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edited Jan 29 2011 at 22:12

Pete L. Clark
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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♦

occurred Oct 19 2010 at 14:11

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edited Sep 27 2010 at 10:55

Pete L. Clark
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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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edited Sep 27 2010 at 6:34

Pete L. Clark
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answered Sep 27 2010 at 6:05

Pete L. Clark
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I agree with the above comments.

The point of my comment-question "What competing definition do you have in mind?" was to emphasize something that seems to be under-emphasized in the question itself: the reason we speak of derivatives as limits is because that's the definition of the derivative, and we want to give a definition of the concept that is going to be discussed for much of the semester.

[It is possible to give other definitions of a derivative, but they are all variations on the same theme and, in particular, all use either the concept of limit or the (equivalent!) concept of continuity. For instance, Caratheodory has a nice definition of the derivative in terms of functions vanishing to first order, but this is not going to be any more palatable to the freshman calculus student.]

If we don't give a definition of the most important concept in the course, then we lose all pretense of developing things in a logical sequence. In particular, it's hard to see how to discuss the derivations of any of the basic rules the students will actually be using to compute derivatives, and thus we would be forced to reduce calculus to a (long!) list of algorithms based on certain unexplained rules.

I would be happy to hear that I am making a false dichotomy between giving the limit definition of the derivative and just giving algorithms to solve problems. I definitely experiment with different kinds of explanation beyond (and instead of!) just a formal proof. Here are some things I have tried:

1) Take the definition of continuity as primary, and define the limit of a function at a point as the value at which one can (re)define the function to make it continuous. I think this should be helpful, since I think most people have an intuitive idea of a "continuous, unbroken curve" and much less of the limit of a function at a point.

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 that rates of change should multiply -- then I often give it.

3) Emphasize "chemical reasoning", i.e., dimensional analysis:. I often give the independent variable and the dependent variable units and emphasize that the units of the derivative are 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. Similarly dimensional analysis should stop you from saying that the volume of a cylinder is $\pi rh$.

Unfortunately none of these things have worked with the portion of the class that doesn't want to hear anything but how to solve the problems.