   Chapter 2.2, Problem 59E

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# (a) Sketch the graph of the function f ( x ) = | x | .(b) For what values of x is f differentiable?(c) Find a formula for f ' .

To determine

To sketch:

(a) fx=x|x|,

(b) values of x for which f is differentiable,

(c) formula for f'.

Explanation

The function is differential if left and right hand derivatives are equal

Formula:

Left limit:

f-'x=limh0-fx+h-f(x)h

Right limit:

f+'x=limh0+fx+h-f(x)h

Given:

fx=xx= x2      if  x0-x2      if x<0

Calculation:

(a)

fx=x|x|

By definition

fx= x2      if  x0-x2      if x<0

(b)

Calculate left and right hand derivatives for x=0

Left hand derivative:

f-'x=limh0-fx+h-f(x)h

f-'0=limh0-f0+h-f(0)h

=limh0- -0+h2--02h

=limh0- h2h

=limh0- h

=0

Right hand derivative:

f+'

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