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Newton's Method will converge to a true solution if you have a good initial approximation. If you don't it may not converge at all. Consider, for example, the equation f(x) = x(x – 1)(x+1) =x' –x = 0. Obviously, the solutions are -1, 0, 1. x= If we start Newton's Method with xo being close to one of these solutions we will get convergence to that solution. On the other hand, note that f'(x) = 3x² – 1 is zero when 1 x=± V3 Thus the tangent will be horizontal in those two cases, and New- ton's can't even be carried out. In this problem we’ll investigate what happens in the contrived case that V5 5 You can enter xo into WeBWorK as sqrt(5)/5. Try it: Xo = Now do your computations using exact arithmetic, and you'll rec- ognize a pattern: X1 = X2 X3 X4 = || |||| ||

Newton's Method will converge to a true solution if you have a good initial approximation. If you don't it may not converge at all. Consider, for example, the equation f(x) = x(x – 1)(x+1) =x' –x = 0. Obviously, the solutions are -1, 0, 1. x= If we start Newton's Method with xo being close to one of these solutions we will get convergence to that solution. On the other hand, note that f'(x) = 3x² – 1 is zero when 1 x=± V3 Thus the tangent will be horizontal in those two cases, and New- ton's can't even be carried out. In this problem we’ll investigate what happens in the contrived case that V5 5 You can enter xo into WeBWorK as sqrt(5)/5. Try it: Xo = Now do your computations using exact arithmetic, and you'll rec- ognize a pattern: X1 = X2 X3 X4 = || |||| ||

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Newton's Method will converge to a true solution if you have a good initial approximation. If you don't it may not converge at all. Consider, for example, the equation f(x) = x(x – 1)(x+1) =x' –x = 0. Obviously, the solutions are -1, 0, 1. If we start Newton's Method with xo being close to one of these solutions we will get convergence to that solution. On the other hand, note that f'(x) = 3x² – 1 is zero when 1 x=± V3 Thus the tangent will be horizontal in those two cases, and New- ton's can't even be carried out. In this problem we’ll investigate what happens in the contrived case that V5 5 You can enter xo into WeBWorK as sqrt(5)/5. Try it: Xo = Now do your computations using exact arithmetic, and you'll rec- ognize a pattern: X1 = X2 X3 X4 = || || || ||
Transcribed Image Text:Newton's Method will converge to a true solution if you have a good initial approximation. If you don't it may not converge at all. Consider, for example, the equation f(x) = x(x – 1)(x+1) =x' –x = 0. Obviously, the solutions are -1, 0, 1. If we start Newton's Method with xo being close to one of these solutions we will get convergence to that solution. On the other hand, note that f'(x) = 3x² – 1 is zero when 1 x=± V3 Thus the tangent will be horizontal in those two cases, and New- ton's can't even be carried out. In this problem we’ll investigate what happens in the contrived case that V5 5 You can enter xo into WeBWorK as sqrt(5)/5. Try it: Xo = Now do your computations using exact arithmetic, and you'll rec- ognize a pattern: X1 = X2 X3 X4 = || || || ||
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