Consider the function on the interval

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

Problem

Consider the function

f(t)=|t-1|

on the interval

[0,3]

. Let the accumulation function be

F(x)=\displaystyle\int_{0}^{x} f(t)\,dt

for

0\le x\le 3

. (a) Derive a piecewise formula for

F(x)

by evaluating the integral directly. Simplify each piece. (b) Determine each interval on

[0,3]

where

F

is differentiable and each interval where

F

is twice differentiable. Justify carefully. (c) On the intervals where

F''(x)

exists, establish the sign of

F''(x)

and hence identify where

F

is concave up and where it is concave down. (d) Classify

x=1

with respect to differentiability, concavity, and inflection. Provide a concise logical argument for each classification.