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

Graphs of Functions & Their Derivatives (College Board AP® Calculus BC): 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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6
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Graph of f'. The derivative of f is made up of 4 line segments, joining the coordinates (-4, -2), (-2, 2), (-1, 0), (1, 1), and (5, -1). The first line segment intersects the x-axis at x = -3 and the last line segment intersects the x-axis at 3x = .

The graph of f apostrophe, the derivative of f, is shown in the figure above. The function f has a critical point of inflection at x equals

  • negative 3

  • negative 2

  • negative 1

  • 3

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7
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What is the x-coordinate of the point of inflection on the graph of y equals 1 half x cubed minus 9 over 2 x squared plus 7?

  • negative 3

  • 0

  • 3

  • 6

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8
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If f is continuous for a less or equal than x less or equal than b and differentiable for a less than x less than b, which of the following is true?

  • f apostrophe open parentheses c close parentheses equals fraction numerator f open parentheses b close parentheses minus f open parentheses a close parentheses over denominator b minus a end fraction for some c such that a less than c less than b.

  • f apostrophe open parentheses x close parentheses not equal to 0 for all values of x on a less than x less than b.

  • f does not have a minimum value on a less than x less than b.

  • integral subscript a superscript b f open parentheses x close parentheses space d x does not exist.

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9
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If f apostrophe apostrophe open parentheses x close parentheses equals x open parentheses x minus 4 close parentheses open parentheses x plus 3 close parentheses squared, then the graph of f has inflection points when x equals

  • negative 3 only

  • 4 only

  • 0 and 4 only

  • negative 3, 0 and 4 only

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10
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Graph starting in the second quadrant, curving down into the third quadrant then back up through the fourth quadrant and into the first quadrant. The graph has a line of symmetry about the y-axis. The x-coordinates of the start and end points of the curve are labeled as 'a' and 'b' respectively.

The graph of f is shown in the figure above. Which of the following could be the graph of the derivative of f?

  • A straight line graph starting in the second quadrant and ending in the fourth quadrant. The line goes through the origin. The x-coordinates of the start and end points are labeled 'a' and 'b' respectively.
  • Graph that starts in the third quadrant and curves up into the second quadrant before curving back down through the first quadrant into the fourth quadrant. There is a line of symmetry about the y-axis. The x-coordinates of the start and end points are labeled 'a' and 'b' respectively.
  • A curve that starts at point 'a' on the negative x-axis and curves down in the third quadrant before curving back up through the origin into the first quadrant before curving back down to end at point 'b' on the positive x-axis.
  • A curve that starts at point 'a' on the negative x-axis and curves up in the second quadrant before curving back down through the origin into the fourth quadrant before curving back up to end at point 'b' on the positive x-axis.

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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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6
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x

negative 4

negative 3

negative 2

negative 1

0

f apostrophe left parenthesis x right parenthesis

26

9

negative 2

negative 7

negative 6

Let f be a polynomial function with values of f apostrophe left parenthesis x right parenthesis at selected values of x given in the table above. Which of the following must be true for negative 4 less than x less than 0?

  • The graph of f has at least two points of inflection.

  • f is decreasing.

  • The graph of f is concave up.

  • The graph of f has a local maximum.

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7
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Curve starting in the second quadrant curving down to the fourth quadrant. The curve intersects the x-axis at x = 1.

The graph of a twice-differentiable function f is shown in the figure above. Which of the following is true?

  • f open parentheses 1 close parentheses equals f apostrophe open parentheses 1 close parentheses equals f apostrophe apostrophe open parentheses 1 close parentheses

  • f open parentheses 1 close parentheses less than f apostrophe open parentheses 1 close parentheses less than f apostrophe apostrophe open parentheses 1 close parentheses

  • f apostrophe open parentheses 1 close parentheses less than f open parentheses 1 close parentheses less than f apostrophe apostrophe open parentheses 1 close parentheses

  • f apostrophe apostrophe open parentheses 1 close parentheses less than f open parentheses 1 close parentheses less than f apostrophe open parentheses 1 close parentheses

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8
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The function f is given by f open parentheses x close parentheses equals 5 minus 3 over 4 x to the power of 4 minus 3 x squared. On which of the following intervals is f increasing?

  • open parentheses negative infinity comma space minus square root of 2 close parentheses

  • open parentheses negative infinity comma space 0 close parentheses

  • open parentheses negative square root of 2 comma space square root of 2 close parentheses

  • open parentheses 0 comma space infinity close parentheses

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9
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The first derivative of the function f is given by f apostrophe open parentheses x close parentheses equals 1 minus x sin squared x. How many critical points does f have on the open interval open parentheses 0 comma space 6 close parentheses?

  • 1

  • 3

  • 4

  • 7

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10
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The absolute minimum value of f left parenthesis x right parenthesis equals 3 x cubed plus 18 x squared minus 7 on the closed interval left square bracket negative 5 comma space 1 right square bracket occurs at x equals

  • negative 5

  • negative 4

  • 0

  • 1

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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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6
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If g is a differentiable function such that g left parenthesis x right parenthesis greater than 0 for all real numbers x and if f apostrophe left parenthesis x right parenthesis equals open parentheses 9 x squared minus 16 close parentheses g open parentheses x close parentheses, which of the following is true?

  • f has a relative maximum at x equals negative 4 over 3 and a relative minimum at x equals 4 over 3.

  • f has a relative minimum at x equals negative 4 over 3 and a relative maximum at x equals 4 over 3.

  • f has relative minima at x equals negative 4 over 3 and at x equals 4 over 3.

  • f has relative maxima at x equals negative 4 over 3 and at x equals 4 over 3.

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7
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x

negative 2

negative 1

1

3

4

f apostrophe open parentheses x close parentheses

negative 2

negative 5

negative 7

negative 8

negative 10

Let f be a twice-differentiable function. Values of f apostrophe, the derivative of f, at selected values of x are given in the table above. Which of the following statements must be true?

  • f is decreasing for negative 2 less or equal than x less or equal than 4.

  • The graph of f is concave up for negative 2 less or equal than x less or equal than 4.

  • There exists c, where negative 2 less or equal than c less or equal than 4, such that f apostrophe open parentheses c close parentheses equals negative 4 over 3.

  • There exists c, where negative 2 less or equal than c less or equal than 4, such that f apostrophe apostrophe open parentheses c close parentheses equals negative 4 over 3.

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8
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The function f is continuous on the closed interval open square brackets negative 2 comma space 1 close square brackets and twice differentiable on the open interval open parentheses negative 2 comma space 1 close parentheses. If f apostrophe open parentheses 0 close parentheses equals 3 and f apostrophe apostrophe open parentheses x close parentheses greater than 0 on the open interval open parentheses negative 2 comma space 1 close parentheses, which of the following could be a table of values for f?

  • x

    f open parentheses x close parentheses

    negative 2

    negative 7

    negative 1

    negative 5

    0

    negative 1

    1

    5

  • x

    f open parentheses x close parentheses

    negative 2

    negative 1

    negative 1

    0

    0

    3

    1

    8

  • x

    f open parentheses x close parentheses

    negative 2

    1

    negative 1

    6

    0

    10

    1

    12

  • x

    f open parentheses x close parentheses

    negative 2

    4

    negative 1

    5

    0

    7

    1

    11

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9
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Let f be the function given by f left parenthesis x right parenthesis equals 3 x e to the power of x. The graph of f is concave up when

  • x less than negative 2

  • x greater than negative 2

  • x less than negative 1

  • x greater than negative 1

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10
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The function f has the property that f open parentheses x close parentheses and f apostrophe apostrophe open parentheses x close parentheses are positive for all real values x, and f apostrophe open parentheses x close parentheses is positive only for x greater than a where a is a finite number. Which of the following could be the graph of f?

  • Graph of a function that starts in the second quadrant and curves up into the first quadrant. There is an asymptote parallel to the x-axis that intersects the positive y-axis.
  • Graph of a function that starts in the second quadrant and curves down into the first quadrant. There is an asymptote parallel to the x-axis that intersects the positive y-axis.
  • Graph of a function that starts in the second quadrant curves down before curving up again into the first quadrant. The curve does not intersect the x-axis.
  • Graph of a function that starts in the second quadrant and curves down into the first quadrant. There are two asymptotes parallel to the x-axis that both intersect the positive y-axis.

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