MT A143-03 (Calculus II)
Fall
2001
Notes
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
f(x) =
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,
the
guessed function, and the slider to see that the function defined as a
series
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
a Fourier
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
to
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
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?
Return to the
Series Function Grapher
Applet page.
Return to the
Calculus Applet
page.
Last updated By
Mike May, S.J.,
October 18, 2004.