B) 5e x y′ = x C) 40e8 x y′ = x 8 D) 4e x y′ = x 8 E) 8e x y′ = x Ans: A

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32. Find the derivative of the following function. 6

p = 2qe q 6 A) = p′ 2e q (6q 6 + 1) 6 B) = p′ 2e q (6q 5 + 1) 6 C)= p′ 2e q (q 6 + 2) 6 D) = p′ 2e q (6q 6 + 2) 6 E) = p′ 2e q (6q 5 + 2)

Ans: A 33.

5 dy if y = x8e x . dx 9 A) 5  x dy x6  = ex  +  dx  9 6  5 B) dy = e x ( x8 + x 5 ) dx 5 C) dy = e x ( 8 x 7 + 5 x12 ) dx 5 D) dy = 8 x 7 e x −1 dx 5 E) dy = 8x7e x dx Ans: C

Find

34. Find the equation of the tangent line to f ( x= ) 3 x + e x at the point (0,1). A) y = –4 x – 4 B) y = 4x – 4 C) = y 4x −1 D) = y 4x +1 E) = y –4 x − 1 Ans: D 35. Find an equation of the tangent line to the graph of y = e8x at the point (0,1) . A) y= x + 1 B)= y ln ( 8 ) x + 1 C) = y 9x +1 D) = y 8x + 1 E) = y 8x − 1 Ans: D

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36. Write the equation of the line tangent to the graph of = y 5 xe − x + 8 at x = 1. A) = y 5e −1 x + 8 B) = y 5e −1 x − 8 C) y = 5e −1 x D) = y 5e −1 + 8 E) = y 5e −1 − 8 Ans: D 37. If x − 5 xe9 y = 8, find dy / dx. A) dy 1 − 5e9 y = dx 90e9 y B) dy 1 + 9e9 y = dx 45 xe9 y C) dy 1 − 5e9 y = dx 45 xe9 y D) dy 1 + 5e9 y = dx 90 xe9 y E) dy 1 − 9e9 y = dx 45e9 y Ans: C 38. If x 7 y = 4e x + y , find dy / dx. A) dy 4e x + y − 7 x 7 y = 7 dx x + 4e x + y B) dy 4e x + y − 7 x 6 y = 7 dx x − 4e x + y C) dy 4e x + y + x 6 y = dx x 7 − 4e x + y D) dy e x + y − 7 x 6 y = 7 x+ y dx x −e E) dy 4e x + y + x 7 y = 7 x+ y dx x −e Ans: B

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39.

Use implicit differentiation to find

dy . dx

5e xy – 2 y 2 = 7 A) dy 5 ye xy = dx 5e xy – 4 y B) dy 5 ye xy = xy dx 5e + 4 y C) dy 5e xy = − xy dx 5e x + 4 y D) dy 5 ye xy = − xy dx 5e x – 4 y E) dy 5 ye xy = − xy dx 5e x – 2 y Ans: D 40. If 7 x + e5 xy = 7, find dy / dx. A) dy 7 + 5 ye5 xy = − dx 5 xe5 xy B) 7 + 5 xe5 xy dy = − 5 ye5 xy dx C) dy 7 + 5 ye5 xy = dx 7 xe5 xy D) dy 7 + 5 xe5 xy = dx 5 ye5 xy E) dy 7 + 5 ye5 xy = dx xe5 xy Ans: A 41. Find f ′′( x) if f ( x= ) ( 6 + 7 x ) e −6 x . A) f ′′= ( x) (132 − 252 x ) e −6 x B) C) D) E)

( –132 − 252 x ) e−6 x −132 ( 6 + 7 x ) e −6 x f ′′( x) = − ( 29 + 42 x ) e −6 x f ′′( x) = f ′′= ( x) (132 + 252 x ) e −6 x ′′( x) f=

Ans: E

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42.

Find the extrema of the function f ( x) = A) B)

1 . 9 − e− x

(0, 1) ( 9, e9 ) , (0, 0)

(1,9 ) , ( 9, e−9 )

D) E)

no relative extrema ( 9, e−9 )

Ans: D 43.

Find the extrema of the function f ( x) =

A) B) C) D) E)

1 by analyzing its graph below. 4 − e− x

(0, 1) no relative extrema ( 4, e4 ) , (0, 0)

(1, 4 ) , ( 4, e−4 )

( 4, e ) −4

Ans: B 44. Solve for the equation e x = e 4 for x . A) x=4 B) x=2 C) x = 64 D) x = 16 E) x = 65 Ans: D

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45. The average typing speed N (in words per minute) after t weeks of lessons is modeled 93 by N = . Find the rate at which the typing speed is changing when t = 20 1 + 7.5e −0.18t weeks. Round your answer to two decimal places. A) 1.75 words/min/week B) 2.36 words/min/week C) 2.85 words/min/week D) 4.59 words/min/week E) 5.38 words/min/week Ans: B 46. Future value. The future value that accrues when $500 is invested at 5%, compounded continuously, is s (t ) = 500e0.05t , where t is the number of years. At what rate is the money in this account growing when t = 7 ? A) $7.10 per year B) $26.81 per year C) $709.53 per year D) $517.81 per year E) $35.48 per year Ans: E 47. A survey of high school seniors from a certain school district who took the SAT has determined that the mean score on the mathematics portion was 500 with a standard deviation of 13.5. Assuming the data can be modeled by a normal probability density function, find a model for these data. A) 1 f ( x) = e − ( x −500) /182.25 13.5 π B) 1 f ( x) = e − ( x −500) / 364.5 13.5 2π 2 C) 1 f ( x) = e − ( x −500) / 364.5 13.5 π 2 D) 1 f ( x) = e − ( x −500) / 364.5 13.5 2π 2 E) 1 f ( x) = e − ( x −500) /182.25 13.5 2π Ans: D

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48. A survey of high school seniors from a certain school district who took the SAT has determined that the mean score on the mathematics portion was 650 with a standard deviation of 13.5. By a normal probability density function the data can 2 1 be modeled as f ( x) = e − ( x −650) / 364.5 . Find the derivative of the model. 13.5 2π 2 A) −2 2( x − 650)e − ( x −650) /182.25 f ′( x) = 4,921 π 2 B) −2 2( x − 650)e − ( x −650) / 364.5 f ′( x) = 9,842 π 2 C) 2( x − 650)e − ( x −650) / 364.5 f ′( x) = 4,921 π 2 D) 2( x − 650)e − ( x −650) / 364.5 f ′( x) = 9,842 π 2 E) 2( x − 650)e − ( x −650) /182.25 f ′( x) = 4,921 π Ans: B 49. Write the logarithmic equation ln 0.7 = –0.3567  as an exponential equation. A) e0.3567 = 0.7 B) e −0.7 = –0.3567  C) e –0.3567 = 0.7 D) e0.7 = –0.3567  E) e –0.3567 = −0.7 Ans: C 50. Write the exponential equation e12 = 162754.7914 as a logarithmic equation. A) ln 162754.7914 = 12 B) ln 162754.7914 = 24 C) ln 12 = 162754.7914 D) ln 12 = 325509.5828 E) ln 325509.5828 = 24 Ans: A

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51. Sketch the graph of the function f ( x) = 1 + ln( x) . A)

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Ans: E

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52. Sketch the graph of the function f ( x) = ln 2 x . A)

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Ans: A

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53. Sketch the graph of the function y= 6 + ln x . A)

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Ans: B 54. Simplify ln e3 x2 . A) 3x 2 B) 6x C) 6x 2 D) 3x E) x2 Ans: A 55. Simplify eln10 x7 . A) − x7 B) −10x 7 C) 10x D) 10x 7 E) x7 Ans: D

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56.

Use the properties of logarithms to approximate ln ln 37 ≈ 3.6109. A) –6.0088 B) –1.2130 C) 0.6641 D) 8.6586 E) 6.0088 Ans: B

57.

Use the properties of logarithms to expand ln ln 4 + ln 21 ln 4 − ln 21 ln 4 ln 21 D) ( ln 4 )( ln 21) E) none of the above Ans: B

11 , given that ln11 ≈ 2.3979 and 37

4 . 21

A) B) C)

58.

 x2 −1  Use the properties of logarithms to expand ln  6  .  x  A) 2 [ ln( x + 1) − ln( x − 1) − 6 ln x ] B) 2 [ ln( x + 1) + ln( x − 1) + 6 ln x ]

2 [ ln( x + 1) − ln( x − 1) − ln x ]

2 [ ln( x + 1) + ln( x − 1) + ln x ]

2 [ ln( x + 1) + ln( x − 1) − 6 ln x ] Ans: E E)

59.

(

Use the properties of logarithms to write the expression ln x 4 x 2 + 6 difference, or multiple of logarithms. A) 1 ln x + ln( x 2 + 4) 6 B) 1 ln( x 2 + 6) + ln x 4 C) 1 ln x + ln( x 2 + 6) 4 D) 1 ln( x 2 + 4) + ln x 6 2 E) ln x + ln( x + 6) Ans: C

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) as a sum,

60. Use the properties of logarithms to write the expression as a single logarithm. ln (3x) − ln (2y ) A)  3x  ln    2y  B) ln ( 2 y − 3 x ) C) ln ( 3x − 2 y )

ln (3 x) ln (2 y )

 3y  ln    2x  Ans: A

61. Write the expression 4 ln( x) + 3ln ( x + 2 ) – 5ln ( x − 2 ) as the logarithm of a single quantity. A) ln ( 2 x + 16 ) B) 12 ( x 2 + x )   ln   5 ( x – 2 )  3 5 C) ln x 4 x + 2 x − 2

( (

)(

))

 x 4 ( x + 2 )3  ln   5  ( x – 2 ) 

  x4 ln   3 5  ( x + 2 ) ( x − 2 ) 

Ans: D

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62.

1 Write the expression 3ln ( 5 ) − ln ( x 2 + 4 ) as the logarithm of a single quantity. 5 A)   125   ln  x2 + 4 5  ) (

(

ln 125 5 x 2 + 4

 243  ln   5 2  x +4

ln 243 5 x 2 + 4

(

)

 125  ln   5 2  x +4 Ans: E

63. Write the following expression as a logarithm of a single quantity. 3ln x − 14 ln ( x 2 + 2 )

  x3  ln   14 ( x 2 + 2 )      x3   ln  ( x 2 + 2 )14   

ln x3 − ( x 2 + 2 )

ln 3 x − 14 ( x 2 + 2 )

( (

)

E) none of the above Ans: B

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64. Write the following expression as a logarithm of a single quantity. ln x − 3ln ( x 2 + 1)

 –3 x   ln  2  ( x + 1)      x  ln   3 ( x 2 + 1)   

(

ln x − 3 ( x 2 + 1)

)

  x   ln  ( x 2 + 1)3    E)  –3 x  ln  2   x +1  Ans: D

65. Solve the following equation for x accurate to three decimal places. eln ( 6 x ) = 7 A) x = 42.000 B) x = 0.857 C) x = 1.167 D) x = 0.324 E) x = –0.324 Ans: C 66. Solve the following equation for x accurate to three decimal places. ln x –3 = 2 A) x = –0.667 B) x = 0.513 C) x = 0.595 D) x = 0.223 E) x = 3.811 Ans: B

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67. Solve the exponential equation. Give the answer correct to 3 decimal places. 12, 000 = 1000e0.5 x A) 0.735 B) 1.471 C) 2.719 D) 1.242 E) 4.970 Ans: E 68. Solve the exponential equation. Give the answer correct to 3 decimal places. = 54 250 − 250e –0.09 x A) –0.243 B) 17.028 C) 2.704 D) –17.028 E) –2.173 Ans: C

69. Solve the exponential equation. Give the answer correct to 3 decimal places. 82 1 + 6e –0.5 x A) –2.734 B) –0.130 C) 1.841 D) –0.921 E) 5.468 Ans: E 59 =

70. Solve the exponential equation. Give answers correct to 3 decimal places. 67 x = 1296 A) 216 B) 0.571 C) 0.774 D) 0.371 E) 108 Ans: B

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71. Solve the following equation for x accurate to three decimal places. 3 ( 45 x−3 ) = 255

A) x = –3.650 B) x = 4.850 C) x = –0.041 D) x = 1.241 E) x = –1.607 Ans: D 72.

0.478   Solve 18 − 30 for t. Round your answer to four decimal places.  = 25   A) 1.1772 B) 0.2942 C) 2.5673 D) 0.2943 E) 2.4502 Ans: D

73. How long (in years) would $400 have to be invested at an annual rate of 10%, compounded continuously, to amount to $530? A) 3.25 years B) 2.95 years C) 0.56 years D) 4.54 years E) 2.81 years Ans: E 74. Find the derivative of the following function. y = ln x 5 A) 1 x B) 5 x C) 1 5x D) 1 x2 E) 1 5x 2 Ans: B

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75. Find the derivative of the following function. y = ln 6 x A) 1 x B) 6 x C) 1 6x D) 1 x2 E) 1 6x 2 Ans: A 76. Find the derivative of the following function. y= 3 + ln 2 x A) 1 x B) 2 x C) 1 2x D) 1 x2 E) 1 2x 2 Ans: A 77. Find the derivative of y= 4 − 19 ln x. A) dy 1 = − dx 19 x B) dy 19 = − dx x C) dy 15 =– dx x D) dy 19 = dx x E) dy 15 = dx x Ans: B

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78. Find the derivative of the following function. = y ln(3 x + 5) A) 5x 3x + 5 B) 3x 3x + 5 C) 1 3x + 5 D) 3 3x + 5 E) 5 3x + 5 Ans: D 79. Find the derivative of the following function. = y 5ln( x 6 − 3) A) 5 6 x −3 B) 6 x5 x6 − 3 C) 6 6 x −3 D) 30 x5 x6 − 3 E) 5 x5 x6 − 3 Ans: D 80. Find the derivative of the following function. y= ln(9 x3 − 4 x) − 8 x A) 27 x 2 − 4 x −8 x(9 x 2 − 4) B) 1 −8 x(9 x 2 − 4) C) 27 x 2 − 4 −8 x(9 x 2 − 4) D) 27 x 2 −8 x(9 x 2 − 4) E) 1 −8 x(9 x 2 − 1) Ans: C

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81. Find the derivative of the function = y ln x 2 − 15 . A) dy 1 = dx 2 x − 15 B) dy x = dx 2 x − 15 C) dy 1 = 2 dx x − 15 D) dy x = 2 dx x − 15 E) dy 2x = dx x 2 − 15 Ans: D 82. Find the derivative of the function = y ln x 2 + 14 . A) 2x 2 x + 14 B) x 2 x + 14 C) 1 D)

x 2 + 14 2x

x 2 + 14 E) x 2 x + 14 Ans: E

83. Find the derivative of the function = y ln x 2 − 7 . A) dy 1 = dx 2 x − 7 B) dy x = dx 2 x − 7 C) dy 1 = 2 dx x − 7 D) dy x = 2 dx x − 7 E) dy 2x = dx x2 − 7 Ans: D

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84. Find y′ . = y ln(9 x + 2)1/ 7 A) 9 7(9 x + 2) B) 2 9(9 x + 2) C) 2 7(9 x + 2) D) 18 7(9 x + 2) E) 63 2(9 x + 2) Ans: A 85. Find y′ . y = 7(ln x) −6 A) 49 − x(ln x)7 B) 36 − x(ln x)7 C) 42 − x(ln x)7 D) 42 − x(ln x)5 E) 36 − x(ln x)5 Ans: C

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86. Find y ' . = y [ln( x 6 + 5)]4 A) 24 x 5 ln( x 6 + 5)3

( x + 5) 6

B) C) D) E)

4 x 4 ln( x 6 + 5)3 ( x6 + 5) −

4 x 5 ln( x 6 + 5)3 ( x6 + 5)

−

6 x 5 ln( x 6 + 5)3 ( x6 + 5)

−

48 x 4 ln( x 6 + 5)3 ( x6 + 5)

Ans: A 87. Find y′ . 7 + ln x y= x5 A) 5ln x + 34 x4 B) 35ln x + 4 − x4 C) 5ln x + 34 − x6 D) 7 ln x + 4 − x6 E) 35ln x − 34 x6 Ans: C

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88.

ds Find = if s ln t 6 (t 5 − 3)  . dt A) 5t 6 − 3 t (t 5 − 3) B) 6t 5 + 5t 4 − 6

C) D)

t (t 5 − 3) 11t 4 − 6 t (t 5 − 3) 11t 5 − 18 t (t 5 − 3)

6t 5 + 5t 4 − 18 t (t 5 − 3) Ans: D E)

89.

Find A)

1 dy = if y ln( x5 ( x + 3) 2 ). dx 5 ( 2 x + 3)

2 x( x + 3) B) 11x + 30 2 x( x + 3) C) 5 ( 2x + 6) x( x + 3) D) 11x + 15 x( x + 3) E) 11x + 10 x( x + 3) Ans: B

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90.

Find A)

dy = if y ln( x 4 ( x 3 − x + 7)) . dx 7 x3 − 5 x + 28 x ( x3 − x + 7 )

7 x3 + 5 x − 28 x ( x3 − x + 7 )

7 x3 + 5 x + 4 x ( x3 − x + 7 )

3x3 − 5 x − 4 x ( x3 − x + 7 )

3 x 3 + 5 x + 28 x ( x3 − x + 7 )

Ans: A 91. Find A) B) C)

 5q 2 − 2  dp if p = ln  . dq  q 

5q 2 − 2 q (5q 2 − 2) 5q 2 + 2 q (5q 2 − 2) 10q 2 + 2 q (5q 2 − 2)

10q 2 − 2 q (5q 2 − 2) E) 10q 2 + 5 q (5q 2 − 2) Ans: B D)

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92.

dy  7x + 4  Find if y = ln  2  . dx  x −2  A) 7 x2 + 8x + 7 1/ 8

8(7 x + 4)( x 2 − 2) B) C) D)

7 x 2 + 4 x + 14 (7 x + 4)( x 2 − 2) 7 x 2 + 4 x + 14 8(7 x + 4)( x 2 − 2)

−

7 x 2 + 8 x + 14 (7 x + 4)( x 2 − 2)

7 x 2 + 8 x + 14 − 8(7 x + 4)( x 2 − 2) Ans: E E)

93. Use a change-of-base formula to rewrite the logarithm in terms of natural logarithms. y = log 5 x A) y = 5 ln x B) = y ln ( x − 5) C) = y ln x − ln 5 D) y = 5x E) ln x y= ln 5 Ans: E 94. Use a calculator to evaluate the logarithm log 2 34 . Round your answer to three decimal places. A) 0.197 B) 2.444 C) 6.360 D) 3.717 E) 5.087 Ans: E

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95. Find y′ . y = 3log 4 x A) 3 x ln 4 B) 4 x ln 3 C) 3 x D) 3ln 4 x E) x 3ln 4 Ans: A 96. Find the derivative of the following function. y = 35 x +1 A) y′ = ( 3ln 5 ) 35 x +1 B) y′ = ( 5ln 3) 35 x +1 C)

y′ = ( ln 5 ) 35 x +1

y′ = ( ln 3) 35 x +1

y′ = ( ln15 ) 35 x +1 Ans: B E)

97. Find y′ . = y log8 (3 − x − x8 ) A) 8 x9 + x

(ln 8)( x8 + x − 3) B) C) D)

x7 + x (ln 8)( x8 + x − 3) 8x7 + x ( x8 + x − 3) 8 x8 + 1 ( x8 + x − 3)

8x7 + 1 (ln 8)( x8 + x − 3) Ans: E E)

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98. Find f ′(t ) if f (t ) = t –5 34t . A) = f ′(t ) –5t –6 34t + 4t –5 34t B) = f ′(t ) –5t –6 34t + 34t −1 C) = f ′(t ) –5t –6 34t + 4 ln ( 3) t –5 34t D) = f ′(t ) –5t –6 34t + 12t –5 34t E) f ′(t ) = –5t –6124t −1 Ans: C 99. For f ( x) = –5 x ln x , calculate f ′ ( 5 ) to three decimal places. A) 1.609 B) –40.236 C) –13.047 D) –1.000 E) –8.047 Ans: C 100. Find an equation of the tangent line to the graph of y = log 4 x at the point (16, 2) . A) 1 y= 2+ ( x − 16 ) ln 4 B) 1 y= 2+ ( x − 16 ) 4 ln16 C) 1 y= 2+ ( x − 16 ) 16 ln 4 D) 1 y= 2 + ( x − 16 ) 16 E) none of the above Ans: C 101. If ln 3 xy = 5, find dy / dx. A) dy y = − dx x B) dy 3y = − dx x C) dy y = − dx 3x D) dy 3y = − dx 5x E) dy 5y = − dx 3x Ans: A

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102. Write the equation of the line tangent to the curve x ln y + 9 xy = 18 at the point (2,1). A) 20 y − 9 x = 38 B) 20 y + 9 x = –38 C) 9 x − 20 y = 38 D) 9 x + 20 y = 38 E) 9 x + 20 y = 0 Ans: D 103. Find the second derivative of the function f ( x)= 6 + x ln x . A) 1 f ′′( x) = 1 + x B) 1 f ′′( x) = x C) f ′′( x) = 1 D) 2 f ′′( x) = x E) 2 f ′′( x) = 1 + x Ans: B 104. Find the second derivative of the function f ( x) = 9 x . A) f ′′( x) = (ln 9)9 x B) f ′′( x) = (ln 9)3 9 x C) f ′′( x) = (ln 9) 2 9 x D) f ′′( x) = (ln 9) 2 92 x E) f ′′( x) = (ln 9)3 93 x Ans: C 105. The relationship between the number of decibels β and the intensity of a sound I in  I  watts per square centimeter is given by β = 10 log10  −16  . Find the rate of change in  10  −3 the number of decibels when the intensity is 10 watt per square centimeter. Round your answer to the nearest decibel. A) 434 decibels per watt per square cm B) 43,429 decibels per watt per square cm C) 4343 decibels per watt per square cm D) 434,294 decibels per watt per square cm E) 4345 decibels per watt per square cm Ans: C

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106. Find the relative minima, and use a graphing utility to check your results. = y 5ln x − 4 x A) 4 − 5 B) 5 − 4 C) 1 − 4 D) 0 E) does not exist Ans: E 107. Find the relative maxima, and use a graphing utility to check your results. = y 7 ln x − 5 x A) y = –5.93 B) y = –4.64 C) y = –12.27 D) y = 0.00 E) does not exist Ans: B 108.

x9 Locate any relative extrema and inflection points of the function = y − ln x . Use a 9 graphing utility to confirm your results. A) 1 relative maximum value y = at x = 1 ; inflection point at x = 0 9 B) relative minimum value y = 9 at x = 1 ; inflection point at x = 0 C) relative minimum value y = 9 at x = 1 ; no inflection points D) 1 relative minimum value y = at x = 1 ; no inflection points 9 E) 1 relative maximum value y = at x = 1 ; no inflection points 9 Ans: D

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109.

Locate any relative extrema and inflection points of the function y = x ln A) B)

C) D)

no relative extrema; inflection point at x = 5e

x6 . 5

–11 30

−1

 5 6 relative maximum at x =   ; inflection point at x = 5e 6 e 1

–11

relative minimum at x = 5e 6 ; inflection point at x = 5e 30 –11

no relative extrema; inflection point at x = 5e 30

 5 6 relative minimum at x =   ; no inflection points e Ans: E 110.

Locate any relative extrema and inflection points of the function y = x3 ln A) B) C) D) E)

−1 3

relative minimum at x = 5e ; inflection point at x = 5e

x . 5

–5 6

−1 3

relative minimum at x = 5e ; no inflection points no relative maximum or minimum; inflection point at x = 5e no relative extrema or inflection points. −1

−1 3

–5

relative maximum at x = 5e 3 ; inflection point at x = 5e 6 Ans: A

111. Find the y-value at the relative minima, and use a graphing utility to check your result. y = 9 x 7 ln x A) 9 y= − 7e B) 7 y= − 9e C) 7 y= − 9 D) y=0 E) does not exist Ans: A

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112. The cost of producing x units of a product is modeled by C =800 + 600 x − 600 ln x, x ≥ 1 . Find the average cost function C . A) 800 600 600 ln x C= + − x x x B) 800 C= + 600 − 600 ln x x C) 800 600 ln x C= + 600 − x x D) 600 ln x C = 800 + 600 − x E) 600 C = 800 + − 600 ln x x Ans: C

113. The cost of producing x units of a product is modeled by C =900 + 200 x − 200 ln x, x ≥ 1 . Find the minimum average cost analytically. Round your answer to two decimal places. A) 200.00 dollars per unit B) 199.40 dollars per unit C) 199.18 dollars per unit D) 201.41 dollars per unit E) 199.28 dollars per unit Ans: C 114. Find the exponential function y = Ce kt that passes through the two given points ( 0,8 ) and ( 7,9 ) . A) y = e0.0168t B) y = 7e −0.0168t C) y = e –0.0168t D) y = 8e0.0168t E) y = 8e –0.0168t Ans: D 115. dy = −4 y, y (t = 0) = 20 dt Use the given information to write an equation for y. A) y = 4e −20t B) y = 20e −4t C) y = e 20t + 4 D) y = 20e 4t E) y = e −4t + 20 Ans: B

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116. Carbon-14(14C) dating assumes that the carbon on the Earth today has the same radioactive content as it did centuries ago. If this is true, then the amount of 14C absorbed by a tree that grew several centuries ago should be the same as the amount of 14 C absorbed by a similar tree today. A piece of ancient charcoal contains only 18% as much of the radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal? (The half-life of 14C is 5715 years.) Round your answer to the nearest integer. A) 2,310 years B) 33,123 years C) 2,315 years D) 14,139 years E) 14,144 years Ans: D 117. The number of a certain type of bacteria increases continuously at a rate proportional to the number present. There are 200 present initially, and 400 present 7 hours later. How many will there be 20 hours after the initial time? Round your answer to the nearest integer. A) 28 bacteria B) 1344 bacteria C) 1449 bacteria D) 41 bacteria E) 36 bacteria Ans: C 118. The effective yield is the annual rate i that will produce the same interest per year as the nominal rate compounded n times per year. For a rate that is compounded n times per n

 r year, the formula for effective yield is given as i = 1 +  − 1 . Find the effective yield  n for a nominal rate of 6%, compounded monthly. Round your answer to two decimal places. A) 0.62% B) 6.41% C) 6.80% D) 1.18% E) 6.17% Ans: E

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119. The cumulative sales (in thousands of units) of a new product after it has been on the market for t years may be modeled by= S 30(1 − e kt ) . During the first year, 5000 units were sold. What is the saturation point for this product? How many units will be sold after 6 years? A) The saturation point for the market is 3000 units and 19,953 units will be sold after 6 years. B) The saturation point for the market is 30,000 units and 27,366 units will be sold after 6 years. C) The saturation point for the market is 30,000 units and 19,953 units will be sold after 6 years. D) The saturation point for the market is 30,000 units and 20,076 units will be sold after 6 years. E) The saturation point for the market is 3000 units and 27,366 units will be sold after 6 years. Ans: C 120. Use the given information to write an exponential equation for y. Does the function represent exponential growth or exponential decay? dy = 2= y, y 10 when = t 0 dt A) y = 10e 2t , exp ontial decay B) y = 10e 2t , exp ontial growth C) y = 10et , exp ontial decay D) y = e 2t , exp ontial growth Ans: B 121. What percent of a present amount of radioactive radium ( 226 Ra ) will remain after 900 years? A) 45% B) 25% C) 65% D) 68%. Ans: D 122. The management of a factory finds that the maximum number of units a worker can produce in a day is 30. The learning curve for the number of units N produced per day after a new employee has worked days is modeled by = N 30(1 − e kt ) After 20 days on the job, a worker is producing 19 units in a day. How many days should pass before this worker is producing 25 units per day? A) about 36 days. B) about 45 days. C) about 30 days. D) about 10 days. Ans: A

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123. Determine the principal P that must be invested at interest rate r compounded continuously, so that $1,000,000 will be available for retirement in years r = 7.5% , t = 40 . A) $49787.07 B) $50787.07 C) $49000.04 D) $40000.06 Ans: A

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Chapter 5: Integration and Its Applications

∫ 12dx and check your result by differentiation.

1. Find the indefinite integral A) 12x B) 12x 2 C) 12 D) 24 E) 24x Ans: A 2. Evaluate the integral

∫ 14x dx . 4

56x3 + C 70x5 + C 14 3 x +C 3 D) 14 5 x +C 5 E) 7 4 x +C 2 Ans: D

A) B) C)

3. Find the indefinite integral A)

∫v

−1/ 4

dv and check your result by differentiation.

3v 4 4 3

v4 3 5

5v 4 3 4v 3

3 4

v4 5 Ans: D

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4. Use algebra to rewrite the integrand; then integrate and simplify. x–5 dx x A) 2 x x – 5 x +C 3 B) 2 x x – 10 x + C 3 C) 2 x x – 15 x + C 3 D) 1 2 x – 15 x + C 2 E) 1 2 x – 10 x + C 2 Ans: B

∫

5. Find the indefinite integral and check the result by differentiation.

∫ ( 4u + 2 ) du A) 2u 2 + 2u + C B) 4u 2 + 2u + C C) 2u 2 + 2u D) 4 E) none of the above Ans: A 6. Evaluate the integral

∫ (7 x – 3x – 7) dx . 2

21x 3 – 6 x 2 – 7 x + C 7 3 3 2 49 x – x + +C 3 2 2 C) 7 3 3 2 49 x – x + x+C 3 2 2 D) 7 3 3 2 x – x – 7x + C 3 2 E) 14 x – 3 + C Ans: D

A) B)

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7. Find the indefinite integral and check the result by differentiation.

∫ ( –20t − 6t + 5) dt 3

A) –5t 4 − 3t 2 + 5 + C B) –5t 4 − 3t + 5t C) –15t 3 − 6t 2 + 5t + C D) –5t 4 − 3t 2 + 5t + C E) –5t 4 − 3t 3 + 5t + C Ans: D 8. Evaluate the integral

∫ ( x – x – 5) dx . 5

1 6 1 5 25 x – x – +C 6 5 2 B) 1 6 1 5 25 x – x – x+C 6 5 2 C) 1 6 1 5 x – x – 5x + C 6 5 D) 1 5 1 4 x – x – 5x + C 5 4 E) 1 4 1 3 x – x – 5x + C 4 3 Ans: C

9. Evaluate the integral A)

∫ (5 + x ) dx . 1/ 4

4 54 x +C 5 B) 5 5 x + x5 4 + C 4 C) 25 4 5 4 + x +C 2 5 D) 1 –3 4 x +C 4 E) 1 54 x +C 4 Ans: A 5x +

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10.

Evaluate the integral ∫

10 dx . x5

10 ln( x5 ) + C 5 +C 3x 6 C) 5 − ln( x 4 ) + C 2 D) 5 − 6 +C 3x E) 5 − 4 +C 2x Ans: E A) B)

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11. The graph of the derivative of a function is given below. Sketch the graphs of two functions that have the given derivative.

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Page 251

Ans: B 12.

Find the particular solution that satisfies the differential equation f ′(= x) initial condition f (12) = –36 . A) 1 2 f= ( x) x − 4x 13 B) 1 2 f= ( x) x + 4 x – 200 15 C) 1 2 f= ( x) x − 4x 12 D) 1 2 f= ( x) x − 4 x – 200 12 E) 1 2 f= ( x) x + 4x 13 Ans: C

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1 x − 4 and 6

13. Find a function that satisfies the conditions = f ′′( x) x 4 ,= f ′(0) 8,= f (0) 3 . A) 1 5 f (= x) x +8 x 5 B) 1 6 f ( x= ) x +8 x + 3 30 C) 1 5 f ( x= ) x +8 x + 3 30 D) 1 6 f= ( x) x +8 x 6 E) 1 5 f ( x= ) x +3 x + 8 30 Ans: B 14.

dC 1 5 Find the cost function for the marginal cost = x + 90 and fixed cost of $2000 dx 20 (for x = 0). A) 1 6 C ( x= ) x +90 x + 2000 100 B) 1 6 C ( x) = x +2000 x + 90 120 C) 1 6 C ( x= ) x +90 x + 2000 120 D) 1 7 C ( x) = x +2000 x + 90 100 E) 1 7 C ( x= ) x +90 x + 2000 120 Ans: C

15. A ball is thrown vertically upwards from a height of 10 ft with an initial velocity of 40 ft per second. How high will the ball go? A) 85.0000 ft B) 28.7500 ft C) 35.0000 ft D) 65.0000 ft E) 88.6000 ft Ans: C

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16. An evergreen nursery sells a certain shrub after 8 years. The growth rate of the shrub is given by dh /= dt 4.5t + 8 , where t is the time in years and h is the height in centimeters. The seedlings are 14 centimeters tall when planted (t = 0). How tall are the shrubs when they are sold? A) 166 centimeters B) 172 centimeters C) 208 centimeters D) 222 centimeters E) 270 centimeters Ans: D 17. Identify u and du / dx for the integral A) B) C) D) E)

u = 1 − x11 and du / dx = −12 x u = 1 − x11 and du / dx = −11x10

= u 1 − x11 and du / dx = −11x10 u = 1 − x11 and du / dx = 11x

= u Ans: B 18.

∫ 1 − x (−11x )dx .

1 − x11 and du / dx = 12 x

1   7  Identify u and du / dx for the integral ∫  6 + 7   − 8  dx . x   x   A) 1 7 u= 6 + 7 and du / dx = 8 x x 3 B) 4 1   = u  6 + 7  and du / dx = − 5 x x   C) 1 7 u= 6 + 7 and du / dx = − 8 x x D) 1 8 u= 6 + 7 and du / dx = − 8 x x 3 E) 4 1   u  6 + 7  and du / dx = 5 = x x   Ans: C

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19. Find the indefinite integral of the following function and check the result by differentiation.

∫ (1 + 5z ) dz 4

A) B)

5(1 + 5 z )5 + C

(1 + 5 z )5 +C 4 C) (1 + 5 z )5 +C 5 D) (1 + 5 z )5 +C 25 E) none of the above Ans: D 20. Evaluate the integral

∫ (9 x – 5) (72 x )dx. 8

1 (9 x8 – 5)8 + C 8 B) 1 (9 x8 – 5)9 + C 9 C) 1 (9 x8 – 5)10 + C 9 D) 1 (9 x8 – 5)10 + C 10 E) 1 (9 x8 – 5)9 + C 10 Ans: D

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21. Find the indefinite integral of the following function and check the result by differentiation.

∫ u ( 2 + u ) du A) (2 + u ) + C 2

3 3 2

2 ( 2 + u3 ) 3 9

3 3 2

(2 + u ) + C 9

2 ( 2 + u3 ) 2

+C 9 E) none of the above Ans: D

22. Evaluate the integral

∫ (4 x – 3) dx. 1/ 6

6 (4 x – 3)7 / 6 + C 7 B) 7 (4 x – 3) –5/ 6 + C 6 C) 24 (4 x – 3) –5/ 6 + C 7 D) 14 (4 x – 3)7 / 6 + C 3 E) 3 (4 x – 3)7 / 6 + C 14 Ans: E

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23. Find the indefinite integral of the following function and check the result by differentiation. 2 3 ∫ t ( 3 + t ) dt 5

A) B)

18(3 + t 3 )6 + C

(3 + t 3 )6 +C 18 C) (3 + t 3 )6 +C 6 D) (3 + t 2 )6 +C 18 E) none of the above Ans: B 24. Evaluate the integral

∫ (4 x – 5) x dx. 5

1 (4 x 5 – 5)6 + C 120 B) 5 (4 x 5 – 5)8 + C 2 C) 20 (4 x5 – 5)7 + C 7 D) 1 (4 x 5 – 5)8 + C 160 E) 1 (4 x 5 – 5)7 + C 140 Ans: D

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25. Find the indefinite integral of the following function and check the result by differentiation.

–5u

∫ u +4

(

)

6 (u + 4) 2

−3

6 (u + 4) 2

Ans: B 26. Find the indefinite integral of the following function and check the result by differentiation.

4t 3

∫ t + 4 dt 4

2 t4 + 4 + C

t4 + 4 + C

2 t3 + 4 + C D) 1 4 t +4 +C 2 E) 1 3 t +4 +C 2 Ans: A

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27.

Use formal substitution to find the indefinite integral ∫

x6 + 1 x7 + 7 x + 2

dx .

2 7 ( x + 7 x + 2) + C 7 B) 1 6 x + 7x + 2 + C 7 C) 2 6 x + 7x + 2 + C 7 D) 2 7 x + 7x + 2 + C 7 E) 1 7 ( x + 7 x + 2) + C 7 Ans: D

28.

 8 Find the equation of the function f whose graph passes through the point  0,  and  3

whose derivative is f ′(= x) x 1 − x 2 . A) 1 9 − (1 − x 2 )1/ 2  f ( x)= 3 B) 1 9 − (1 − x 2 )3/ 2  f ( x)= 3 C) 1 11 + (1 − x 2 ) 2 / 3  f ( x= ) 3 D) 1 11 + (1 − x 2 )1/ 2  f ( x= ) 3 E) 1 11 − (1 − x 2 ) 2 / 3  f ( x= ) 3 Ans: B 29.

The marginal cost of a product is modeled by

dC = dx

2 , when x = 3, C = 90. Find x +1

the cost function. A) C ( x= ) 2 x + 1 + 82 B) C)

C ( x)= 4( x + 1) + 84

C ( x= ) 4 x + 1 + 82

C ( x= ) 2 x + 1 + 82 E) C ( x)= 2( x + 1) + 84 Ans: C

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30.

dx Find the supply function x = f ( p ) that satisfies= p p 2 −25 and the initial dp condition x = 600 when p = $13 . 3/ 2 A) 1 x = ( p 2 − 25 ) + 24 3 1/ 2 B) 1 x = ( p 2 − 25 ) + 596 3 C) 1 x= ( p − 5 ) + 24 3 3/ 2 D) 1 x = ( p 2 − 25 ) + 27 5 1/ 2 E) 1 = x p 2 − 25 ) + 599 ( p −1 Ans: A

31. Evaluate the integral

∫ e dx. 15 x

1 15 x e +C 15 B) 15e15 x + C C) 1 16 x e +C 16 D) 15e14 x + C E) 1 14 x e +C 14 Ans: A

32. Evaluate the integral

∫ 760e

0.4 x

dx.

A) 304e0.4 x + C B) 542.9e1.4 x + C C) 1900e0.4 x + C D) 304e –0.6 x + C E) 304e1.4 x + C Ans: C

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33. Find the indefinite integral.

∫e

−6 y

−6e −6 y + C −6e –7 y + C 1 − e –7 y + C 6 D) 1 −6 y − e +C 6 E) 1 −1 y − e 6 +C 6 Ans: D

A) B) C)

34.

Evaluate the integral ∫ x8e7 x dx. A)

1 7 x9 e +C 9

B) C)

63e7 x + C 1 7 x9 e +C 63

9e 7 x + C E) 1 9 7 x9 x e +C 63 Ans: C 35.

Use the Log Rule to find the indefinite integral for ∫

1 dx . 6 − 5x

1 − ln 6 + 5 x + C 5 B) 1 ln 6 − 5 x + C 6 C) 1 − ln 6 − 5 x + C 5 D) 1 ln 6 + 5 x + C 6 E) 1 ln 6 − 5 x + C 5 Ans: C

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36. Find the indefinite integral. x

∫ –5 x + 4 dx 2

A) B) C) D) E)

1 +C –10 x ln –5 x 2 + 4 + C –1 ln –5 x 2 + 4 + C 10 ln –5 x 2 + 4 +C –5 x 2 + 4 ln –5 x 2 − 4 + C

Ans: C 37. Find the indefinite integral. –9 x 2 ∫ 5 – 4 x3 dx A) 4 – ln 5 – 4 x3 + C 3 B) ln 5 – 4x 3 + C C)

–9 ln 5 – 4x3 + C

3 ln 5 – 4 x3 + C 4 E) 1 – ln 5 – 4 x3 + C 12 Ans: D

38. Find the indefinite integral. x2 ∫ 3x3 + 4 dx A) 1 ln 3 x 3 + 4 + C 9 B) ln 3 x3 + 4 + C C)

x3 +C 3x 4 + 4 x D) integral does not exist E) none of the above Ans: A

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39. Find the indefinite integral. x2 + 4x + 8 ∫ x3 + 6 x 2 + 24 x dx A) 1 ln x3 + 6 x 2 + 24 x + C 3 B) 1 − ln x3 + 6 x 2 + 24 x + C 3 C) ln x3 + 6 x 2 + 24 x + C D)

−3ln x3 + 6 x 2 + 24 x + C

− ln x3 + 6 x 2 + 24 x + C

Ans: A 40. Find the indefinite integral.

( ln x ) dx 6

∫

( ln x ) + C

6 5 6 ( ln x ) + C

B) C)

7 ln x +C x

( ln x ) + C 7

7 E) none of the above Ans: D 41. Use any basic integration formula or formulas to find the indefinite integral

∫e

9 − e x dx .

3/ 2 3 9 − ex ) + C ( 2 2/3 B) 3 11 − e x ) + C ( 2 3/ 2 C) 3 − (11 − e x ) + C 2 2/3 D) 2 9 − ex ) + C ( 3 3/ 2 E) 2 − (9 − ex ) + C 3 Ans: E

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42.

Find the equation of the function whose derivative is f ′( x) =

x2 + 8x + 3 and whose x −1

graph passes through the point (2,14) . A) x2 + 9 x + 12 ln x − 1 − 7 2 B) x2 + 8 x + 12 ln x − 1 − 6 2 C) x2 + 9 x + 12 ln x − 1 − 6 2 D) x2 + 8 x + 3ln x − 1 − 6 2 E) x2 + 9 x + 3ln x − 1 − 7 2 Ans: C 43. Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. 9

∫ 6t dt 5

A) –318 B) 636 C) 336 D) 168 E) 12 Ans: D 44. Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. 5

∫ 25 − s ds 2

25π 25π 2 C) 25 4 D) 25π 4 E) none of the above Ans: D

A) B)

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45.

Use the values ∫ f ( x)dx = 6 and ∫ g ( x)dx = 5 to evaluate the definite 3

integral ∫ [ f ( x) − 3 g ( x) ] dx . 0

A) 21 B) –9 C) 1 D) 11 E) –7 Ans: B 46. Determine the area of the given region. = y 2 x(1 − x)

5 3 B) 1 3 C) 3 7 D) 1 2 E) None of the above Ans: B

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47. Evaluate the definite integral of the algebraic function. 5

∫ ( –2s + 6 ) ds 2

Use a graphing utility to verify your results. A) –6 B) –6 C) 13 D) 18 E) –3 Ans: E 48.

Evaluate the definite integral ∫ ( x − 2)5 dx . 1

1 5

−

1 7 C) 1 − 5 D) 1 6 E) 1 − 6 Ans: E

49. Evaluate the definite integral of the algebraic function. 9

∫ 4

3  23  2 u − u   du  

Use a graphing utility to verify your results. A) –67.0832 B) –115.0555 C) 101.7168 D) 17.3168 E) –182.1386 Ans: A

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50. Evaluate the following definite integral. 3

∫ 1

1 dx 4x +1

Use a graphing utility to check your answer. A) 13 − 5 B) C) D) E)

4 13 − 5 2 13 + 5 2 13 + 5 4 5 − 13 2

Ans: B 51. Find the area between the curve y = 5 x 2 + 3 x – 6 and the x-axis from = x –2 = to x 1 . A) 15 2 B) 105 2 C) 25 2 D) 75 2 E) 15 4 Ans: A 52. Find the average value of the function over the given interval. f ( x= ) 18 − x 2 on [-3,3] A) 15 B) 52.5 C) 90 D) 10 E) 50 Ans: A

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53. Find the average value of the function over the given interval. f ( x) = 5 3 x on [0,1] A) 5 2 B) 5 14 C) 15 4 D) 8 21 E) 4 3 Ans: C

54. The rate of depreciation of a building is given by = D '(t ) 3, 700(25 − t ) dollars per year, 0 ≤ t ≤ 25. Use the definite integral to find the total depreciation over the first 25 years. A) $1,156, 250 B) $46, 250 C) $578,125 D) $330,357 E) $2,312,500 Ans: A

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55. Determine the graph whose area (the shaded region) is represented by the integral. 4

∫ ( 5 x − 20 x + 25) − ( 5 x + 5) dx 2

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Ans: C

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56. Find the area of the shaded region.

13 6 B) 37 12 C) 37 6 D) 13 12 E) 13 7 Ans: B

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57. Set up the definite integral that gives the area of the region bounded by the graphs. f ( x= ) ( x − 1)3 g ( x)= x − 1

A) B) C) D) E)

∫ (( x − 1) − ( x − 1) )dx + ∫ (( x − 1) − ( x − 1))dx 3

0 0

1 1

∫ (( x − 1) + ( x − 1) )dx + ∫ (( x − 1) + ( x − 1))dx 3

−1

∫ (( x − 1) − ( x − 1))dx + ∫ (( x − 1) − ( x − 1) )dx 3

−1 1

0 2

∫ (( x − 1) − ( x − 1))dx + ∫ (( x − 1) − ( x − 1) )dx 3

0 1

1 2

∫ (( x − 1) + ( x − 1))dx + ∫ (( x − 1) + ( x − 1) )dx 3

Ans: D

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58. The integrand of the following definite integral is a difference of two functions.

Sketch the graph of the two functions and shade the region whose area is represented by the integral. A)

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Ans: B

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59. Find the area of the region bounded by the graphs of the algebraic functions. f ( x= ) x 2 − 20 x g ( x) = 0 A) 667 A= 2 B) 2000 A= 3 C) 4001 A= 6 D) 4000 A= 3 E) 1001 A= 3 Ans: D 60. Find the area of the region bounded by the graphs of the algebraic functions. f ( x) =x 2 + 30 x + 225 g= ( x) 17( x + 15) A) 4913 A= 6 B) 4913 A= 3 C) 4913 A= 12 D) 5363 A= 6 E) 6263 A= 6 Ans: A

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61. Find the consumer and producer surpluses by using the demand and supply functions, where p is the price (in dollars) and x is the number of units (in millions). Demand Function = P 975 − 23 x A) a. $2587.50 B) C) D) E)

Supply Function 42x

b. $3725.00 a. $5587.50 b. $4725.00 a. $2587.50 b. $1725.00 a. $1587.50 b. $4725.00 a. $3587.50 b. $4725.00

Ans: C 62. The demand function for a product is = p 100 − 2 x , where p is the number of dollars and x is the number of units. If the equilibrium price is $40 , what is the consumer’s surplus? A) $ 825 B) $ 900 C) $ 1025 D) $ 870 E) $ 990 Ans: B 63. Two models, = R1 9.21 + 0.60t and = R2 9.21 + 0.45t , are given for revenue (in billions of dollars per year) for a large corporation. Both models are estimates of revenues for 2007 through 2011, with t = 7 corresponding to 2007. Which model is projecting the greater revenue? How much more total revenue does that model project over the five-year period? A) The model R1 projects greater revenue than R2 . $8.75 billion B) The model R2 projects greater revenue than R1 . $7.75 billion C) The model R1 projects greater revenue than R2 . $6.75 billion D) The model R1 projects greater revenue than R2 . $10.75 billion E) The model R2 projects greater revenue than R1 . $16.75 billion Ans: C

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64. The revenue from a manufacturing process (in millions of dollars per year) is projected to follow the model = R 400 + 0.06t for 10 years. Over the same period of time, the cost (in millions of dollars per year) is projected to follow the model C = 80 + 0.5t 2 , where t is the time (in years). Approximate the profit over the 10-year period, beginning with t = 0. Round your answer to two decimal places. A) $3036.33 million B) $3178.00 million C) $2953.00 million D) $3035.33 million E) $3185.33 million Ans: A 65. Use the Midpoint Rule with n = 4 to approximate the area of the following region.

A) 3 B) 8 C) 2 D) 1 E) 6 Ans: C

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66. Use the rectangles to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. f ( x) = −2 x + 3,[0,1]

A) B) C) D) E)

a. The approximate area: 3 b. The exact area: 2 a. The approximate area: 2 b. The exact area: 3 a. The approximate area: 2 b. The exact area: 1 a. The approximate area: 2 b. The exact area: 2 a. The approximate area: 1 b. The exact area: 2

Ans: D 67. Use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of f ( x)= 5 − x 2 and the x-axis over the interval [ −3,3 ]. A) 13.5671 B) 13.1273 C) 13.3364 D) 13.1250 E) 14.1250 Ans: D 68. Use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of f ( x= ) 7 x − x5 and the x-axis over the interval [0,1]. A) 3.7882 B) 3.3484 C) 3.5575 D) 4.3461 E) 3.3461 Ans: E

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69. Use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of and the -axis over the interval. Sketch the region. f ( x) =( x 2 − 4) 2 ,[−2, 2] A) The approximate area is: ≈ 30.25

The approximate area is: ≈ 24.25

The approximate area is: ≈ 34.25

The approximate area is: ≈ 14.25

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The approximate area is: ≈ 34.99

Ans: C 70. Use the Midpoint Rule n = 4 to approximate the area of the following region.

A) 2.5 B) 1.2 C) 1.5 D) 1.9 E) 2.3 Ans: C

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71. Estimate the surface area of the pond shown in the figure using the Midpoint Rule.

A) ≈ 990 ft 2 B) ≈ 9920 ft 2 C) ≈ 9020 ft 2 D) ≈ 9990 ft 2 E) ≈ 920 ft 2 Ans: B 72. Estimate the surface area of the oil spill shown in the figure using the Midpoint Rule.

A) ≈ 481.6 mi 2 B) ≈ 301.6 mi 2 C) ≈ 311.6 mi 2 D) ≈ 431.6 mi 2 E) ≈ 381.6 mi 2 Ans: E

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73.

4 dx . Then use 2 1 x + 0 a graphing utility to evaluate the definite integral. Compare your results. A) a. Midpoint Rule: ≈ 0.146801 b. Graphing utility: ≈ 3.141593 B) a. Midpoint Rule: ≈ 3.146801 b. Graphing utility: ≈ 0.141593 C) a. Midpoint Rule: ≈ 1.146801 b. Graphing utility: ≈ 3.141593 D) a. Midpoint Rule: ≈ 3.146801 b. Graphing utility: ≈ 3.141593 E) a. Midpoint Rule: ≈ 3.146801 b. Graphing utility: ≈ 1.141593 Ans: D

Use the Midpoint Rule with n = 4 to approximate π where π = ∫

74. Estimate the surface area of the golf green shown in the figure using the midpoint rule.

A) 780 B) 156 C) 1404 D) 1502 E) 524 Ans: A

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Chapter 6: Techniques of Integration 1. Identify u and dv for finding the integral using integration by parts. 5 4x ∫ x e dx 5 A) = u x= ; dv e 4 x dx B)= u ∫= x5 ; dv ∫ e 4 x dx

x dx; dv e dx ∫= D) = u ∫= x dx; dv ∫ e dx E) = u x= dx; dv ∫ e dx

C) = u

Ans: A 2. Identify u and dv for finding the integral using integration by parts. 5 ∫ x ln 4 x dx A)

u = ∫ ln 4 x dx; dv = ∫ x5 dx

B) = u ln= 4 x; dv x5 dx C) = u ∫= ln 4 x; dv x5 dx

∫ x dx E) = u ∫= ln 4 x; dv ∫ x dx D) = u ln= 4 x; dv

Ans: B 3. Use integration by parts to evaluate A)

−

( 7 x + 1) ( 6e –7 x ) 49

∫ 6 xe dx. –7 x

e –7 x − +C 7 C) 6e –7 x ) ( − +C 49 D) x ( 6e –7 x ) − +C 7 E) e –7 x +C 7 Ans: A

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4. Find the indefinite integral. 6x2 ∫ e x dx A) 6 ( x 2 − 2 x + 2 ) e− x + C

(

)

−x

− x 2 + 2 x + 2 e− x + C

( x + 2x + 2) e + C –6 x ( x + 2 x + 2 ) e + C –6 ( x + 2 x + 2 ) e + C

C) D)

−x

Ans: E 5. Use integration by parts to evaluate x 2 e –11x dx . Note that evaluation may require ∫ integration by parts more than once. A) 2 + 22 x + 121x 2 ) e –11x ( – +C 1331 B) (1 + 22 x – 11x 2 ) e –11x + C 121 C) 2 + 33 x + 121x 2 ) e –11x ( – +C 1331 D) ( 2 – 11x + 121x 2 ) e –11x + C 121 E) 1 – 11x + 121x 2 ) e –11x ( – +C –1331 Ans: A 6. Use integration by parts to find the integral below.

∫ ln x dx 9

A) ln x10 − 9 x10 + C B) x ln x10 − 10 x10 + C C) ln x10 − x10 + C D) x ln x 9 − 9 x + C E) 10 x ln x10 − 10 x + C Ans: D

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7. Find the indefinite integral. ∫ y ln ( y – 4 ) dy A) B) C) D)

 y 2 – 16  y2 + 8 y +C   ln ( y – 4 ) − 4  2   y 2 – 16  y2 + 8 y ln y – 4 + +C ( )   4  2   y 2 + 16  y2 – 4 y +C   ln ( y – 4 ) − 4  2   y 2 – 16  y2 + 8 y ln y – 4 − +C ( )   2  2 

 y 2 – 16  y2 + 4 y +C   ln ( y – 4 ) + 4  2  Ans: A E)

8. Use integration by parts to evaluate A)

–5 x 7 ( 7 ln ( x ) + 1)

7 6 5 x ( 6 ln ( x ) + 1)

6 7 5 x ( 7 ln ( x ) + 1)

49 6 5 x ( 6 ln ( x ) − 1)

36 7 5 x ( 7 ln ( x ) − 1) 49

∫ 5 x ln x dx . 6

+C +C +C +C

Ans: E

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9. Find the indefinite integral.

∫

9 ( ln x ) x2

(

) +C

9 ( ln x ) − 2 ln x + 2

9 ( ln x ) + 2 ln x + 2

−

C) D) E)

(

) +C

( ) 9 ( 2 ( ln x ) + 2 ) − +C

−9 ( ln x ) + 2 ln x + 2 + C 2

(

9 2 ( ln x ) + 2 2

) +C

Ans: B 10. Find the indefinite integral. ln m ∫ 6m3 dm A) 1 − ( 2 ln m + 1) + C 12m 2 B) 1 ( 2 ln m + 1) + C ) 12m 2 C) 1 ( 2 ln m − 1) + C 24m 2 D) 1 − ( 2 ln m − 1) + C 24m 2 E) 1 − ( 2 ln m + 1) + C 24m 2 Ans: E

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11. Find the indefinite integral. ∫ m 2m – 1 dm A)

( 2m – 1) ( 3m – 1) + C 32

( 2m – 1) ( 3m + 2 ) + C

( 2m – 1) ( 3m + 1) + C

( 2m + 1) ( 3m – 1) + C 32

( 2m – 1) ( 3m – 2 ) + C 32

Ans: C 12. Find the indefinite integral. p ∫ 1 + 3 p dp A) (3 p − 2) 3 p + 1 + C 9 B) 2 (3 p − 2) 3 p + 1 +C 27 C) 2(2 − 3 p) 3 p +1 +C 27 D) 2 (3 p − 2) 3 p + 1 +C 9 E) none of the above Ans: B 13.

Evaluate the definite integral ∫ x 2 e7 x dx . Round your answer to three decimal places. 0

A) 828.111 B) 207.823 C) 118.290 D) 207.811 E) 79.245 Ans: C

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14. Find the definite integral. 4

∫ x ln x dx 3

2048ln 4 – 255 16 B) 1024 ln 4 – 255 16 C) −1024 ln 4 – 255 16 D) 1024 ln 4 + 255 16 E) none of the above Ans: B

15. Use integration by parts to find the integral below.

∫ 3x ln ax dx (a ≠ 0, n ≠ −1) n

A) B) C) D)

3x n 3 ln ax − 2 x n + C n n n +1 4x 4 ln ax − x n +1 + C (n + 1) 2 n +1 3x n 3 ln ax + C − n (n + 1) 2

3 x n +1 3 ln ax − x n +1 + C 2 (n + 1) n +1

4 x n +1 4 − ln ax + C n + 1 n2 Ans: D E)

16. A model for the ability M of a child to memorize, measured on a scale from 0 to 10, is M = 1 + 1.6t ln t , 0 < t ≤ 3 where t is the child’s age in years. Find the average value predicted by the model for a child’s ability to memorize between first and second birthdays. Round your answer to three decimal places. A) 3.318 B) 2.218 C) 4.118 D) 1.318 E) 2.018 Ans: E

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17. Present Value of a Continuous Stream of Income. An electronics company generates a continuous stream of income of 4t million dollars per year, where t is the number of years that the company has been in operation. Find the present value of this stream of income over the first 9 years at a continuous interest rate of 12%. Round answer to one decimal place. A) $143.7 million B) $81.6 million C) $182.7 million D) $343.2 million E) $85.8 million Ans: B 18. Use a table of integrals to find the indefinite integral A)

x  2 − 2 ln(5 x) + (ln 5 x) 2  + C

x  2 ln(5 x) + (ln 5 x) 2  + C

5 x  2 + 2 ln(5 x) + (ln 5 x) 2  + C

x [ 2 − 2 ln(5 x) − (ln 5 x) ] + C

∫ (ln 5 x) dx . 2

5 x [ 2 ln(5 x) + (ln 5 x)] + C Ans: A

19. Use a table of integrals with forms involving eu to find the integral. 7 ∫ 1 + e –9 x dx A) 7 –7 x + ln(1 − e –9 x ) + C 9 B) 7 –7 x + ln(1 + e –9 x ) + C 9 C) 7 7 x – ln(1 + e –9 x ) + C 9 D) 7 7 x – ln(1 − e –9 x ) + C 9 E) 7 7 x + ln(1 + e –9 x ) + C 9 Ans: E

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20. Find the integral below using an integral table. 1

∫ 36 − x dx 2

1 6+ x ln +C 12 6 − x B) 1 36 + x ln +C 12 36 − x C) 1 36 − x ln +C 12 36 + x D) 1 6− x ln +C 6 6+ x E) 1 6− x ln +C 12 6 + x Ans: E

21.

Use a table of integrals with forms involving a + bu to find ∫

x2 dx. 8 + 11x

1 (11x − 8ln 8 + 11x ) + C 121 B) 1  64  − 16 ln 8 + 11x  + C 11x − 1331  8 + 11x  C) 1  11x  (11x − 16) + 64 ln 8 + 11x  + C  1331  2  D) 1  11x  (11x − 16) + 64 ln 8 + 11x  + C  121  2  E) 1  64  − 16 ln 8 + 11x  + C 11x − 121  8 + 11x  Ans: C

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22.

Use a table of integrals to find the indefinite integral ∫ A) B)

x2 dx . (5 + 4 x)5

  −1 10 25  2(5 + 4 x) 2 + 3(5 + 4 x)3 − 4(5 + 4 x) 4  + C    1  −1 10 25 + − +C  2 3 4 64  2(5 + 4 x) 3(5 + 4 x) 4(5 + 4 x) 

 1  −1 10 25 + + +C  2 3 4 64  2(5 + 4 x) 3(5 + 4 x) 4(5 + 4 x) 

 1  −1 10 25 − + +C  2 3 4 64  2(5 + 4 x) 3(5 + 4 x) 4(5 + 4 x) 

  −1 10 25  2(5 + 4 x) 2 − 3(5 + 4 x)3 + 4(5 + 4 x) 4  + C   Ans: B E)

23.

Use a table of integrals with forms involving A) B) C)

–6 x

6 25 − x 2 +C 25 x 25 − x 2 − +C 25 x 6 5 + 25 − x 2 ln +C 5 x

– E)

a 2 − u 2 to find ∫

6 25 − x 2 +C 25 x

6 5 + 25 − x 2 – ln +C 5 x

Ans: A

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25 − x 2

dx.

24. Use a table of integrals to find the indefinite integral

∫ x e dx . 2 x3

1 x3 e +C 2 B) 1 x2 e +C 3 C) 1 x3 e +C 3 D) 1 x2 e +C 2 E) 1 x e +C 3 Ans: C

25.

Use a table of integrals to find the indefinite integral ∫ A) B)

ln x dx . x(5 + 6 ln x)

1 6 ln x − 5ln 5 + 6 ln x  + C 36  5ln x − 6 ln 5 + 6 ln x  + C

1 6 ln x + 5ln 5 + 6 ln x  + C 36  D) ln11x + C E) 1 ln x + C 11 Ans: A

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26. Use the Trapezoidal Rule and Simpson’s Rule to approximate the value of the definite integral for the indicated value of n. Compare these results with the exact value of the definite integral. Round your answers to four decimal places.

 x2  4 ∫0  2 + 1 dx, n = A) a. Exact: ≈ 1.9667 b. Trapezoidal Rule: ≈ 1.1719 c. Simpson’s Rule: ≈ 1.1667 B) a. Exact: ≈ 1.1667 b. Trapezoidal Rule: ≈ 1.1719 c. Simpson’s Rule: ≈ 1.9667 C) a. Exact: ≈ 1.1667 b. Trapezoidal Rule: ≈ 1.9719 c. Simpson’s Rule: ≈ 1.1667 D) a. Exact: ≈ 2.1667 b. Trapezoidal Rule: ≈ 1.1719 c. Simpson’s Rule: ≈ 1.1667 E) a. Exact: ≈ 1.1667 b. Trapezoidal Rule: ≈ 1.1719 c. Simpson’s Rule: ≈ 1.1667 Ans: E 1

27. Approximate the value of the definite integral using (a) the Trapezoidal Rule and (b) Simpson’s Rule for the indicated value of n. Round your answers to three significant digits. 2

∫ e dx, n = 4 − x2

a. Trapezoidal Rule: ≈ 1.881 b. Simpson’s Rule: ≈ 0.882 B) a. Trapezoidal Rule: ≈ 0.881 b. Simpson’s Rule: ≈ 0.882 C) a. Trapezoidal Rule: ≈ 0.881 b. Simpson’s Rule: ≈ 1.882 D) a. Trapezoidal Rule: ≈ 0.081 b. Simpson’s Rule: ≈ 0.882 E) a. Trapezoidal Rule: ≈ 0.881 b. Simpson’s Rule: ≈ 0.082 Ans: B A)

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28. The rate of change in the number of subscribers S to a newly introduced magazine is dS modeled= by 1000t 2 e −1 , 0 ≤ t ≤ 6 where t is the time in years. Use Simpson’s dt Rule n = 12 with to estimate the total increase in the number of subscribers during the first 6 years. A) ≈ 1870 subscribers B) ≈ 1780 subscribers C) ≈ 1800 subscribers D) ≈ 1878 subscribers E) ≈ 1987 subscribers Ans: D 29. A body assimilates a 12-hour cold tablet at a rate modeled by dC / dt = 8 − ln(t 2 − 2t + 4), 0 ≤ t ≤ 12 where dC / dt is measured in milligrams per hour and t is the time in hours. Use Simpson’s Rule with n = 8 to estimate the total amount of the drug absorbed into the body during the 12 hours. A) ≈ 58.915 mg B) ≈ 68.915 mg C) ≈ 38.915 mg D) ≈ 48.915 mg E) ≈ 78.915 mg Ans: A 30.

Evaluate the definite integral ∫ 8 + x 2 dx . Round your answer to three decimal places. 3

A) 20.210 B) 16.873 C) 32.580 D) 26.395 E) 4.504 Ans: E 31.

Evaluate the definite integral ∫ x 5 ln x dx. Round your answer to three decimal places. 1

A) 6.144 B) 7.207 C) 4.541 D) 5.644 E) 4.881 Ans: D

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32. The probability of recall in an experiment is modeled by b 75  x  = P ( a ≤ x ≤ b) ∫  dx, 0 ≤ x ≤ 1 where x is the percent of recall. What is the 14  4 + 5 x  a probability of recalling between 60% and 90%? Round your answer to three decimal places. A) 0.432 B) 0.270 C) 0.710 D) 0.219 E) 0.936 Ans: A 33. Use the table of integrals to find the average value of the growth function 325 over the interval [ 23, 28] , where N the size of a population and t is the N= 1 + e5.2−0.25t time in days. Round your answer to three decimal places. A) 248.346 B) 1057.983 C) 246.346 D) 680.477 E) 682.451 Ans: C 34. The revenue (in dollars per year) for a new product is modeled by   1 where t the time in years. Estimate the total revenue from R 10, 000 1 − = 2 12   (1 + 0.1t )  sales of the product over its first 3 years on the market. Round your answer to nearest dollar A) $6579 B) $3291 C) $10,821 D) $15,830 E) $1138 Ans: B 35. Approximate the definite integral "by hand," using the Trapezoidal Rule with n = 4 trapezoids. Round answer to three decimal places. 4

∫ x dx 1

A) 11.425 B) 11.381 C) 5.691 D) 8.569 E) 15.175 Ans: D

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36. Use the Trapezoidal Rule to approximate the value of the definite integral 2

6 . Round your answer to three decimal places. ∫ 1 + x dx, n = 0

A) 6.7643 B) 2.7931 C) 2.7955 D) 4.6552 E) 4.6615 Ans: C 37. Decide whether the integral is proper or improper. 5

∫ e dx −x

A) The integral is improper. B) The integral is proper. Ans: B 38. Determine the amount of money required to set up a charitable endowment that pays the amount P each year indefinitely for the annual interest rate compounded continuously. = P $12, = 000, r 6% A) $210,000 B) $200,000 C) $220,000 D) $240,000 E) $230,000 Ans: B

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39.

The capitalized cost C of an asset is given by C = C0 + ∫ C (t )e − rt dt where C0 is the 0

original investment, t is the time in years, r is the annual interest rate compounded continuously, and C (t ) is the annual cost of maintenance (in dollars). Find the capitalized cost of an asset (a) for 5 years, (b) for 10 years, and (c) forever. = C0 $300, 000, = C (t ) 15, = 000, r 6% A) a. For n = 5, C ≈ $253,901.30 b. For n = 10, C ≈ $807,922.43 c. For n = ∞, C ≈ $4,466,666.67 B) a. For n = 5, C ≈ $453,901.30 b. For n = 10, C ≈ $807,922.43 c. For n = ∞, C ≈ $1,466,666.67 C) a. For n = 5, C ≈ $453,901.30 b. For n = 10, C ≈ $2807,922.43 c. For n = ∞, C ≈ $4,466,666.67 D) a. For n = 5, C ≈ $453,901.30 b. For n = 10, C ≈ $807,922.43 c. For n = ∞, C ≈ $4,466,666.67 E) a. For n = 5, C ≈ $453,901.30 b. For n = 10, C ≈ $807,922.43 c. For n = ∞, C ≈ $466,666.67 Ans: D 40.

x dx , n = 6. Round your 2 + x + x2 0

Approximate the integral using Simpson's Rule: ∫ answer to three decimal places. A) 1.161 B) 1.284 C) 0.850 D) 0.652 E) 1.017 Ans: D

41. Use Simpson's Rule to approximate the revenue for the marginal revenue function dR = 5 8000 − x 3 with n = 4. Assume that the number of units sold, x, increases from dx 12 to 16. Round your answer to one decimal place. A) $1602.40 B) $678.36 C) $1346.14 D) $1439.03 E) $1230.54 Ans: D

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42. Use the error formulas to find n such that the error in the approximation of the definite 5 1 integral ∫ dx is less than 0.0001 using the Trapezoidal Rule. x 3 A) 25 B) 26 C) 24 D) 22 E) 23 Ans: E 43.

dC =7 − ln(t 2 − 2t + 4) , dt 0 ≤ t ≤ 16 where t is measured in milligrams per hour and t is the time in hours. Use Simpson’s Rule with n = 16 to estimate the total amount of the drug absorbed into the body during the 16 hours. A) 58.88 B) 34.88 C) 54.33 D) 46.88 E) 64.90 Ans: C

A body assimilates a 16-hour cold tablet at a rate modeled by

44. Decide whether the following integral is improper.

A) no B) yes Ans: B 45. Evaluate the improper integral if it converges, or state that it diverges. ∞

∫ x dx A)

1 6 B) 6 C) 5 D) 1 4 E) diverges Ans: D

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46. Evaluate the improper integral if it converges, or state that it diverges. ∞

1 dx x A) 3 B) 1 C) 2 D) 3 E) diverges Ans: E

∫

47. Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. 0

∫ e dx 3x

−∞

A) B) C)

converges to 0 converges to 3 1 converges to 3 D) diverges to −∞ E) diverges to ∞ Ans: C 48. Suppose the mean height of American women between the ages of 30 and 39 is 68.5 inches, and the standard deviation is 2.7 inches. Use a symbolic integration utility to approximate the probability that a 30-to 39-year-old woman chosen at random is between 5 and 6 feet tall. A) 0.1772 B) 0.9017 C) 0.5707 D) 0.8547 E) 0.4257 Ans: B 49. A business is expected to yield a continuous flow of profit at the rate of $1,000,000 per year. If money will earn interest at the nominal rate of 8% per year compounded continuously, what is the present value of the business forever? A) $12,600,000 B) $12,510,000 C) $12,500,000 D) $1,250,000 E) $1,350,000 Ans: C

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50. Find the capitalized cost C of an asset forever. The capitalized cost is given by n

C = C0 + ∫ c(t )e − rt dt where C0 = $900, 000 is the original investment, t is the time in 0

years, r = 12% is the annual interest rate compounded continuously, n is the total time in years over which the asset is capitalized, and = c(t ) 25, 000(1 + 0.08t ) is the annual cost of maintenance (measured in dollars). Round your answer to two decimal places. A) $1,525,000.00 B) $1,275,000.00 C) $1,108,333.33 D) $1,299,218.75 E) $1,247,222.22 Ans: E

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Chapter 7: Functions of Several Variables 1. Find the coordinates of the point that is located two units behind of the yz-plane, seven units to the left of the xz-plane, and four units above of the xy-plane. A) ( –2, –7, 4 ) B) ( 2, –7, 4 ) C) D) E)

( 2, 7, 4 ) ( –2, 7, 4 ) ( –2, 7, –4 )

Ans: A 2. Find the the distance between the two points ( 2, 4, 4 ) and (1,1,5 ) . A) 1 units B) 11 units C) 11 units D) 3 units E) 5 units Ans: C 3. Find ( x, y, z ) if the midpoint of the line segment joining the two points ( x, y, z ) and

( 9, –9, –4 ) is (8, 6, 7 ) . A) ( 25,3,10 ) B) C)

 17 3 3   ,– ,  2 2  2 (1, –15, –11)

 1 15 11  – , ,   2 2 2 E) ( 7, 21,18) Ans: E

4. Find the lengths of the sides of the triangle with the given vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither.

( 0, 0, 0 ) , ( 2, 2,1) , ( 2, −4, 4 ) A)

3, 5,5 ; obtuse triangle

3,3 5, 6 ; right triangle

6,3,5 ; right triangle 2, 4,3 ; acute triangle

D) E)

2, 4, 3 ; acute triangle Ans: B

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5. Find the center and radius of the sphere. x2 + y 2 + z 2 − 5x = 0 A) 5  Center:  , 0, 0  2  5 Radius: 2 B)  3  Center:  0, , 0   2  3 Radius: 2 C)  3 1 Center:  0, ,   2 2 10 Radius: 2 D) 1  Center:  , 0, 0  2  1 Radius: 2 E) 5  Center:  , 0,1 2  29 Radius: 2 Ans: A 6. Find the standard equation of the sphere whose center is ( 6, –6, –1) and whose radius is 4. A) ( x – 6) 2 + ( y + 6) 2 + ( z + 1) 2 = 16 2 2 2 B) ( x + 6) + ( y – 6) + ( z – 1) = 16 C) ( x – 6) + ( y + 6) + ( z + 1) = 4 D) ( x – 6) 2 + ( y + 6) 2 + ( z + 1) 2 = 4 E) ( x + 6) + ( y – 6) + ( z – 1) = 4 Ans: A

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7. Find the equation of the sphere that has the points ( 2,8, 6 ) and ( 4, 4, 4 ) as endpoints of a diameter. A) 12 ( x + 3) + ( y + 6 ) + ( z + 5 ) = B) C) D) E)

6 ( x + 3) + ( y + 6 ) + ( z + 5 ) = 2 2 2 12 ( x − 3) + ( y − 6 ) + ( z − 5 ) = 2 2 2 6 ( x − 3) + ( y − 6 ) + ( z − 5 ) = 6 ( x − 3) + ( y − 6 ) + ( z − 5 ) = 2

Ans: D 8. Find the center and radius of the sphere whose equation is 3 x 2 + 3 y 2 + 3 z 2 + 24 x – 42 y + 6 z + 173 = 0 . Round your answer to two decimal places, where applicable. A) center: ( 4, –7,1) ; radius: 2.89 B) center: ( –4, 7, –1) ; radius: 2.89 C)

center: ( –4, 7, –1) ; radius: 8.35

center: ( 4, 7, –1) ; radius: 2.89

center: ( 4, –7,1) ; radius: 8.35 Ans: B E)

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9. Sketch the yz-trace of the equation: A)

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Ans: B

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10. Sketch the trace of the intersection of plane z = 4 with the sphere: x 2 + y 2 + z 2 =. 25 A)

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Ans: D

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11. Find the intercepts of the plane given by 4 x + 5 y + 2 z = 20 . A) The x -intercept is (5, 0, 0) . The y -intercept is (0, 4, 0) . The z -intercept is (0, 0,10) . B) The x -intercept is (0,5, 0) . The y -intercept is (0, 0, 4) . The z -intercept is (10, 0, 0) . C) The x -intercept is (0, −5, 0) . The y -intercept is (0, 0, 4) . The z -intercept is (10, 0, 0) . D) The x -intercept is (−5, 0, 0) . The y -intercept is (0, −4, 0) . The z -intercept is (0, 0, −10) . E) The x -intercept is (0, 0,5) . The y -intercept is (0, 4, 0) . The z -intercept is (10, 0, 0) . Ans: A

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12. Sketch the graph of the plane given by y = 5. A)

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Ans: B 13. Find the intercepts of the plane given by 8 x − 6 z = 24 . A) The x -intercept is (0, 0,3) . The z -intercept is (–4, 0, 0) . B) The x -intercept is (0,3, 0) . The z -intercept is (–4, 0, 0) . C) The x -intercept is (0, −3, 0) . The z -intercept is (–4, 0, 0) . D) The x -intercept is (−3, 0, 0) . The z -intercept is (0, 0, –4) . E) The x -intercept is (3, 0, 0) . The z -intercept is (0, 0, –4) . Ans: E 14. The two planes 4 x – 3 y + z = 3 and x + 6 y + 8 z = 3 are perpendicular. A) false B) true Ans: A 15. The two planes 6 x – 2 y – 8 z = 2 and 6 x – 20 y – 6 z = 2 are parallel. A) true B) false Ans: B 16. Describe the trace of the surface given by the function below in the xy-plane.

A) circle B) ellipse C) parabola D) hyperbola E) line Ans: B 17. Describe the trace of the surface given by the function below xz-plane. A) circle B) parabola C) line D) ellipse E) hyperbola Ans: E

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in the

18. Identify the quadric surface.

A) The graph is a hyperboloid of one sheet. B) The graph is hyperboloid of two sheets. C) The graph is an elliptic cone. D) The graph is an elliptic paraboloid. E) The graph is an ellipsoid. Ans: B 19. Identify the quadric surface.

A) The graph is an elliptic cone. B) The graph is hyperboloid of two sheets. C) The graph is a hyperboloid of one sheet. D) The graph is an ellipsoid. E) The graph is an elliptic paraboloid. Ans: D 20. Because of the forces caused by its rotation, a planet is actually an oblate ellipsoid rather than a sphere. The equatorial radius is 3963 miles and the polar radius is 3953 miles. Find an equation of the ellipsoid. Assume that the center of a planet is at the origin and the xy- trace ( z = 0) corresponds to the equator. A) x2 y2 z2 1 + − = 39632 39632 39532 B) x2 y2 z2 1 + + = 39632 39532 39632 C) x2 y2 z2 1 + + = 39632 39632 39532 D) x2 y2 z2 1 − − = 39632 39632 39532 E) x2 y2 z2 1 + + = 39632 39532 39532 Ans: C

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21.

Use the function q ( p1 , p2 ) = A)

7 p1 – 9 p2 to find q ( 6, –10 ) . 9 p1 + 9 p2

3 11 B) 3 q ( 6, –10 ) = 11 C) 11 q ( 6, –10 ) = – 3 D) 11 q ( 6, –10 ) = 3 E) 6 q ( 6, –10 ) = – 11 Ans: C q ( 6, –10 ) = –

22. Evaluate the function at the given values of the independent variables. = z –3 x 3 – xy + 7 y 2 = ; x –4, = y 3 A) z = 267 B) z = 27 C) z = 259 D) z = 252 E) z = 393 Ans: A 23.

Use the function f ( x, y ) = A)

ln ( 7 xy ) to find f ( 9, 6 ) . 8x2 + 3 y 2

ln 378 90 B) ln 378 f ( 9, 6 ) = 531 C) ln 63 f ( 9, 6 ) = 756 D) ln 42 f ( 9, 6 ) = 936 E) ln 378 f ( 9, 6 ) = 756 Ans: E f ( 9, 6 ) =

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24.

Evaluate the function w = A)

x 2 + 10 yz at ( 4, 3, – 4 ) . xyz

29 12 B) 23 w=– 24 C) 17 w=– 6 D) 1 w=– 12 E) 13 w= 6 Ans: E w=

25. Find the domain and range of the function. x y

f ( x, y ) = e A) Domain: all point ( x, y ) such that y ≠ 0 Range: ( −∞, 0 ) B)

Domain: all point ( x, y ) such that y ≠ 0 Range: ( 0, ∞ )

Domain: all point ( x, y ) such that y ≠ 0,1 Range: ( 0, ∞ )

Domain: all point ( x, y ) such that y ≠ 0,1 Range: ( −∞, ∞ )

Domain: all point ( x, y ) such that y ≠ 0

Range: ( −∞, ∞ ) Ans: B 26. A manufacturer estimates the Cobb-Douglas production function to be given by

f ( x, y ) = 100 x 0.75 y 0.25 . Estimate the production levels when x = 1500 and y = 1000 . A) 135,540 units B) 122,560 units C) 131,601 units D) 145,330 units E) 112,745 units Ans: A

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27. Describe the domain and range of the function. f ( x, y ) = 16 − x 2 − y 2 A) domain: The disk x 2 + y 2 < 16

range: The interval (0,4) domain: The disk x 2 + y 2 < 16 range: The interval [0,4] C) domain: The disk x 2 + y 2 ≤ 16 range: The interval [0,4) D) domain: The disk x 2 + y 2 ≤ 16 range: The interval [0,4] E) domain: The disk x 2 + y 2 < 16 range: The interval [0,4) Ans: D B)

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28. Describe the level curves of the function. Sketch the level curves for the given c-values. z = 9 – 3 x – 4 y , c = 0, 2, 4, 6 A)

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Ans: B

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29. Sketch the level curves for the function below for the given c − values c = 0,1, 2,3, 4,5 . A)

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Ans: E

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30. Describe the level curves for the function f ( x, y= ) x 2 + y 2 for the c-values given by c = 0, 2, 4, 6,8 . A) c=0 = 8 x2 + y 2 c=2 = 6 x2 + y 2 c=4 = 4 x2 + y 2 c=6 = 2 x2 + y 2 c =8 = 0 x2 + y 2 B) c=0 = 0 x2 + y 2 c=2 = 2 x2 + y 2 c=4 = 4 x2 + y 2 c=6 = 6 x2 + y 2 c =8 = 8 x2 + y 2 C) c=0 = 2 x2 + y 2 c=2 = 4 x2 + y 2 c=4 = 6 x2 + y 2 c=6 c =8

c=0 c=2 c=4 c=6 c =8

c=0 c=2 c=4 c=6

c =8 Ans: B

= 8 x2 + y 2 = 0 x2 + y 2 = 4 x2 + y 2 = 6 x2 + y 2 = 8 x2 + y 2 = 0 x2 + y 2 = 2 x2 + y 2 = 0 x2 + y 2 = 4 x2 + y 2 16 = x2 + y 2 36 = x2 + y 2 64 = x2 + y 2

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31. If f x, y = x8 – 9 y 4 , find f and f . ( ) y x A) = fx = 8x7 – 9 y 4 , f y x8 – 36 y 3 B) 1 1 , fy = fx = 2 8x7 – 9 y 4 2 x8 – 36 y 3 C) 8x7 –36 y 3 = fx = , fy – x8 – 9 y 4 x8 – 9 y 4 D) 4 x7 18 y 3 = fx = , fy – x8 – 9 y 4 x8 – 9 y 4 E) = f 8= x 7 x8 – 9 y 4 , f –36 y 3 x8 – 9 y 4 x

Ans: D 32.

If f ( x, y ) = ln (11xy 4 + 4 ) , find

A) ∂f = ∂x B) ∂f = ∂x C) ∂f = ∂x D) ∂f = ∂x E) ∂f = ∂x Ans: B

∂f ∂f and . ∂x ∂y

11x ∂f 11 y 4 , = 11xy 4 + 4 ∂y 11xy 4 + 4 11 y 4 ∂f 44 xy 3 , = 11xy 4 + 4 ∂y 11xy 4 + 4 1 ∂f 1 = , 4 11 y ∂y 44 xy 3 11xy 4 + 4 ∂f 44 xy 4 + 4 , = 11y 4 ∂y 11xy 3  1  ∂f  1  ln= , ln   4  3   11 y  ∂y  44 xy 

33. Evaluate f x and f y for the function f ( x, y ) = 2 x 2 + xy − y 3 at the point (–5, –5) . A) f x (–5, –5) = –25 and f y (–5, –5) = –50 B)

f x (–5, –5) = –45 and f y (–5, –5) = –80

f x (–5, –5) = –25 and f y (–5, –5) = –80

f x (–5, –5) = 100 and f y (–5, –5) = –30

f x (–5, –5) = –25 and f y (–5, –5) = 70 Ans: C E)

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34.

Evaluate f x and f y for the function f ( x, y ) =

7 xy x2 + y 2

at the point (6,9) . Round

your answer to two decimal places. A) f x (6,9) = 4.03 and f y (6,9) = 1.19 B)

f x (6,9) = 3.53 and f y (6,9) = 1.19

f x (6,9) = 3.88 and f y (6,9) = 5.82

f x (6,9) = 4.03 and f y (6,9) = 0.69

f x (6,9) = 3.88 and f y (6,9) = 5.82 Ans: A E)

35. Find the first partial derivatives with respect to x, y, and z. 7 xz 4 x + 11 y A) ∂w 77 xz ∂w 77 yz ∂w 7x = , = − , = 2 2 ∂x (4 x + 11 y ) ∂y (4 x + 11 y ) ∂z 4 x + 11 y B) ∂w 77 xz ∂w 77 yz ∂w 7x = = − = , , 4 x + 11 y ∂z (4 x + 11 y ) 2 ∂x 4 x + 11 y ∂y C) ∂w 77 yz ∂w 77 xz ∂w 7x = = − = , , 4 x + 11 y ∂z (4 x + 11 y ) 2 ∂x 4 x + 11 y ∂y D) ∂w 77 yz ∂w 77 xz ∂w 7x = , = − , = 2 2 ∂x (4 x + 11 y ) ∂y (4 x + 11 y ) ∂z 4 x + 11 y E) ∂w 77 yz ∂w 77 xz ∂w 7x = , = − , = 2 2 ∂x (4 x + 11 y ) ∂y (4 x + 11 y ) ∂z (4 x + 11 y ) 2 Ans: D w=

36. For f ( x, y ) , find all values of x and y such that f x ( x, y ) = 0 and f y ( x, y ) = 0 simultaneously. f ( x, y ) = 6 x 3 – 2 xy + 6 y 3 A)  1 1 − ,−   9 9 B) 1 1  ,  9 9 C) (0,0) D) Both B and C E) Both A and B Ans: D

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37. Find the slopes of the surface h( x,= y ) 2 y 2 − x 2 in the x- and y- directions at the point

( 3, –2, –1) . A)

slope in x-direction: 2 slope in y-direction: –17 B) slope in x-direction: –6 slope in y-direction: –8 C) slope in x-direction: –17 slope in y-direction: 2 D) slope in x-direction: –14 slope in y-direction: –14 E) slope in x-direction: –8 slope in y-direction: –6 Ans: B 38. Find the four second partial derivatives. Observe that the second mixed partials are equal. z =x 2 + 5 xy + 4 y 2 A) ∂2 z ∂2 z = 2, = ∂x 2 ∂y 2 B) ∂2 z ∂2 z = 0, = ∂x 2 ∂y 2 C) ∂2 z ∂2 z = 2, = ∂x 2 ∂y 2 D) ∂2 z ∂2 z = 0, = ∂x 2 ∂y 2 E) ∂2 z ∂2 z = 0, = ∂x 2 ∂y 2 Ans: C

∂2 z ∂2 z 4, = = 0 ∂x∂y ∂y∂x ∂2 z ∂2 z 4, = = 5 ∂x∂y ∂y∂x ∂2 z ∂2 z 8, = = 5 ∂x∂y ∂y∂x ∂2 z ∂2 z 0, = = 5 ∂x∂y ∂y∂x ∂2 z ∂2 z 0, = = 0 ∂x∂y ∂y∂x

39. A company manufactures two types of wood-burning stoves: a freestanding model and a fireplace-insert model. The cost function for producing x freestanding and y fireplace-insert stoves is C 30 xy +185 x + 205 y + 1050 . Find the marginal costs = ( ∂C ∂x and ∂C ∂y ) when x = 90 and y = 40 . Round your answers to two decimal places. A) ∂C= ∂x 185.25, ∂C= ∂y 205.25 B) ∂C= ∂x 205.00, ∂C= ∂y 250.00 C) ∂C= ∂x 186.58, ∂C= ∂y 207.37 D) ∂C= ∂x 195.00, ∂C= ∂y 227.50 E) ∂C= ∂x 374.74, ∂C= ∂y 489.60 Ans: D

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40. The value A(t , r ) of an investment of $18,000 after t years in an account for which the interest rate 100r% is compounded continuously is given by the function ∂A . A(t , r ) = 18, 000e rt dollars. Write the partial derivative ∂t A) ∂A = 18, 000re rt ∂t B) ∂A = 18, 000e rt ∂t C) ∂A = 18, 000te rt ∂t D) ∂A = 18, 000re r (t −1) ∂t E) ∂A = 18, 000ret ∂t Ans: A 41. The utility function U = f ( x, y ) is a measure of utility (or satisfaction) derived by a person from the consumption of two products x and y. Suppose the utility function is U = –8 x 2 + 5 xy – 5 y 2 . Determine the marginal utility of product x. A) 5 x –10 y B) –16 x + 5 y C) –16 x + 5 y – 5 y 2 D) –8 x 2 + 5 x –10 y E) –16 x + 5 –10 y Ans: B 42. Test for relative extrema and saddle points. = z x 2 + 8 xy + y 2 + 60 x A) saddle point at ( –2, 8, –180 ) B)

saddle point at ( 2, – 8, 60 )

saddle point at ( 0, 0, 0 )

relative minimum at ( 2, 8, 316 )

relative minimum at ( –24, 64, – 9056 ) Ans: B E)

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43. Examine the function f ( x, y ) = –4 x 2 – y 2 – 9 x + 3 y – 6 for relative extrema. A)  9 3 relative maximum at  – ,   8 2 B)  9 3 saddle point at  – ,   8 2 C) 9 3 relative maximum at  ,  8 2 D)  9 3 relative minimum at  – , –   8 2 E) 9 3 relative maximum at  , –  8 2 Ans: A 44. Examine the function f ( x, y ) = 2 x 2 – 3 xy + 2 y 2 + 7 x + 14 y for relative extrema. A) relative minimum at ( –10, –11) B)

relative maximum at ( –10, –11)

relative minimum at ( –10,11)

relative maximum at ( –10,11) E) no relative extrema Ans: A D)

45. Examine the function f ( x, y ) = x 3 − 6 xy + y 3 + 3 for relative extrema and saddle points. A) saddle point: (0, 0,3) ; relative minimum: (2, 2, –5) B) relative minimum: (0, 0,3) ; relative maximum: (2, 2, –5) C) saddle points: (0, 0,3) , (2, 2, –5) D) saddle point: (2, 2, –5) ; relative minimum: (0, 0,3) E) relative minimum: (2, 2, –5) ; relative maximum: (0, 0,3) Ans: A 46. Examine the function given below for relative extrema and saddle points. f ( x, y ) = A) The function has a relative maximum at ( 0, 0,3) .

The function has a relative minimum at ( 0, 0,3) .

The function has a saddle point at ( 0, 0,3) .

The function has a relative maximum at ( 0, 0, 0 ) .

The function has a relative minimum at ( 0, 0, 0 ) . Ans: A E)

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47. Examine the function given below for relative extrema and saddle points. f ( x, y ) = A) The function has a saddle point at ( 0, 0, 0 ) . B)

The function has a relative maximum at ( 0, 0, 0 ) .

The function has a relative minimum at ( 0, 0, 0 ) .

The function has a saddle point at ( 0, 0, 2 ) .

The function has a relative maximum at ( 0, 0, 2 ) . Ans: A E)

48. Determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function f ( x, y ) at the critical point ( x0 , y0 ) . f xx ( x0 , y0 ) = –1 Given:

f yy ( x0 , y0 ) = –6 f xy ( x0 , y0 ) = 3

A) relative minimum at ( x0 , y0 ) B) saddle point at ( x0 , y0 ) C) relative maximum at ( x0 , y0 ) D) insufficient information to determine the nature of the function at ( x0 , y0 ) Ans: B 2 2 2 49. Find the critical points of the function f ( x, y= , z ) ( x – 2 ) + ( 8 − y ) + ( z – 5 ) , and, from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point. A) relative minimum at ( –2, –8, –5 ) B) relative maximum at ( –2, –8, –5 )

relative minimum at ( 2,8,5 )

relative maximum at ( 2,8,5 ) E) no relative extrema Ans: C D)

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50. Find the critical points of the function f ( x, y, z )= 4 − ( ( x + 9)( y + 4)( z – 7) )2 , and, from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point. A) relative minima at ( –9, a, b ) , ( c, –4, d ) , ( m, n, 7 ) , where a, b, c, d , m, and n are arbitrary real numbers B) relative maxima at ( –9, a, b ) , ( c, –4, d ) , ( m, n, 7 ) , where a, b, c, d , m, and n are arbitrary real numbers C) relative minimum at ( –9, –4, 7 ) D) relative maximum at ( –9, –4, 7 ) E) no relative extrema Ans: B 51. Find three positive numbers x, y, and z whose sum is 15 and product is a maximum. A) x = 2.5, y = 2.5, z = 10 B) x = y = z = 5 C) x = 3.75, y = 3.75, z = 7.5 D) x = 2, y = 3, z = 5 E) x = 1, y = 6, z = 8 Ans: B 52. Find three positive numbers x, y, and z whose sum is 33 and the sum of the squares is a maximum. A) x = y = z = 11 B) x = 5.5, y = 5.5, z = 22 C) x = 8.25, y = 8.25, z = 16.5 D) x = 4.4, y = 6.6, z = 11 E) x = 2.2, y = 13.2, z = 17.6 Ans: A 53. The sum of the length (denote by z) and the girth (perimeter of a cross section) of packages carried by a delivery service cannot exceed 72 inches. Find the dimensions of the rectangular package of largest volume that may be sent. A) x = 9, y = 9, z = 24 B) x = 6, y = 6, z = 48 C) x = 12 , y = 12 , z = 24 D) x = 4.8, y = 7.2, z = 48 E) x = 2.4, y = 14.4, z = 38.4 Ans: C

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54. A company manufactures two types of sneakers: running shoes and basketball shoes. The total revenue from x1 units of running shoes and y1 units of basketball shoes is: R = –3 x12 − 9 x2 2 − 2 x1 x2 + 48 x1 + 98 x2 ,

where x1 and x2 are in thousands of units. Find x1 and x2 so as to maximize the revenue. A) 167 123 = x1 = , x2 26 26 B) 123 167 = x1 = , x2 26 26 C) 334 246 = x1 = , x2 53 53 D) 246 334 = x1 = , x2 53 53 E) 123 167 = x1 = , x2 26 52 Ans: A 55. Use Lagrange multipliers to maximize the function f ( x, y )= the following constraint: x + y − 6 = 0

28 − x 2 − y 2 subject to

Assume that x, y, and z are positive. A) 46 B) 46 C) 10 D) 10 E) no absolute maximum Ans: C 56. Use Lagrange multipliers to find the given extremum. In each case, assume that x and y are positive. Maximize f ( x, y ) = xy Constraint x + y = 10 A) f (7,3) = 21 B) f (5,5) = 25 C) f (6, 4) = 24 D) f (2,8) = 16 E) f (1,9) = 9 Ans: B

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57. Use Lagrange multipliers to find the given extremum. Assume that x and y are positive. Minimize f ( x, y ) = e xy Constraint x 2 + y 2 − 8 = 0 4 A) f ( 2, 2 ) = e B) f (1,1) = e1 C)

f ( 3,1) = e3

f ( 4, 2 ) = e8

f ( 2,1) = e 2 Ans: A E)

58. Use Lagrange multipliers to find the given extremum. In each case, assume that x, y, and z are positive. Maximize f ( x, y, z ) = x + y + z Constraints x 2 + y 2 + z 2 = 1 A)  2 2 2 f  , ,  = 2 3 3 3   B)  5 5 5 f  , ,  = 5 3 3 3   C)  3 3 3 f  , ,  = 3 3 3 3   D) f 3, 3, 3 = 3 3

(

)

 1 1 1  f , , = 3  3 3 3 Ans: C E)

59. A rectangular box is resting on the xy -plane with one vertex at the origin. The opposite lies in the plane Find the dimensions that maximize the volume. (Hint: Maximize V = xyz subject to the constraint 2 x + 3 y + 5 z − 90 = 0 ). A) 15 units × 12 units × 5 units B) 11 units × 9 units × 5 units C) 10 units × 9 units × 6 units D) 15 units × 10 units × 6 units E) 12 units × 11 units × 7 units Ans: D

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60. A microbiologist must prepare a culture medium in which to grow a certain type of bacteria. The percent of salt contained in this medium is given by S = 12 xyz where x, y, and z are the nutrient solutions to be mixed in the medium. For the bacteria to grow, the medium must be 13% salt. Nutrient solutions x, y, and z cost $1, $2, and $3 per liter, respectively. How much of each nutrient solution should be used to minimize the cost of the culture medium? A) = x 3 0.065 ≈ 0.402 L

= y

0.065 ≈ 0.201L

= z B) = x

0.065 ≈ 0.134 L 0.035 ≈ 0.327 L

y =

0.165 ≈ 0.548 L

= z C) = x

0.015 ≈ 0.2466 L 0.065 ≈ 0.402 L

= y

= z D) = x

= y

0.035 ≈ 0.327 L

= z E)= x

0.115 ≈ 0.486 L 0.165 ≈ 0.548 L

= y

0.165 ≈ 0.548 L

= z Ans: A

0.265 ≈ 0.642 L

0.165 ≈ 0.548 L

0.055 ≈ 0.380 L 3 0.025 ≈ 0.292 L

61. Use Lagrange multipliers to minimize the function f ( x, y, z ) = x 2 + y 2 + z 2 subject to the following constraint. x + y + z − 27 = 0

Assume that x, y, and z are positive. A) 81 B) 243 C) 162 D) 486 E) 729 Ans: B

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62. Use Lagrange multipliers to find the minimum distance from the circle x 2 + ( y − 5 )2 = 9 to the point ( –10,1) . Round your answer to the nearest tenth. A) 13.6 B) 107.0 C) 184.4 D) 7.8 E) 60.4 Ans: D 63. A manufacturer has an order for 800 units of fine paper that can be produced at two locations. Let x1 and x2 be the numbers of units produced at the two plants. Find the number of units that should be produced at each plant to minimize the cost if the cost function is given by C = 0.4 x12 + 20 x1 + 0.1x2 2 + 10 x2 . A) x1 = 300 units and x2 = 650 units B) x1 = 650 units and x2 = 150 units C) x1 = 150 units and x2 = 650 units D) x1 = 150 units and x2 = 1300 units E) x1 = 1300 units and x2 = 300 units Ans: C

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64. A rancher plans to use an existing stone wall and the side of a barn as a boundary for two adjacent rectangular corrals. Fencing for the perimeter costs 25 per foot. To separate the corrals, a fence that costs 20 per foot will divide the region. The total area of the two corrals is to be 7000 square feet. Use Lagrange multipliers to find the dimensions that will minimize the cost of the fencing.

A) dimensions: 50 feet by 70 feet B) dimensions: 70 feet by 50 feet C) dimensions: 25 feet by 70 feet D) dimensions: 35 feet by 50 feet E) dimensions: 25 feet by 35 feet Ans: A 65. Find the sum of the squared errors for the linear model f ( x ) and the quadratic model

g ( x ) using the given points. f ( x ) = 1.6 x + 6, g ( x ) = 0.29 x 2 + 2.2 x + 6

( −3, 2 ) , ( −2, 2 ) , ( −1, 4 ) , ( 0, 6 ) , (1,8 ) A) S = 1.5 ; S = 0.7159 B) S = 1.6 ; S = 0.8259 C) S = 1.2 ; S = 0.8623 D) S = 1.3 ; S = 0.4160 E) S = 1.1 ; S = 0.7621 Ans: B

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66. Find the least squares regression line for the given points. Then plot the points and sketch the regression line.

( −2, −1) , ( 0, 0 ) , ( 2,3) A)

y= x +

2 3 2 x− 3 1 x− 3 1 x− 2 1 x+ 3

Ans: A 67. Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points.

( −4, −1) , ( −2, 0 ) , ( 2, 4 ) , ( 4,5) A) = y 1.8 x + 5 B) = y 1.2 x + 2 C) = y 0.5 x + 1 D) = y 0.8 x + 2 E) = y 0.2 x + 1 Ans: D 68. An agronomist used four test plots to determine the relationship between the wheat yield y (in bushels per acre) and the amount of fertilizer x (in pounds per acre). The results are shown in the table. Fertilizer , x 100 150 200 250 Yield , y 35 44 50 56

(a) Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the data. (b) Estimate the yield for a fertilizer application of 160 pounds per acre. A) = y 0.15 x + 15.1 B)= y 0.155 x + 21.1 C)= y 0.138 x + 22.1 D)= y 0.052 x + 34 E)= y 0.234 x + 17.5 Ans: C

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69. Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Data that are modeled by = y 3.29 x − 4.17 have a positive correlation. A) True B) False; The data modeled by = y 3.29 x − 4.17 have a positive correlation. Ans: B 70. Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. When the correlation coefficient is r ≈ −0.98781 , the model is a good fit. A) False B) True Ans: B 71. Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

A linear regression model with a positive correlation will have a slope that is greater than 0.

A) True B) False Ans: A 72. Find the least squares regression line for the points (1,0) , (3,3) , (8,6). A) 21 3 y= x– 26 13 B) 21 3 y= x– 26 61 C) 21 3 y= x– 122 13 D) 21 3 y= x– 122 61 E) none of the above Ans: A 73. Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the points (10, 2), (7,8), (4,1), (6,10), (11, 6), (14, 6) . Round your answer to three decimal places. A) y = 0.049 x + 4.989 B) y = 0.059 x + 4.979 C) y = 0.059 x + 4.989 D) y = 0.069 x + 4.989 E) y = 0.069 x + 4.979 Ans: C

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74. A store manager wants to know the demand y for an energy bar as a function of price x. The daily sales for three different prices of the energy bar are shown in the table.

Price, x Demand, y

$ 1.00 450

$ 1.25 335

$ 1.54 300

(i) Use the regression capabilities of a graphing utility to find the least squares regression line for the data. (ii) Use the model to estimate the demand when the price is $1.39. A) (i) y = –28.012762 x + 397.056123 ; (ii) 357.97832 B) (i) y = –28.012762 x + 397.056123 ; (ii) –436.133926 C) (i) y = 397.056123 x – 28.012762 ; (ii) 525.880529 D) (i) y = –28.012762 x – 397.056123 ; (ii) –436.133926 E) none of the above Ans: E 75. Evaluate the following integral. x6

–6 y dy x 3x A) – 3 ( x11 − 9 x )

∫

B) C) D)

– 6 ( x11 − 9 x ) – 3 ( x11 − 3 x )

– 3 ( 9x − x11 )

E) none of the above Ans: A 76.

4 3

Evaluate the double integral ∫ ∫ ( 2 x + 3 y )dydx . 0 0

A) 92.00 B) 102.00 C) 112.00 D) 29.50 E) 17.00 Ans: B

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77.

3 y

Evaluate the double integral ∫ ∫ ( 3 x + 5 y )dxdy . Round your answer to two decimal 0 0

places, where applicable. A) 48.50 B) 68.50 C) 49.50 D) 58.50 E) 24.00 Ans: D 78.

3 2y

Evaluate the double integral ∫ ∫ ( 2 + 3 x 2 + 2 y 2 )dxdy . Round your answer to two 0 y

decimal places, where applicable. A) 69.75 B) 184.25 C) 192.25 D) 191.25 E) 184.75 Ans: D 79.

4 x

2 dydx . x +7 0 0

Evaluate the double integral ∫ ∫

A) ln 22 B) ln 22 − ln 7 C) ln 23 D) ln 23 + ln 7 E) ln 23 − ln 7 Ans: E 80.

∞∞

Evaluate the double integral ∫ ∫ xye

−(5 x2 +8 y2 )

dxdy .

0 0

1 200 B) 1 23 C) 1 160 D) 1 − 80 E) 1 − 120 Ans: C

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81. Sketch the region R whose area is given by the following double integral.

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Ans: D 82.

Use a double integral to find the area of the region bounded by the graphs of y = x 2 and y = x . A) 2 B) 9 C) 11 2 D) 11 13 E) 9 26 Ans: E

83. Use a double integral to find the area of the region bounded by the graphs of y 2 ( x + 1) . y = x 2 + 2 x + 1 and= A) 4 3 B) 5 2 C) −3 D) 2 E) 3 − 2 Ans: A

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84. Use a symbolic integration utility to evaluate the double integral. 2

∫ ∫ e dydx xy

A) 8.1747 B) 9.1211 C) 6.2031 D) 7.88.7522 E) 9.4362 Ans: A

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85. Sketch the region of integration A)

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Ans: C 86. Set up an integral for both orders of integration, and use the more convenient order to evaluate the integral below over the region R. y

∫∫ x + y dA 2

by y 5= R : triangle bounded = x, y 6 x, and x 5 = A) 2 ( log ( 37 ) − log ( 26 ) ) 5 B) 5 ( log ( 26 ) − log ( 37 ) ) 2 C) 5 ( log ( 37 ) − log ( 26 ) ) D) 5 ( log ( 37 ) − log ( 26 ) ) 2 E) 1 ( log ( 37 ) − log ( 26 ) ) 3 Ans: D

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87. Use a double integral to find the volume of the indicated solid.

z = 25 − y 2 , z > 0, x > 0, 5 x < y < 5 A) 4 125 B) 623 18 C) 125 4 D) 18 623 E) none of the above Ans: C 88. Use a double integral to find the volume of the solid bounded by the graphs of the equations. = z xy= , z 0,= y 2 x= , y 0,= x 0,= x 3 A) 71 3 B) 83 3 C) 79 2 D) 81 2 E) 62 5 Ans: D

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89. A firm’s weekly profit (in dollars) in marketing two products is given by

P = 192 x1 + 576 x2 − x12 − 5 x22 − 2 x1 x2 − 5000 where x1 and x2 represent the numbers of units of each product sold weekly. Estimate the average weekly profit when x1 varies between 40 and 50 units and

x2 varies between 45 and 50 units.

A) $12,300 B) $11,100 C) $13, 400 D) $21, 760 E) $16, 450 Ans: C 90. Use a double integral to find the volume of the solid bounded by the graphs of the equations= z x 2= , z 0,= x 0,= x 2,= y 0,= y 1. A) 10 3 B) 8 5 C) 8 3 D) 8 E) 3 Ans: C 91. The population density (in people per square mile) for a coastal town can be modeled by 130, 000 where x and y are measured in miles. What is the population f ( x, y ) = (2 + x + y )3 inside the rectangular area defined by the vertices ( 0, 0 ) , ( 2, 0 ) , ( 0, 2 ) , and ( 2, 2 ) ? Round to the nearest integer. A) 12,833 people B) 32,500 people C) 21,667 people D) 11,833 people E) 10,833 people Ans: E

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92. Find the average value of f ( x, y= ) x 2 + y 2 over the region R: square with vertices (0, 0), (7, 0), (7, 7), (0, 7) . A) 98 3 B) 4802 3 C) 686 5 D) 49 3 E) 98 5 Ans: A 93. A company sells two products whose demand functions are given by = x1 500 − 3 p1 and = x2 750 − 2.4 p2 . So, the total revenue is given by= R x1 p1 + x2 p2 . Estimate the average revenue if the price p1 varies between $45 and $70 and the price p2 varies between $ 45 and $ 70. A) $ 52,725 B) $ 54,875 C) $ 52,223 D) $ 53,740 E) $ 55,285 Ans: D 94. The Cobb-Douglas production function for an automobile manufacturer is f ( x, y ) = 100 x 0.6 y 0.4 where x is the number of units of labor and y is the number of units of capital. Estimate the average production level if the number of units of labor x varies between 250 and 300 and the number of units of capital y varies between 250 and 300. A) 20.99 B) 21.10 C) 10.99 D) 31.44 E) 31.24 Ans: C

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Chapter 8: Trigonometric Functions 1. Determine two coterminal angles (one positive and one negative) for each angle. Give the answers in radians.

positive: negative:

8π 3 7π 4 5π − 3 7π 3 7π − 3 5π 3 7π 3 5π − 3 7π 4 5π − 2

Ans: D

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2. Find the radian measure of the given angle. 150o A) 5π 3 B) 5π 2 C) 15π 2 D) 5π 6 E) 5π 12 Ans: D 3. Find the radian measure of the given angle. 750o A) 25π 3 B) 25π 6 C) 75π 2 D) 25π 2 E) 25π 12 Ans: B 4. Find the degree measure of the given angle. 7π 4 A) 50.0o B) 315.0o C) 51.4o D) 102.9o E) 100.3o Ans: B

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5. Solve the triangle for the indicated side and angle.

angle θ : side c : B) angle θ : side c : C) angle θ : side c : D) angle θ : side c : E) angle θ : side c : Ans: B A)

60 ° 30 ° 45 ° 60 ° 40 °

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6. Solve the triangle for the indicated angle.

A) angle θ : B) angle θ : C) angle θ : D) angle θ : E) angle θ : Ans: D

40 ° 90 ° 140 ° 50 ° 130 °

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7. Solve the triangle for the indicated side.

side h :

7 18 25 7 18 7 7 25 9 7

Ans: C 8. Find the area of the equilateral triangle with sides of length s = 4 in. Round your answer to two decimal places. A) 6.00 square inches B) 27.71 square inches C) 6.93 square inches D) 8.00 square inches E) 24.00 square inches Ans: C

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9. A guy wire is stretched from a broadcasting tower at a point 300 feet above the ground to an anchor 175 feet from the base (see figure). How long is the wire?

A) 347.31 feet B) 173.66 feet C) 243.67 feet D) 121.83 feet E) 237.50 feet Ans: A 10. A compact disc can have an angular speed up to 3160 radians per minute. At this angular speed, how many revolutions per minute would the CD make? Round your answer to the nearest integer. A) 123 B) 144 C) 923 D) 503 E) 72 Ans: D

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11. A sector of a circle is the region bounded by two radii of the circle and their intercepted arc (see figure).

For a circle of radius r , the area A of a sector of the circle with central angle θ (measured 1 in radians) is given by A = r 2θ . 2 A sprinkler system on a farm is set to spray water over a distance of 60 feet and rotates through an angle of 90° . Use the above given information to find the area of the region. Round your answer to two decimal places. A) 2828.57 B) 257.14 C) 2700.00 D) 1414.29 E) 9900.00 Ans: A

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12. Find the cosine of θ .

8 17 B) 8 15 C) 15 17 D) 17 8 E) 17 15 Ans: A

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13. Find csc θ from the given graph.

35 37 B) 12 – 37 C) 37 – 12 D) 37 12 E) 37 – 35 Ans: C

14. Find the cosine of θ .

20 21 B) 20 – 29 C) 21 29 D) 29 – 20 E) 29 21 Ans: B –

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15. Find the cosine of θ .

A) B)

4 3

5 3 C) 5 – 4 D) 3 – 5 E) 4 – 5 Ans: E –

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16.

From the given function cos θ =

4 , find the following trigonometric function. 5

5 3 B) 5 4 C) 3 5 D) 4 5 E) 3 9 Ans: C

17.

Find sin θ given that sec θ = 8 and 0 < θ < A) B) C)

π 2

8 3 7

3 7 8 D) 1 3 7 E) 1 64 Ans: C

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18. Determine the quadrant in which θ lies if sin q > 0 and cos q < 0. A) third quadrant B) second quadrant C) fourth quadrant D) first quadrant E) first or third quadrants Ans: B 19. Evaluate without using a calculator. tan

3π 4

A) B) C)

–1 undefined 3 2 D) 2 3 E) 0 Ans: A

20. Evaluate without using a calculator, leaving the answers in exact form. sin

π 3

3 2 B) 1 2 C) 0 D) –1 E) 3 2 Ans: A

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21. Evaluate without using a calculator, leaving the answers in exact form. cos

7π 3

3 2 2 2

B) C) D)

0 –

1 2

1 2 Ans: E

22. Approximate using a calculator (set for radians). Round answers to two decimal places. sin 3 A) 0.78 B) 0.14 C) –0.62 D) –0.99 E) –1.00 Ans: B 23. Use a calculator to evaluate the trigonometric function cos 350° to four decimal places. A) –1.3504 B) –5.6713 C) –0.7405 D) 0.9848 E) –0.1763 Ans: D

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24. Find two values of q that satisfy the equation below. Give values of q in radians (0 ≤ θ ≤ 2π ) . Do not use a calculator. 2 2

cos q = A) B) C)

7π 4 7 7π π θ= , θ= 4 4

θ=

,θ =π

5π 6 6 E) π 4π θ= , θ= 7 4 Ans: B

θ=

, θ=

25. Find two values of q that satisfy the equation below. Give values of q in radians (0 ≤ θ ≤ 2π ) . Do not use a calculator. 2 2

sin q = − A) B)

θ=

π 4

θ=

, θ=

π 4

3π 4

,θ =π

7π 5π , θ= 4 4 D) 5π π θ= , θ= 6 6 E) 7π π θ= , θ= 4 4 Ans: C C)

θ=

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26. Solve the equation for θ (0 ≤ θ ≤ 2π ) . For some of the equations you should use the trigonometric identities listed in this section. Use the trace feature of a graphing utility to verify your results. A) B) C) D) E) Ans: A 27. Solve the equation below for θ (0 ≤ θ ≤ 2π ) . For some of the equations you should use the trigonometric identities listed in this section. Use the trace feature of a graphing utility to verify your results. A) B) C) D) E) Ans: D

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28. Solve the equation below for θ (0 ≤ θ ≤ 2π ) . For some of the equations you should use the trigonometric identities listed in this section. Use the trace feature of a graphing utility to verify your results. A) B) C) D) E) Ans: B 29. A 21-foot ladder leaning against the side of a house makes a 78 ° angle with the ground (see figure). How far up the side of the house does the ladder reach? Round your answer to four decimal places.

A) 20.5411 feet B) 21.4692 feet C) 4.3661 feet D) 4.4637 feet E) 101.0044 feet Ans: A

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30. In traveling across flat land, you notice a mountain directly in front of you. Its angle of elevation (to the peak) is 4.5 ° . After you drive 13 miles closer to the mountain, the angle of elevation is 11 ° . Approximate the height of the mountain. Round your answer to two decimal places.

A) 45.50 miles B) 8.84 miles C) 1.94 miles D) 1.72 miles E) 17.69 miles Ans: D 31. Find the period and amplitude of the function y = 3cos 2 x .

A) B) C) D) E)

period: 2π ; amplitude: 6 period: 2π ; amplitude: 3 period: π ; amplitude: 3 period: π ; amplitude: 6 period:

; amplitude: 3

Ans: C

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32.

3 πx  Find the period and amplitude of the function y = cos  . 2  2 

Period: 2π ; Amplitude:3 3 Period: 2π ; Amplitude: 2 C) 3 Period: 2 ; Amplitude: 2 D) Period: 4 ; Amplitude: 3 E) 3 Period: 4 ; Amplitude: 2 Ans: E A) B)

33. Find the derivative of the trigonometric function. = y cos 3 x + sin 2 x A) −3sin 3 x + 2sin x cos x B) 3sin 3 x − 2sin x cos x C) 3sin 3 x + 2sin 2 x cos x D) − sin 3 x + 2sin 2 x cos 2 x E) −3sin 2 x + sin 2 x cos 2 x Ans: A

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34. Find the period of the trigonometric function. y = 3sec 5 x A) π 5 B) 2π 3 C) π 3 D) 2π 5 E) 3π 5 Ans: D

35. Find the period of the trigonometric function. πx y = cot 6 A) π 6 B) 6 C) π 3 D) π 2 π E) Ans: B

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36. Sketch the graph of the function y = cot 2 x . A)

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Ans: B

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37. Sketch the graph of the function y = cot π x . A)

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Ans: A

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38. Sketch the graph of the function y = 2 csc 2 x . A)

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Ans: C 39. Find a and d for = f ( x) a cos x + d such that the graph of f matches the figure.

A) = a 1;= d 2 B) = a 3;= d 1 C) = a 2;= d 3 D) = a 2;= d 1 E) = a 3;= d 3 Ans: D

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40. Match the function below with the correct graph. A)

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41. For a person at rest, the velocity v (in liters per second) of air flow into and out of the πt lungs during a respiratory cycle is given by 0.1sin , where t is the time in seconds. 9 Inhalation occurs when v > 0 and exhalation occurs when v < 0 . Find the time for one full respiratory cycle. A) 18π seconds B) π seconds C) 18 seconds D) 2π seconds E) 9 seconds Ans: C 42. Find the derivative of the function. y = 2 cos 2 x A) y ' = –4sin 2 x B) y ' = 4sin 2 x C) y ' = –2sin 2 x D) y ' = –4 cos 2 x E) y ' = –2sin 2 x Ans: A

43. Find the derivative of the function. 4 f (θ ) = sin 2 4θ 5 A) 4 sin 4θ cos 4θ f ′(θ ) = 5 B) 32 sin 4θ cos 4θ f ′(θ ) = 5 C) 32 cos 4θ f ′(θ ) = 5 D) 32 sin 4θ cos 4θ f ′(θ ) = − 5 E) 32 sin 4θ f ′(θ ) = 5 Ans: B

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44. Find the derivative of the function and simplify your answer by using the trigonometric identities y = cos 2 x A) −2 cos 2 x sin 2 x = 2sin 2 x B) 2 cos x sin x = sin 2 x C) −2 cos x sin x = − sin 2 x 2 D) 2 cos x sin x = 2sin x E) 2 cos 2 x sin 2 x = 2sin 2 x Ans: C 45. Find an equation of the tangent line to the graph of the function at the given point.  3π   , −1  4  A) 3 y= −2 x + π − 1 2 B) 1 y =2 x + π − 1 2 C) 1 y= −2 x − π 2 D) 5 y =− x − π − 1 2 E) 3 y =2 x − π + 1 2 Ans: A

y = cot x

46. Find the derivative of the function y = ln(sec 2 x) and simplify your answer by using the trigonometric identities. A) B) C) D) E) Ans: C

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47. Determine the relative extrema of the function = y 2sin x + sin 2 x on the interval (0, 2π ) . A)  5π 3 3  relative minimum:  ,  2   3  5π 3 3  relative maximum:  , −  2   3 B)  5π 3 3  relative minimum:  , −  2   3 π 3 3  relative maximum:  ,  3 2  C) π 3 3  relative minimum:  , −  2  3  5π 3 3  relative maximum:  ,  2   3 D) π 3 3  relative minimum:  ,  3 2   5π 3 3  relative maximum:  , −  2   3 E)  5π 3 3  relative minimum:  ,  2   3 π 3 3  relative maximum:  , −  2  3 Ans: B

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48. Determine the relative extrema of the function e5 x cos x on the interval (0, 2π ) . A)  2 254π 5π  relative minimum:  − e ,  2 4    2 54π π  relative maximum:  e ,  2 4  B) π 2 54π  relative minimum:  , − e  4 2   25π  5π 2 4  relative maximum:  , e  4 2   25π C)  5π 2 4  relative minimum:  , − e  4 2   5π π 2 4  relative maximum:  , e  4 2   5π D)  5π 2 4  relative minimum:  , e  4 2   25π π 2 4  relative maximum:  , − e  4 2   5π E)  2 4 5π  relative minimum:  e ,  2 4    2 254π π  relative maximum:  − e ,  2 4  Ans: C 49. The normal average daily temperature in degrees Fahrenheit for a city is given by 5π (t − 32) where t is the time in days, with t = 1 corresponding to January 52 − 20 cos 365 1. Find the warmest day. A) March 16 B) March 15 C) April 16 D) April 15 E) April 14 Ans: D

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50. Suppose that the numbers W (in thousands) of construction workers employed in the United States during 2006 can be modeled by W = 9094 + 455.2sin(0.6t − 1.763) where t is the time in months, with t = 1 corresponding to January 1. Approximate the month t in which the number of construction workers employed was a maximum. What was the maximum number of construction workers employed? Round your answer to nearest hundredth. A) July; The maximum number of construction workers employed is 9559. B) May; The maximum number of construction workers employed is 9539. C) June; The maximum number of construction workers employed is 9539. D) May; The maximum number of construction workers employed is 9549. E) June; The maximum number of construction workers employed is 9549. Ans: E 51. Find the indefinite integral of the following function.

∫ cos 4s ds

cos 4s + C sin 4s + C 4 sin 4s sin 4 s +C 4 E) sin 4 s 5 Ans: D

A) B) C) D)

52. Find the indefinite integral of the following function.

∫ 8s cos s ds 7

A) B) C) D)

cos s 8 + C sin s 8 + C sin s 7

sin s 8 +C 8 E) sin s 8 Ans: B

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53.

Find the indefinite integral of ∫ A)

ln 5sec x + 3 + C

ln 5sec x − 3 + C

ln 3cos x + 5sec x + C

ln 5cos x − 3 + C

5sec x tan x dx . 5sec x − 3

ln 5cos x + 3 + C Ans: B E)

54. Find the indefinite integral of

∫ e cos e dx . x

A) e x – sin e x + C B) e x – cos e x + C C) sin e x + C D) – sin e x + C E) cos e x + C Ans: C 55. Find the indefinite integral.

∫ tan x sec xdx 3

1 − tan 3 x sec 2 x + C 6 B) 1 tan 4 x sec3 x + C 12 C) 1 tan 4 x + C 4 D) 1 tan 4 x sec 4 x + C 4 E) 1 4 sec x + C 4 Ans: C

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56. Find the indefinite integral. x

∫ sec 4 tan 4 dx 6

2 6x x sec tan + C 3 4 4 B) 2 6x sec + C 3 4 C) 1 7x sec + C 6 4 D) 1 6x x sec tan + C 4 4 4 E) 1 x tan + C 3 4 Ans: B

57. Use integration by parts to find the indefinite integral.

∫ x cos 2 xdx A)

1 1 x cos 2 x − sin 2 x + C 2 4 B) 1 1 x sin 2 x − cos 2 x + C 2 4 C) 1 1 sin 2 x + cos 2 x + C 2 2 D) 1 1 x sin 2 x + cos 2 x + C 2 4 E) 1 1 x cos 2 x + sin 2 x + C 4 2 Ans: D

58. The average monthly precipitation P (in inches), including rain, snow, and ice, for Sacramento, California can be modeled by = P 2.47 sin ( 0.40t + 1.80 ) + 2.08, 0 ≤ t ≤ 12 where t is the time (in months), with t = 1 corresponding to January. Find the total annual precipitation for Sacramento. A) 18.02 in. B) 17.69 in. C) 14.52 in. D) 16.57 in. E) 18.90 in. Ans: B

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59. Evaluate the definite integral.

∫

π 4

4sec 2 tdt

A) –4 B) 5 C) 0 D) 4 E) undefined Ans: D 60.

Evaluate the definite integral ∫ π6 csc 3 x cot 3 xdx . 18

A) B) C)

3 –3 1 − 3 D) 1 3 E) ∞ Ans: D

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Chapter 9: Probability and Calculus 1. A coin is tossed five times. Describe the event A that at least four tails occur. A) {HHHHH, TTTTH, TTTHT, TTHTT, THTTT, HTTTT} B) {THHHH, TTTTH, TTTHT, TTHTT, THTTT, HTTTT} C) D) E)

{TTTTT, TTTTH, TTTHT, TTHTT, THTTT, HTTTT} {TTTTTT, TTTTTH, TTTTHT, TTTHTT, THTTTT, HTTTTT} {TTT, TTTH, TTTTH, TTTHTT, THTTTT, HTTTTT}

Ans: C 2. Three people are asked their opinions on a political issue. They can answer "Opposed" (O) or "Undecided" (U). Find the sample space S. A) {OOO, OOU, OUO, UOO, UUU, UUO, UOU} B) {OOO, OOU, UOO, UUU, UUO, UOU, OUU} C) D) E)

{OOO, OOU, OUO, UOO, UUO, UOU, OUU} {OOO, OUO, UOO, UUO, UOU, OUU} {OOO, OOU, OUO, UOO, UUU, UUO, UOU, OUU}

Ans: E 3. A card is chosen at random from three 52-card decks of standard playing cards . What is the probability that the card will be black and a face card? A face card is a king, a queen, or a jack. A) 1 13 B) 1 26 C) 3 13 D) 2 13 E) 4 13 Ans: B 4. Determine whether the table represents a probability distribution.

A) no B) yes Ans: A

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5. Sketch a graph of the probability distribution. x 0 1 2 3 4 8 6 3 2 1 P( x) 20 20 20 20 20 A)

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Ans: D 6. Find P ( x ≤ 4) given the probability distribution. x 0 1 2 3 4 5 P ( x) 0.047 0.185 0.240 0.325 0.159 0.044 A) 0.159 B) 0.797 C) 0.203 D) 0.956 E) 0.044 Ans: D 7. Find P ( x ≤ 2) given the probability distribution. x 0 1 2 3 P ( x) 0.029 0.181 0.440 0.350 A) 0.440 B) 1.000 C) 0.790 D) 0.650 E) 0.250 Ans: D 8. Estimate the variance for the following probability distribution to two decimal places. x 0 1 2 3

P(x) 1/30

1/6

1/5

3/5

A) 2.37 B) 0.77 C) 0.87 D) 0.59 E) 1.00 Ans: B

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9. Find the standard deviation σ for the following probability distribution. Round your answer to three decimal places. x 1 2 3 4 5 2 1 1 1 1 P( x) 5 10 10 5 5 A) 2.600 B) 1.562 C) 5.954 D) 2.440 E) 1.612 Ans: B 10. Find the expected value E ( x) for the following probability distribution. Round your answer to three decimal places. x − 5,000 − 2,500 300 0.062 0.924 P ( x) 0.014 A) 7.225 B) 904.398 C) 2,724.840 D) 817,935.160 E) 52.200 Ans: E 11. A publishing company introduces a new weekly magazine that sells for $4.69 on the newsstand. The marketing group of the company estimates that sales x (in thousands) will be approximated by the following probability function. Find the expected revenue. Round your answer to the nearest dollar. x 10 15 20 30 40 0.29 0.23 0.10 0.14 P ( x) 0.24 A) $ 93,565 B) $ 94 C) $ 456 D) $ 19,950 E) $ 456,091 Ans: A

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12. An insurance company needs to determine the annual premium required to break even on fire protection policies with a face value of $ 90,000. If x is the claim size on these policies and the analysis is restricted to the losses $ 30,000, $ 50,000, and $ 90,000, then the probability distribution of x is as shown in the table. What premium should customers be charged for the company to break even? Round your answer to the nearest dollar. x 0 30,000 50,000 90,000 P ( x) 0.9910 0.0036 0.0009 0.0045 A) $ 553 B) $ 24 C) $ 42,500 D) $ 558 E) $ 62,000 Ans: D 13. If x is the net gain to a player in a game of chance, then E(x) is usually negative. This value gives the average amount per game the player can expect to lose over the long run. A service organization is selling $2 raffle tickets as part of a fundraising program. The first prize is a boat valued at $2920, and the second prize is a camping tent valued at $600. In addition to the first and second prizes, there are twenty-two $20 gift certificates to be awarded. The number of tickets sold is 3000. Find the expected net gain to the player for one play of the game. Round your answer to the nearest cent. A) $ 1178.00 B) $ 1.47 C) $ 0.68 D) –$0.68 E) –$1178.00 Ans: D 14. A baseball fan examined the record of a favorite baseball player’s performance during his last 50 games. The numbers of games in which the player had zero, one, two, three, and four hits are recorded in the table shown below. Find the standard deviation σ . Round your answer to two decimal places. Number of hits 0 1 2 3 4 Frequency 10 22 5 12 1 A) 1.44 B) 1.12 C) 1.55 D) 1.25 E) 1.20 Ans: B

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15. Find the constant k such that the function f ( x) = kx is a probability density function over the interval [1, 6] . A) 5 B) 2 35 C) 2 5 D) 35 2 E) 5 2 Ans: B

16. Find the value of the constant a that makes the given function a probability density function on the stated interval.

= f ( x) ax 2 ( 8 − x ) on [ 0,1] A) 6 29 B) 29 12 C) 12 29 D) 29 6 E) 1 Ans: C 17.

3 − x

Find the constant k such that the function f ( x) = ke 2 is a probability density function

over the interval [ 0, ∞ ] . A) 1 B) 2 3 C) 2 − 3 D) 3 − 2 E) 3 2 Ans: E

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18.

Sketch the graph of the probability density function f ( x) =

1 over the interval [ 0, 7 ] . 7

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Ans: B 19.

x on the interval [ 0,18] , find the 162 probability that x ≥ 3 . Round your answer to the nearest hundredth. A) 0.07 B) 0.83 C) 0.97 D) 0.09 E) 0.06 Ans: C

For the probability density function f ( x) =

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20.

For the probability density function f ( x) =

21.

1 −t For the probability density function f (t ) = e 3 on the interval [0, ∞) , find the 3 probability that t ≥ 4 . Round your answer to the nearest thousandth. A) 0.088 B) 0.245 C) 0.224 D) 0.250 E) 0.264 Ans: E

7 on the interval [ 0, 6] find the 6( x + 1) 2 probability that x ≤ 2 . Round your answer to the nearest hundredth. A) 0.15 B) 0.67 C) 0.78 D) 0.33 E) 0.12 Ans: C

22. Buses arrive and depart from a college every 40 minutes. The probability density function for the waiting time t (in minutes) for a person arriving at the bus stop is 1 on the interval [ 0, 40] . Find the probability that the person will wait no f (t ) = 40 longer than 20 minutes. A) 1 40 B) 1 20 C) 1 2 D) 1 2 E) 1 800 Ans: C

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23. The daily demand for gasoline (in millions of gallons) in a city is described by the 1 1 probability density function f ( x)= − x over the interval [ 0, 6] . Find the probability 3 18 that the daily demand for gasoline will be at least 3 million gallons. A) 0.444 B) 0.111 C) 0.174 D) 0.340 E) 0.250 Ans: E 24. The time t (in hours) required for a new employee to successfully learn to operate a machine in a manufacturing process is described by the probability density function 5 f (t ) t 9 − t over the interval [0,9] . Find the probability that a new employee = 324 will learn to operate the machine in more than 2 hours but less than 5 hours. A) 0.4498 B) 0.1264 C) 0.3714 D) 0.0981 E) 0.2733 Ans: C 25. A meteorologist predicts that the amount of rainfall (in inches) expected for a certain coastal community during a hurricane has the probability density π πx function f ( x) = sin , 0 ≤ x ≤ 15 . Find and interpret the 30 15 probability P(0 ≤ x ≤ 13) . A) 90% probability of receiving up to 13 inches of rain B) 96% probability of receiving up to 13 inches of rain C) 98% probability of receiving up to 13 inches of rain D) 4% probability of receiving up to 13 inches of rain E) 10% probability of receiving up to 13 inches of rain Ans: B

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26. Sketch the graph of the following probability density function and locate the mean on the graph.

mean :

1.5

mean :

2.0

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mean :

4.0

mean :

3.0

mean :

4.0

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Ans: C

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27. Sketch the graph of the following probability density function and locate the mean on the graph.

mean : 1.600 B)

mean : 2.012 C)

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mean : 0.555 D)

mean : 1.848 E)

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mean : 0.714 Ans: B 28. For the given probability density function, find Var ( X ) . 1 2 x on [ 0,12] 576 A) 12.000 B) 6.400 C) 9.000 D) 5.400 E) 28.800 Ans: D f ( x) =

29.

7 − 74t Find the median of the exponential probability density function f (t ) = e over the 4 interval [0, ∞) . A) 1.213 B) 0.693 C) 0.396 D) 1.061 E) 0.875 Ans: C

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30.

Find the mean, variance, and standard deviation of the uniform distribution f ( x) = over the interval [0, 8] without using integration. A) expected value ( mean ) : 4.000 variance: 5.333 standard deviation: 2.309 B) expected value ( mean ) : 5.333 variance: 4.000 standard deviation: 2.309 C) expected value ( mean ) : 5.333 variance: 2.309 standard deviation: 4.000 D) expected value ( mean ) : 2.309 variance: 4.000 standard deviation: 5.333 E) expected value ( mean ) : 4.000 variance: 2.309 standard deviation: 5.333 Ans: A

31. If X is an exponential random variable with the probability density function = f ( x ) 37e −37 x on [0,∞), find E ( X ) . A) 74 B) 37 C) 1 37 D) 2 37 E) 1 Ans: C

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1 8

32. Find the mean, variance, and standard deviation of the normal density function 2 1 f ( x) = e −( x − 4) 72 over ( −∞, ∞ ) . Do not use integration. 6 2π A) expected value ( mean ) : 4 variance: 36 standard deviation: 6 B) expected value ( mean ) : 36 variance: 4 standard deviation: 6 C) expected value ( mean ) : 36 variance: 6 standard deviation: 4 D) expected value ( mean ) : 6 variance: 4 standard deviation: 36 E) expected value ( mean ) : 4 variance: 6 standard deviation: 36 Ans: A 33. Let x be a random variable that is normally distributed with the given mean µ = 50 and standard deviation σ = 10 . Find the probability P ( x > 55 ) using a symbolic integration utility. Round your answer to four decimal places. A) 0.1908 B) 0.3085 C) 0.2525 D) 0.1695 E) 0.0763 Ans: B 34. Let x be a random variable that is normally distributed with the given mean µ = 48 and standard deviation σ = 8 . Find the probability P ( 30 < x < 55 ) using a symbolic integration utility. A) 0.5889 B) 0.6687 C) 0.7970 D) 0.6302 E) 0.5292 Ans: C

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35. The arrival time t of a bus at a bus stop is uniformly distributed between 08 : 00 A.M. and 08 : 08 A.M. What is the probability that you will miss the bus if you arrive at the bus stop at 08 : 02 A.M.? Round your answer to two decimal places. A) 0.25 B) 0.67 C) 0.36 D) 0.33 E) 0.57 Ans: A 36. The daily demand x for a certain product (in hundreds of pounds) is a random 3 variable with the probability density function = f ( x) x ( 4 − x ) over the interval 32 [0, 4] . Find the probability that x is within one standard deviation of the mean. Express your answer as a percent. A) 96.37 % B) 49.27 % C) 62.61 % D) 13.34 % E) 72.72 % Ans: C 37.

Find the mean and median of f ( x) = A)

mean: median: B) mean: median: C) mean: median: D) mean: median: E) mean: median: Ans: D

1 , [ 0,18] . 18

36.00 36.00 36.00 0.11 6.00 0.50 9.00 9.00 0.50 0.50

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Chapter 10: Series and Taylor Polynomials 1. Write the first five terms of the sequence. n

 2 an =  −   3 A) 2 4 8 16 32 ,− , ,− , 3 9 27 81 243 B) 2 4 8 16 32 − , ,− , ,− 3 9 27 81 243 C) 2 4 8 16 32 , , , , 3 9 27 81 243 D) 2 4 8 16 32 − ,− ,− ,− ,− 3 9 27 81 243 E) none of the above Ans: B

2. Write the first five terms of the sequence. 3 7 – n n2 A) 7 29 61 103 –5, , , , 4 9 16 25 B) 3 1 1 3 –5, – , – , , 2 3 4 5 C) 1 11 25 43 –5, , , , 4 9 16 25 D) 1 11 25 43 –5, – , , – , 4 9 16 25 E) 7 29 61 103 –5, – , , – , 4 9 16 25 Ans: A

an = 5 –

Find the limit of the sequence an = ∞ 1 0 1 2 E) The sequence diverges. Ans: C

5 . n5 2

A) B) C) D)

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4. Find the limit of the following sequence.

A) B) C) D) E) The sequence diverges. Ans: A 5. Find the limit of the following sequence. A) ∞ B) 1 C) 2 D) −∞ E) The sequence diverges. Ans: E 6. Find the limit of the following sequence.

A) –1 B) 1 C) 0 D) 2 E) The sequence diverges. Ans: C 7.

Determine the convergence or divergence of the sequence 7 −

1 . If the sequence 2n

converges, use a symbolic algebra utility to find its limit. A) 7 B) 1 C) 2 D) −∞ E) The sequence diverges. Ans: A 8. Write an expression for the nth term of the sequence 2, 8, 26, 80, .... A) an = 1 − 3n B) a= 4n + 1 n C)

a= 3n − 1 n

a= 4n − 5 n

an = 1 + 3n Ans: C

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9. Write an expression for the nth term of the sequence. 9 9 9 9, − , , − , 4 9 16 n +1 A) 9 −1 n2 B) 9(−1) n n2 C) 9(−1) n −1 n2 D) −1n −1 9n 2 E) (−1) n 9n 2 Ans: C 10.

Write an expression for the nth term of the sequence

1 6 36 216 , , , , . 5 25 125 625

6n −1 5n B) 6n +1 an = n 5 C) 6n an = 5 D) 3n +1 an = n −1 4 E) 3n −1 an = n 4 Ans: A A)

an =

11. What are the next three terms in the arithmetic sequence –5, –1,3,... ? A) 7,11,15 B) –9, –13, –17 C) 28,112, 448 D) –20, –25, –30 E) 3, 7,11 Ans: A

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12. Find the next three terms of the geometric sequence. 2, 6,18,... A) 22, 26,30,... B) 54,162, 486,... C) 30, 42,54,... D) 21, 24, 27,... E) 18,54,162,... Ans: B 13.

Give an example of a sequence that converges to

7 . 5

5n 2 − 4 7n2 − 9 B) 7 + 4n 2 an = 5 + 9n 2 C) 8n 2 − 4 an = 2 6n − 9 D) 7n2 − 4 an = 2 5n − 9 E) 8 + 4n 2 an = 6 + 9n 2 Ans: D A)

14.

an =

r  Consider the sequence (An) whose nth term is given by A= P 1 +  where P is the n  12  principal, An is the amount of compound interest after n months, and r is the annual percentage rate. Write the first four terms of the sequence for P = $ 9,500 and r = 0.04. Round your answer to two decimal places. A) 9531.67, 9573.51, 9620.26, 9627.30 B) 9532.67, 9573.51, 9620.26, 9660.34 C) 9531.67, 9563.44, 9595.32, 9627.30 D) 9532.67, 9573.51, 9595.32, 9627.30 E) 9532.67, 9563.44, 9595.32, 9660.34 Ans: C

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15. A deposit of $ 800 is made each month in an account that earns 8.4% interest, compounded monthly. The balance in the account after n months is given by An 800(201) (1.007) n − 1 . Find the balance after 20 years by computing the 240th = term of the sequence. Round your answer to two decimal places. A) $696,946.54 B) $1,018,546.54 C) $24,073.84 D) $343,954.07 E) $1,125.60 Ans: A 16.

A ball is dropped from a height of 10 feet, and on each rebound it rises to preceding height. Write an expression for the height of the nth rebound. n A) 2   5 hn =   10 n B) 2 hn = 10   5 n C) 5 hn = 10   2 D) 10 hn = n 2   5 n E)  5 hn = 10   2 Ans: B

17. Write the first five terms of the sequence of partial sums. 3 3 3 3 3 + + + + + 4 9 16 25 A) 5 147 205 15807 3, , , , 2 32 44 3200 B) 3 1 3 3 3, , , , 4 3 16 25 C) 15 49 205 5269 3, , , , 4 12 48 1200 D) 21 55 215 53 3, , , , 4 12 48 12 E) 3 135 50 315 3, , , , 2 32 11 64 Ans: C

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2 its 5

18. Find the sum of the convergent series. ∞

1 7  ∑ n =0  2  A) 14 B) 0 C) 7 D) 7 E) 1 Ans: A

19. Find the sum of the convergent series. ∞

 7 6 −  ∑ 8 n =0  A) 18 5 B) 14 5 C) 48 13 D) 16 5 E) 54 13 Ans: D

20. Determine the convergence or divergence of the following series. Use a symbolic algebra utility to verify your result.

A) The series converges. B) The series diverges. Ans: B 21.

∞

Determine the convergence or divergence of the series ∑ ( 0.100 ) . Use a symbolic n

n =0

algebra utility to verify your result. A) The series converges. B) The series diverges. Ans: A

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22.

∞

9n . Use a symbolic algebra n =1 200

Determine the convergence or divergence of the series ∑ utility to verify your result. A) The series converges. B) The series diverges. Ans: B

23. The repeating decimal 0.4 is expressed as a geometric series 0.4 + 0.04 + 0.004 + 0.0004 + ... . Write the decimal 0.4 as the ratio of two integers. A) 4 99 B) 2 11 C) 4 9 D) 4 11 E) 9 4 Ans: C

24. Express the value of the given repeating decimal as a fraction. [Hint: Write as an infinite series.] 0.30 A) 31 99 B) 31 100 C) 1 2 D) 10 33 E) 3 10 Ans: D

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25. A company produces a new product for which it estimates the annual sales to be 7000 units. Suppose that in any given year 10 % of the units (regardless of age) will become inoperative. How many units will be in use after n years? A) 70, 000 (1 − 0.9n ) B) C) D) E)

70, 000 ( 0.1n )

7000 (1 − 0.9n )

7000 ( 0.9n )

70, 000 (1 − 0.1n )

Ans: A 26. Bouncing Ball. A ball dropped from a height of 38 feet bounces to 3 4 of its former height with each bounce. Find the total vertical distance that the ball travels. A) 266 feet B) 304 feet C) 146 feet D) 104 feet E) 152 feet Ans: A 27. The annual spending by tourists in a resort city is 300 million dollars. Approximately 50% of that revenue is again spent in the resort city, and of that amount approximately 50% is again spent in the resort city. If this pattern continues, write the geometric series that gives the total amount of spending generated by the 300 million dollars (including the initial outlay of 300 million dollars) and find the sum of the series. ∞ A) The geometric series is ∑ 300(50) n +1 . n =1

The sum of the series is $ 600.00 million. B)

∞

The geometric series is ∑ 300(0.50) n . n =1

The sum of the series is $ 15,000 million. C)

∞

The geometric series is ∑ 300(0.50) n . n =0

The sum of the series is $ 600.00 million. D)

∞

The geometric series is ∑ 300(0.50) n . n =0

The sum of the series is $ 15,000 million. E)

∞

The geometric series is ∑ 300(50) n +1 . n =1

The sum of the series is $ 150.00 million. Ans: C

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28. You accept a job that pays a salary of $ 70,000 the first year. During the next 39 years, you will receive a 6% raise each year. What would be your total compensation over the 40-year period? Round your answer to the nearest integer. A) $ 10,833,338 B) $ 1,166,667 C) $ 65,800 D) $ 420,000 E) $ 4,200 Ans: A 29. A factory is polluting a river such that at every mile down river from the factory an environmental expert finds 15% less pollutant than at the preceding mile. If the pollutant’s concentration is 500 ppm (parts per million) at the factory, what is its concentration 15 miles down river? A) 75.00 ppm B) 225.00 ppm C) 43.68 ppm D) 588.24 ppm E) 51.38 ppm Ans: C 30.

∞

Determine whether the series ∑ n –2 / 3 is a p-series. n =1

∞

∑n

–2 / 3

is not a p − series.

–2 / 3

is a p − series.

n =1

∞

∑n n =1

Ans: B 31.

∞

1 . 3/ 8 n =1 n

Determine the convergence or divergence of the p-series ∑ A) The series diverges. B) The series converges. Ans: A

32. Determine the convergence or divergence of the following p-series.

A) The series converges. B) The series diverges. Ans: B

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33. Use the Ratio Test to determine the convergence or divergence of the series. ∞

5 n  ∑ n =1  3  A) Ratio Test is inconclusive B) diverges C) converges Ans: B

34. Use the Ratio Test to determine the convergence or divergence of the series. ∞

n5 ∑ n n =1 2 A) converges B) diverges C) Ratio Test is inconclusive Ans: A 35. Use the Ratio Test to determine the convergence or divergence of the series ∞ (−6) n 4n . ∑ n! n =0 A) The series converges. B) The series diverges. Ans: A 36.

∞

1 using three terms. Estimate the 3 n =1 n maximum error of your approximation. Round your answers to four decimal places. A) The approximate value is 2.1854. The maximum error of your approximation is 0.0588. B) The approximate value is 1.1620. The maximum error of your approximation is 0.0556. C) The approximate value is 2.1676. The maximum error of your approximation is 0.0721. D) The approximate value is 3.1591. The maximum error of your approximation is 1.0527. E) The approximate value is 1.1644. The maximum error of your approximation is 0.0573. Ans: B Approximate the sum of the convergent series ∑

37. Determine the convergence or divergence of the following series.

A) converges B) diverges Ans: A

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38. Determine the convergence or divergence of the series. ∞

∑ n⋅ n n =1

A) diverges B) converges C) inconclusive Ans: B 39. Determine the convergence or divergence of the series. ∞

2 ∑   n =0  3  A) inconclusive B) diverges C) converges Ans: C

40.

∞

n 3 5n for convergence or divergence using any appropriate test. n =1 n !

Test the series ∑ A) diverges B) converges Ans: B

41.

∞

Test the series ∑ n(1.9) n for convergence or divergence using any appropriate test. n =1

A) diverges B) converges Ans: A 42.

(−1) n ( x − 5) n . 7n n =1 ∞

Write the first five terms of the power series ∑

( x − 5) ( x − 5) 2 ( x − 5)3 ( x − 5) 4 ( x − 5)5 − + − + 7 72 73 74 75 B) ( x − 5) ( x − 5) 2 ( x − 5)3 ( x − 5) 4 ( x − 5)5 − − − − − 7 72 73 74 75 C) ( x − 5) ( x − 5) 2 ( x − 5)3 ( x − 5) 4 ( x − 5)5 − + − + − 7 72 73 74 75 D) ( x − 5) ( x − 5) 2 ( x − 5)3 ( x − 5) 4 ( x − 5)5 + + + + 7 72 73 74 75 E) ( x − 5) ( x − 5) 2 ( x − 5)3 ( x − 5) 4 ( x − 5)5 − − + + − 7 72 73 74 75 Ans: C A)

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43.

∞

x Find the radius of convergence of the series ∑   . n =0  7  A) 14 B) 16 C) 1 D) 8 E) 7 Ans: E

44. Find the radius of convergence of the power series. ∞

(10 x) 2 n ∑ n = 0 (2n)! A) 0 B) 10 C) 20 D) 100 E) ∞ Ans: E 45.

(−1) n +8 ( x − 1) n +8 Find the radius of convergence of the series ∑ . n+8 n =0 A) 14 B) 5 C) 1 D) 9 E) 8 Ans: C ∞

46. Find the radius of convergence of the power series. ∞ ( x − 8) n −1 ∑ 8n −1 n =1 A) –8 B) 1 C) 8 D) -1 E) 0 Ans: C

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47.

(−1) n x 4 n + 2 . 4n + 2 n =1 ∞

Find the radius of convergence of the series ∑ A) 1 B) 4 C) 6 D) 8 E) 5 Ans: A

48. Apply Taylor’s Theorem to find the power series centered at c = 3 for the function f ( x) = e x . ∞ A) ( x) n e3 ∑ n =0 n ! ∞ B) ( x − 3) n ∑ n =1 ( n + 1)! ∞ C) ( x − 3) n e3 ∑ n! n =0 ∞ D) ( x) n +1 3 e ∑ n = 0 ( n + 1)! ∞ E) ( x − 3) 2 n ∑ 2n ! n =0 Ans: C 49. Apply Taylor’s Theorem to find the power series centered at c = 0 for the function f ( x) = e 2 x . ∞ A) (2 x) n +1

∑ (n + 1)! n =0

∞

(2 x) n

∑ (n + 1)! n =0

C) D)

∞

(2 x) n ∑ n! n =0 ∞ (2 x) n +1 ∑ n! n =1

(2 x) n +1 ∑ n! n =0 Ans: C E)

∞

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50. Find the radius of convergence centered at c = 0 for the following function. A) 3 B) 2 C) 0 D) 1 E) ∞ Ans: D 51.

∞

x Find the radius of convergence of f ′( x), where f ( x) = ∑   . n =0  8  A) 1 B) ∞ C) 1 8 D) 8 E) 16 Ans: D

52. Find the power series for the function f ( x) = e x3 using the power series for e x . ∞ A) x3( n +1)

∑ (n + 1)! n =1

B) C) D)

∞

x3n ∑ n =0 n ! ∞ x3( n +1) ∑ n = 0 ( n + 1)! ∞

x3n ∑ n = 0 ( n + 1)!

x3( n +1) ∑ n! n =1 Ans: B E)

∞

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53.

Find the power series for the function f ( x) = A)

∞

∑ (−1) 2 x n +1

2x 1 using the power series for . x +1 x +1

n +1

n =0

∞

∑ (−1) 2 x n

n −1

n =0

∞

∑ (−1) 2 x n −1

n −1

n =1

∞

∑ (−1) 2 x n

n +1

n =0

∞

∑ (−1) 2 x n

n +1

n =1

Ans: D 54.

Integrate the series for

1 ( x) ln ( x + 6 ) . to find the power series for the function f= x+6

1 ∞  −1  ( x) n ∑  7 n =0  7  n + 1

(−1) n (6 x − 1) n −1 ∑ n −1 n =0

1 ∞ ( x) n +1 ∑ 6 n =0 n + 1

(−1) n (6 x − 1) n ∑ n −1 n =1

∞

1 ∞  −1  ( x) n +1 ∑  6 n =0  6  n + 1 Ans: E n

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55.

Differentiate the series for −

9 1 to find the power series for the function f ( x) = 2 . x x

∞

9∑ (−1) n +1 n( x − 1) n −1

∞

n =0

∑ (−9) n( x − 1) n

n −1

n =1

∞

9∑ (−1) n +1 n( x − 1) n −1

∞

n =1

∑ (−9) n( x − 1) n −1

∑ (−9) n( x − 1)

n +1

n =0

∞

n −1

n =0

Ans: C 56. Find the third Taylor polynomial at x = 0 for the given function.

f ( x ) = e4 x A) 64 3 1 + 8x + 8x2 + x 3 B) 64 3 1 + 4 x + 16 x 2 + x 3 C) 64 3 1 + 4 x + 8x2 + x 3 D) 32 1 + 8 x + 16 x 2 + x 3 3 E) 32 1 + 4 x + 8 x 2 + x3 3 Ans: E 57. Find the third degree Taylor polynomial centered at c = 4 for the function. f ( x) = x A) 1 1 1 2 + ( x − 4) − ( x − 4) 2 + ( x − 4)3 4 64 512 B) 1 1 1 2 − ( x − 4) + ( x − 4) 2 − ( x − 4)3 4 64 512 C) 1 1 1 2 − ( x − 4) − ( x − 4) 2 − ( x − 4)3 4 64 512 D) 1 1 1 2 + ( x + 4) − ( x + 4) 2 + ( x + 4)3 4 64 512 E) 1 1 1 2 − ( x + 4) + ( x + 4) 2 − ( x + 4)3 4 64 512 Ans: A

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58.

A Taylor polynomial approximation of f ( x) = e utility to graph both functions.

−

x2 2

is given below. Use a graphing

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Ans: A 59. Find the Taylor polynomials (centred at zero) of degree (a) 1, (b) 2, (c) 3, and (d) 4. f ( x= ) x +1 A) x x x2 x x 2 x3 x x 2 x3 S1 ( x ) =1 + , S 2 ( x ) =1 + − , S3 ( x ) =1 + − + , S 4 ( x ) =1 + − + − 2 2 8 2 8 16 2 8 16 B)

x x x2 x x 2 x3 x x 2 x3 S1 ( x ) =1 − , S 2 ( x ) =1 − + , S3 ( x ) =1 − + − , S 4 ( x ) =1 − + − + 2 2 8 2 8 16 2 8 16

x x x2 x x 2 x3 x x 2 x3 S1 ( x ) =1 + , S 2 ( x ) =1 + + , S3 ( x ) =1 + + + , S 4 ( x ) =1 + + + + 2 2 8 2 8 16 2 8 16

x x x2 x x 2 x3 x x 2 x3 S1 ( x ) =1 − , S 2 ( x ) =1 − − , S3 ( x ) =1 − − − , S 4 ( x ) =1 − − − − 2 2 8 2 8 16 2 8 16

x x x2 x x 2 x3 x x 2 x3 S1 ( x ) =1 − , S 2 ( x ) =1 + − , S3 ( x ) =1 + + + , S 4 ( x ) =1 + + + − 2 2 8 2 8 16 2 8 16

Ans: A

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60. Use a symbolic differentiation utility to find the Taylor polynomials (centred at zero) of degrees (a) 2, (b) 4, (c) 6, (d) 8. 1 f ( x) = 1 + x2 A) S 2 ( x ) =1 − x 2 , S 4 ( x ) =1 − x 2 + x 4 , S6 ( x ) =1 − x 2 + x 4 − x 6 , S8 ( x ) =1 − x 2 + x 4 − x 6 + B)

S 2 ( x ) =1 + x 2 , S 4 ( x ) =1 + x 2 + x 4 , S6 ( x ) =1 + x 2 + x 4 + x 6 , S8 ( x ) =1 + x 2 + x 4 + x 6 +

S 2 ( x ) =1 − x 2 , S 4 ( x ) =1 − x 2 − x 4 , S6 ( x ) =1 − x 2 − x 4 − x 6 , S8 ( x ) =1 − x 2 − x 4 − x 6 − x

S 2 ( x ) =1 + x 2 , S 4 ( x ) =1 + x 2 − x 4 , S6 ( x ) =1 + x 2 − x 4 + x 6 , S8 ( x ) =1 + x 2 − x 4 + x 6 −

S 2 ( x ) =1 − x 2 , S 4 ( x ) =1 − x 2 + x 4 , S6 ( x ) =1 − x 2 + x 4 + x 6 , S8 ( x ) =1 − x 2 + x 4 + x 6 +

Ans: A 61. Use a symbolic differentiation utility to find the fourth-degree Taylor polynomials (centred at zero) . 1 f ( x) = 3 x +1 A) x 2 x 2 14 x 3 35 x 4 S 4 ( x ) =1 − + − + 3 9 81 243 2 3 B) x 2 x 14 x 35 x 4 S 4 ( x ) =1 + + + + 3 9 81 243 2 3 C) x 2 x 14 x 35 x 4 S 4 ( x ) =1 − − − − 3 9 81 243 2 3 D) x 2 x 14 x 35 x 4 S 4 ( x ) =1 + − + − 3 9 81 243 2 3 E) x 2 x 14 x 35 x 4 S 4 ( x ) =1 − + + − 3 9 81 243 Ans: A 62. Find the Taylor polynomials (centred at zero) of degree (a) 1, (b) 2, (c) 3, and (d) 4. x f ( x) = x +1 A) S1 ( x ) =x, S 2 ( x ) =x − x 2 , S3 ( x ) =x − x 2 + x 3 , S 4 ( x ) =x − x 2 + x 3 − x 4 B) S1 ( x ) =x, S 2 ( x ) =x + x 2 , S3 ( x ) =x + x 2 + x 3 , S 4 ( x ) =x + x 2 + x 3 + x 4 C)

S1 ( x ) =x, S 2 ( x ) =x − x 2 , S3 ( x ) =x − x 2 − x 3 , S 4 ( x ) =x − x 2 − x 3 − x 4

S1 ( x ) =x, S 2 ( x ) =x + x 2 , S3 ( x ) =x + x 2 − x 3 , S 4 ( x ) =x + x 2 − x 3 + x 4

S1 ( x ) =x, S 2 ( x ) =x + x 2 , S3 ( x ) =x + x 2 − x 3 , S 4 ( x ) =x + x 2 + x 3 − x 4 Ans: A E)

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63. Use the fourth-degree Taylor polynomial centered at c = 0 for the function f ( x) = e − x 1 to approximate f   . Round your answer to nearest thousandth. 2 A) 0.610 B) 0.607 C) 0.732 D) 0.718 E) 0.619 Ans: B

64. Use the fifth-degree Taylor polynomial centered at c = 2 for the function f ( x) = ln x to 3 approximate f   . Round your answer to nearest ten thousandth. 2 A) 0.5165 B) 0.5267 C) 0.5067 D) 0.4055 E) 0.4306 Ans: D 65. Use the sixth-degree Taylor polynomial centered at zero for the function 5

f ( x) =

1 + x2 ten thousandth. A) 0.7620 B) 0.7720 C) 1.0170 D) 0.7670 E) 0.5170 Ans: D

to approximate the integral

∫ 1 + x . Round your answer to nearest 2

66. Determine the maximum error guaranteed by Taylor’s Theorem with Remainder when the seventh-degree Taylor polynomial is used to approximate e − x in the interval [ 0,1] centered at 0. Round your answer to five decimal places. A) 0.02002 B) 0.00202 C) 0.00020 D) 0.00002 E) 0.05002 Ans: D

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67. Complete two iterations of Newton´s Method for the function f ( x= ) x 2 − 3 using the initial guess 1.7. A) 1.732463, 1.731851 B) 1.732493, 1.732251 C) 1.732493, 1.732151 D) 1.732353, 1.732051 E) 1.732143, 1.731951 Ans: D 68. Use Newton´s Method to approximate the zero(s) of the function f ( x) = x 3 + 3 x + 2 accurate to three decimal places. A) –0.596 B) 0.603 C) 0.596 D) –0.604 E) 0.591 Ans: C 69. Use Newton's Method to approximate the zero(s) of the function f ( x) = x 5 + x + 1 accurate to three decimal places. A) 0.755 B) 0.759 C) –0.755 D) –0.759 E) 0.748 Ans: C 70. Use Newton's Method to approximate the zero(s) of the function f ( x) = x − 4 x +1 accurate to three decimal places. A) 16.944 B) 16.948 C) –16.944 D) –16.948 E) 16.938 Ans: A

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71. Use Newton's Method to approximate the x-value of the indicated point intersection of the two graphs accurate to three decimal places. f ( x= ) 3x + 1 g ( x= )

x+4

A) 0.371 B) 0.363 C) 0.356 D) 0.359 E) 0.352 Ans: B

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72. Approximate, to three decimal places, the x-value of the point of intersection of the graphs of f(x) and g(x). Round your answer to three decimal places.

A) 0.572 B) 0.562 C) 0.567 D) 0.517 E) 0.617 Ans: C 73. Use a graphing utility to approximate all the real zeros of the function f ( x) = x 3 + 3.973 x 2 + 5.235 x + 2.287 by Newton’s Method. A) –2.031, 3.236, 9.571 B) –1.271, 2.028, 5.333 C) –1.351, –1.471, –1.151 D) –1.471, 3.236, 9.707 E) –1.991, 2.970, 9.571 Ans: C

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74. The value for which Newton's method fails for the function below is shown in the graph. Give the reason why the method fails.

A) B) C) D)

f ′′ ( x2 ) = 0

E) Ans: E 75. Use Newton's Method to approximate 5 7 . Round your answer to three decimal places. A) 0.714 B) 1.476 C) 2.597 D) 1.627 E) 2.000 Ans: B 76. Use Newton's Method to find the point on the graph of f ( x)= 4 − x 2 that is closest to the point (1, 0 ) . Round your answer to three decimal places. A) (1.926, 0.291) B) C) D) E)

( 2.485, 0.291) (1.926,1.526 ) ( 0.670, 0.291) (1.926, –2.27 )

Ans: A Larson, Calculus: An Applied Approach (+Brief), 9e

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77. You are in a boat 2 miles from the nearest point on the coast (see figure). You are to go to a point Q, which is 3 miles down the coast and 1 mile inland. You can row at 8 miles per hour and walk at 9 miles per hour. Toward what point on the coast should you row in order to reach point Q in the least time? Round your answer to three decimal places.

A) approximately 2.286 miles B) approximately 2.722 miles C) approximately 1.606 miles D) approximately 1.831 miles E) approximately 2.486 miles Ans: D 78. The ordering and transportation cost C of the components used in manufacturing a x   300 product is given by C 200  2 + =  where C is measured in thousands of x + 20   x dollars and x is the order size in hundreds. Find the order size that minimizes the cost. Round your answer to the nearest unit. A) 58 hundreds B) 56 hundreds C) 53 hundreds D) 55 hundreds E) 60 hundreds Ans: B

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Chapter 11: Differential Equations 1.

Determine whether y  2 x 3 is a solution of the differential equation y 

3 y  0. x

3 2 x3   0  x B) 3 y  6 x 2 and 2 x 2   2 x 3   0 x 4 4 C) x x 3 y  and   2 x3   0 4 4 x D) 2 2 2 3 y  x and x 2   2 x 3   0 3 3 x E) 2 2 3 y  x and 2 x 2   2 x 3   0 3 x Ans: A

y  6 x 2 and 6 x 2 

2. Determine whether y  x ln x  Cx  4 is a solution of the differential equation x  y  1   y  4   0 .

y  ln x  1  C and x  y  1   y  4   x  ln x  1  C  1   x ln x  Cx  4  4   0

y  ln x  x  C and x  y  1   y  4   x  ln x  1  C  1   x ln x  Cx  4  4   0

y  ln x  1  C and x  y  1   y  4   x  ln x  x  C  1   x ln x  Cx  4  4   0

y  ln x  1  C and x  y  1   y  4   x  ln x  1  C  1   x ln x  C  4  4   0

y  ln x  x 2  C and x  y  1   y  4   x  ln x  x 2  C  1   x ln x  Cx  4  4  

Ans: A 3. Determine whether y  C1 sin x  C2 cos x is a solution of the differential equation y  y  0 . A) y  C1 sin x  C2 cos x and y  y  C1 sin x  C2 cos x  C1 sin x  C2 cos x  0 B) y  C1 sin x  C2 cos x and y  y  C1 sin x  C2 cos x  C1 sin x  C2 cos x  0 C) y  C1 sin x  C2 cos x and y  y  C1 sin x  C2 cos x  C1 sin x  C2 cos x  0 D) y  C1 sin x  C2 cos x and y  y  C1 sin x  C2 cos x  C1 sin x  C2 cos x  0 E) y  C1 sin x  C2 cos x and y  y  C1 sin x  C2 cos x  C1 sin x  C2 cos x  0 Ans: A

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4. Determine whether y  Ce4 x is a solution of the differential equation y  4 y . A) y  4Ce4 x  4 y B) y  Ce4 x  4 y C) y  4Ce3 x  4 y D) y  3Ce3 x  4 y E) Ce 4 x y   4y 4 Ans: A 5. Which of the following is a solution of the differential equation 3 y  3 y  0 ? A) – C1 sin x – C2 cos x B) e –4 x  e x C) – C1e x sin x – C2 e x cos x D) x2e x  4 x2 E)  sin x ln sec x  tan x Ans: A

6. Which of the following is a solution of the differential equation y (4)  1296 y  0 ? A) y  x 2 (6  e x ) B) y  5ln x C) y  e –6 x y  –6 x cos x D) E) y  3e –6 x  x sin x Ans: C 7. Find the particular solution of the differential equation 15 x  4 yy  0 that satisfies the initial condition y = 8 when x = 3, where 15 x 2  4 y 2  C is the general solution. A) 15 x 2  4 y 2  996 B) 15 x 2  4 y 2  391 C) 15 x 2  4 y 2  199 D) 15 x 2  4 y 2  167 E) 15 x 2  4 y 2  265 Ans: B

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8. Use integration to find a general solution of the differential equation. dy  5 x3 – 3x dx A) 5 x 4 – 3x 2  C B) 15 x 2 – 3  C C) 5 4 3 2 x – x C 4 2 2 D) 5x – 3  C E) 5 4 3 2 x – x  xC 4 2 Ans: C 9.

Use integration to find a general solution of the differential equation A) B)





22 ln 15  x 2  C 2 x 11x 2 C ln 15  x 2







x ln 15  x 2 D)

dy 11x  . dx 15  x 2

 

C



11 ln 15  x 2  C 2 E) 11 ln 26  x 2  C 2x Ans: D



10. Determine whether the function y  e2 x is a solution of the differential equation y  4  16 y  0 . A) Solution B) Not a solution Ans: A 11.

Determine whether the function y  y  4  16 y  0 . A) Solution B) Not a solution Ans: B

4 is a solution of the differential equation x

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12.

Determine whether the function y  y  3 y  2 y  0 . A) Solution B) Not a solution Ans: B

2 2 x xe is a solution of the differential equation 9

13. Determine whether the function y  xe x is a solution of the differential equation y  3 y  2 y  0 . A) Solution B) Not a solution Ans: B 14. Find the general solution satisfies the differential equation. Then find the particular solution that satisfies the initial condition. General Solution: y  Ce 2 x Differential equation: y  2 y  0 Initial condition: y  3 when x  0 A) y  2Ce 2 x , so y  2 y  0; y  3e2 x B) y  2Ce 2 x , so y  2 y  0; y  3e2 x C) y  2Ce 2 x , so y  2 y  0; y  e 2 x D) y  2Ce 2 x , so y  2 y  0; y  3e2 x E) y  2Ce 2 x , so y  2 y  0; y  e2 x Ans: A 15. Find the general solution satisfies the differential equation. Then find the particular solution that satisfies the initial condition. General Solution: y  C1  C2 ln x Differential equation: xy  y  0 Initial condition: y  5 and y  0.5 when x  1 A) 1 y  C2 1 x  and y  C2 1 x 2  , so xy  y  0; y  5  ln x 2 B) 1 y  C2 1 x  and y  C2 1 x 2  , so xy  y  0; y  5  x ln x 2 C) 1 y  C2 1 x  and y  C2 1 x 2  , so xy  y  0; y  5  ln x 2 D) 1 y  C2 1 x  and y  C2 1 x 2  , so xy  y  0; y  5  x ln x 2 E) 5 1 y  C2 1 x  and y  C2 1 x 2  , so xy  y  0; y   ln x x 2 Ans: A

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16.

Use the integration to find the general solution of the differential equation

dy  3x 2 . dx

A) y  x3  C y  6x  C B) C) y  x 3  C D) y  6 x3  C E) y  6 x  C Ans: A 17.

Use the integration to find the general solution of the differential equation A)

y  ln 1  x  C

y

dy 1  . dx 1  x

1 C 1  x 

1

1  x 

C

y  x ln x  C

y   1  x   C

Ans: A 18.

Use the integration to find the general solution of the differential equation A) B) C) D) E)

1 y  sin  4 x  4 y  4sin  4 x  1 y  sin  3x  4 1 y   sin  4 x  4 y  4sin  4 x 

Ans: A

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dy  cos 4 x . dx

19. Some of the curves corresponding to different values of C in general solution of the differential equation y 2  Cx 3 , 2 xy  3 y  0 are shown in the figure given below. Find the particular solution that passes through the points plotted on the graph.

y  5 x 2  16  17

y  x 2  16  17

y  5 x 2  16 Ans: A

20. Some of the curves corresponding to different values of C in general solution of the differential equation y  Ce x , y  y  0 are shown in the figure given below. Find the particular solution that passes through the points plotted on the graph.

A) y  3e x B) y  3e x C) y  3e  x D) y  e3 x E) y  e 3 x Ans: A

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21. Write the the differential equation by separating the variables. dy x  dx y  3 A) ( y  3)dy  xdy ( y  3)dx  xdy B) C) ( y  3)dy  xdx D) ( x  3)dy  ydx ( x  3)dy  xdy E) Ans: C 22. Write the the differential equation by separating the variables. dy 1  1 dx x A) 1  dy    1 dx y  B) 1  dx    1 dy x  xdy  1  dx C) D) 1  dy    1 dx x  E) 1  dy    1 dx x  Ans: D

23. Write the the differential equation by separating the variables. dy  x y dx A) 1  dy    1 dx y  B) 1  dx    1 dy x  C) xdy  1  dx D) No, the variables cannot be separated E) 1  dy    1 dx x  Ans: D

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24. Use separation of variables to find the general solution of the differential equation. dy  2x dx A) y  x3  C B) y  x 2  C C) y  x2  C D) y  x 3  C y  xC E) Ans: C 25. Use separation of variables to find the general solution of the differential equation. dy x  1  3 dx y A) y4  2x2  4x  C B) y 4  x2  x  C C) y  2x2  4x  C D) y  2 x 2  4 x  C E) y  2 x2  4 x  C Ans: A 26. Use separation of variables to find the general solution of the differential equation. dy 3 y2 1 dx A) y  xC B) dy 3  xC dx C) y  3 xC D) y  3 x2  C E)

dy  ( x  C )3 dx Ans: C

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27. Use separation of variables to find the general solution of the differential equation. dy x2  4 y 0 dx A) 1 y  x3  C 6 B) dy 1   xC 6 dx C) 1 y 2   x3  C 6 D) dy 1   x3  C 6 dx E) 1 y2  x  C 3 Ans: C 28. Use separation of variables to find the general solution of the differential equation. y ' xy  0 2 A) y  Ce x B)

2 dy  Ce x / 2 dx 2 C) y  Ce x / 2 2 D) dy  Ce x dx 2 E) dy  e x / 2 Ans: C

29. Use separation of variables to find the general solution of the differential equation. dy ey  3t 2  1 dt A) y  In t 2  t  C B)

y  In t 3  t  C

y  In t 3  t  C

y  In t 2  t  C

y  In t 2  t  C

Ans: B

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30. Use separation of variables to find the general solution of the differential equation. dy  1 y dx A) x  y  1  C   2  B) x  y  1  C   2  C) y  1  C  x  D)

y  1  C  x 

x  y  1  C   2  Ans: E

31. Use separation of variables to find the general solution of the differential equation. (2  x) y '  2 y y  C (2  x) A) B) y  C (2  x) 2 C) y  C (2  x) 2 y  C (2  x) D) E) y  C (4  x) 2 Ans: C 32. Use separation of variables to find the general solution of the differential equation. dy y  sin x dx A) y  2sin x  C B) y  cos x  C C) y 2  2 cos x  C D) y 2  2 cos x  C E) y 2  cos x  C Ans: C

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33. Find the general solution of the differential equation dy x dx A) 1 y  x C 2 B) 1 y  x2  C 2 C) y  x2  C D) 1 y  C  x2 2 E) y  C  x2 Ans: B 34. The isotope 14 C has a half-life of 5,715 years. Given an initial amount of 11 grams of the isotope, how many grams will remain after 1,000 years? After 10,000 years? Round your answers to four decimal places. A) 6.8205 gm, 2.2896 gm B) 3.8974 gm, 1.3083 gm C) 9.7436 gm, 3.2708 gm D) 11.6923 gm, 3.9250 gm E) 5.8462 gm, 1.9625 gm Ans: C 35. Use the initial condition to find the particular solution of the differential equation. yy ' e x  0 , y  4 when x  0 A) y 2  e x  12 B) y 2  2e x  14 C) y 2  2e x  14 D) y 2  2e 2 x  14 E) y 2  2 ye x  14 Ans: C 36. Use the initial condition to find the particular solution of the differential equation. x  y  4   y '  0 , y  5 when x  0 A)

y  4  e  x / 2

y  4  e x / 2

y  4  e  x / 2

y  4  e x / 2

y  4  ex Ans: A

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37. Use the initial condition to find the particular solution of the differential equation. x 2  16 y '  5 x , y  2 when x  5 A) y  5 x 2  17  16

y  5 x  17  16

y  5 x 2  16  17

y  5 x  16  17

y  2 x 2  16  17 Ans: C

38. Use the initial condition to find the particular solution of the differential equation. dy  y cos x , y  1 when x  0 dx A) y  e 2cos x B) y  esin x C) y  e 2sin x D) y  ecos x E) y  e  sin x Ans: B

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39. Find an equation of the graph that passes through the point and has the specified slope. Then graph the equation. Point:  1,1 ,

Slope: y '   A)

9x 16 y

16 y 2  9 x 2  25 2.5

2 1.5 1 0.5

-2.5

-2

-1.5

-1

-0.5 -0.5

x 0.5

1.5

2.5

-1 -1.5 -2 -2.5

9 y 2  16 x 2  25

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2.5

2 1.5 1 0.5

-2.5

-2

-1.5

-1

-0.5 -0.5

0.5

1.5

2.5

-1 -1.5 -2 -2.5

16 y 2  9 x 2  25 2.5

2 1.5 1 0.5

-2.5

-2

-1.5

-1

-0.5 -0.5

x 0.5

1.5

2.5

-1 -1.5 -2 -2.5

9 y 2  16 x 2  25

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2.5

2 1.5 1 0.5

-2.5

-2

-1.5

-1

-0.5 -0.5

x 0.5

1.5

2.5

-1 -1.5 -2 -2.5

E) None of the Above Ans: A 40. Solve the differential equation to find velocity v as a function of time t if v  0 when t  0 The differential equation models the motion of two people on a toboggan after consideration of the forces of gravity, friction, and air resistance. dv 12.5  43.2  1.25v dt A) v  34.56 1  e  t  B) C) D) E)

v  24.56 1  e 0.1 

v  34.25 1  e 0.1t 

v  24.56 1  e 0.1t 

v  34.56 1  e 0.1t 

Ans: E

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41. Write the first-order linear differential equation in standard form. x3  2 x 2 y  3 y  0 A) x 3 y  2 y  2x 2 B) x 3 y  y 2x 2 C) 3 y  y  x 2x D) 3 x y  y 2x 2 E) 3 y  2 y   x 2x Ans: A 42. Write the first-order linear differential equation in standard form. xy  y  xe x A) 1 y  y  e x 2 B) y  xy  e  x C) 1 y  y  e x x D) 1 y  xy  e x 2 E) 1 y  y  2e  x 2 Ans: C 43. Write the first-order linear differential equation in standard form.

y  1  ( x  1) y A) 1 2 y  y 1 x 1 x B) 1 1 y  y 1 x x 1 C) 1 2 y  y 1 x x 1 D) 1 1 y  y 1 x x 1 E) 1 1 y  y 1 x x 1 Ans: E

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44. Find the general solution of the first-order linear differential equation. dy  3y  6 dx A) y  1  Ce 3 x B) y  2  Ce 3 x C) y  2  Ce3 x D) y  1  Ce3 x E) y  3  Ce3 x Ans: B 45. Find the general solution of the first-order linear differential equation. dy  y  e4 x dx A) 1 y  (e 3 x  C ) 3 B) 1 3x y  (e  C ) 3 C) 1 y   (e 3 x  C ) 3 D) 1 y  e x (e3 x  C ) 3 E) 1 y   (e3 x  C ) 3 Ans: D 46. Find the general solution of the first-order linear differential equation. dy x 2  3  dx x A) 1 1 y  x  ln x  C 2 3 B) 1 1 y   x  ln x  C 2 3 C) 1 y  x  3ln x  C 2 D) 1 y  x 2  3ln x  C 2 E) 1 2 y  x  3ln x  C 2 Ans: D

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47. Find the general solution of the first-order linear differential equation. y  2 xy  10 x 2 2 A) 5e  x (e x  C ) B)

5e x (e x  C )

10e x (e  x  C )

10e x (e  x  C )

5e  x (e  x  C ) Ans: A

48. Find the general solution of the first-order linear differential equation. ( x  1) y  y  x 2  1 A) x2  2 x  C y ( x  1) 2 B) x  2x  C y 3( x  1) 3 C) x  3x  C y 3( x  1) D) x 2  3x  C y 3( x  1) 3 E) x  3x  C y 3( x  1) Ans: C 49. Find the general solution of the first-order linear differential equation. 1

x3 y  2 y  e x 1 A) 1 2   y  ex  2  C   2x  1 B) 1 2   y  ex   2  C   2x  1 C)  2  1  y  e x  2 C  2x  1 D) 1 2   y  ex  3  C   2x  1 E) 1 2   y  ex   3  C   2x  Ans: B 2

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50.

dw  k (1100  w), where dt w is weight in pounds and t is time in years. Use a computer algebra system to solve the differential equation for k  1.2 . A) w  1100et  1020e 1.2t B) w  80e 1.2t C) w  1020  1.2e t D) w  1100  1020e 1.2t E) w  1020e 1.2t Ans: D A calf that weighs 80 pounds at birth gains weight at the rate

51. Find the particular solution of the differential equation y  (2 x – 4) y  0 that satisfies the boundary condition y  4   2 . y  2e 4 x – x

y  2e –4 x + 2 x

y  2e 4 x + 2 x

y  4e –4 x – x

y  4e 4 x – x Ans: A 52.

Find the particular solution of the differential equation

dy  5 x3 y  x 3 passing through dx

 13  the point  0,  .  2 4 A) 1 63 y   e –1.25 x 5 10 4 B) 1 3 y   e –1.25 x 5 10 4 C) 1 67 y   e –1.25 x 5 10 4 D) 1 63 y   e –1.25 x 10 10 4 E) 1 67 y   e –1.25 x 10 10 Ans: A

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53. Solve for y in two ways.

y  y  4 A) y  Ce  x  4 B) y  Ce x  4 C) y  Ce x  4 D) y  Ce  x  4 E) y  Ce x  4e  x Ans: A 54. Solve for y in two ways. y  2 xy  2 x 2 A) y  Ce x  2 B)

y  Ce  x  1

y  Ce x  1

y  Ce  x  2

y  Ce  x  2 Ans: C

55. Find the solution of the differential equation y  2 x  0 , without solving it. A) y  2 x 2  C B) y  2 x2  C y  2x  C C) D) y  x2  C y  2 x  C E) Ans: D 56. Find the solution of the differential equation y  2 xy  0 , without solving it. 2 A) 2 x  Ce x 2 B) 2 x  Ce x C) Ce x 2 D) x 2  Ce x 2 E) Ce x Ans: E

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57. Find the particular solution that satisfies the initial condition. y   y  6e x ; Initial Condition: y  3 when x  0 A) y  e x B) y  2e x C) y  2e  x D) y  3e x E) y  3e  x Ans: D 58. Find the particular solution that satisfies the initial condition.

xy  y  0 ; Initial Condition: y  2 when x  2 A) x  2 y B) xy  4 C) xy  2 xy  2 D) E) x  4 y Ans: B 59. Find the particular solution that satisfies the initial condition. y  3 x 2 y  3x 2 ; Initial Condition: y  6 when x  0 2 A) y  3  3e x B)

y  1  5e  x

y  1  3e  x

y  3  3e  x Ans: B

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60. Find the particular solution that satisfies the initial condition. xy  2 y  3 x 2  5 x ; Initial Condition: y  3 when x  1 A) 3 5 5 y  x2  x  4 4 12 x 2 B) 3 5 1 y  x2  x  4 4 12 x 2 C) 3 5 7 y  x2  x  4 4 12 x 2 D) 3 5 7 y  x2  x  4 3 12 x 2 E) 3 5 1 y  x2  x  4 4 12 x Ans: D

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61. Use the chemical reaction model described to find the amount y (in grams) as a function of time t (in hours). Then use a graphing utility to graph the function. y  45 grams when t  0 ; y  4 grams when t  2

y

360 8  4t

y

350 8  4t

y

360 7  3t

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y

360 8  2t

y

350 8  5t

Ans: A

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62. Use the advertising awareness model described to find the number of people y (in millions) aware of the product as a function of time t (in years). y  0 when t  0 ; y  0.75 when t 1

A) y  1  e 1.386 B) y  1  e1.386 C) y  1  e 1.386t D) y  1  e1.386t E) y  1  e 1.386t Ans: E

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63. Use the Gompertz growth model described to find the population y as a function of time t (in years). y 100 when t  0 ; y 150 when t  2

y  200e0.6931e

y  200e0.6931

y  200e 0.6931e

y  20e0.6931e

0.4397 t

0.4397

0.4397 t

0.4397

y  2000e 0.6931e Ans: C

0.4397 t

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64. Use the hybrid selection model described to find the percent y (in decimal form) of the population that has the indicated characteristics as a function of time t (in generations). y  0.1 when t  0 ; y  0.4 when t  4

y (2  y ) 19 0.50634t  e (1  y ) 2 81 B) y (1  y ) 18 0.50634t  e (1  y ) 2 81 C) y (2  y ) 19 0.1t  e (1  y ) 2 81 D) y (1  y ) 19 0.50634t  e (2  y ) 2 81 E) y (2  y ) 81 0.50634t  e (1  y ) 2 19 Ans: A A)

65. Assume that the rate of change in y is proportional to y. Solve the resulting differential equation dy / dx  ky and find the particular solution that passes through the points (0,1), (3, 2) A) y  x ( x ln 2) / 8  e0.2310 x B) y  e( x ln 2) /10  e0.2310 y C) y  e( y ln10) / 3  e0.2310 x D) y  e( x ln 2) / 3  e0.2310 x E) y  e( x ln 2)  e0.2310 Ans: D

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66. Assume that the rate of change in y is proportional to y. Solve the resulting differential equation dy / dx  ky and find the particular solution that passes through the points (0, 4), (4,1) A) y  4e  ( x ln 4) /10  40.3466 x B) y  e  ( x ln 4) / 4  4e 0.3466 x C) y  4e  ( x ln 4) / 4  4e 0.3466 x D) y  4e( y ln 4) / 4  4e 0.3466 x E) y  2e  ( x ln 4) / 4  8e 0.3466 x Ans: C 67. Assume that the rate of change in y is proportional to y. Solve the resulting differential equation dy / dx  ky and find the particular solution that passes through the points (2, 2), (3, 4) A) 1 1 y  e(ln 2) x  e0.6931x 3 3 B) 1 (ln 2) x 1 0.6931x y e  e 2 2 C) 1 1 y  e(ln 2) x  e0.6931x 4 4 D) 1 1 y  e(ln10) x  e0.6931x 4 4 E) 1 (ln10) x 1 0.6931x y e  e 2 2 Ans: B 68. During a chemical reaction, a compound changes into another compound at a rate proportional to the unchanged amount y . Write the differential equation for the chemical reaction model. Find the particular solution when the initial amount of the original compound is 20 grams and the amount remaining after 1 hour is 16 grams. A) dy  y, y  200.2231t dt B) dx  kx, y  20e 0.2231t dt C) dy  k , y  2e0.2231t dt D) dy  ky, y  20e 0.2231t dt E) dy  ky, y  20e dt Ans: D

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69. The rate of change of the population of a city is proportional to the population P at any time (in years). In 2000, the population was 200,000, and the constant of proportionality was 0.015. Estimate the population of the city in the year 2020. A) 260,972 people B) 268,972 people C) 263,972 people D) 266,972 people E) 269,972 people Ans: E 70. A wet towel hung from a clothesline to dry loses moisture through evaporation at a rate proportional to its moisture content. After 1 hour, the towel has lost 40% of its original moisture content. How long will it take the towel to lose 80% of its original moisture content? A)  2.15 h B)  4.95 h C)  3.95 h D)  3.15 h E)  4.15 h Ans: D 71. The rate of change in sales S (in thousands of units) of a new product is proportional to the difference between L and S at any time t (in years), where L is the maximum number of units of the new product available. When t  0, S  0 Write and solve the differential equation for this sales model. A) S  L(1  e  kt ) B) S  L(1  e  k ) C) S  (1  e  kt ) D) L  S (1  e  kt ) E) L  S (1  e  k ) Ans: A 72. A population of eight beavers has been introduced into a new wetlands area. Biologists estimate that the maximum population the wetlands can sustain is 60 beavers. After 3 years, the population is 15 beavers. The population follows a Gompertz growth model. How many beavers will there be in the wetlands after 10 years? A) 34 beavers B) 30 beavers C) 33 beavers D) 31 beavers E) 39 beavers Ans: A

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73. At any time (in years), the rate of growth of the population N of deer in a state park is proportional to the product of N and L  N , where L  500 is the maximum number of deer the park can maintain. (a) Use a symbolic integration utility to find the general solution. (b) Find the particular solution given the conditions N  100 when t  0 and N  200 when t  4 (c) Find N when t  1 (d) Find t when N  350 A) 50 a. N  1  Ce  kt 500 b. N  1  4e 0.2452t c. 121 deer d.  9.1yr B) 500 a. N  1  Ce  kt 500 b. N  1  4e 0.2452t c. 121 deer d.  9.1yr C) 500 a. N  1  Ce  kt 50 b. N  1  4e 0.2452t c. 121 deer d.  9.1yr D) 500 a. N  1  Ce  kt 500 b. N  1  4e 0.2452t c. 131 deer d.  9.1yr E) 500 a. N  1  Ce  kt 500 b. N  1  4e 0.2452t c. 121 deer d.  10.1yr Ans: B

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74. A 100-gallon tank is full of a solution containing 25 pounds of a concentrate. Starting at time t  0 distilled water is admitted to the tank at the rate of 5 gallons per minute, and the well-stirred solution is withdrawn at the same rate. (a) Find the amount Q of the concentrate in the solution as a function of by solving the  Q  differential equation Q '  5    100  (b) Find the time required for the amount of concentrate in the tank to reach 15 pounds. A) a. Q  25e 0.05t b.  1.22 min B) a. Q  15e 0.05t b.  10.22 min C) a. Q  25e 0.05t b.  8.22 min D) a. Q  5e 0.05t b.  10.22 min E) a. Q  25e 0.05t b.  10.22 min Ans: E 75. When predicting population growth, demographers must consider birth and death rates as well as the net change caused by the difference between the rates of immigration and emigration. Let P be the population at time t and let N be the net increase per unit time due to the difference between immigration and emigration. So, the rate of growth dP of the population is given by  kP  N , N is constant. Solve the differential equation dt to find P as a function of t . A) N p  Ce kt  k B) N p  Ce kt k C) N p  e kt  k D) N p  Ce kt  k E) N p  Ce  k Ans: A

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76. A large corporation starts at time t  0 to invest part of its profit at a rate of P dollars per year in a fund for future expansion. Assume that the fund earns r percent interest per year compounded continuously. The rate of growth of the amount A in the fund is dA given by  rA  P where A  0 when t  0 and r is in decimal form. Solve this dt differential equation for A as a function of t . A) P A   (e rt  1) r B) P A   (ert  1) r C) P rt A  (e  1) r D) P A  (ert  1) r E) P A  (e  1) r Ans: D 77.

P rt (e  1) and Find A for each situation. r (a) P  $100, 000 , r  0.12 and t  5 years. (b) P  $250, 000 , r  0.15 and t  10 years. A) a. 685099.00 b. 5802815.12 B) a. 585099.00 b. 5802815.12 C) a. 685099.00 b. 6802815.12 D) a. 585099.00 b. 4802815.12 E) a. 485099.00 b. 5802815.12 Ans: A

78.

P rt (e  1) and Find P if the corporation needs $120,000,000 in r 8 years and the fund earns 8% interest compounded continuously. A) $110,708,538.49 B) $10,500,538.49 C) $10,708,538.49 D) $100,708,538.49 E) $1,708,538.49 Ans: C

Use this equation A 

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79. A medical researcher wants to determine the concentration C (in moles per liter) of a tracer drug injected into a moving fluid with flow R (in liters per minute). Solve this problem by considering a single-compartment dilution model (see figure). Assume that the fluid is continuously mixed and that the volume V (in liters) of fluid in the compartment is constant.

If the tracer is injected instantaneously at time t  0 , then the concentration of the fluid in the compartment begins diluting according to the differential equation dC R     C , C  C0 when t  0 dt V  (a) Solve this differential equation to find the concentration as a function of time. (b) Find the limit of C as t   A) a. C  C0 e Rt / V b. 1 B) a. C  C0 e  Rt / V b. 0 C) a. C  e  Rt / V b. 2 D) a. C  C0e  Rt b. 0 E) a. C  C0  Rt /V b. 1 Ans: B 80. A 300-gallon tank is full of a solution containing 35 pounds of concentrate. Starting at time t  0, distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the amount of concentrate Q in the solution as a function of t. A) Q  35e 1/10t B) Q  30e 1/10t C)

Q  e1/ 30t

Q  e 1/ 35t

Q  300e 1/10t Ans: A

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