   Chapter 3.8, Problem 9E

Chapter
Section
Textbook Problem

# Use Newton’s method with initial approximation x 1 = − 1 to find x 2 , the second approximation to the root of the equation x 3 + x + 3 = 0 . Explain how the method works by first graphing the function and its tangent line at (−1,1).

To determine

To find:

The second approximation x2 and explain how the method works by graphing

Explanation

1) Concept:

We will 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

iii. ddxconstant=0

3) Given:

The equation x3+x+3=0 with initial approximation x1=-1

4) Calculation:

Given f(x)=x3+x+3

Differentiate f(x)=x3+x+3 by power rule

f'x=3·x3-1+1·x1-1+0

=3x2+1

Substitute x

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