MATH 163, Spring 2001 Due Date: Name(s):
Project 4.9c: Newton's Method
Objective
To get some experience applying Newton's Method using Maple.
Narrative
If you have not already done so, read Section 4.9 in the text. There you will learn Newton's Method for
finding an approximation to a root x of f(x) = 0 (or a zero of f) given an initial approximation x1 to x.
This method uses the iterative equation
xn+1 = xn -
f(xn)
f (xn)
, n = 1, 2, 3, . . . ,
x1 being given, to obtain successive approximations to x.
Task
1. a) Type the command lines in the left-hand column below into Maple in the order in which they are
listed. These commands apply Newton's Method to finding the root of f(x) = 8x3
- 6x + 1 = 0 given
an initial estimate of x1 = 1.0. The effect of each command is described in the right-hand column for
your reference. Note that Maple uses a period to denote a subscript.
> # Project 4.9c: Newton's Method
> # Task 1
> restart; Clear Maple's memory.
> f := x -> 8*x^3-6*x+1; Define f.
> plot(f(x),x=-2..2,y=-2..4); Plot the graph of f.
> g := x -> x - f(x)/D(f)(x); Let g(x) = x -
f(x)
f (x)
.
> x.1 := 1.0; Let x1 = 1.0. (Make sure you type 1.0,
not 1: it makes a difference to Maple!
> for n from 1 to 6 do x.(n+1) := g(x.n) od; This computes xn+1 = g(xn) for
n = 1, 2, . . .. (Since g(x) = x -
f(x)
f (x)
,
it follows that xn+1 = g(xn).)
At this point, make a hard-copy of your typed input and Maple's responses. Then ...
b) To 8 decimal places of accuracy, what number x does it appear that xn is converging to?
c) Plot and label the point P on the x-axis whose x-coordinate is x.
2. Use Newton's Method to estimate the solutions to the equation sin x = x2
- 1 by proceeding as follows:
a) Let f(x) = sin x - (x2
- 1) and plot the graph of f for x  [-5, 5] and y  [-10, 2]. (The graph will
help you estimate the roots of f(x) = 0.)
b) Estimate each zero of f(x) by a number x1, and iterate Newton's Method enough times to obtain
an approximation to the root to 8 decimal places of accuracy. (Note: Your initial estimate of each root
must be close enough to the root for Newton's Method to converge: if it's not close enough, the values
x2, x3, . . . might not converge to the value you're interested in!)
Complete this task by again making a hard-copy of your typed input and Maple's responses (both text
and graphics).
Your lab report will be the hard copies of your typed input and Maple's responses (both text and graphics).