Timeline for Why do we teach calculus students the derivative as a limit?
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| Sep 15, 2013 at 11:09 | comment | added | Pietro Majer | @Mark Meckes: I don't think so; actually what I'm saying is that applying de l'Hopital's rule to compute that limit would be cheating. Recognising that this limit is the derivative of $\sin(x)$ at $0$ is just calling a thing by its name, which is quite a fair fact -especially if one already knows the name. But I agree that one should also keep in mind the way these identities were introduced, to avoid circular or redundant argments. | |
| Apr 4, 2011 at 6:29 | comment | added | Pietro Majer | Well for $\lim_{x\to0+} x\log x$ I guess they would make a substitution $x=e^{-t}$ to get $\lim_{t\to+\infty} -te^{-t}=0$ (and for the latter, they know how to use inequalities such as $e^t\ge t^2/2$ for $t>0$ to conclude). | |
| Apr 3, 2011 at 23:02 | comment | added | Toby Bartels | Without L'Hôpital's Rule, how do your students evaluate limits such as $\lim_{x \searrow 0} x^x$ (or $\lim_{x \searrow 0} x \ln x$)? Not that I can't think of a way to do it, but I'd like to know what you (those who wouldn't teach the rule) would suggest to their students as the method of attack. | |
| Oct 19, 2010 at 14:11 | history | made wiki | Post Made Community Wiki by S. Carnahan♦ | ||
| Sep 28, 2010 at 14:27 | comment | added | Mark Meckes | About that funny remark: even saying $\lim_{x\to 0} \sin(x)/x$ is the derivative of $\sin(x)$ at 0 may be viewed as cheating, since the typical textbook approach is to use a geometric argument to prove $\lim_{x\to 0} \sin(x)/x = 1$ and then use that limit to prove that $\frac{d}{dx} \sin(x) = \cos(x)$. | |
| Sep 27, 2010 at 20:34 | comment | added | Pietro Majer | @Pete: This is a good observation, and I certainly agree with it. There's also a vague and soft irony in it, that at the moment I'm not able to answer otherwise than upvoting your comment. | |
| Sep 27, 2010 at 20:18 | comment | added | Thierry Zell | @Andrew: the prof's point though was that you never need l'Hopital's rule. | |
| Sep 27, 2010 at 19:49 | comment | added | Pete L. Clark | @Pietro: what you give is essentially Caratheodory's definition, as alluded to in my answer. It's so close to the usual definition that I don't really believe that students have a significantly easier time with it. However, I believe that when you teach calculus, this definition inspires you and you do a very good job teaching it, more so than you would with the standard definition. I suspect that most "the students find it easier when..." statements are like this, but that's fine -- finding the version that you can get behind enthusiastically and explain well is part of good teaching. | |
| Sep 27, 2010 at 19:13 | comment | added | Pietro Majer | @Andrew: and if I was the chairman in another maths department, I'd immediately engage him with double salary. De'LHopital himself would be embarassed to know somebody's still wasting time with such an awkward theorem like that thing that brings his name. Theorems, like cakes, don't always come out well; that thing came out very badly, and left a mess in the oven. Today, it may be at most of some historical interest. Teach the Landau notation instead! (Btw, as you probably know, Edmund Landau was fired from Göttingen in 1933, with the pretext of his way of teaching a calculus course.) | |
| Sep 27, 2010 at 17:17 | comment | added | Simon Rose | J.M. - If you can make them believe that $e^ix = \cos(x) + i\ \sin(x)$ then it is pretty easy once they know the derivative of $e^x$. But that's a pretty tall order there | |
| Sep 27, 2010 at 16:53 | comment | added | The Mathemagician | @Thierry If I was chairman of the mathematics department when your professor was teaching and I'd heard that,he'd have been fired on the spot unless he had tenure-and if he had tenure,I'd have made sure he never taught it again. He should have been ashamed of himself.Really. If you think this material is usually presented so lousily,do something about it! | |
| Sep 27, 2010 at 15:57 | comment | added | Benoît Kloeckner | Yes! I think that introducing the differential of a function of several variables would be much easier for students if they had this point of vue on derivative. | |
| Sep 27, 2010 at 12:42 | comment | added | Thierry Zell | From my prof back in the days: "Some of you might have heard of a thing called L'Hopital's rule. It has a lot of hypotheses that no-one ever checks, and students always apply it when the quotient is in the wrong form, so I won't teach it and you'd better not use it." And now I do the same... (I didn't mind when he said that because I was one of the ones who'd never heard of it.) | |
| Sep 27, 2010 at 12:40 | comment | added | J. M. isn't a mathematician | +1. This reminds me of the following question: without making use of the fact that $\lim_{x\to 0}\frac{\sin x}{x}=1$, are there any other ways of developing the formulae for the derivatives of the trigonometric functions? It seems there's no way to escape this (which makes it even funnier that people actually use l'Hôpital for this). | |
| Sep 27, 2010 at 12:28 | history | edited | Pietro Majer | CC BY-SA 2.5 |
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| Sep 27, 2010 at 12:10 | history | answered | Pietro Majer | CC BY-SA 2.5 |