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 \(\scriptsize[a,b]\).
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