Answer The approximation formula is x n + 1 = x n − f ( x n ) f ′ ( x n ) Explanation Given information : The given statement is that derive the formula for finding the ( n + 1 ) st approximation x n + 1 from the nth approximation x n . Calculation : The graph of the curve of f ( x ) and its tangent. The point-slope equation of the tangent line to the curve at ( x n , f ( x ) ) y − f ( x n ) = f ′ ( x n ) ( x − x n ) Substitute y = 0 0 − f ( x n ) = f ′ ( x n ) ( x − x n ) f ′ ( x n ) ( x − x n ) = − f ( x n ) ( x − x n ) = − f ( x n ) f ′ ( x n ) [ divide each side by f ′ ( x n ) ] x = − f ( x n ) f ′ ( x n ) + x n [ add x n to each side ] x = x n − f ( x n ) f ′ ( x n ) The next to x n is x n + 1 Therefore you can write that x n + 1 = x n − f ( x n ) f ′ ( x n )

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

4th Edition

ISBN: 9780133178579

Author: Ross L. Finney

Publisher: PEARSON

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Question

Chapter 5.5, Problem 72E

To determine

To derive the formula for finding the (n+1)st approximation xn+1 from the nth approximation xn .

Expert Solution & Answer

Answer to Problem 72E

The approximation formula is

xn+1=xn−f(xn)f′(xn)

Explanation of Solution

Given information:

The given statement is that derive the formula for finding the (n+1)st approximation xn+1 from the nth approximation xn .

Calculation :

The graph of the curve of f(x) and its tangent.

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy), Chapter 5.5, Problem 72E

The point-slope equation of the tangent line to the curve at (xn,f(x))

y−f(xn)=f′(xn)(x−xn)

Substitute y=0

0−f(xn)=f′(xn)(x−xn)f′(xn)(x−xn)=−f(xn)(x−xn)=−f(xn)f′(xn) [ divide each side byf′(xn)]x=−f(xn)f′(xn)+xn [add xn to each side]x=xn−f(xn)f′(xn)

The next to xn is xn+1

Therefore you can write that

xn+1=xn−f(xn)f′(xn)

Chapter 5.5, Problem 71E

Chapter 5.6, Problem 1QR

Chapter 5 Solutions

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

Ch. 5.1 - Prob. 1QR Ch. 5.1 - Prob. 2QR Ch. 5.1 - Prob. 3QR Ch. 5.1 - Prob. 4QR Ch. 5.1 - Prob. 5QR Ch. 5.1 - Prob. 6QR Ch. 5.1 - Prob. 7QR Ch. 5.1 - Prob. 8QR Ch. 5.1 - Prob. 9QR Ch. 5.1 - Prob. 10QR

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To derive the formula for finding the ( n + 1 ) st approximation x n + 1 from the nth approximation x n .