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partial fractions
Posted:
Apr 24, 2001 12:17 AM


I guess I'll take my own bait:
> Assume for the sake of argument that > > (i) all your Calc II students have TI89s 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/(1x^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 TI89 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 destress 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+2x2) , 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 5076463414 office 1520 St Olaf Avenue 5076463116 fax Northfield Minnesota 550571098 **************************************************************

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