One day a graduate student submitted some writing to me, in which she was explaining rates of change at the high school level. She made an interesting statement:
The slope of a secant line joining two points $(a, f(a))$ and $(b, f(b))$ on the graph of a differentiable function $f$ is the average of the instantaneous rates of change of the function on the interval $(a, b)$.
In all my many years of teaching calculus and writing about it, this thought had never occurred to me. Sure, we call the slope of the secant line the average rate of change, but I had never connected it to the average of the instantaneous rates of change. I doubted whether it was true, and said so. To her it was obvious.
My subsequent thoughts were as follows: OK, we have a well-known procedure for calculating the average value of a function, involving integration. In this case, the function for which we wish to calculate the average value is the instantaneous rate of change, that is the derivative $f^{\prime}(x)$. Let’s apply the averaging procedure to $f^{\prime}(x)$ on the interval $(a, b)$ and see what happens.
The average value of $f^{\prime}(x)$ on the interval $(a, b)$ is the integral of $f^{\prime}(x)$ from a to b, divided by the length of the interval, which is $[b – a]$:
$\displaystyle{\textrm{average value of} \; f^{\prime}(x) \; \textrm{on} \; [a, b] = \dfrac{1}{b – a} \int_a^b f^{\prime} (x) \, {\rm d}x }$
But there it is! Applying the fundamental theorem of calculus to the integral in the previous line, we get
$\displaystyle{\textrm{average value of} \; f^{\prime}(x) \; \textrm{on} \; [a, b] = \dfrac{f(b) – f(a)}{b – a} }$
But this is exactly the slope of the secant line joining the points $(a, f(a))$ and $(b, f(b))$. So the student’s statement is correct, and the proof is quite direct, a simple consequence of the fundamental theorem of calculus.
I found the whole exchange delightful. This is one of the things I love about teaching. No matter how long you’ve taught a subject, there is always another perspective hiding around the corner, waiting to be found, and it’s particularly nice when the finding is stimulated by a question from a student. I was very happy to learn this new (for me) perspective on the fundamental theorem of calculus, and to learn that the slope of the secant line joining two points on the graph of a differentiable function is the average value of the instantaneous rates of change on the interval between the two endpoints.
As an undergraduate student many years ago, I had the privilege of being a TA for a course taught by the late A. J. Coleman [a beautiful eulogy by Peter Taylor is here] at Queen’s University at Kingston. I vividly remember something he very solemnly told us TAs at the time: “There is an ancient truth: The teacher learns more than the student.”
So true.
(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 25 January 2021.)