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[apcalculus] Re: Power Series
Posted:
Feb 23, 2006 10:52 AM


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 TI89 (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 xaxis (i.e. has a positive angle of intersection with the xaxis) and the latter will look like a parabola tangent to the xaxis (i.e. has a zero angle of intersection with the xaxis). 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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