   Chapter 3.8, Problem 15E

Chapter
Section
Textbook Problem

# 15-16 Use Newton’s method to approximate the indicated root of the equation correct to six decimal places.The positive root of sin x = x 2

To determine

To use:

The Newton’s method to approximation the indicated root 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

3) Given:

The positive root of   sinx=x2

4) Calculation:

For Newton’s method, you need to make an estimate to use as the starting point, x1

Since the positive root of sinx=x2 is near 1

Let x1=1 as an initial approximation

As   sinx=x2 this can be written as

x2-sinx=0

Use f(x)=x2-sinx

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

f'x=2x2-1-cosx

=2x-cosx

So Newton’s formula for nth approximation becomes

xn+1=xn-xn2-sinxn2xn-cosxn

To find x2

Substitute x1=1 in formula xn+1=xn-xn2-sinxn2xn-cosxn

x2=1-12-sin12·1-cos1

0

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