   Chapter 3.8, Problem 26E

Chapter
Section
Textbook Problem

# 23-26 Use Newton’s method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. cos ( x 2 − x ) = x 4

To determine

Find:

All the solutions of the equation correct to eight decimal places using Newton’s method.

Explanation

1) Concept:

Start with initial approximation x1  which is obtained from graph. Evaluate fx1and f'(x1) and apply the Newton’s formula. After the first iteration we get x2. Repeat the process with x1  replaced by x2. Calculate the approximations x1, x2, x3,  until the successive approximations xn, xn+1  agree the eight decimal places.

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

iii) Chain rule of differentiation: [fgx'=f'(gx·g'(x)

iv)

ddx cosx= -sin(x)

3) Calculation:

cosx2-x=x4  is same as cosx2-x-x4=0

Consider the function fx=cosx2-x-x4

We find the x1  value using the graph.

From the graph, one root ix x=1

Also, the graph intersect at approximately 0.7.

So x1=-0.7

fx=cosx2-x-x4

Use chain rule, power rule and derivative of cos(x),

f'x=-2x-1sinx2-x-4x3

xn+1=xn-fxnf'xn

xn+1=xn-cosxn2-xn-xn4-2xn-1sinxn2-xn-4xn3

x1= -0.7

x2=x1-cosx12-x1-x14-2x1-1sinx12-x1-4x13

x2=-0.7-cos(--0.7)2-(-0.7)-(-0.7)4-2*(-0

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