Math 1312 - Calculus II - Spring 2001                                         
"Challenge 98" Project
Exercise 2: Approximating Functions By Their Taylor Polynomials

This assignment is due Wednesday, March 28, 2001.

The Mathematics Computer Lab is located at Bell Hall 130, and open during usual business hours, at least. All the Macintosh computers including the IMacs should habe Mathematica installed.  If Mathematica is available to you somewhere else, you don't have to work in Bell Hall.
Start by opening a web browser (preferably Netscape). Go to my  homepage at http://helmut.knaust.info/classes.html
Under our class heading, find "Exercise 2" of the Challenge 98 Project. Click on it to download this notebook.

Step 1

Fill in your names and execute the command below.

\(name[1] = "Jane A. Doe"; 
name[2] = "John B. Doe"; 

    name[3] = ""; 

    Print["Problem ", 
      m1 = 1 + Mod[
            Apply[Plus, ToCharacterCode[name[1]]] + 
				
              Apply[Plus, ToCharacterCode[name[2]]] + 
              Apply[Plus, 
				ToCharacterCode[name[3]]], 4], 
      "	Problem ", m1 + 4, "	Problem ", m1 + 8, "	Problem ", 
      m1 + 12]; \)

You will work with these four problems from our Worksheet 2. This worksheet is also available as a PDF file at http://www.math/Faculty/helmut/011_1312/ws2.pdf

Point your browser at http://helmut.knaust.info/challenge98/toc.html, and look for the file "TaylorPolys.nb". Download the notebook by clicking on the link to the notebook.
This notebook graphs a function and its Taylor polynomial. "appcenter" is the center of the Taylor polynomial, "appdegree" is the degree of the approximating Taylor polynomial, "viewradius" allows you to set the viewing rectangle: You will see the graphs of the function and its Taylor polynomial from x = appcenter - viewradius to x = appcenter + viewradius.
For each of the 4 functions selected above do the following:
	Graph the funtion and its Taylor polynomial (with the center indicated on the worksheet) of degree n, where n=12, 25, 50. You may try n=100, if the wait for n=50 was not too long.
	In each case estimate over which interval the function and its Taylor polynomial "coincide". Let us agree that the function and its Taylor polynomial "coincide" if it appears that their graphs lie "on top of each other".
	Does the "interval of coincidence" increase as n gets bigger? Can you make a prediction for the interval of coincidence if n=1600? (Do not run the graphing command for n=1600). Does there seem to be a "maximal" interval of coincidence, which one cannot exceed even when n is very large?

Step 2

You should turn in the following items:
1.) A printout of the graphs of the function and its Taylor polynomials for one function only. Pick the most interesting of the four.
2.) Your observations and speculations about the interval of coincidence for all four functions.
3.) Good luck! I am available at Bell Hall 219, tel. 747-7002, email: helmut@math.utep.edu