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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem
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The left-hand and right-hand derivatives of f at a are defined by

f ' ( a ) = lim h 0 f ( a + h ) f ( a ) h

and f ' + ( a ) = lim h 0 + f ( a + h ) f ( a ) h

if these limits exist. Then f ' ( a ) exists if and only if these one-sided derivatives exist and are equal.

(a) Find f ' ( 4 )  and  f ' + ( 4 ) for the function

f ( x ) = { 0        if  x 0 5 x        if  0 < x < 4 1 5 x        if  x 4

(b) Sketch the graph of f.

(c) Where is f discontinuous?

(d) Where is f not differentiable?

To determine

(a)To find: f-'4 and f+'4.

Explanation

Formula:

f-'a= lim h  0-fa+h-f(a)h and f+'a= lim h  0+fa+h-f(a)h

Given:

fx=0,                    if x0    5-x,                 if 0<x<415-x                  if x4       

Calculation:

The left hand derivative at x = 4 is given by,

f-'4= lim h  0-f4+h-f(4)h

As x approaches  4 from left side, f(x) = 5  x,

f-'4= lim h  0-5-4+h

To determine

(b)To sketch: Graph of f

To determine

(c)To check: The points at which f is discontinuous.

To determine

(d)To check:

The points at which f  is not differentiable.

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