Students often confuse the several concepts that have the word “average” or “mean” in their title. This may be partly because not just the names, but the formulas associated with each are very similar, but I think the main reason may be that they are keying in on the word “average” rather than the full name.

Here are the three items. We will assume that the function *f* is continuous on the closed interval [*a, *b] and differentiable on the open interval (*a, b*):

1. The

average rate of changeof a function over the interval is simply the slope of the line from one endpoint of the graph to the other.

2. The

mean (or average) value theoremsay that somewhere in the open interval (a, b) there is a numbercsuch that the derivative (slope) atx = cis equal to theaverage rate of changeover the interval.3. The

average valueof a function is literally the average of all they-coordinates on the interval. It is the vertical side of a rectangle whose base extends on thex-axis fromx = atox =band whose area is the same as the area between the graph and thex-axis and the function over the same interval.Notice that when you evaluate the integral, the result looks very much like the ones above. This formula is also called the

mean value theorem for integralsor theintegral form of the mean value theorem. No wonder people get confused.

The three are closely related. Consider a position-velocity-acceleration situation. The *average rate of change of position* (#1 above) is the *average value of the velocity* (#3) and somewhere the velocity must equal this number (#2). Similarly, the *average rate of change of velocity *(#1) is the *average acceleration* (#3) and somewhere in the interval the acceleration (derivative of velocity) must equal this number (#2).

These ideas are tested on the AP calculus exams sometimes in the same question. See for example 2004 AB 1 parts c and d.

So help your students concentrate on the entire name of the concepts, not just the “average” part.

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