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[SOUND] Here's an application of the intermediate value theorem I'm very fond

of. Here's how it goes.

Suppose you've got a function f. f is continuous in the closed interval

between zero and one. And whenever x is between zero and one, f

of x is between zero and one. So that's what you've got to suppose when

you get out of this. Then, there is a point x, x between zero

and one, and f of x = x. It's telling you that there exists an

input. So that if you apply f to that input

value, you get the output which is the same as the thing you plugged in.

You call these things fixed points. You imagine that f is moving around the

points between zero and one, and you're finding a point x which

doesn't move a fixed point. Why is something like this true?

Let's see. Let me use the Intermediate Value Theorem

but I'm not going to apply the Intermediate Value Theorem to f but I'm

going to make up a new function. g(x) = f(x) - x.

What do I know about g? g's continuous.

f is continuous by assumption. The identity function x is continuous and

differences of continuous functions are continuous.

So, g is a continuous function and the closed interval 01.

I also know some of the values of g. What's g of zero?

Well, g(0) is by definition f(0) - 0. f(0) is between zero and one.

So I've got a number between zero and one minus zero,

that number's at least zero. I also know something about g(1).

g(1) is by definition f(1) - 1 f(1) is between zero and one, it could be zero,

could be one. But, a number between zero and one minus

one is less than or equal to zero. What do I know altogether?

I've got a continuous function g, its value at zero is bigger than equal to

zero. Its value at one is less than equal to

zero. By the intermediate value theorem, this

gives me a point x so that g(x) = 0. Why do I care about finding a point x so

that g(x) is equal to zero? Well, if I take a look at what g(x) is

equal to, g(x) is f(x) - x. So, I found a point x so that f(x) - x is

equal to zero. I found a point x so that f(x) = x, I

found the fixed point. So, we've seen why this is true.

Let'd try to apply this to a specific example life f(x) equals cosine x.

2:39

Now, here I've got a graph of cosine and that the graph of y equals cosine x.

Now, what do I know about cosine? I know cosine's continuous and cosine of

a number between zero and one is a number between zero and one.

So, our statement applies to it. There's a fixed point for cosine.

There's some value of x between zero and one so cosine of x equals x.

Let's see if we can find that on a computer.

3:07

Okay. I've got cosine loaded up on to my

computer. Cosine of one is about 0.54.

Let me decrease the input. And I notice that cosine of 0.68 is 0.77.

Here the input is smaller than the output.

Let me increase the input. Now, notice what happened.

Now the input is 0.74 but the output is smaller than 0.74.

Cosine of 0.74 is just 0.73. Now, it decreased the input.

Oh, the opposite thing happened. Now the input is smaller than the output.

Now, I'll increase the input. I'll decrease the input.

3:50

Increase the input. And you can see that I, I can sort of

zoom in or narrow in on the correct value of of x so that cosine of x is equal to

x. Sees in the computer I found that this

point here is at about 0.739. So, without a number between zero and

one, cosine of that number is itself. It's a fixed point for cosine.

These kinds of fixed point theorem statements that assert the existence of

fixed points, they pervade mathematics.

Here's another example. I've got a map of Ohio right here.

Let me apply a continuous function to this map.

[SOUND] I'm going to throw it down to the ground,

stomp on it. And now let's take a look at it.