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
Sme Calculator
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
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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6
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.

How did you do?