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Topic: more bait
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Paul Zorn

Posts: 325
Registered: 12/6/04
more bait
Posted: Apr 24, 2001 11:04 AM
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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 under-appreciated 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


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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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