WEBVTT 00:00:00.040 --> 00:00:01.940 NARRATOR: The following content is provided under a 00:00:01.940 --> 00:00:03.690 Creative Commons license. 00:00:03.690 --> 00:00:06.630 Your support will help MIT OpenCourseWare continue to 00:00:06.630 --> 00:00:09.990 offer high-quality educational resources for free. 00:00:09.990 --> 00:00:12.830 To make a donation or to view additional materials from 00:00:12.830 --> 00:00:16.760 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:16.760 --> 00:00:18.010 ocw.mit.edu. 00:00:39.230 --> 00:00:40.020 PROFESSOR: Hi. 00:00:40.020 --> 00:00:44.140 I'm Herb Gross, and welcome to Calculus Revisited. 00:00:44.140 --> 00:00:46.280 I guess the most difficult lecture to give with any 00:00:46.280 --> 00:00:48.440 course is probably the first one. 00:00:48.440 --> 00:00:51.260 And you're sort of tempted to look at your audience and say 00:00:51.260 --> 00:00:54.120 you're probably wondering why I called you all here. 00:00:54.120 --> 00:00:57.740 And in this sense, I have elected to entitle our first 00:00:57.740 --> 00:01:03.150 lecture simply Preface to give a double overview, an overview 00:01:03.150 --> 00:01:07.160 both of the hardware and the software that will make up 00:01:07.160 --> 00:01:08.680 this course. 00:01:08.680 --> 00:01:13.240 To begin with, we will have a series of lectures of which 00:01:13.240 --> 00:01:15.080 this is the first. 00:01:15.080 --> 00:01:19.150 In our lectures, our main aim will be to give an overview of 00:01:19.150 --> 00:01:22.880 the material being covered, an insight as to why various 00:01:22.880 --> 00:01:26.260 computations are done, and insights as to how 00:01:26.260 --> 00:01:29.710 applications of these concepts will be made. 00:01:29.710 --> 00:01:35.430 The heart of our course will consist of a regular textbook. 00:01:35.430 --> 00:01:37.630 You see, we have our lectures. 00:01:37.630 --> 00:01:40.390 We have a textbook. 00:01:40.390 --> 00:01:44.810 The textbook is designed to supply you with deeper 00:01:44.810 --> 00:01:48.610 insights than what we can give in a lecture. 00:01:48.610 --> 00:01:53.000 In addition, recognizing the fact that the textbook may 00:01:53.000 --> 00:01:56.430 leave gaps, places where you may want some additional 00:01:56.430 --> 00:02:00.030 knowledge, we also have supplementary notes. 00:02:00.030 --> 00:02:05.360 And finally, at the backbone of our package is what we call 00:02:05.360 --> 00:02:06.630 the study guide. 00:02:06.630 --> 00:02:10.520 The study guide consists of a breakdown of the course. 00:02:10.520 --> 00:02:14.990 It tells us what the various lectures will be, the units. 00:02:14.990 --> 00:02:19.290 There are pretests to help you decide how well prepared you 00:02:19.290 --> 00:02:21.090 are for the topic that's coming up. 00:02:21.090 --> 00:02:23.640 There is a final examination at the end of 00:02:23.640 --> 00:02:25.300 each block of material. 00:02:25.300 --> 00:02:28.640 And perhaps most importantly, especially from an engineer's 00:02:28.640 --> 00:02:33.450 point of view, in each unit that we study, the study guide 00:02:33.450 --> 00:02:37.970 will consist of exercises primarily called learning 00:02:37.970 --> 00:02:42.970 exercises, exercises which hopefully will turn you on 00:02:42.970 --> 00:02:46.060 towards wanting to be able to apply the material, and at the 00:02:46.060 --> 00:02:50.100 same time, serve as a springboard by which we can 00:02:50.100 --> 00:02:54.050 highlight why the theory and many about our lecture points 00:02:54.050 --> 00:02:56.780 are really as important as they are. 00:02:56.780 --> 00:03:00.140 So much for the hardware of our course. 00:03:00.140 --> 00:03:03.540 And now let's turn our attention to the software. 00:03:03.540 --> 00:03:05.920 Just what is calculus? 00:03:05.920 --> 00:03:09.870 In a manner of speaking, calculus can be viewed as 00:03:09.870 --> 00:03:13.770 being high school mathematics with one additional concept 00:03:13.770 --> 00:03:16.150 called the limit concept thrown in. 00:03:16.150 --> 00:03:18.770 If you recall back to your high school days, remember 00:03:18.770 --> 00:03:21.250 that we're always dealing with things like 00:03:21.250 --> 00:03:23.390 average rate of speed. 00:03:23.390 --> 00:03:26.400 Notice I say average or constant rate of speed. 00:03:26.400 --> 00:03:29.800 The old recipe that distance equals rate times time 00:03:29.800 --> 00:03:33.200 presupposes that the rate is constant, because if the rate 00:03:33.200 --> 00:03:36.750 is varying, which rate is it that you use to multiply the 00:03:36.750 --> 00:03:39.200 time by to find the distance? 00:03:39.200 --> 00:03:43.760 You see, in other words, roughly speaking, we can say 00:03:43.760 --> 00:03:46.660 that at least one branch of calculus known as differential 00:03:46.660 --> 00:03:49.170 calculus deals with the subject of 00:03:49.170 --> 00:03:50.780 instantaneous speed. 00:03:50.780 --> 00:03:53.730 And instantaneous speed is a rather easy thing to talk 00:03:53.730 --> 00:03:55.240 about intuitively. 00:03:55.240 --> 00:03:59.740 Imagine an object moving along this line and passing the 00:03:59.740 --> 00:04:03.520 point P. And we say to ourselves how fast was the 00:04:03.520 --> 00:04:06.260 object moving at the instant that we're at the point P? 00:04:06.260 --> 00:04:09.700 Now, you see, this is some sort of a problem. 00:04:09.700 --> 00:04:13.560 Because at the instant that you're at P, you're not in a 00:04:13.560 --> 00:04:17.010 sense moving at all because you're at P. 00:04:17.010 --> 00:04:20.230 Of course, what we do to reduce this problem to an old 00:04:20.230 --> 00:04:24.190 one is we say, well, suppose we have a couple of observers. 00:04:24.190 --> 00:04:26.940 Let's call them O1 and O2. 00:04:26.940 --> 00:04:30.500 Let them be stationed, one on each side of P. Now, certainly 00:04:30.500 --> 00:04:33.030 what we could do physically here is we can measure the 00:04:33.030 --> 00:04:37.910 distance between O1 and O2. 00:04:37.910 --> 00:04:43.090 And we can also measure the time that it takes to 00:04:43.090 --> 00:04:46.420 go from O1 to O2. 00:04:46.420 --> 00:04:50.270 And what we can do is divide that distance by the time, and 00:04:50.270 --> 00:04:54.670 that, you see, is our old high school concept of the average 00:04:54.670 --> 00:04:58.285 speed of the particle as it moves from O1 to O2. 00:04:58.285 --> 00:05:01.330 Now, you see, the question is, somebody says gee, that's a 00:05:01.330 --> 00:05:03.780 great answer, but it's the wrong problem. 00:05:03.780 --> 00:05:05.920 We didn't ask what was the average speed as we 00:05:05.920 --> 00:05:07.420 went from O1 to O2. 00:05:07.420 --> 00:05:09.730 We asked what was the instantaneous speed. 00:05:09.730 --> 00:05:11.700 And the idea is we say, well, lookit. 00:05:11.700 --> 00:05:14.880 The average speed and the instantaneous speed, it seems, 00:05:14.880 --> 00:05:17.740 should be pretty much the same if the observers were 00:05:17.740 --> 00:05:19.800 relatively close together. 00:05:19.800 --> 00:05:22.670 The next observation is it seems that if we were to move 00:05:22.670 --> 00:05:26.820 the observers in even closer, there would be less of a 00:05:26.820 --> 00:05:31.440 discrepancy between O1 and O2 in the sense that-- not a 00:05:31.440 --> 00:05:34.345 discrepancy, but in the sense that the average speed would 00:05:34.345 --> 00:05:36.900 now seem like a better approximation to the 00:05:36.900 --> 00:05:39.480 instantaneous speed because there was less distance for 00:05:39.480 --> 00:05:41.725 something to go wrong in. 00:05:41.725 --> 00:05:45.220 And so we get the idea that maybe what we should do is 00:05:45.220 --> 00:05:48.480 make the observers gets closer and closer together. 00:05:48.480 --> 00:05:50.810 That would minimize the difference between the average 00:05:50.810 --> 00:05:54.380 speed and the instantaneous rate of speed, and maybe the 00:05:54.380 --> 00:05:56.480 optimal thing would happen when the two 00:05:56.480 --> 00:05:58.390 observers were together. 00:05:58.390 --> 00:05:59.870 But the strange part is-- 00:05:59.870 --> 00:06:02.080 and this is where calculus really begins. 00:06:02.080 --> 00:06:04.140 This is what calculus is all about. 00:06:04.140 --> 00:06:07.370 As soon as the observers come together, notice that what you 00:06:07.370 --> 00:06:10.420 have is that the distance between them is 0. 00:06:10.420 --> 00:06:13.430 The time that it takes to get from one to the other is 0. 00:06:13.430 --> 00:06:17.020 And therefore, it appears that if we divide distance by time, 00:06:17.020 --> 00:06:21.550 we are going to wind up with 0/0. 00:06:21.550 --> 00:06:25.630 Now, my claim is that 0/0 should be called-- 00:06:25.630 --> 00:06:30.190 well, I'll call it undefined, but actually, I think 00:06:30.190 --> 00:06:36.920 indeterminate would be a better word. 00:06:36.920 --> 00:06:38.415 Why do I say that? 00:06:38.415 --> 00:06:40.330 Well, here's an interesting thing. 00:06:40.330 --> 00:06:43.900 When we do arithmetic with small numbers, observe that if 00:06:43.900 --> 00:06:46.980 you add two small numbers, you expect the result to be a 00:06:46.980 --> 00:06:47.960 small number. 00:06:47.960 --> 00:06:51.230 If you multiply two small numbers, you expect the result 00:06:51.230 --> 00:06:52.890 to be a small number. 00:06:52.890 --> 00:06:57.280 Similarly, for division, for subtraction, the difference of 00:06:57.280 --> 00:06:59.740 two small numbers is a small number. 00:06:59.740 --> 00:07:02.960 On the other hand, the quotient of two small numbers 00:07:02.960 --> 00:07:04.450 is rather deceptive. 00:07:04.450 --> 00:07:08.000 Because it's a ratio, if one of the very small numbers 00:07:08.000 --> 00:07:12.390 happens to be very much larger compared with the other small 00:07:12.390 --> 00:07:15.200 number, the ratio might be quite large. 00:07:15.200 --> 00:07:19.960 Well, for example, visualize, say, 10 to the minus 6, 00:07:19.960 --> 00:07:27.470 1/1,000,000, 0.000001, which is a pretty small number. 00:07:27.470 --> 00:07:32.310 Now, divide that by 10 to the minus 12th. 00:07:32.310 --> 00:07:35.210 Well, you see, 10 to the minus 12th is a small number, so 00:07:35.210 --> 00:07:38.220 small that it makes 10 to the minus sixth appear large. 00:07:38.220 --> 00:07:43.440 In fact, the quotient is 10 to the sixth, which is 1,000,000. 00:07:43.440 --> 00:07:46.210 And here we see that when you're dealing with the ratio 00:07:46.210 --> 00:07:49.180 of small numbers, you're a little bit in trouble, because 00:07:49.180 --> 00:07:53.310 we can't tell whether the ratio will be small, or large, 00:07:53.310 --> 00:07:54.930 or somewhere in between. 00:07:54.930 --> 00:07:57.430 For example, if we reverse the role of numerator and 00:07:57.430 --> 00:08:00.020 denominator here, we would still have the quotient of two 00:08:00.020 --> 00:08:03.880 small numbers, but 10 to the minus 12th divided by 10 to 00:08:03.880 --> 00:08:11.930 the minus sixth is a relatively small number, 10 to 00:08:11.930 --> 00:08:13.810 the minus 6. 00:08:13.810 --> 00:08:16.600 Of course, this is the physical way of looking at it. 00:08:16.600 --> 00:08:18.910 Small divided by small is indeterminate. 00:08:18.910 --> 00:08:21.400 We have a more rigorous way of looking at this if you want to 00:08:21.400 --> 00:08:24.090 see it from a mathematical structure point of view. 00:08:24.090 --> 00:08:28.540 Namely, suppose we define a/b in the traditional way. 00:08:28.540 --> 00:08:33.059 Namely, a/b is that number such that when we multiply it 00:08:33.059 --> 00:08:36.020 by b we get a. 00:08:36.020 --> 00:08:40.200 Well, what would that say as far as 0/0 was concerned? 00:08:40.200 --> 00:08:41.419 It would say what? 00:08:41.419 --> 00:08:45.060 That 0/0 is that number such that when we multiply 00:08:45.060 --> 00:08:48.030 it by 0 we get 0. 00:08:48.030 --> 00:08:51.410 Now, what number has the property that when we multiply 00:08:51.410 --> 00:08:53.330 it by 0 we get 0? 00:08:53.330 --> 00:08:55.890 And the answer is any number. 00:08:55.890 --> 00:08:58.380 This is why 0/0 is indeterminate. 00:08:58.380 --> 00:09:01.630 If we say to a person, tell me the number I must multiply by 00:09:01.630 --> 00:09:05.570 0 to get 0, the answer is any number. 00:09:05.570 --> 00:09:08.850 Well, the idea then is that we must avoid the expression 0/0 00:09:08.850 --> 00:09:10.310 at all costs. 00:09:10.310 --> 00:09:14.570 What this means then is that we say OK, let the observers 00:09:14.570 --> 00:09:17.940 get closer to closer together, but never touch. 00:09:17.940 --> 00:09:20.410 Now, the point is that as long as the observers get closer 00:09:20.410 --> 00:09:23.970 and closer together and never touch, let's ask the question 00:09:23.970 --> 00:09:26.300 how many pairs of observers do we need? 00:09:26.300 --> 00:09:28.790 And the answer is that theoretically we need 00:09:28.790 --> 00:09:31.400 infinitely many pairs of observers. 00:09:31.400 --> 00:09:32.740 Well, why is that? 00:09:32.740 --> 00:09:35.610 Because as long as there's a distance between a pair of 00:09:35.610 --> 00:09:38.520 observers, we can theoretically fit in another 00:09:38.520 --> 00:09:40.380 pair of observers. 00:09:40.380 --> 00:09:44.700 This is why in our course we do not begin with this idea, 00:09:44.700 --> 00:09:48.340 but looking backwards now, we say ah, we had better find 00:09:48.340 --> 00:09:51.140 some way of giving us the equivalent of having 00:09:51.140 --> 00:09:53.290 infinitely many pairs of observers. 00:09:53.290 --> 00:09:59.140 And to do this, the idea that we come up with is the concept 00:09:59.140 --> 00:10:01.990 called a function. 00:10:01.990 --> 00:10:05.400 Consider the old Galileo freely falling body problem, 00:10:05.400 --> 00:10:09.520 where the distance that the body falls s equals 16t 00:10:09.520 --> 00:10:12.950 squared, where t is in seconds and s is in feet. 00:10:12.950 --> 00:10:17.760 Notice that this apparently harmless recipe gives us a way 00:10:17.760 --> 00:10:21.250 for finding s for each given t. 00:10:21.250 --> 00:10:24.650 In other words, to all intents and purposes, this recipe 00:10:24.650 --> 00:10:28.992 gives us an observer for each point of time. 00:10:28.992 --> 00:10:32.260 For each time, we can find the distance, which is physically 00:10:32.260 --> 00:10:36.330 equivalent to knowing an observer at every point. 00:10:36.330 --> 00:10:40.430 In turn, the study of functions lends itself to a 00:10:40.430 --> 00:10:43.220 study of graphs, a picture. 00:10:43.220 --> 00:10:47.590 Namely, if we look at s equals 16t squared again, notice that 00:10:47.590 --> 00:10:49.910 we visualize a recipe here. 00:10:49.910 --> 00:10:54.250 t can be viewed as being an input, s as the output. 00:10:54.250 --> 00:10:58.560 For a given input t, we can compute the output s. 00:10:58.560 --> 00:11:03.090 In general, if we now elect to plot the input along a 00:11:03.090 --> 00:11:07.770 horizontal line and the output at right angles to this, we 00:11:07.770 --> 00:11:11.270 now have a picture of our relationship, a picture which 00:11:11.270 --> 00:11:14.200 is called a graph. 00:11:14.200 --> 00:11:17.980 You see, we can talk about this more explicitly as far as 00:11:17.980 --> 00:11:20.830 this particular problem is concerned, just by taking a 00:11:20.830 --> 00:11:22.650 look at a picture like this. 00:11:22.650 --> 00:11:25.810 In other words, in this particular problem, the input 00:11:25.810 --> 00:11:30.970 is time t, the output is distance s. 00:11:30.970 --> 00:11:35.270 For each t, we locate a height called s by squaring t and 00:11:35.270 --> 00:11:37.450 multiplying by 16. 00:11:37.450 --> 00:11:40.390 And now, what average speed means in terms of this kind of 00:11:40.390 --> 00:11:42.640 a diagram is the following. 00:11:42.640 --> 00:11:46.410 To find the average speed, all we have to do is on a given 00:11:46.410 --> 00:11:49.960 time interval find the distance traveled, which I 00:11:49.960 --> 00:11:53.510 call delta s, the change in distance, and divide that by 00:11:53.510 --> 00:11:54.860 the change in time. 00:11:54.860 --> 00:11:58.030 That's the average speed, which, by the way, from a 00:11:58.030 --> 00:12:02.150 geometrical point of view, becomes known as the slope of 00:12:02.150 --> 00:12:03.820 this particular straight line. 00:12:03.820 --> 00:12:08.130 In other words, average speed is to functions what slope of 00:12:08.130 --> 00:12:10.830 a straight line is to geometry. 00:12:10.830 --> 00:12:14.190 At any rate, knowing what the average rate of speed is, we 00:12:14.190 --> 00:12:18.570 sort of say why couldn't we define the instantaneous speed 00:12:18.570 --> 00:12:19.500 to be this. 00:12:19.500 --> 00:12:24.040 We will take the change in distance divided by the change 00:12:24.040 --> 00:12:26.740 in time and see what happens. 00:12:26.740 --> 00:12:28.430 And we write this this way. 00:12:28.430 --> 00:12:30.450 Limit as delta t approaches 0. 00:12:30.450 --> 00:12:33.450 Let's see what happens as that change in time becomes 00:12:33.450 --> 00:12:37.370 arbitrarily small, but never equaling 0 because we don't 00:12:37.370 --> 00:12:39.820 want a 0/0 form here. 00:12:39.820 --> 00:12:44.260 You see, this then becomes the working definition of what we 00:12:44.260 --> 00:12:46.670 call differential calculus. 00:12:46.670 --> 00:12:50.390 The point is that this particular definition does not 00:12:50.390 --> 00:12:53.240 depend on s equaling 16t squared. 00:12:53.240 --> 00:12:57.430 s could be any function of t whatsoever. 00:12:57.430 --> 00:13:00.210 We could have a more elaborate type of situation. 00:13:00.210 --> 00:13:02.170 The important point is what? 00:13:02.170 --> 00:13:05.420 The basic definition stays the same. 00:13:05.420 --> 00:13:09.980 What changes is the amount of arithmetic that's necessary to 00:13:09.980 --> 00:13:13.970 handle the particular relationship between s and t. 00:13:13.970 --> 00:13:18.010 This will be a major part of our course, the strange thing 00:13:18.010 --> 00:13:20.970 being that even at the very end of our course when we've 00:13:20.970 --> 00:13:24.200 gone through many, many things, our basic definition 00:13:24.200 --> 00:13:26.790 of instantaneous rate of change will have never 00:13:26.790 --> 00:13:27.960 changed from this. 00:13:27.960 --> 00:13:30.440 It will always stay like this. 00:13:30.440 --> 00:13:34.320 But what will change is how much arithmetic and algebra 00:13:34.320 --> 00:13:38.250 and geometry and trigonometry, et cetera, we will have to do 00:13:38.250 --> 00:13:40.350 in order to compute these things from a 00:13:40.350 --> 00:13:42.170 numerical point of view. 00:13:42.170 --> 00:13:45.620 Well, so much for the first phase of calculus called 00:13:45.620 --> 00:13:47.150 differential calculus. 00:13:47.150 --> 00:13:50.610 A second phase of calculus, one which was developed by the 00:13:50.610 --> 00:13:54.510 Ancient Greeks by 600 BC, the subject that ultimately 00:13:54.510 --> 00:13:58.200 becomes known as integral calculus, concerns problem of 00:13:58.200 --> 00:14:00.954 finding area under a curve. 00:14:00.954 --> 00:14:07.200 Here, I've elected to draw the parabola y equals x squared on 00:14:07.200 --> 00:14:10.840 the interval from 0, 0 to 1, 0. 00:14:10.840 --> 00:14:17.290 And the question basically is what is the area bounded by 00:14:17.290 --> 00:14:19.990 this sort of triangular region? 00:14:19.990 --> 00:14:24.330 Let's call that region R, and what we would like to find is 00:14:24.330 --> 00:14:26.420 the area of the region R. 00:14:26.420 --> 00:14:30.310 And the Ancient Greeks had a rather interesting title for 00:14:30.310 --> 00:14:32.300 this type of approach for finding the area. 00:14:32.300 --> 00:14:35.400 It is both figurative and literal, I guess. 00:14:35.400 --> 00:14:36.810 It's called the method of exhaustion. 00:14:40.670 --> 00:14:40.754 What they did was to -- 00:14:40.754 --> 00:14:42.610 They would divide the interval, say, 00:14:42.610 --> 00:14:44.530 into n equal parts. 00:14:44.530 --> 00:14:47.570 And picking the lowest point in each interval, they would 00:14:47.570 --> 00:14:50.940 inscribe a rectangle. 00:14:50.940 --> 00:14:53.500 Knowing that the area of the rectangle was the base times 00:14:53.500 --> 00:14:57.110 the height, they would add up the area of each of these 00:14:57.110 --> 00:15:00.880 rectangles, and know that whatever that area was, that 00:15:00.880 --> 00:15:04.540 would have to be too small to be the right answer because 00:15:04.540 --> 00:15:07.040 that region was contained in R. And that would be 00:15:07.040 --> 00:15:08.780 labeled A sub n-- 00:15:08.780 --> 00:15:10.120 lower bar, say-- 00:15:10.120 --> 00:15:13.130 to indicate that this was a sum of rectangles which was 00:15:13.130 --> 00:15:15.780 too small to be the right answer. 00:15:15.780 --> 00:15:20.210 Similarly, they would then find the highest point in each 00:15:20.210 --> 00:15:24.430 rectangle, get an overapproximation by adding up 00:15:24.430 --> 00:15:28.240 the sum of those areas, which they would call A sub n upper 00:15:28.240 --> 00:15:31.050 bar, and now know that the area of the regions they were 00:15:31.050 --> 00:15:34.260 looking for was squeezed in between these two. 00:15:34.260 --> 00:15:37.530 Then what they would do is make more and more divisions, 00:15:37.530 --> 00:15:40.320 and hopefully, and I think you can see this sort of 00:15:40.320 --> 00:15:42.840 intuitively happening here, each of the lower 00:15:42.840 --> 00:15:45.910 approximations gets bigger and fills out 00:15:45.910 --> 00:15:47.970 the space from inside. 00:15:47.970 --> 00:15:51.730 Each of the upper approximations gets smaller 00:15:51.730 --> 00:15:54.770 and chops off the space from outside here. 00:15:54.770 --> 00:16:01.530 And hopefully, if both of these bounds sort of converge 00:16:01.530 --> 00:16:05.520 to the same value L, we get the idea that the area of the 00:16:05.520 --> 00:16:08.130 region R must be L. 00:16:08.130 --> 00:16:09.780 This is not anything new. 00:16:09.780 --> 00:16:12.700 In other words, this is a technique that is some 2,500 00:16:12.700 --> 00:16:16.820 years old, used by the Ancient Greeks. 00:16:16.820 --> 00:16:19.030 Of course, what happens with engineering students in 00:16:19.030 --> 00:16:22.620 general is that one frequently says, but I'm not interested 00:16:22.620 --> 00:16:23.920 in studying area. 00:16:23.920 --> 00:16:25.630 I am not a geometer. 00:16:25.630 --> 00:16:26.720 I am a physicist. 00:16:26.720 --> 00:16:28.370 I am an engineer. 00:16:28.370 --> 00:16:30.990 What good is the area under a curve? 00:16:30.990 --> 00:16:35.510 And the interesting point here becomes that if we label the 00:16:35.510 --> 00:16:39.450 coordinate axis rather than x and y, give them physical 00:16:39.450 --> 00:16:43.490 labels, it turns out that area under a curve has a physical 00:16:43.490 --> 00:16:44.880 interpretation. 00:16:44.880 --> 00:16:46.390 Consider the same problem. 00:16:46.390 --> 00:16:50.270 Only now, instead of talking about y equals x squared, 00:16:50.270 --> 00:16:53.340 let's talk about v, the velocity, equaling the square 00:16:53.340 --> 00:16:54.460 of the time. 00:16:54.460 --> 00:16:57.080 And say that the time goes to 0 to 1. 00:16:57.080 --> 00:17:00.870 In other words, if we plot v versus t, we get a 00:17:00.870 --> 00:17:02.380 picture like this. 00:17:02.380 --> 00:17:05.670 And the question that comes up is what do we mean by the area 00:17:05.670 --> 00:17:07.010 under the curve here? 00:17:07.010 --> 00:17:09.890 And again, without belaboring this point, not because it's 00:17:09.890 --> 00:17:12.980 not important, but because this is just an overview and 00:17:12.980 --> 00:17:15.680 we'll come back to all of these topics later in our 00:17:15.680 --> 00:17:19.849 course, the point I just want to bring out here is, notice 00:17:19.849 --> 00:17:23.490 that the area under the curve here is the distance that this 00:17:23.490 --> 00:17:28.820 particle would travel moving at this speed if the time goes 00:17:28.820 --> 00:17:30.460 from 0 to 1. 00:17:30.460 --> 00:17:32.180 And notice what we're saying here. 00:17:32.180 --> 00:17:36.670 Again, suppose we divide this interval into n equal parts 00:17:36.670 --> 00:17:39.760 and inscribe rectangles. 00:17:39.760 --> 00:17:43.100 Notice that each of these rectangles 00:17:43.100 --> 00:17:44.580 represents a distance. 00:17:44.580 --> 00:17:53.730 Namely, if a particle moved at the speed over this length of 00:17:53.730 --> 00:17:57.220 time, the area under the curve would be the distance that it 00:17:57.220 --> 00:17:59.400 traveled during that time interval. 00:17:59.400 --> 00:18:02.040 In other words, what we're saying is that if the particle 00:18:02.040 --> 00:18:05.850 moved at this speed from this time to this time, then moved 00:18:05.850 --> 00:18:10.200 at this speed from this time to this time, the sum of these 00:18:10.200 --> 00:18:12.830 two areas would give the distance that the particle 00:18:12.830 --> 00:18:17.110 traveled, which obviously is less than the distance that 00:18:17.110 --> 00:18:19.640 the particle truly traveled, because notice that the 00:18:19.640 --> 00:18:22.957 particle was moving at a speed which at every instance from 00:18:22.957 --> 00:18:26.770 here to here was greater than this and at every instant from 00:18:26.770 --> 00:18:28.690 here to here was greater than this. 00:18:28.690 --> 00:18:32.180 In other words, in the same way as before, that area of 00:18:32.180 --> 00:18:39.580 the region R was whittled in between A sub n upper bar and 00:18:39.580 --> 00:18:43.250 A sub n lower bar, notice that the distance traveled by the 00:18:43.250 --> 00:18:48.280 particle can now be limited or bounded in the same way. 00:18:48.280 --> 00:18:52.470 And in the same way that we found area as a limit, we can 00:18:52.470 --> 00:18:55.390 now find distance as a limit. 00:18:55.390 --> 00:18:58.740 And these two things, namely, what? 00:18:58.740 --> 00:19:02.750 Instantaneous speed and area under a curve are the two 00:19:02.750 --> 00:19:05.960 essential branches of calculus, differential 00:19:05.960 --> 00:19:10.000 calculus being concerned with instantaneous rate of speed, 00:19:10.000 --> 00:19:12.790 integral calculus with area under a curve. 00:19:12.790 --> 00:19:17.560 And the beauty of calculus, surprisingly enough, in a way 00:19:17.560 --> 00:19:19.530 is only secondary as far as these 00:19:19.530 --> 00:19:21.020 two topics are concerned. 00:19:21.020 --> 00:19:24.690 The true beauty lies in the fact that these apparently two 00:19:24.690 --> 00:19:27.430 different branches of calculus, one of which was 00:19:27.430 --> 00:19:30.850 invented by the Ancient Greeks as early as 600 BC, 00:19:30.850 --> 00:19:32.030 the other of which-- 00:19:32.030 --> 00:19:33.070 differential calculus-- 00:19:33.070 --> 00:19:37.370 was not known to man until the time of Isaac Newton in 1690 00:19:37.370 --> 00:19:41.070 AD are related by a rather remarkable thing. 00:19:41.070 --> 00:19:44.480 That remarkable thing, which we will emphasize at great 00:19:44.480 --> 00:19:49.440 length during our course, is that areas and rates of change 00:19:49.440 --> 00:19:52.245 are related by area under a curve. 00:19:52.245 --> 00:19:54.600 Now, I don't know how to draw this so that you see this 00:19:54.600 --> 00:19:58.630 thing as vividly as possible, but the idea is this. 00:19:58.630 --> 00:20:04.150 Think of area being swept out as we take a line and move it, 00:20:04.150 --> 00:20:07.400 tracing out the curve this way towards the right. 00:20:07.400 --> 00:20:13.580 Notice that if we have a certain amount of area, if we 00:20:13.580 --> 00:20:18.110 now move a little bit further to the right, notice that the 00:20:18.110 --> 00:20:24.840 new area somehow depends on what the height of this curve 00:20:24.840 --> 00:20:26.570 is going to be. 00:20:26.570 --> 00:20:30.130 That somehow or other, it seems that the area under the 00:20:30.130 --> 00:20:35.190 curve must be related to how fast the height of this line 00:20:35.190 --> 00:20:36.320 is changing. 00:20:36.320 --> 00:20:40.670 Or to look at it inversely, how fast the area is changing 00:20:40.670 --> 00:20:44.670 should somehow be related to the height of this line. 00:20:44.670 --> 00:20:47.000 And just what that relationship is will be 00:20:47.000 --> 00:20:49.730 explored also in great detail in the course. 00:20:49.730 --> 00:20:53.210 And we will show the beautiful marriage between this 00:20:53.210 --> 00:20:56.390 differential and integral calculus through this 00:20:56.390 --> 00:20:59.150 relationship here, which becomes known as the 00:20:59.150 --> 00:21:02.550 fundamental theorem of integral calculus. 00:21:02.550 --> 00:21:06.420 At any rate then, what this should show us is that 00:21:06.420 --> 00:21:07.900 calculus hinges-- 00:21:07.900 --> 00:21:12.410 whether it's differential calculus or integral calculus, 00:21:12.410 --> 00:21:13.960 that calculus hinges on something 00:21:13.960 --> 00:21:15.730 called the limit concept. 00:21:15.730 --> 00:21:18.930 Again, by way of a very quick review, one 00:21:18.930 --> 00:21:19.910 of the limit concepts-- 00:21:19.910 --> 00:21:22.380 and I think it's easy to see geometrically rather than 00:21:22.380 --> 00:21:23.300 analytically. 00:21:23.300 --> 00:21:26.410 Imagine that we have a curve, and we want to find the 00:21:26.410 --> 00:21:30.080 tangent of the curve at the point P. What we can do is 00:21:30.080 --> 00:21:34.120 take a point Q and draw the straight line that joins P to 00:21:34.120 --> 00:21:38.690 Q. We could then find the slope of the line PQ. 00:21:38.690 --> 00:21:41.940 The trouble is that PQ does not look very much like the 00:21:41.940 --> 00:21:42.930 tangent line. 00:21:42.930 --> 00:21:48.300 So we say OK, let Q move down so it comes closer to P. We 00:21:48.300 --> 00:21:50.740 can then find the slopes of PQ1. 00:21:50.740 --> 00:21:53.390 We could find the slope of PQ2. 00:21:53.390 --> 00:21:57.160 But in each case, we still do not have the slope of the line 00:21:57.160 --> 00:22:01.370 tangent to the curve at P. But we get the idea that as Q gets 00:22:01.370 --> 00:22:05.710 closer and closer to P, the slope, or the secant line that 00:22:05.710 --> 00:22:10.040 joins P to Q, becomes a better and better approximation to 00:22:10.040 --> 00:22:13.190 the line that would be tangent to the curve at P. 00:22:13.190 --> 00:22:16.270 In fact, it's rather interesting that in the 16th 00:22:16.270 --> 00:22:20.280 century, the definition that was given of a tangent line 00:22:20.280 --> 00:22:25.390 was that a tangent line is a line which passes through two 00:22:25.390 --> 00:22:27.250 consecutive points on a curve. 00:22:27.250 --> 00:22:29.560 Now, obviously, a curve does not have 00:22:29.560 --> 00:22:31.270 two consecutive points. 00:22:31.270 --> 00:22:32.650 What they really meant was what? 00:22:32.650 --> 00:22:37.600 That as Q gets closer and closer to P, the secant line 00:22:37.600 --> 00:22:40.040 becomes a better and better approximation for the tangent 00:22:40.040 --> 00:22:43.970 line, and that in a way, if the two points were allowed to 00:22:43.970 --> 00:22:47.120 coincide, that should give us the perfect answer. 00:22:47.120 --> 00:22:51.585 The trouble is, just like you can't divide 0 by 0, if P and 00:22:51.585 --> 00:22:54.560 Q coincide, how many points do you have? 00:22:54.560 --> 00:22:55.990 Just one point. 00:22:55.990 --> 00:22:59.550 And it takes two points to determine a straight line. 00:22:59.550 --> 00:23:03.140 No matter how close Q is to P, we have two distinct points. 00:23:03.140 --> 00:23:06.050 As soon as Q touches P, we lose this. 00:23:06.050 --> 00:23:10.060 And this is what was meant by ancient man or medieval man by 00:23:10.060 --> 00:23:12.530 his notion of two consecutive points. 00:23:12.530 --> 00:23:16.130 And I should put this in double quotes because I think 00:23:16.130 --> 00:23:19.940 you can see what he's begging to try to say with the word 00:23:19.940 --> 00:23:22.990 "consecutive," even though from a purely rigorous point 00:23:22.990 --> 00:23:26.280 of view, this has no geometric meaning. 00:23:26.280 --> 00:23:30.580 Now, the other form of limit has to do with adding up areas 00:23:30.580 --> 00:23:32.480 of rectangles under curves. 00:23:32.480 --> 00:23:35.370 Namely, we divided the curve up into n parts. 00:23:35.370 --> 00:23:38.600 We inscribed n rectangles, and then we let n 00:23:38.600 --> 00:23:40.320 increase without bound. 00:23:40.320 --> 00:23:44.700 In other words, this is sort of a discrete type of limit. 00:23:44.700 --> 00:23:49.190 Namely, we must add up a whole number of areas, but the sum 00:23:49.190 --> 00:23:52.590 is endless in the sense that the number of rectangles 00:23:52.590 --> 00:23:56.390 becomes greater than any number we want to preassign. 00:23:56.390 --> 00:24:02.350 And the basic question that we must contend with here is how 00:24:02.350 --> 00:24:04.230 big is an infinite sum? 00:24:04.230 --> 00:24:07.060 You see, when we say infinite sum, that just tells you how 00:24:07.060 --> 00:24:08.540 many terms you're combining. 00:24:08.540 --> 00:24:11.800 It doesn't tell you how big each term, how big 00:24:11.800 --> 00:24:12.930 the sum will be. 00:24:12.930 --> 00:24:15.410 For example, look at the following sum. 00:24:15.410 --> 00:24:17.310 I will start with 1. 00:24:17.310 --> 00:24:19.640 Then I'll add 1/2 on twice. 00:24:19.640 --> 00:24:23.040 Then I'll add 1/3 on three times. 00:24:23.040 --> 00:24:25.990 And without belaboring this point, let me then say I'll 00:24:25.990 --> 00:24:30.960 had on 1/4 four times, 1/5 five times, 1/6 00:24:30.960 --> 00:24:33.710 six times, et cetera. 00:24:33.710 --> 00:24:37.890 Notice as I do this that each time the terms gets smaller, 00:24:37.890 --> 00:24:41.190 yet the sum increases without any bound. 00:24:41.190 --> 00:24:43.980 Namely, notice that this adds up to 1. 00:24:43.980 --> 00:24:45.400 This adds up to 1. 00:24:45.400 --> 00:24:47.640 The next four terms will add up to 1. 00:24:47.640 --> 00:24:51.050 And as I go out further and further, notice that this sum 00:24:51.050 --> 00:24:53.470 can become as great is I want, just by me 00:24:53.470 --> 00:24:55.320 adding on enough 1's. 00:24:55.320 --> 00:24:57.850 On the other hand, let's look at this one. 00:24:57.850 --> 00:25:05.160 1 plus 1/2 plus 1/4 plus 1/8 plus 1/16 plus 1/32. 00:25:05.160 --> 00:25:09.300 In other words, I start with 1 and each time add on half the 00:25:09.300 --> 00:25:10.430 previous number. 00:25:10.430 --> 00:25:13.310 See, 1 plus 1/2 plus 1/4 plus 1/8. 00:25:13.310 --> 00:25:17.320 You may remember this as being the geometric series whose 00:25:17.320 --> 00:25:20.240 ratio is 1/2. 00:25:20.240 --> 00:25:25.140 The interesting thing is that now this sum gets as close to 00:25:25.140 --> 00:25:28.230 2 as you want without ever getting there. 00:25:28.230 --> 00:25:30.790 And rather than prove this right now, let's just look at 00:25:30.790 --> 00:25:33.690 the geometric interpretation here. 00:25:33.690 --> 00:25:37.130 Take a line which is 2 inches long. 00:25:37.130 --> 00:25:39.290 Suppose you first go halfway. 00:25:39.290 --> 00:25:40.500 You're now here. 00:25:40.500 --> 00:25:42.620 Now go half the remaining distance. 00:25:42.620 --> 00:25:43.080 That's what? 00:25:43.080 --> 00:25:44.080 1 plus 1/2. 00:25:44.080 --> 00:25:45.600 That puts you over here. 00:25:45.600 --> 00:25:47.860 Now go half the remaining distance. 00:25:47.860 --> 00:25:49.990 That means add on 1/4. 00:25:49.990 --> 00:25:51.900 Now go half the remaining distance. 00:25:51.900 --> 00:25:53.580 That means add on on 1/8. 00:25:53.580 --> 00:25:55.420 Now go half the remaining distance. 00:25:55.420 --> 00:25:57.720 Add up this on 1/16, you see. 00:25:57.720 --> 00:25:59.260 And ultimately, what happens? 00:25:59.260 --> 00:26:02.150 Well, no matter where you stop, you've become closer and 00:26:02.150 --> 00:26:04.570 closer to 2 without ever getting there. 00:26:04.570 --> 00:26:06.970 And as you go further and further, you can get as close 00:26:06.970 --> 00:26:08.500 to 2 as you want. 00:26:08.500 --> 00:26:11.550 In other words, here are infinitely many terms whose 00:26:11.550 --> 00:26:13.450 infinite sum is 2. 00:26:13.450 --> 00:26:18.020 Here are infinitely many terms whose infinite sum is 00:26:18.020 --> 00:26:19.410 infinity, we should say, because it 00:26:19.410 --> 00:26:20.790 increases without bound. 00:26:20.790 --> 00:26:23.760 And this was the problem that hung up the Ancient Greek. 00:26:23.760 --> 00:26:26.370 How could you do infinitely many things in a 00:26:26.370 --> 00:26:27.830 finite amount of time? 00:26:27.830 --> 00:26:31.440 In fact, at the same time that the Greek was developing 00:26:31.440 --> 00:26:36.020 integral calculus, the famous greek philosopher Zeno was 00:26:36.020 --> 00:26:39.190 working on things called Zeno's paradoxes. 00:26:39.190 --> 00:26:42.780 And Zeno's paradoxes are three in number, of which I only 00:26:42.780 --> 00:26:44.340 want to quote one here. 00:26:44.340 --> 00:26:47.720 But it's a paradox which shows how Zeno could not visualize 00:26:47.720 --> 00:26:49.500 quite what was happening. 00:26:49.500 --> 00:26:52.810 You see, it's called the Tortoise and the Hare problem. 00:26:52.810 --> 00:26:56.870 Suppose that you give the Tortoise a 1 yard head start 00:26:56.870 --> 00:26:58.360 on the Hare. 00:26:58.360 --> 00:27:00.920 And suppose for the sake of argument, just to mimic the 00:27:00.920 --> 00:27:03.560 problem that we were doing before, suppose it's a slow 00:27:03.560 --> 00:27:06.840 Hare and a fast Tortoise so that the Hare only runs twice 00:27:06.840 --> 00:27:08.660 as fast as the Tortoise. 00:27:08.660 --> 00:27:11.680 You see, Zeno's paradox says that the Hare can never catch 00:27:11.680 --> 00:27:12.480 the Tortoise. 00:27:12.480 --> 00:27:13.360 Why? 00:27:13.360 --> 00:27:16.680 Because to catch the Tortoise, the Hare must first go the 1 00:27:16.680 --> 00:27:18.960 yard head start that the Tortoise had. 00:27:18.960 --> 00:27:22.010 Well, by the time the Hare gets here, the Tortoise has 00:27:22.010 --> 00:27:25.750 gone 1/2 yard because the Tortoise travels half as fast. 00:27:25.750 --> 00:27:27.650 Now, the Hare must make up the 1/2 yard. 00:27:27.650 --> 00:27:30.860 But while the Hare makes up the 1/2 yard, the Tortoise 00:27:30.860 --> 00:27:32.730 goes 1/4 of a yard. 00:27:32.730 --> 00:27:36.300 When the Hare makes up the 1/4 of a yard, the Tortoise goes 00:27:36.300 --> 00:27:37.470 1/8 of a yard. 00:27:37.470 --> 00:27:40.930 And so, Zeno argues, the Hare gets closer and closer to the 00:27:40.930 --> 00:27:43.830 Tortoise but can't catch him. 00:27:43.830 --> 00:27:46.140 And this, of course, is a rather strange thing because 00:27:46.140 --> 00:27:49.300 Zeno knew that the Tortoise would catch the Hare. 00:27:49.300 --> 00:27:50.940 That's it's called a paradox. 00:27:50.940 --> 00:27:54.000 A paradox means something which appears to be true yet 00:27:54.000 --> 00:27:56.010 is obviously false. 00:27:56.010 --> 00:27:59.380 Now, notice that we can resolve Zeno's paradox into 00:27:59.380 --> 00:28:01.370 the example we were just talking about. 00:28:01.370 --> 00:28:03.880 For the sake of argument, notice what's happening here 00:28:03.880 --> 00:28:04.760 with the time. 00:28:04.760 --> 00:28:07.160 For the sake of argument, let's suppose that the 00:28:07.160 --> 00:28:09.800 Tortoise travels at 1 yard per second. 00:28:09.800 --> 00:28:10.900 Then what you're saying is-- 00:28:10.900 --> 00:28:12.890 I mean, the Hare travels at 1 yard per second. 00:28:12.890 --> 00:28:15.460 What you're saying is it takes the Hare 1 00:28:15.460 --> 00:28:17.850 second to go this distance. 00:28:17.850 --> 00:28:22.190 Then it takes him 1/2 a second to go this distance, then 1/4 00:28:22.190 --> 00:28:24.760 of a second to go this distance. 00:28:24.760 --> 00:28:27.210 And what you're saying is that as he's gaining on the 00:28:27.210 --> 00:28:29.630 Tortoise, these are the time intervals which are 00:28:29.630 --> 00:28:30.910 transpiring. 00:28:30.910 --> 00:28:34.710 And this sum turns out to be 2. 00:28:34.710 --> 00:28:38.050 Now, of course, those of us who had eighth grade algebra 00:28:38.050 --> 00:28:40.690 know an easier way of solving this problem. 00:28:40.690 --> 00:28:44.285 We say lookit, let's solve this problem algebraically. 00:28:44.285 --> 00:28:49.390 Namely, we say give the Tortoise a 1 yard head start. 00:28:49.390 --> 00:28:55.170 Now call x the distance of a point at which the Hare 00:28:55.170 --> 00:28:56.780 catches the Tortoise. 00:28:56.780 --> 00:28:59.610 Now, the Hare is traveling 1 yard per second. 00:28:59.610 --> 00:29:05.380 The Tortoise is traveling 1/2 yard per second, OK? 00:29:05.380 --> 00:29:12.400 So if we take the distance traveled and divided by the 00:29:12.400 --> 00:29:16.890 rate, that should be the time. 00:29:16.890 --> 00:29:19.600 And since they both are at this point at the same time, 00:29:19.600 --> 00:29:24.800 we get what? x/1 equals x minus 1 divided by 1/2. 00:29:24.800 --> 00:29:28.410 And assuming as a prerequisite that we have had algebra, it 00:29:28.410 --> 00:29:32.270 follows almost trivially that x equals 2. 00:29:32.270 --> 00:29:36.790 In other words, what this says is, in reality, that the Hare 00:29:36.790 --> 00:29:39.540 will not overtake the Tortoise until he catches 00:29:39.540 --> 00:29:41.900 him, which is obvious. 00:29:41.900 --> 00:29:43.740 But what's not so obvious is what? 00:29:43.740 --> 00:29:46.270 That these infinitely many terms can add 00:29:46.270 --> 00:29:48.370 up to a finite sum. 00:29:48.370 --> 00:29:52.420 Well, at any rate, this complete the overview of what 00:29:52.420 --> 00:29:53.430 our course will be like. 00:29:53.430 --> 00:29:58.890 And to help you focus your attention on what our course 00:29:58.890 --> 00:30:03.450 really says, what we shall do computationally is this. 00:30:03.450 --> 00:30:06.610 In review, we shall start with functions, and functions 00:30:06.610 --> 00:30:09.800 involve the modern concept of sets because they're 00:30:09.800 --> 00:30:12.230 relationships between sets of objects. 00:30:12.230 --> 00:30:16.680 We'll talk about limits, derivatives, rate of change, 00:30:16.680 --> 00:30:18.980 integrals, area under curves. 00:30:18.980 --> 00:30:22.080 This will be our fundamental building block. 00:30:22.080 --> 00:30:25.230 Once this is done, these things will never change. 00:30:25.230 --> 00:30:28.540 But the remainder of our course will be to talk about 00:30:28.540 --> 00:30:31.470 applications, which is the name of the game as far as 00:30:31.470 --> 00:30:33.220 engineering is concerned. 00:30:33.220 --> 00:30:36.300 More elaborate functions, namely, how do we handle 00:30:36.300 --> 00:30:37.990 tougher relationships. 00:30:37.990 --> 00:30:41.130 Related to the tougher relationships will come more 00:30:41.130 --> 00:30:43.560 sophisticated techniques. 00:30:43.560 --> 00:30:46.900 And finally, we will conclude our course with the topic that 00:30:46.900 --> 00:30:50.020 we were just talking about: infinite series, how do we get 00:30:50.020 --> 00:30:53.690 a hold of what happens when you add up infinitely many 00:30:53.690 --> 00:30:56.060 things, each of which gets small. 00:30:56.060 --> 00:31:00.750 At any rate, that concludes our lecture for today. 00:31:00.750 --> 00:31:04.820 We will have a digression in the sense that the next few 00:31:04.820 --> 00:31:09.540 lessons will consist of sets, things that you can read about 00:31:09.540 --> 00:31:12.160 at your leisure in our supplementary notes. 00:31:12.160 --> 00:31:15.400 Learn to understand these because the concept of a set 00:31:15.400 --> 00:31:18.970 is the building block, the fundamental language of modern 00:31:18.970 --> 00:31:20.300 mathematics. 00:31:20.300 --> 00:31:24.470 And then we will return, once we have sets underway, to talk 00:31:24.470 --> 00:31:26.680 about functions. 00:31:26.680 --> 00:31:29.310 And then we will build gradually from there. 00:31:29.310 --> 00:31:31.840 Hopefully, when our course ends, we will have in slow 00:31:31.840 --> 00:31:35.310 motion gone through today's lesson. 00:31:35.310 --> 00:31:37.620 This completes our presentation for today. 00:31:37.620 --> 00:31:39.630 And until next time, goodbye. 00:31:45.570 --> 00:31:48.110 NARRATOR: Funding for the publication of this video is 00:31:48.110 --> 00:31:52.820 provided by the Gabriella and Paul Rosenbaum Foundation. 00:31:52.820 --> 00:31:56.990 Help OCW continue to provide free and open access to MIT 00:31:56.990 --> 00:32:01.190 courses by making a donation at ocw.mit.edu/donate.