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Topic: [ap-calculus] Power Series
Replies: 1   Last Post: Feb 23, 2006 10:52 AM

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Dave L. Renfro

Posts: 2,165
Registered: 11/18/05
[ap-calculus] Re: Power Series
Posted: Feb 23, 2006 10:52 AM
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Carol Hart wrote:

http://mathforum.org/kb/thread.jspa?threadID=1338880

> My BC students are just beginning Power Series .. and
> I have to say how much fun is to teach the concept
> with the Foerster text. I used to sort of dread this
> relatively difficult concept .. I'm not sure I ever
> really understood it completely before I started
> using this text. I love the way it is approached.
> We also use the "Explorations" text - Advanced
> Placement Calculus with the TI-89 (Chapter 9 Infinite
> Sequences and Series) from Texas Instruments for
> very concise instructions on how to calculate and
> graph the problems.


I don't know if this is in Foerster's text, but here's
something neat that you can do with Taylor expansions.
I first saw the idea on this list, back in Fall 2000.
I tried it out the next day in a college calculus 2
class (we were in the middle of Taylor series at the
time) where each student had a computer equipped with
Mathematica, and it went over very well, but even with
graphing calculators I've found it's nice (but not as
neat as with CAS's).

If you graph e^x - (1 + x), the behavior near x=0
will look like a parabola.

If you graph e^x - (1 + x + (x^2)/2), the behavior
near x=0 will look like a cubic.

If you graph sin(x) - x, the behavior near x=0
will look like a cubic.

Moreover, these aren't just any parabolas or cubics,
they appear to have the form (constant)*x^2 or
(constant)*x^3.

In general, the behavior of f(x) - P_n(x), where
P_n is the n'th order Taylor polynomial about x=0,
will look like the graph of (constant)*x^m near x=0,
where x^m is the first nonzero term in the Taylor
expansion of f(x) about x=0 that isn't in P_n.

This provides a nice graphical illustration for
the remainder term of the Taylor series.

You can also do this for expansions about other
points besides x=0. Moreover, you can do this
near the beginning of a calculus course (and I
have done so many times) to illustrate that the
tangent line is the *best* linear approximation
in this way: Let L be the equation of a line that
isn't the tangent line and let TL be the equation
of the tangent line. Then the graph of y = f(x) - L
and the graph of y = f(x) - TL will differ near
the tangency point in a way that is readily
visible on a graphing calculator. When you zoom
in, the former will look like a line through the
x-axis (i.e. has a positive angle of intersection
with the x-axis) and the latter will look like
a parabola tangent to the x-axis (i.e. has a zero
angle of intersection with the x-axis). The idea
is that BOTH L and TL can be used to approximate
f(x) arbitrarily closely near the point in question
(since both lines intersect the graph of y = f(x)
at that point), but the error involved with TL is
*qualitatively* better than the errors associated
with any of the L's.

This is such a neat idea that I did some searching
to see who brought it up. Here are the posts in
this list where I learned about this graphical
idea:

Richard Sisley (November 22, 2000)
http://mathforum.org/kb/thread.jspa?messageID=662138

Mark Howell (November 26, 2000)
http://mathforum.org/kb/thread.jspa?messageID=662142

Incidentally, "angles zero measure" are called
horn angles, and they were apparently even studied
by Euclid:

http://aleph0.clarku.edu/~djoyce/java/elements/bookV/defV4.html

"The result on horn angles in proposition III.16
excludes ratios between horn angles and rectilinear
angles. That proposition states that a horn angle
is less than any rectilinear angle, hence no
multiple of a horn angle is greater than a
rectilinear angle."

Dave L. Renfro

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