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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

(a) Given an initial approximation x 1 to a root of the equation f ( x ) = 0 , explain geometrically, with a diagram, how the second approximation x 2 in Newton’s method is obtained.

(b) Write an expression for x 2 in terms of x 1 , f ( x 1 ) , and f ' ( x 1 ) .

(c) Write an expression for x n + 1 in terms of x n , f ( x n ) , and f ' ( x n ) .

(d) Under what circumstances is Newton’s method likely to fail or to work very slowly?

To determine

a)

To explain:

Geometrically, with diagram, how the second approximation in Newton’s method is obtained

Explanation

1) Concept:

The x-intercept of the tangent line L of the graph y=f(x) at the point (x1, fx1) is the second approximation x2

To determine

b)

To write:

An expression for x2 in terms of x1,fx1,f'(x1)

To determine

c)

To write:

An expression for xn+1 in terms of xn,fxn,f'(xn)

To determine

d)

To explain:

Under what circumstances is Newton’s method likely to fail or to work very slowly?

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