They were not told beforehand that anything like this would be on the test.Īnother thing which they were fairly good at was using differentials, since the differential was not some mysterious formal symbol to them but an actual infinitessimal value. Nearly everybody in the class, even the C-students, did this derivation properly. The next question on the final was “prove the quotient rule”. Almost uniformly, the students could create correct computations of the derivative. ![]() On the final, I asked them to compute the derivatives of, , and from the definition. The derivative of, with no more than two steps left out: I was surprised that they saw this trick so quickly, until I realized that they had been doing the exact same thing with complex numbers for several years already. When a number like appeared in the denominator, the students quickly figured out that they could multiply on the top and bottom by to “realify” the denominator. They used them every day in nearly every problem, and they remembered. Even the weaker students at the end of the course could all compute the derivative of as a routine computation:īut even better, they remembered or because these manipulations had become common to them. I never allowed formula sheets or calculators or anything like that in the class or on the tests - the kids were genuinely remembering the formulas and the derivations. By the end of the third day, there were homework problems (generally solved correctly) like “use the definition of the derivative to find the derivative of ”:īy asking them to compute the derivative of things like or by hand, they rapidly re-learned (and remembered!) the angle-sum formulas for sine and cosine and the algebraic rules involving exponents. From here, it is easy to compute derivatives of relatively complex functions by hand. Then by looking at a right triangle of height, we decided that and. On the second day, I gave them a few useful tidbits: as an axiom, I said. So if you are judging mathematical truth by “is relevant to things I know in the real world”, it seems that nilsquare infinitesimals aren’t such a strange idea. I also pointed out that computers are using a number system which is closer to this than it is to : since floating point processors have finite precision there is a smallest positive denormal. They thought this was a little funny, but came around to it after I drew an analogy to and : even if they had philosophical objections to thinking about or as “numbers”, their recent experience with complex numbers in high school put them in a good place for accepting as a useful formal symbol, if not an honest-to-goodness number.We could then quickly move on to computing the derivative of some elementary functions like : In the first serious lecture, I introduced as a positive number so small that. This may have seriously affected the outcome of the class. ![]() So I had a fantastically clean slate to work with. Maybe they had been told how to formally take derivatives of polynomials, but not much more. Most students had not seen much beyond precalculus. Now, a bit more about the class I taught. The (serious) tradeoff is that we lose the transfer principal when we use non-nonstandard analysis. ![]() But the main idea is this: we want infinitesimals, but we don’t want to bring in the heavy machinery if we can avoid it. I’d like to say more about this version of non-nonstandard analysis sometime in the future, but not just this moment. The idea was to get infinitesimals into calculus without the big ultrafilter-style logic machinery of standard nonstandard analysis. I learned nonstandard analysis from him one summer, but also learned what he called “non-nonstandard analysis”. Maybe a word about the name before we begin: I think that a lot of my philosophy on mathematics came from taking several analysis classes at Smith College from logician Jim Henle. I think I have had enough time to process the experience - let’s talk non-nonstandard calculus. I’ve been promising Greg and Jim that I’d write up some of my experiences with the course. The class was a six-week, 5-day-per-week intensive course covering the usual material of a college Calculus I course. I alluded in one of my very first posts here to a calculus class that I was teaching using the ring.
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