MT A143-03 (Calculus II) Fall 2001
on §8.6 (Fourier Polynomials)
Background idea In lots of applications on mathematics (i.e. in the disciplines that require the non-math majors to take this course) approximation is a powerful tool. The idea of approximation is to start with an arbitrary function f(x) and to find the “nice function” that is “closest” to it. The application will determine what we mean by “nice function” and what “closest” means. Taylor polynomials (§4.3 and chapter 11) are one example of an approximation method and Fourier polynomials (§8.6 and differential equations) are another approximation technique.
Things you need to know that are clear from the book An approximation technique is a technique. Thus you need to be able to do the mechanics of the approximation. (Given a function f(x) and a degree n, be able to find the approximation.)
The formula for the nth degree Taylor polynomial approximation to f(x) at the point x = a is
You should be able to sketch a graph of the approximation, evaluate the approximation at a point and find the error between the function and the approximation. In class we constructed the approximating polynomial from the values of and .
The standard approximating tool for approximating periodic functions or functions that are only defined on a finite interval is a Fourier polynomial, which is really a linear combination of sine and cosine functions. For this course, we are only concerned with approximating functions that are continuous on the interval . In that case the degree n Fourier polynomial has the formula
where , and , for .
You should be able to sketch a graph of the approximation, evaluate the approximation at a point and find the error between the function and the approximation.
Deep background (Why this works) By deep results beyond the scope of this course, integration on a closes interval works like the dot product on , and the basic functions work like the coordinates of . If we measure with the integral they are perpendicular to each other. (This is what exercises 3, 6, and 7 were showing.) One of the rules for defining a good approximation system is that an approximation of an approximation give back the approximation. (This is what the last half of page 488 and exercise 5 are showing.)
An important observation to make Besides the mechanics that you should learn in constructing approximating functions, it is worthwhile to note that the approximations pass the naive test for approximations, that if you take better and better approximations of a function, the approximation eventually becomes indistinguishable from the function we are approximating. (That is the point of exercises 1 and 4, as well as 9c and 10c.)
This can be visualized with the series grapher applet. To use the applet, launch Internet Explorer and go to http://www.slu.edu/classes/maymk and choose the link for the Series Function Grapher. You want to see the graphs of both the series function (green) and the guessed function (red), so check the appropriate boxes.
1) The default is set for the Taylor series of
sin(2x). Evaluating the nth
derivative at 0, we see that
is 0 if n is even,
if n is of the form 4m+1, and
if n is of the form 4m-1. Thus we
only look at odd terms of the
Taylor polynomial. These terms
have the form
. Move the slider to see that the
function defined as a series is a good approximation. Click the
boxes to show f(x) and the error term. How high should the degree
of the Taylor polynomial be for the polynomial to be visually
indistinguishable from f(x) between -5 and 5? For a 17th degree
approximation what is the error at 4? (Click on the axis at 4 and
read off the error.)
2) To approximate f(x) = cos(3x) we note that
is 0 if n is odd,
if n is of the form 4m, and
if n is of the form 4m+2. Thus we
only look at even terms of the
Taylor polynomial. These terms
have the form
. Change the series term definition,
guessed function, and the slider to see that the function defined as a
is a good approximation. What degree approximation do we need for
the approximation to be visually indistinguishable from the function
from -5 to 5?
3) Exercise 8 asks you to do the work of producing
approximation for f(x)=x. We note
by symmetry that
for all k. The exercise
asks you to show that
for k>0. The term we need in
the series grapher is
. Note that the lower k should be set
1 or the function is not defined.
Verify that this is a good approximation within a particular
range. What is the interval where the approximation works?
What is the error at x0=2 with a 40th degree approximation?
4) Exercise 9 looks for a Fourier approximation
for f(x) =
π-|x|. For technical reasons, we
will look at f(x) = π/2 - |x|.
(This should only change the first coefficient.) Then by symmetry
for all k, and
. The term we need for the grapher is
. Note that the minimal k should be 0
again. Verify that this is a good
approximation. At what degree is the approximation visibly the
same as the function between -2 and 2? Why do you think the
approximation technique works better for this function rather than the
function in problem 2?