The Mean Value Theorem (MVT) is a central piece of a first class in calculus. However, the role it plays in the class is often not clearly defined. In many classes and books, students are told that the MVT is the “backbone of calculus,” or some such thing: it is the foundational result on which many important things are built. From the standpoint of analysis, this is a reasonable claim: the MVT and extensions of it are essential to proving everything from the Increasing Function Theorem (IFT) to some of the more general versions of L’hospital’s rule. But how does this come across to a typical calculus student? Such a student has not yet taken (and may never take) a course in analysis and lacks the context to see why the MVT is so manifestly important.
One of my favorite start-of-the-year activities is to have students produce a growth rate graph from the CDC growth chart for children. Here is a link to a student-facing version of the activity, which you are free to copy and adapt for use in your class1. And here is a quick-start guide to using this activity in class.
This activity has students compute slopes using both difference quotients and tangent lines. It asks them to abstract the difference quotient (in spreadsheet notation), so that it can be computed quickly for hundreds of data points. It gets them warmed up to the idea of viewing slope as a function, illustrating a situation where graphing slope as its own function is quite illuminating. It helps them see how a function relates to its derivative and prepares them for interpreting derivatives in the future, both in applied contexts and in curve sketching.
The graphs below2 give a quick sense of how this activity turns out. Students start with the graph on the left (data from the CDC), and end up producing the graph on the right. Features that are barely visible on the left become obvious on the right, for example the incredibly rapid growth of 2-year-olds and the growth spurt around age 12. Truly, the growth rate graph gives a much richer sense of how human growth varies throughout childhood.
The right edge of the growth rate graph also illustrates how a function can continue to increase even if its derivative decreases. Every year, I say to my students after they do this: “the rate of growth graph drops off pretty fast after age 12. Does that mean the person is shrinking?” And they immediately say, “no, it means that they’re growing more slowly.” In this context, students see that this makes sense: the declining growth rate corresponds to a flattening slope, not to a decreasing height. Later in the year, I remind them of this every time someone gets confused about the distinction between decreasing f’ and decreasing f.
The Increasing Function Theorem (IFT), which connects positive derivatives and increasing functions, is absolutely foundational to a first course in calculus: for example, it justifies the use of the first derivative test for extreme values. The theorem feels intuitively obvious (positive slope means increasing, right?) but proving it is harder than one might expect. For this reason, many books “prove” the IFT using the Mean Value Theorem (MVT). But this can hardly be called rigorous if the MVT is left unproved? Even worse is to prove the MVT using the Extreme Value Theorem, but then leave that unproved. If we’re going to tell students to be suspicious of their intuition, we need to give them something better in its place.
One can simply say to students “the IFT is true, but its proof is harder than you might expect. It is not easy to relate the properties of a function across a whole interval to its properties at individual points in that interval.” Then there is no need to muddy the waters using unproved theorems to justify a claim that students probably found intuitive in the first place.
