Chapter 2.2, Problem 62E

### Single Variable Calculus

8th Edition
James Stewart
ISBN: 9781305266636

Chapter
Section

### Single Variable Calculus

8th Edition
James Stewart
ISBN: 9781305266636
Textbook Problem

# 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?

(a)

To determine

To find: The values of f(4) and f+(4).

Explanation

Result Used: The left-hand and right-hand derivative is of f at x=a

are given by fâ€²âˆ’(a)=limhâ†’0âˆ’f(a+h)âˆ’f(a)h and fâ€²+(a)=limhâ†’0+f(a+h)âˆ’f(a)h.

Given:

The function f(x)={0ifÂ xâ‰¤05âˆ’xifÂ 0<x<415âˆ’xifÂ xâ‰¥4

Calculation:

Calculate the left-hand derivative of f at x=4.

fâ€²âˆ’(4) =limhâ†’0âˆ’f(4+h)âˆ’f(4)h

Since h<0, f(4+h)=5âˆ’(4+h).

fâ€²âˆ’(4) =limhâ†’0âˆ’5âˆ’(4+h)âˆ’15âˆ’4h =limhâ†’0âˆ’5âˆ’4âˆ’hâˆ’1h =limhâ†’0âˆ’(âˆ’hh) =âˆ’1

Thus, the value of fâˆ’(4) is âˆ’1

(b)

To determine

To sketch: The graph of f(x)={0if x05xif 0<x<415xif x4

(c)

To determine

To find: The points at which f is discontinuous.

(d)

To determine

To find: The points at which f is not differentiable.

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