   Chapter 2.9, Problem 5E

Chapter
Section
Textbook Problem

# Find the linear approximation of the function f ( x ) = 1 − x at a = 0 and use it to approximate the numbers 0.9 and 0.99 . Illustrate by graphing f and the tangent line.

(I)

To determine

To find:

The linear approximation of the function.

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 multiple rule:

ddxc.fx=c.ddxfx where c is constant.

iv. Difference rule:

ddxfx-gx=ddxfx-ddxg(x)

2) Given:

fx=1-x, a=0

3) Calculation:

Substitute x = a, to find f(a)

fa=1-a

Now substitute a = 0, in f(a)

f0=1-a=1

Differentiate f(x) with respect to x,

f'x=ddx1-x

By using power rule combined with chain rule,

f'x=121-x

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