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…)