   Chapter 4.8, Problem 9E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Use Newton’s method with initial approximation x1 = −1 to find x2, the second approximation to the root of the equation x3 + 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.

Explanation

Formula used:

The Newton’s formula is,xn+1=xnf(xn)f(xn) where f(x) is the given function.

Given:

The initial approximation is x1=1.

The function is f(x)=x3+x+3.

Calculation:

Calculate the derivative of f(x).

f(x)=ddx(x3+x+3)=2ddx(x3)+ddx(x)+ddx(3)=3x2+1+0=3x2+1

The value of f(x) at x1=1 is,

f(1)=3(1)2+1=3(1)+1=3+1=4

Calculate the value of f(x) at x1=1

f(1)=(1)3+(1)+3=11+3=1

Calculate x2 by using Newton’s method

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