   Chapter 2.R, Problem 83E

Chapter
Section
Textbook Problem

# (a) Find the linearization of f ( x ) = 1 + 3 x 3 at a = 0 .State the corresponding linear approximation and use it to give an approximate value for 1.03 3 .(b) Determine the values of x for which the linear approximation given in part (a) is accurate to within 0.1.

(a)

To determine

To find:

The linearization of function at given point a and approximate value for 1.033

Explanation

1. Concept:

Use of Linear Approximations and Differentials, Derivative of function

2. Formula:

(i)  Linearization Formula:

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

(ii)  Power rule:

ddx xn=n xn-1

(iii) Chain Rule:

dydx= dydu dudx

(iv)  Absolute Value:

If ‘a’ is any positive integer, then | x | < a is equivalent to -a < x < a

3. Given:

fx= 1+3x3

4. Calculation:

fx= 1+3x3

fx=(1+3x)13

Let, u= 1+3x

Differentiating u with respect to  x

dudx=3

So function becomes,

fx=u13

Now, differentiating with respect to x.

ddxfx=ddxu13

By using power rule and Chain Rule

f'(x) =13u13-1dudx

Simplifying and substituting the value of u and dudx,

f'(x)=131+3x-23  (3)

f'(x)=

(b)

To determine

The values of x for which the linear approximation is accurate to within 0.1

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