   Chapter 3.8, Problem 7E

Chapter
Section
Textbook Problem

# 6-8 Use Newton’s method with the specified initial approximation x 1 to find x 3 , the third approximation to the root of the given equation. (Give your answer to four decimal places.) 2 x − x 2 + 1 = 0 ,    x 1 = 2

To determine

To find:

The third approximation x3

Explanation

1) Concept:

Use Newton’s formula to find for nth approximation

2) Formula:

i. Newton’s formula for nth approximation is xn+1=xn-fxnf'xn for n=1,2,3,

ii. Power rule of differentiation ddxxn=nxn-1

iii. ddxconstant=0

3) Given:

The equation 2x-x2+1=0 with initial approximation x1=2

4) Calculation:

Given the equation f(x)=2x-x2+1

Differentiate f(x)=2x-x2+1 with respect to x.

f'x=-2x2-2x

xn+1=xn-fxnf'xn

x2=x1-fx1f'x1

Substitute x1=2 in f(x)=2x-x2+1

f2=22-22+1

=-2

Now substitute x1

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