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AP®MathsCollege BoardCalculus ABExam QuestionsUnit 5: Analytical Applications of DifferentiationGraphs of Functions & Their Derivatives

Graphs of Functions & Their Derivatives (College Board AP® Calculus AB)

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Graphs of Functions & Their Derivatives

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Graphs of Functions & Their Derivatives

1
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Graph of a polynomial function intersecting the x-axis at points A, C, and E, with local maximum at B and local minimum at D, labeled x and y axes.
Graph of f''

The graph of the cubic function f to the power of apostrophe apostrophe end exponent, the second derivative of the function f, is shown above. The letters A comma space B comma space C comma space D comma space E represent the x-coordinates at each of the labeled points.

What are all the intervals on which the graph of the function f is concave down?

  • A less than x less than C

  • B less than x less than D

  • C less than x less than E only

  • x less than A and C less than x less than E

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2
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Let f be a polynomial function with degree greater than 2. If f open parentheses 1 close parentheses equals f open parentheses 6 close parentheses equals 5, which of the following must be true for at least one value of x between 1 and 6?

I. f open parentheses x close parentheses equals 0

II. f to the power of apostrophe open parentheses x close parentheses equals 0

III. f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses equals 0

  • None

  • I only

  • II only

  • I and II only

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3
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Let f be the function given by f open parentheses x close parentheses equals 600 x minus 8 x cubed. On which of the following intervals is the function f increasing?

  • (negative infinity comma space minus 5] and [5 comma space infinity)

  • open square brackets negative 5 comma space 5 close square brackets

  • open square brackets 0 comma space 5 close square brackets only

  • (negative infinity comma space minus 5 square root of 3] and [5 square root of 3 comma space infinity)

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4
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Let g be a function with first derivative given by g to the power of apostrophe open parentheses x close parentheses equals integral subscript 0 superscript x e to the power of negative 2 t end exponent italic d t. Which of the following must be true on the interval 0 less than x less than 3?

  • g is increasing, and the graph of g is concave up.

  • g is increasing, and the graph of g is concave down.

  • g is decreasing, and the graph of g is concave up.

  • g is decreasing, and the graph of g is concave down.

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5
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A graph of a polynomial function with labeled points A, B, C, and D along the curve, intersecting the x-axis and y-axis. The x- and y-axes are marked with arrows.
Graph of f

The graph of the function f is shown in the figure above. At which of the labeled points is f to the power of apostrophe open parentheses x close parentheses positive and increasing?

  • A

  • B

  • C

  • D

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6
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The volume of a cylindrical plastic container with a top and a bottom is to be 128 pi cubic centimeters. If a minimum amount of plastic is to be used to construct the container, what must be the height, in centimeters, of the container?

  • 2

  • cube root of 128

  • 4

  • 8

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7
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Graph of a mathematical function with x and y axes, showing points A, B, C, D, and E on the curve at various positions along the axes.

The graph of f to the power of apostrophe, the derivative of the function f, is shown above.

Which of the following statements must be true?

I. f has a relative minimum at x equals D.

II. The graph of f has a point of inflection at x equals C.

III. The graph of f is concave down for A less than x less than 0.

  • I and II only

  • I and III only

  • II and III only

  • All three statements are true

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8
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Let f be the function defined by f open parentheses x close parentheses equals fraction numerator ln open parentheses 1 over x close parentheses over denominator x squared end fraction comma space space x greater than 0.

What is the absolute minimum value of f?

  • f does not have an absolute minimum value.

  • e to the power of 1 half end exponent

  • negative e over 2

  • negative fraction numerator 1 over denominator 2 e end fraction

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