AP CALCULUS AB Test Booklet

Unit 4

1. A differentiable function f has the property that and . What is the estimate for using
the local linear approximation for f at x= 5 ?
(A) 2.2
(B) 2.8
(C) 3.4
(D) 3.8
(E) 4.6

2.

A particle moves along the x-axis so that at time t ≥ 0 its position is given by x(t) = 2t3 - 21t2 + 72t - 53. At what
time t is the particle at rest?
(A) t = 1 only
(B) t = 3 only
(C) t = 7/2 only
(D) t = 3 and t = 7/2
(E) t = 3 and t = 4

3. If the position of a particle on the x-axis at time t is , then the average velocity of the particle for 0 ≤ t ≤ 3 is
(A) -45
(B) -30
(C) -15
(D) -10
(E) -5

4. For , the velocity of a particle moving along the x-axis is given by . At

what time t does the direction of motion of the particle change from right to left?
(A) 0.586
(B) 1.184
(C) 2.000
(D) 2.816

5. Two particles start at the origin and move along the x-axis. For , their respective position functions
are given by and . For how many values of t do the particles have the same velocity?
(A) None
(B) One
(C) Two
(D) Three
(E) Four

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6. People are entering a building at a rate modeled by f (t) people per hour and exiting the building at a rate modeled
by g(t) people per hour, where t is measured in hours. The functions f and g are nonnegative and differentiable for
all times t. Which of the following inequalities indicates that the rate of change of the number of people in the
building is increasing at time t ?
(A) f (t) > 0
(B) f'(t) > 0
(C) f (t) − g(t) > 0
(D) f'(t) − g'(t) > 0

7. If is the size of a population at time t, which of the following differential equations describes linear growth in
the size of the population?
(A)
(B)
(C)
(D)
(E)

8. The velocity v, in meters per second, of a certain type of wave is given by v(h) = , where h is the depth, in
meters, of the water through which the wave moves. What is the rate of change, in meters per second per meter, of
the velocity of the wave with respect to the depth of the water, when the depth is 2 meters?
(A)

(B)

(C)

(D)
(E)

9. A particle moves along the x-axis so that at time the position of the particle is given by
. What is the velocity of the particle at the first instance the particle is at
the origin?
(A) -4.071
(B) -2.048
(C) 0
(D) 5.153
(E) 6

10. A particle moves along the x-axis so that its position at time t > 0 is given by x(t) and
. The acceleration of the particle is zero when t =

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Unit 4

(A) 0.387
(B) 0.831
(C) 1.243
(D) 1.647
(E) 8.094

11. A particle moves along a straight line with velocity given by for time t ≥ 0. What is the
acceleration of the particle at time t =4?
(A) 0.422
(B) 0.698
(C) 1.265
(D) 8.794
(E) 28.381

12. A particle moves along the x-axis so that its position at time t is given by x(t)=t2-6t+5. For what value of t is the
velocity of the particle zero?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

13. The position of a particle moving along a straight line at any time t is given by . What is the
acceleration of the particle when ?
(A) 0
(B) 2
(C) 4
(D) 8
(E) 12

14. A particle moves along the x-axis so that at any time t ≥ 0, its velocity is given by v(t) = 3 + 4.1cos(0.9t). What
is the acceleration of the particle at time t = 4
(A) -2.016
(B) -0.677
(C) 1.633
(D) 1.814
(E) 2.978

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Unit 4

15. A particle moves along a line so that at time t, where , its position is given by
. What is the velocity ofthe particle when its acceleration is zero?
(A) -5.19
(B) 0.74
(C) 1.32
(D) 2.55
(E) 8.13

16.

A bug begins to crawl up a vertical wire at time t = 0. The velocity v of the bug at time t, 0 ≤ t ≤ 8, is given by the
function whose graph is shown above.

At what value of t does the bug change direction?

(A) 2
(B) 4
(C) 6
(D) 7
(E) 8

17. The position of a particle moving along the x-axis is given by a twice-differentiable function x(t). If x(2) < 0, x'(2) <
0, and x''(2) <0, which of the following statements must be true about the particle at time t = 2 ?
(A) The particle is moving toward the origin at a decreasing speed.
(B) The particle is moving toward the origin at an increasing speed.
(C) The particle is moving away from the origin at a decreasing speed.
(D) The particle is moving away from the origin at an increasing speed.
(E) The particle is moving away from the origin at a constant speed.

18. The velocity of a particle moving along the x-axis is given by for t ≥ 0. Which of the
following statements describes the motion of the particle at t = 1?

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(A) The particle is moving to the left with positive acceleration.

(B) The particle is moving to the right with positive acceleration.
(C) The particle is moving to the left with negative acceleration.
(D) The particle is moving to the right with negative acceleration.

19.

The table above gives the distance in miles, that a car has traveled at various times t, in hours, during a 6-hour
trip. The graph of the function s is increasing and concave up. Based on the information, which of the following
could be the velocity of the car, in miles per hour, at time
(A) 37
(B) 49
(C) 58
(D) 65
(E) 92

20. Let f be a differentiable function such that f(3)=2 and f'(3)=5. If the tangent line to the graph of f at x=3 is used to
find an approximation to a zero of f, that approximation is
(A) 0.4
(B) 0.5
(C) 2.6
(D) 3.4
(E) 5.5

21. If a and b are positive constants, then

(A) 0
(B)
(C)
(D) 2
(E)

22. An isosceles right triangle with legs of length s has area . At the instant when centimeters, the
area of the triangle is increasing at a rate of 12 square centimeters per second. At what rate is the length of the
hypotenuse of the triangle increasing, in centimeters per second, at that instant?

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(A)
(B) 3
(C)
(D) 48

23. A particle moves along the x-axis so that at any time t ≥ 0 its velocity is given by . What is
the acceleration of the particle at time t = 6?
(A) 1.500
(B) 20.453
(C) 29.453
(D) 74.860
(E) 133.417

24. A cube with edges of length x centimeters has volume V(x) = x3 cubic centimeters. The volume is increasing at a
constant rate of 40 cubic centimeters per minute. At the instant when x = 2, what is the rate of change of x, in
centimeters per minute, with respect to time?
(A) 10/3

(B)

(C) 5
(D) 10

25. is

(A) -2
(B) 0
(C) 1
(D) 2
(E) nonexistent

26. is

(A) 0
(B)
(C)
(D) 1
(E) infinite

27. The position of a particle moving along the x-axis is for time t≥0. When t=π, the
acceleration of the particle is

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(A) 9
(B) 1/9
(C) 0
(D)
(E) -9

28.

A particle moves along a straight line. The graph of the particle’s position x(t) at time t is shown above for 0 < x <6.
The graph has horizontal tangents at t = 1 and t = 5 and a point of inflection at t = 2. For what values of t is the
velocity of the particle increasing?
(A) 0 < t < 2
(B) 1< t<5
(C) 2< t<6
(D) 3 < t < 5 only
(E) 1< t < 2 and 5 < t < 6

29.

(A) 0
(B) 1
(C) 3
(D)

30. The acceleration of a particle moving along the x-axis is given by for .
At what value of t is the particle’s velocity decreasing most rapidly?
(A) 0
(B) 1.420
(C) 3.142
(D) 4.439

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31. A file is downloaded to a computer at a rate modeled by the differentiable function f(t), where t is the time in
seconds since the start of the download and f(t) is measured in megabits per second. Which of the following is the
best interpretation of f'(5) = 2.8 ?
At time t = 5 seconds, the rate at which the file is downloaded to the computer is 2.8 megabits per second.
(A)
(B) (C)(D)
At time t = 5 seconds, the rate at which the file is downloaded to the computer is increasing at a rate of 2.8
(B)
megabits per second per second.
(C) Over the time interval seconds, 2.8 megabits of the file are downloaded to the computer.
Over the time interval seconds, the average rate at which the file is downloaded to the computer
(D)
is 2.8 megabits per second.

32. For t ≥ 0 hours, H is a differentiable function of t that gives the temperature, in degrees Celsius, at an Arctic
weather station. Which of the following is the best interpretation of ?
(A) The change in temperature during the first day
(B) The change in temperature during the 24th hour
(C) The average rate at which the temperature changed during the 24th hour
(D) The rate at which the temperature is changing during the first day
(E) The rate at which the temperature is changing at the end of the 24th hour

33. The function P(t) models the population of the world, in billions of people, where t is the number of years since
January 1, 2010. Which of the following is the best interpretation of the statement P'(1) = 0.076?
(A) On February 1, 2010, the population of the world was increasing at a rate of 0.076 billion people per year.
(B) On January 1, 2011, the population of the world was increasing at a rate of 0.076 billion people per year.
(C) On January 1, 2011, the population of the world was 0.076 billion people.
From January 1, 2010 to January 1, 2011, the population of the world was increasing at an average rate of
(D)
0.076 billion people per year.
When the population of the world was 1 billion people, the population of the world was increasing at a rate
(E)
of 0.076 billion people per year.

34. For a car driven kilometers at a constant speed, the amount of fuel used as a function of the speed is modeled
by a differentiable function . Fuel is measured in liters, and speed is measured in kilometers per hour ( ). In
this context, which of the following is a correct interpretation of the statement ?
(A) Driving at uses more fuel than driving at .
(B) The rate at which liters of fuel is used is greater than the rate at which liters of fuel is used.
(C) The rate of change of the speed is greater when driving at than when driving at .
The rate of change of liters of fuel used with respect to the speed is greater when driving at than
(D)
when driving at .

35. The cost, in dollars, to paint square feet of a house is modeled by the differentiable function . Which of the
following is the best interpretation of the statement ?

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(A) The cost of painting square feet is .

(B) The cost of painting square feet is .
(C) The cost of painting square feet is per square foot.
(D) When square feet have been painted, the cost to paint the next square foot is approximately .

36. Let be the function defined by , and let be a differentiable function with derivative given
by . It is known that . The value of is

(A) 0
(B)
(C) 1
(D) nonexistent

37.

(A)
(B) 0
(C)
(D) 1

38. is

(A)
(B) 0
(C)
(D) nonexistent

39. Let be the function defined by , and let be a differentiable function with derivative given
by . It is known that . The value of is

(A) 0
(B)
(C) 1
(D) nonexistent

40.

(A) 2
(B) 4
(C) 9
(D) 18

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41. is

(A)
(B) 0
(C)
(D) nonexistent

42. The equation is a particular solution to which of the following differential equations?
(A)
(B)
(C)
(D)

43. A tube is being stretched while maintaining its cylindrical shape. The height is increasing at the rate of 2 millimeters
per second. At the instant that the radius of the tube is 6 millimeters, the volume is increasing at the rate of
cubic millimeters per second. Which of the following statements about the surface area of the tube is true at this
instant? (The volume of a cylinder with radius and height is . The surface area of a cylinder, not
including the top and bottom of the cylinder, is .)
(A) The surface area is increasing by square millimeters per second.
(B) The surface area is decreasing by square millimeters per second.
(C) The surface area is increasing by square millimeters per second.
(D) The surface area is decreasing by square millimeters per second.

44. A particle moves on the hyperbola for time seconds. At a certain instant, and .
Which of the following is true about at this instant?
(A) is decreasing by 4 units per second.
(B) is increasing by 4 units per second.
(C) is decreasing by 1 unit per second.
(D) is increasing by 1 unit per second.

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45.

The figure above shows a Ferris wheel with radius 5 meters as Jalen, whose eye level is at point , watches
his friend, Ashanti, ride in one of the cars as the wheel turns. Let denote the distance from Jalen to Ashanti’s car.
The diagram indicates the center of the Ferris wheel at the point and the position of Ashanti’s car at the
point . If and are functions of time , in seconds, what is the rate of change of when , ,
and ? (The equation of a circle with radius and center is .)

(A) , so Ashanti is moving toward Jalen at a rate of approximately 0.47 meter per second.

(B) , so Ashanti is moving away from Jalen at a rate of approximately 0.36 meter per second.

(C) , so Ashanti is moving away from Jalen at a rate of approximately 0.47 meter per second.

(D) , so Ashanti is moving away from Jalen at a rate of 13.5 meters per second.

46. A piece of rubber tubing maintains a cylindrical shape as it is stretched. At the instant that the inner radius of the
tube is 2 millimeters and the height is 20 millimeters, the inner radius is decreasing at the rate of 0.1 millimeter per
second and the height is increasing at the rate of 3 millimeters per second. Which of the following statements about
the volume of the tube is true at this instant? (The volume of a cylinder with radius and height is
.)
(A) The volume is increasing by cubic millimeters per second.
(B) The volume is decreasing by cubic millimeters per second.
(C) The volume is increasing by cubic millimeters per second.
(D) The volume is decreasing by cubic millimeters per second.

47. A particle moves on the hyperbola for time seconds. At a certain instant, and .
Which of the following is true about at this instant?

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(A) is decreasing by 10 units per second.

(B) is increasing by 10 units per second.
(C) is decreasing by 5 units per second.
(D) is increasing by 5 units per second.

48.

A model car travels around a circular track with radius 5 feet. Let denote the distance between the model car
and a fixed point that is 20 feet to the left of the center of the circular track. The diagram above indicates the fixed
point at the origin, the center of the circular track at the point , and the position of the car at the point .
is the length of the line segment from the origin to the point . If and are functions of time , in seconds,
what is the rate of change of when , , and ? (The equation of a circle with radius and
center is .)
(A) , so the distance between the model car and the fixed point is constant.
, so the model car is moving away from the fixed point at a rate of approximately 1.7 feet per
(B)
second.
, so the model car is moving toward the fixed point at a rate of approximately 1.7 feet per
(C)
second.
(D) , so the model car is moving away from the fixed point at a rate of 80 feet per second.

49.

(A) 0
(B)
(C) 8
(D) non existent

50. is

(A) 0
(B) 1/3
(C) 1/2
(D) nonexistent

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51. is

(A)
(B) e-1
(C) 1
(D) 0
(E) ex

52. is

(A) -2
(B)
(C) 0
(D)
(E) nonexistent

53.

(A) 0
(B) 1/8
(C) 1/4
(D) 1
(E) nonexistent

54. is

(A)
(B) 0
(C)
(D) 1
(E) nonexistent

55. If f′(x) = cos x and g′(x) = 1 for all x, and if , then is

(A)
(B) 1
(C) 0
(D) −1
(E) nonexistent

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56.

The third derivative of the function f is continuous on the interval (0,4). Values for f and its first three derivatives at
x=2 are given in the table above. What is ?

(A) 0
(B)
(C) 5
(D) 7
(E) The limit does not exist.

57.
Let g be a continuously differentiable function with and . What is ?

(A) 0
(B)
(C) 1
(D) 2
(E) The limit does not exist.

58. is

(A) 0
(B)
(C) 1
(D) 2
(E) nonexistent

59. is

(A)
(B) 0
(C) 1
(D)
(E) nonexistent

60. is

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(A) 0
(B)
(C) 1
(D) e
(E) nonexistent

61.

(A) 0
(B) 1
(C) 2e
(D)
(E)

62. is

(A)
(B) 0
(C) 1
(D)
(E) nonexistent

63. is

(A) −1
(B) 0
(C) 1
(D)
(E) nonexistent

64. A particle moves along the x-axis so that its acceleration at any time t is . If the initial velocity of the
particle is 6, at what time t during the interval is the particle farthest to the right?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

65. On a certain day, the total number of pieces of candy produced by a factory since it opened is modeled by C, a
differentiable function of the number of hours since the factory opened. Which of the following is the best
interpretation of C'(3) = 500 ?

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(A) The factory produces 500 pieces of candy during its 3rd hour of operation.
(B) The factory produces 500 pieces of candy in the first 3 hours after it opens.
(C) The factory is producing candy at a rate of 500 pieces per hour, 3 hours after it opens.
The rate at which the factory is producing candy is increasing at a rate of 500 pieces per hour per hour, 3
(D)
hours after it opens.

66. The height above the ground of a passenger on a Ferris wheel t minutes after the ride begins is modeled by the
differentiable function H, where H(t) is measured in meters. Which of the following is an interpretation of the
statement H'(7.5) = 15.708?
The Ferris wheel is turning at a rate of 15.708 meters per minute when the passenger is 7.5 meters above the
(A)
ground.
(B) The Ferris wheel is turning at a rate of 15.708 meters per minute 7.5 minutes after the ride begins.
The passenger’s height above the ground is increasing by 15.708 meters per minute when the passenger is
(C)
7.5 meters above the ground.
The passenger’s height above the ground is increasing by 15.708 meters per minute 7.5 minutes after the ride
(D)
begins.
(E) The passenger is 15.708 meters above the ground 7.5 minutes after the ride begins.

67. The number of insects in a certain population at time t days is modeled by the function P with first derivative
. At time , the number of insects in the population is 40. Which of the
following statements are true?

I At time , the number of insects in the population is 2840.

II. At time , the number of insects in the population is increasing at a rate of 360 insects per day.

III. At time , the rate of change of the number of insects in the population is increasing at a rate of 18
insects per day per day.
(A) I only
(B) II only
(C) III only
(D) I, II, and III

68. The rate at which water leaks from a tank, in gallons per hour, is modeled by R, a differentiable function of the
number of hours after the leak is discovered. Which of the following is the best interpretation of R'(3)?
The amount of water, in gallons, that has leaked out of the tank during the first three hours after the leak is
(A)
discovered
The amount of change, in gallons per hour, in the rate at which water is leaking during the three hours after
(B)
the leak is discovered
(C) The rate at which water leaks from the tank, in gallons per hour, three hours after the leak is discovered
The rate of change of the rate at which water leaks from the tank, in gallons per hour per hour, three hours
(D)
after the leak is discovered

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69. The rate at which water leaks from a tank, in gallons per hour, is modeled by R, a differentiable function of the
number of hours after the leak is discovered. Which of the following is the best interpretation of ?
The amount of water, in gallons, that has leaked out of the tank during the first three hours after the leak is
(A)
discovered
The amount of change, in gallons per hour, in the rate at which water is leaking during the three hours after
(B)
the leak is discovered
(C) The rate at which water leaks from the tank, in gallons per hour, three hours after the leak is discovered
The rate of change of the rate at which water leaks from the tank, in gallons per hour per hour, three hours
(D)
after the leak is discovered

70. A particle moves along the x-axis so that at time t > 0 its position is given by x(t) = 12e−tsin t. What is the first time
t at which the velocity of the particle is zero?
(A)
(B)
(C)
(D)

71. A particle moves along the x-axis so that at any time its position is given by x(t) = t3 − 3t2 − 9t + 1 . For
what values of t is the particle at rest?
(A) No values
(B) 1 only
(C) 3 only
(D) 5 only
(E) 1 and 3

72. A particle moves along a straight line with velocity given by at time . What is the
acceleration of the particle at time ?
(A) −0.914
(B) 0.055
(C) 5.486
(D) 6.086
(E) 18.087

73.

The position of a particle moving along a line is given by s(t) = 2t3 - 24t2 + 90t + 7 for t ≥ 0. For what values of t is
the speed of the particle increasing?

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(A) 3 < t < 4 only

(B) t > 4 only
(C) t > 5 only
(D) 0 < t < 3 and t > 5
(E) 3 < t < 4 and t > 5

74.

A particle moves along a straight line. The graph of the particle’s velocity at time t is shown above for
where j, k, l, and m are constants. The graph intersects the horizontal axis at and
and has horizontal tangents at and . For what values of t is the speed of the particle decreasing?
(A)
(B)
(C) and
(D) and
(E) and

75. For the position of a particle moving along the x-axis is given by . What is the
acceleration of the particle at the point where the velocity is first equal to 0 ?
(A)
(B) -1
(C) 0
(D) 1
(E)

76. A particle moves along the x-axis so that at time its position is given by . What is the
velocity of the particle at the first instance the particle is at the origin?
(A) -1
(B) -0.624
(C) -0.318
(D) 0
(E) 0.065

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77. A particle moves along a line so that its velocity is given by for . For what
values of t is the speed of the particle increasing?
(A) and
(B) only
(C) only
(D) only
(E) and

78.

A particle moves along the y-axis. The graph of the particle’s position y(t) at time t is shown above for .
For what values of t is the velocity of the particle negative and the acceleration positive?
(A)
(B)
(C)
(D)
(E)

79. A particle moves on the x-axis so that at any time t,0≤t≤1 its position is given by . For
what value of t is the particle at rest?
(A) 0
(B)
(C)
(D)
(E) 1

80. A particle moves along the x-axis with its position at time t given by where a and b are
constants and . For which of the following values of t is the particle at rest?

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(A)
(B)
(C)
(D)
(E) and

81.

A particle moves on the x-axis with velocity given by v(t) = 3t4 − 11t2 + 9t − 2 for −3 ≤ t ≤ 3 . How many times
does the particle change direction as t increases from to −3 to 3?
(A) Zero
(B) One
(C) Two
(D) Three
(E) Four

82.

A particle moves along a straight line. The graph of the particle’s position x(t) at time t is shown above for 0 < t < 6.
The graph has horizontal tangents at t = 1 and t = 5 and a point of inflection at t = 2 For what values of t is the
velocity of the particle increasing?
(A) 0 < t < 2
(B) 1<t<5
(C) 2<t<6
(D) 3 < t < 5 only
(E) 1 < t < 2 only 5 < t < 6

83. A particle starts from rest at the point (2,0) and moves along the x-axis with a constant positive acceleration for
time . Which of the following could be the graph of the distance s(t) of the particle from the origin as a
function of time t ?

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(A)

(B)

(C)

(D)

(E)

84. Let be the function defined by . For how many values of in the open
interval is the instantaneous rate of change of equal to the average rate of change of on the closed
interval ?
(A) Zero
(B) One
(C) Three
(D) Four

85. Let be the function with first derivative for . If , what is the value of
?
(A)
(B)
(C)
(D)

86. ⅆ
is

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(A)
(B)
(C)
(D) nonexistent

87. ⅆ
is

(A)
(B)
(C)
(D) nonexistent

88.
Let be a differentiable function with and . Using the line tangent to the graph of at
as a local linear approximation for , what is the estimate for ?
(A)
(B)
(C)
(D)

89. Let be a function such that at each point on the graph of , the slope is given by . The graph
of passes through the point and is concave down on the interval . Let be the approximation
for found by using the locally linear approximation of at . Which of the following statements about
is true?
(A) and is an overestimate for .
(B) and is an underestimate for .
(C) and is an overestimate for .
(D) and is an underestimate for .

90.

Selected values of the derivative of the function are given in the table above. It is known that . What is
the approximation for found using the line tangent to the graph of at ?
(A) 12.44
(B) 12.40
(C) 12.36
(D) 11.60

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91. Let be a differentiable function such that and . The graph of is concave down on the
interval . Which of the following is true about the approximation for found using the line tangent to
the graph of at ?
(A) and this approximation is an overestimate of the value of .
(B) and this approximation is an underestimate of the value of .
(C) and this approximation is an overestimate of the value of .
(D) and this approximation is an underestimate of the value of .

92.

Selected values of the derivative of the function are given in the table above. It is known that . What is
the approximation for found using the line tangent to the graph of at ?
(A) 16.76
(B) 16.80
(C) 16.84
(D) 17.40

93. Let be a differentiable function such that and . The graph of is concave up on the interval
. Which of the following is true about the approximation for found using the line tangent to the graph
of at ?
(A) and this approximation is an overestimate of the value of .
(B) and this approximation is an underestimate of the value of .
(C) and this approximation is an overestimate of the value of .
(D) and this approximation is an underestimate of the value of .

94. Let be a function such that at each point on the graph of , the slope is given by .
The graph of passes through the point and is concave up on the interval . Let be the
approximation for found by using the locally linear approximation of at . Which of the following
statements about is true?
(A) and is an underestimate for .
(B) and is an overestimate for .
(C) and is an underestimate for .
(D) and is an overestimate for .

95. The function models the time, in hours, for a sample of water to evaporate as a function of the size of
the sample, measured in milliliters. What are the units for ?

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(A) hours per milliliter

(B) milliliters per hour
(C) hours per milliliter per milliliter
(D) milliliters per hour per hour

96. The function gives the production cost for a bakery to produce cakes of a certain type, where is the cost, in
dollars, to produce of the cakes. The function defined by gives the marginal
cost, in dollars, to produce cake number . Which of the following gives the best estimate for the marginal
cost, in dollars, to produce the 40th cake?
(A)

(B)
(C)
(D)

97.

The function defined above models the depth, in feet, of the water hours after 12 A.M. in a certain harbor.
Which of the following presents the method for finding the instantaneous rate of change of the depth of the water, in
feet per hour, at 6 A.M. ?
(A)
(B)
(C)
(D)

98. The function models the time, in minutes, for a chemical reaction to occur as a function of the amount
of catalyst used, measured in milliliters. What are the units for ?
(A) minutes per milliliter
(B) milliliters per minute
(C) minutes per milliliter per milliliter
(D) milliliters per minute per minute

99. The function gives the cost, in dollars, to produce a particular product, where is the cost, in dollars, to
produce units of the product. The function defined by gives the marginal cost,
in dollars, to produce unit number . Which of the following gives the best estimate for the marginal cost, in
dollars, to produce the 57th unit of the product?
(A)

(B)
(C)
(D)

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100.

The function defined above models the electric charge, measured in coulombs, inside a lightbulb seconds after
it is turned on. Which of the following presents the method for finding the instantaneous rate of change of the
lightbulb’s electric charge, in coulombs per second, at time ?
(A)
(B)
(C)
(D)

101. At time , , the velocity of a particle moving along the -axis is given by . Let
be the time at which the particle changes direction from moving left to moving right. What is the total
distance traveled by the particle during the time interval ?
(A)
(B)
(C)
(D)

102. A particle moves along the curve for . The -coordinate of the particle changes at a
constant rate of units per second. At the instant when the -coordinate of the particle is , what is the rate of
change of the -coordinate of the particle, in units per second?
(A)
(B)
(C)
(D)

103. Paint spills onto a floor in a circular pattern. The radius of the spill increases at a constant rate of inches per
minute. How fast is the area of the spill increasing when the radius of the spill is inches?
(A)
(B)
(C)
(D)

104. A particle moves along the -axis so that at any time its position is given by , where is a
positive constant. At what time is the particle’s position farthest to the right?
(A)
(B)
(C)
(D) There is no such value of .

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105.

A particle traveling on the -axis has position at time . The graph of the particle’s velocity is shown
above for . Which of the following expressions gives the total distance traveled by the particle over the
time interval ?
(A)
(B)
(C)
(D)

106. An object moves along a straight line so that at any time , , its position is given by
. For what value of is the object at rest?
(A)
(B)
(C)
(D)

107. Let be a function with and . What is the approximation of obtained by using the
line tangent to the graph of at ?
(A)
(B)
(C)
(D)

108. The derivative of the function is given by . If , what is the approximation for
found by using the line tangent to the graph of at ?

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(A)
(B)
(C)
(D)

109. The number of gallons of water in a storage tank at time t, in minutes, is modeled by w(t) = 25 − t2 for .
At what rate, in gallons per minute, is the amount of water in the tank changing at time t = 3 minutes?
(A) 66
(B) 16
(C) -3
(D) -6

110. Let f and g be functions that are differentiable for all real numbers, with for . If
and exists, then is

(A) 0
(B)

(C)

(D)

(E) nonexistent

111. The volume of a certain cone for which the sum of its radius, r, and height is constant is given by
. The rate of change of the radius of the cone with respect to time is 6. In terms of r, what is
the rate of change of the volume of the cone with respect to time?
(A)
(B)
(C)
(D)
(E)

112. At time , a cube has volume and edges of length . If the volume of the cube decreases at a rate
proportional to its surface area, which of the following differential equations could describe the rate at which the
volume of the cube decreases?
(A)
(B)
(C)
(D)
(E)

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113. The radius of a circle is increasing. At a certain instant, the rate of increase in the area of the circle is numerically
equal to twice the rate of increase in its circumference. What is the radius of the circle at that instant?
(A)
(B) 1
(C)
(D) 2
(E) 4

114.

A person whose height is 6 feet is walking away from the base of a streetlight along a straight path at a rate of 4 feet
per second. If the height of the streetlight is 15 feet, what is the rate at which the person’s shadow is lengthening?
(A) 1.5 ft/sec
(B) 2.667 ft/sec
(C) 3.75 ft/sec
(D) 6 ft/sec
(E) 10 ft/sec

115.

A container has the shape of an open right circular cone, as shown in the figure above. The container has a
radius of 4 feet at the top, and its height is 12 feet. If water flows into the container at a constant rate of 6 cubic feet
per minute, how fast is the water level rising when the height of the water is 5 feet? (The volume V of a cone with
radius r and height h is .)

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(A) 0.358 ft/min

(B) 0.688 ft/min
(C) 2.063 ft/min
(D) 8.727 ft/min
(E) 52.360 ft/min

116. Sand is deposited into a pile with a circular base. The volume V of the pile is given by , where r is the
radius of the base, in feet. The circumference of the base is increasing at a constant rate of feet per hour. When
the circumference of the base is feet, what is the rate of change of the volume of the pile, in cubic feet per hour?
(A)
(B) 16
(C) 40
(D)
(E)

117. A cup has the shape of a right circular cone. The height of the cup is 12 cm, and the radius of the opening is 3 cm.
Water is poured into the cup at a constant rate of 2cm3/sec What is the rate at which the water level is rising when
the depth of the water in the cup is 5 cm? (The volume of a cone of height h and radius r is given by )
(A) cm/sec
(B) cm/sec
(C) cm/sec
(D) cm/sec
(E) cm/sec

118. The volume of a sphere is increasing at a rate of 6π cubic centimeters per hour. At what rate, in centimeters per
hour, is its diameter increasing with respect to time at the instant the radius of the sphere is 3 centimeters?

(Note: The volume of a sphere with radius r is given by .)

(A) 1/3
(B) 1
(C)
(D) 6

119. The positive variables p and c change with respect to time t. The relationship between p and c is given by the
equation . At the instant when and what is the value of

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(A)
(B)

(C)

(D)

120. The height h, in meters, of an object at time t is given by h(t) = 24t + 24t3/2 - 16t2. What is the height of the
object at the instant when it reaches its maximum upward velocity?
(A) 2.545 meters
(B) 10.263 meters
(C) 34.125 meters
(D) 54.889 meters
(E) 89.005 meters

121.
Let be the function given by ⅆ . What is the -coordinate of the point of inflection
of the graph of ?
(A)
(B)
(C)
(D)

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122.

The graph of on the interval is shown above. Which of the following could be the graph of
?

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(A)

(B)

(C)

(D)

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123.

The graph of , the derivative of the function , is shown above. If , what is the approximation for
using the line tangent to the graph of at ?
(A)
(B)
(C)
(D)

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124.

An amusement park ride reaches a maximum height of feet above the ground. The graph shows the height above
the ground, in feet, of a passenger on the ride for times seconds. At which of the following points is
the height of the passenger above the ground changing the fastest?
(A)
(B)
(C)
(D)

125. If is a function that has a removable discontinuity at , which of the following could be the graph of ?

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(A)

(B)

(C)

(D)

126. Let f be a twice-differentiable function such that for all x. The graph of is the secant
line passing through the points and . The graph of is the line tangent to the
graph of f at . Which of the following is true?
(A)
(B)
(C)
(D)

127. Let be a differentiable function such that and . What is the approximation of
using the line tangent to the graph of f at ?

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(A) 0.4
(B) 2.025
(C) 2.4
(D) 6

128. Let f be the function defined by . What is the approximation for found by using the line
tangent to the graph of f at the point ?
(A)
(B)
(C)
(D)

129. Let f be a differentiable function such that f (2) = 4 and f (2) = − 1/2 . What is the approximation for f (2.1) found
by using the line tangent to the graph of f at x = 2 ?
(A) 2.95
(B) 3.95
(C) 4.05
(D) 4.1

130. Let f be the function given by . The tangent line to the graph of f at x=2 is used to
approximate values of f(x). Which of the following is the greatest value of x for which the error resulting from this
tangent line approximation is less than 0.5?
(A) 2.4
(B) 2.5
(C) 2.6
(D) 2.7
(E) 2.8

131. For the function and . What is the approximation for found by using the line
tangent to the graph of f at ?
(A) 0.6
(B) 3.4
(C) 4.2
(D) 4.6
(E) 4.64

132. Let f be the function given by f(x) = 2 cos x + 1. What is the approximation for f(1.5) found by using the line
tangent to the graph of f at ?

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(A) -2
(B) 1
(C)
(D)

133. The function f is twice differentiable with f(2) = 1 ,f′(2) = 4 , and f″(2) = 3 . What is the value of the approximation
of f(1.9) using the line tangent to the graph of f at x = 2 ?
(A) 0.4
(B) 0.6
(C) 0.7
(D) 1.3
(E) 1.4

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