Timeline for Why do we teach calculus students the derivative as a limit?
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| Oct 6, 2012 at 2:47 | comment | added | user21349 | I appreciate your point, but your examples are all discontinuous functions, so in fact the notion of a limit doesn't quite work. For example, the US deficit can only change in steps of one cent, the amount of water only in steps of one molecule. These could all be used as examples to show that the Cauchy-Weierstrass limit does not actually connect to reality. Perhaps a better point to make with these examples would be that students could benefit by understanding discrete/numerical calculus as well as the classical calculus of continuous real functions. | |
| Mar 12, 2011 at 1:25 | comment | added | Thierry Zell | @Pete: I have been thinking a lot about how the calculus we teach is too neat and not applied enough these days, and I've been thinking about it not because of my calculus classes, but because of my higher-level courses where I've had to use "messy" or "exotic" stuff like Taylor expansions pretty often to motivate some avenues of investigation. I wish my students were more comfortable with this, and expected less of the neat closed-form formula results. Though of course, we should teach them to appreciate when closed-form stuff happens too! | |
| Jan 29, 2011 at 22:44 | comment | added | Pete L. Clark | Once or twice I tried to remedy this by asking exam questions like: "Write down an explicit function which has local minima at $\pm 2$ and approaches infinity as $x$ approaches $\pm infinity$." I was hoping that a student would see that a simple function satisfying these conditions is a fourth degree polynomial with double roots at $\pm 2$, thus $f(x) = (x-2)^2 (x+2)^2$. But they had a lot of trouble with this, and the answer to my question "Does teaching the standard curriculum give them the tools to answer this question?" was "No." But wouldn't it be great if students could do this? | |
| Jan 29, 2011 at 22:38 | comment | added | Pete L. Clark | @Anna: I agree heartily. To take it a step further: I have often thought that the traditional calculus sequence lacks an applied component which severely limits its usefulness to those who are not going on to physics and math. As you say, when given a real world function of interest, you are generally not given an algebraic expression for it. Rather, in order to apply the methods of calculus in a quantitative way, there needs to be a step where you create a mathematical model of the function. I was never taught how to do this step myself, and it seems not to be at all trivial... | |
| Jan 29, 2011 at 19:25 | history | answered | Anna Varvak | CC BY-SA 2.5 |