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more bait
Posted:
Apr 24, 2001 11:04 AM


Dear colleagues,
Lots of good ideas about partial fractions ... the subject seems to be a good catalyst for thoughtful reactions (none exothermic, so far) on larger issues. This is exactly the kind of thing that keeps me (for one) interested in this discussion group.
A note on just one part of Arch's recent (interesting!) posting:
> ... Quadratic equations: a thing of the past ...
I'm not sure how far Arch would take this, and don't want to put words in his ... uh ... fingers.
I agree with it *if* what's meant here is that there's more to mathematical life than the busywork of factoring expanded equations and expanding factored ones.
But in other ways I think quadratic functions are excellent, accessible, and generally underappreciated tools for helping students understand key ideas that should matter in calculus courses.
1. They're the simplest functions on which (in some sense) calculus can actually be ``done''.
2. They illustrate about as simply as possible the interplay between algebraic forms and analytic properties  completing the square is a *great* example (and a look ahead) of altering algebraic form to reveal structure.
3. They're the local prototype for the DIFFERENCE between ``any'' calculus function and its LINEAR approximation.
4. They're the local prototype for QUADRATIC approximation of ``any'' calculus function. ``Tangent parabolas'' can help make sense of concavity and of the second derivative test in single variable settings. In the multivariate context the idea of quadratic approximation seems almost essential to making any sense of classifying stationary points  at least, I've never really understood it any other way.
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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