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Topic: partial fractions
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Paul Zorn

Posts: 325
Registered: 12/6/04
partial fractions
Posted: Apr 24, 2001 12:17 AM
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I guess I'll take my own bait:

> Assume for the sake of argument that
>
> (i) all your Calc II students have TI-89s or other technology that
> plots, expands, and antidifferentiates rational functions;
>
> (ii) the next section in the book is on partial fractions.
>
> What do you do tomorrow? Why?


I might start by plotting both f1=1/(1+x^2) and f2=1/(1-x^2) ---
typographically similar but otherwise radically different, with the
first bounded and the second having two poles. Then I might have
students GRAPHICALLY antidiffi'ate both functions. The second
antiderivative wouldn't be obvious from its graph, but I'd try to
elicit or just announce that some logs are involved. and we'd
rediscover that antidifferentiating 1/linear rational functions gives
logarithmic results. Then we'd ask the TI-89 for help, see what it
says, and take it from there.

I'd stress a few main points:

(1) the divide and conquer idea ... try to *restructure* the original
fraction as a sum of objects that are simpler in a
sense appropriate to the task at hand;

(2) finding numerical coefficients that do the right things;

(3) that systems of linear equations come up naturally;

(4) that answers involve just a few predictable ingredients:
logs, arctangents, and some basic rational functions.

I'd de-stress hand manipulations, but I'd force students (by hand)
through a few simple examples. I'd also force students to
complete a square (e.g., in antidiff'ing 1/(x^2+2x+2) and
1/(x^2+2x-2) , less as a hand technique than to reveal
important but somewhat hidden structural differences.

Paul

**************************************************************
Paul Zorn zorn@stolaf.edu
Department of Mathematics http://www.stolaf.edu/people/zorn/
St Olaf College 507-646-3414 office
1520 St Olaf Avenue 507-646-3116 fax
Northfield Minnesota 55057-1098
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