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1. Use the graph of to identify at which of the indicated points the derivative 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 to identify at which of the indicated points the derivative 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 is increasing or decreasing.

A) decreasing: ; increasing:

B) increasing: ; decreasing:

C) increasing on

D) decreasing on

E) none of the above

Ans: A

4. Both a function and its derivative are given. Use them to find all critical numbers.

A)

B)

C)

D)

E)

Ans: D

5. Identify the open intervals where the function is increasing or decreasing.

A) increasing: ; decreasing:

B) decreasing: ; increasing:

C) increasing on

D) decreasing on

E) none of the above

Ans: A

6. For the given function, find all critical numbers.

A)

B) and

C) and

D) and

E) and

Ans: D

7. Find any critical numbers of the function , t < 5.
A) 0
B)
C)
D) both A and B
E) both A and C
Ans: C
8. Identify the open intervals where the function is increasing or decreasing.
A) decreasing: ; increasing:
B) increasing: ; decreasing:
C) increasing: ; decreasing:
D) increasing: ; decreasing:
E) decreasing for all x
Ans: B
9. For the given function, find the critical numbers.
A)
B)
C)
D)
E)
Ans: A
10. Find the open intervals on which the function is increasing or decreasing.
A) The function is increasing on the interval , and decreasing on the intervals and .
B) The function is increasing on the interval , and decreasing on the intervals and .
C) The function is increasing on the interval , and decreasing on the intervals and .
D) The function is decreasing on the interval , and increasing on the intervals and .
E) The function is decreasing on the interval , and increasing on the intervals and .
Ans: A
11. Find the open intervals on which the function is increasing or decreasing.
A) The function is increasing on the interval and decreasing on the interval .
B) The function is increasing on the interval and decreasing on the interval .
C) The function is increasing on the interval and decreasing on the interval .
D) The function is increasing on the interval and decreasing on the interval .
E) The function is increasing on the interval and decreasing on the interval .
Ans: E
12. Suppose the number y of medical degrees conferred in the United States can be modeled by for , where t is the time in years, with 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
13. A fast-food restaurant determines the cost model, and revenue model, for where x is the number of 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 and decreasing on the interval .
B) The profit function is increasing on the interval and decreasing on the interval .
C) The profit function is increasing on the interval and decreasing on the interval .
D) The profit function is increasing on the interval and decreasing on the interval .
E) The profit function is increasing on the interval and decreasing on the interval .
Ans: C
14. For the given function, find the relative minima.
A)
B)
C)
D)
E) no relative minima
Ans: B
15. Find the x-values of all relative maxima of the given function.
A)
B)
C)
D)
E) no relative maxima
Ans: D
16. For the function :
(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: ; decreasing:
(c) relative max: ; relative min:
B) (a) x = 0 , 6
(b) decreasing: ; increasing:
(c) relative min: ; relative max:
C) (a) x = 0 , 2
(b) increasing: ; decreasing:
(c) relative max: ; relative min:
D) (a) x = 0 , 2
(b) decreasing: ; increasing:
(c) relative min: ; relative max:
E) (a) x = 0 , 2
(b) increasing: ; decreasing:
(c) relative max: ; no relative min.
Ans: A
17. Find all relative maxima of the given function.
A)
B)
C)
D) ,
E) no relative maxima
Ans: B
18. Find all relative minima of the given function.
A)
B)
C)
D) ,
E) no relative minima
Ans: D
19. Locate the absolute extrema of the function on the closed interval .
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 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].
A)
B)
C)
D)
E)
Ans: A
22. Locate the absolute extrema of the given function on the closed interval [36,36].
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
23. Find the absolute extrema of the function on the closed interval . 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.
A) The critical number yields an absolute maximum and the critical number yields an absolute minimum..
B) Both the critical numbers & yield an absolute maximum.
C) The critical number yields an absolute minimum and the critical number yields an absolute maximum.
D) Both the critical numbers and yield an absolute minimum.
E) The critical number yields a relative minimum and the critical number yields a relative maximum.
Ans: C
25. Find the absolute extrema of the function on the interval .
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
26. Graph a function on the interval having the following characteristics.
Absolute maximum at
Absolute minimum at
Relative maximum at
Relative minimum at
A)
B)
C)
D)
E)
Ans: A
27. Medication. The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by . Find the t-value at which 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 after a dose is taken can be modeled by . Find the maximum value 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
29. Suppose the resident population P(in millions) of the United States can be modeled by , where corresponds to 1800. Analytically find the minimum and maximum populations in the U.S. for .
A) The population is minimum at and maximum at .
B) The population is minimum at and maximum at .
C) The population is minimum at and maximum at .
D) The population is minimum at and maximum at .
E) The population is minimum at and maximum at .
Ans: D
30. Determine the open intervals on which the graph of is concave downward or concave upward.
A) concave upward on ; concave downward on
B) concave downward on
C) concave upward on
D) concave downward on ; concave upward on
E) concave upward on ; concave downward on
Ans: C
31. Determine the open intervals on which the graph of is concave downward or concave upward.
A) concave downward on
B) concave downward on ; concave upward on
C) concave upward on ; concave downward on
D) concave downward on ; concave upward on
E) concave upward on ; concave downward on
Ans: E
32. Find all relative extrema of the function . Use the Second Derivative Test where applicable.
A) relative min:
B) relative max:
C) no relative max
D) no relative min
E) both A and C
F) both B and D
Ans: E
33. Find all relative extrema of the function Use the Second Derivative Test where applicable.
A) relative max: ; no relative min
B) relative max: ; no relative min
C) no relative max or min
D) relative min: ; no relative max
E) relative min: ; no relative max
Ans: E
34. Find all relative extrema of the function . 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 . Use the Second-Derivative Test when applicable.
A) The relative minimum is and the relative maximum is .
B) The relative maximum is .
C) The relative minimum is .
D) The relative maximum is and the relative minima are and .
E) The relative minimum is and the relative maximum is .
Ans: B
36. Find all relative extrema of the function . Use the Second-Derivative Test when applicable.
A) The relative maximum is .
B) The relative minimum is .
C) The relative maximum is .
D) The relative minimum is .
E) The relative maximum is .
Ans: C
37. State the signs of and on the interval (0, 2).
A) = 0
> 0

B) < 0
< 0
C) > 0

> 0

D) < 0
> 0

E) > 0

< 0
Ans: B
38. Find the x-value at which the given function has a point of inflection.
A)
B)
C)
D)
E) no point of inflection
Ans: C
39. Find the points of inflection and discuss the concavity of the function.
A) inflection point at ; concave upward on ; concave downward on
B) inflection point at ; concave downward on ; concave upward on
C) inflection point at ; concave upward on ; concave downward on
D) inflection point at ; concave downward on ; concave upward on
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 . Check these results against the graph shown.
A)
B)
C)
D) ,
E) , ,
Ans: D
41. Sketch a graph of a function f having the following characteristics.
A)
B)
C)
D)
E)
Ans: C
42. The graph of f is shown in the figure. Sketch a graph of the derivative of f.
A)
B)
C)
D)
E)
Ans: E
43. The graph of f is shown in the figure. Sketch a graph of the derivative of f.
A)
B)
C)
D)
E)
Ans: C
44. The graph of f is shown in the figure. Sketch a graph of the derivative of f.
A)
B) The derivative of f does not exist.
C)
D)
E)
Ans: D
45. The graph of f is shown. Graph f, f' and f'' on the same set of coordinate axes.
A)
B)
C)
D)
E) none of the above
Ans: C
46. The graph of f is shown in the figure. Sketch a graph of the derivative of f.
A)
B)
C)
D)
E)
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 . Find the number of hours before the rate of production is maximized. That is, find the point of diminishing returns.
A)
B)
C)
D)
E)
Ans: B
48. The profit P (in thousands of dollars) for a company spending an amount s (in thousands of dollars) on advertising is The point of diminishing returns 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
donors, where t is the number of years after 1975. Find the inflection point(s) from through , if any exist.
A) There are no inflection points from through .
B) There is one inflection point at .
C) There are inflection points at and .
D) There is one inflection point at .
E) There are inflection points at , , and .
Ans: B
50. Find the length and width of a rectangle that has perimeter meters and a maximum area.
A) 1, 3
B) 1, 3
C) 2, 2
D) 3, 1
E) 6, 2
Ans: C
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) and
B) and
C) and
D) and
E) and
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) square base side ; height
B) square base side ; height
C) square base side ; height
D) square base side ; height
E) square base side ; height
Ans: D
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 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 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)
B) 15, 15
C) 13, 13
D) 16, 16
E) 14, 14
Ans: A
56. Find the dimensions of the rectangle of maximum area bounded by the x-axis and y-axis and the graph of .
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 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
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 per foot and the fence for the middle costs per foot. If each lot contains 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
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: where 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 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 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 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
64. If the total cost function for a product is 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 dollars. Find the minimum average cost.
A)
B)
C)
D)
E)
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 , where x is the number of machines used. How many 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 , where x is the number of machines used. What is the minimum average cost?
A) $0
B) $10
C) $100
D) $50
E) $505
Ans: B
68. A firm can produce 100 units per week. If its total cost function is dollars, and its total revenue function is 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 dollars, and its total revenue function is 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 or larger. If the group contains exactly people, the price is per person. However, each persons price is reduced by for each additional person above the . If the travel agency incurs a price of 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
72. Find the speed v, in miles per hour, that will minimize costs on a 105-mile delivery trip. The cost per hour for fuel is dollars, and the driver is paid dollars 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 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 at the price .
A) 5.40
B) 1.00
C) 5.40
D) 3.00
E) 3.00
Ans: C
75. A function and its graph are given. Use the graph to find the vertical asymptotes, if they exist, where Confirm your results analytically.
A)
B)
C)
D)
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 Confirm your results analytically.
A)
B)
C)
D)
E) no horizontal asymptotes
Ans: B
77. A function and its graph are given. Use the graph to find the vertical asymptotes, if they exist. Confirm your results analytically.
A)
B)
C)
D)
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.
A)
B)
C)
D)
E) no horizontal asymptotes
Ans: C
79. Analytically determine the location(s) of any vertical asymptote(s).
A)
B)
C) ,
D)
E) no vertical asymptotes
Ans: C
80. Analytically determine the location(s) of any horizontal asymptote(s).
A)
B)
C) ,
D)
E) no horizontal asymptotes
Ans: D
81. This problem contains a function and its graph, where 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.
A)
B)
C)
D)
E) no vertical asymptote
Ans: D
82. This problem contains a function and its graph, where 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.
A)
B)
C)
D)
E)
Ans: C
83. Find any horizontal asymptotes for the given function.
A)
B)
C)
D)
E) no horizontal asymptotes
Ans: D
84. Find any vertical asymptotes for the given function.
A)
B)
C)
D)
E) no vertical asymptotes
Ans: C
85. Find any horizontal asymptotes for the given function.
A)
B)
C)
D)
E) no horizontal asymptotes
Ans: D
86. Find all vertical asymptotes for the given function.
A)
B)
C)
D)
E) no vertical asymptotes
Ans: B
87. Analytically determine the location of any vertical asymptotes.
A)
B)
C)
D)
E) no vertical asymptotes
Ans: E
88. Analytically determine the location(s) of any horizontal asymptote(s).
A)
B)
C)
D)
E) no horizontal asymptotes
Ans: A
89. Find any horizontal asymptotes for the given function.
A)
B)
C)
D)
E) no horizontal asymptotes
Ans: E
90. Match the function with one of the following graphs.
A)
B)
C)
D)
E)
Ans: D
91. Match the function with one of the following graphs.
A)
B)
C)
D)
E)
Ans: B
92. Find the limit:
A)
B)
C) 0
D) 1
E) 1
Ans: B
93. Find the limit:
A) 0
B)
C) 1
D) 1
E)
Ans: E
94. For the function , use a graphing utility to complete the table and estimate the limit as x approaches infinity.
x 100 101 102 103 104 105 106
f(x)
A) 0.525
B) 1.375
C) 0.125
D) 2.555
E) does not exist
Ans: C
95. Use a table utility with x-values larger than 10,000 to investigate .What does the table indicate about
A)
B)
C)
D) 0
E) does not exist
Ans: A
96. For the function , use a graphing utility to complete the table and estimate the limit as x approaches infinity.
x 100 101 102 103 104 105 106
f(x)
A) 0.6
B) 1.666667
C) 2.666667
D) 1.6
E) 0.4
Ans: B
97. Use analytic methods to find the limit as for the given function.
A)
B)
C)
D)
E) does not exist
Ans: D
98. Use analytic methods to find the limit as for the given function.
A)
B)
C)
D)
E) does not exist
Ans: A
99. Find the limit.
A)
B)
C) 1
D) 0
E) does not exist
Ans: B
100. Find the limit.
A)
B) 1
C) 0
D)
E)
Ans: C
101. Find the limit.
A)
B) 7
C) 7
D)
E)
Ans: A
102. Find the limit.
A)
B)
C)
D)
E)
Ans: E
103. Sketch the graph of the function using any extrema, intercepts, symmetry, and asymptotes.
A)
B)
C)
D)
E)
Ans: E
104. Sketch the graph of the function using any extrema, intercepts, symmetry, and asymptotes.
A)
B)
C)
D)
E)
Ans: A
105. Sketch the graph of the relation using any extrema, intercepts, symmetry, and asymptotes.
A)
B)
C)
D)
E)
Ans: B
106. Sketch the graph of the relation using any extrema, intercepts, symmetry, and asymptotes.
A)
B)
C)
D)
E)
Ans: E
107. Sketch the graph of the equation given below. Use intercepts, extrema, and asymptotes as sketching aids.
A)
B)
C)
D)
E)
Ans: A
108. A business has a cost (in dollars) of for producing x units. What is the limit of 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 , for . Find the limit of C as .
A) 580
B)
C)
D)
E)
Ans: D
110. Analyze and sketch a graph of the function .
A)
B)
C)
D)
E)
Ans: E
111. Analyze and sketch a graph of the function .
A)
B)
C)
D)
E)
Ans: C
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.
A)
B)
C)
D)
E)
Ans: A
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.
A)
B)
C)
D)
E)
Ans: B
114. Analyze and sketch a graph of the function .
A)
B)
C)
D)
E)
Ans: C
115. Analyze and sketch a graph of the function .
A)
B)
C)
D)
E) none of the above
Ans: E
116. Analyze and sketch a graph of the function .
A)
B)
C)
D)
E)
Ans: A
117. Analyze and sketch a graph of the function .
A)
B)
C)
D)
E)
Ans: D
118. Analyze and sketch a graph of the function .
A)
B)
C)
D)
E)
Ans: B
119. Use the graph to sketch the graph of .
A)
B)
C)
D)
E)
Ans: C
120. Use the graph to sketch the graph of .
A)
B)
C)
D)
E)
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 s (in miles per hour, with ), the van gets miles per gallon. Determine 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
122. Find the differential dy of the function .
A)
B)
C)
D)
E)
Ans: E
123. Find the differential dy of the function .
A)
B)
C)
D)
E)
Ans: D
124. Compare dy and for at x = 1 with dx = 0.05. Give your answers to four decimal places.
A)
B)
C)
D)
E)
Ans: B
125. Compare dy and for at x = 1 with dx = 0.07. Give your answers to four decimal places.
A)
B)
C)
D)
E)
Ans: B
126. Compare dy and for at x = 0 with dx = 0.09. Give your answers to four decimal places.
A) dy = 0.0200 ; = 0.0002
B) dy = 0.0100 ; = 0.0001
C) dy = 0.0300 ; = 0.0000
D) dy = 0.0000 ; = 0.0001
E) dy = 0.0200 ; = 0.0000
Ans: D
127. Complete the table for the function . Let x = 4.
4.00000
2.00000
0.40000
A)
4.00000 0.25 0.125 0.125 2
2.00000 0.125 0.08333 0.04167 1.50006
0.40000 0.0561 0.02273 0.00227 1.10208
B)
4.00000 0.25 0.125 0.125 2
2.00000 0.125 0.22667 0.04167 1.50427
0.40000 0.025 0.02273 0.00227 1.09987
C)
4.00000 0.25 0.125 0.125 2
2.00000 0.125 0.08333 0.04167 1.50006
0.40000 0.025 0.02273 0.00227 1.09987
D)
4.00000 0.1989 0.125 0.125 2
2.00000 0.125 0.08333 0.04167 1.50006
0.40000 0.0561 0.02273 0.00227 1.10208
E)
4.00000 0.1989 0.125 0.125 2
2.00000 0.125 0.22667 0.04167 1.50427
0.40000 0.0561 0.02273 0.00227 1.10208
Ans: C
128. Complete the table for the function . Let x = 3.
3.00000
1.50000
0.30000
A)
3.00000 0.86603 0.71744 0.14859 1.20711
1.50000 0.43301 0.38927 0.04374 1.11236
0.30000 0.0555 0.08454 0.00206 1.02658
B)
3.00000 0.86603 0.71744 0.14859 1.20711
1.50000 0.43301 0.69927 0.04374 1.11657
0.30000 0.0866 0.08454 0.00206 1.02437
C)
3.00000 0.91713 0.71744 0.14859 1.20711
1.50000 0.43301 0.69927 0.04374 1.11657
0.30000 0.0555 0.08454 0.00206 1.02658
D)
3.00000 0.91713 0.71744 0.14859 1.20711
1.50000 0.43301 0.38927 0.04374 1.11236
0.30000 0.0555 0.08454 0.00206 1.02658
E)
3.00000 0.86603 0.71744 0.14859 1.20711
1.50000 0.43301 0.38927 0.04374 1.11236
0.30000 0.0866 0.08454 0.00206 1.02437
Ans: E
129. The revenue R for a company selling x units is . Use differentials to approximate the change in revenue if sales increase from to 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 , 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
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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