   Chapter 3.8, Problem 22E

Chapter
Section
Textbook Problem

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

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:

xn+1=xn-fxnf'xn for n=1,2,3,

ii. Power rule of differentiation ddxxn=nxn-1

3) Given:

sinx=x2-2

4) Calculations:

Given sinx=x2-2

Subtract x2-2

sinx-x2+2=0

Let fx=sinx-x2+2

Differentiate f(x)=sinx-x2+2 using power rule,

f'x=cosx-2x2-1

=cosx-2x

Newton’s formula for nth approximation becomes,

xn+1=xn-sin(xn)-xn2+2cos(xn)-2xn

Looking at graph, it intersects the x-axis two times so we start with a first guess of x1=-1

To find x2

Substitute x1=-1 in formula xn+1=xn-sin(xn)-xn2+2cos(xn)-2xn

x2=-1-sin-1--12+2cos-1-2-1

-1.062406

To find x3

Substitute x2=-1.062406 in formula xn+1=xn-sin(xn)-xn2+2cos(xn)-2xn

x3=-1.062406-sin-1.062406--1.0624062+2cos-1.062406-2-1.062406

-1.061550

To find x4

Substitute x3=-1

### 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

#### Find more solutions based on key concepts 