11.2.A.1Multiple Choice
The function f is given by f(x) = x² + 1. What is the average rate of change of f over the interval [1, 3]?
Select A, B, C, or D to check your answer.
Directions: Work each problem on your own paper first. For multiple choice, select A, B, C, or D. You will see right away whether you are correct. If you miss, you will see the most likely reason, and you can try again. For short response, type your answer, then compare it with the model solution. Questions 1–12 cover 1.2.A; questions 13–20 cover 1.2.B.
The function f is given by f(x) = x² + 1. What is the average rate of change of f over the interval [1, 3]?
Select A, B, C, or D to check your answer.
The table gives values of a function g. What is the average rate of change of g over the interval [5, 9]?
| x | 2 | 5 | 9 |
| g(x) | 11 | 20 | 40 |
Select A, B, C, or D to check your answer.
A drone takes off. The table gives its height h(t), in meters, t seconds after launch. Find the average rate of change of h over [0, 2] and over [2, 6], with units. On which interval is the height changing faster?
| t (s) | 0 | 2 | 6 |
| h (m) | 0 | 12 | 24 |
The function p is given by p(x) = x³ − 2x. What is the average rate of change of p over the interval [−1, 2]?
Select A, B, C, or D to check your answer.
Over the interval [2, 6], the average rate of change of a function f is 5, and f(2) = 7. If a function changed at the constant rate of 5 over this interval, starting from the output 7 at x = 2, what would its output be at x = 6?
Select A, B, C, or D to check your answer.
The table gives values of a function f near x = 3. Which value is the best approximation of the rate of change of f at x = 3, using the interval [2.9, 3.1]?
| x | 2.9 | 3 | 3.1 |
| f(x) | 8.41 | 9 | 9.61 |
Select A, B, C, or D to check your answer.
Which method gives a good approximation of the rate of change of a function f at the input x = a?
Select A, B, C, or D to check your answer.
The table gives values of a function g near x = 3. Use the interval [2.9, 3.1] to estimate the rate of change of g at x = 3. Then check your estimate with the intervals [2.9, 3] and [3, 3.1].
| x | 2.9 | 3 | 3.1 |
| g(x) | −3.19 | −3 | −2.79 |
The table shows the temperature T(t), in °F, of a pan t minutes after it leaves the oven. Using the interval [9.8, 10.2], which is the best estimate of the rate of change of T at t = 10, and what does it mean?
| t (min) | 9.8 | 10 | 10.2 |
| T (°F) | 140.4 | 140.0 | 139.7 |
Select A, B, C, or D to check your answer.
The table gives values of f(x) = x³ near x = 1 and near x = 2. Using the intervals [0.9, 1.1] and [1.9, 2.1], which statement is best supported?
| x | 0.9 | 1.1 | 1.9 | 2.1 |
| f(x) | 0.729 | 1.331 | 6.859 | 9.261 |
Select A, B, C, or D to check your answer.
For p(x) = x³ − 3x, the table gives values near x = −2 and x = 1. Use intervals of length 0.2 centered at each point to estimate the rate of change at x = −2 and at x = 1. At which point is the rate of change greater?
| x | −2.1 | −1.9 | 0.9 | 1.1 |
| p(x) | −2.961 | −1.159 | −1.971 | −1.969 |
The average rate of change of a function f over [4.9, 5.1] is −3, and over [7.9, 8.1] it is 5. Which conclusion is best supported?
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The function C gives the cost, in dollars, of producing n widgets. The average rate of change of C over [100, 150] is 12. Which statement is the correct interpretation?
Select A, B, C, or D to check your answer.
Let V(t) be the volume of water, in liters, in a tank t minutes after a valve opens. V(2) = 40 and V(10) = 120. (a) Find the average rate of change of V over [2, 10]. (b) Write a sentence that interprets your answer, with units.
The table shows the number of riders, in thousands, on a city bus system. On which interval is the average rate of change greater?
| year | 2010 | 2014 | 2020 |
| riders (thousands) | 12 | 24 | 36 |
Select A, B, C, or D to check your answer.
The average rate of change of a function f over the interval [3, 7] is 4.2. Which statement must be true?
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The yield Y, in bushels, of a farm depends on the amount F of fertilizer, in pounds. The average rate of change of Y over [20, 50] is positive. Which statement is correct?
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The table gives values of a function f. What is the average rate of change of f over [1, 6]?
| x | 1 | 4 | 6 |
| f(x) | 25 | 19 | 15 |
Select A, B, C, or D to check your answer.
Which situation has a negative average rate of change of the output with respect to the input?
Select A, B, C, or D to check your answer.
Let f(x) = x² − 4x. (a) Find the average rate of change of f over [0, 2], [2, 5], and [0, 5]. (b) Estimate the rate of change of f at x = 3 using the interval [2.9, 3.1]. (c) Using your answers, describe how f varies on [0, 2].