AP Calculus BC 8.1 Average Value Overview

Review average-value problems by dividing a definite integral by interval length and interpreting the result in context.

[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 average number of acres affected by the invasive species from time t=0 to time

t=4 weeks. Show the setup for your calculations.


$\frac{1}{4-0} \int_{0}^{4} C(t) d t$	Average value formula	Point 1 (P1)
$=\frac{1}{4}(11.112896)=2.778224$	Answer	Point 2 (P2)
From time t=0 to t=4 weeks, the average number of acres affected by the invasive species was 2.778 acres.

Scoring Notes for Part A

- P1 is earned for the correct integral, with or without the differential, along with evidence of division

by 4. In the presence of the correct integral, the correct answer will suffice as evidence of division

by 4. These may appear all in one step, as in the model solution, or in multiple steps.

- P2 is earned for the correct answer, with or without supporting work. 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.

- Incorrect or unclear communication between the correct integral and the correct answer is treated as

scratch work and is not considered in scoring. For example:

○ $\int_{0}^{4} C(t) d t=11.112896$ so the average velocity is 2.778224.

Note: This response earns P1 for the correct integral with the correct answer as evidence of

division by 4. It also earns P2 for the correct answer.

○ $\int_{0}^{4} C(t) d t=\frac{11.112896}{4}=2.778224$

Note: This response earns P1 for the correct integral with the correct answer as evidence of

division by 4. It also earns P2 for the correct answer. (In this instance, incorrect linkage is not

considered in scoring.)

○ $\int_{0}^{4} C(t) d t=2.778224$

Note: This response earns P1 for the correct integral with the correct answer as evidence of

division by 4. It also earns P2 for the correct answer. (In this instance, incorrect linkage is not

considered in scoring.)

- Note that the values $\frac{1}{4}(11.112)$ and $\frac{1}{4}(11.113)$ are accurate to three digits after the decimal and

therefore earn P2.

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Covered syllabus points

8.1 Finding the Average Value of a Function on an Interval question 1