   Chapter 3.R, Problem 49E

Chapter
Section
Textbook Problem

# Use Newton’s method to find the root of the equation x 5 − x 4 + 3 x 2 − 3 x − 2 = 0 in the interval [1, 2] correct to six decimal places.

To determine

To find:

The roots of the given equation in the interval 1, 2 correct to six decimal places

Explanation

1) Concept:

Working rule of Newton’s method is

To find a root of y=f(x) start with an initial approximation x1 . After the first iteration of Newton’s method, we will get x2. This x2 is actually the x-intercept of the tangent at that point (x1,f(x1)). Now we draw a tangent at that point (x2,f(x2)) and the x-intercept at that tangent will be x3. Continue to do this till the value of xn tends to converge.

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.

ddxconstant=0

3) Given:

x5-x4+3x2-2=0

4) Calculation:

Here  x5-x4+3x2-2=0

It can be written as, fx= x5-x4+3x2-2

Differentiate f(x) with respect to x, using power rule,

f'x= 5x4-4x3+6x

So Newton’s formula for nth approximation becomes

xn+1=xn-xn5-8xn4+3xn2-25xn4-24xn3+6xn

Set x1=1

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