AP®MathsCollege BoardPrecalculusExam QuestionsPolynomial & Rational FunctionsFunctions & Rates of Change

Functions & Rates of Change (College Board AP® Precalculus): Exam Questions

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Functions & Rates of Change

1
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1 mark

A musician released a new song on a streaming service. A streaming service is an online entertainment source that allows users to play music on their computers and mobile devices.

Several months later, the musician began using an app (at time t equals 0) that counts the total number of plays for the song since its release. A “play” is a single stream of the song on the streaming service. The table gives the total number of plays, in thousands, for selected times t months after the musician began using the app. At t equals 0, the total number of plays was 25 thousand. At t equals 2, the total number of plays was 30 thousand. At t equals 4, the total number of plays was 34 thousand.

Months after the musician began using the app

0

2

4

Total number of plays for the song since its release (thousands)

25

30

34

Use the given data to find the average rate of change of the total number of plays for the song, in thousands per month, from t equals 0 to t equals 4 months. Express your answer as a decimal approximation. Show the computations that lead to your answer.

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2
2 marks

For a guitar to make a sound, the strings need to vibrate, or move up and down or back and forth, in a motion that can be modeled by a periodic function.

At time t equals 0 seconds, point X on one vibrating guitar string starts at its highest position, 2 millimeters above its resting position. Then it passes through its resting position and moves to its lowest position, 2 millimeters below the resting position. Point X then passes through its resting position and returns to 2 millimeters above the resting position.

The sinusoidal function h models how far point X is from its resting position, in millimeters, as a function of time t, in seconds. A positive value of h open parentheses t close parentheses indicates the point is above the resting position; a negative value of h open parentheses t close parentheses indicates the point is below the resting position.

The graph of h and its dashed midline for two full cycles is shown. Five points, F, G, space J, K, and P, are labeled on the graph.

Graph of a sinusoidal wave showing peaks labelled F and P, a valley labelled J, and with points G and K on the midline. Horizontal lines indicate maximum and minimum levels, and a dashed line shows the midline.

The t-coordinate of G is t subscript 1, and the t-coordinate of space J is t subscript 2.

(i) On the interval open parentheses t subscript 1 comma space t subscript 2 close parentheses, which of the following is true about h?

space space space space spacea. space h is positive and increasing.
space space space space spaceb. space h is positive and decreasing.
space space space space spacec. space h is negative and increasing.
space space space space spaced. space h is negative and decreasing.

(ii) On the interval open parentheses t subscript 1 comma space t subscript 2 close parentheses, describe the concavity of the graph of h and determine whether the rate of change of h is increasing or decreasing.

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3
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1 mark

On the initial day of sales (t equals 0) for a new video game, there were 40 thousand units of the game sold that day. Ninety-one days later (t equals 91), there were 76 thousand units of the game sold that day.

Use the given data to find the average rate of change of the number of units of the video game sold, in thousands per day, from t equals 0 to t equals 91 days. Express your answer as a decimal approximation. Show the computations that lead to your answer.

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4
Sme Calculator
2 marks
Woman pushing a tire, indicated by her posture and an arrow indicating the tyre is rolling forwards. A point W is marked on the circumference of the tire. Note reads: "Figure not drawn to scale."

The tire of a car has a radius of 9 inches, and a person rolls the tire forward at a constant rate on level ground, as shown in the figure.

The sinusoidal function h models the height of point W above the ground, in inches, as a function of time t, in seconds.

The graph of h and its dashed midline for two full cycles is shown. Five points, F, G, J, K, and P, are labeled on the graph.

Graph of a sinusoidal wave showing peaks labelled F and P, a valley labelled J, and with points G and K on the midline. Horizontal lines indicate maximum and minimum levels, and a dashed line shows the midline.

The t-coordinate of K is t subscript 1, and the t-coordinate of P is t subscript 2.

(i) On the interval open parentheses t subscript 1 comma space t subscript 2 close parentheses, which of the following is true about h?

space space space space spacea. space h is positive and increasing.
space space space space spaceb. space h is positive and decreasing.
space space space space spacec. space h is negative and increasing.
space space space space spaced. space h is negative and decreasing.

(ii) Describe how the rate of change of h is changing on the interval open parentheses t subscript 1 comma space t subscript 2 close parentheses.

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5
1 mark

The function space f is defined for all real numbers. The table gives values for space f open parentheses x close parentheses at selected values of x.

space x

- 2

- 1

0

1

2

space f open parentheses x close parentheses

- 3

2

5

7

4

Find the value of space f open parentheses negative 1 close parentheses, or indicate that it is not defined.

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6
1 mark

The function space f is defined for negative 2 less or equal than x less or equal than 6. The table gives values for space f open parentheses x close parentheses at selected values of x. Assume that between any two consecutive values of x listed in the table, space f is either always increasing or always decreasing.

space x

- 2

0

2

4

6

space f open parentheses x close parentheses

9

5

2

8

4

Identify all intervals on which space f is increasing.

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7
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1 mark

The function space f is defined for all real numbers by space f open parentheses x close parentheses equals x cubed plus x minus 5.

Find all values of x, as decimal approximations, for which space f open parentheses x close parentheses equals 0, or indicate that there are no such values.

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8
Sme Calculator
2 marks

The table gives values for a function g at selected values of x.

x

0

3

6

9

g \left(x\right)

20

14

10

8

Use the average rates of change of g over the consecutive equal-length intervals open square brackets 0 comma 3 close square brackets, open square brackets 3 comma 6 close square brackets, and open square brackets 6 comma 9 close square brackets to determine whether the graph of g is concave up or concave down on the interval open square brackets 0 comma 9 close square brackets. Give a reason for your answer. Show the work that leads to your answer.

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9
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1 mark

The function space f is given by space f open parentheses x close parentheses equals square root of x plus 4 end root minus 2 x.

Find all values of x for which space f open parentheses x close parentheses equals 0, or indicate that there are no such values. Express your answer as a decimal approximation.

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10
Sme Calculator
1 mark

An electric utility company begins a phased installation of smart meters. At time t equals 0, the company begins tracking the total number of smart meters installed, in thousands.

The table gives the total number of smart meters installed, in thousands, for selected times t months after tracking began.

Months after tracking began, t

0

2

Total number of smart meters installed (thousands)

3.655

4.375

Use the given data to find the average rate of change of the total number of smart meters installed, in thousands per month, from t equals 0 to t equals 2 months. Express your answer as a decimal approximation. Show the computations that lead to your answer.

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11
2 marks

A surfboard floating in the sea moves up and down as waves pass beneath it. The height of the surfboard above the sea floor varies periodically.

The function h models the height of the surfboard above the sea floor, in feet, as a function of time t, in seconds.

A graph of h and its dashed midline for two full cycles is shown. Five points F, G, J, K and P are labeled on the graph.

A black-and-white sketch of a cosine wave showing two full cycles. the graph has a horizontal dashed midline and solid horizontal lines at the maximum and minimum values. Five points are labeled in italics: F at a peak, G where the graph crosses the midline while decreasing, J at a trough, K where the graph crosses the midline while increasing, and P at the next peak. the axes are not shown.

The t-coordinate of F is t subscript 1, and the t-coordinate of G is t subscript 2. The axes are chosen so that h equals 0 is the midline.

(i) On the interval open parentheses t subscript 1 comma space t subscript 2 close parentheses, which of the following is true about h?

a. h is positive and increasing.

b. h is positive and decreasing.

c. h is negative and increasing.

d. h is negative and decreasing

(ii) On the interval open parentheses t subscript 1 comma space t subscript 2 close parentheses, describe the concavity of the graph of h and determine whether the rate of change of h is increasing or decreasing.

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12
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2 marks

A software company tracks the number of active users for a new application over its first year to model user retention. At the launch of the app t equals 0, there were 120 thousand active users. After 4 months t equals 4, the number of active users dropped to 102.5 thousand.

(i) Use the given data to find the average rate of change of the number of users, in thousands of users per month, from t equals 0 to t equals 4 months. Express your answer as a decimal approximation. Show the computations that lead to your answer.

(ii) Interpret the meaning of your answer from part B(i) in the context of the problem.

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13
Sme Calculator
2 marks
Diagram of a Ferris wheel with center 15 meters above the ground and radius 10 meters. The lowest point is 5 meters above the ground. A passenger starts at the bottom (lowest point) at t = 0 and the wheel rotates counterclockwise.

A Ferris wheel at an amusement park has a radius of 10 meters. The center of the Ferris wheel is 15 meters above the ground. The Ferris wheel rotates at a constant speed in a counterclockwise direction. As the wheel rotates, the height of point P above the ground periodically increases and decreases.

The sinusoidal function h models the height of point P above the ground, in meters, as a function of time t, in seconds.

A graph of h and its dashed midline for two full cycles is shown. Five points F, G, J, K and P are labeled on the graph.

A black-and-white sketch of a sine wave showing two full cycles. the graph has a horizontal dashed midline and solid horizontal lines at the maximum and minimum values. Five points are labeled in italics: F at a trough, G where the graph crosses the midline while increasing, J at a peak, K where the graph crosses the midline while decreasing, and P at the next trough. The axes are not shown.

The t-coordinate of G is t subscript 1, and the t-coordinate of space J is t subscript 2. The axes are chosen so that h equals 0 is the midline.

(i) On the interval open parentheses t subscript 1 comma space t subscript 2 close parentheses, which of the following is true about h?

a. h is positive and increasing.

b. h is positive and decreasing.

c. h is negative and increasing.

d. h is negative and decreasing

(ii) On the interval open parentheses t subscript 1 comma space t subscript 2 close parentheses, describe the concavity of the graph of h and determine whether the rate of change of h is increasing or decreasing.

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14
Sme Calculator
3 marks

In a long-term study, the mean test score of a group of students is recorded each month after they enroll in a tutoring program. The mean score S open parentheses t close parentheses, on a scale from 0 to 100, is modeled by

S open parentheses t close parentheses equals 12 ln open parentheses t plus 1 close parentheses plus 50, for 0 \leq t \leq 12.

(i) Use the model to find the average rate of change of S, in points per month, from t equals 0 to t equals 6. Express your answer as a decimal approximation. Show the computations that lead to your answer.

(ii) Interpret the meaning of your answer from (i) in the context of the problem.

(iii) Consider the average rates of change of S from t equals 6 to t equals q months, where q greater than 6. Are these average rates of change less than or greater than the average rate of change from t equals 0 to t equals 6 found in (i)? Explain your reasoning. Your explanation should include a reference to the graph of S.

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15
3 marks

A polynomial function space f is defined for all real numbers. The function space f is increasing on open parentheses negative infinity comma 1 close parentheses and on open parentheses 3 comma infinity close parentheses, and decreasing on open parentheses 1 comma 3 close parentheses. Selected values of space f are given in the table.

space x

- 2

1

3

5

space f open parentheses x close parentheses

- 7

5

1

17

(i) Find the average rate of change of space f from x equals 1 to x equals 5. Show the computations that lead to your answer.

(ii) Determine whether space f has a zero on the interval 1 less or equal than x less or equal than 5. Give a reason for your answer.

(iii) Determine the number of real zeros of space f on the interval negative 2 less or equal than x less or equal than 5. Justify your answer based on the values in the table and the given intervals of increase and decrease.

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16
3 marks

The function g is defined for all real numbers and is concave up on its entire domain. The table below gives selected values of g.

x

0

2

4

6

8

g open parentheses x close parentheses

1

3

7

13

21

(i) Find the average rate of change of g from x equals 0 to x equals 4 AND from x equals 4 to x equals 8. Show the computations that lead to your answer.

(ii) Consider the average rate of change of g from x equals 8 to x equals 12. Is this average rate of change less than, greater than, or equal to the average rate of change from x equals 4 to x equals 8 found in (i)? Give a reason for your answer.

(iii) Determine whether g open parentheses 3 close parentheses is less than, greater than, or equal to the average of g open parentheses 2 close parentheses and g open parentheses 4 close parentheses, that is, fraction numerator g open parentheses 2 close parentheses plus g open parentheses 4 close parentheses over denominator 2 end fraction.

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