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.

Suppose that you find some number, a, at which the value of the function is negative. And suppose you find some number, b, at which the value of the function is positive. (This means we can go through the function and replace every appearance of x with the number a, or replace every appearance of x with the number b.) The theorem tells us that somewhere between a and b there must be at least one number, which we can call c, at which the value of the function is 0. That is, the function goes from being negative-valued to being positive-valued by some point where its value is zero. We can prove that, but, it’s hard to imagine how it could possibly not be true.

The sharp-eyed might object: Chiaroscuro was looking for a number which, raised to the fifth power, was 1/6000, and he went about this by trying out different numbers and seeing if once they were raised to the fifth power they were above or below 1/6000. And yes, technically, they are correct, unless I misunderstood what he was doing. That looks a lot like looking at a function, which takes the input number and raises it to the fifth power, and seeing if it’s equal to a set number, since it is.

But if we want to solve the problem “this function equals this number”, using a tool that helps us find “this function equals zero”, we can use a common enough gimmick: create a new function based on the old, one that we can use our old tools for. This new function gives as output the old function minus the set number. In this case, the new function is, take the number, raise it to the fifth power, and then subtract 1/6000. If we find some number c which makes the new function equal to zero, then, we’ve also found a number which makes the old function equal to 1/6000. It’s a good gimmick, because it does one of the things mathematicians like generally to do, which is find a way to use something they already know how to do, in this case find where a function is zero, to do something new, in this case find where it’s some number other than zero.

He also used another of mathematics’s great and reliable tools for solving problems, and that’ll be another essay.