Consider the derivative defined on by f'(x)= cases 2,&0 x 2, 6-2x,&2< x 4

MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Consider the derivative

f'(x)

defined on

0\le x\le4

f'(x)=\begin{cases}2,&0\le x\le2,\\6-2x,&2< x\le4.\end{cases}

You are given that

f(0)=1

. (a) Determine a formula for

f(x)

on each subinterval

[0,2]

and

(2,4]

. Include any constants required by the initial condition. (b) Compute the absolute minimum value of

f

[0,4]

. Explain how you identify and compare all relevant candidates. (c) Evaluate

\displaystyle\int_{0}^{4}f(x)\,dx

and interpret its meaning in terms of accumulated change of the original quantity modeled by

f