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Newton's Method and Iterative Functions
Posted:
Oct 31, 1997 2:59 PM


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: yf(x0) = f'(x0)(xx0) 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 TI8x 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 = Un1 + 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 = Un1  f(Un1)/f'(Un1) = Un1 (n^3+n1)/(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 **************************************************************************** Teacher/Professor AppliedMathematics/ComputerScience BaldwinSHS/NassauCC http://freenet.buffalo.edu/~aj317 mailto:aj317@freenet.buffalo.edu  President SFF/StarTrek/Wars, Magic/NetRunner/DragonDice, Romance WWWBookClub http://ourworld.compuserve.com/homepages/sffbookclub mailto:sffbookclub@compuserve.com ****************************************************************************



