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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): Exam Questions

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Graphs of Functions & Their Derivatives

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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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If f apostrophe left parenthesis x right parenthesis equals 2 x minus x cubed plus square root of x to the power of 4 plus 2 end root minus 1, then f has a local minimum at x equals

  • negative 1.103

  • negative 0.712

  • negative 0.212

  • 1.865

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5
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For negative 2 less than x less than 2, let f be a function with first derivative given by f apostrophe left parenthesis x right parenthesis equals cube root of x left parenthesis x minus x cubed right parenthesis plus 2 x squared minus 2. Which of the following are all intervals on which the graph of f is concave up?

  • open parentheses negative 1.372 comma space 1.372 close parentheses

  • open parentheses negative 2 comma space minus 1.372 close parentheses and open parentheses 0 comma space 1.372 close parentheses

  • open parentheses negative 1.682 comma space minus 1 close parentheses and open parentheses 1 comma space 1.682 close parentheses

  • open parentheses negative 2 comma space minus 1.682 close parentheses, open parentheses negative 1 comma space 1 close parenthesesand open parentheses 1.682 comma space 2 close parentheses

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1
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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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2
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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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3
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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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4
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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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5
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Let f be the function f left parenthesis x right parenthesis equals square root of x e to the power of k x end exponent, where k is a constant. For what value of k does f have a critical point at x equals 1 fourth?

  • negative 2

  • negative 1 half

  • 0

  • 2

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1
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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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2
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Graph of y=f'(x), a straight line going between -3 on the x-axis and 9 on the y-axis.

The graph of f apostrophe, the derivative of f, is the line shown in the figure above. If f open parentheses 0 close parentheses equals 4, then f open parentheses negative 2 close parentheses equals

  • negative 20

  • negative 12

  • negative 8

  • negative 2

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3
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Let f be a twice-differentiable function with f apostrophe open parentheses x close parentheses less than 0 and f apostrophe apostrophe open parentheses x close parentheses less than 0 for all real numbers x, such that f open parentheses 2 close parentheses equals 4 and f open parentheses 4 close parentheses equals negative 4. Of the following, which is a possible value for f open parentheses 6 close parentheses.

  • negative 14

  • negative 12

  • negative 10

  • negative 8

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4
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Graph of the movement of a particle, x(t), over time, t. The horizontal axis is labeled t and goes from 0 to 18. The vertical axis is labeled x(t) and goes from -2 to 3. The curve starts at (0, 0), goes up to a local maximum at x=4.5 and down to a local minimum at x=11 before rising again and finishing above the x axis.

A particle moves along a straight line. The graph of the particle's position x left parenthesis t right parenthesis at time t is shown above for 0 less than t less than 18. The graph has horizontal tangents at t equals 4.5 and t equals 11 and points of inflection at t equals 8 and t equals 15. For what values of t is the velocity of the particle decreasing?

  • 4.5 less than t less than 11

  • 8 less than t less than 15

  • 9 less than t less than 11 only

  • 0 less than t less than 8 and 15 less than t less than 18

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5
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The function f is continuous for negative 5 less or equal than x less or equal than 3 and f open parentheses negative 5 close parentheses equals f open parentheses 3 close parentheses equals 7. If there is no c, where negative 5 less than c less than 3, for which f apostrophe open parentheses c close parentheses equals 0, which of the following statements must be true?

  • For negative 5 less than k less than 3 comma space space f apostrophe open parentheses k close parentheses greater than 0.

  • For negative 5 less than k less than 3 comma space space f apostrophe open parentheses k close parentheses less than 0.

  • For negative 5 less than k less than 3 comma space space f open parentheses k close parentheses exists.

  • For some k, where negative 5 less than k less than 3 comma space space f apostrophe open parentheses k close parentheses does not exist.

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