Timeline for Why do we teach calculus students the derivative as a limit?
Current License: CC BY-SA 2.5
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| Jan 29, 2011 at 22:12 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
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| Nov 6, 2010 at 18:05 | comment | added | Pete L. Clark | To belatedly reply to Timothy Chow: when I said that I couldn't teach a nonstandard calculus course if I wanted to, what I meant was that I alone couldn't do it. It would have to be a decision made at the departmental level, and I don't think it would be an easy sell. | |
| Oct 19, 2010 at 14:11 | history | made wiki | Post Made Community Wiki by S. Carnahan♦ | ||
| Sep 27, 2010 at 22:45 | comment | added | Nick S | A much simpler version of your problem is the following (I actually used this problem in the past): Consider $f(x)= x^2 \sin( \frac{1}{x})$ for $x \neq 0$ and $f(0)=0$. Find $f'(0)$. This function is differentiable at zero but $f'$ is not eve continuous at 0 (so no power series representation), so I doubt that any other approach than the limit would work. I think this is one of the simplest examples which explains why the limit is needed. Also, what if one discovers a completelly new great function $f$, how does one find $f'$? | |
| Sep 27, 2010 at 20:36 | comment | added | dvitek | @Thierry: I like your quote of "any problem with two letters in it is too hard." | |
| Sep 27, 2010 at 19:33 | comment | added | Pete L. Clark | @Mariano: no, I was dead serious. The point was that this person was far too bright to realize that this was a ridiculously hard question for freshman calculus. | |
| Sep 27, 2010 at 17:12 | comment | added | Simon Rose | I really like the unit analysis example. I might keep that in mind next time I teach. | |
| Sep 27, 2010 at 16:55 | comment | added | Mariano Suárez-Álvarez | Was your «who was a tenured professor of mathematics, hence a very brilliant person» parenthesis ironic? :P | |
| Sep 27, 2010 at 14:50 | comment | added | Timothy Chow | Teaching undergraduate calculus using nonstandard analysis is not out of the question. I haven't done it but I know others who have, using for example Henle and Kleinberg's Infinitesimal Calculus. | |
| Sep 27, 2010 at 11:41 | comment | added | Thierry Zell | The freshman-level example is especially biased because students tend to believe that any problem with two letters in it is very hard. But I've had a reasonable degree of success with a problem of this type, using a value for a (disclaimer: at a good school, though no Harvard). But it's also because I'd spent some time on this in class; you can't spring this on students out of the blue like that professor did and expect they'll do well. | |
| Sep 27, 2010 at 10:55 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
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| Sep 27, 2010 at 8:37 | comment | added | Gjergji Zaimi | +1. Differentiation can be done without limits too, en.wikipedia.org/wiki/Formal_derivative I interpreted the question as distinguishing between derivatives in analysis and "generating functions". Even most trigonometric functions have combinatorial meaning and so their derivatives can be computed formally. But as you say that misses the point of calculus (continuity, physical reasoning etc.). | |
| Sep 27, 2010 at 6:34 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
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| Sep 27, 2010 at 6:18 | comment | added | Steven Sam | Thanks for the answer and sharing your experiences with this. Of course as a mathematician I understand that we give the definition because otherwise we have no logical foundation to work with, but I also understand that there is a huge disconnect between the mathematician's mentality and what students expect out of a freshman course. As a young person I really know nothing about good teaching so it's good to hear about what kind of balance is possible. | |
| Sep 27, 2010 at 6:05 | history | answered | Pete L. Clark | CC BY-SA 2.5 |