   Chapter 4.3, Problem 3E

Chapter
Section
Textbook Problem

# Let g ( x ) = ∫ 0 x f ( t ) d t , where f is the function whose graph is shown.(a) Evaluate g ( 0 ) , g ( 1 ) , g ( 2 ) , g ( 3 ) , and g ( 6 ) .(b) On what interval is g increasing?(c) Where does g have a maximum value?(d) Sketch a rough graph of g. To determine

a)

To evaluate:

i. g0

ii. g1

iii. g2

iv. g3

v. g6

Explanation

Given:

gx=0xf(t)dt

Calculation:

i. To evaluate g0

The integral of a function over a trivial interval is zero so

g0=00f(t)dt =0

ii. To evaluate g1

g1=01f(t)dt

Geometrically g(1)  is the area under the graph from 0 to 1.

The area is two whole squares

g1=2

iii. To evaluate g2

The area is four whole squares and triangle with base 1 and height 2

g2=02f(t)dt=01f(t)dt+12f(t)dt

g2=2+2+12·2·1=5

iv

To determine

b)

To find:

On what interval is g increasing

To determine

c)

To find:

Where does  g have maximum value

To determine

d)

To sketch:

A rough graph of g

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