The Intermediacy That Was Overused


However I may sulk, Chiaroscuro did show off a use of the Intermediate Value Theorem that I wanted to talk about because normally the Intermediate Value Theorem occupies a little spot around Chapter 2, Section 6 of the Intro Calculus textbook and it gets a little attention just before the class moves on to this theorem about there being some point where the slope of the derivative equals the slope of a secant line which is very testable and leaves the entire class confused.

The theorem is pretty easy to state, and looks obviously true, which is a danger sign. One bit of mathematics folklore is that the only things one should never try to prove are the false and the obvious. But it’s not hard to prove, at least based on my dim memories of the last time I went through the proof. One incarnation of the theorem, one making it look quite obvious, starts off with a function that takes as its input a real number — since we need a label for it we’ll use the traditional variable name x — and returns as output a real number, possibly a different number. And we have to also suppose that the function is continuous, which means just about what you’d expect from the meaning of “continuous” in ordinary human language. It’s a bit tricky to describe exactly, in mathematical terms, and is where students get hopelessly lost either early in Chapter 2 or early in Chapter 3 of the Intro Calculus textbook. We’ll worry about that later if at all. For us it’s enough to imagine it means you can draw a curve representing the function without having to lift your pen from the paper.

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