WEBVTT 00:00:24.206 --> 00:00:25.800 PROFESSOR: Hi. 00:00:25.800 --> 00:00:31.460 Well, this is exponential day, the day for the function that 00:00:31.460 --> 00:00:33.850 only calculus could create, y is e to the x. 00:00:36.450 --> 00:00:41.700 And it couldn't have come from algebra because, however we 00:00:41.700 --> 00:00:47.620 approach e to the x, there's some limiting step. 00:00:47.620 --> 00:00:49.190 Something goes to 0. 00:00:49.190 --> 00:00:51.400 Something goes to infinity. 00:00:51.400 --> 00:00:55.170 I've got different ways to reach e to the x, but all of 00:00:55.170 --> 00:00:58.810 them involve that limiting process, which we haven't 00:00:58.810 --> 00:01:01.160 discussed in full. 00:01:01.160 --> 00:01:06.100 Let me come back at a later time to the whole theory, 00:01:06.100 --> 00:01:11.630 discussion of limits and just go forward here with this 00:01:11.630 --> 00:01:14.360 highly important function. 00:01:14.360 --> 00:01:18.930 And I'd like to start with its most important 00:01:18.930 --> 00:01:22.320 property, which is-- 00:01:22.320 --> 00:01:27.350 so it has this remarkable property that its slope is 00:01:27.350 --> 00:01:28.600 equal to itself. 00:01:33.500 --> 00:01:37.250 That's what is special about e to the x. 00:01:41.110 --> 00:01:43.550 The slope is equal to the function. 00:01:43.550 --> 00:01:51.486 Now, I have to admit that if we had a function like that, y 00:01:51.486 --> 00:01:55.950 equals e to the x, then 2e to the x, x would work just as 00:01:55.950 --> 00:01:58.450 well, or 10e to the x. 00:01:58.450 --> 00:01:59.720 Those would-- 00:01:59.720 --> 00:02:04.600 the factor 2 or the factor 10 would be in y and it would 00:02:04.600 --> 00:02:09.600 also be in the slope and it would cancel and-- 00:02:09.600 --> 00:02:14.310 this is a differential equation, our first 00:02:14.310 --> 00:02:16.360 differential equation. 00:02:16.360 --> 00:02:21.060 A differential equation is an equation that involves, as 00:02:21.060 --> 00:02:23.910 this one does, the function and the slope. 00:02:23.910 --> 00:02:24.720 It connects them. 00:02:24.720 --> 00:02:29.040 And that's the fantastic description of nature, is by 00:02:29.040 --> 00:02:30.580 differential equations. 00:02:30.580 --> 00:02:34.180 So it's great to see this one early and it's the most 00:02:34.180 --> 00:02:36.170 important one. 00:02:36.170 --> 00:02:38.210 When you get this one, you've got a whole 00:02:38.210 --> 00:02:40.510 lot of others solved. 00:02:40.510 --> 00:02:41.210 OK. 00:02:41.210 --> 00:02:46.080 But I needed to give it a starting point so that the 00:02:46.080 --> 00:02:50.220 solution would be e to the x and not 10e to the x. 00:02:50.220 --> 00:02:51.960 So where should I start it? 00:02:51.960 --> 00:02:58.700 Well if I want it to be an e to the x, then when x is 0, e 00:02:58.700 --> 00:03:05.490 to the 0 power, some number to the 0 power, is always 1. 00:03:05.490 --> 00:03:13.980 So let me start this y equals 1 at x equals 0. 00:03:13.980 --> 00:03:17.390 Differential equations, you have to tell where they begin. 00:03:17.390 --> 00:03:19.800 So that's our starting point. 00:03:19.800 --> 00:03:22.070 And do you see what this means? 00:03:22.070 --> 00:03:27.720 This means that it starts at 1. 00:03:27.720 --> 00:03:31.760 And what's it's slope at the starting point? 00:03:31.760 --> 00:03:33.200 The slope is also 1. 00:03:33.200 --> 00:03:35.720 So it's climbing. 00:03:35.720 --> 00:03:38.680 As it climbs-- 00:03:38.680 --> 00:03:42.330 so y gets larger because it's got a positive slope. 00:03:42.330 --> 00:03:46.940 As y gets larger, the slope gets larger. 00:03:46.940 --> 00:03:49.080 So it climbs faster. 00:03:49.080 --> 00:03:54.560 And then it's gone higher, y is bigger, the slope is equal, 00:03:54.560 --> 00:03:57.350 so the slope is also bigger, so it climbs even faster. 00:03:57.350 --> 00:03:58.600 It just takes off. 00:04:02.150 --> 00:04:05.590 It climbs much faster than x to the 100th power. 00:04:05.590 --> 00:04:09.150 You might think x to the 100th, that's 00:04:09.150 --> 00:04:10.760 climbing pretty well. 00:04:10.760 --> 00:04:14.115 2 to the 100th, 10 to the 100th, but now way. 00:04:14.115 --> 00:04:20.203 It doesn't come close to keeping up with y 00:04:20.203 --> 00:04:21.453 equals e to the x. 00:04:23.550 --> 00:04:25.390 OK. 00:04:25.390 --> 00:04:27.300 I've got several things to do. 00:04:27.300 --> 00:04:31.530 And one more thing I have to do, this is a key property, 00:04:31.530 --> 00:04:37.310 but there's another key property that is true for any 00:04:37.310 --> 00:04:40.740 2 to the x, 3 to the x, e to the x. 00:04:40.740 --> 00:04:44.370 And that key property is also to show-- 00:04:44.370 --> 00:04:52.713 I have to show this, that my function, e to the x, times e 00:04:52.713 --> 00:04:58.892 to the possibly a different x is equal to-- 00:04:58.892 --> 00:05:02.020 do you know what we want here? 00:05:02.020 --> 00:05:03.830 This has got to come out of the 00:05:03.830 --> 00:05:06.930 construction, out of this property. 00:05:06.930 --> 00:05:08.260 It's got to come-- 00:05:08.260 --> 00:05:12.950 but we want this to deserve, to be called some number to 00:05:12.950 --> 00:05:14.360 the x power. 00:05:14.360 --> 00:05:19.450 If we take some number x times multiplied by that same number 00:05:19.450 --> 00:05:22.190 capital x times, then we've got that 00:05:22.190 --> 00:05:24.100 number how many times? 00:05:24.100 --> 00:05:25.860 x plus capital x. 00:05:28.920 --> 00:05:32.220 So that's a key property to be proved. 00:05:32.220 --> 00:05:35.150 So what will I do? 00:05:35.150 --> 00:05:40.050 Let me summarize in advance, outline in advance. 00:05:40.050 --> 00:05:47.660 I'm going to construct this function from its property. 00:05:47.660 --> 00:05:54.030 Then I'm going check that it's got this property, that 00:05:54.030 --> 00:05:56.690 important equality there. 00:05:56.690 --> 00:05:58.160 Then, of course, I'll graph it. 00:05:58.160 --> 00:06:02.830 I'll figure out what e is, and I'll say something about cases 00:06:02.830 --> 00:06:04.770 where this comes up. 00:06:04.770 --> 00:06:07.380 I could even say something right away about, where does 00:06:07.380 --> 00:06:14.620 this happen that growth is equal or proportional to the 00:06:14.620 --> 00:06:15.870 function itself? 00:06:18.540 --> 00:06:24.010 It happens with interest, with money in a bank. 00:06:24.010 --> 00:06:29.260 When you get interest, the interest is proportional, of 00:06:29.260 --> 00:06:32.050 course, the amount there. 00:06:32.050 --> 00:06:35.680 And if they add that interest in, if you don't take it out 00:06:35.680 --> 00:06:39.440 and spend it but you compound it, put it in there, then you 00:06:39.440 --> 00:06:41.170 have more money. 00:06:41.170 --> 00:06:44.990 When they compute the interest again, it's computed on that 00:06:44.990 --> 00:06:50.470 larger amount and is more interest than the first time. 00:06:50.470 --> 00:06:52.820 And so it goes. 00:06:52.820 --> 00:07:00.400 So money in the bank is a case of exponential growth. 00:07:00.400 --> 00:07:05.820 A hedge fund grows faster than our bank account does, but all 00:07:05.820 --> 00:07:07.410 following e to the x. 00:07:07.410 --> 00:07:11.400 If you just hang on long enough, you're way up there. 00:07:11.400 --> 00:07:12.770 OK. 00:07:12.770 --> 00:07:16.120 So here's my job. 00:07:16.120 --> 00:07:19.120 Follow this rule and start at y equals 1. 00:07:19.120 --> 00:07:22.990 So can I just do it this way? 00:07:22.990 --> 00:07:26.610 Here is my function, y of x, that I want to construct. 00:07:26.610 --> 00:07:28.410 I want to build that function. 00:07:28.410 --> 00:07:31.120 And I know that it starts at 1. 00:07:31.120 --> 00:07:34.760 But it's going to have some more things. 00:07:34.760 --> 00:07:38.200 Now, this has to equal dy dx. 00:07:41.430 --> 00:07:43.720 These have to be the same. 00:07:43.720 --> 00:07:45.870 That's my rule. 00:07:45.870 --> 00:07:52.780 So dy dx is going to start with a 1. 00:07:52.780 --> 00:08:00.370 But now I can't stop because if the derivative is a 1, I 00:08:00.370 --> 00:08:01.150 better put-- 00:08:01.150 --> 00:08:06.260 I have to put an x up here so that its 00:08:06.260 --> 00:08:08.280 derivative will be 1, right? 00:08:08.280 --> 00:08:11.680 Its slope will be 1. 00:08:11.680 --> 00:08:16.340 That's that steadily climbing x whose slope is 1. 00:08:16.340 --> 00:08:21.650 But now, these are supposed to be equal again. 00:08:21.650 --> 00:08:26.100 So I have to put this x also here. 00:08:26.100 --> 00:08:31.380 But now, I've got to add something more on the top so 00:08:31.380 --> 00:08:34.669 that the slope will be 1 plus x. 00:08:34.669 --> 00:08:36.220 The slope of the x was 1. 00:08:36.220 --> 00:08:41.460 What do I need here to give the slope to be x? 00:08:41.460 --> 00:08:47.660 Remember, x squared had the slope 2x, so I need half of x 00:08:47.660 --> 00:08:49.710 squared so that I'll have 1x. 00:08:49.710 --> 00:08:54.310 So I need a half of x squared. 00:08:54.310 --> 00:08:55.730 Good. 00:08:55.730 --> 00:08:57.880 The slope of that is this. 00:08:57.880 --> 00:09:01.740 But I'm also trying to get the 2 to be equal. 00:09:01.740 --> 00:09:02.960 So I better-- 00:09:02.960 --> 00:09:03.950 I have no choice. 00:09:03.950 --> 00:09:07.070 I have to put in the 1/2 x squared there. 00:09:07.070 --> 00:09:09.980 You see, I'm never going to catch up. 00:09:09.980 --> 00:09:13.840 Or only if I go forever. 00:09:13.840 --> 00:09:14.560 That's the point. 00:09:14.560 --> 00:09:16.750 I'll have to go forever. 00:09:16.750 --> 00:09:18.640 And what will the next one be? 00:09:18.640 --> 00:09:19.640 Oh yeah. 00:09:19.640 --> 00:09:22.620 If you see the next one, then we can see the pattern. 00:09:27.030 --> 00:09:27.930 Now what am I doing? 00:09:27.930 --> 00:09:33.870 This one has to have this slope. 00:09:33.870 --> 00:09:37.180 I'm fixing the top line now. 00:09:37.180 --> 00:09:40.660 If I'm aiming for a slope of x squared, then I need some 00:09:40.660 --> 00:09:43.010 number of x cubes. 00:09:43.010 --> 00:09:45.000 So how many x cubes do I need? 00:09:47.700 --> 00:09:51.420 Well, I need to know, what's the slope of x cube? 00:09:51.420 --> 00:09:56.360 The rule for powers of x, x to the n, is n 00:09:56.360 --> 00:09:58.330 times one smaller power. 00:09:58.330 --> 00:10:05.040 The slope of x cube is 3 times x squared. 00:10:05.040 --> 00:10:11.540 So I had better divide by that 3 so that the 3 cancels the 3. 00:10:11.540 --> 00:10:16.120 Now the slope of that, the 3x squared, the threes would 00:10:16.120 --> 00:10:17.740 cancel and I would get x squared. 00:10:17.740 --> 00:10:20.080 But I'm looking for 1/2 x squared. 00:10:20.080 --> 00:10:24.300 I need also a 2. 00:10:24.300 --> 00:10:28.410 Do you see that it's 1/6 of x cube that's 00:10:28.410 --> 00:10:29.740 going to do the job. 00:10:29.740 --> 00:10:33.520 1/6 of x cubed because the slope-- 00:10:33.520 --> 00:10:37.600 the 3 cancels the 3 and I wanted to end up with a 2. 00:10:37.600 --> 00:10:40.700 And now, do you know what's coming? 00:10:40.700 --> 00:10:42.040 These are supposed to be equal. 00:10:42.040 --> 00:10:45.140 I have to have this 1/6 x cubed down here too. 00:10:48.290 --> 00:10:49.775 And I never get to stop. 00:10:54.170 --> 00:10:58.570 We have to see, OK, what is a typical-- 00:10:58.570 --> 00:11:03.260 after I've done this, say, n times, I'd like to have some 00:11:03.260 --> 00:11:10.660 idea of what is it when I get up to x to some nth power, 00:11:10.660 --> 00:11:14.860 then it's multiplied by some fraction and I'm looking to 00:11:14.860 --> 00:11:17.840 see, what is that fraction? 00:11:17.840 --> 00:11:19.950 What is that fraction? 00:11:19.950 --> 00:11:24.050 And then, of course, they'll all show up down there again. 00:11:24.050 --> 00:11:30.780 Well, if you see this pattern, this was 3 times 2-- you could 00:11:30.780 --> 00:11:33.380 say 3 times 2 times 1. 00:11:33.380 --> 00:11:35.250 This one was 2 times 1. 00:11:35.250 --> 00:11:36.940 This one was just 1. 00:11:36.940 --> 00:11:40.200 It's n factorial. 00:11:40.200 --> 00:11:42.420 n factorial is what I need. 00:11:42.420 --> 00:11:46.620 I need n times n minus 1. 00:11:46.620 --> 00:11:50.100 I need all these numbers all the way and I'll throw in the 00:11:50.100 --> 00:11:51.420 1 at the end. 00:11:51.420 --> 00:11:58.580 And I have to put the mathematicians take it away 00:11:58.580 --> 00:12:01.870 symbol, the little three dots that mean 00:12:01.870 --> 00:12:04.230 don't stop, keep going. 00:12:04.230 --> 00:12:07.090 But do you see that this will be OK? 00:12:07.090 --> 00:12:12.670 This is called n factorial, x to the nth over n factorial, 00:12:12.670 --> 00:12:15.830 because when I take the slope of x to the nth, 00:12:15.830 --> 00:12:17.220 an n will come down. 00:12:17.220 --> 00:12:19.810 Cancel that n. 00:12:19.810 --> 00:12:23.180 x, I'll have one lower power. 00:12:23.180 --> 00:12:26.720 You see, when I take the slope of this, I'll have the n will 00:12:26.720 --> 00:12:27.960 cancel the n. 00:12:27.960 --> 00:12:33.900 So I'll still have these other guys down below. 00:12:33.900 --> 00:12:36.990 And I'll have x to the n minus 1. 00:12:36.990 --> 00:12:39.470 And that will be x to the n minus 1 00:12:39.470 --> 00:12:40.960 over n minus 1 factorial. 00:12:40.960 --> 00:12:42.750 That will be the previous one. 00:12:42.750 --> 00:12:50.860 But now I have to add in the x to the nth over n factorial 00:12:50.860 --> 00:12:54.850 because y and dy to the x have to be the same, so I have to 00:12:54.850 --> 00:12:55.610 keep going. 00:12:55.610 --> 00:12:56.640 OK. 00:12:56.640 --> 00:13:03.010 So you might say, well, you're going to blow up. 00:13:03.010 --> 00:13:06.590 Not personally, the series. 00:13:06.590 --> 00:13:09.830 But what saves you? 00:13:09.830 --> 00:13:15.690 What saves you is the fact that these n factorials, those 00:13:15.690 --> 00:13:19.840 fractions, that n factorial gets to be really large really 00:13:19.840 --> 00:13:25.650 fast, faster than this x to nth could grow. 00:13:25.650 --> 00:13:31.370 So altogether, these terms, x to the nth over n factorial, 00:13:31.370 --> 00:13:36.020 they get extremely, extremely small. 00:13:36.020 --> 00:13:42.790 And then this series of things, it comes to a limit. 00:13:42.790 --> 00:13:46.270 It doesn't keep going, getting bigger, and bigger, and bigger 00:13:46.270 --> 00:13:48.620 as I had more terms, because what I'm adding is 00:13:48.620 --> 00:13:50.935 so small, so small. 00:13:53.810 --> 00:13:59.200 And that's the point where we have to discuss limits later. 00:13:59.200 --> 00:13:59.980 OK. 00:13:59.980 --> 00:14:04.600 So that's my construction. 00:14:04.600 --> 00:14:05.850 Construction complete. 00:14:09.050 --> 00:14:11.380 The exponential function e to the x-- 00:14:11.380 --> 00:14:13.390 this is e to the x-- 00:14:13.390 --> 00:14:19.470 is being defined by 1 plus x plus 1/2 x squared plus 6 x 00:14:19.470 --> 00:14:21.350 cubed, and so on. 00:14:21.350 --> 00:14:22.600 OK. 00:14:24.430 --> 00:14:26.480 I've got a function. 00:14:26.480 --> 00:14:30.510 Now, its property. 00:14:30.510 --> 00:14:34.950 And the key property is this one. 00:14:34.950 --> 00:14:37.780 Can I move to the next board? 00:14:37.780 --> 00:14:40.580 So the next step is, check-- 00:14:40.580 --> 00:14:43.370 well, I've asked you. 00:14:43.370 --> 00:14:44.620 I've got e to the x. 00:14:46.562 --> 00:14:48.715 And let me write again what it is. 00:14:48.715 --> 00:14:57.810 1 plus x plus 1/2 x squared plus 1/6 x cubed and so on. 00:14:57.810 --> 00:15:01.880 And then I've got e to the any other power, or even the same 00:15:01.880 --> 00:15:03.990 power, 1 plus-- 00:15:03.990 --> 00:15:08.970 I'll just use capital x four this power. 00:15:08.970 --> 00:15:15.240 1/6 of capital x cubed plus so on. 00:15:15.240 --> 00:15:18.620 And I want to multiply those and see what I get. 00:15:21.200 --> 00:15:21.580 OK. 00:15:21.580 --> 00:15:23.840 I apologize. 00:15:23.840 --> 00:15:29.360 Here I ask you to believe in this infinite series, and 00:15:29.360 --> 00:15:34.560 yeah, a little dodgy, but it works. 00:15:34.560 --> 00:15:38.730 And now I ask you to multiply two of the things. 00:15:38.730 --> 00:15:41.380 You might say, OK, you're asking a lot here. 00:15:41.380 --> 00:15:45.030 But just hang on. 00:15:45.030 --> 00:15:47.430 Let's multiply these. 00:15:47.430 --> 00:15:52.850 e to the x times e to the capital x, because that's what 00:15:52.850 --> 00:15:55.560 I'm interested in knowing. 00:15:55.560 --> 00:15:56.810 OK. 00:15:59.760 --> 00:16:01.470 Just do all the multiplications. 00:16:05.590 --> 00:16:07.750 And we'll see what we get. 00:16:07.750 --> 00:16:09.350 OK, so 1 times 1 is 1. 00:16:09.350 --> 00:16:10.580 No problem. 00:16:10.580 --> 00:16:13.000 1 times x is the x. 00:16:13.000 --> 00:16:16.210 1 times this x is the big x. 00:16:16.210 --> 00:16:17.490 Now can I keep going? 00:16:17.490 --> 00:16:22.640 All right, well, 1 times 1/2 x squared is-- 00:16:22.640 --> 00:16:27.620 and now I have x times a big x. 00:16:27.620 --> 00:16:30.870 And now I have a 1 times 1/2 big x squared. 00:16:34.050 --> 00:16:36.570 And more, of course. 00:16:36.570 --> 00:16:42.370 Notice the way I'm doing is like I'm keeping all the 00:16:42.370 --> 00:16:45.600 things that have two x's together. 00:16:45.600 --> 00:16:48.020 And then I would keep all the things that have three x's 00:16:48.020 --> 00:16:50.160 together, and so on. 00:16:50.160 --> 00:16:54.140 Now what is it that I'm hoping? 00:16:54.140 --> 00:17:04.890 I'm hoping that this is the same as the series for x plus 00:17:04.890 --> 00:17:08.240 capital x, OK? 00:17:08.240 --> 00:17:09.750 What's that? 00:17:09.750 --> 00:17:12.099 That's my exponential series. 00:17:12.099 --> 00:17:16.030 And every time, I have to put in x plus capital x. 00:17:16.030 --> 00:17:19.180 In other words, of course, it starts with 1. 00:17:19.180 --> 00:17:22.210 Then it has the x plus capital x. 00:17:22.210 --> 00:17:27.380 And then it has the 1/2 of x plus capital x squared. 00:17:27.380 --> 00:17:28.630 And it keeps going. 00:17:32.030 --> 00:17:35.170 And I just wanted you to say, yes. 00:17:35.170 --> 00:17:40.920 I guess I hope you say yes when I ask, is this big 00:17:40.920 --> 00:17:44.080 multiplication the same as this one? 00:17:44.080 --> 00:17:45.760 Well, I think it is. 00:17:45.760 --> 00:17:48.530 Let's just start to check, anyway. 00:17:48.530 --> 00:17:50.020 The ones are good. 00:17:50.020 --> 00:17:50.890 The x and the x-- 00:17:50.890 --> 00:17:56.680 I'm really just putting parentheses around all the-- 00:17:56.680 --> 00:17:59.440 now I'm going to put parentheses around all the 00:17:59.440 --> 00:18:05.360 second degree terms and say, is that the same as that? 00:18:05.360 --> 00:18:06.330 Yeah. 00:18:06.330 --> 00:18:09.080 This is the critical point here. 00:18:09.080 --> 00:18:13.690 Do we, at least, start out correctly? 00:18:13.690 --> 00:18:16.730 So we have to remember, how do you do-- but, of course, you 00:18:16.730 --> 00:18:22.000 do remember how to multiply x plus capital x by itself. 00:18:22.000 --> 00:18:24.790 You just do the multiplications. 00:18:24.790 --> 00:18:29.990 x, when I multiply that by itself, I get x squared. 00:18:29.990 --> 00:18:32.060 With 1/2, I get that. 00:18:32.060 --> 00:18:33.900 And then, you remember? 00:18:33.900 --> 00:18:37.360 How many x times x's do I get? 00:18:37.360 --> 00:18:41.380 Little x times big x, there'd be two of those. 00:18:41.380 --> 00:18:44.700 But then the 1/2 factor leaves me with 1, and 00:18:44.700 --> 00:18:47.030 that's what I want. 00:18:47.030 --> 00:18:51.940 And then, finally, this guy by himself squared is the 1/2 00:18:51.940 --> 00:18:54.520 capital x squared that I also want. 00:18:54.520 --> 00:18:55.770 So far, so good. 00:18:59.430 --> 00:19:03.530 Do you want to see the cubed terms? 00:19:07.150 --> 00:19:12.550 Well, I'd rather you did it, but I should at least show 00:19:12.550 --> 00:19:16.260 that I'm willing to try. 00:19:16.260 --> 00:19:19.230 So what do I mean by the cubed terms? 00:19:19.230 --> 00:19:24.800 I mean that here, I want to get-- the next one should be 00:19:24.800 --> 00:19:29.360 1/6 of x plus x cubed. 00:19:29.360 --> 00:19:35.650 And from the multiplication, I get some separate pieces. 00:19:35.650 --> 00:19:38.750 I get 1 times-- 00:19:38.750 --> 00:19:43.690 when I do that multiplication, I get 1/6 x cubed. 00:19:43.690 --> 00:19:49.750 And then I maybe get some 1/2 x squared times x. 00:19:49.750 --> 00:19:56.040 You see why I would rather you did this. 00:19:56.040 --> 00:19:58.210 But I'll finish this little line. 00:19:58.210 --> 00:20:04.500 There's also an x times 1/2 x squared. 00:20:04.500 --> 00:20:08.780 So that's 1/2 of x times the big x squared. 00:20:08.780 --> 00:20:12.775 And then there is the 1 times the 1/6 x cubed. 00:20:19.270 --> 00:20:23.560 So those are the four pieces that come, third degree, when 00:20:23.560 --> 00:20:25.360 I do the big multiplication. 00:20:25.360 --> 00:20:27.570 And they have to match the third degree 00:20:27.570 --> 00:20:30.820 term in the last line. 00:20:30.820 --> 00:20:32.270 And they do match. 00:20:35.200 --> 00:20:40.530 Do you remember the right words to say now? 00:20:40.530 --> 00:20:42.970 Binomial theorem. 00:20:42.970 --> 00:20:48.680 The binomial theorem tells you how to take the nth power all 00:20:48.680 --> 00:20:54.180 a sum like x plus capital x to the nth power. 00:20:54.180 --> 00:20:58.200 It tells you all the many pieces you get. 00:20:58.200 --> 00:21:02.960 And those many pieces are exactly the pieces that we get 00:21:02.960 --> 00:21:10.490 directly by multiplying that line by that line. 00:21:10.490 --> 00:21:15.920 So the binomial theorem, at long last, pays off and 00:21:15.920 --> 00:21:19.780 confirms our great property here. 00:21:19.780 --> 00:21:22.020 So this is a big deal. 00:21:25.800 --> 00:21:27.050 OK. 00:21:30.560 --> 00:21:33.880 So let me now come back here, having checked that. 00:21:36.760 --> 00:21:41.950 I wanted to say something about this series, 1 plus x 00:21:41.950 --> 00:21:45.430 plus 1/2 x squared, where the typical term is x to the nth 00:21:45.430 --> 00:21:46.900 over n factorial. 00:21:46.900 --> 00:21:53.740 This is the, I would say, the second most important infinite 00:21:53.740 --> 00:21:58.510 series in mathematics, the exponential series. 00:21:58.510 --> 00:22:06.010 And it's the way I wanted to construct e to the x by 00:22:06.010 --> 00:22:11.460 matching term by term and seeing that these n 00:22:11.460 --> 00:22:13.260 factorials show up. 00:22:13.260 --> 00:22:17.900 You might want to know, what's the most important series? 00:22:17.900 --> 00:22:20.220 Reasonable question. 00:22:20.220 --> 00:22:25.380 For me, the most important series would be the one 00:22:25.380 --> 00:22:30.060 looking like this, except it doesn't have the fractions. 00:22:30.060 --> 00:22:33.230 For me, the most important series would be the one-- 00:22:33.230 --> 00:22:35.410 I'll slip it up here-- 00:22:35.410 --> 00:22:43.500 1 plus x plus x squared, without the 1/2, plus x cubed, 00:22:43.500 --> 00:22:49.570 without the 1/6, plus so on, plus x to the n without this n 00:22:49.570 --> 00:22:52.445 factorial that's making it so small. 00:22:55.310 --> 00:23:00.200 Can you see this 1 plus x plus x squared plus x cubed 00:23:00.200 --> 00:23:02.640 plus x to the n? 00:23:02.640 --> 00:23:06.560 That, I think it's called the geometric series. 00:23:06.560 --> 00:23:08.620 Powers of x. 00:23:08.620 --> 00:23:14.260 Now, it's simpler because it doesn't have these fractions. 00:23:14.260 --> 00:23:19.320 But it's riskier because those fractions were making the 00:23:19.320 --> 00:23:22.420 exponential series succeed. 00:23:22.420 --> 00:23:26.160 Whereas here, with the geometric series, well, look 00:23:26.160 --> 00:23:29.330 what happens when x is 1. 00:23:29.330 --> 00:23:35.550 When x is 1, we have 1 plus 1 plus 1 plus 1 plus 1 forever. 00:23:35.550 --> 00:23:36.270 All ones. 00:23:36.270 --> 00:23:38.710 It blows up. 00:23:38.710 --> 00:23:40.960 And when x is bigger than 1, that series 00:23:40.960 --> 00:23:44.130 blows up even faster. 00:23:44.130 --> 00:23:47.350 So in this series, the geometric series, this most 00:23:47.350 --> 00:23:54.260 important one, does succeed but only when x is below 1. 00:23:54.260 --> 00:23:58.050 x equal 1 is the cutoff and it fails after that. 00:23:58.050 --> 00:24:03.760 There is no cutoff for the exponential series because of 00:24:03.760 --> 00:24:07.940 dividing by these bigger and bigger numbers. 00:24:07.940 --> 00:24:09.910 This works for all x. 00:24:09.910 --> 00:24:12.290 OK, so those are the two series. 00:24:12.290 --> 00:24:13.540 OK. 00:24:15.410 --> 00:24:21.470 So let me ask you, what happens if I put x equal 1 in 00:24:21.470 --> 00:24:25.550 the exponential series? 00:24:25.550 --> 00:24:31.630 That gives me e to the first power, which is e. 00:24:31.630 --> 00:24:35.770 So finally, you may say, it's rather late in the day. 00:24:35.770 --> 00:24:41.100 i'm going to figure out what e is from this series. 00:24:41.100 --> 00:24:47.770 Put in set x equal 1 and you learn that e to the first 00:24:47.770 --> 00:24:51.770 power, which is e, is-- 00:24:51.770 --> 00:24:53.060 can I just put it in? 00:24:53.060 --> 00:25:02.180 1 plus x is 1 plus 1/2 of 1 squared plus 1/6 of 1 cubed. 00:25:02.180 --> 00:25:06.470 What's the next term in this? 00:25:06.470 --> 00:25:09.160 So these are numbers now, and I'm getting a number. 00:25:09.160 --> 00:25:14.290 I'm getting this incredible number e, named after Euler. 00:25:14.290 --> 00:25:16.360 Euler was a fantastic mathematician. 00:25:16.360 --> 00:25:21.110 I think he wrote more important papers then any 00:25:21.110 --> 00:25:22.360 mathematician in history. 00:25:25.450 --> 00:25:31.620 So he was allowed to name this number after himself, e. 00:25:31.620 --> 00:25:34.890 E-U-L-E-R, his name is spelled. 00:25:34.890 --> 00:25:36.180 OK, what's the next term? 00:25:38.930 --> 00:25:40.910 This is 3 factorial, right? 00:25:40.910 --> 00:25:41.990 3 times 2 times 1. 00:25:41.990 --> 00:25:43.920 The next term will be 4 factorial. 00:25:43.920 --> 00:25:45.570 I'll multiply that by 4. 00:25:45.570 --> 00:25:47.140 It'll be 1/24. 00:25:47.140 --> 00:25:48.750 And then times 5. 00:25:48.750 --> 00:25:51.090 1/120, and so on. 00:25:51.090 --> 00:25:54.310 They're getting small. 00:25:54.310 --> 00:25:58.090 What can I tell you about this number? 00:25:58.090 --> 00:25:59.355 It will be a definite number. 00:26:02.300 --> 00:26:04.230 And is more than-- 00:26:04.230 --> 00:26:06.840 well, it's certainly more than 2 1/2, because I 00:26:06.840 --> 00:26:08.250 start with 2 1/2 here. 00:26:08.250 --> 00:26:09.720 And then I add these. 00:26:09.720 --> 00:26:12.510 Well, I could even throw in 1/6. 00:26:12.510 --> 00:26:16.350 That's more than 2 2/3, would that be? 00:26:16.350 --> 00:26:19.090 If I quit here, I'd have 2 2/3. 00:26:19.090 --> 00:26:21.460 And then I get a little more. 00:26:21.460 --> 00:26:22.740 It's easy to show. 00:26:22.740 --> 00:26:25.790 No way you would reach as far as 3. 00:26:25.790 --> 00:26:30.450 These later terms are dropping too fast. And actually, the 00:26:30.450 --> 00:26:32.590 number turns out to be-- 00:26:32.590 --> 00:26:34.850 so it's 2 point something. 00:26:34.850 --> 00:26:36.360 2 point-- 00:26:36.360 --> 00:26:39.820 let's see, a little more than 2 2/3, so it's around 2.7. 00:26:43.690 --> 00:26:47.290 But it's it's not exactly 2.7. 00:26:47.290 --> 00:26:50.530 In fact, it's not exactly any fraction 00:26:50.530 --> 00:26:53.690 or any finite decimal. 00:26:53.690 --> 00:26:54.980 It goes on and on. 00:26:54.980 --> 00:26:59.800 1, 8, 2, 8, something. 00:26:59.800 --> 00:27:04.280 I think there are more eights than you'd expect right here 00:27:04.280 --> 00:27:10.370 at the beginning, but then, in the long run, not. 00:27:10.370 --> 00:27:13.010 So that's the number, e. 00:27:13.010 --> 00:27:14.025 OK. 00:27:14.025 --> 00:27:16.330 Oh, so now we know e. 00:27:16.330 --> 00:27:17.480 We know e to the x. 00:27:17.480 --> 00:27:18.340 We know e. 00:27:18.340 --> 00:27:19.810 We know this thing. 00:27:19.810 --> 00:27:21.750 I should draw a graph, right? 00:27:21.750 --> 00:27:23.080 That's the other thing you do with a 00:27:23.080 --> 00:27:25.860 function is draw a graph. 00:27:25.860 --> 00:27:26.610 OK. 00:27:26.610 --> 00:27:29.740 So here's a graph. 00:27:29.740 --> 00:27:31.410 This is x. 00:27:31.410 --> 00:27:36.040 Let me put in x equals 0 here and x equal 1 here. 00:27:36.040 --> 00:27:39.460 And this is going to be a graph of e to the x. 00:27:39.460 --> 00:27:44.820 And at x equals 0, what is it? 00:27:44.820 --> 00:27:46.060 We started with that. 00:27:46.060 --> 00:27:46.950 It should be-- 00:27:46.950 --> 00:27:48.090 so this is y. 00:27:48.090 --> 00:27:50.060 I'm graphing y. 00:27:50.060 --> 00:27:53.430 And it starts at 1. 00:27:53.430 --> 00:27:54.210 That's what we said. 00:27:54.210 --> 00:27:58.280 At x equals 0, I've started at 1 with a slope of 1. 00:27:58.280 --> 00:28:00.920 So I have a slope of 1, but the slope, the slope, the 00:28:00.920 --> 00:28:02.260 slope is climbing up. 00:28:02.260 --> 00:28:05.530 And it reaches here. 00:28:05.530 --> 00:28:06.890 That height is what-- 00:28:09.490 --> 00:28:10.830 e. 00:28:10.830 --> 00:28:12.080 That height is e. 00:28:14.410 --> 00:28:18.140 Because when we said x equal 1 here, we got e. 00:28:18.140 --> 00:28:20.410 So it's climbing, climbing, climbing. 00:28:20.410 --> 00:28:23.130 And now what about on the other side? 00:28:23.130 --> 00:28:28.470 That had a slope of 1, so it was more like that. 00:28:28.470 --> 00:28:32.890 Now what about when x is negative? 00:28:32.890 --> 00:28:38.770 When x is negative, this is a highly useful fact. 00:28:38.770 --> 00:28:44.010 Suppose I want to think about e to the minus x. 00:28:44.010 --> 00:28:47.240 Well now, let me just take capital x to be 00:28:47.240 --> 00:28:48.420 minus little x. 00:28:48.420 --> 00:28:52.410 So I get e to the x times e to the minus x. 00:28:52.410 --> 00:28:54.220 What is that? 00:28:54.220 --> 00:28:57.110 What does that equal if I multiply e to the x times e to 00:28:57.110 --> 00:29:00.090 the minus x? 00:29:00.090 --> 00:29:03.310 As usual, I'm supposed to add these. 00:29:03.310 --> 00:29:07.100 I get 0, so I get e to the 0, which is 1. 00:29:07.100 --> 00:29:13.030 In other words, e to the minus x is 1 over e to the x, which 00:29:13.030 --> 00:29:15.670 we fully expected. 00:29:15.670 --> 00:29:22.430 So that at x equal minus 1 here, I'm down to 1 over e, 00:29:22.430 --> 00:29:24.770 1/3, approximately. 00:29:24.770 --> 00:29:28.120 So it's going down. 00:29:28.120 --> 00:29:32.100 In this way, it's decaying very fast. It almost touches 00:29:32.100 --> 00:29:34.480 that line, but never quite. 00:29:34.480 --> 00:29:35.730 This way, it's climbing. 00:29:38.860 --> 00:29:41.600 It's growing, growing really-- 00:29:41.600 --> 00:29:43.243 well, it's growing exponentially. 00:29:46.650 --> 00:29:49.710 And that's what this graph looks like. 00:29:49.710 --> 00:29:55.690 And now I would like to connect back, at the end of 00:29:55.690 --> 00:30:01.650 this lecture, to the insurance business-- 00:30:01.650 --> 00:30:07.680 sorry, the interest business, the bank compounding interest. 00:30:07.680 --> 00:30:15.940 Can I take your time with that important example of the 00:30:15.940 --> 00:30:17.130 exponential function? 00:30:17.130 --> 00:30:20.430 And we'll see a new way to reach e. 00:30:20.430 --> 00:30:22.300 I like this way. 00:30:22.300 --> 00:30:26.320 I like the way we did it with the infinite series. 00:30:26.320 --> 00:30:29.280 But here's another way. 00:30:29.280 --> 00:30:37.540 So suppose you're getting 100% interest. Generous bank. 00:30:37.540 --> 00:30:38.870 OK. 00:30:38.870 --> 00:30:43.870 And you start with $1 at 100% now. 00:30:43.870 --> 00:30:46.696 It's 100%. 00:30:46.696 --> 00:30:49.450 And the bank gives you interest at the 00:30:49.450 --> 00:30:51.770 end of every year. 00:30:51.770 --> 00:30:57.020 So at the end of the first year, you had $1 dollar in the 00:30:57.020 --> 00:31:00.890 bank, it adds in 100%. 00:31:00.890 --> 00:31:02.780 It adds in another dollar. 00:31:02.780 --> 00:31:08.360 So now you've got $2 in the bank after the first year. 00:31:08.360 --> 00:31:13.390 At the end of the second year, it gives you 100% of what 00:31:13.390 --> 00:31:14.360 you've got in the bank. 00:31:14.360 --> 00:31:15.500 So it gives you 2 more. 00:31:15.500 --> 00:31:18.030 It give you 4. 00:31:18.030 --> 00:31:22.050 At the end of the third year, it gives you an additional 4. 00:31:22.050 --> 00:31:24.220 You're up 50 to 8. 00:31:24.220 --> 00:31:26.190 And you see what's happening. 00:31:26.190 --> 00:31:29.130 It's the powers of two. 00:31:29.130 --> 00:31:33.770 Well, that's pretty good growth. 00:31:33.770 --> 00:31:37.140 But it's not calculus. 00:31:37.140 --> 00:31:41.660 Calculus doesn't do things in steps of a year. 00:31:41.660 --> 00:31:44.160 Calculus says cut that step down. 00:31:44.160 --> 00:31:48.120 You would want to ask your bank, couldn't you just, like, 00:31:48.120 --> 00:31:51.430 figure the interest a little more often and put it in 00:31:51.430 --> 00:31:53.890 there-- like, figure it every month? 00:31:53.890 --> 00:31:55.550 So what would happen if you figured the 00:31:55.550 --> 00:31:58.100 interest every month? 00:31:58.100 --> 00:32:01.590 Of course, you wouldn't get 100% interest in a month. 00:32:01.590 --> 00:32:06.710 You'd get 100% divided by 12, because we're only talking 00:32:06.710 --> 00:32:08.540 about one month. 00:32:08.540 --> 00:32:12.900 So if it was months, you start with 1. 00:32:12.900 --> 00:32:18.350 You have 1 plus 1/12. 00:32:18.350 --> 00:32:21.880 That's what you'd have after a month. 00:32:21.880 --> 00:32:24.970 Now, what would you have after 2 months and what would you 00:32:24.970 --> 00:32:29.640 have after 12 months? 00:32:29.640 --> 00:32:31.935 Well, we're going to follow the rule. 00:32:35.670 --> 00:32:39.030 They gave you the 1/12 at the end of January. 00:32:39.030 --> 00:32:41.730 So through all of February, you've got 1 00:32:41.730 --> 00:32:43.680 plus 1/12 in there. 00:32:43.680 --> 00:32:51.420 At the end of February, they take 1/12 of that, add it in. 00:32:51.420 --> 00:33:00.400 What you get the next time is 1 plus 1/12 squared. 00:33:00.400 --> 00:33:04.860 That's what you have. 00:33:04.860 --> 00:33:08.310 Essentially every time, they're going to multiply what 00:33:08.310 --> 00:33:10.860 you've got by this number 1 plus 1/12. 00:33:10.860 --> 00:33:13.140 1 to give you-- 00:33:13.140 --> 00:33:14.320 leave the money in. 00:33:14.320 --> 00:33:15.560 You have to leave your money. 00:33:15.560 --> 00:33:17.790 I'm sorry. 00:33:17.790 --> 00:33:23.060 Plus 1/12 of it for the interest. And then twice, and 00:33:23.060 --> 00:33:27.060 after 1 year, it's done this. 00:33:27.060 --> 00:33:30.360 You see what happens after 1 year, it's 00:33:30.360 --> 00:33:34.540 multiplied 12 times. 00:33:34.540 --> 00:33:38.010 1 plus 1/12 to the 12th power. 00:33:38.010 --> 00:33:40.720 And that's better than 2, right? 00:33:40.720 --> 00:33:45.000 You've got the 2 only when they put the interest in just 00:33:45.000 --> 00:33:45.630 once a year. 00:33:45.630 --> 00:33:51.380 Now we're speeding up the bank and getting more out of it. 00:33:51.380 --> 00:33:55.170 So I don't know exactly what 1 plus 1/12 to the 12th power 00:33:55.170 --> 00:33:57.700 is, but I know it's more than 2. 00:33:57.700 --> 00:34:01.080 And actually, I'm sure it's not more than 3. 00:34:04.160 --> 00:34:09.420 In fact, yeah, I'm claiming that it's not 00:34:09.420 --> 00:34:12.699 as much as e, 2.7. 00:34:12.699 --> 00:34:18.610 But it was worth doing, to get them to compound every month. 00:34:18.610 --> 00:34:21.790 But, of course, you think, okay, I'm on to a good thing. 00:34:21.790 --> 00:34:22.600 Every day. 00:34:22.600 --> 00:34:24.120 Why not? 00:34:24.120 --> 00:34:26.770 So what would every day be? 00:34:26.770 --> 00:34:31.830 1 plus 1/365. 00:34:31.830 --> 00:34:34.620 That's the interest you would get for just that day. 00:34:34.620 --> 00:34:39.810 But then they would compound it 365 times. 00:34:39.810 --> 00:34:44.090 So that would be a little more than this because they're 00:34:44.090 --> 00:34:47.600 adding the interest in more frequently. 00:34:47.600 --> 00:34:51.880 And, in general, I'm going to divide the 00:34:51.880 --> 00:34:54.540 year up into n pieces. 00:34:54.540 --> 00:34:59.730 In every piece, they multiply my wealth by 1 plus 1 over n. 00:34:59.730 --> 00:35:01.685 And they do it n times in a year. 00:35:05.020 --> 00:35:11.110 And the beautiful thing is that as n goes to infinity, 00:35:11.110 --> 00:35:15.150 and calculus comes in, because we're asking them to compound 00:35:15.150 --> 00:35:20.450 interest continuously, not just every month, not every 00:35:20.450 --> 00:35:26.100 day, every second even, but all the time. 00:35:26.100 --> 00:35:28.850 You don't get an infinite amount out of this. 00:35:28.850 --> 00:35:31.430 You get e. 00:35:31.430 --> 00:35:35.140 As n gets bigger, that approaches this number e. 00:35:35.140 --> 00:35:39.330 That's another way to construct e, as the limit-- 00:35:39.330 --> 00:35:45.000 you see, as n gets bigger, it's like 1 to the infinity, 00:35:45.000 --> 00:35:48.400 which is kind of meaningless. 00:35:48.400 --> 00:35:50.460 I don't want to say that 1 to the-- 00:35:50.460 --> 00:35:53.990 I had an email the other day that said, well, 1 to the 00:35:53.990 --> 00:35:56.350 infinity is e. 00:35:56.350 --> 00:35:57.140 What's happening? 00:35:57.140 --> 00:35:58.890 That's not true. 00:35:58.890 --> 00:36:04.190 It's this thing that's going to 1, this thing that's going 00:36:04.190 --> 00:36:05.610 to infinity. 00:36:05.610 --> 00:36:08.780 Then the combination goes to e. 00:36:08.780 --> 00:36:09.610 OK. 00:36:09.610 --> 00:36:16.030 So that's the application that shows the 00:36:16.030 --> 00:36:18.850 number e appearing again. 00:36:18.850 --> 00:36:20.100 OK. 00:36:23.140 --> 00:36:25.240 You've got the essence of e to the x. 00:36:28.280 --> 00:36:31.780 I just would like to say one thing, coming back to the very 00:36:31.780 --> 00:36:33.310 beginning here. 00:36:33.310 --> 00:36:37.730 The great differential equation, dy dx equal y. 00:36:37.730 --> 00:36:39.480 That was beautiful. 00:36:39.480 --> 00:36:41.620 Which we've now solved. 00:36:41.620 --> 00:36:46.560 Now I want to ask, what if the differential equation was dy 00:36:46.560 --> 00:36:51.570 dx is some multiple of y? 00:36:51.570 --> 00:36:53.620 How would that come up? 00:36:53.620 --> 00:36:57.360 Well, up to now, c was 1. 00:36:57.360 --> 00:37:02.840 We were getting 100% interest per year. 00:37:02.840 --> 00:37:06.260 But now, if c is sort of the interest rate, the growth 00:37:06.260 --> 00:37:10.580 rate, or the decay rate of c is negative, we may be losing 00:37:10.580 --> 00:37:15.890 money in this bank. 00:37:15.890 --> 00:37:22.090 So can I just tell you what is the solution to this 00:37:22.090 --> 00:37:24.750 differential equation? 00:37:24.750 --> 00:37:27.960 When I tell you, and we learned about taking 00:37:27.960 --> 00:37:30.650 derivatives, you'll see, of course, that's all it is. 00:37:30.650 --> 00:37:33.550 It's just the solution to this one. 00:37:33.550 --> 00:37:35.800 I'll also start at one. 00:37:35.800 --> 00:37:40.010 The solution to that one is y of x is e-- 00:37:40.010 --> 00:37:41.710 e is coming in again-- 00:37:41.710 --> 00:37:42.960 to the cx. 00:37:45.430 --> 00:37:49.790 What I'm doing is like changing the rate at-- 00:37:49.790 --> 00:37:52.490 I've made the rate of chance c. 00:37:52.490 --> 00:37:56.850 And then that c is going to come up there and in the 00:37:56.850 --> 00:38:03.030 derivative, the slope of this guy, that c will come down. 00:38:03.030 --> 00:38:07.530 The slope of this will be c e to the cx, which is cy, which 00:38:07.530 --> 00:38:09.680 is what that second differential 00:38:09.680 --> 00:38:11.540 equation tells us. 00:38:11.540 --> 00:38:16.460 So that's just a comment looking ahead, that we've 00:38:16.460 --> 00:38:19.290 solved not only the most important differential 00:38:19.290 --> 00:38:21.380 equation with the most important function that 00:38:21.380 --> 00:38:26.680 calculus creates but a whole collection of related 00:38:26.680 --> 00:38:34.390 equations in which the rate can be any fixed number, c. 00:38:34.390 --> 00:38:35.050 OK. 00:38:35.050 --> 00:38:35.780 Thank you. 00:38:35.780 --> 00:38:38.010 FEMALE SPEAKER: This has been a production of MIT 00:38:38.010 --> 00:38:40.390 OpenCourseWare and Gilbert Strang. 00:38:40.390 --> 00:38:42.670 Funding for this video was provided by the Lord 00:38:42.670 --> 00:38:43.890 Foundation. 00:38:43.890 --> 00:38:47.020 To help OCW continue to provide free and open access 00:38:47.020 --> 00:38:50.090 to MIT courses, please make a donation at 00:38:50.090 --> 00:38:51.650 ocw.mit.edu/donate.