   Chapter 3.8, Problem 10E

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 4 − x − 1 = 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:

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 x4-x-1=0 with initial approximation x1=1

4) Calculation:

Given f(x)=x4-x-1

Differentiate f(x)=x4-x-1 with respect to x

f'x=4·x4-1-1·x1-1+0

=4x3-1

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