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
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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
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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