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Limiting process
Posted:
Jun 12, 1997 4:04 PM
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I am following the list with great interest. As a community college
math teacher, I teach many courses "below" calculus, but I try to inject
the concept of a limiting process wherever I can - so that when they
arrive in calculus they are prepared for a formal derivation.
As a few examples:
1. The formula for the area of a circle can be justified by disecting
the circle into wedges which can be set down in alternating orientations
to approximate the shape of a square. The smaller the wedges the, the
better the approximation - and if the wedges are infinitely thin, then
the approximation becomes exact.
2. The Babylonian algorithm which converges to the root of a number. A
nice example of digit doubling as well.
3. Other feedback loops with iterative function.
4. The Koch curve (snowflake). This one is nice since it shows that,
when the limiting process is complete, you have a new and marvelous
object with qualities unlike any of the approximations.
5. In developing the formula for natural growth we have the limit of
(1+1/n)^n as n->inf, numerical data are used to justify the definition
of e.
If a student can digest and come to appreciate these, then they are ripe
for appreciating what it really means for a curve to be smooth.
Geoff Hagopian
Palm Desert, CA
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