   Chapter 3.8, Problem 16E

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 3 sin x = x

To determine

To use:

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:

3sinx=x

4) Calculation:

Given that 3sinx=x

Subtract by x

3sinx-x=0

Let fx=3sinx-x

Differentiate f(x)=3sinx-x with respect to x,

f'x=3cosx-1

So Newton’s formula for nth approximation becomes

xn+1=xn-3sinxn-xn3cosxn-1

Since positive root of 3sinx=x is near 2

Let x1=2

To find x2

Substitute x1=2 in formula xn+1=xn-3sinxn-xn3cosxn-1

x2=2-3sin2-23cos2-1

2

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