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.
(more…)Tag: MVT
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An Alternative Proof of the Increasing Function Theorem
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.
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