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 a course in analysis and lacks the context to see why the MVT is so manifestly important.
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A common way of illustrating the importance of the MVT is to show how it can be used to prove the IFT (which relates a positive derivative to an increasing function) and related results such as the constant difference theorem for antiderivatives. But these “proofs” have a critical weakness, which is that the MVT itself is not proved. Or, perhaps it is proved using Rolle’s theorem, which is proved using the extreme value theorem (EVT), which is…not proved.
When we do this, we are proving a common-sense theorem most students would never even bother to question (the IFT) using a sophisticated theorem whose proof is beyond the scope of the course (the EVT). If “proof by picture” is good enough for the EVT, why not apply the same standard to the IFT? This approach to the IFT teaches students to mistrust their intuition (“you can’t say the IFT is true just because it seems like it should be”) in a situation where there intuition is actually right. If we want to teach students that intuition can be wrong, we should focus on claims that are actually wrong (“the derivative equals 0 at any extreme value” is a good example). And, while cautionary notes on “proof by picture” are important, it is just as important to nourish whatever kernel of intuition students are starting to develop: although intuition can be wrong, it is often right and it is almost always a good place to start coming up with conjectures.
Using the MVT to prove the IFT not only teaches students that their intuition is suspect, it does so without replacing their intuition with any real rigor. If proving the EVT is beyond the scope of your course, then anything proved from it is also beyond the scope of your course. What mathematician would accept a “proof” based on an unproved lemma? Especially if the unproved lemma seems less intuitive and probable than the theorem we are trying to prove?
This critique raises two questions:
- If, in the name of rigor, we think it is important to prove the IFT, is there another way? Any proof will be challenging, but can we find a proof that doesn’t lean on the EVT and is a bit more accessible? See this post for a constructive proof of the IFT.
- How do we justify the importance of the MVT to students? Using it to prove the IVT just muddies the waters. And classic examples like “a cop can give you a ticket if you go 80 miles in 1 hour” feel contrived. If we want to teach the MVT, is there a better way of communicating its importance? See this post for a connection between the MVT and numerical derivatives.