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Topic: Topic-motivated Calculus (which topics best?)
Replies: 2   Last Post: Jan 22, 2002 5:47 PM

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Kirby Urner

Posts: 4,709
Registered: 12/6/04
Topic-motivated Calculus (which topics best?)
Posted: Jan 21, 2002 4:41 PM
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I'm trying to think of topics which motivate study of calculus and
at the same time necessitate exposure to a healthy cross-section of
its core ideas, i.e. the exposure isn't too narrow.

An example I've encountered a few times recently is Fourier
Analysis. I've read two excursions into that topic recently,
one for language students [1], another for music students [2].

Both get into a derivation of e, a discussion of logs and In, of
derivatives as limits, of Taylor and Maclaurin series, of
Eulers e^j(pi) = -1 and CIS. Plus there's adding functions,
integral notation, derivative of trig functions, background
on complex numbers etc. etc.

My interest is in curricula which start with some overview challenge
or topic which drives or motivates delving into X, where X is some
technical domain, e.g. calculus. A parallel example would be
using cryptography to drive a study of group theory. You start with
an application, and use that as a guiding outline for fishing out
what you need from a broader, more generic theory.

Aside from Fourier Analysis, which other topic areas motivate
coverage of a lot of interconnected calculus topics? We usually
see the classical mechanics stuff a lot, starting with velocity and
acceleration. Maybe it's used too much? Electromagnetism is another
such topic, and gives us more of a workout with vectors, gets to the
div, grad, curl stuff, plus partial differentiation. The problem
here is fewer majors seem to need a "semi-casual" understanding of
Maxwell's Equations (they're either science and engineering majors,
and therefore have a lot of calculus) or they don't study them at
all. So maybe that's why it's harder to find these kinds of "from
the ground up" approaches to electromagnetism, although I'm not
saying they don't exist (recommendations?).

Not that any of these topics are mutually exclusive of course -- I'm
just wondering what's a good motivating topic that might be used with
students who presumably start with not a lot of background, and
therefore need several puzzle pieces explained in order to fit a
coherent system together. It seems that Fourier Analysis is already
proving itself a winner in this role, but I was wondering what other
topics might be leading contenders.

Kirby

[1] http://www.lexlrf.org/pub/math.html
[2] http://www-ccrma.stanford.edu/~jos/r320/

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