1 00:00:00,000 --> 00:00:06,487 [SOUND] Here's an application of the intermediate value theorem I'm very fond 2 00:00:06,487 --> 00:00:08,317 of. Here's how it goes. 3 00:00:08,317 --> 00:00:14,222 Suppose you've got a function f. f is continuous in the closed interval 4 00:00:14,222 --> 00:00:19,378 between zero and one. And whenever x is between zero and one, f 5 00:00:19,378 --> 00:00:25,533 of x is between zero and one. So that's what you've got to suppose when 6 00:00:25,533 --> 00:00:30,440 you get out of this. Then, there is a point x, x between zero 7 00:00:30,440 --> 00:00:34,513 and one, and f of x = x. It's telling you that there exists an 8 00:00:34,513 --> 00:00:37,051 input. So that if you apply f to that input 9 00:00:37,051 --> 00:00:41,243 value, you get the output which is the same as the thing you plugged in. 10 00:00:41,243 --> 00:00:45,670 You call these things fixed points. You imagine that f is moving around the 11 00:00:45,670 --> 00:00:49,389 points between zero and one, and you're finding a point x which 12 00:00:49,389 --> 00:00:53,049 doesn't move a fixed point. Why is something like this true? 13 00:00:53,049 --> 00:00:55,963 Let's see. Let me use the Intermediate Value Theorem 14 00:00:55,963 --> 00:01:00,090 but I'm not going to apply the Intermediate Value Theorem to f but I'm 15 00:01:00,090 --> 00:01:03,610 going to make up a new function. g(x) = f(x) - x. 16 00:01:03,610 --> 00:01:06,409 What do I know about g? g's continuous. 17 00:01:06,409 --> 00:01:11,712 f is continuous by assumption. The identity function x is continuous and 18 00:01:11,712 --> 00:01:15,469 differences of continuous functions are continuous. 19 00:01:15,469 --> 00:01:19,667 So, g is a continuous function and the closed interval 01. 20 00:01:19,667 --> 00:01:23,571 I also know some of the values of g. What's g of zero? 21 00:01:23,571 --> 00:01:29,611 Well, g(0) is by definition f(0) - 0. f(0) is between zero and one. 22 00:01:29,611 --> 00:01:33,147 So I've got a number between zero and one minus zero, 23 00:01:33,147 --> 00:01:38,160 that number's at least zero. I also know something about g(1). 24 00:01:38,160 --> 00:01:44,920 g(1) is by definition f(1) - 1 f(1) is between zero and one, it could be zero, 25 00:01:44,920 --> 00:01:48,526 could be one. But, a number between zero and one minus 26 00:01:48,526 --> 00:01:53,127 one is less than or equal to zero. What do I know altogether? 27 00:01:53,127 --> 00:01:57,884 I've got a continuous function g, its value at zero is bigger than equal to 28 00:01:57,884 --> 00:02:00,701 zero. Its value at one is less than equal to 29 00:02:00,701 --> 00:02:03,455 zero. By the intermediate value theorem, this 30 00:02:03,455 --> 00:02:10,059 gives me a point x so that g(x) = 0. Why do I care about finding a point x so 31 00:02:10,059 --> 00:02:14,871 that g(x) is equal to zero? Well, if I take a look at what g(x) is 32 00:02:14,871 --> 00:02:20,520 equal to, g(x) is f(x) - x. So, I found a point x so that f(x) - x is 33 00:02:20,520 --> 00:02:24,565 equal to zero. I found a point x so that f(x) = x, I 34 00:02:24,565 --> 00:02:32,624 found the fixed point. So, we've seen why this is true. 35 00:02:32,624 --> 00:02:38,380 Let'd try to apply this to a specific example life f(x) equals cosine x. 36 00:02:39,920 --> 00:02:44,755 Now, here I've got a graph of cosine and that the graph of y equals cosine x. 37 00:02:44,755 --> 00:02:49,797 Now, what do I know about cosine? I know cosine's continuous and cosine of 38 00:02:49,797 --> 00:02:54,148 a number between zero and one is a number between zero and one. 39 00:02:54,148 --> 00:02:58,569 So, our statement applies to it. There's a fixed point for cosine. 40 00:02:58,569 --> 00:03:02,920 There's some value of x between zero and one so cosine of x equals x. 41 00:03:02,920 --> 00:03:05,960 Let's see if we can find that on a computer. 42 00:03:07,740 --> 00:03:11,028 Okay. I've got cosine loaded up on to my 43 00:03:11,028 --> 00:03:13,905 computer. Cosine of one is about 0.54. 44 00:03:13,905 --> 00:03:19,083 Let me decrease the input. And I notice that cosine of 0.68 is 0.77. 45 00:03:19,083 --> 00:03:22,782 Here the input is smaller than the output. 46 00:03:22,782 --> 00:03:29,496 Let me increase the input. Now, notice what happened. 47 00:03:29,496 --> 00:03:34,415 Now the input is 0.74 but the output is smaller than 0.74. 48 00:03:34,415 --> 00:03:39,720 Cosine of 0.74 is just 0.73. Now, it decreased the input. 49 00:03:39,720 --> 00:03:44,253 Oh, the opposite thing happened. Now the input is smaller than the output. 50 00:03:44,253 --> 00:03:49,540 Now, I'll increase the input. I'll decrease the input. 51 00:03:50,620 --> 00:03:55,663 Increase the input. And you can see that I, I can sort of 52 00:03:55,663 --> 00:04:03,319 zoom in or narrow in on the correct value of of x so that cosine of x is equal to 53 00:04:03,319 --> 00:04:11,145 x. Sees in the computer I found that this 54 00:04:11,145 --> 00:04:19,844 point here is at about 0.739. So, without a number between zero and 55 00:04:19,844 --> 00:04:24,525 one, cosine of that number is itself. It's a fixed point for cosine. 56 00:04:24,525 --> 00:04:29,844 These kinds of fixed point theorem statements that assert the existence of 57 00:04:29,844 --> 00:04:32,540 fixed points, they pervade mathematics. 58 00:04:32,540 --> 00:04:36,583 Here's another example. I've got a map of Ohio right here. 59 00:04:36,583 --> 00:04:39,916 Let me apply a continuous function to this map. 60 00:04:39,916 --> 00:04:43,179 [SOUND] I'm going to throw it down to the ground, 61 00:04:43,179 --> 00:04:46,300 stomp on it. And now let's take a look at it. 62 00:04:48,120 --> 00:04:58,442 [SOUND] The question is, is there a point on the map which is exactly above its 63 00:04:58,442 --> 00:05:05,498 corresponding point in the ground? And indeed there is. 64 00:05:05,498 --> 00:05:11,378 I'll use this pin to stab through that point. 65 00:05:11,378 --> 00:05:12,424 [SOUND]