Average Value of a Continuous Function
One of the application topics covered in our integral calculus course is the use of the integral to define the average value of an integrable function over an interval , i.e., . Here’s a ConcepTest I cooked up to see whether the students knew and understood the definition we had briefly discussed a few days earlier.
The graphic is a screen-shot from Google Finance, edited with MS Paint putting in letters A,B,C,D for the classroom vote. It took me 3 minutes to draft this question and it involves current, real world data, and a function of interest to many people which is not given by a formula in terms of the elementary functions. I didn’t need to engineer some exotic formula for a sufficiently rough modeling function and then waste hours monkeying around with a plotting program to get it to work. Just point, click, copy, and paste. Fortunately for me, the averages on Thursday and Monday were really close, so the question stirred up good discussion.
The most important reason to cover this, in my opinion, is that you can use it to give students an intuitive feeling of what integrating a function over an interval does. After all, if is then . In other words, when you integrate a function over an interval, what you are doing is averaging that function (then multiplying by the length of the interval). This point of view helps you to understand that indefinite integration is a smoothing operation on functions. Indeed, by the Fundamental Theorem of Calculus, forming the indefinite integral of a continuous function yields a differentiable function. Integrating again gives a twice-differentiable function, and so on.