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Added: by the way, it's not as though the above question is "bad" in the sense that it's not testing mathematical competence and depth of understanding of calculus. I think it absolutely is, just at a level way above that which one should be testing in a freshman class for non math majors. For the next few years, when the story came up in a social setting involving mathematical hotshots, after telling it I would press them for an answer to part c) on the spot. Most people I asked did not get it. (Note that I would not of course give them pen and paper and a quiet spot to think about the problem for some period of time. I generally required an answer after a minute or so. Let's hold PhD mathematicians to higher standards than freshman non-majors after all!) For instance, I watched a cloud pass over one Fields Medalist's face as he got very confused. After a while though I stopped using this as a pop quiz in addition to a story: I can't explicitly remember why, but I'd like to think it dawned me how obnoxious it was to put people on the spot like that...

Post Made Community Wiki by S. Carnahan
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[Added: I admit that I forgot about nonstandard analysis when I wrote the above paragraph. That indeed has a somewhat different feel from the usual limits and continuity. One the one hand, although I have never taught calculus this way, I rather doubt that doing so would suddenly make the difficult concepts of continuity and differentiability go over easily. On the other hand, I certainly couldn't decide to teach a nonstandard approach to calculus because it would be...nonstandard. The curriculum among different sections, different classes and different departments has to have a certain minimal level of coherence, and at the moment the majority of the grad students and faculty in every math department I have ever seen are not familiar enough with nonstandard analysis to field questions from students who have learned calculus by this approach.]

Nevertheless I take your question seriously, since I have taught a fair amount of freshman calculus in recent years. It is absolutely correct that a lot of students get impatient, angry and/or confused at the limit definition of the derivative (or really, at anything having to do with limits and/or continuity). I do derivations of things like the product rule and the power rule rather quickly in class, because I know that something like half the class isn't following and doesn't care to follow. And yet I do them anyway (not all of them, but more than half) because, to me, not to do them makes the course something I could not bring myself to teach (and, by the way, would put it well below the level of the AP calculus class I had in high school: I feel somewhat honorbound to give to my calculus students not too much less than what was given to me). Thus there is a real disconnect between the calculus class that I want to teach and the calculus class that something like half of the students want to take. It's discouraging.

2) Emphasize physical reasoning. The last time I taught freshman calculus, I spent the entire first day talking about velocities: first average velocity, then instantaneous velocity. If a differentiation rule has a plausible physical interpretation -- e.g. the chain rule says that rates of change should multiply -- then I often give it.

different from the units of the original function. In this way one can see that the conjectured product rule $(fg)' = f'g'$ is dimensionally wrong and thus nonsense. (And again, the chain rule is "obvious" from a unit conversion perspective.) Similarly dimensional analysis should stop you from saying that the volume of a cylinder is $\pi rh$.

Consider the function $f(x)$ defined as $x^a \sin(\frac{1}{x^2})$ for $x \neq 0$ and $f(0) = 0$. What is the smallest integer value of $a$ such that $f$ is (i) continuous, (ii) differentiable, (iii) twice differentiable. ?

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