Day: September 23, 2010

Intermediate Value Theorem

I wish we could do “baby posts” that were formatted different. Little asides that don’t warrant full posts.  Like this one.

Anyway, today, the AP Calculus BC teacher (and all around awesome person) asked me if I had any good ways to introduce the intermediate value theorem.

That’s the most boring theorem ever. Saying that if you have a continuous function f(x) on [a,b], and u is between f(a) and f(b), then there exists a c in [a,b] such that f(c)=u.

In other words, if you have a continuous curve that goes from point (x_1,y_1) to (x_2,y_2), then at some point along the curve’s journey from the first point to the second point, it’s going to pass through every y value between y_1 and y_2.

If you still don’t see it, just draw two points on a coordinate plane and try to connect them with a continuous function. You’ll see it then.

Anyway, it’s boring. So she was right to ask for ideas. I searched and found none.

So I suggested a warm-up for the class — before they know anything about this theorem. I asked her to throw this up on the board:

INDIVIDUAL CHALLENGE: I am so wise. I have drawn a function f(x) on [1,5], with 10 between f(1) and f(5), such that then there does not exist a c on [1,5] such that f(c)=10. Are you as clever?

And then I wrapped up some Jolly Ranchers for her to give to the first student who could do it.

She said it went really well. And it took a few minutes (read that: minutes) before the first student got it. Perfect warm-up.

The reason I really liked this idea, and wanted to share it, is because: (a) kids were motivated by it, (b) kids were forced to grapple with complex mathematical language, (c) kids got to play around (by drawing different graphs — a puzzle-y thing), and (d) kids discovered the Intermediate Value Theorem on their own.

Let’s think about the last point. The first 5 or so graphs students would draw would not satisfy the challenge. And they’d see the problem: that the graphs they were drawing were continuous. So the only way to satisfy the challenge would be to make their function discontinuous. So not only would they learn the IVT, but they’d really remember the restriction: you need a continuous function for the IVT to hold.

I’m sure many of you probably introduce the IVT this way. It’s certainly not new or revolutionary. But I am now excited to when I get to teach the IVT.

PS. I also am really impressed by this consequence (click link to see proof). The consequence of total boringness happens not to be boring at all!:

The theorem implies that on any great circle around the world, the temperature, pressure, elevation, carbon dioxide concentration, or any other similar quantity which varies continuously, there will always exist two antipodal points that share the same value for that variable.