   Chapter 2.R, Problem 82E

Chapter
Section
Textbook Problem

# (a) Find the linear approximation to f ( x ) = 25 − x 2 near 3.(b) Illustrate part (a) by graphing f and the linear approximation.(c) For what values of x is the linear approximation accurate to within 0.1?

To determine

(a)

To Find:

Linearization of the function near 3

Explanation

Concept:

Use of Linear Approximations and Differentials, Derivative of function

Formula:

(i) Linearization Formula:

Lx=fa+f'a(x-a)

(ii)  Power rule:

ddxxn=n xn-1

(iii)  Chain Rule:

dydx= dydududx

(iv) Use distributive property:

a (b+ c) = a*b + a*c

x=-b±b2-4ac2a

Given:

fx= 25-x2, a=3

Calculation:

Consider,

fx= 25-x2

fx=(25-x2)12

Let, u= 25-x2

Differentiating u with respect to  x

dudx=-2x

So, function becomes,

fx=u1/2

Now, differentiating with respect to x.

ddxfx=ddxu1/2

By using power rule and Chain Rule

f'(x) =12u12-1dudx

f'(x) =12u-12dudx

Simplifying and substituting the value of u and

dudx

f'(x)=1225-x2-12

To determine

(b)

To illustrate:

Linearization of function f=25-x2 near 3 is Lx=6.25-34x

by graphing function and linear approximation

To determine

(c)

To find:

The value of x for which the approximation is good

### 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

#### Find more solutions based on key concepts 