   Chapter 3.8, Problem 20E

Chapter
Section
Textbook Problem

# 17-22 Use Newton’s method to find all solutions of the equation correct to six decimal places. ( x − 1 ) 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-12=x

4) Calculations:

Given that x-12=x

Subtract x

x-12-x=0

Let fx=x-12-x

Differentiate f(x)=x-12-x with respect to x,

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

=2x-1-12x

So Newton’s formula for nth approximation becomes

xn+1=xn-xn-12-xn2xn-1-12xn

Looking at graph, it intersects the x-axis 2 times so we start with a first guess of x1=0.27

To find x2

Substitute x1=0.27 in formula xn+1=xn-xn-12-xn2xn-1-12xn

x2=0.27-0.27-12-0.2720.27-1-120.27

0.275484

To find x3

Substitute x2=0.275484  in formula xn+1=xn-xn-12-xn2xn-1-12xn

x3=0

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started 