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Topic: Newton's Method and Iterative Functions
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Alvar J. Garcia

Posts: 304
Registered: 12/6/04
Newton's Method and Iterative Functions
Posted: Oct 31, 1997 2:59 PM
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Hello to ALL ListServ Members:

As you know, the College Board has changed the subjects that they
require in AP Calculus AB & BC (see Acorn book). I am of the opinion
that, just because a topic is not listed, doesn't mean it doesn't deserve
our attention - and the attention of our students! Newton's Method is a
case in point. It is no longer listed in the course, but not teaching
it is a disservice. This one topic is so rich, it leads into discussions
on so many important points of The Calculus, that I simply cannot ignore
it in my class!

I start with the equation of a tangent line:
y-f(x0) = f'(x0)(x-x0) and derive the rule xn+1 = xn - f(xn)/f'(xn).
So the concept of a tangent line at a point and the value of the
derivative is reinforced.

Then we apply this recursive rule to a particular f(x). I like to
start with a sample AP Calculus Part I question (93 AB I #45) where
f(x) is the cubic y = x^3 + x -1 which has only one positive root near
x=1. So we let xn+1=xn-(xn^3 + xn -1)/(3xn^2 + 1) and x0=1. We then find
x1 = x0 - f(x0)/f'(x0) = 1 - 1/4 = 0.750. The question asks to find x3 if
x1=1. The problem soon becomes evident: if f(x) is sufficiently painfull
and n is even moderatly large, this is a very tedious process
to complete by hand.

Now I begin to introduce the concepts behind programming the
TI-8x series of calculators. We develop a program to calculate
successive values of xn. If anyone would like a copy of our prgmNEWT, let
me know and I'll write it up the next time I'm posting to the ListServs.

Enter the concept of convergence and convergence tables. The
students use the program (to which we added a pause every 7th line of
output for long tables that would scroll off the screen and graphics
showing convergence of the tangent line roots to the root in question) to
create convergence tables for whatever desired accuracy. Discussion of
convergence here and earlier with limits is a great way to get the
students used to this key concept of The Calculus in general and BC
Calculus in particular.

Enter sequence mode. I define a sequence {an}. For example, let
an=2^n, {an} for n=0 to inf = {1, 2, 4, 8, 16, ...}. I define an infinite
series sigma(an) for n=0 to inf = 1 + 2 + 4 + 8 + 16 + .... We discuss
convergence and divergence of infinite series, etc. Then we write it in
sequence mode on the TI: Un = 2^n with appropriate window for the sequence
and Un = Un-1 + 2^n with appropriate window for the series. If the series
represents the pnny problem (how much money do you make at a job where you
are paid a penny the first day, double that the next, double again the
next, etc - you're a millionaire before a month is up!), the students
easily recognize why divergence of this series is desirable.

Then in sequence mode we rewrite Newton's Method:
xn+1 = xn - f(xn)/f'(xn)
as
Un = Un-1 - f(Un-1)/f'(Un-1)
= Un-1 -(n^3+n-1)/(3n^2+1)
and graph the results for f(x) above and find the same answers (if U0=1
then U2=0.686 to three decimal place accuracy).

Is this what was meant by 'iterative functions on the TI' in a
previous post? If so, I suppose I just answered that person's question
of how to plot one! However, I don't think of this as iteration but as
recursion! So this is how to graph a recursive function or difference
equation (wait until we get to ODEs!).

What do you think? Comments are welcome!

Regards,
A. Jorge Garcia
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http://freenet.buffalo.edu/~aj317
mailto:aj317@freenet.buffalo.edu
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