Average Value and Average Rate of Change

One connection I like to make in my class is that, on any interval, the average rate of change of a differentiable function is equal the the average value of its derivative. This follows directly from the Fundamental Theorem, since the average value of \(\scriptsize f’\) on \(\scriptsize[a,b]\) is $$\frac{\int_a^bf'(x){\rm d}x}{b-a}=\frac{f(b)-f(a)}{b-a} .$$From my point of view, this connection is what justifies the use of the word “average” in describing a rate of change. What are we averaging? The derivative, which is to say the rate of change, of f. What are we averaging over? The entire interval \(\scriptsize[a,b]\).

In my experience, students are often confused by this use of the word “average.” And no wonder: we start using the phrase “average rate of change” long before defining the word “average” in the context of a continuous variable. I have had students who think that the average rate of change on \(\scriptsize[a,b]\) should be \(\frac{f'(a)+f'(b)}{2}\), and I don’t blame them: this is a totally reasonable thing to think if your concept of average is “add a list of n numbers and then divide by n.” Students have this confusion because we start using “average” in this different way, before explaining what we mean by it.

It is for this reason that I begin my calculus class by using the phrase “rate of change over an interval” or “rate of change between two points” instead of using the word “average.” This helps students focus on the challenge posed by finding slope at a point (as opposed to slope between two points), rather than getting hung up on a confusing piece of jargon. And I switch to using the phrase “average rate of change” after defining average value and proving (as above) that \(\scriptsize \frac{f(b)-f(a)}{b-a}\) is equal to the average value of f’ on \(\scriptsize [a,b]\)

This connection also fits with my approach of always looking at everything from the perspective of both derivatives and antiderivatives: rather than siloing average value and average rate of change in different parts of the class, I always want my students to think about calculus from both derivative and integral points of view.

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