   Chapter 3.8, Problem 18E

Chapter
Section
Textbook Problem

# 17-22 Use Newton’s method to find all solutions of the equation correct to six decimal places. x + 1 = x 2 − x

To determine

To use:

Newton’s method to find all solutions of the equation correct to six decimal places

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.

ddxx=12x

3) Given:

x+1=x2-x

4) Calculations:

Given x+1=x2-x

Subtract x+1

x2-x-x+1=0

Let fx=x2-x-x+1

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

f'x=2x2-1-1-12x+1

=2x-1-12x+1

So Newton’s formula for nth approximation becomes

xn+1=xn-xn2-xn-xn+12xn-1-12xn+1

Looking at graph, it intersects the x-axis 2 times so start with a guess of x1=-0.5

To find x2,

Substitute x1=-0.5 in formula xn+1=xn-xn2-xn-xn+12xn-1-12xn+1

x2=-0.5--0.52--0.5--0.5+12·-0.5-1-12-0.5+1

-0.48415533

To find x3

Substitute x2=-0.48415533 in formula xn+1=xn-xn2-xn-xn+12xn-1-12xn+1

x3=-0.48415533--0.484155332--0.48402831--0.48415533+12·-0.48415533-1-12-0.48415533+1

-0.484028

To find x4

Substitute x3=-0.484028  in formula xn+1=xn-xn2-xn-xn+12xn-1-12xn+1

x4=-0

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