   Chapter 2.2, Problem 62E

Chapter
Section
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 f(a)=limh0f(a+h)f(a)h and f+(a)=limh0+f(a+h)f(a)h.

Given:

The function f(x)={0if x05xif 0<x<415xif x4

Calculation:

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

f(4) =limh0f(4+h)f(4)h

Since h<0, f(4+h)=5(4+h).

f(4) =limh05(4+h)154h =limh054h1h =limh0(hh) =1

Thus, the value of f(4) is 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.

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

## Additional Math Solutions

#### Find more solutions based on key concepts 