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Chapter 3 1. Find all intervals where the derivative of the function shown below is negative. Ans: 2 < x < 3 Difficulty: easy Section: 3.1 2. Find all intervals where the derivative of the function shown below is negative. Ans: –5 < x < 0 and 0 < x < 3 Difficulty: easy Section: 3.1 Page 54 Chapter 3 3. The displacement of an object is s(t ) t 4 t 3 2 . Is the velocity increasing or decreasing at t = 5? Ans: Increasing Difficulty: hard Section: 3.1 4. A company has profit defined by p(t ) 4t 2 30t . Does the profit increase or decrease at t = 2? Ans: Decreases Difficulty: hard Section: 3.1 5. Find the intervals of increase and decrease for the function f ( x) x2 1 . Ans: Decreasing for x < 0; increasing for x > 0 Difficulty: hard Section: 3.1 6. x 1 Find the intervals of increase and decrease for the function f ( x) . x2 3 Ans: Decreasing for x < –1 and x > 3; increasing for –1 < x < 3 Difficulty: hard Section: 3.1 7. Find the intervals of increase and decrease for the function f ( x) x5 x3 4 x 10 . Ans: Decreasing for all x Difficulty: moderate Section: 3.1 8. x 1 Find all critical numbers of the function f ( x) . x2 3 Ans: –1; 3 Difficulty: hard Section: 3.1 9. Find all critical numbers of the function f ( x) x 3 3x 4 . Ans: 0; 1/4 Difficulty: easy Section: 3.1 10. An open box with square base and vertical sides is constructed out of 100 m 2 of tin. What should the dimensions of the box be if its volume is to be as large as possible? Ans: 10 3 10 3 5 3 ; ; 3 3 3 Difficulty: hard Section: 3.1 11. Find all the critical numbers of the function f ( x) 2 x 4 4 x 2 1 . 1 A) 0, 1, –1 B) –1 C) D) None 2 Ans: A Difficulty: moderate Section: 3.1 Page 55 Chapter 3 12. Find all the critical numbers of the function f ( x) x3 12 x 5 . A) None B) –2, 2 C) 0, –2, 2 D) 3 5 Ans: B Difficulty: moderate Section: 3.1 13. Find all the critical numbers of the function f ( x) 2 x 2 8 x 7 . 7 A) –7 B) C) 2 D) None 2 Ans: C Difficulty: moderate Section: 3.1 14. Find the intervals of increase and decrease for the function f ( x) x 2 5 x 3 . A) 5 5 Decreasing for x ; increasing for x 2 2 B) 5 5 Decreasing for x ; increasing for x 2 2 C) Decreasing for all x D) Increasing for all x Ans: A Difficulty: moderate Section: 3.1 15. Find the intervals of increase and decrease for f ( x) 6 x3 54 x 2 288 x 7 . A) Increasing on x –2 and x 8 , decreasing on – 2 x 8 B) Increasing on x < –8 and x > 2, decreasing on –8 < x < 2 C) Increasing on –8 < x < 2, decreasing on x < –8 and x > 2 D) Increasing on x < –2 and x > 8, decreasing on –2 < x < 8 Ans: B Difficulty: moderate Section: 3.1 16. Determine the critical points of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. f ( x) 6 x 4 16 x3 12 x 2 1 A) (0, 1) relative minimum; (1, 3) neither B) (1, 3) relative minimum; (0, 2) neither C) (0, 2) relative minimum; (1, 4) neither D) (1, 3) relative maximum; (0, 2) relative minimum Ans: A Difficulty: moderate Section: 3.1 17. The revenue derived from the production of x units of a particular commodity is 80 x x 2 R( x) 2 million dollars. What level of production results in maximum x 80 revenue? What is the maximum revenue? A) Maximum at x = 8 and maximum revenue is R(8) = 32 (million dollars) B) Maximum at x = 8 and maximum revenue is R(8) = 2.67 (thousand dollars) C) Maximum at x = 8 and maximum revenue is R(8) = 4 (million dollars) D) Maximum at x = 9 and maximum revenue is R(9) = 2.67 (million dollars) Ans: C Difficulty: moderate Section: 3.1 Page 56 Chapter 3 18. Find constants a, b, and c so that the graph of the function f ( x) ax 2 bx c has a relative maximum at (3, 7) and crosses the y-axis at (0, 1). A) 2 C) 2 a , b = –4, c = –1 a , b = 4, c = 1 3 3 B) 2 D) 2 a , b = 4, c = 1 a , b = –4, c = 1 3 3 Ans: B Difficulty: moderate Section: 3.1 19. 8x 3 Find the intervals of increase and decrease for f ( x) . 2 x 10 A) Increasing on x < 5.00, decreasing on x > 5.00 B) Increasing on x < 5.00 and x > 5.00 C) Increasing on x 0.38 and on x > 5.00, decreasing on 0.38 < x 5.00 D) Increasing on 0.38 < x 5.00 , decreasing on x 0.38 and on x > 5.00 Ans: B Difficulty: moderate Section: 3.1 20. Determine the critical points of the given function and classify each critical point as a 8 relative maximum, a relative minimum, or neither. f ( x) 2 x 9 x 20 A) (4.5, –32) relative maximum; x = 5 neither; x = 4 neither, B) (4.5, –32) relative minimum; (5, 8) relative maximum; (4, 8) relative maximum; C) (5, 8) relative maximum; (4, 8) relative maximum D) (5, 8) relative minimum; (4, 8) relative maximum Ans: A Difficulty: moderate Section: 3.1 21. Two ships are anchored in the water along a straight shoreline. Ship A is 8 miles from shore, and 12 miles down the shoreline ship B is 16 miles from shore. A small boat must travel from the first boat to the second, making a stop on land between the ships. Assume the shoreline corresponds with the x-axis and ship A lies along the line x = 0, while ship B lies along x = 12. Assume the small boat touches land at point x. Produce a function that gives the distance the small boat travels as a function of x. What are the critical values of x for this function? A) x = –4, x = 4 C) x = –8, x = 8 B) x = –12, x = 4 D) x = 0, x = 4, x = 8, x = 12 Ans: B Difficulty: moderate Section: 3.1 22. Find all the critical numbers of the function f ( x) x3 48 x 2 A) None B) –4, 4 C) 0, –4, 4 D) 3 2 Ans: B Difficulty: moderate Section: 3.1 23. Find all the critical numbers of the function f ( x) 5 x 2 5 x 3 A) –3 B) 0.50 C) –3 / 5 D) None Ans: B Difficulty: moderate Section: 3.1 Page 57 Chapter 3 24. Find the intervals of increase and decrease for the function f ( x) x 2 7 x 8 A) Decreasing for x 27 and increasing for x 27 B) Decreasing for x 27 and increasing for x 27 C) Decreasing for all x D) Increasing for all x Ans: B Difficulty: moderate Section: 3.1 25. Find all critical points of the function f (t ) t 2 2t 2 . Ans: (1, 1) Difficulty: hard Section: 3.1 26. A truck is 250 kilometers due east of a car and is traveling west at a constant speed of 60 km/hr. Meanwhile, the car is going north at 80 km/hr. In how many hours will the truck and the car be closest to each other? Ans: 1.5 Difficulty: hard Section: 3.1 27. The displacement of an object is s(t ) t 5 t 3 9 . Is the velocity increasing or decreasing at t = 2? Ans: Increasing. Difficulty: hard Section: 3.1 28. A company has profit defined by p(t ) 3t 2 50t . Does the profit increase or decrease at time t = 2? Ans: The profit decreases. Difficulty: hard Section: 3.1 29. Find all critical numbers of the function f (t ) t 2 2t 6 . Ans: 1 Difficulty: hard Section: 3.1 30. Find all critical numbers of the function f ( x) x 4 4 x5 . Ans: 0, 15 Difficulty: easy Section: 3.1 31. Find the intervals of increase and decrease for the function f ( x) x2 6 . Ans: Decreasing for x < 0, increasing for x > 0. Difficulty: hard Section: 3.1 Page 58 Chapter 3 32. Find the intervals of increase and decrease for the function f ( x) x 5 . x2 21 Ans: Decreasing for x < 3 and x > 7, increasing for 3 < x < 7. Difficulty: hard Section: 3.1 33. Find the intervals of increase and decrease for the function f ( x) x 7 x3 4 x 10 . Ans: Decreasing for all x. Difficulty: hard Section: 3.1 34. True or False: The function shown below is decreasing on the interval shown. A) True B) False Ans: B Difficulty: easy Section: 3.1 Page 59 Chapter 3 35. True or False: The function graphed below has a positive second derivative everywhere. A) True B) False Ans: B Difficulty: hard Section: 3.1 36. True or False: The first derivative of a function is graphed below. The function has a relative maximum. A) True B) False Ans: B Difficulty: moderate Section: 3.1 Page 60 Chapter 3 37. True or False: The first derivative of a function is graphed below. The function has at least one relative extrema. A) True B) False Ans: B Difficulty: moderate Section: 3.1 38. True or False: The first derivative of a function is graphed below. The function is increasing everywhere. A) True B) False Ans: B Difficulty: hard Section: 3.1 Page 61 Chapter 3 39. 2x True or False: The absolute maximum of the function f ( x) on the interval x 1 0 x 1 is 1. A) True B) False Ans: A Difficulty: moderate Section: 3.1 40. True or False: If f and g are increasing on an interval I, then f – g is also increasing on I. A) True B) False Ans: B Difficulty: easy Section: 3.1 41. True or False: If f is decreasing on an interval I, then –f is increasing on I. A) True B) False Ans: A Difficulty: easy Section: 3.1 42. True or False: If f (0) 0 , f 1 0 , and f 1 0 , then f has a relative 2 2 minimum at the point where x = 0. A) True B) False Ans: B Difficulty: hard Section: 3.1 43. A small manufacturing company estimates that the total cost in dollars of producing x radios per day is given by the formula C 0.1x 2 20 x 500 . Find the number of units that will minimize the average cost. A) 100 B) 147 C) 36 D) 71 Ans: D Difficulty: moderate Section: 3.1 44. True or False: If f and g are both differentiable and f + g has a relative extreme at x = a, then f (a) g (a) 0 . A) True B) False Ans: B Difficulty: easy Section: 3.1 Page 62 Chapter 3 45. Determine where the second derivative in the function graphed below is positive. Ans: x < 0 Difficulty: hard Section: 3.2 46. Determine where the second derivative in the function graphed below is positive. Ans: x < 0 Difficulty: hard Section: 3.2 47. Determine where the graph of f ( x) x 4 6 x3 24 x 2 26 is concave up and concave down. Ans: Concave up for x < –4 and x > 1; concave down for –4 < x < 1 Difficulty: hard Section: 3.2 Page 63 Chapter 3 48. Determine where the graph of f ( x) x 2 / 3 is concave up and concave down. Ans: Concave up for all x Difficulty: moderate Section: 3.2 49. 1 Determine where the graph of g ( x) is concave up and concave down. x 3 2 Ans: Concave up for x < –1 and x > 1; concave down for –1 < x < 1 Difficulty: hard Section: 3.2 50. Let f ( x) 2 x3 3x 2 12 x 13 . Find all critical points of f and use the second derivative test to classify each as a relative maximum, a relative minimum, or neither. Ans: Maximum at (–1, 20); minimum at (2, –7) Difficulty: moderate Section: 3.2 51. Find all critical points of f ( x) 3x 4 2 x3 12 x 2 18 x , and use the second derivative test to classify each as a relative maximum, a relative minimum, or neither. Ans: 3 513 Relative minimum at ( , ) ; point of inflection at (1, 7) 2 16 Difficulty: hard Section: 3.2 52. The second derivative test reveals that f ( x) x 4 4 x 2 1 has A) a relative maximum only B) a relative minimum only C) both a relative maximum and a relative minimum D) neither a relative maximum nor a relative minimum Ans: C Difficulty: easy Section: 3.2 53. The second derivative test reveals that f ( x) x 2 3 has A) a relative maximum at x 3 C) a relative minimum at x = 0 B) a point of inflection at x 3 D) nothing significant at x = 3 Ans: C Difficulty: easy Section: 3.2 54. Determine where the graph of f ( x) x 4 6 x 2 is concave up. A) for x < –1 and x > 1 B) for –1 < x < 1 C) Everywhere D) Nowhere Ans: A Difficulty: hard Section: 3.2 55. Determine where the graph of f ( x) x3 3x 2 9 x 1 is concave down. A) x > 1 B) x < 1 C) x > –1 D) x < –1 Ans: B Difficulty: hard Section: 3.2 Page 64 Chapter 3 56. x Locate all inflection points of f ( x) . x 1 2 A) 3 3 C) (0, 0) 3, , (0, 0), 3, 4 4 0, 0 , 1, , 1, 0, 0 , 1, B) 1 1 D) 1 2 2 2 Ans: A Difficulty: moderate Section: 3.2 57. Locate all inflection points of f ( x) x 4 6 x3 24 x 2 26 . A) (1, 9) and (–4, –486) B) (1, 9) C) None D) (0, 26) Ans: A Difficulty: moderate Section: 3.2 58. Use the second derivative test to find the relative maxima and minima of the function f ( x) 10 x3 45 x 2 1200 x 3 . A) Relative maximum at (5, 7,363); relative minimum at (–8, –3,622) B) Relative maximum at (–8, –3,622); relative minimum at (–5, 5,878) C) Relative maximum at (–8, –3,621); relative minimum at (5, 7,363) D) Relative maximum at (–8, 7,363); relative minimum at (5, –3,622) Ans: D Difficulty: moderate Section: 3.2 59. A 5-year projection of population trends suggests that t years from now, the population of a certain community will be P(t ) t 3 12t 2 144t 55 thousand. 1) At what time during the 5-year period will the population be growing most rapidly? 2) At what time during the 5-year period will the population be growing least rapidly? 3) At what time is the rate of population growth changing most rapidly? A) t = 4 years; t = 0 years; t = 0 years C) t = 4 years; t = 3 year; t = 5 years B) t = 0 years; t = 0 years; t = 4 years D) t = 4 years; t = 0 years; t = 4 years Ans: A Difficulty: moderate Section: 3.2 60. A local tavern expects to use 1,200 bottles of Golden Globe wine this year. Each bottle costs the tavern $ 4. The ordering cost is $ 3 per shipment, and the cost of storage is 25 cents per bottle per year. The wine is consumed at a constant rate throughout the year, and each shipment arrives just as the preceding shipment has been used up. Let x be the 1, 200 1 number of shipments per year. bottles will be used over a time period of x x year. Determine a function that represents the total cost as a function of x. Over what positive interval is the function increasing? What does the second derivative test tell you about the function value at the critical point? A) x > 3.53, minimum at critical point C) x < 1.18, maximum at critical point B) x > 7.07, minimum at critical point D) x > 1.77, minimum at critical point Ans: B Difficulty: moderate Section: 3.2 Page 65 Chapter 3 61. The second derivative test reveals that f ( x) x 4 2 x 2 6 has: A) both a relative maximum and a relative minimum B) a relative maximum only C) a relative minimum only D) neither a relative maximum nor a relative minimum Ans: A Difficulty: easy Section: 3.2 62. The second derivative test reveals that f(x) x2 6 has: A) a relative maximum at x 6 C) a relative minimum at x = 0 B) a point of inflection at x 6 D) nothing significant at x = 6 Ans: C Difficulty: easy Section: 3.2 63. Determine where the graph of f ( x) x 4 24 x 2 8 is concave up. A) For x < –2 and x > 2 B) For –2 < x < 2 C) Everywhere D) Nowhere Ans: A Difficulty: hard Section: 3.2 64. Determine where the graph of f ( x) x3 3x 2 2 x 4 is concave down. A) For x < 1 B) For x > 1 C) For x < –1 D) For x > –1 Ans: A Difficulty: hard Section: 3.2 65. A manufacturer estimates that if he produces x units of a particular commodity, the total cost will be C ( x) x3 24 x 2 350 x 400 dollars. For what value of x does the marginal cost M ( x) C ( x) satisfy M ( x) 0 ? Ans: 8 Difficulty: hard Section: 3.2 66. Determine where the graph of f ( x) x 2 /(2) is concave up and concave down. Ans: Concave up for all x. Difficulty: moderate Section: 3.2 Page 66 Chapter 3 67. True or False: The function graphed below has a negative second derivative everywhere. A) True B) False Ans: A Difficulty: moderate Section: 3.2 68. True or False: The function f ( x) x 6 5 x3 2 has a relative maximum at x = 1. A) True B) False Ans: B Difficulty: easy Section: 3.2 69. 2 True or False: The function f ( x) x 2 has a relative minimum at x = 1. x A) True B) False Ans: A Difficulty: easy Section: 3.2 70. 2 True or False: The graph of f ( x) 3x 4 2 x3 12 x 2 18 x 5 is concave up for x 3 . A) True B) False Ans: B Difficulty: moderate Section: 3.2 71. True or False: The graph of g (t ) t 4 2t 3 is concave up everywhere. A) True B) False Ans: B Difficulty: moderate Section: 3.2 Page 67 Chapter 3 72. 1 True or False: The graph of f ( x) 2 x3 3x 2 12 x 5 is concave down for x . 2 A) True B) False Ans: A Difficulty: moderate Section: 3.2 73. True or False: The inflection point of f ( x) x 3 6 x 2 13 is (–2, 3). A) True B) False Ans: A Difficulty: moderate Section: 3.2 74. x3 1 Evaluate the limit: lim x x2 x 2 Ans: Difficulty: moderate Section: 3.3 75. Sketch the graph of a function f that has all of the given properties. f ( x) 0 when x 5 and 1 x 3 f ( x) 0 when 5 x 1 and x 3 f ( x) 0 when 3 x 1 f ( x) 0 when x 3 and x 1 Ans: One possible graph is below. Difficulty: easy Section: 3.3 Page 68 Chapter 3 76. Sketch the graph of a function f that has all of the given properties. f ( x) 0 when x 4 and 4 x 0 f ( x) 0 when x 0 f ( x) 0 when x 4 f ( x) 0 when x 4 f (4) is undefined; f (0) 0; f (0) 0 Ans: One possible graph is below. Difficulty: easy Section: 3.3 77. Sketch the graph of a function f that has all of the given properties. f ( x) 0 when x 0 f ( x) 0 when x 0 f ( x) 0 when 10 x 10 f ( x) 0 when x 10 and x 10 f (0) 0; lim f ( x) 20 x Ans: One possible graph is below. Difficulty: moderate Section: 3.3 Page 69 Chapter 3 78. Sketch the graph of a function f that has all of the given properties. f ( x) 0 when x 0 f ( x) 0 when x 0 f ( x) 0 for all x f (0) is undefined; f (0) 5; lim x Ans: One possible graph is below. Difficulty: moderate Section: 3.3 79. Graph f ( x) x 2 4 x 5 . Ans: See graph below. Difficulty: hard Section: 3.3 Page 70 Chapter 3 80. Graph f ( x) 1.2 0.6 x 0.2 x 2 . Ans: See graph below. Difficulty: hard Section: 3.3 81. Graph f ( x) 2 x3 4 x 2 x . Ans: See graph below. Difficulty: hard Section: 3.3 Page 71 Chapter 3 82. 2 x3 5 x Evaluate the limit: lim x 2 x3 A) –2 B) 2 C) 0 D) Ans: A Difficulty: moderate Section: 3.3 83. x4 2 x2 3 Evaluate the limit: lim x x5 A) 1 B) –1 C) 0 D) Ans: C Difficulty: easy Section: 3.3 84. x3 2 x 3 Evaluate the limit: lim x x3 3x 2 A) –5 B) 1 C) 0 D) Ans: B Difficulty: moderate Section: 3.3 85. Evaluate the limit: 1 1 lim x x2 x 1 x3 A) 0 B) 2 C) D) 1 Ans: C Difficulty: hard Section: 3.3 86. 1 The function f ( x) has x2 A) a vertical asymptote at x = –2 and no horizontal asymptote B) a vertical asymptote at x = 2 and no horizontal asymptote C) a vertical asymptote at x = 2 and a horizontal asymptote at y = 0 D) no asymptote Ans: C Difficulty: moderate Section: 3.3 87. x2 The function f ( x) has x 1 A) a vertical asymptote at x = 1 and no horizontal asymptote B) no asymptote C) a horizontal asymptote at y = 0 and no vertical asymptote D) a vertical asymptote at x = –1 and a horizontal asymptote at y = 0 Ans: A Difficulty: hard Section: 3.3 Page 72 Chapter 3 88. The graph below shows A) no asymptotes B) a horizontal asymptote, but no vertical asymptote C) a horizontal asymptote and a vertical asymptote D) a vertical asymptote, but no horizontal asymptote Ans: C Difficulty: hard Section: 3.3 89. x 1 How many vertical asymptotes does f ( x) have? x2 1 A) 2 B) 1 C) 0 D) It cannot be determined Ans: B Difficulty: hard Section: 3.3 90. 3x 2 2 x 1 Find lim x f ( x) . f ( x) 8x 1 A) 3 B) C) 0 D) 1/8 Ans: B Difficulty: moderate Section: 3.3 91. 9 x2 9 x 9 Find lim x f ( x) . f ( x) 5 x 4 10 A) 9 B) 1.80 C) 0 D) Ans: B Difficulty: moderate Section: 3.3 Page 73 Chapter 3 92. 7 4 2 Find lim x f ( x) . f ( x) x2 x 11 4 x2 A) 0.64 B) –0.50 C) 1.57 D) Ans: B Difficulty: moderate Section: 3.3 93. Evaluate the limit: lim 4 x3 8 x . x 4 x3 A) 4 B) 0 C) –4 D) Ans: C Difficulty: moderate Section: 3.3 94. Evaluate the limit: lim x4 7 x2 4 . x x5 A) 1 B) –1 C) 0 D) Ans: C Difficulty: moderate Section: 3.3 95. Evaluate the limit: lim x3 2 x 1 . x x3 2 x 8 A) –1 / 8 B) 0 C) 1 D) Ans: C Difficulty: moderate Section: 3.3 96. Evaluate the limit: 2 2 lim x x2 x . 5 x3 A) 4 B) 0 C) 5 D) 2 Ans: D Difficulty: hard Section: 3.3 97. The function f ( x) x 7 has 1 A) a vertical asymptote at x = –7 and no horizontal asymptote B) a vertical asymptote at x = 7 and no horizontal asymptote C) no asymptote D) a vertical asymptote at x = 7 and a horizontal asymptote at y = 0 Ans: D Difficulty: moderate Section: 3.3 98. The function f ( x) x2 has x 7 A) a horizontal asymptote at y = 0 and no vertical asymptote B) a vertical asymptote at x = 7 and no horizontal asymptote C) no asymptote D) a vertical asymptote at x = 7 and a horizontal asymptote at y = 0 Ans: B Difficulty: hard Section: 3.3 Page 74 Chapter 3 99. How many vertical asymptotes does f ( x) x 6 have? x2 7 A) 2 B) 1 C) 0 D) It cannot be determined Ans: A Difficulty: hard Section: 3.3 100. 1 Ax Find A and B so that the graph of f ( x) has y = 7 as a horizontal asymptote and Bx 7 x = 2 as a vertical asymptote. Ans: 49 7 A , B 2 2 Difficulty: moderate Section: 3.3 101. x 2 3x 2 Evaluate the limit: lim x 1 x3 Ans: 0 Difficulty: moderate Section: 3.3 102. x2 5 Evaluate the limit: lim x x3 1 Ans: 0 Difficulty: moderate Section: 3.3 103. x 2 3x 3 Evaluate the limit: lim x 2 x3 Ans: 0 Difficulty: moderate Section: 3.3 104. x2 4 Evaluate the limit: lim x x3 2 Ans: 0 Difficulty: moderate Section: 3.3 105. x3 2 Evaluate the limit: lim x x2 x 8 Ans: Difficulty: moderate Section: 3.3 106. x3 2 x 8 True or False: lim x 4 x2 2 x A) True B) False Ans: A Difficulty: easy Section: 3.3 Page 75 Chapter 3 107. 3x 4 3x True or False: lim x 3 x4 A) True B) False Ans: A Difficulty: moderate Section: 3.3 108. x3 4 True or False: lim x 0 x3 8 A) True B) False Ans: B Difficulty: moderate Section: 3.3 109. 1 2 x3 True or False: lim x 2 x3 4 A) True B) False Ans: A Difficulty: moderate Section: 3.3 110. Find the absolute maximum and minimum of f ( x) x 4 8 x 2 12 on the interval – 1x2. Ans: Maximum is 12 at x = 0, minimum is –4 at x = 2 Difficulty: easy Section: 3.4 111. Determine the absolute maximum and minimum of f ( x) x 4 2 x 2 5 on the interval –2x1. Ans: Maximum is 13 at x = –2, minimum is 4 at x = 1 or –1 Difficulty: easy Section: 3.4 112. Suppose the total cost of producing x units of a certain commodity is C ( x) 2 x 4 10 x3 18 x 2 200 x 167 . Determine the largest and smallest values of the marginal cost for 0 x 5 . Ans: Maximum is 270 at x = 5, minimum is 38 at x = 3 Difficulty: hard Section: 3.4 113. The total cost of producing x units of a certain commodity is C ( x) x3 5 x 2 8 x . Determine the minimum average cost of the commodity. Ans: 7 5 at x 4 2 Difficulty: hard Section: 3.4 Page 76 Chapter 3 114. 1 3 Find the absolute maximum and minimum of the function f ( x) ( x 6 x 2 9 x 1) 6 on the interval 0 x 2 . Ans: 5 1 Maximum is at x = 1; minimum is at x = 0 6 6 Difficulty: easy Section: 3.4 115. Find the absolute maximum and minimum of the function f ( x) 2 x3 7 x 2 8 x 2 on the interval 0 x 3 . Ans: Maximum is 17 at x = 3; minimum is 2 at x = 0 Difficulty: easy Section: 3.4 116. Find the absolute maximum of the function f ( x) x5 x 4 on the interval – 1 x 1 . A) 0 B) 1 C) –1 D) –2 Ans: A Difficulty: easy Section: 3.4 117. 1 Find the absolute maximum of the function f ( x) x3 on the interval x 1 . 2 1 A) 0 B) C) –1 D) 1 8 Ans: D Difficulty: easy Section: 3.4 118. Find the absolute minimum of the function f ( x) x 3 3x 2 on the interval – 1 x 3 . A) –1 B) –4 C) 0 D) 3 Ans: B Difficulty: easy Section: 3.4 119. The cost of producing x units of a certain commodity is C ( x) 3x 2 6 x 9 dollars. If the price is p(x) = (45 – x) dollars per unit, determine the level of production that maximizes profit. A) x = 1 B) x = 2 C) x = 3 D) x = 5 Ans: B Difficulty: moderate Section: 3.4 120. An apartment complex has 240 units. When the monthly rent for each unit is $360, all units are occupied. Experience indicates that for each $16 per month increase in rent, 3 units will become vacant. Each rented apartment costs the owner of the complex $46 per month to maintain. What monthly rent should be charged to maximize profit? A) $168.6 B) $337.2 C) $505.8 D) $843 Ans: B Difficulty: moderate Section: 3.4 121. Find the absolute maximum of the function f ( x) x9 x8 on the interval – 1 x 1 . A) 0 B) 1 C) –1 D) –2 Ans: A Difficulty: easy Section: 3.4 Page 77 Chapter 3 122. Find the absolute maximum of the function f ( x) x 4 on the interval 1 7 x 1 . A) 1 B) 0 C) 714 D) –1 Ans: A Difficulty: easy Section: 3.4 123. Find the absolute minimum of the function f ( x) x 3 3x 2 on the interval –1 x 5 . A) –4 B) 0 C) 5 D) –1 Ans: A Difficulty: easy Section: 3.4 124. Find the absolute maximum and absolute minimum of f ( x) x 4 8 x 2 12 on the interval – 1 x 9 Ans: Maximum is 12 at x = 0 and minimum is –4 at x = 2 Difficulty: easy Section: 3.4 125. Find the absolute maximum and absolute minimum of f ( x) x 4 2 x 2 10 on the interval – 4 x 2 Ans: Maximum is 234 at x = 4 and minimum is 9 at x = –1 or 1. Difficulty: easy Section: 3.4 126. Find the absolute maximum and absolute minimum of f ( x) 1 ( x 3 6 x 2 9 x 7) on 4 the interval 0 x 2 . Ans: Maximum is 11 at x = 1 and minimum is 7 at x = 0. 4 4 Difficulty: easy Section: 3.4 127. Find the absolute maximum and absolute minimum of f ( x) (2 x3 7 x 2 8 x 7) on the interval 0 x 5 . Ans: Maximum is 122 at x = 5 and minimum is 7 at x = 0. Difficulty: easy Section: 3.4 128. True or False: The absolute maximum of the function f ( x) x 4 2 x 2 3 on the interval – 1 x 2 is 11. A) True B) False Ans: A Difficulty: easy Section: 3.4 129. 9 True or False: The absolute minimum of f ( x) x 3 on the interval 1 x 9 is x 1. A) True B) False Ans: B Difficulty: easy Section: 3.4 Page 78 Chapter 3 130. True or False: The absolute minimum of the function f ( x) 8 x3 4 x 2 72 x on the interval 0 x 4 is 0. A) True B) False Ans: A Difficulty: easy Section: 3.4 131. True or False: The demand function for a certain commodity is D(p) = 150 – 3p, where p is the price at which the commodity is sold. The price at which the total consumer expenditure is maximized is p = 25. A) True B) False Ans: A Difficulty: hard Section: 3.4 132. 1 2 If the cost of a commodity is C ( x) x 3x 98 dollars when x units are produced 8 1 and the selling price is p( x) (75 x) dollars per unit, find the level of production 3 where profit is maximized. Ans: x = 24 Difficulty: hard Section: 3.5 133. If the demand for a commodity is D( p ) 146 2 p 4 p 2 and the average cost is 1 A( x) 1 , find the maximum profit. x Ans: 207 at p = 3 Difficulty: hard Section: 3.5 134. Which non-negative number exceeds its own fourth power by the greatest amount? Ans: 1 3 4 Difficulty: hard Section: 3.5 135. Find the dimensions of the rectangle of largest area that can be inscribed in a semi-circle of radius R, assuming that one side of the rectangle lies on a diameter of the semi-circle. Ans: R 2 R; 2 Difficulty: hard Section: 3.5 136. Find two positive numbers whose sum is 100 and whose product is as large as possible. Ans: 50; 50 Difficulty: hard Section: 3.5 Page 79 Chapter 3 137. Find two non-negative numbers whose sum is 8 if it is required that the product of their squares is to be as large as possible. A) 2 and 6 B) 1 and 7 C) 0 and 8 D) 4 and 4 Ans: D Difficulty: hard Section: 3.5 138. A commuter train carries 600 passengers each day from a suburb to a city. It now costs $1 per person to ride the train. A study shows that 50 additional people will ride the train for each 5 cent reduction in fare. What fare should be charged in order to maximize total revenue? A) 78 cents B) 79 cents C) 80 cents D) 85 cents Ans: C Difficulty: hard Section: 3.5 139. A manufacturer receives an order for 5,000 items. He owns 12 machines, each of which can produce 25 items per hour. The cost of setting up a machine for a production run is $20. Once the machines are in operation, the procedure is fully automated and can be supervised by a single worker earning $4.90 per hour. Find the number of machines that should be used in order to minimize the total cost of filling the order. A) 6 B) 7 C) 8 D) 9 Ans: B Difficulty: hard Section: 3.5 140. Find the length of one leg of a right triangle when the hypotenuse is 40 miles and perimeter is maximal. A) 50 miles B) 75 miles C) 2 5 miles D) 102 miles Ans: C Difficulty: hard Section: 3.5 141. Find two non-negative numbers whose sum is 10 if it is required that the product of one number and the square of the other is to be as large as possible. 10 20 A) and B) 10 and 20 C) 5 and 5 D) 9 and 1 3 3 Ans: A Difficulty: hard Section: 3.5 142. A manufacturer knows that when he charges p dollars per unit for his product, he will be able to sell x = 380 – 20p units. He also estimates that at this level of production, his x average cost is A( x) 5 . What price should the manufacturer charge in order to 50 maximize profit? A) $13 B) $14 C) $15 D) $16 Ans: B Difficulty: hard Section: 3.5 143. A poster is to contain 108 cm 2 of printed matter with additional margins 6 cm each at top and bottom and 2 cm on the sides. What is the minimal cost of the poster if it is to be made out of material costing 20 cents per square centimeter? A) $59 B) $60 C) $61 D) $62 Ans: B Difficulty: hard Section: 3.5 Page 80 Chapter 3 144. 1 2 If the total cost of manufacturing a commodity is C ( x) x 4 x 200 dollars when 8 x units are produced, for what value of x is the average cost the least? A) 37 B) 38 C) 39 D) 40 Ans: D Difficulty: hard Section: 3.5 145. 3 Find the elasticity n of the demand function q . 1 2 p2 6 4 p2 A) n B) n C) n = 4p D) n 4 p 3 1 2 p2 1 2 p2 Ans: B Difficulty: hard Section: 3.5 146. 300 p 2 The demand function for a certain commodity is x . For what values of p is 60 the demand elastic? A) p = 100 B) p > 100 C) p < 100 D) p > 0 Ans: B Difficulty: hard Section: 3.5 147. 30 The demand function for a certain commodity is x . For what values of p is the p5 demand inelastic? A) p > 0 B) p < 0 C) p > –5 D) p < –5 Ans: A Difficulty: hard Section: 3.5 148. A store uses 900 cases of electronic parts each year. The cost of storing one case for a year is 80 cents and the ordering fee is $40 per shipment. How many cases should the store order each time to minimize total cost? Assume the orders are planned so that a new shipment arrives just as the number of cases in the store reaches zero. Also assume the parts are consumed at a constant rate. A) 300 cases B) 600 cases C) 900 cases D) 1,000 cases Ans: A Difficulty: hard Section: 3.5 149. A Florida citrus grower estimates that if 80 orange trees are planted, the average yield per tree will be 500 oranges. The average yield will decrease by 5 oranges per tree for each additional tree planted on the same acreage. How many trees should the grower plant to maximize the total yield? A) 90 trees B) 40 trees C) 10 trees D) 85 trees Ans: A Difficulty: moderate Section: 3.5 Page 81 Chapter 3 150. A cable is to be run from a power plant on one side of a river 900 meters wide to a factory on the other side, 4,000 meters downstream. The cost of running cable under the water is $5 per meter, while over land is $4 per meter. What is the most economical route over which to run the cable? A) The cable reaches the opposite bank directly across stream. B) The cable reaches the opposite bank 1,200 meters downstream. C) The cable reaches the opposite bank 4,000 meters downstream. D) The cable reaches the opposite bank 2,000 meters downstream. Ans: B Difficulty: moderate Section: 3.5 151. An open box is to be made from a square piece of cardboard, 30 inches by 30 inches, by removing a small square from each corner and folding up the flaps to form the sides. What are the dimensions of the box of greatest volume that can be constructed in this way? A) 10 inches long by 10 inches wide by 5 inches deep B) 15 inches long by 15 inches wide by 5 inches deep C) 5 inches long by 5 inches wide by 20 inches deep D) 20 inches long by 20 inches wide by 5 inches deep Ans: D Difficulty: moderate Section: 3.5 152. To raise money, a service club has been collecting used bottles that it plans to deliver to a local glass company for recycling. Since the project began 50 days ago, the club has collected 25,000 pounds of glass for which the glass company currently offers 1 cent per pound. However, because bottles are accumulating faster than they can be recycled, the company plans to reduce by 1 cent each day the price it will pay for 100 pounds of used glass. Assume that the club can continue to collect bottles at the same rate and that transportation costs make more than one trip to the glass company unfeasible. What is the most advantageous time for the club to conclude its project and deliver the bottles? A) Today B) 75 days from now C) 35 days from now D) 25 days from now Ans: D Difficulty: moderate Section: 3.5 153. The owner of an appliance store expects to sell 600 toasters this year. Each toaster costs her $5 dollars to purchase, and each time she orders a shipment of toasters, it costs $40. In addition, it costs $1 a year to store each toaster. Assuming the toasters sell out at a uniform rate and that the owner never allows herself to run out of toasters, how many toasters should be ordered in each shipment to minimize the annual cost? (Round any fractional amounts.) A) 40 B) 80 C) 4 D) 3 Ans: B Difficulty: moderate Section: 3.5 Page 82 Chapter 3 154. The owner of a novelty store can obtain joy buzzers from the manufacturer for 70 cents apiece. He estimates he can sell 60 buzzers when he charges $1.00 apiece for them and that he will be able to sell 14 more buzzers for every 8 cent decrease in price. What price should he charge in order to maximize profit? A) $0.92 B) $1.02 C) $0.82 D) $1.12 Ans: B Difficulty: moderate Section: 3.5 155. The personnel manager of a department store estimates that if she hires n temporary salespersons for the holiday season, the total net revenue derived from their efforts will be R(n) 3n 4 40n3 96n 2 11 hundred dollars for 0 n 12 . How many salespersons should be hired in order to maximize total net revenue? A) 2 B) 8 C) 12 D) 0 Ans: B Difficulty: moderate Section: 3.5 156. A house is located in the desert, 12 miles from a long, straight road. A man in a town 15 miles down the road arranges to be transported to the house by a company that charges $3 per mile on the road and $4 per mile on the desert. What is the minimum cost of the trip? A) $93.48 B) $228 C) $118.56 D) $273.6 Ans: B Difficulty: moderate Section: 3.5 157. Find two non-negative numbers whose sum is 10 for which the product of their squares is as large as possible. A) 5 and 5 B) 0 and 10 C) 1 and 9 D) 3 and 7 Ans: A Difficulty: hard Section: 3.5 158. Find the length of one leg of a right triangle when the hypoteneuse is 30 miles and perimeter is maximal. A) 15 miles B) 40 miles C) 10 miles D) 15 miles Ans: A Difficulty: hard Section: 3.5 159. Find two non-negative numbers whose sum is 20 if it is required that the product of one number with the square of the other number is to be as large as possible. A) 20 and 40 B) 20 and 40 C) 10 and 10 D) 1 and 19 3 3 Ans: A Difficulty: hard Section: 3.5 160. Which non-negative number exceeds its own 6th power by the greatest amount? Ans: 51 . 6 Difficulty: hard Section: 3.5 Page 83 Chapter 3 161. An open box with square base and vertical sides is constructed out of 100m2 of tin. What should the dimensions of the box be if its volume is to be as large as possible? Ans: 10 3 m by 10 3 m by 5 3 m . 3 3 3 Difficulty: hard Section: 3.5 162. True or False: A dune buggy is on the desert at a point A located 40 km from the nearest point B on a long, straight road. The driver can travel at 45 km/hr on the desert and 75 km/hr on the road. He has an appointment in a town 28 km down the road from B in exactly one hour. He can make it. A) True B) False Ans: B Difficulty: hard Section: 3.5 163. True or False: Through physical experiments, it is known that a quantity of water that occupies 1 liter at 0 degrees C will occupy V (t ) 1 6.42 105 T 8.51 106 T 2 6.79 108 T 3 liters when the temperature is T degrees C. The temperature at which V(T) is minimized is 60 degrees C. A) True B) False Ans: B Difficulty: hard Section: 3.5 164. True or False: If the demand for a commodity is D(p) = 28 – 5p, where p is the price, and the total cost is C ( p) p 2 4 p , the maximum profit is $24. A) True B) False Ans: A Difficulty: moderate Section: 3.5 165. True or False: Average cost is minimized at the level of production where marginal cost equals average cost. A) True B) False Ans: A Difficulty: easy Section: 3.5 166. True or False: The function f ( x) x 6 9 x3 6 has a relative maximum at x = 1 A) True B) False Ans: B Difficulty: easy Section: 3.5 167. True or False: The function f ( x ) x 6 6 has a relative minimum at x = 1. x A) True B) False Ans: A Difficulty: easy Section: 3.5 Page 84 Chapter 3 168. A company that distributes landscape materials buys 4,000 tons of pine mulch a year. The ordering fee is $30 per shipment, the mulch costs them $20 per ton, and annual storage costs are $1.50 per ton. How many tons should be ordered in each shipment to minimize the total annual cost? A) 2,000 tons B) 400 tons C) 200 tons D) 500 tons Ans: B Difficulty: hard Section: 3.5 Page 85