AP Calculus AB Unit 5.1: Mean Value Theorem

Practice AP Calculus Unit 5.1 questions on applying the Mean Value Theorem to connect average and instantaneous rates of change.

[Maximum number: 2]

An invasive species of plant appears in a fruit grove at time t=0 and begins to spread. The

function C defined by $C(t)=7.6 \arctan (0.2 t)$ models the number of acres in the fruit grove

affected by the species t weeks after the species appears. It can be shown that $C^{\prime}(t)=\frac{38}{25+t^{2}}$ .

(Note: Your calculator should be in radian mode.)

Find the time t when the instantaneous rate of change of C equals the average rate of change

of C over the time interval $0 \leq t \leq 4$ . Show the setup for your calculations.

B Find the time t when the instantaneous rate of change of C equals the average rate of change of C over

the time interval $0 \leq t \leq 4$ . Show the setup for your calculations.

$\frac{C(4)-C(0)}{4-0}=1.282008$	Uses average rate of change	Point 3 (P3)
$C^{\prime}(t)=\frac{38}{25+t^{2}}=1.282008 \Rightarrow t=2.154298$ <br>The instantaneous rate of change of C equals the average rate of change of C over the interval $0 \leq t \leq 4$ at time t=2.154.	Answer with supporting work	Point 4 (P4)

Scoring Notes for Part B

- P3 may be earned by presenting the expression or value for the average rate of change. Note that

because C(0)=0 and the interval is $0 \leq t \leq 4$ , any of the following will earn $\mathbf{P 3}: \frac{\int_{0}^{4} C^{\prime}(t) d t}{4}$ ,

$\frac{C(4)-C(0)}{4-0}, \frac{C(4)}{4}, \frac{5.128-0}{4-0}, \frac{5.128}{4}$ , or 1.282. However, neither P3 nor P4 is earned by just

presenting t=1.282.

- P4 is earned for the correct answer supported by the appropriate equation. A reported answer should

be accurate to three places after the decimal point, rounded or truncated. An inappropriately rounded

answer does not earn the point, unless an earlier point was not earned due to inappropriate rounding.

The following response, for example, earns both $\mathbf{P 3}$ and $\mathbf{P 4}: C^{\prime}(t)=\frac{C(4)-C(0)}{4}$ when t=2.154.

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Covered syllabus points

5.1 Using the Mean Value Theorem question 1