AP Calculus AB 8.3 Accumulation & Net Change Review

Review AP Calculus AB 8.3 by linking rate functions, definite integrals and net change, then return to focused questions on the accumulation skills you need to improve.

Covered syllabus points

CHA-4.D—Interpret the meaning of a definite integral in accumulation problems CHA-4.E—Determine net change using definite integrals in applied contexts

[Maximum number: 1]

A function f(t) gives the rate of evaporation of water, in liters per hour, from a pond, where t is measured in hours since 12 noon. Which of the following gives the meaning of $\int_{4}^{10} f(t) d t$ in the context described?

The total volume of water, in liters, that evaporated from the pond during the first 10 hours after 12 noon

The total volume of water, in liters, that evaporated from the pond between 4 P.M. and 10 P.M.

The net change in the rate of evaporation, in liters per hour, from the pond between 4 P.M. and 10 P.M.

The average rate of evaporation, in liters per hour, from the pond between 4 P.M. and 10 P.M.

The average rate of change in the rate of evaporation, in liters per hour per hour, from the pond between 4 P.M. and 10 P.M.

[Maximum number: 3]

A student starts reading a book at time t=0 minutes and continues reading for the next 10

minutes. The rate at which the student reads is modeled by the differentiable function R, where

R(t) is measured in words per minute. Selected values of R(t) are given in the table shown.

Table for Question 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts question 2 — AP Calculus AB

A teacher also starts reading at time t=0 minutes and continues reading for the next 10

minutes. The rate at which the teacher reads is modeled by the function W defined by

$W(t)=-\frac{3}{10} t^{2}+8 t+100$ , where W(t) is measured in words per minute. Based on the model,

how many words has the teacher read by the end of the 10 minutes? Show the work that

leads to your answer.

D A teacher also starts reading at time t=0 minutes and continues reading for the next 10 minutes. The

rate at which the teacher reads is modeled by the function W defined by $W(t)=-\frac{3}{10} t^{2}+8 t+100$ ,

where W(t) is measured in words per minute. Based on the model, how many words has the teacher

read by the end of the 10 minutes? Show the work that leads to your answer.

$\int_{0}^{10} W(t) d t=\int_{0}^{10}\left(-\frac{3}{10} t^{2}+8 t+100\right) d t$	Integrand	Point 7 (P7)
$=\left.\left(-\frac{1}{10} t^{3}+4 t^{2}+100 t\right)\right\|_{0} ^{10}$	Antiderivative	Point 8 (P8)
$\begin{aligned} & =\left(-\frac{1}{10} \cdot 1000+4 \cdot 100+100 \cdot 10\right)-\left(-\frac{1}{10} \cdot 0+4 \cdot 0+100 \cdot 0\right)<br>& =1300 \end{aligned}$ <br>Based on the model, the teacher has read 1300 words by the end of the 10 minutes.	Answer	Point 9 (P9)
Scoring Notes for Part D

- P7 is earned for an indefinite or definite integral with integrand W(t), with or without the

differential d t.

- P8 is earned for the correct antiderivative, with or without the constant of integration.

- To be eligible for P9, a response must have earned P8.

- A response of $\left(-\frac{1}{10} \cdot 1000+4 \cdot 100+100 \cdot 10\right)-\left(-\frac{1}{10} \cdot 0+4 \cdot 0+100 \cdot 0\right)$ or equivalent banks P9

(i.e., subsequent errors in simplification will not be considered in scoring for P9).

Part A (AB or BC): Graphing calculator not allowed

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Covered syllabus points

8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts question 1

8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts question 2