livingcalculus.com

Tag: average value

  • Visualizing the Mean Value Theorem

    If your students have seen that (for a differentiable function on a closed interval) the average rate of change of f always equals the average value of f’, then you are ready to show them a nice way of visualizing the Mean Value Theorem. I think almost every calculus teacher illustrates the MVT with a picture something like this one:

    Here, we can see the point(s) at which the average rate of change matches the instantaneous rate of change. If we view the same situation from the perspective of f’, then we get this picture:

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  • 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 \(\scriptsize[a,b]\).

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