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

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

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Rates of Change & Related Rates

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Rates of Change & Related Rates

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 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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3
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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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4
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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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5
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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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