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Today's complaint about the new teacher's guide.
Posted:
Oct 24, 1997 8:10 PM


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 ultrahighfrequency noise, so in practice we must use deltax/deltat 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



