   Chapter 3.R, Problem 14E

Chapter
Section
Textbook Problem

# 13-15 Sketch the graph of a function that satisfies the given conditions. f ( 0 ) = 0 ,      f  is continuous and even, f ′ ( x ) = 2 x  if  0 < x < 1 ,     f ′ ( x ) = − 1  if  1 < x < 3 , f ′ ( x ) = 1  if  x > 3

To determine

To sketch:

The graph of a function that satisfies the given conditions

Explanation

1) Concept:

i. From the given conditions, we plot the graph by getting the piecewise function for f(x).

ii. f is continuous means that there is no hole, gap, or sudden changes that appears in the function. It is a smooth curve.

iii. f is even means that it is symmetric with respect to the y-axis

2) Given:

f0=0,f is continues and even

f'x=2x if0<x<1, f'x=-1 if1<x<3

f'x=1if x>3

3) Calculation:

Given that:

f'x=2x when 0<x<1

f'x=-1 when 1<x<3

f'x=1when x>3

After getting antiderivative,

fx=x2+c1  when 0<x<1

fx=-x+c2 when 1<x<3

fx=x+c3 when x>3

Given that f(x) is continuous, therefore limx0+f(x)=f(0)

Therefore,

f0=02+c1  when 0<<

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