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AP®MathsCollege BoardCalculus ABExam QuestionsUnit 4: Contextual Applications of DifferentiationRates of Change & Related Rates

Rates of Change & Related Rates (College Board AP® Calculus AB): Exam Questions

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Rates of Change & Related Rates

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1
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The radius of a circle is decreasing at a constant rate of 2 centimeters per second. In terms of the circumference C, what is the rate of change of the area of the circle, in square centimeters per second?

  • 2 C

  • negative 2 C

  • negative 4 C

  • C

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2
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A particle moves along a straight line with velocity given by v open parentheses t close parentheses equals 9 minus 2 t cubed at time t greater or equal than 0. What is the acceleration of the particle at time t equals 2?

  • -15

  • 24

  • -24

  • 10

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3
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A particle moves along a straight line with a position given by s open parentheses t close parentheses equals 8 minus open parentheses 1.02 close parentheses to the power of t at time t greater or equal than 0. What is the velocity of the particle at time t equals 4?

  • 6.92

  • -4.24

  • 0.0214

  • -0.0214

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4
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The edges of a cube are increasing at a uniform rate of 0.1 inches per second. At the instant when the total surface area becomes 54 square inches, what is the rate of increase, in cubic inches per second, of the volume of the cube?

  • 2.7

  • 3.6

  • 1.134

  • 0.9

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5
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When the area of an expanding square, in square units, is increasing 5 times as fast as its side length is increasing, the length of the side in linear units is

  • 2 over 5

  • 5 over 2

  • 1 over 10

  • 10

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6
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A particle moves along the x-axis so that its position at time t is given by x open parentheses t close parentheses equals t squared minus 9 t plus 20. For what value of t is the velocity of the particle zero?

  • 3

  • 4

  • 4.5

  • 5

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7
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A particle moves along the x-axis so that at any time t greater or equal than 0 its position is given by x open parentheses t close parentheses equals 2 over 3 t cubed minus 3 t squared minus 20 t. For what values of t is the particle at rest?

  • 0 only

  • 2 only

  • 5 only

  • 2 and 5

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8
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The velocity of a particle moving along a straight line is given by v open parentheses t close parentheses equals 2.2 t cross times ln open parentheses 0.05 t plus 0.4 close parentheses for time t greater or equal than 0. What is the acceleration of the particle at time t equals 12?

  • -38.107

  • 0

  • 1.076

  • 1.32

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9
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A particle moves along the x-axis so that at time t greater or equal than 0 its position is given by x open parentheses t close parentheses equals 2 t cubed minus 15 t squared plus 36 t minus 20. At what time t is the particle at rest?

  • t equals 2 only

  • t equals 2.5 only

  • t equals 2 and t equals 3

  • t equals 3 and t equals 4

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1
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A particle moves along the x-axis with its position at time t given by x open parentheses t close parentheses equals open parentheses 2 t minus a close parentheses open parentheses t minus b close parentheses, where a and b are non-zero constants and a over 2 not equal to b.

For which of the following values of t is the particle at rest?

  • t equals fraction numerator a plus b over denominator 2 end fraction

  • t equals fraction numerator a plus 2 b over denominator 4 end fraction

  • t equals a over 2 and t equals b

  • t equals fraction numerator a minus 2 b over denominator 4 end fraction

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2
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For t greater or equal than 0, the position of a particle moving along the x-axis is given by x open parentheses t close parentheses equals cos space t space minus space sin space t. What is the velocity of the particle at the point where the acceleration is first equal to 0?

  • negative square root of 2

  • square root of 2

  • -1

  • 0

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3
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The radius of a sphere is changing at a rate directly proportional to the radius, where the constant of proportionality is k.

At the instance when the radius of the sphere is 4 centimeters, what is the rate of change, in terms of k, of the surface area of the sphere? (The surface area S of a sphere with radius r is S equals 4 pi r squared.)

  • 128 k pi

  • 32 pi

  • 32 k pi

  • 128 k

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4
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The radius of a circle is increasing at a constant rate of 0.2 meters per second. What is the rate of increase in the area of the circle at the instant when the circumference of the circle is 25 pi meters?

  • 25 pi

  • 10 pi

  • 5 pi

  • fraction numerator 2 pi over denominator 5 end fraction

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5
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Let y equals sin space x cross times ln space x. Both x and y vary with time in such a way that x increases at the constant rate of 2 pi units per second. The rate at which y is changing when x equals pi over 2 is

  • 0

  • 1

  • 4

  • negative 4

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6
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A security light on the wall of a bank shines on a fleeing criminal . The light is at a height of 4 meters above the ground, the criminal is 1.75 meters tall and is running at a speed of 6 meters per second. What is the rate at which the criminal's shadow is lengthening?

Silhouette of a person running under a light, casting a shadow on the ground, with dashed lines illustrating the light beam angle.

  • 21 over 8 meters per second

  • 14 over 3 meters per second

  • 6 meters per second

  • 54 over 7 meters per second

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7
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The maximum acceleration attained on the interval 0 less or equal than t less or equal than 4 by the particle whose velocity is given by v open parentheses t close parentheses equals negative 1 third t cubed plus t squared plus 8 t plus 5 occurs when t equals

  • -2

  • 1

  • 4

  • 9

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8
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The graph below shows the displacement, x, of a particle in meters in the interval 0 less or equal than t less or equal than 4 seconds.

Which of the following tables could be the table of values for the velocity, v, in meters per second for the particle in the interval 1 less or equal than t less or equal than 3?

Graph showing a curve with an m-shape peaking at 1 and 3 on the horizontal axis, and passing through the origin and (4, 0).

  • t

    1

    1.5

    2

    2.5

    3

    v open parentheses t close parentheses

    3

    -2

    0

    -2

    3

  • t

    1

    1.5

    2

    2.5

    3

    v open parentheses t close parentheses

    0

    -3

    0

    6

    0

  • t

    1

    1.5

    2

    2.5

    3

    v open parentheses t close parentheses

    0

    -3

    0

    3

    0

  • t

    1

    1.5

    2

    2.5

    3

    v open parentheses t close parentheses

    0

    3

    0

    -3

    0

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1
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If the base b of a triangle is increasing at a rate of 12 centimeters per second while its height h is decreasing at a rate 12 centimeters per second, which of the following must be true about the area A of the triangle?

  • A is always increasing.

  • A is always decreasing.

  • A is decreasing only when b less than h.

  • A is decreasing only when b greater than h.

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2
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The radius of a right circular cylinder is increasing at a rate of 4 units per second. The height of the cylinder is decreasing at a rate of 6 units per second.

Which of the following expressions gives the rate at which the volume of the cylinder is changing with respect to time in terms of the radius r and height h of the cylinder?

(The volume of a cylinder with radius r and height h is V equals pi r squared h.)

  • negative 12 pi r

  • 8 pi r

  • 8 pi r h minus 6 pi r squared

  • 8 pi r h plus 6 pi r squared

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3
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A particle moves along the x-axis so that at any time t greater or equal than 0, its velocity is given by v open parentheses t close parentheses equals 4 t plus 4 cos open parentheses 2 t close parentheses. At what time, where t greater than 0, is the acceleration equal to zero for the second time?

  • pi over 12

  • fraction numerator 5 pi over denominator 12 end fraction

  • pi over 6

  • fraction numerator 5 pi over denominator 6 end fraction

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4
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The diagonal c of the rectangle shown is increasing at the rate of 3 centimeters per second. The rate of change of length a is twice the rate of change of length b. At what rate is the length b increasing when a equals 5 cm and b equals 12 cm ?

Rectangle with a diagonal dividing it into two right triangles, labelled sides a, b, and hypotenuse c.

  • 2.690

  • 2.294

  • 1.773

  • 0.886

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5
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Let y equals 3 e to the power of cos open parentheses 4 x close parentheses end exponent. Both x and y vary with time in such a way that y increases at the constant rate of 6 units per second. The rate at which x is changing when x equals pi over 8 is

  • negative 2

  • negative 72

  • 1 half

  • negative 1 half

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