Define for all real

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

Problem

Define

f(x)=|x|

for all real

x

. A function

G

is proposed:

G(x)=\tfrac12 x|x|+C,

where

C

is a constant. (a) Compute

G'(x)

separately for the cases

x>0

and

x<0

. Show all steps in each region. (b) Combine your work from part (a) to verify that

G'(x)=f(x)

for every

x\neq0

. Explain how this establishes that

G

is an antiderivative of

f

on each open interval

(-\infty,0)

and

(0,\infty)

. (c) Discuss briefly what happens at

x=0

. Does the conclusion in part (b) extend to all real numbers? Support your answer.