Selling completeness, extreme value theorem, etc.?

Question

There is a set of related topics in a freshman calc course that includes the completeness axiom for the reals, the intermediate value theorem, extreme value theorem, Rolle's theorem, and mean value theorem.

How can one interest a freshman calc student in these topics, when they almost certainly have zero interest in foundational issues?

This material seems to me to have a fairly tenuous connection to the mainstream of a freshman calc course. I'm not an expert in the history of mathematics, but it seems that Dedekind cuts and such were not invented until about 1860. That leaves a period of about 150 years during which the world's greatest mathematicians happily did calculus without feeling a need for these foundational issues to be addressed. They would probably have felt that a result like the extreme value theorem was geometrically obvious. (Some of the theorems were stated earlier, e.g., Rolle's theorem dates to 1691, but I don't see how Rolle could have given a meaningful proof 150 years before the arithmetization of the reals.)

Further adding to the difficulties in selling this to a class of students are the facts that:

1. These theorems are not particularly useful in solving problems (because in most problems where they were needed, the theorems' claims would have been intuitively obvious anyway).

2. In a freshman calculus course we don't really attempt to present this material in detail. E.g., textbooks tend to omit most of the proofs.

(Just to put this in perspective, I recently asked a class of students who had all already had a year of calculus to describe in general terms how the derivative was defined -- I wasn't asking them to regurgitate the definition, just to describe its flavor. Despite a lot of poking, prodding, and hinting, not a single student volunteered that it was defined using a limit.)

The best idea I've been able to come up with so far is just to give the class a series of tasks in which they come up with counterexamples to the various theorems when one of the assumptions is omitted, e.g., give an example of a function that is continuous everywhere in its domain but that never attains a maximum value. This might at least help to convince them that the results make sense and have a geometrically visualizable interpretation.

Related: https://mathoverflow.net/questions/184358/what-did-rolle-prove-when-he-proved-rolles-theorem

Answer 1

In the preamble to your question you gave several great arguments against dwelling on foundational issues in a freshman calculus class, and indeed I think it is inappropriate to focus too much on these issues. In my view, the point of a calculus class is to teach students how to use a particular set of tools to solve problems, and we should give them time to understand how to use the tools before expecting them to appreciate why the tools work. It's better to teach someone how to ride a bike before teaching them how to build one.

That said, you were asking how, not if, you should sell foundational concepts to your students. My suggestion is to integrate them into your examples as much as possible. If you go over the intermediate value theorem in one class toward the beginning of the semester and then never revisit it, the theorem will appear as an esoteric curiosity that they can safely dismiss. But suppose that later on you do an example such as:

Let $f(x) = e^x + 5x^2 + 3x$. Is there a point on the graph of $f$ where the tangent line has slope $2$?

This comes down to solving the equation $f'(x) = e^x + 10x + 3 = 2$. This can't be done algebraically, but you can observe that $f'(0) = 4 > 2$ and $f'(-1) = e^{-1} - 7 < 2$, so indeed $f'(x) = 2$ for some $x$ between $-1$ and $0$ by the intermediate value theorem. This makes the intermediate value theorem feel more like a technique than just an idle fact.

With a little thought, you can find ways to sneak other theorems into your examples. For instance, after you've done a bit of integration you can give the following example:

Find the average value of $f(x) = x \sin x$ on the interval $[-\pi, \pi]$. Does $f$ attain its average at some point in the interval?

After you find that the average is $1$, you can remark that the equation $x \sin x = 1$ can't be solved algebraically, and a direct application of the intermediate value theorem is no help since $f(-\pi) = f(\pi) = 0$. But $f$ attains its minimum and maximum values at points $x_{min}$ and $x_{max}$ by the extreme value theorem, and you can observe that $$f(x_{min}) \leq \frac{1}{2\pi} \int_{-\pi}^\pi f(x)\, dx \leq f(x_{max})$$ So by the intermediate value theorem $f$ has to attain its average. (Notice that this actually sneaks the proof of the mean value theorem for integrals into an example. The word "proof" shuts off students' brains, but anything you say after the word "example" commands their full attention.)