WEBVTT 00:00:00.058 --> 00:00:05.248 Derivatives and integrals. One of them measures how fast something is changing. 00:00:05.352 --> 00:00:08.392 The other measures how much of something has piled up. 00:00:08.475 --> 00:00:11.935 Two completely different questions, and this whole lecture is about the fact 00:00:11.935 --> 00:00:15.709 that they are one question, asked from opposite ends. 00:00:17.012 --> 00:00:18.952 So let me start with a tank. 00:00:19.404 --> 00:00:21.167 Somebody has opened a tap, 00:00:21.250 --> 00:00:26.069 and I know the rate, at every instant, exactly how fast the water is arriving. 00:00:26.265 --> 00:00:28.098 The question is simple enough. 00:00:28.181 --> 00:00:30.051 How much water is in there? 00:00:31.354 --> 00:00:32.865 Let's draw the rate. 00:00:33.061 --> 00:00:34.778 Time runs along the bottom, 00:00:34.861 --> 00:00:38.980 and the height of this graph is how fast the water is arriving at that moment. 00:00:39.063 --> 00:00:42.663 This tap is being opened steadily, so the line climbs. 00:00:43.367 --> 00:00:47.269 So at time one, the water is arriving at one litre a second. 00:00:47.466 --> 00:00:51.704 Move out to time three, and it is arriving at three litres a second. 00:00:52.919 --> 00:00:56.844 Now hold that thought, and imagine a different tap for a moment. 00:00:57.040 --> 00:00:58.908 Suppose the rate were constant. 00:00:58.991 --> 00:01:03.183 Say two litres a second, flat, the whole way across. 00:01:04.779 --> 00:01:07.926 Held there for three seconds, the answer is easy. 00:01:08.239 --> 00:01:12.570 Two litres a second, for three seconds, is six litres. 00:01:12.767 --> 00:01:14.381 Rate times time. 00:01:14.578 --> 00:01:19.234 And on this picture, rate times time is the area of one rectangle. 00:01:19.543 --> 00:01:21.716 But our tap is not that tap. 00:01:22.028 --> 00:01:26.638 Our rate climbs the whole way, so there is no single rectangle to draw. 00:01:27.779 --> 00:01:30.718 So we chop the time up into slivers instead. 00:01:30.914 --> 00:01:33.722 Over a stretch short enough the rate barely changes, 00:01:33.805 --> 00:01:36.499 so each sliver is rate times time again. 00:01:36.696 --> 00:01:38.029 One thin rectangle. 00:01:38.112 --> 00:01:40.133 Then we add them all up. 00:01:40.616 --> 00:01:44.088 Now watch what happens as we use more and more of them. 00:01:44.401 --> 00:01:48.267 The staircase closes on the region underneath the curve, and in the limit 00:01:48.267 --> 00:01:52.366 the water that has arrived is exactly the area under the rate graph. 00:01:53.070 --> 00:01:55.683 So here is the object I care about. 00:01:55.879 --> 00:02:00.662 The amount of water that has arrived by time x is the area under the graph, 00:02:00.662 --> 00:02:02.649 from the start out to x. 00:02:02.752 --> 00:02:03.995 Give it a name. 00:02:04.099 --> 00:02:07.188 A of x, the accumulation function. 00:02:07.892 --> 00:02:10.818 And notice what kind of object A is. 00:02:10.922 --> 00:02:12.420 It is a function. 00:02:12.617 --> 00:02:17.726 You hand it a time, here, and it hands you back a number, the area up to there. 00:02:18.778 --> 00:02:21.670 Watch what happens when I slide that edge along. 00:02:21.774 --> 00:02:25.304 More of the region is swept in, and the number gets bigger. 00:02:25.756 --> 00:02:30.294 Every stopping point gives you one number, so this really is a function of x, 00:02:30.377 --> 00:02:32.677 built out of nothing but area. 00:02:33.729 --> 00:02:35.390 Hold on to that picture. 00:02:35.493 --> 00:02:38.603 The area under a rate graph is itself a function, 00:02:38.686 --> 00:02:42.007 and everything that follows comes from one question about it. 00:02:42.203 --> 00:02:45.978 So let me put that question on the board, in as many words. 00:02:46.774 --> 00:02:48.783 How fast does A grow? 00:02:48.980 --> 00:02:51.848 Answering that is the whole of the rest of this lecture. 00:02:52.954 --> 00:02:56.356 A moment ago we ended with the accumulation function. 00:02:56.460 --> 00:03:01.047 A of x is the area under the rate graph, from zero up to x. 00:03:01.150 --> 00:03:05.468 For this particular tap we can work that area out exactly, and pleasingly, 00:03:05.551 --> 00:03:08.013 we need no calculus at all to do it. 00:03:08.810 --> 00:03:10.749 First, one word about the letters, 00:03:10.749 --> 00:03:13.603 because there are two of them and they are not the same letter. 00:03:13.686 --> 00:03:15.867 Along the bottom the clock is called t, 00:03:15.950 --> 00:03:19.257 since t is what sweeps across the region while we shade it in. 00:03:19.340 --> 00:03:22.092 The letter x is saved for where we stop. 00:03:22.833 --> 00:03:25.353 So stop the clock at x. 00:03:25.968 --> 00:03:29.649 There is the mark, on the axis the clock runs along. 00:03:29.753 --> 00:03:32.795 And look at what is shaded: it is a triangle. 00:03:33.336 --> 00:03:36.852 Its base runs from zero out to x, so the base is x. 00:03:36.935 --> 00:03:43.159 Its height is the rate at time x, and for this tap the rate at time x is x as well. 00:03:43.770 --> 00:03:47.170 The area of a triangle is a half, base, height. 00:03:47.253 --> 00:03:50.772 Here that is a half times x times x. 00:03:50.968 --> 00:03:53.558 Which comes to one half x squared. 00:03:54.471 --> 00:03:55.676 There it is. 00:03:55.759 --> 00:03:59.708 The accumulation function for this tap, in closed form. 00:03:59.904 --> 00:04:04.398 Now let us put the two of them side by side, because this is where it gets good. 00:04:05.795 --> 00:04:07.641 On the left, the rate. 00:04:07.838 --> 00:04:10.161 A straight line, climbing steadily. 00:04:10.357 --> 00:04:12.692 On the right, the accumulation. 00:04:12.888 --> 00:04:15.258 A parabola, bending upward. 00:04:15.710 --> 00:04:19.808 As the tap opens further, the water piles up faster and faster. 00:04:20.860 --> 00:04:23.357 Now a question about the curve on the right. 00:04:23.461 --> 00:04:24.925 How steep is it? 00:04:25.121 --> 00:04:29.984 The steepness of A at a point is exactly what the symbol A prime of x will mean, 00:04:30.067 --> 00:04:34.433 so take the tangent line at x equals two and read off its slope. 00:04:35.346 --> 00:04:38.446 Rise over run, the slope there is two. 00:04:38.643 --> 00:04:40.757 Now look across at the rate graph. 00:04:40.860 --> 00:04:44.425 At time two, the height of the rate line is, two. 00:04:45.222 --> 00:04:46.477 Coincidence? 00:04:46.673 --> 00:04:50.145 Let's slide x along and watch both pictures at once. 00:04:51.342 --> 00:04:53.804 Carry on out to x equals three. 00:04:54.117 --> 00:04:57.575 The tangent on the accumulation curve now has slope three. 00:04:57.658 --> 00:05:00.956 And the rate line at time three is sitting at height three. 00:05:01.152 --> 00:05:05.530 The slope of the accumulation is the height of the rate, every single time. 00:05:05.843 --> 00:05:08.096 That deserves a line of its own. 00:05:09.399 --> 00:05:14.056 Two things to write down, and the first one is what the symbol is going to mean. 00:05:14.252 --> 00:05:19.777 A is a function of x, so A prime of x is its rate of change: 00:05:19.860 --> 00:05:23.727 how fast the shaded area grows as x moves off to the right. 00:05:24.431 --> 00:05:27.241 And the second is what we just watched. 00:05:27.554 --> 00:05:30.710 That rate of change is the height of the rate graph at x. 00:05:30.793 --> 00:05:34.369 So A prime of x equals f of x. 00:05:35.166 --> 00:05:38.682 The slope of the accumulation curve is the height of the rate curve, 00:05:38.765 --> 00:05:43.431 at every single point. We got there from one triangle and one parabola, 00:05:43.514 --> 00:05:45.325 but it has nothing to do with triangles, 00:05:45.325 --> 00:05:49.435 and I want to show you why it has to be true for absolutely any rate at all. 00:05:50.490 --> 00:05:54.377 That was one particular tap, and one particularly friendly shape. 00:05:54.460 --> 00:05:57.651 But the fact we landed on has nothing to do with triangles. 00:05:57.734 --> 00:06:00.101 So here is a rate that wanders. 00:06:00.184 --> 00:06:03.714 No nice formula, and no geometry that will help you. 00:06:04.711 --> 00:06:10.014 Same set-up as before. A of x is the area underneath it, from zero up to x. 00:06:10.097 --> 00:06:11.305 We stop the clock here, 00:06:11.305 --> 00:06:15.891 at x, and everything shaded to the left of that line is A of x. 00:06:16.833 --> 00:06:23.054 Now nudge it. Push x forward by a small amount, and give that amount a name: h. 00:06:23.137 --> 00:06:28.630 There is h, on the picture, the little step from x across to x plus h. 00:06:29.633 --> 00:06:31.524 The area grows by this sliver, 00:06:31.607 --> 00:06:35.854 the water that arrives between time x and time x plus h. 00:06:35.937 --> 00:06:38.876 Call that extra area delta A. 00:06:39.254 --> 00:06:40.496 And here is the whole trick, 00:06:40.496 --> 00:06:43.644 so let me put it on the screen rather than leave it in the air. 00:06:43.747 --> 00:06:46.349 That sliver is very nearly a rectangle. 00:06:47.961 --> 00:06:51.643 There it is, standing on the sliver it is pretending to be. 00:06:51.839 --> 00:06:55.137 Its width is h, the step we just marked. 00:06:55.334 --> 00:07:00.209 Its top sits level with the curve at x, at a height of f of x, 00:07:00.292 --> 00:07:04.321 because over a stretch that short the curve has not had time to change. 00:07:04.665 --> 00:07:08.123 So delta A is about f of x times h. 00:07:08.206 --> 00:07:10.538 Now divide both sides by h, 00:07:10.621 --> 00:07:14.253 and the left-hand side turns into something you have met before. 00:07:14.336 --> 00:07:17.690 Change in A, divided by change in x. 00:07:17.773 --> 00:07:19.968 The average rate of change of A. 00:07:20.346 --> 00:07:22.318 And now let h shrink. 00:07:22.401 --> 00:07:25.978 Watch the sliver, and watch the rectangle standing on it. 00:07:26.082 --> 00:07:29.691 The thinner they both get, the better the rectangle fits, 00:07:29.774 --> 00:07:32.990 and the closer those two sides come to being equal. 00:07:34.387 --> 00:07:38.193 In the limit, the left-hand side is the derivative of A at x. 00:07:38.276 --> 00:07:41.574 That is precisely the definition of a derivative. 00:07:41.678 --> 00:07:44.314 And the right-hand side never moved. 00:07:44.511 --> 00:07:49.202 So A prime of x equals f of x, for any rate you like. 00:07:50.115 --> 00:07:53.201 Said out loud, it is almost a tautology. 00:07:53.284 --> 00:07:55.920 How fast is the water in the tank going up? 00:07:56.372 --> 00:07:58.263 At the rate the water is coming in. 00:07:58.346 --> 00:08:00.402 That is the whole theorem. 00:08:01.016 --> 00:08:05.905 The surprise is not that it is true, the surprise is what it lets you do. 00:08:06.966 --> 00:08:11.472 Before we cash this in, we need the notation everybody actually writes. 00:08:11.924 --> 00:08:16.162 The area under f, from a up to x, has a symbol of its own. 00:08:17.075 --> 00:08:19.247 A long, stretched S. 00:08:19.351 --> 00:08:22.416 The function goes in the middle, and dee t closes it off. 00:08:22.868 --> 00:08:28.372 Out loud: the integral of f of t, dee t, from a to x. 00:08:29.285 --> 00:08:32.685 And let me say what those two ends are, because nobody ever does. 00:08:32.768 --> 00:08:35.532 The little a at the bottom is where we start counting. 00:08:35.635 --> 00:08:37.549 It is the left-hand edge of the region, 00:08:37.632 --> 00:08:40.814 chosen once and then left exactly where it is. 00:08:41.811 --> 00:08:44.247 The x on the top is where we stop. 00:08:44.330 --> 00:08:46.790 So the whole symbol is a function of where you stop, 00:08:46.873 --> 00:08:49.648 which is exactly what the shaded area was. 00:08:50.352 --> 00:08:54.207 And the S is for sum, because that is precisely what it is. 00:08:54.311 --> 00:08:59.699 All those thin rectangles, added up, with dee t as the width of one of them. 00:09:00.751 --> 00:09:04.836 So in this language, what we proved with the thin sliver reads like this. 00:09:04.919 --> 00:09:08.052 Differentiate an integral with respect to its upper limit, 00:09:08.135 --> 00:09:10.283 and you get the integrand straight back. 00:09:11.680 --> 00:09:16.462 Now the payoff. Here is a rate again, and here is the job in front of us: 00:09:16.545 --> 00:09:19.808 the area underneath it, between two fixed times. 00:09:20.511 --> 00:09:22.483 The left-hand end we call a. 00:09:22.566 --> 00:09:24.529 That is where the counting starts. 00:09:24.633 --> 00:09:28.245 The right-hand end we call b, and that is where it stops. 00:09:28.557 --> 00:09:32.561 Everything shaded between those two lines is the area we are trying to find, 00:09:32.644 --> 00:09:35.362 and we have no formula for a shape like that. 00:09:36.465 --> 00:09:40.980 Now suppose somebody hands you a function big F whose derivative is little f. 00:09:41.063 --> 00:09:43.432 Any such function at all will do. 00:09:43.606 --> 00:09:48.425 There it is, written down: F prime of x equals f of x. 00:09:49.198 --> 00:09:51.892 And we already have one function like that. 00:09:51.984 --> 00:09:58.429 Our own area function, A. Last time we proved that A prime of x is f of x too. 00:09:59.202 --> 00:10:05.226 So big F and our area function A have exactly the same derivative, everywhere, 00:10:05.309 --> 00:10:07.365 whatever big F happens to be. 00:10:08.562 --> 00:10:10.185 Which raises a fair question. 00:10:10.268 --> 00:10:13.959 How different can two functions with the same derivative actually be? 00:10:14.042 --> 00:10:15.782 Let me draw one of them over here. 00:10:15.865 --> 00:10:17.035 This is big F, 00:10:17.118 --> 00:10:20.649 and its slope at every point is the height of the curve beside it. 00:10:21.445 --> 00:10:24.859 Now take a copy of it, and lift the whole copy upward. 00:10:25.172 --> 00:10:29.434 Every point goes up by the same amount, so the curve keeps its shape exactly. 00:10:29.630 --> 00:10:32.554 At every x it is still leaning the way it was leaning. 00:10:32.637 --> 00:10:34.890 Its slope has not changed anywhere. 00:10:35.594 --> 00:10:38.695 Push a copy downward and you get the same story again. 00:10:38.891 --> 00:10:43.152 All three of these are antiderivatives of the same little f. There is a whole family 00:10:43.152 --> 00:10:47.379 of them stacked up the board, and every one of them is parallel to the others. 00:10:48.776 --> 00:10:51.212 And here is where the constant comes from. 00:10:51.295 --> 00:10:54.428 Measure the gap between two of them at x equals a. 00:10:54.511 --> 00:10:58.747 Now measure it again over at x equals b. It is the same gap. 00:10:58.830 --> 00:11:03.451 It is the same gap everywhere, and that gap is the number we call C. 00:11:03.876 --> 00:11:07.845 So two antiderivatives of one function differ by a constant, 00:11:07.928 --> 00:11:09.903 and only by a constant. 00:11:10.006 --> 00:11:15.415 Big F of x is our area function A of x, plus some fixed number C. 00:11:15.498 --> 00:11:18.877 Now watch what that does the moment we subtract. 00:11:20.483 --> 00:11:22.863 Start from the line we just earned. 00:11:23.025 --> 00:11:26.126 Big F of x is A of x, plus C. 00:11:27.197 --> 00:11:31.747 That is true at every single x, so in particular it is true at b. 00:11:31.830 --> 00:11:34.872 Big F of b is A of b, plus C. 00:11:35.586 --> 00:11:37.708 And it is just as true at a. 00:11:37.791 --> 00:11:40.855 Big F of a is A of a, plus C. 00:11:40.938 --> 00:11:46.209 The very same C both times, because it is the very same big F both times. 00:11:47.306 --> 00:11:51.509 Now take the second of those two lines away from the first, exactly the way you 00:11:51.509 --> 00:11:55.932 would take one number off another. So rule a line underneath them both. 00:11:57.274 --> 00:11:58.642 And there is the whole trick, 00:11:58.725 --> 00:12:01.440 standing one above the other at the end of those two lines. 00:12:01.523 --> 00:12:04.226 Plus C on the top, plus C underneath. 00:12:04.309 --> 00:12:07.826 The same unknown number added on and then taken straight back off. 00:12:07.909 --> 00:12:12.553 So slash the pair of them out, and whatever C was, it has gone. 00:12:14.350 --> 00:12:17.878 So what survives on the right is A of b minus A of a. 00:12:17.961 --> 00:12:20.803 The constant has gone, and it was always going to go, 00:12:20.886 --> 00:12:24.080 whichever antiderivative you had picked up in the first place. 00:12:25.084 --> 00:12:31.003 And A of b minus A of a is exactly the area under f between a and b. 00:12:31.086 --> 00:12:33.859 Which, in the notation we built at the start of this scene, 00:12:33.942 --> 00:12:36.753 is the integral of f from a to b. 00:12:36.949 --> 00:12:38.515 So there is the bottom line. 00:12:38.598 --> 00:12:41.664 Let me write it out properly, because that is the theorem. 00:12:43.060 --> 00:12:45.079 So there it is, in a single line. 00:12:45.162 --> 00:12:50.652 The integral of f from a to b equals big F of b, minus big F of a, 00:12:50.735 --> 00:12:53.789 where big F is any antiderivative of f. 00:12:54.272 --> 00:12:58.022 That is the Fundamental Theorem of Calculus, and it is the reason 00:12:58.022 --> 00:13:01.146 integration is something you can actually sit down and do. 00:13:01.342 --> 00:13:04.501 To find an area, you no longer add up rectangles. 00:13:04.814 --> 00:13:07.749 You find a function whose derivative is the one you started with, 00:13:07.832 --> 00:13:09.772 and subtract at the two ends. 00:13:10.839 --> 00:13:13.891 Let's spend it on a problem you cannot do with geometry. 00:13:13.974 --> 00:13:18.512 Here is y equals x squared, and here is the piece of it we want: everything 00:13:18.595 --> 00:13:21.010 under the curve between zero and two. 00:13:21.551 --> 00:13:25.523 The old route uses thin bars like these, then takes a limit. 00:13:25.627 --> 00:13:29.340 That is real work, because this is not a triangle, a rectangle, 00:13:29.423 --> 00:13:32.570 a circle, or any other shape with a school formula. 00:13:33.367 --> 00:13:36.247 By the new route you need exactly one thing. 00:13:36.350 --> 00:13:39.230 Some function whose derivative is x squared. 00:13:39.427 --> 00:13:41.158 So let us go and find one. 00:13:42.410 --> 00:13:43.873 Start by guessing. 00:13:43.977 --> 00:13:47.948 Differentiate x cubed, and you get three x squared. 00:13:48.145 --> 00:13:50.955 Write that down, because it is nearly right. 00:13:51.752 --> 00:13:53.680 It is three times too big. 00:13:53.877 --> 00:13:57.302 So divide the whole thing by three before you differentiate it. 00:13:57.499 --> 00:14:00.690 Now the three on top and the three underneath are the same three, 00:14:00.773 --> 00:14:03.537 and what comes out is exactly x squared. 00:14:04.148 --> 00:14:05.748 So there is our big F. 00:14:05.831 --> 00:14:08.302 F of x is x cubed over three, 00:14:08.385 --> 00:14:11.428 and its derivative is the x squared we started with. 00:14:12.541 --> 00:14:17.218 Now the theorem does the rest. The integral from zero to two of x squared, 00:14:17.301 --> 00:14:21.121 dee x, is F at two minus F at zero. 00:14:21.918 --> 00:14:24.774 F at two is two cubed over three. 00:14:24.971 --> 00:14:26.931 F at zero is zero. 00:14:27.014 --> 00:14:29.244 Take the second one away from the first. 00:14:29.518 --> 00:14:33.185 And what is left is eight thirds, which is about two point six seven. 00:14:33.268 --> 00:14:35.902 That is the exact area under the parabola, 00:14:35.985 --> 00:14:38.305 and we got it by differentiating backwards. 00:14:38.388 --> 00:14:41.535 No rectangles, no limits, three lines of algebra. 00:14:42.076 --> 00:14:43.993 Does that number look sensible? 00:14:44.189 --> 00:14:46.964 The region sits inside this two by four rectangle, 00:14:46.964 --> 00:14:51.304 which has area eight, and by eye the shaded part is about a third of it. 00:14:51.387 --> 00:14:53.454 Two point six seven out of eight. 00:14:53.651 --> 00:14:55.080 That checks out. 00:14:56.384 --> 00:14:59.217 So here are the two halves of one idea. 00:14:59.414 --> 00:15:00.352 On the left, 00:15:00.435 --> 00:15:05.138 differentiate an accumulation, and you get back the rate you were accumulating at. 00:15:05.335 --> 00:15:08.561 On the right, to total up a rate across an interval, 00:15:08.644 --> 00:15:12.754 find something that differentiates to it, and subtract at the two ends. 00:15:13.969 --> 00:15:17.845 Area and slope are not two separate subjects that happen to share a course. 00:15:17.928 --> 00:15:20.735 They are one machine, run in the two directions. 00:15:20.818 --> 00:15:23.362 That is the Fundamental Theorem of Calculus.