This is an update with (unexpected) good news. My collaborator Miles Wheeler and I were given the Paul R. Halmos – Lester R. Ford Award for our paper Perturbing the mean value theorem: implicit functions, the morse lemma, and beyond1 1The arxiv version of this paper is called When are there continuous choices for the mean value abscissa, and is slightly longer than the final published form.
This was an unexpected honor. Partly, this is due to the fact that we first wrote this paper years ago and it was accepted in 2019. But the publication backlog meant that it wasn't published until January of 2021, and thus eligible for the 2022 award. But also it's always sort of a nice surprise to hear that people read what I write.
I described this paper before, and wrote an additional note on how we chose our functions and made the figures.
When my wife learned that this paper won an award, she asked if this "was that paper you really liked that you wrote wtih Miles during grad school"? Yes! It is that paper. I really do like this paper.
Origin story
The string of ideas leading to this paper began when I first began to TA calculus at Brown. I was becoming aware of the fact that I would soon be teaching calculus courses, and I began to really think about why we structured the courses the way we do.2 2I don't have a completely satisfactory explanation for everything, especially for the "integration bag of tricks" or the "series convergence bag of tricks" portion. But upon reflection, I can understand the purpose of most portions of the calculus sequence.
One of my least favorite questions was the verify the mean value theorem for the function $f(x)$ on the interval $[1, 4]$ by... sort of question. The problem is that this question is really just a way to check that one understands the statement of the mean value theorem — and this statement feels very unimportant.
But it turns out that the mean value theorem is extremely important. The mean value theorem and intermediate value theorems are the two sneaky abstractions that encapsulate underlying topological ideas that we typically brush aside in introductory calculus courses.
We don't do calculus on real valued functions over the rationals
We illustrate this with two examples in the analogous case of functions \begin{equation*} f: \mathbb{Q} \longrightarrow \mathbb{R}. \end{equation*} I would expect that many introductory students would think these functions feel intuitively about the same as functions from $\mathbb{R}$ to $\mathbb{R}$. But in fact both the intermediaet value theorem and mean value theorem are false for real-valued functions defined on the rationals.
The intermediate value theorem is false here. Consider the function \begin{align*} f(x): \mathbb{Q} &\longrightarrow \mathbb{R}\\ x &\mapsto x^2 - 2. \end{align*} On the interval $[0, 2]$, for example, we see that $f(0) = -2$ and $f(2) = 2$. But there is no $x \in \mathbb{Q} \cap [0, 2]$ such that $f(x) = 0$.
The mean value theorem is false too. Consider the function \begin{align*} f(x): \mathbb{Q} &\longrightarrow \mathbb{R} \\ x &\mapsto \begin{cases} 0 & \text{if } x^2 > 2, \\ 1 & \text{if } x^2 > 2. \end{cases} \end{align*} For this function, $f'(x) = 0$ everywhere, even though the function isn't constant. (But it is locally constant). We try to avoid pathological examples initially.3 3It is a funny thing. Initially, we pretend that everything is as nice as possible to build intuition. Then we see that there are pathological counterexamples and use these to sharpen intuition. Seeing these too early is potentially misleading. Implicit within the order is an idea of what usually hapens and what behavior is exceptional.
It turns out that the topological space the functions are defined on really matter.
Mean value theorem as abstraction
I don't talk about this in a typical calculus class because topological concerns are almost entirely ignored. We work with the practical case of real-valued functions defined over the reals. And the key tools we use to hide the underlying topological details are the intermediate and mean value theorems.
I didn't fully realize this until I wondered whether we could teach calculus without covering these two theorems.
It might be possible. But I decided that it was a bad idea.
Instead, I think it's a good idea to give students a better idea of how these two theorems are two of the most important ideas in calculus. I kept this idea in mind when I wrote An intuitive introduction to calculus for my students in 2013.
And more broadly, I began to investigate just how many things we could reduce, as directly as possible, to the mean value theorem. Miles realized we could investigate the continuity of the mean value abscissa with a low dimnsional implicit function theorem and baby version of Morse's lemma, and the rest is history.
I've collected so many random facts and questions tangentially related to the mean value theorem over the years. But I could always collect more!
MathFest
I attended MathFest for the first time to receive this award in person. MathFest was an extraordinary and interesting experience. I was particularly impressed by how many sessions and talks focused on actionable ways to improve math education in the classroom now.
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