11.3.A.1Multiple Choice
The function f is given by f(x) = −4x + 9. What is the average rate of change of f over the interval [−3, 5]?
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 1–9 cover 1.3.A; questions 10–19 cover 1.3.B; question 20 covers both.
The function f is given by f(x) = −4x + 9. What is the average rate of change of f over the interval [−3, 5]?
Select A, B, C, or D to check your answer.
The table gives values of a linear function f. What is f(11)?
| x | 2 | 5 | 9 |
| f(x) | 1 | 10 | 22 |
Select A, B, C, or D to check your answer.
A tank holds V(t) = 200 − 12t liters of water, t minutes after a drain opens. (a) Find the average rate of change of V over [1, 4] and over [6, 10]. (b) Explain why your two answers are equal, and interpret the rate with units.
Let f(x) = x² − 4x. Which list gives the average rates of change of f over [0, 1], [1, 2], and [2, 3], in that order?
Select A, B, C, or D to check your answer.
The table gives values of a quadratic function g at equally spaced inputs. What is g(8)?
| x | 0 | 2 | 4 | 6 |
| g(x) | 1 | 3 | 9 | 19 |
Select A, B, C, or D to check your answer.
Let f(x) = x² + 2x. (a) Find the average rate of change of f over [0, 1], [1, 2], and [2, 3]. (b) Show that these rates are given by a linear function of the interval midpoints, and find that function.
Let f(x) = x² − x. What is the slope of the secant line through the points on the graph of f at x = 2 and x = 5?
Select A, B, C, or D to check your answer.
Let h(x) = x² − 6x. What is the average rate of change of h over [1, 5]?
Select A, B, C, or D to check your answer.
Which of the following is equal to the average rate of change of a function f over the closed interval [a, b]?
Select A, B, C, or D to check your answer.
For a linear function, the average rates of change over three consecutive equal-length intervals are 7, 7, and 7. At what rate are these average rates changing?
Select A, B, C, or D to check your answer.
Four functions each have their average rates of change listed over three consecutive intervals of length 2. Which list could belong to a linear function?
Select A, B, C, or D to check your answer.
A cyclist's distance from home is D(t) = 12t + 5 miles, t hours after starting. (a) Find the average rate of change of D over [0, 1], [1, 2], and [2, 3]. (b) How fast are these rates changing? Include units. (c) What does this tell you about the cyclist?
A quadratic function has average rates of change of 3, 7, and 11 over three consecutive intervals of length 1. How are these average rates changing?
Select A, B, C, or D to check your answer.
Let f(x) = −3x² + 5x. The average rates of change of f are found over consecutive intervals of length 1 starting at x = 0. By how much does the average rate change from one interval to the next?
Select A, B, C, or D to check your answer.
Let f(x) = ax² + bx + c. The average rates of change of f over three consecutive intervals of length 2 are −5, −1, and 3. What is a?
Select A, B, C, or D to check your answer.
The average rates of change of a function over consecutive equal-length intervals are 12, 9, 5, and 0. Which statement is correct?
Select A, B, C, or D to check your answer.
The table gives values of a function f at equally spaced inputs. Which statement is correct?
| x | 0 | 1 | 2 | 3 | 4 |
| f(x) | 30 | 20 | 13 | 9 | 7 |
Select A, B, C, or D to check your answer.
Which of the following graphs is concave down on the interval [0, 4]?
Select A, B, C, or D to check your answer.
Let f(x) = −x² + 6x. (a) Find the average rates of change of f over [0, 2], [2, 4], and [4, 6]. (b) Is the graph concave up or concave down on [0, 6]? Justify your answer using the rates. (c) f increases and then decreases on [0, 6]. Does that change your answer to (b)?
A ball's height is s(t) = −5t² + 30t meters, t seconds after it is thrown upward. (a) Find the average rate of change of s over each of [0, 1], [1, 2], [2, 3], and [3, 4]. (b) By how much do the rates change from one second to the next, and how does that compare with 2ah? (c) Is the graph concave up or concave down? Describe what that means for the ball, with units.