   Chapter 2.9, Problem 9E

Chapter
Section
Textbook Problem

# 7-10 Verify the given linear approximation at a = 0 . Then determine the values of x for which the linear approximation is accurate to within 0.1. 1 / ( 1 + 2 x ) 4 ≈ 1 − 8 x

To determine

I)

To verify: Linear approximation at a = 0

Explanation

1) Formula:

i. Lx=fa+f '(a)(x-a)

ii. Power rule combined with chain rule :

ddxf(x)n=nfxn-1.f'(x)

iii. Constant function rule:

ddxc=0  where c is constant.

iv. Sum rule:

ddxfx+gx=ddxfx+ddxg(x)

2) Given:

fx=11+2x4=1+2x-4 , a=0

3) Calculation:

Substitute x = a, to find f(a)

fa=1+2a-4

Now substitute a = 0, in f(a)

f0=1+0-4=1

Differentiate f(x) with respect to x,

f'(x)=ddx1+2x-4

By using power rule combined with chain rule,

f'x=(-4)1+2x-5

To determine

II)

To find: x for which linear approximation is within 0.1.

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