Mathematics

Problem

Consider the function

h

defined on

[0,4]

h(x)=|x-2|-1

. a. Determine the set of all

x

[0,4]

for which

h(x)\ge 0

and the set for which

h(x)<0

. Justify your answer algebraically. b. Compute

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

by evaluating separate integrals over appropriate sub-intervals. c. Without using technology, describe the geometric shapes whose areas you added or subtracted in part (b), and verify that their areas are consistent with your computed value. d. Let

m(x)=h(x)+c

, where

c

is a real constant. Determine the value of

c

for which

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

, and explain why this value of

c

produces that result.

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