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Topic: Today's complaint about the new teacher's guide.
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Joshua Zucker

Posts: 710
Registered: 12/4/04
Today's complaint about the new teacher's guide.
Posted: Oct 24, 1997 8:10 PM
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Don't get me wrong -- I think there are a lot of good revisions in the
AP calc syllabus. But there are two places in particular, directed
oppositely (one toward less rigor, the other toward more) that I think
are not so good.

The first is that they suggest using a table of values (e.g. when h is
0.1, 0.01, 0.001) to "compute" the derivative. This is not terrible,
but it's not the mathematics, either. They need to make it clear that
we're talking about the 'best estimate from the given data' of the
derivative, and not CALCULATING the derivative. dx/dt is a bad thing,
because for empirical data you can never measure ultra-high-frequency
noise, so in practice we must use delta-x/delta-t instead, as an
approximation -- but they are not the same, and the approach in the
new syllabus seems to confuse the two. I think that teachers need to
be very careful to distinguish between the two ideas, and to make
clear when the substitution is OK -- and I think it's very hard to be
precise about when the approximation is OK, and when it isn't (e.g. if
you're looking for a local max, it might be a bad idea to use a
"derivative" interpolated from a table of values...)

The second is that they say

"no matter what it might mean to a student intuitively,
integral from 1 to infinity of x^(-2) = -x^(-1) from 1 to infinity
= -1/infinity - (-1) = 0+1 = 1
is {\bf incorrect mathematics}"

which is the extreme the other way, of insisting on rigor when rigor
is stupid. This implies to me that graders will take off points for
this kind of work ... do students have to explicitly write it as a
limit? Or do they have to just say "and since 1/x -> 0 as
x->infinity, the contribution from the upper limit -> 0, so ..."

The second case is at least somewhat reasonable, but still excessive.
The ability to facilely throw away infinities like this one
(1/infinity = 0) is very useful in physics, I know. And physicists do
far worse things with infinity sometimes, too. Yet the AP syllabus is
moving us away from that kind of thinking. Sigh.

--Joshua Zucker

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