WEBVTT 00:00:24.970 --> 00:00:29.150 PROFESSOR: Hi, I'm Gilbert Strang, and this is the very 00:00:29.150 --> 00:00:35.870 first in a series of videos about highlights of calculus. 00:00:35.870 --> 00:00:40.200 I'm doing these just because I hope they'll be helpful. 00:00:40.200 --> 00:00:47.790 It seems to me so easy to be lost in the big calculus 00:00:47.790 --> 00:00:53.300 textbooks and the many, many problems and in the details. 00:00:53.300 --> 00:00:56.660 But do you see the big picture? 00:00:56.660 --> 00:01:00.370 Well, I hope this will help. 00:01:00.370 --> 00:01:04.150 For me, calculus is about the 00:01:04.150 --> 00:01:09.110 relation between two functions. 00:01:09.110 --> 00:01:13.990 And one example for those two functions, one good example, 00:01:13.990 --> 00:01:20.760 is function 1, the distance, distance traveled, what you 00:01:20.760 --> 00:01:23.890 see on a trip meter in a car. 00:01:23.890 --> 00:01:30.790 And function 2, the one that goes with distance, is speed, 00:01:30.790 --> 00:01:34.720 how quickly you're going, how fast you're traveling. 00:01:34.720 --> 00:01:37.330 So that's one pair of functions. 00:01:37.330 --> 00:01:40.570 Let me give another pair. 00:01:40.570 --> 00:01:44.450 I could get more and more, but I think if we get these two 00:01:44.450 --> 00:01:49.190 pairs, we can move forward. 00:01:49.190 --> 00:01:54.610 So in this second pair, height is function 1, 00:01:54.610 --> 00:01:57.860 how high you've climbed. 00:01:57.860 --> 00:02:03.540 If it's a graph, how far the graph goes above the axis. 00:02:03.540 --> 00:02:06.060 Up, in other words. 00:02:06.060 --> 00:02:11.600 So that's height, and then the other one tells you 00:02:11.600 --> 00:02:13.460 how fast you climb. 00:02:13.460 --> 00:02:15.730 The height tells how far you climbed. 00:02:15.730 --> 00:02:18.070 It could be a mountain. 00:02:18.070 --> 00:02:22.300 And then the slope tells you how quickly you're climbing at 00:02:22.300 --> 00:02:23.050 each point. 00:02:23.050 --> 00:02:25.630 Are you going nearly straight up? 00:02:25.630 --> 00:02:26.620 Flat? 00:02:26.620 --> 00:02:28.790 Possibly down? 00:02:28.790 --> 00:02:34.400 So distance and speed, height and slope will serve as good 00:02:34.400 --> 00:02:36.660 examples to start with. 00:02:36.660 --> 00:02:42.480 And let me give you some letters, some algebra letters 00:02:42.480 --> 00:02:45.090 that you might use. 00:02:45.090 --> 00:02:53.490 Distance, maybe I would call that f of t. 00:02:53.490 --> 00:03:01.480 So f for how far or for function, and the idea is that 00:03:01.480 --> 00:03:04.990 t is the input. 00:03:04.990 --> 00:03:07.970 It's the time when you're asking for the distance. 00:03:07.970 --> 00:03:10.060 The output is the distance. 00:03:10.060 --> 00:03:16.580 Or in the case of height, maybe y of x would 00:03:16.580 --> 00:03:18.740 be the right one. 00:03:18.740 --> 00:03:22.270 x is how far you go across. 00:03:22.270 --> 00:03:23.870 That's the input. 00:03:23.870 --> 00:03:28.830 And at each x, you have an output y how far up? 00:03:28.830 --> 00:03:31.860 So f is telling you how far. 00:03:31.860 --> 00:03:36.660 y is telling you the height of a graph. 00:03:36.660 --> 00:03:41.340 That's function 1, two examples of function 1. 00:03:41.340 --> 00:03:43.060 Now, what about slope? 00:03:43.060 --> 00:03:46.560 Well, luckily, speed and slope start with the same letter, so 00:03:46.560 --> 00:03:52.990 I'll often use s for the speed or the slope for this second-- 00:03:52.990 --> 00:03:56.960 oh, it even stands for a second function. 00:03:56.960 --> 00:04:00.590 But let me tell you also the right-- 00:04:00.590 --> 00:04:01.620 the official-- 00:04:01.620 --> 00:04:06.670 letters that make the connection between function 2 00:04:06.670 --> 00:04:09.030 and function 1. 00:04:09.030 --> 00:04:15.240 If my function is a function of time, the distance, how far 00:04:15.240 --> 00:04:19.589 I go, then the speed is-- 00:04:19.589 --> 00:04:27.050 the right letters are df dt. 00:04:27.050 --> 00:04:29.840 Everybody uses those letters. 00:04:29.840 --> 00:04:32.830 So let me say again how to pronounce: df dt. 00:04:35.710 --> 00:04:40.100 And Leibniz came up with that notation, and 00:04:40.100 --> 00:04:42.850 he just got it right. 00:04:42.850 --> 00:04:44.920 And what would this one be? 00:04:44.920 --> 00:04:56.760 Well, corresponding to this, it looks the same, or dy dx. 00:04:56.760 --> 00:05:01.060 Again, I'll just repeat how to say that: dy dx. 00:05:01.060 --> 00:05:06.940 And that is the slope, and we have to understand what those 00:05:06.940 --> 00:05:07.820 symbols mean. 00:05:07.820 --> 00:05:09.730 Right now, I'm just writing them down as symbols. 00:05:13.070 --> 00:05:15.740 May I begin with the most important and the simplest 00:05:15.740 --> 00:05:17.650 example of all? 00:05:17.650 --> 00:05:19.660 Let me take that case. 00:05:19.660 --> 00:05:25.590 OK, so the key example here, the one to get completely 00:05:25.590 --> 00:05:29.450 straight is the case of constant 00:05:29.450 --> 00:05:32.820 speed, constant slope. 00:05:32.820 --> 00:05:34.900 I'll just graph that. 00:05:34.900 --> 00:05:36.225 So here I'm go to graph. 00:05:39.890 --> 00:05:43.220 Shall I make it the speed? 00:05:43.220 --> 00:05:45.000 Yeah, let's say speed. 00:05:45.000 --> 00:05:48.600 So time is going along that way. 00:05:48.600 --> 00:05:51.410 Speed is up this way. 00:05:51.410 --> 00:05:56.740 And I'm going to say in this first example that the speed 00:05:56.740 --> 00:05:57.690 is the same. 00:05:57.690 --> 00:06:00.250 We're traveling at the constant speed 00:06:00.250 --> 00:06:03.610 of let's say 40. 00:06:03.610 --> 00:06:06.920 So it stays at the height of 40. 00:06:06.920 --> 00:06:13.040 Oh, properly, I should add units like miles per hour or 00:06:13.040 --> 00:06:17.520 kilometers per hour or meters per second or whatever. 00:06:21.600 --> 00:06:24.680 For now, I'll just write 40. 00:06:24.680 --> 00:06:32.330 OK, now if we're traveling at a speed of 40 miles per hour, 00:06:32.330 --> 00:06:34.780 what's the distance? 00:06:34.780 --> 00:06:42.300 Well, let me start with the trip meter at zero. 00:06:42.300 --> 00:06:45.790 so this is time again, and now this is 00:06:45.790 --> 00:06:47.040 going to be the distance. 00:06:49.820 --> 00:06:55.000 After one hour, my distance is 40. 00:06:55.000 --> 00:06:59.990 So if I mark t equal to 1, I've reached 40. 00:06:59.990 --> 00:07:01.940 That's height of 40. 00:07:01.940 --> 00:07:06.175 At t equal to 2, I've reached 80. 00:07:06.175 --> 00:07:10.420 At t equal to 1/2, half an hour, I've reached 20. 00:07:10.420 --> 00:07:14.830 Those points lie on a line. 00:07:14.830 --> 00:07:21.120 The graph of distance covered when you're just traveling at 00:07:21.120 --> 00:07:27.890 a steady rate, constant rate, constant speed is just a 00:07:27.890 --> 00:07:30.070 straight line. 00:07:30.070 --> 00:07:32.660 And now I can make the connection. 00:07:32.660 --> 00:07:37.500 I've been speaking here about distance and speed. 00:07:37.500 --> 00:07:40.170 But now let me think of this as the height-- 00:07:40.170 --> 00:07:41.450 40 is that height. 00:07:41.450 --> 00:07:45.090 80 is that height-- 00:07:45.090 --> 00:07:46.740 and ask about slope. 00:07:46.740 --> 00:07:48.180 What is slope? 00:07:48.180 --> 00:07:52.810 So let's just remember what's the connection here. 00:07:52.810 --> 00:07:57.720 What's the slope if that's the distance if I look at my trip 00:07:57.720 --> 00:08:02.970 meter and I know I'm traveling along at that constant speed, 00:08:02.970 --> 00:08:05.700 how do I find that speed? 00:08:05.700 --> 00:08:13.670 Well, slope, it's the distance up, which would be 40 after 00:08:13.670 --> 00:08:15.980 one hour, divided by the distance 00:08:15.980 --> 00:08:23.820 across, 40/1, or 80/2. 00:08:23.820 --> 00:08:26.760 Doesn't matter, because we're traveling at constant speed, 00:08:26.760 --> 00:08:39.110 so the slope, which is up, over, across is 40/1, 00:08:39.110 --> 00:08:45.570 80/2, 20 over 1/2. 00:08:45.570 --> 00:08:51.360 I'll put 80/2 as one example: 40. 00:08:51.360 --> 00:08:55.780 Oh, let me do it-- 00:08:55.780 --> 00:08:56.960 that's arithmetic. 00:08:56.960 --> 00:08:59.240 Let me do it with algebra. 00:08:59.240 --> 00:09:02.000 We don't need calculus yet, by the way. 00:09:02.000 --> 00:09:05.060 Calculus is coming pretty quickly. 00:09:05.060 --> 00:09:08.090 This is the step we can take. 00:09:08.090 --> 00:09:15.370 Because the speed is constant, we can just divide the 00:09:15.370 --> 00:09:22.760 distance by the time to find-- 00:09:22.760 --> 00:09:26.020 and this slope, let me right speed also. 00:09:28.870 --> 00:09:33.620 Up, over, across, distance over time, f/t, 00:09:33.620 --> 00:09:35.280 that gives us s. 00:09:35.280 --> 00:09:36.530 This is s. 00:09:39.500 --> 00:09:43.590 OK, what about-- 00:09:43.590 --> 00:09:46.400 calculus goes both ways. 00:09:46.400 --> 00:09:48.210 We can go both ways here. 00:09:48.210 --> 00:09:50.500 We already have practically. 00:09:50.500 --> 00:09:56.140 Here I went in the direction from 1 to 2. 00:09:56.140 --> 00:09:58.090 Now, I want to go in the direction-- 00:09:58.090 --> 00:10:00.090 suppose I know the speed. 00:10:00.090 --> 00:10:03.490 How do I recover the distance? 00:10:03.490 --> 00:10:08.560 If I know my speed is 40 and I know I started at zero, what's 00:10:08.560 --> 00:10:09.810 my distance? 00:10:11.780 --> 00:10:14.740 Distance or height, either one. 00:10:19.970 --> 00:10:24.830 So these are like both. 00:10:24.830 --> 00:10:26.280 Now, I'm just going the other way. 00:10:26.280 --> 00:10:27.670 Well, you see how. 00:10:27.670 --> 00:10:29.060 How do I find f? 00:10:32.294 --> 00:10:36.630 It's s times t, right? 00:10:36.630 --> 00:10:39.560 Your algebra automatically says if you see a t there, you 00:10:39.560 --> 00:10:41.490 can put it there. 00:10:41.490 --> 00:10:43.250 So it's s times t. 00:10:43.250 --> 00:10:45.640 It's a straight line. 00:10:45.640 --> 00:10:50.150 s times t, s times x, y equal sx. 00:10:50.150 --> 00:10:51.400 Let me put another-- 00:10:55.050 --> 00:10:59.310 the same idea with my y, x letters. 00:10:59.310 --> 00:11:00.610 It's that line. 00:11:00.610 --> 00:11:05.030 In other words, if that one is constant, this one is a 00:11:05.030 --> 00:11:07.280 straight line. 00:11:07.280 --> 00:11:12.400 OK, straightforward, but very, very fundamental. 00:11:12.400 --> 00:11:15.290 In fact, can I call your attention to something a 00:11:15.290 --> 00:11:17.780 little more? 00:11:17.780 --> 00:11:27.280 Suppose I measured between time 2 and time 1. 00:11:27.280 --> 00:11:31.330 So I'm looking between time 2 and time 1, and I look how far 00:11:31.330 --> 00:11:32.860 I went in that time. 00:11:36.760 --> 00:11:39.150 But what I'm trying is-- 00:11:39.150 --> 00:11:41.710 I'm going to put in another little symbol because it's 00:11:41.710 --> 00:11:44.720 going to be really worth knowing. 00:11:44.720 --> 00:11:52.430 It's really the change in f divided by the change in t. 00:11:52.430 --> 00:11:58.210 I use that letter delta to indicate 00:11:58.210 --> 00:12:00.800 a difference between-- 00:12:00.800 --> 00:12:05.110 the difference between time 2 and time 1 was 1, and the 00:12:05.110 --> 00:12:11.130 difference between height 2 and height 1 was 40. 00:12:11.130 --> 00:12:14.390 You see, I'm looking at this little piece. 00:12:14.390 --> 00:12:17.710 And, of course, the slope is still 40. 00:12:17.710 --> 00:12:21.350 It's still the slope of that line. 00:12:21.350 --> 00:12:26.800 Yeah, so that really what I'm measuring in speed there, I 00:12:26.800 --> 00:12:30.670 don't always have to be starting at t equals 0, and I 00:12:30.670 --> 00:12:33.390 don't always have to be starting at f equals 0. 00:12:33.390 --> 00:12:34.880 Oh, let me draw that. 00:12:34.880 --> 00:12:38.100 Suppose I started at f equals 40. 00:12:38.100 --> 00:12:40.720 My trip meter happened to start at 40. 00:12:40.720 --> 00:12:43.300 After an hour, I'd be up to 80. 00:12:43.300 --> 00:12:46.270 After another hour, I'd be up to 120. 00:12:46.270 --> 00:12:50.430 Do you see that this starting the trip meter, who cares 00:12:50.430 --> 00:12:53.160 where the trip meter started? 00:12:53.160 --> 00:12:55.450 It's the change in the trip meter that tells how 00:12:55.450 --> 00:12:57.520 long the trip was. 00:12:57.520 --> 00:12:58.560 Clear. 00:12:58.560 --> 00:13:01.383 OK, so that's that example. 00:13:06.450 --> 00:13:14.020 We come back to it because it's the basic one where the 00:13:14.020 --> 00:13:15.950 speed is constant. 00:13:15.950 --> 00:13:20.170 And even if now I have to move to a changing speed, you have 00:13:20.170 --> 00:13:24.210 to let me bring calculus into these lectures. 00:13:24.210 --> 00:13:28.290 OK, I'm going to draw another picture, and you 00:13:28.290 --> 00:13:32.692 tell me about the-- 00:13:32.692 --> 00:13:38.310 yeah, let me draw function 1, another example of function 1. 00:13:38.310 --> 00:13:40.600 So again I have time. 00:13:40.600 --> 00:13:42.280 I have distance. 00:13:42.280 --> 00:13:47.280 I'm going to start at zero, but I'm not going to keep the 00:13:47.280 --> 00:13:48.610 speed constant. 00:13:48.610 --> 00:13:53.520 I'm going to start out at a good speed, but I'm 00:13:53.520 --> 00:13:55.805 going to slow down. 00:13:55.805 --> 00:13:59.590 Do you see me slowing down there? 00:13:59.590 --> 00:14:01.340 I don't mean slowing down with the chalk. 00:14:01.340 --> 00:14:05.650 I mean slowing down with slope. 00:14:05.650 --> 00:14:08.690 The slope started out steep. 00:14:08.690 --> 00:14:12.800 By here, by that point, the slope was zero. 00:14:12.800 --> 00:14:16.050 What was the car doing here? 00:14:16.050 --> 00:14:21.860 The car is certainly moving forward because the distance 00:14:21.860 --> 00:14:23.970 is increasing. 00:14:23.970 --> 00:14:26.770 Here it's increasing faster. 00:14:26.770 --> 00:14:30.480 Here it's increasing barely. 00:14:30.480 --> 00:14:33.810 In other words, we're putting on the brakes. 00:14:33.810 --> 00:14:35.130 The car is slowing down. 00:14:35.130 --> 00:14:36.540 We're coming to a red light. 00:14:36.540 --> 00:14:40.860 In fact, there is the red light right at that time. 00:14:40.860 --> 00:14:48.030 Now, just stay with it to think what would the speed 00:14:48.030 --> 00:14:49.840 look like for this problem? 00:14:53.430 --> 00:14:55.850 If that's a picture of the function, just 00:14:55.850 --> 00:14:57.410 let's get some idea. 00:14:57.410 --> 00:15:03.210 I'm not going to have a formula yet. 00:15:03.210 --> 00:15:05.170 I'm not putting in all the details. 00:15:05.170 --> 00:15:11.230 Well, actually, I don't plan to put in all the details of 00:15:11.230 --> 00:15:18.070 calculus of every possible step we might take. 00:15:18.070 --> 00:15:21.070 It's the important ones I'm hoping to show you and I'm 00:15:21.070 --> 00:15:24.890 hoping for you to see that they are important. 00:15:24.890 --> 00:15:27.430 OK, what is important? 00:15:27.430 --> 00:15:29.820 Roughly, what does the graph looks like? 00:15:29.820 --> 00:15:33.060 Well, the speed-- 00:15:33.060 --> 00:15:34.100 the slope-- 00:15:34.100 --> 00:15:37.800 started out somewhere up there. 00:15:37.800 --> 00:15:45.040 Yeah, it started out at a good speed and slowed down. 00:15:45.040 --> 00:15:48.030 And by this point, ha! 00:15:48.030 --> 00:15:51.040 Let's mark that time here on that graph. 00:15:51.040 --> 00:15:55.930 Do you see what is the speed at that moment? 00:15:55.930 --> 00:16:00.770 The speed at that moment is zero. 00:16:00.770 --> 00:16:02.420 The car has stopped. 00:16:02.420 --> 00:16:04.500 The speed is decreasing. 00:16:04.500 --> 00:16:08.280 Let me make it decrease, decrease, decrease, decrease, 00:16:08.280 --> 00:16:13.860 and at that moment, the speed is zero right there. 00:16:13.860 --> 00:16:15.680 That's that point. 00:16:15.680 --> 00:16:21.400 See, two different pictures, two different functions. 00:16:21.400 --> 00:16:24.730 but same information. 00:16:24.730 --> 00:16:31.190 So calculus has the job of given one of those functions, 00:16:31.190 --> 00:16:32.300 find the other one. 00:16:32.300 --> 00:16:34.690 Given this function, find that one. 00:16:34.690 --> 00:16:36.380 This way is called-- 00:16:36.380 --> 00:16:41.040 from function one to function two, that's called 00:16:41.040 --> 00:16:43.690 differential calculus. 00:16:43.690 --> 00:16:46.230 Big, impressive word anyway. 00:16:46.230 --> 00:16:51.800 That's function one to two, finding the speed. 00:16:51.800 --> 00:16:57.860 Going the other direction is called integral calculus. 00:16:57.860 --> 00:17:03.500 The step is called integration when you take the speed over 00:17:03.500 --> 00:17:09.510 that period of time, and you recover the distance. 00:17:09.510 --> 00:17:12.569 So it's differential calculus in one direction, integral 00:17:12.569 --> 00:17:13.930 calculus in the other. 00:17:13.930 --> 00:17:15.880 Now, here's a question. 00:17:18.940 --> 00:17:22.230 Let me continue that curve a little longer. 00:17:22.230 --> 00:17:24.609 I got it to the red light. 00:17:24.609 --> 00:17:32.040 Now imagine that the distance starts going 00:17:32.040 --> 00:17:36.160 down from that point. 00:17:36.160 --> 00:17:39.080 What's happening? 00:17:39.080 --> 00:17:42.690 The distance is decreasing. 00:17:42.690 --> 00:17:45.360 The car is going backwards. 00:17:45.360 --> 00:17:48.020 It's going in reverse. 00:17:48.020 --> 00:17:51.700 The speed, what's the speed? 00:17:51.700 --> 00:17:53.860 Negative. 00:17:53.860 --> 00:18:01.430 The speed, because distance is going from higher to lower, 00:18:01.430 --> 00:18:03.320 that counts for negative speed. 00:18:03.320 --> 00:18:08.245 The speed curve would be going down here. 00:18:08.245 --> 00:18:13.730 Do you see that that's a not brilliantly drawn picture, but 00:18:13.730 --> 00:18:15.230 you're seeing the-- 00:18:15.230 --> 00:18:19.240 that's the farthest it went. 00:18:19.240 --> 00:18:24.660 Then the car started backwards, and the speed curve 00:18:24.660 --> 00:18:27.820 reflected that by going below zero. 00:18:27.820 --> 00:18:30.990 You see, two different curves, but same information. 00:18:35.950 --> 00:18:38.290 I'm remembering an old movie. 00:18:38.290 --> 00:18:41.250 I don't know if you saw an old B movie called Ferris 00:18:41.250 --> 00:18:43.920 Bueller's Day Off. 00:18:43.920 --> 00:18:45.440 Did you see that? 00:18:45.440 --> 00:18:49.080 So the kid had borrowed his father's-- 00:18:49.080 --> 00:18:55.570 not borrowed, but lifted his father's good car and drove it 00:18:55.570 --> 00:19:00.740 a lot like so and put on a lot of mileage. 00:19:00.740 --> 00:19:03.260 The trip meter was way up, and he knew his father was going 00:19:03.260 --> 00:19:05.390 to notice this. 00:19:05.390 --> 00:19:12.620 So he had the idea to put the car up on a lift, put it in 00:19:12.620 --> 00:19:16.820 reverse, and go for a while, and the trip 00:19:16.820 --> 00:19:18.160 meter would go backwards. 00:19:22.940 --> 00:19:24.755 I don't know if trip meters do go backwards. 00:19:29.380 --> 00:19:36.000 It's kind of tough to watch them while going in reverse. 00:19:36.000 --> 00:19:43.780 But if whoever made the car understood calculus, as you 00:19:43.780 --> 00:19:47.520 do, the speedometer-- 00:19:47.520 --> 00:19:49.680 now that I think of it, speedometers don't have a 00:19:49.680 --> 00:19:50.450 below zero. 00:19:50.450 --> 00:19:55.540 They should have. And trip meters should go backwards. 00:19:55.540 --> 00:19:59.690 I mean, that movie was just made for a calculus person. 00:20:05.830 --> 00:20:07.950 Maybe I'm remembering more. 00:20:07.950 --> 00:20:10.330 I think it didn't work or something. 00:20:10.330 --> 00:20:15.030 And the kid got mad and kicked the car, and it fell off the 00:20:15.030 --> 00:20:20.190 lift, went through the glass window. 00:20:20.190 --> 00:20:23.370 Anyway, calculus would have saved him if 00:20:23.370 --> 00:20:27.480 only the car had been-- 00:20:27.480 --> 00:20:31.500 or the meters in the car had been made correctly. 00:20:31.500 --> 00:20:36.320 All right, that's one pair. 00:20:36.320 --> 00:20:42.650 That's our first real pair in which the speed changes. 00:20:42.650 --> 00:20:44.580 OK. 00:20:44.580 --> 00:20:54.460 I thought in this first video, later, even today, I'll get to 00:20:54.460 --> 00:20:59.340 a case where we have formulas. 00:20:59.340 --> 00:21:03.200 That's what calculus moves into. 00:21:03.200 --> 00:21:08.420 When f of t is given by some formula, well, here it's given 00:21:08.420 --> 00:21:10.210 by a formula: s times t. 00:21:13.110 --> 00:21:15.020 A simple formula. 00:21:15.020 --> 00:21:20.350 And then, knowing that, we know that the speed is s. 00:21:20.350 --> 00:21:23.340 Later, we got more functions. 00:21:23.340 --> 00:21:29.030 But let me take an example, just because these pairs of 00:21:29.030 --> 00:21:32.030 functions are everywhere. 00:21:32.030 --> 00:21:33.970 What could I take? 00:21:33.970 --> 00:21:38.920 Maybe height of a person. 00:21:38.920 --> 00:21:41.060 Height of a person. 00:21:41.060 --> 00:21:47.430 OK, so this is now another example, just to get practice 00:21:47.430 --> 00:21:51.940 in the relation between the height of a person and the 00:21:51.940 --> 00:21:54.470 rate of change of the height. 00:21:54.470 --> 00:21:57.120 So this is the height. 00:21:57.120 --> 00:21:58.990 Maybe I'll call it y. 00:21:58.990 --> 00:22:04.040 Let me write height of a person. 00:22:04.040 --> 00:22:06.840 And what is this going to be? 00:22:06.840 --> 00:22:09.950 What is function two? 00:22:09.950 --> 00:22:12.510 Well, slope doesn't seem quite right. 00:22:12.510 --> 00:22:18.120 The point about function two is it tells how fast function 00:22:18.120 --> 00:22:20.000 one changes. 00:22:20.000 --> 00:22:25.720 It's the rate of change of the height. 00:22:25.720 --> 00:22:28.610 It's the rate of change. 00:22:28.610 --> 00:22:34.890 So let me call it s, and it'll be the rate of change. 00:22:34.890 --> 00:22:36.620 Good if I use those words. 00:22:43.738 --> 00:22:47.120 Yeah, so I want to think just how we grow, a 00:22:47.120 --> 00:22:49.540 typical person growing. 00:22:49.540 --> 00:22:56.910 In fact, as I wrote this on the board, I thought of 00:22:56.910 --> 00:22:58.060 another pair. 00:22:58.060 --> 00:23:01.310 Can I just say it in words, this other pair, and then I'll 00:23:01.310 --> 00:23:03.330 come back to this one? 00:23:03.330 --> 00:23:05.430 Here's another pair. 00:23:05.430 --> 00:23:11.880 This could be money in a bank. 00:23:15.000 --> 00:23:16.360 Wealth sounds better. 00:23:16.360 --> 00:23:17.610 Let's call it wealth. 00:23:19.990 --> 00:23:23.340 That's zippier. 00:23:23.340 --> 00:23:25.490 And then what is this one? 00:23:25.490 --> 00:23:28.840 If this is your wealth, your total 00:23:28.840 --> 00:23:32.890 assets, what's your worth? 00:23:32.890 --> 00:23:38.730 This would be the rate of change, how 00:23:38.730 --> 00:23:41.510 quickly you're saving. 00:23:41.510 --> 00:23:42.970 s could be for saving. 00:23:42.970 --> 00:23:50.450 Or if you're down here, s is for spending, right? 00:23:50.450 --> 00:23:54.550 If s is positive, that means you're wealth is increasing, 00:23:54.550 --> 00:23:55.800 you're saving. 00:23:58.010 --> 00:24:01.940 Negative s means you're spending, and your wealth goes 00:24:01.940 --> 00:24:07.210 whatever, maybe-- 00:24:07.210 --> 00:24:08.760 I hope-- up. 00:24:08.760 --> 00:24:12.470 Height is mostly up, right? 00:24:12.470 --> 00:24:14.270 So let me come back to height of a person. 00:24:17.010 --> 00:24:18.360 Now, where-- 00:24:18.360 --> 00:24:22.780 oh, and this is time in years. 00:24:22.780 --> 00:24:28.340 This is t in years, and this, too, of course. 00:24:34.620 --> 00:24:40.340 Actually, I realize you started at t 00:24:40.340 --> 00:24:43.120 equals zero: birth. 00:24:43.120 --> 00:24:44.835 You do start at a certain-- 00:24:47.790 --> 00:24:49.330 actually, what do I know? 00:24:52.640 --> 00:24:53.680 You don't say tall. 00:24:53.680 --> 00:24:54.530 You say long. 00:24:54.530 --> 00:24:56.930 But then as soon as you can stand up, it's tall, 00:24:56.930 --> 00:24:58.180 so let's say tall. 00:25:02.230 --> 00:25:03.780 Shall we guessed 20 inches? 00:25:03.780 --> 00:25:06.720 If that's way off, I apologize to everybody. 00:25:06.720 --> 00:25:11.060 Let me just say 20, 20 inches. 00:25:11.060 --> 00:25:13.270 OK, at year zero. 00:25:13.270 --> 00:25:17.270 OK, and then presumably you grow. 00:25:17.270 --> 00:25:19.680 OK, so you grow a little. 00:25:19.680 --> 00:25:20.790 What are we headed for? 00:25:20.790 --> 00:25:24.450 About 60, 70 inches or something. 00:25:24.450 --> 00:25:25.900 Anyway, you grow. 00:25:30.230 --> 00:25:34.230 Let's say that's 10 years old and here is 20 years old. 00:25:34.230 --> 00:25:35.780 OK, so you grow. 00:25:35.780 --> 00:25:37.700 Maybe you grow faster than that. 00:25:37.700 --> 00:25:40.990 Let's say you're a healthy person here. 00:25:40.990 --> 00:25:43.970 OK, up you grow. 00:25:43.970 --> 00:25:55.200 And then at about maybe age 12 or 13, there's a growth spurt. 00:25:55.200 --> 00:25:59.030 And maybe the point is, how do we see that growth spurt on 00:25:59.030 --> 00:26:00.210 the two graphs? 00:26:00.210 --> 00:26:03.320 Differently, but it's the same growth spurt. 00:26:03.320 --> 00:26:07.340 OK, so here your height suddenly jumps up. 00:26:07.340 --> 00:26:09.330 Boy, yeah, you catch up with everybody. 00:26:12.070 --> 00:26:16.500 And then at about 12 or 13 well, then unfortunately, it 00:26:16.500 --> 00:26:22.190 doesn't do that forever, and it kind of levels off here. 00:26:22.190 --> 00:26:24.960 It levels off, and actually you don't 00:26:24.960 --> 00:26:28.300 grow a whole lot more. 00:26:28.300 --> 00:26:34.930 In fact, I think when you get to about-- oh, I don't know. 00:26:34.930 --> 00:26:35.380 Whatever. 00:26:35.380 --> 00:26:37.200 We won't discuss this point. 00:26:37.200 --> 00:26:43.900 I say when you get too old, you probably lose some. 00:26:43.900 --> 00:26:47.100 Let's not emphasize that. 00:26:47.100 --> 00:26:51.230 OK, so here is the-- 00:26:51.230 --> 00:26:55.460 now, what's happening over here? 00:26:55.460 --> 00:26:58.950 Well, it's the slope of that graph. 00:26:58.950 --> 00:27:01.080 So the slope might be-- 00:27:01.080 --> 00:27:07.520 this is time zero, but you're growing right away. 00:27:07.520 --> 00:27:12.230 The s graph, the rate of growth graph, 00:27:12.230 --> 00:27:13.990 doesn't start at zero. 00:27:13.990 --> 00:27:17.280 It starts how fast you're growing, whatever you're 00:27:17.280 --> 00:27:21.040 growing, whatever that slope is. 00:27:21.040 --> 00:27:24.710 It's fantastic that when we draw graphs of things, the 00:27:24.710 --> 00:27:27.870 word "slope" is suddenly the right word. 00:27:27.870 --> 00:27:32.370 OK, so you're growing, maybe at a pretty good rate here. 00:27:32.370 --> 00:27:36.960 And let me mark out 10 and 20 years. 00:27:36.960 --> 00:27:43.500 And OK, you're doing well, you're coming along here, and 00:27:43.500 --> 00:27:46.050 then the growth spurt. 00:27:46.050 --> 00:27:49.295 OK, so then suddenly, your rate of growth takes off. 00:27:52.190 --> 00:27:58.090 But it doesn't stay that way, right? 00:27:58.090 --> 00:28:01.070 Your rate of growth levels off, in fact, 00:28:01.070 --> 00:28:03.990 levels way off, levels-- 00:28:03.990 --> 00:28:09.650 you'll come down to here, and you probably don't grow a lot. 00:28:09.650 --> 00:28:11.190 Do you see the two? 00:28:11.190 --> 00:28:15.000 This was the growth curve. 00:28:15.000 --> 00:28:16.940 This was the fast growth. 00:28:16.940 --> 00:28:19.090 But then it stopped. 00:28:19.090 --> 00:28:21.160 Up here, it slowed down. 00:28:21.160 --> 00:28:23.090 Here, it dropped. 00:28:23.090 --> 00:28:33.120 And oh, if we allow for this person who lived too long, 00:28:33.120 --> 00:28:36.540 height actually drops. 00:28:36.540 --> 00:28:41.640 OK, there is an example in which I don't-- 00:28:41.640 --> 00:28:48.550 also I'm sure people have devised approximate formulas 00:28:48.550 --> 00:28:53.480 for average growth rates, but you see, I'm not-- 00:28:53.480 --> 00:28:57.520 it's the idea of the relation between function one and 00:28:57.520 --> 00:29:00.770 function two that I'm emphasizing. 00:29:00.770 --> 00:29:11.040 Now, my last example, let me take one more example, one 00:29:11.040 --> 00:29:12.900 more example for this first lecture. 00:29:16.050 --> 00:29:21.990 So let me take a case in which the speed is-- 00:29:21.990 --> 00:29:23.483 so here will be my two-- 00:29:29.020 --> 00:29:30.210 let's use speed. 00:29:30.210 --> 00:29:34.230 Let's use this as distance. 00:29:34.230 --> 00:29:41.840 This is distance again, and graph two, as 00:29:41.840 --> 00:29:44.045 always, will be speed. 00:29:49.670 --> 00:29:53.720 And I'm going to take a case in which 00:29:53.720 --> 00:29:55.630 it's given by a formula. 00:29:55.630 --> 00:30:02.030 I'm going to let the speed be increasing steadily. 00:30:02.030 --> 00:30:07.480 OK, so my speed graph this time is going to go up at a 00:30:07.480 --> 00:30:09.810 constant rate. 00:30:09.810 --> 00:30:12.690 So this is the speed s. 00:30:12.690 --> 00:30:14.780 This is the time t. 00:30:14.780 --> 00:30:18.530 So this would be s equals-- 00:30:18.530 --> 00:30:20.070 s is proportional to t. 00:30:20.070 --> 00:30:22.180 That's where you get a straight line. 00:30:22.180 --> 00:30:26.460 s is let's say a times t. 00:30:26.460 --> 00:30:30.690 That a, a physicist, if we were physicists, would say 00:30:30.690 --> 00:30:32.570 acceleration. 00:30:32.570 --> 00:30:34.260 You're accelerating. 00:30:34.260 --> 00:30:40.560 You're keeping your foot on the gas, steadily speeding up, 00:30:40.560 --> 00:30:47.260 and so then s is proportional to t. 00:30:47.260 --> 00:30:54.580 Now, think about the distance. 00:30:54.580 --> 00:30:56.600 What's happening with distance? 00:30:56.600 --> 00:31:00.540 If this is accelerating, you're 00:31:00.540 --> 00:31:01.780 going faster and faster. 00:31:01.780 --> 00:31:05.210 You're covering more and more speed, more and more distance, 00:31:05.210 --> 00:31:08.420 more and more quickly. 00:31:08.420 --> 00:31:11.770 If this is slope, the slope is increasing. 00:31:11.770 --> 00:31:12.870 Look, the graph-- 00:31:12.870 --> 00:31:15.820 let's start the trip meter at zero. 00:31:15.820 --> 00:31:17.920 So you started with a speed of zero. 00:31:17.920 --> 00:31:24.120 You were not really increasing distance until you got 00:31:24.120 --> 00:31:25.820 slightly beyond zero, and then it 00:31:25.820 --> 00:31:27.920 slightly started to increase. 00:31:27.920 --> 00:31:36.225 But then it increases faster and faster, right? 00:31:36.225 --> 00:31:47.730 It never gets infinitely fast, but it keeps going upwards. 00:31:47.730 --> 00:31:50.780 And the calculus question would be 00:31:50.780 --> 00:31:52.550 can we give a formula-- 00:31:52.550 --> 00:31:53.810 an equation-- 00:31:53.810 --> 00:31:55.060 for the distance? 00:31:57.820 --> 00:32:01.970 Because in this case, I guess I started with function two, 00:32:01.970 --> 00:32:05.710 and therefore, it's function one that I want to look at. 00:32:05.710 --> 00:32:08.550 It's always pairs of functions. 00:32:08.550 --> 00:32:17.260 OK, now, let's think where this would actually happen. 00:32:17.260 --> 00:32:22.040 If we were leaning over the Tower of Pisa or whatever, 00:32:22.040 --> 00:32:25.220 like Galileo, and drop something, or even just drop 00:32:25.220 --> 00:32:31.700 something anywhere, that would be-- we drop it. 00:32:31.700 --> 00:32:35.310 At the beginning, it has no speed, but of course, 00:32:35.310 --> 00:32:39.310 instantly it picks up speed. 00:32:39.310 --> 00:32:42.580 The a would have something to do with the gravitational 00:32:42.580 --> 00:32:45.820 constant for the Earth, whatever, and then maybe-- 00:32:45.820 --> 00:32:47.880 yeah. 00:32:47.880 --> 00:32:50.580 And what would be the distance? 00:32:50.580 --> 00:32:56.810 OK, now can I just mention a small miracle of calculus? 00:32:56.810 --> 00:32:59.810 A small miracle. 00:32:59.810 --> 00:33:03.270 I'm going this direction now from speed to distance so I'm 00:33:03.270 --> 00:33:05.600 doing integral calculus. 00:33:05.600 --> 00:33:11.730 And we'll get to that later. 00:33:11.730 --> 00:33:13.500 In the first lectures, we're almost always 00:33:13.500 --> 00:33:15.320 going from one to two. 00:33:15.320 --> 00:33:20.190 But here is a neat fact about going from two to one, that if 00:33:20.190 --> 00:33:26.580 this is the time t, then, of course, this height here will 00:33:26.580 --> 00:33:31.600 be a times t. 00:33:31.600 --> 00:33:40.710 And the amazing fact is that this graph tells you the area 00:33:40.710 --> 00:33:42.830 under this one. 00:33:42.830 --> 00:33:44.550 Graph one-- 00:33:44.550 --> 00:33:45.630 function one-- 00:33:45.630 --> 00:33:50.970 tells you the area under the graph two. 00:33:50.970 --> 00:33:56.640 And in this example with a nice constant acceleration, 00:33:56.640 --> 00:34:00.550 steady increase in speed, we know this. 00:34:00.550 --> 00:34:02.000 This is a triangle. 00:34:02.000 --> 00:34:03.890 It has a base of t. 00:34:03.890 --> 00:34:06.470 It has a height of at. 00:34:06.470 --> 00:34:12.610 And the area of a triangle, of course, the area, and my point 00:34:12.610 --> 00:34:16.050 is that the area is function one-- 00:34:16.050 --> 00:34:22.130 amazing; that's just terrific-- 00:34:22.130 --> 00:34:29.090 will be-- the area of this triangle here is 1/2 of the 00:34:29.090 --> 00:34:31.560 base times the height. 00:34:36.469 --> 00:34:39.310 That's the area, and calculus will tell us 00:34:39.310 --> 00:34:41.360 that's function one. 00:34:41.360 --> 00:34:50.860 So this function is 1/2 of a times t squared. 00:34:50.860 --> 00:34:59.180 So there is a function one, and here is df dt. 00:34:59.180 --> 00:35:03.470 If I go back to the first letters that I mentioned, if 00:35:03.470 --> 00:35:09.250 this is my function f, then this is my function that-- 00:35:09.250 --> 00:35:11.650 and notice what kind of a curve that is. 00:35:11.650 --> 00:35:14.250 Do you recognize that with a square? 00:35:14.250 --> 00:35:19.060 That tells me it's a parabola, a famous and important curve. 00:35:19.060 --> 00:35:20.540 And, of course, it's important because it 00:35:20.540 --> 00:35:22.730 has such a neat formula. 00:35:22.730 --> 00:35:32.680 OK, so we have found the function one. 00:35:32.680 --> 00:35:36.710 We've recovered the information in that lost black 00:35:36.710 --> 00:35:43.260 box, the distance box, the trip meter, from what we did 00:35:43.260 --> 00:35:50.110 find in black box two, the speed, the record of speed. 00:35:50.110 --> 00:35:52.710 And notice, I'm using the speed all the way 00:35:52.710 --> 00:35:54.560 from here to here. 00:35:54.560 --> 00:35:58.770 The speed kind of tells me how the distance is piled up. 00:35:58.770 --> 00:36:03.700 The distance is kind of a running total, where the speed 00:36:03.700 --> 00:36:07.610 at that moment is an instant thing. 00:36:07.610 --> 00:36:11.060 Oh, we have to do that in future lectures. 00:36:11.060 --> 00:36:16.480 The difference between a running total of total 00:36:16.480 --> 00:36:21.780 distance covered and a speed that's telling me at a moment, 00:36:21.780 --> 00:36:25.120 at an instant how distance is changing. 00:36:25.120 --> 00:36:31.050 The slope at this very point t, that slope is 00:36:31.050 --> 00:36:33.440 this height, is at. 00:36:33.440 --> 00:36:36.160 OK, so there you have the first-- 00:36:36.160 --> 00:36:37.850 well, I'll say the second. 00:36:37.850 --> 00:36:42.050 The first pair of calculus was this one. 00:36:42.050 --> 00:36:46.500 f equals st, and it's derivative was s. 00:36:46.500 --> 00:36:50.220 Our second pair is f is this, and will you allow 00:36:50.220 --> 00:36:52.210 me to write df dt? 00:36:55.100 --> 00:37:05.134 If f is 1/2 of at squared, then df dt is at. 00:37:08.250 --> 00:37:10.250 You'll see this rule again. 00:37:10.250 --> 00:37:15.680 The power two dropped to a power one. 00:37:15.680 --> 00:37:20.790 But the two multiplied the thing so it canceled the 1/2 00:37:20.790 --> 00:37:24.390 and just left the a. 00:37:24.390 --> 00:37:27.580 OK, that's a start on the highlights of calculus. 00:37:27.580 --> 00:37:28.810 Thanks. 00:37:28.810 --> 00:37:30.610 NARRATOR: This has been a production of MIT 00:37:30.610 --> 00:37:33.000 OpenCourseWare and Gilbert Strang. 00:37:33.000 --> 00:37:35.270 Funding for this video was provided by the Lord 00:37:35.270 --> 00:37:36.490 Foundation. 00:37:36.490 --> 00:37:39.620 To help OCW continue to provide free and open access 00:37:39.620 --> 00:37:42.690 to MIT courses, please make a donation at 00:37:42.690 --> 00:37:44.250 ocw.mit.edu/donate.