Objective:
The purpose of this demo is to use a graph of the function y = f(x) and its n^{th} Taylor Polynomial, p_{n}(x) to illustrate the approximation of y =
f(x) by a Taylor Polynomial centered at x = a. We include an option for the visualization of the error function R_{n}(x)
= f(x)  p_{n}(x).
Level: This demo
is appropriate for any course in which Taylor Polynomials are discussed.
Prerequisites:
Students should be familiar with computing derivatives of a function at x =
a and with the definition of n^{th} Taylor Polynomial centered at x = a.
Platforms:
(1) Browser Based:
(a) A Javascript slide show for several example functions. The
Javascript
codes have been tested using Internet Explorer 5+ and
Netscape
4.5+.
(b) Animated gifs (for
browsers that are not Javascript enabled.)
(2) Mathematica Notebook
(3) MATLAB Mfiles
(4) Maple 6
(6) Mathcad
(7) TI89
(8) Derive
Instructor's Notes: An important area in
mathematics is the computation of approximate values for functions at particular
points. One of the first encounters students have with such approximations
is using the slope of a secant line to a graph to estimate the slope of a
tangent line. Then the equation of a tangent line at a point is used for a
linear approximation to the function in a neighborhood of the point. As
they study Taylor Polynomials the more general problem of approximating a
function by a polynomial is encountered.
Suppose
we are interested in approximating a function y = f(x) near x = a by a
polynomial of degree n:
.
The
strategy we use to find the coefficients is to require a high degree of
"match" at x = a. Specifically, if we require that the
polynomial and its first n derivatives at x = a match the function and its first
n derivatives at x = a, the result of these requirements is that we construct a formula for the
n^{th} Taylor Polynomial for f,
centered at x = a:
.
When
a = 0, the polynomial is called an n^{th} Maclaurin polynomial for f.
Students
spend so much time learning this rather complicated formula, they tend to lose sight of why they would want to
use a polynomial to approximate a function. They also never seem to really grasp
exactly what it is they have found or how good the approximation might be.
Once I have introduced Taylor
Polynomials in class, I show this demonstration to students. We calculate the
terms of the Taylor Polynomial in class, and use a Javascript slide show to get
a picture of what we are calculating. Click on the following links to view slide
shows for the following functions and their approximation by Maclaurin and/or
Taylor Polynomials.
Other issues involved in
approximation are "How GOOD is the approximation?" and "Over what
interval can I expect the approximation to be good?"
To investigate these issues, the
Javascript slide shows have an option to display the error associated with the
nth Taylor polynomial. The error at any value for x is defined to be
R_{n}(x) = f(x)  p_{n}(x).
R_{n}(x) is sometimes called the n^{th} remainder of f.
By plotting the error function, we can
visualize the "goodness" of the approximation for various values of n
as well as the interval over which the approximation could be considered
"good." These ideas lead to a discussion of the interval of
convergence for a Taylor series. In Examples 16 above, it is not
hard to convince students that the interval of convergence is ,
while in Example 7 the interval of convergence is (1,1). More details about
error analysis may be appropriate in a numerical analysis class, however, the
pictures supply a visual foundation and hence an intuitive idea about what we mean when we say that the Taylor
Series converges to f(x) (in an appropriate interval) if and only if the n^{th}
remainder tends to 0 as n increases without bound.
Approximation to f(x) = cos(x) by Maclaurin
Polynomials and the error.

Approximation to f(x) = log(x+1) by Maclaurin Polynomials and the error.

My experience has been that this demo plants a
visual image of the idea of Taylor approximation that students readily
recall. As a result they seem to remember that
the n^{th} Taylor Polynomial at x = a agrees with the function and its first n
derivatives at x = a. I surveyed my Calculus 3 students before we began a
discussion of Taylor Series. The students who had seen the demo in the previous
year all remembered much more clearly and could put into words what a Taylor
Polynomial is used for, and why we might want to use one.
Additional Resources:
Taylor
Polynomial Web Site: by Cathy Frey. Click
here
to download Cathy Frey's Mathematica file.
Mathematica Notebook: Preview
and download an interactive Mathematica notebook
here.
MATLAB MFiles: Two MATLAB Mfiles,
sinmovie.m and logmovie.m, illustrate the approximation of f(x) = sin(x) and f(x)
= log(x),respectively, by Taylor Polynomials. These files were developed
by David R. Hill. Preview the animations and download from
here.
Maple 6 Worksheet: Preview and
download an interactive Maple 6 worksheet from
here.
Mathcad Worksheet: Preview and
download an interactive Mathcad worksheet from
here.
TI89 Calculator Program: TI89
animation of Taylor Polynomial approximations. View and download from
here.
The TI89 program was developed by Lila F. Roberts.
Gallery
of Animations for Examples 1 7.
Credits: This demo was submitted by
Cathy
Frey
Associate Professor of Mathematics
Norwich University
158 Harmon Drive
Northfield, VT 05663
and is included in Demos with Positive Impact
with her permission.