WEBVTT 00:00:00.000 --> 00:00:05.928 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:05.928 --> 00:00:21.282 align:middle line:90% 00:00:21.282 --> 00:00:23.240 align:middle line:84% CASEY RODRIGUEZ: --differentiable as many times 00:00:23.240 --> 00:00:23.765 align:middle line:90% as you like. 00:00:23.765 --> 00:00:28.920 align:middle line:90% 00:00:28.920 --> 00:00:36.846 align:middle line:84% And the derivative at 0 equals 0 for all n. 00:00:36.846 --> 00:00:39.430 align:middle line:90% 00:00:39.430 --> 00:00:40.640 align:middle line:90% Why do I bring this up? 00:00:40.640 --> 00:00:48.270 align:middle line:84% Because then the Taylor polynomial for this function 00:00:48.270 --> 00:00:52.180 align:middle line:90% that I've written here at 0-- 00:00:52.180 --> 00:00:56.020 align:middle line:84% so this is the Taylor polynomial at 0-- 00:00:56.020 --> 00:01:00.100 align:middle line:84% just equals, again, the sum of the derivatives evaluated 00:01:00.100 --> 00:01:03.610 align:middle line:90% at 0 times x minus 0 to the k. 00:01:03.610 --> 00:01:05.155 align:middle line:84% But all of the derivatives are 0. 00:01:05.155 --> 00:01:10.980 align:middle line:90% 00:01:10.980 --> 00:01:14.190 align:middle line:84% And thus, the function is, in fact, 00:01:14.190 --> 00:01:18.180 align:middle line:84% equal to the remainder term near 0. 00:01:18.180 --> 00:01:21.053 align:middle line:84% So you see, the remainder term carries all of f, 00:01:21.053 --> 00:01:22.845 align:middle line:84% so it doesn't necessarily need to be small. 00:01:22.845 --> 00:01:26.630 align:middle line:90% 00:01:26.630 --> 00:01:29.990 align:middle line:84% And what I'm trying to say is that in general, you can't just 00:01:29.990 --> 00:01:32.420 align:middle line:84% throw away the remainder term and expect 00:01:32.420 --> 00:01:37.145 align:middle line:84% that to be even near the point x, some sort 00:01:37.145 --> 00:01:42.730 align:middle line:84% of faithful representation of the function 00:01:42.730 --> 00:01:44.170 align:middle line:90% just by the Taylor polynomial. 00:01:44.170 --> 00:01:45.820 align:middle line:84% Because as we see for this function, 00:01:45.820 --> 00:01:48.010 align:middle line:84% the Taylor polynomial is identically 0. 00:01:48.010 --> 00:01:51.280 align:middle line:84% If I throw away the remainder term, I would be saying f is 0, 00:01:51.280 --> 00:01:53.485 align:middle line:84% but it most certainly is not near x equals 0. 00:01:53.485 --> 00:01:57.310 align:middle line:90% 00:01:57.310 --> 00:02:04.030 align:middle line:84% So that was the point of that discussion. 00:02:04.030 --> 00:02:05.425 align:middle line:90% Now, let's give the proof. 00:02:05.425 --> 00:02:10.913 align:middle line:90% 00:02:10.913 --> 00:02:13.580 align:middle line:84% And as you see, we're just going to kind of apply the mean value 00:02:13.580 --> 00:02:18.860 align:middle line:84% theorem repeatedly to higher derivatives of f, 00:02:18.860 --> 00:02:22.340 align:middle line:84% but not necessarily of f, but of a function we cook up out of f. 00:02:22.340 --> 00:02:30.720 align:middle line:90% 00:02:30.720 --> 00:02:35.010 align:middle line:84% Let's take two points, x0 and x, not equal to each other. 00:02:35.010 --> 00:02:37.230 align:middle line:84% Of course, if they're equal to each other, 00:02:37.230 --> 00:02:42.030 align:middle line:84% we can take C to be whatever we want, because then, f of-- 00:02:42.030 --> 00:02:43.920 align:middle line:84% because then what we pick up is f of x 00:02:43.920 --> 00:02:45.760 align:middle line:84% equals f of x on the right-hand side. 00:02:45.760 --> 00:02:50.070 align:middle line:84% So we can just consider the case that x0 does not equal x. 00:02:50.070 --> 00:02:54.076 align:middle line:90% 00:02:54.076 --> 00:02:58.680 align:middle line:90% And let M be-- 00:02:58.680 --> 00:03:11.090 align:middle line:84% this is just a number depending on x and f0 over x minus x0 00:03:11.090 --> 00:03:12.230 align:middle line:90% to the n plus 1. 00:03:12.230 --> 00:03:17.020 align:middle line:90% 00:03:17.020 --> 00:03:21.010 align:middle line:84% P sub n This is, again, the Taylor polynomial of degree n. 00:03:21.010 --> 00:03:24.220 align:middle line:84% f of x minus P sub n of x over x minus x sub 0. 00:03:24.220 --> 00:03:26.470 align:middle line:84% So this is just a number depending on x and x0. 00:03:26.470 --> 00:03:29.330 align:middle line:90% 00:03:29.330 --> 00:03:35.240 align:middle line:84% Then just rewriting this, this means that f of x 00:03:35.240 --> 00:03:39.050 align:middle line:84% is equal to P sub n of x, the Taylor polynomial of n 00:03:39.050 --> 00:03:41.930 align:middle line:90% at x, plus Mx times-- 00:03:41.930 --> 00:03:48.710 align:middle line:84% Mx x0 times x minus x sub 0 over n to the n plus 1. 00:03:48.710 --> 00:03:52.800 align:middle line:84% Now, the goal is to show that, in fact, this number can 00:03:52.800 --> 00:03:57.360 align:middle line:84% be written as the n plus 1 derivative evaluated 00:03:57.360 --> 00:03:59.730 align:middle line:84% at some point over n plus 1 factorial. 00:03:59.730 --> 00:04:04.410 align:middle line:90% 00:04:04.410 --> 00:04:15.260 align:middle line:84% Now, the goal-- show there exists a C and a B 00:04:15.260 --> 00:04:25.250 align:middle line:84% such that Mx equals f n plus 1 over C over n plus 1 factorial. 00:04:25.250 --> 00:04:32.310 align:middle line:90% 00:04:32.310 --> 00:04:37.480 align:middle line:84% Now, what is this defining characteristic 00:04:37.480 --> 00:04:40.540 align:middle line:90% of this Taylor polynomial at-- 00:04:40.540 --> 00:04:43.030 align:middle line:84% this n-th order Taylor polynomial? 00:04:43.030 --> 00:04:47.170 align:middle line:90% Evaluate with respect to x0. 00:04:47.170 --> 00:04:57.450 align:middle line:84% Well, the point is that this Taylor polynomial 00:04:57.450 --> 00:05:03.840 align:middle line:84% agrees with f at x0 up to n-th order, up to n derivatives. 00:05:03.840 --> 00:05:10.420 align:middle line:84% In other words, if I take the k-th derivative of f 00:05:10.420 --> 00:05:13.240 align:middle line:84% and evaluate it at 0, this is the same 00:05:13.240 --> 00:05:17.980 align:middle line:84% as taking the k-th derivative of the Taylor polynomial 00:05:17.980 --> 00:05:21.010 align:middle line:90% and evaluating it at 0. 00:05:21.010 --> 00:05:25.390 align:middle line:84% So the Taylor polynomial agrees with f up to n derivatives 00:05:25.390 --> 00:05:28.840 align:middle line:90% at the point x0. 00:05:28.840 --> 00:05:32.500 align:middle line:84% Again, this is the whole point of Taylor polynomials, 00:05:32.500 --> 00:05:34.930 align:middle line:84% is that they, at least at the point, 00:05:34.930 --> 00:05:41.330 align:middle line:90% agree with f up to n-th order. 00:05:41.330 --> 00:05:43.460 align:middle line:84% Does it mean they agree with f, or even 00:05:43.460 --> 00:05:47.910 align:middle line:84% are a good representation of f, away from x0 like we just saw? 00:05:47.910 --> 00:05:51.890 align:middle line:84% But at least at x0, they agree with f. 00:05:51.890 --> 00:05:59.360 align:middle line:90% 00:05:59.360 --> 00:06:02.720 align:middle line:84% Now, I'm going to define a new function, which 00:06:02.720 --> 00:06:06.442 align:middle line:84% I'm going to start applying the mean value theorem to, 00:06:06.442 --> 00:06:16.510 align:middle line:84% and hopefully come up with this C. It's g of s 00:06:16.510 --> 00:06:20.410 align:middle line:84% equals f of s minus P sub n of s minus this number 00:06:20.410 --> 00:06:24.760 align:middle line:84% from earlier times s minus 0 to the n plus 1. 00:06:24.760 --> 00:06:27.350 align:middle line:90% 00:06:27.350 --> 00:06:32.270 align:middle line:84% And something to note is that this function here, 00:06:32.270 --> 00:06:38.420 align:middle line:90% this whole function, so g-- 00:06:38.420 --> 00:06:41.330 align:middle line:84% first off, f is n plus 1 times differentiable. 00:06:41.330 --> 00:06:44.700 align:middle line:84% This is a polynomial, so it's n plus 1 times differentiable. 00:06:44.700 --> 00:06:46.980 align:middle line:84% And this is just a polynomial, also. 00:06:46.980 --> 00:06:47.900 align:middle line:90% And s. 00:06:47.900 --> 00:06:52.431 align:middle line:84% So it's n plus 1 times differentiable. 00:06:52.431 --> 00:06:57.250 align:middle line:90% 00:06:57.250 --> 00:06:58.120 align:middle line:90% So ns. 00:06:58.120 --> 00:07:05.400 align:middle line:90% 00:07:05.400 --> 00:07:07.755 align:middle line:90% Let me draw a picture. 00:07:07.755 --> 00:07:12.640 align:middle line:90% 00:07:12.640 --> 00:07:14.440 align:middle line:90% We have x0 and x. 00:07:14.440 --> 00:07:16.540 align:middle line:84% At least in the picture, x is bigger than x0, 00:07:16.540 --> 00:07:19.690 align:middle line:90% but that doesn't really matter. 00:07:19.690 --> 00:07:23.350 align:middle line:90% What do we know about g of x0? 00:07:23.350 --> 00:07:25.270 align:middle line:90% Well, this is equal to f. 00:07:25.270 --> 00:07:29.880 align:middle line:90% 00:07:29.880 --> 00:07:34.260 align:middle line:84% And now, when I stick in x0 here, I get 0. 00:07:34.260 --> 00:07:36.950 align:middle line:90% 00:07:36.950 --> 00:07:42.350 align:middle line:84% And now f of x0 minus P n of x0, again, by this first thing here 00:07:42.350 --> 00:07:43.430 align:middle line:90% for k equals 0. 00:07:43.430 --> 00:07:44.345 align:middle line:90% This is equal to 0. 00:07:44.345 --> 00:07:48.600 align:middle line:90% 00:07:48.600 --> 00:07:51.435 align:middle line:84% And now, what do I know about g evaluated at x? 00:07:51.435 --> 00:07:56.080 align:middle line:90% 00:07:56.080 --> 00:07:59.590 align:middle line:84% This is equal to f of x minus P n of x. 00:07:59.590 --> 00:08:05.890 align:middle line:84% Remember, the variable that I'm changing-- or at least, 00:08:05.890 --> 00:08:07.390 align:middle line:90% the free variable there, is s. 00:08:07.390 --> 00:08:11.740 align:middle line:84% So if I stick in x, I get f of x minus P n of x 00:08:11.740 --> 00:08:18.170 align:middle line:84% minus M, this constant from earlier, which 00:08:18.170 --> 00:08:20.630 align:middle line:90% I chose depending on x and x0. 00:08:20.630 --> 00:08:23.555 align:middle line:84% But using this relation here, this is 0. 00:08:23.555 --> 00:08:27.680 align:middle line:90% 00:08:27.680 --> 00:08:34.820 align:middle line:84% I have that-- the function f at x0 and at x is 0. 00:08:34.820 --> 00:08:37.414 align:middle line:84% By the mean value theorem, or Rolle's theorem-- 00:08:37.414 --> 00:08:40.780 align:middle line:90% 00:08:40.780 --> 00:08:53.860 align:middle line:84% so by mean value theorem, there exists a point x1 between x0, 00:08:53.860 --> 00:09:00.640 align:middle line:84% x such that g prime of x1 equals 0. 00:09:00.640 --> 00:09:01.140 align:middle line:90% Yeah? 00:09:01.140 --> 00:09:04.430 align:middle line:90% 00:09:04.430 --> 00:09:06.440 align:middle line:90% Now, remember, at x0-- 00:09:06.440 --> 00:09:08.300 align:middle line:90% or g prime of x1 equals 0. 00:09:08.300 --> 00:09:13.500 align:middle line:90% Now, at x0, we have that-- 00:09:13.500 --> 00:09:14.000 align:middle line:90% OK. 00:09:14.000 --> 00:09:22.140 align:middle line:84% So at g prime of x1, at x1, g prime is 0. 00:09:22.140 --> 00:09:30.420 align:middle line:84% But also, if I look at the derivative of g at x0, 00:09:30.420 --> 00:09:37.950 align:middle line:84% this is equal to f prime of x0 minus P n prime of x0 minus-- 00:09:37.950 --> 00:09:40.620 align:middle line:84% now, here, I'm working under the assumption, 00:09:40.620 --> 00:09:43.620 align:middle line:84% just for illustration purposes, I'm 00:09:43.620 --> 00:09:50.133 align:middle line:84% assuming n is, say, bigger than or equal to 2, 00:09:50.133 --> 00:09:52.050 align:middle line:84% at least from what I'm writing down right now. 00:09:52.050 --> 00:09:57.210 align:middle line:84% But if I take the derivative of this and plug in x0, 00:09:57.210 --> 00:09:59.590 align:middle line:90% then I will also get 0 here. 00:09:59.590 --> 00:10:03.630 align:middle line:84% So I just get f prime of x0 minus P n prime of x0. 00:10:03.630 --> 00:10:05.640 align:middle line:90% So this equals 0. 00:10:05.640 --> 00:10:17.190 align:middle line:84% So I have g prime of x1 equals 0, g prime of x0 equals 0, 00:10:17.190 --> 00:10:24.610 align:middle line:84% and therefore, by the mean value theorem applied again, 00:10:24.610 --> 00:10:36.570 align:middle line:84% there exists a point x2 between x and x0 such 00:10:36.570 --> 00:10:48.600 align:middle line:84% that the second derivative of g evaluated at x2 equals 0. 00:10:48.600 --> 00:10:50.370 align:middle line:90% And now, I just iterate this. 00:10:50.370 --> 00:10:55.740 align:middle line:84% Because I still know, at x0, the second derivative of g 00:10:55.740 --> 00:11:04.020 align:middle line:84% is also 0 as long as n is bigger than or equal to 2. 00:11:04.020 --> 00:11:09.500 align:middle line:84% And then I'll get that there's x1 here-- 00:11:09.500 --> 00:11:13.900 align:middle line:90% 00:11:13.900 --> 00:11:17.410 align:middle line:84% let me write here, at this point, we know g equals 0, 00:11:17.410 --> 00:11:20.290 align:middle line:90% g prime equals 0, and so on. 00:11:20.290 --> 00:11:23.380 align:middle line:90% At this point, g prime equals 0. 00:11:23.380 --> 00:11:24.265 align:middle line:90% And then at x2-- 00:11:24.265 --> 00:11:27.830 align:middle line:90% 00:11:27.830 --> 00:11:31.580 align:middle line:84% then the fact that g double prime here at this point is 0. 00:11:31.580 --> 00:11:35.680 align:middle line:90% 00:11:35.680 --> 00:11:38.240 align:middle line:84% And here, we apply the mean value theorem again. 00:11:38.240 --> 00:11:40.430 align:middle line:84% And we get a point, x3, in between them 00:11:40.430 --> 00:11:42.490 align:middle line:84% where, now, the third derivative equals 0. 00:11:42.490 --> 00:11:46.690 align:middle line:84% And we can keep going on, up until a certain point. 00:11:46.690 --> 00:11:47.710 align:middle line:90% And what point is that? 00:11:47.710 --> 00:11:50.630 align:middle line:84% That's when I've taken away n derivatives here, 00:11:50.630 --> 00:11:52.710 align:middle line:84% and all that I have left here is s minus x0. 00:11:52.710 --> 00:11:55.720 align:middle line:90% 00:11:55.720 --> 00:11:57.250 align:middle line:90% Let me summarize. 00:11:57.250 --> 00:12:01.780 align:middle line:90% 00:12:01.780 --> 00:12:12.545 align:middle line:84% Continuing in this way, we see there exists-- 00:12:12.545 --> 00:12:15.540 align:middle line:90% 00:12:15.540 --> 00:12:23.980 align:middle line:84% we see for all k between 0 and n, 00:12:23.980 --> 00:12:35.430 align:middle line:84% there exists an x sub k between x0 and x such 00:12:35.430 --> 00:12:43.655 align:middle line:84% that the k-th derivative at x sub k equals 0. 00:12:43.655 --> 00:12:52.870 align:middle line:90% 00:12:52.870 --> 00:12:57.730 align:middle line:84% In particular, at the k equals n stage, what do I have? 00:12:57.730 --> 00:13:03.380 align:middle line:90% 00:13:03.380 --> 00:13:10.740 align:middle line:90% x0, x, and this is x sub n. 00:13:10.740 --> 00:13:13.410 align:middle line:84% I'm just now going to repeat this argument one last time, 00:13:13.410 --> 00:13:14.940 align:middle line:84% and we'll see where that leads us. 00:13:14.940 --> 00:13:18.090 align:middle line:90% 00:13:18.090 --> 00:13:22.690 align:middle line:84% Since g-- the n-th derivative of g evaluated 00:13:22.690 --> 00:13:25.930 align:middle line:84% at x sub 0 equals 0-- again, this 00:13:25.930 --> 00:13:30.900 align:middle line:84% is coming from this relation here. 00:13:30.900 --> 00:13:32.460 align:middle line:84% Let me, in fact, write that again. 00:13:32.460 --> 00:13:35.380 align:middle line:90% 00:13:35.380 --> 00:13:42.010 align:middle line:84% This is equal to f of x0 minus P n of x0. 00:13:42.010 --> 00:13:54.030 align:middle line:84% And I'll even write out, this is equal to Mx x0 n plus 1 00:13:54.030 --> 00:13:58.200 align:middle line:90% factorial times x0 minus x0. 00:13:58.200 --> 00:14:02.760 align:middle line:84% This is what happens when I take n derivatives of ns, 00:14:02.760 --> 00:14:07.270 align:middle line:90% of this monomial here. 00:14:07.270 --> 00:14:15.795 align:middle line:90% This, of course, is 0 equals 0. 00:14:15.795 --> 00:14:20.020 align:middle line:90% 00:14:20.020 --> 00:14:26.470 align:middle line:84% Since we have that and we have, it's 00:14:26.470 --> 00:14:33.340 align:middle line:84% equals 0 at this other point, there exists, 00:14:33.340 --> 00:14:38.440 align:middle line:84% by the mean value theorem, now applied to g-- 00:14:38.440 --> 00:14:40.810 align:middle line:90% the n-th derivative of g. 00:14:40.810 --> 00:14:42.610 align:middle line:84% When I write mean value theorem here, 00:14:42.610 --> 00:14:44.300 align:middle line:84% I'm not applying it to the function g. 00:14:44.300 --> 00:14:49.450 align:middle line:84% I'm applying it to the derivative. 00:14:49.450 --> 00:14:52.090 align:middle line:84% Here, I was applying it just to g. 00:14:52.090 --> 00:14:54.370 align:middle line:84% Here, I was applying the mean value theorem 00:14:54.370 --> 00:14:55.715 align:middle line:90% to the derivative of g. 00:14:55.715 --> 00:14:59.280 align:middle line:90% 00:14:59.280 --> 00:15:01.200 align:middle line:84% And, here I'm now applying the mean value 00:15:01.200 --> 00:15:05.550 align:middle line:84% theorem to the n-th derivative of g. 00:15:05.550 --> 00:15:13.450 align:middle line:90% 00:15:13.450 --> 00:15:14.825 align:middle line:84% Let me make that perfectly clear. 00:15:14.825 --> 00:15:17.370 align:middle line:90% 00:15:17.370 --> 00:15:28.740 align:middle line:84% There exists a number, C, between x and x0 such that 00:15:28.740 --> 00:15:32.190 align:middle line:84% the n plus 1-- the derivative of g, 00:15:32.190 --> 00:15:35.250 align:middle line:84% the derivative of the n-th order derivative of g, 00:15:35.250 --> 00:15:38.130 align:middle line:84% so the n plus 1st derivative of g-- 00:15:38.130 --> 00:15:40.280 align:middle line:90% of C equals 0. 00:15:40.280 --> 00:15:41.280 align:middle line:90% But what does this mean? 00:15:41.280 --> 00:15:45.590 align:middle line:90% 00:15:45.590 --> 00:15:49.100 align:middle line:84% Now, if I take n plus 1 derivatives with respect 00:15:49.100 --> 00:15:58.825 align:middle line:90% to s of this over here, I get f. 00:15:58.825 --> 00:16:02.490 align:middle line:90% 00:16:02.490 --> 00:16:08.400 align:middle line:84% Now, if I take n plus 1 derivatives of an n-th degree 00:16:08.400 --> 00:16:10.230 align:middle line:90% polynomial, I get 0. 00:16:10.230 --> 00:16:13.860 align:middle line:84% If I take two derivatives of a degree 1 polynomial, which 00:16:13.860 --> 00:16:15.510 align:middle line:90% is just x, I get 0. 00:16:15.510 --> 00:16:21.390 align:middle line:84% If I take three derivatives of a degree 2 polynomial, I get 0. 00:16:21.390 --> 00:16:27.720 align:middle line:84% So I get 0 for when I differentiate n plus 1 times 00:16:27.720 --> 00:16:34.800 align:middle line:84% an n-th degree polynomial minus this constant again times n 00:16:34.800 --> 00:16:41.080 align:middle line:84% plus 1 derivatives of this monomial here in s. 00:16:41.080 --> 00:16:43.723 align:middle line:90% 00:16:43.723 --> 00:16:45.140 align:middle line:84% Remember, all of these derivatives 00:16:45.140 --> 00:16:47.540 align:middle line:84% I'm writing down here, these are all in terms of s. 00:16:47.540 --> 00:16:50.450 align:middle line:90% 00:16:50.450 --> 00:16:53.050 align:middle line:90% Times n plus 1 factorial. 00:16:53.050 --> 00:16:55.300 align:middle line:90% And this equals 0. 00:16:55.300 --> 00:17:00.310 align:middle line:84% This is just this here expressed here. 00:17:00.310 --> 00:17:04.280 align:middle line:90% 00:17:04.280 --> 00:17:05.900 align:middle line:90% And it should be C, I'm sorry. 00:17:05.900 --> 00:17:09.180 align:middle line:90% Because we're plugging in C. 00:17:09.180 --> 00:17:23.180 align:middle line:84% But that means precisely that, which is 00:17:23.180 --> 00:17:27.290 align:middle line:90% what I wanted to show existed. 00:17:27.290 --> 00:17:35.420 align:middle line:84% At this point C, this constant from earlier, which, remember, 00:17:35.420 --> 00:17:41.740 align:middle line:84% was defined in this way, is actually 00:17:41.740 --> 00:17:46.120 align:middle line:84% equal to the n plus 1 derivative of f evaluated at some C. 00:17:46.120 --> 00:17:50.980 align:middle line:84% And therefore, f of x is equal to P n of x plus-- 00:17:50.980 --> 00:17:58.080 align:middle line:90% 00:17:58.080 --> 00:18:04.080 align:middle line:84% x C times-- where C is between x and x0. 00:18:04.080 --> 00:18:14.110 align:middle line:90% 00:18:14.110 --> 00:18:22.190 align:middle line:84% Again, Taylor's theorem says a couple of things, 00:18:22.190 --> 00:18:26.410 align:middle line:84% but it doesn't say certain things. 00:18:26.410 --> 00:18:29.020 align:middle line:84% The mean value theorem, as it's written, 00:18:29.020 --> 00:18:34.420 align:middle line:84% says there exists some point in between so that the secant line 00:18:34.420 --> 00:18:38.230 align:middle line:84% from f of b to f of a is equal to the derivative 00:18:38.230 --> 00:18:40.540 align:middle line:84% of the function, the tangent to the graph, 00:18:40.540 --> 00:18:42.730 align:middle line:90% at some point in between. 00:18:42.730 --> 00:18:48.520 align:middle line:84% But it doesn't tell you that the function near a point 00:18:48.520 --> 00:18:49.600 align:middle line:90% can necessarily-- 00:18:49.600 --> 00:18:52.700 align:middle line:90% 00:18:52.700 --> 00:18:53.720 align:middle line:90% what am I trying to say? 00:18:53.720 --> 00:19:00.700 align:middle line:90% 00:19:00.700 --> 00:19:03.160 align:middle line:84% What Taylor's theorem does say is 00:19:03.160 --> 00:19:06.370 align:middle line:84% that you can iterate the mean value theorem 00:19:06.370 --> 00:19:08.290 align:middle line:90% for higher-order derivatives. 00:19:08.290 --> 00:19:12.880 align:middle line:84% But what it doesn't say is that this polynomial that you get 00:19:12.880 --> 00:19:15.310 align:middle line:84% over here, which you interpret kind 00:19:15.310 --> 00:19:19.720 align:middle line:84% of as an approximation of the function f near x0, 00:19:19.720 --> 00:19:22.630 align:middle line:84% it doesn't say that approximation 00:19:22.630 --> 00:19:25.228 align:middle line:90% is necessarily good. 00:19:25.228 --> 00:19:26.770 align:middle line:84% Because we just saw from this example 00:19:26.770 --> 00:19:30.565 align:middle line:84% that that remainder term may end up being the entire function. 00:19:30.565 --> 00:19:33.590 align:middle line:90% 00:19:33.590 --> 00:19:43.370 align:middle line:84% But still, that doesn't make it any less 00:19:43.370 --> 00:19:45.470 align:middle line:90% useful in applications. 00:19:45.470 --> 00:20:00.800 align:middle line:90% 00:20:00.800 --> 00:20:20.150 align:middle line:84% Let's give a simple application of Taylor's theorem, which 00:20:20.150 --> 00:20:30.210 align:middle line:84% perhaps you had endless homework problems or exam 00:20:30.210 --> 00:20:33.900 align:middle line:84% problems on back when you first took calculus 00:20:33.900 --> 00:20:37.950 align:middle line:84% and were finding critical points and trying to characterize them 00:20:37.950 --> 00:20:40.950 align:middle line:84% as relative minimums or relative maximums. 00:20:40.950 --> 00:20:42.610 align:middle line:84% We have the second derivative test, 00:20:42.610 --> 00:20:44.460 align:middle line:90% which says the following-- 00:20:44.460 --> 00:20:51.990 align:middle line:84% which states that, suppose I have a function 00:20:51.990 --> 00:20:59.630 align:middle line:84% from the open interval a, b to R. And suppose 00:20:59.630 --> 00:21:12.210 align:middle line:84% this has two continuous derivatives 00:21:12.210 --> 00:21:14.010 align:middle line:90% on this open interval a, b. 00:21:14.010 --> 00:21:18.680 align:middle line:90% 00:21:18.680 --> 00:21:30.360 align:middle line:84% If, at a point in a, b, the derivative equals 0, 00:21:30.360 --> 00:21:37.110 align:middle line:84% and the second derivative of f, you evaluated it at 0, 00:21:37.110 --> 00:21:53.125 align:middle line:84% x sub 0 is positive, then f has a relative min at x0. 00:21:53.125 --> 00:21:58.130 align:middle line:90% 00:21:58.130 --> 00:22:01.460 align:middle line:84% And I should say that this is a strict relative min. 00:22:01.460 --> 00:22:08.080 align:middle line:90% 00:22:08.080 --> 00:22:11.500 align:middle line:84% What's the difference between a strict relative min and just 00:22:11.500 --> 00:22:12.250 align:middle line:90% a relative min? 00:22:12.250 --> 00:22:16.180 align:middle line:84% A strict relative min, I mean that if I'm at any point 00:22:16.180 --> 00:22:20.350 align:middle line:84% other than x0 and I'm nearby, then 00:22:20.350 --> 00:22:22.930 align:middle line:90% f of x is bigger than f of x0. 00:22:22.930 --> 00:22:27.560 align:middle line:90% 00:22:27.560 --> 00:22:28.970 align:middle line:90% Let me just write that here. 00:22:28.970 --> 00:22:32.725 align:middle line:90% That means near x0-- 00:22:32.725 --> 00:22:43.570 align:middle line:90% 00:22:43.570 --> 00:22:44.070 align:middle line:90% OK. 00:22:44.070 --> 00:22:48.360 align:middle line:90% 00:22:48.360 --> 00:22:53.360 align:middle line:84% In fact, let's just briefly recall 00:22:53.360 --> 00:22:57.350 align:middle line:84% what the definition of relative min is. 00:22:57.350 --> 00:22:59.630 align:middle line:84% And this will allow me to state what it means 00:22:59.630 --> 00:23:02.300 align:middle line:90% to be a strict relative min. 00:23:02.300 --> 00:23:10.570 align:middle line:84% This means there exists a delta positive such 00:23:10.570 --> 00:23:17.295 align:middle line:90% that for all x, n-- 00:23:17.295 --> 00:23:31.200 align:middle line:90% 00:23:31.200 --> 00:23:44.150 align:middle line:84% such that for all x, x minus x0 implies 00:23:44.150 --> 00:23:47.690 align:middle line:90% f of x is bigger than f of x0. 00:23:47.690 --> 00:23:59.050 align:middle line:90% 00:23:59.050 --> 00:24:02.770 align:middle line:84% This is the definition of strict relative min. 00:24:02.770 --> 00:24:04.810 align:middle line:84% A strict relative min is a relative min 00:24:04.810 --> 00:24:07.900 align:middle line:84% because, what's the only thing missing 00:24:07.900 --> 00:24:10.090 align:middle line:84% from the definition of relative min 00:24:10.090 --> 00:24:12.940 align:middle line:84% is, what happens if I evaluate at x0? 00:24:12.940 --> 00:24:16.690 align:middle line:84% And then at x0, we get f of x0 equals f of x0. 00:24:16.690 --> 00:24:25.838 align:middle line:84% so a strict relative min is a relative min, 00:24:25.838 --> 00:24:27.130 align:middle line:90% but it's a little bit stronger. 00:24:27.130 --> 00:24:30.250 align:middle line:84% Because it's saying that as long as x is not equal to x0, 00:24:30.250 --> 00:24:32.740 align:middle line:84% meaning this thing is bigger than 0, f of x 00:24:32.740 --> 00:24:36.070 align:middle line:84% is bigger than f of x0, not bigger than or equal to. 00:24:36.070 --> 00:24:45.090 align:middle line:90% 00:24:45.090 --> 00:24:48.150 align:middle line:84% I hate doing this, but the theorem's stated over there, 00:24:48.150 --> 00:24:50.610 align:middle line:84% and now we need to go across the room to do the proof. 00:24:50.610 --> 00:25:01.490 align:middle line:90% 00:25:01.490 --> 00:25:07.190 align:middle line:84% f has two continuous derivatives on a, b. 00:25:07.190 --> 00:25:10.055 align:middle line:84% And therefore, the second derivative is continuous at 0. 00:25:10.055 --> 00:25:17.690 align:middle line:90% 00:25:17.690 --> 00:25:20.420 align:middle line:84% Since the second derivative is continuous, 00:25:20.420 --> 00:25:24.140 align:middle line:84% we get that the limit, as x goes to x0-- 00:25:24.140 --> 00:25:25.370 align:middle line:90% or, let me put-- 00:25:25.370 --> 00:25:31.940 align:middle line:84% instead of x, say C. This equals f double prime of x0, 00:25:31.940 --> 00:25:34.280 align:middle line:84% which is positive, by assumption. 00:25:34.280 --> 00:25:35.765 align:middle line:90% That's what we're assuming. 00:25:35.765 --> 00:25:38.990 align:middle line:90% 00:25:38.990 --> 00:25:49.930 align:middle line:84% And therefore, by an exercise in one of the assignments, 00:25:49.930 --> 00:25:58.490 align:middle line:84% since this limit, this implies that there 00:25:58.490 --> 00:26:09.010 align:middle line:84% exists a delta 0 positive such that for all 0 bigger than C, 00:26:09.010 --> 00:26:11.250 align:middle line:90% bigger than x0-- 00:26:11.250 --> 00:26:14.070 align:middle line:90% and in fact, we can include-- 00:26:14.070 --> 00:26:28.740 align:middle line:90% 00:26:28.740 --> 00:26:31.080 align:middle line:90% let me see. 00:26:31.080 --> 00:26:37.360 align:middle line:84% There exists a delta 0 positive such that, for all C, 00:26:37.360 --> 00:26:38.160 align:middle line:90% satisfying-- 00:26:38.160 --> 00:26:48.800 align:middle line:90% 00:26:48.800 --> 00:26:56.010 align:middle line:84% we get that f prime of C is positive. 00:26:56.010 --> 00:26:59.790 align:middle line:84% All I'm saying is we have this point, x0. 00:26:59.790 --> 00:27:03.210 align:middle line:90% x0 plus delta 0. 00:27:03.210 --> 00:27:05.190 align:middle line:90% x0 minus delta 0. 00:27:05.190 --> 00:27:13.170 align:middle line:84% And then on this interval, f double prime of C is positive. 00:27:13.170 --> 00:27:15.940 align:middle line:90% 00:27:15.940 --> 00:27:19.030 align:middle line:84% You proved that, in fact, in an assignment. 00:27:19.030 --> 00:27:22.120 align:middle line:84% If the limit of a function as I approach a point 00:27:22.120 --> 00:27:26.110 align:middle line:84% equals l, which is positive, then near the point, 00:27:26.110 --> 00:27:27.610 align:middle line:90% the function has to be positive. 00:27:27.610 --> 00:27:36.300 align:middle line:90% 00:27:36.300 --> 00:27:39.270 align:middle line:90% Now, I have to verify that-- 00:27:39.270 --> 00:27:40.300 align:middle line:90% what am I trying to do? 00:27:40.300 --> 00:27:43.410 align:middle line:84% I'm trying to verify that I have a strict relative minimum so 00:27:43.410 --> 00:27:44.910 align:middle line:90% that there exists a delta-- 00:27:44.910 --> 00:27:47.910 align:middle line:84% I have this delta 0, which ensures 00:27:47.910 --> 00:27:50.520 align:middle line:84% that the second derivative is positive on this interval. 00:27:50.520 --> 00:27:53.140 align:middle line:90% 00:27:53.140 --> 00:27:56.340 align:middle line:84% So I say choose delta to be this delta 0. 00:27:56.340 --> 00:27:58.560 align:middle line:84% And now, I have to show that this delta works, 00:27:58.560 --> 00:28:05.040 align:middle line:84% meaning for all x satisfying that inequality, 00:28:05.040 --> 00:28:10.850 align:middle line:84% I have f of x is bigger than f of x0. 00:28:10.850 --> 00:28:18.585 align:middle line:90% So take an x between delta-- 00:28:18.585 --> 00:28:21.190 align:middle line:90% 00:28:21.190 --> 00:28:26.860 align:middle line:84% I mean, within delta distance to x0. 00:28:26.860 --> 00:28:30.818 align:middle line:90% Here's x, say. 00:28:30.818 --> 00:28:34.170 align:middle line:90% 00:28:34.170 --> 00:28:52.200 align:middle line:84% Then by Taylor's theorem, there exists a C between x and x0-- 00:28:52.200 --> 00:28:54.360 align:middle line:90% so here's x. 00:28:54.360 --> 00:28:58.200 align:middle line:84% There's this point C between x and x0, 00:28:58.200 --> 00:29:03.770 align:middle line:84% which I can always choose strictly in between them, such 00:29:03.770 --> 00:29:14.420 align:middle line:84% that I have that f of x equals f of x0 00:29:14.420 --> 00:29:23.060 align:middle line:84% plus f prime of x0 times C minus x0 plus-- 00:29:23.060 --> 00:29:25.100 align:middle line:90% no, that should be x, sorry-- 00:29:25.100 --> 00:29:32.480 align:middle line:84% plus f double prime of C over 2 times x minus m 0 squared. 00:29:32.480 --> 00:29:37.570 align:middle line:90% 00:29:37.570 --> 00:29:43.330 align:middle line:84% Now, at x0, the derivative is assumed to be 0. 00:29:43.330 --> 00:29:46.540 align:middle line:84% We're assuming the derivative vanishes at x0 00:29:46.540 --> 00:29:49.900 align:middle line:84% and the second derivative is positive there. 00:29:49.900 --> 00:29:54.850 align:middle line:84% So this equals f of x0 plus f prime 00:29:54.850 --> 00:30:01.970 align:middle line:84% of C over 2 times x minus x0 squared. 00:30:01.970 --> 00:30:07.110 align:middle line:84% Now, on this whole interval, which is where I'm looking at, 00:30:07.110 --> 00:30:09.730 align:middle line:90% f double prime of C is positive. 00:30:09.730 --> 00:30:11.640 align:middle line:90% So this thing here is positive. 00:30:11.640 --> 00:30:16.650 align:middle line:84% And as long as x minus x0 is not 0-- 00:30:16.650 --> 00:30:19.620 align:middle line:84% as long as x is not equal to x0-- this thing is positive. 00:30:19.620 --> 00:30:21.120 align:middle line:90% This is a square. 00:30:21.120 --> 00:30:25.707 align:middle line:84% So this is strictly bigger than f of x0, which 00:30:25.707 --> 00:30:26.790 align:middle line:90% is what I wanted to prove. 00:30:26.790 --> 00:30:29.580 align:middle line:90% 00:30:29.580 --> 00:30:36.080 align:middle line:84% So I have proven that f of x is bigger than f of x0 00:30:36.080 --> 00:30:38.600 align:middle line:90% on this interval here. 00:30:38.600 --> 00:30:42.510 align:middle line:90% 00:30:42.510 --> 00:30:45.060 align:middle line:84% Of course, the picture that goes along 00:30:45.060 --> 00:30:52.950 align:middle line:84% with this is something like, let's say, the point x0, 0, 00:30:52.950 --> 00:30:59.870 align:middle line:84% at least near this point, the derivative is 0. 00:30:59.870 --> 00:31:01.470 align:middle line:84% The second derivative is positive. 00:31:01.470 --> 00:31:03.725 align:middle line:84% So this is how the function should look. 00:31:03.725 --> 00:31:07.460 align:middle line:90% 00:31:07.460 --> 00:31:09.500 align:middle line:84% CASEY RODRIGUEZ: So that concludes 00:31:09.500 --> 00:31:14.750 align:middle line:84% what we're going to say about differentiation. 00:31:14.750 --> 00:31:19.700 align:middle line:84% I have put in the assignment the most useful version 00:31:19.700 --> 00:31:24.020 align:middle line:84% of L'Hopital's rule, which is kind of the only 00:31:24.020 --> 00:31:27.440 align:middle line:84% other main thing we're missing right now from just 00:31:27.440 --> 00:31:29.240 align:middle line:90% the theory of differentiation. 00:31:29.240 --> 00:31:33.120 align:middle line:90% 00:31:33.120 --> 00:31:45.310 align:middle line:84% But remember, differentiation is a bit of a miracle, 00:31:45.310 --> 00:31:48.610 align:middle line:84% as I've said before, because there 00:31:48.610 --> 00:31:53.200 align:middle line:84% exist continuous functions that never have a derivative. 00:31:53.200 --> 00:31:55.800 align:middle line:90% 00:31:55.800 --> 00:31:58.980 align:middle line:84% Integration, which is what we're moving on to now, 00:31:58.980 --> 00:32:00.210 align:middle line:90% is not so much of a miracle. 00:32:00.210 --> 00:32:03.360 align:middle line:90% 00:32:03.360 --> 00:32:07.050 align:middle line:84% Because as we'll show, every continuous function 00:32:07.050 --> 00:32:16.070 align:middle line:84% has a Riemann integral, which is a different limiting process. 00:32:16.070 --> 00:32:18.190 align:middle line:84% So all of these things we're talking about, 00:32:18.190 --> 00:32:20.320 align:middle line:84% all of these notions-- continuity 00:32:20.320 --> 00:32:27.370 align:middle line:84% is a notion that involves limits. 00:32:27.370 --> 00:32:32.530 align:middle line:84% Differentiation is a process involving limits, 00:32:32.530 --> 00:32:35.140 align:middle line:84% and integration is a process involving limits. 00:32:35.140 --> 00:32:38.530 align:middle line:84% But somehow, integration is not as harsh a process 00:32:38.530 --> 00:32:39.820 align:middle line:90% as differentiation. 00:32:39.820 --> 00:32:50.690 align:middle line:90% 00:32:50.690 --> 00:33:03.600 align:middle line:84% We're moving on, now, to Riemann-- 00:33:03.600 --> 00:33:06.110 align:middle line:90% 00:33:06.110 --> 00:33:11.450 align:middle line:84% I should say the Riemann integral, 00:33:11.450 --> 00:33:14.080 align:middle line:84% but I'll say Riemann integration. 00:33:14.080 --> 00:33:17.160 align:middle line:90% 00:33:17.160 --> 00:33:18.720 align:middle line:90% What is Riemann integration? 00:33:18.720 --> 00:33:22.130 align:middle line:90% 00:33:22.130 --> 00:33:25.625 align:middle line:84% It is-- you were told this in calculus, but maybe 00:33:25.625 --> 00:33:27.680 align:middle line:90% in not so careful a way. 00:33:27.680 --> 00:33:30.710 align:middle line:84% This is a theory of what it means-- 00:33:30.710 --> 00:33:38.130 align:middle line:84% or this is a number that we associate to a function 00:33:38.130 --> 00:33:43.500 align:middle line:84% that you interpret as the area underneath the curve. 00:33:43.500 --> 00:33:50.320 align:middle line:84% It is not, as maybe you're told, somehow 00:33:50.320 --> 00:33:53.560 align:middle line:84% magically equal to the area underneath the curve. 00:33:53.560 --> 00:33:57.760 align:middle line:84% There is no notion of area underneath the curve. 00:33:57.760 --> 00:34:01.150 align:middle line:84% The Riemann integral is a number which 00:34:01.150 --> 00:34:03.700 align:middle line:84% you interpret as the area underneath the curve 00:34:03.700 --> 00:34:05.800 align:middle line:84% because it agrees with what you think 00:34:05.800 --> 00:34:09.610 align:middle line:84% the area underneath the curve should be for simple examples. 00:34:09.610 --> 00:34:13.719 align:middle line:84% For example, a half-circle or just a box. 00:34:13.719 --> 00:34:15.429 align:middle line:90% These two notions agree. 00:34:15.429 --> 00:34:18.889 align:middle line:84% And therefore, you interpret the Riemann integral, 00:34:18.889 --> 00:34:21.699 align:middle line:84% which is a number obtained by a limiting process, 00:34:21.699 --> 00:34:23.949 align:middle line:84% as the area underneath the curve. 00:34:23.949 --> 00:34:28.000 align:middle line:84% It is not somehow, out in the universe, 00:34:28.000 --> 00:34:30.370 align:middle line:84% there is this notion of area underneath the curve, 00:34:30.370 --> 00:34:35.570 align:middle line:84% and the Riemann integral magically 00:34:35.570 --> 00:34:37.080 align:middle line:90% coincides with that notion. 00:34:37.080 --> 00:34:37.580 align:middle line:90% No. 00:34:37.580 --> 00:34:42.409 align:middle line:84% It is a theory, if you like, of assigning a number 00:34:42.409 --> 00:34:44.840 align:middle line:84% that we interpret as the area underneath the curve. 00:34:44.840 --> 00:34:50.170 align:middle line:90% 00:34:50.170 --> 00:34:51.880 align:middle line:90% And it's good for-- 00:34:51.880 --> 00:34:55.030 align:middle line:84% very good for, especially once we 00:34:55.030 --> 00:34:56.770 align:middle line:84% get to the real miracle of calculus, 00:34:56.770 --> 00:34:59.170 align:middle line:84% the fundamental theorem of calculus, which connects 00:34:59.170 --> 00:35:01.900 align:middle line:84% the derivative to integration-- it's 00:35:01.900 --> 00:35:09.460 align:middle line:84% fantastic in being able to, in its ease of computing. 00:35:09.460 --> 00:35:11.020 align:middle line:84% Hopefully, at some point, you go on 00:35:11.020 --> 00:35:14.950 align:middle line:84% to learn about Lebesgue integration, which 00:35:14.950 --> 00:35:19.645 align:middle line:84% is a much more versatile notion of area underneath the curve. 00:35:19.645 --> 00:35:23.630 align:middle line:90% 00:35:23.630 --> 00:35:25.745 align:middle line:90% And a little bit more robust. 00:35:25.745 --> 00:35:29.630 align:middle line:84% We have better theorems that you can then use and prove-- 00:35:29.630 --> 00:35:33.380 align:middle line:90% prove, then use, of course. 00:35:33.380 --> 00:35:37.880 align:middle line:84% But Riemann integration is a place to start. 00:35:37.880 --> 00:35:45.640 align:middle line:84% And in fact, in some treatments of Lebesgue integration, 00:35:45.640 --> 00:35:49.750 align:middle line:84% Lebesgue immigration is treated as the completion of Riemann 00:35:49.750 --> 00:35:52.120 align:middle line:84% integration, just as the real numbers 00:35:52.120 --> 00:35:58.120 align:middle line:84% are the completion, in some sense, of the rational numbers. 00:35:58.120 --> 00:36:05.140 align:middle line:84% Let's set up some definitions and notions that we'll need. 00:36:05.140 --> 00:36:07.390 align:middle line:84% I'm just going to be talking about Riemann integration 00:36:07.390 --> 00:36:08.470 align:middle line:90% of continuous functions. 00:36:08.470 --> 00:36:11.890 align:middle line:90% This is the simplest way to go. 00:36:11.890 --> 00:36:15.610 align:middle line:84% Why not for some general functions or something 00:36:15.610 --> 00:36:17.240 align:middle line:90% like that? 00:36:17.240 --> 00:36:19.420 align:middle line:84% Because in general, a function does not 00:36:19.420 --> 00:36:21.770 align:middle line:90% have a Riemann integral. 00:36:21.770 --> 00:36:25.990 align:middle line:84% So you could try to ask, can you characterize what functions 00:36:25.990 --> 00:36:28.480 align:middle line:90% do have a Riemann integral? 00:36:28.480 --> 00:36:32.970 align:middle line:84% And the answer to that is functions which are continuous, 00:36:32.970 --> 00:36:35.160 align:middle line:90% in a sense, almost everywhere. 00:36:35.160 --> 00:36:39.750 align:middle line:84% Almost everywhere, though, we don't have the machinery 00:36:39.750 --> 00:36:40.740 align:middle line:90% to describe that. 00:36:40.740 --> 00:36:43.950 align:middle line:90% That's a measure theory course. 00:36:43.950 --> 00:36:47.980 align:middle line:84% Because you cannot-- or at least, 00:36:47.980 --> 00:36:51.460 align:middle line:84% because we don't have the machinery to fully state what 00:36:51.460 --> 00:36:56.950 align:middle line:84% it means, a precise "if and only if" statement about when 00:36:56.950 --> 00:37:00.640 align:middle line:84% a function is Riemann integral, I'm just going to do the ribbon 00:37:00.640 --> 00:37:03.040 align:middle line:84% in a rule for continuous functions, 00:37:03.040 --> 00:37:05.260 align:middle line:84% which is nice and simple enough-- 00:37:05.260 --> 00:37:06.025 align:middle line:90% and still pretty. 00:37:06.025 --> 00:37:09.520 align:middle line:90% 00:37:09.520 --> 00:37:13.450 align:middle line:84% Let me just introduce, first, some 00:37:13.450 --> 00:37:16.840 align:middle line:84% notation that I'll be using a lot. 00:37:16.840 --> 00:37:20.530 align:middle line:90% C of a, b. 00:37:20.530 --> 00:37:26.410 align:middle line:84% This is going to be the set of all continuous functions 00:37:26.410 --> 00:37:34.370 align:middle line:84% from a, b to R. So f from a, b to R. f is continuous. 00:37:34.370 --> 00:37:37.980 align:middle line:90% 00:37:37.980 --> 00:37:54.260 align:middle line:84% Now, as I said, we're going to associate to an interval 00:37:54.260 --> 00:37:56.330 align:middle line:90% and a function-- a number-- 00:37:56.330 --> 00:38:05.750 align:middle line:84% which we will later interpret as a notion of area 00:38:05.750 --> 00:38:08.300 align:middle line:90% underneath the curve. 00:38:08.300 --> 00:38:10.880 align:middle line:84% This process is a limiting process where 00:38:10.880 --> 00:38:12.990 align:middle line:84% we're going to be taking the domain 00:38:12.990 --> 00:38:15.410 align:middle line:84% and cutting it up into smaller and smaller pieces, 00:38:15.410 --> 00:38:20.430 align:middle line:84% and somehow writing down a number that we think 00:38:20.430 --> 00:38:24.455 align:middle line:84% approximates the area, is a good approximate area 00:38:24.455 --> 00:38:25.330 align:middle line:90% underneath the curve. 00:38:25.330 --> 00:38:28.840 align:middle line:90% 00:38:28.840 --> 00:38:31.600 align:middle line:84% I'm going to assign some words to this breaking down process. 00:38:31.600 --> 00:38:37.350 align:middle line:90% 00:38:37.350 --> 00:38:42.120 align:middle line:90% Partition of the interval a, b. 00:38:42.120 --> 00:38:52.592 align:middle line:84% This is just a finite set x underline, 00:38:52.592 --> 00:38:53.800 align:middle line:90% which I'll write in this way. 00:38:53.800 --> 00:38:57.460 align:middle line:90% 00:38:57.460 --> 00:39:04.960 align:middle line:84% It's a finite set, which I write x0, x1, x2, up to xn, 00:39:04.960 --> 00:39:08.860 align:middle line:84% with the property that x0 is equal to a is less than x1, 00:39:08.860 --> 00:39:09.595 align:middle line:90% less than x2. 00:39:09.595 --> 00:39:23.710 align:middle line:90% 00:39:23.710 --> 00:39:29.740 align:middle line:84% The norm of a partition, which I denote 00:39:29.740 --> 00:39:34.630 align:middle line:84% with these two vertical lines on either side of x underline, 00:39:34.630 --> 00:39:39.850 align:middle line:84% is by definition, the max of the differences 00:39:39.850 --> 00:39:42.760 align:middle line:90% between these partition points. 00:39:42.760 --> 00:39:46.250 align:middle line:84% I refer to these points that are in the partition as partition 00:39:46.250 --> 00:39:46.750 align:middle line:90% points. 00:39:46.750 --> 00:39:52.300 align:middle line:84% This is x1 minus x0, and so on, xn minus xn minus 1. 00:39:52.300 --> 00:39:59.960 align:middle line:90% 00:39:59.960 --> 00:40:14.780 align:middle line:84% A tag for partition x bar is a finite set xi. 00:40:14.780 --> 00:40:16.760 align:middle line:84% Get used to some Greek letters in your life. 00:40:16.760 --> 00:40:21.710 align:middle line:90% Xi equals C1 up to Cn. 00:40:21.710 --> 00:40:24.378 align:middle line:84% As before, in the partition, we started off with a 0 here. 00:40:24.378 --> 00:40:25.295 align:middle line:90% We started off with 1. 00:40:25.295 --> 00:40:28.830 align:middle line:90% 00:40:28.830 --> 00:40:35.980 align:middle line:84% Such that each of these xis lie between partition points. 00:40:35.980 --> 00:40:41.460 align:middle line:84% In other words, x0 is less than xi 1 is less than x1. 00:40:41.460 --> 00:40:55.370 align:middle line:90% 00:40:55.370 --> 00:41:07.420 align:middle line:84% And the pair is referred to as a tagged partition. 00:41:07.420 --> 00:41:15.270 align:middle line:90% 00:41:15.270 --> 00:41:19.110 align:middle line:84% Although maybe it looks a little bit fancy, it's not. 00:41:19.110 --> 00:41:21.630 align:middle line:84% A partition is, you take your interval a, b, 00:41:21.630 --> 00:41:25.830 align:middle line:84% and you cut it up into pieces, with your first point always 00:41:25.830 --> 00:41:31.080 align:middle line:84% being a and your last point always being b. 00:41:31.080 --> 00:41:43.300 align:middle line:84% So x1, x2, and because I can't draw n points, 00:41:43.300 --> 00:41:49.020 align:middle line:90% I'm going to draw four points. 00:41:49.020 --> 00:41:52.900 align:middle line:90% x3, x4. 00:41:52.900 --> 00:41:56.320 align:middle line:84% There's a partition of a, b, into-- 00:41:56.320 --> 00:41:59.770 align:middle line:84% think of these points as being the endpoints 00:41:59.770 --> 00:42:01.930 align:middle line:84% of little intervals that I've broken up the bigger 00:42:01.930 --> 00:42:04.270 align:middle line:90% interval into. 00:42:04.270 --> 00:42:09.460 align:middle line:84% And the xis are just points in each of these little intervals. 00:42:09.460 --> 00:42:12.520 align:middle line:90% C1 has to land there. 00:42:12.520 --> 00:42:14.785 align:middle line:90% xi 2 could land in the next one. 00:42:14.785 --> 00:42:16.660 align:middle line:84% It could actually be the endpoint if we like. 00:42:16.660 --> 00:42:20.100 align:middle line:90% 00:42:20.100 --> 00:42:21.348 align:middle line:90% xi 3. 00:42:21.348 --> 00:42:22.515 align:middle line:90% Let's say it's the midpoint. 00:42:22.515 --> 00:42:26.021 align:middle line:90% 00:42:26.021 --> 00:42:28.585 align:middle line:90% We'll say xi 4 is there as well. 00:42:28.585 --> 00:42:32.260 align:middle line:90% 00:42:32.260 --> 00:42:37.150 align:middle line:84% The tagged points are just lying in these smaller intervals. 00:42:37.150 --> 00:42:42.610 align:middle line:84% And at least in this picture, the biggest separation 00:42:42.610 --> 00:42:47.410 align:middle line:84% between partition points would be something 00:42:47.410 --> 00:42:54.850 align:middle line:90% like x3 minus x2 here. 00:42:54.850 --> 00:43:05.120 align:middle line:84% The norm of a partition is the length 00:43:05.120 --> 00:43:06.690 align:middle line:90% of the largest subinterval. 00:43:06.690 --> 00:43:17.030 align:middle line:90% 00:43:17.030 --> 00:43:22.350 align:middle line:84% I drew kind of something abstract here. 00:43:22.350 --> 00:43:23.830 align:middle line:90% Let's make this more concrete. 00:43:23.830 --> 00:43:27.960 align:middle line:90% Let's say I'm looking at-- 00:43:27.960 --> 00:43:32.740 align:middle line:84% just to write down a few examples, 00:43:32.740 --> 00:43:38.510 align:middle line:84% let's say my interval is 1, 3, and then 00:43:38.510 --> 00:43:52.850 align:middle line:84% my partition are the points 1, 3/2, 2, 3. 00:43:52.850 --> 00:44:00.740 align:middle line:84% And then my set of tags are 5/4-- 00:44:00.740 --> 00:44:03.780 align:middle line:90% 00:44:03.780 --> 00:44:08.140 align:middle line:90% just a midpoint-- 7/4, 5/2. 00:44:08.140 --> 00:44:12.470 align:middle line:90% So my partition is 1. 00:44:12.470 --> 00:44:14.075 align:middle line:90% There's 2. 00:44:14.075 --> 00:44:14.575 align:middle line:90% 3/2. 00:44:14.575 --> 00:44:17.990 align:middle line:90% 00:44:17.990 --> 00:44:21.340 align:middle line:90% Those are my partition points. 00:44:21.340 --> 00:44:23.785 align:middle line:84% And meanwhile, my tags are the midpoint. 00:44:23.785 --> 00:44:34.260 align:middle line:90% 00:44:34.260 --> 00:44:39.620 align:middle line:84% And then the norm of this partition 00:44:39.620 --> 00:44:45.170 align:middle line:84% is the maximum of the lengths of these smaller subintervals 00:44:45.170 --> 00:44:48.330 align:middle line:84% here-- not the ones using C, but the one with the partition 00:44:48.330 --> 00:44:48.830 align:middle line:90% points. 00:44:48.830 --> 00:45:03.700 align:middle line:84% So max of 3/2 minus 1, 2 minus 3/2, 3 minus 2, and is 1-- 00:45:03.700 --> 00:45:08.030 align:middle line:90% the length of this subinterval. 00:45:08.030 --> 00:45:39.990 align:middle line:90% 00:45:39.990 --> 00:45:45.760 align:middle line:84% Now, given a tagged partition, we're 00:45:45.760 --> 00:45:50.350 align:middle line:84% going to associate a number to this tagged partition, which 00:45:50.350 --> 00:45:52.795 align:middle line:84% we interpret as an approximate area. 00:45:52.795 --> 00:46:05.170 align:middle line:90% 00:46:05.170 --> 00:46:18.319 align:middle line:84% Let f be a continuous function, xi, a tagged partition 00:46:18.319 --> 00:46:26.265 align:middle line:90% the Riemann sum associated to-- 00:46:26.265 --> 00:46:31.030 align:middle line:90% 00:46:31.030 --> 00:46:32.475 align:middle line:90% I should say, of f-- 00:46:32.475 --> 00:46:38.270 align:middle line:90% 00:46:38.270 --> 00:46:53.800 align:middle line:84% associated to the tag partition xi is the number 00:46:53.800 --> 00:46:56.500 align:middle line:90% s sub f of x bar-- 00:46:56.500 --> 00:46:59.050 align:middle line:84% I mean, x underline, xi underline, 00:46:59.050 --> 00:47:05.260 align:middle line:84% which is the sum from k equals 0 to k equals 1 to n 00:47:05.260 --> 00:47:12.430 align:middle line:84% of f of xi k times xk minus xk minus 1. 00:47:12.430 --> 00:47:20.080 align:middle line:90% 00:47:20.080 --> 00:47:27.260 align:middle line:84% Again, what we interpret this number as-- 00:47:27.260 --> 00:47:28.620 align:middle line:90% how do we interpret this number? 00:47:28.620 --> 00:47:30.050 align:middle line:90% We interpret it as somehow-- 00:47:30.050 --> 00:47:33.230 align:middle line:90% 00:47:33.230 --> 00:47:36.860 align:middle line:84% we give meaning to this number as an approximate area. 00:47:36.860 --> 00:47:45.580 align:middle line:84% If this is a, b, and there's a function f, 00:47:45.580 --> 00:47:49.890 align:middle line:84% and let's say those are the partition points. 00:47:49.890 --> 00:47:56.820 align:middle line:90% So x1, x2, x3, x4, x0. 00:47:56.820 --> 00:48:07.310 align:middle line:84% And let's say the tags are just the right endpoints of each 00:48:07.310 --> 00:48:10.390 align:middle line:90% of these smaller intervals. 00:48:10.390 --> 00:48:16.070 align:middle line:84% Then what is this number, at least in terms of this picture? 00:48:16.070 --> 00:48:28.705 align:middle line:90% 00:48:28.705 --> 00:48:30.080 align:middle line:84% That's a little off, but anyways. 00:48:30.080 --> 00:48:34.370 align:middle line:90% 00:48:34.370 --> 00:48:42.720 align:middle line:84% What I've shaded in, this area, this 00:48:42.720 --> 00:48:53.560 align:middle line:84% equals this Riemann sum of f associated to this tag here. 00:48:53.560 --> 00:48:59.230 align:middle line:84% And let me go over, again, the graph of f. 00:48:59.230 --> 00:49:02.680 align:middle line:90% 00:49:02.680 --> 00:49:08.160 align:middle line:84% This number here which we've come up with we 00:49:08.160 --> 00:49:11.790 align:middle line:84% interpret as somehow being an approximate area. 00:49:11.790 --> 00:49:14.910 align:middle line:84% Again, I don't like saying area underneath the curve, 00:49:14.910 --> 00:49:16.980 align:middle line:84% because that presupposes that there 00:49:16.980 --> 00:49:19.530 align:middle line:84% is a notion of the area underneath the curve 00:49:19.530 --> 00:49:21.760 align:middle line:84% independent of what we're doing here. 00:49:21.760 --> 00:49:23.500 align:middle line:90% But that's not the case. 00:49:23.500 --> 00:49:27.030 align:middle line:84% We are, in fact, giving a theory-- 00:49:27.030 --> 00:49:32.910 align:middle line:84% a mathematical theory-- of area underneath the curve. 00:49:32.910 --> 00:49:35.160 align:middle line:84% We are prescribing a number which 00:49:35.160 --> 00:49:38.160 align:middle line:84% we interpret as the area underneath the curve. 00:49:38.160 --> 00:49:44.075 align:middle line:84% These Riemann sums we interpret as being approximate areas. 00:49:44.075 --> 00:49:46.640 align:middle line:90% 00:49:46.640 --> 00:49:48.410 align:middle line:84% What we would like to do is somehow 00:49:48.410 --> 00:49:53.390 align:middle line:84% take a limit as the lengths of these subintervals 00:49:53.390 --> 00:49:58.340 align:middle line:84% get smaller and smaller, as the norm of the partitions go to 0. 00:49:58.340 --> 00:50:01.430 align:middle line:84% And what we would like to say is that these approximate areas-- 00:50:01.430 --> 00:50:03.380 align:middle line:90% these are just numbers-- 00:50:03.380 --> 00:50:05.570 align:middle line:84% converge to some limiting number-- 00:50:05.570 --> 00:50:07.750 align:middle line:90% a, say. 00:50:07.750 --> 00:50:13.790 align:middle line:84% That number we refer to as the Riemann integral of f, 00:50:13.790 --> 00:50:20.090 align:middle line:84% and we interpret as the area underneath the graph of f. 00:50:20.090 --> 00:50:23.090 align:middle line:84% Now, for this to work, we have to show 00:50:23.090 --> 00:50:29.980 align:middle line:84% that as we take partitions with smaller and smaller norm, where 00:50:29.980 --> 00:50:32.660 align:middle line:84% the intervals get smaller and smaller, 00:50:32.660 --> 00:50:38.890 align:middle line:84% these approximate areas actually do converge to some number. 00:50:38.890 --> 00:50:42.190 align:middle line:84% And that's going to be the content of the next lecture, 00:50:42.190 --> 00:50:44.320 align:middle line:84% in which we'll prove the existence of the Riemann 00:50:44.320 --> 00:50:49.210 align:middle line:84% integral and do some properties about. 00:50:49.210 --> 00:50:53.230 align:middle line:84% So the take-home point is, again, there 00:50:53.230 --> 00:50:57.220 align:middle line:84% is no definition of area underneath the curve 00:50:57.220 --> 00:50:59.140 align:middle line:90% independent of what we're doing. 00:50:59.140 --> 00:51:03.040 align:middle line:84% It's not like, out there in the universe, 00:51:03.040 --> 00:51:06.200 align:middle line:84% there's a notion of the area underneath the curve, 00:51:06.200 --> 00:51:08.290 align:middle line:84% and when we compute the Riemann integral, 00:51:08.290 --> 00:51:10.210 align:middle line:90% magically, those two things-- 00:51:10.210 --> 00:51:12.490 align:middle line:90% those two numbers-- coincide. 00:51:12.490 --> 00:51:13.610 align:middle line:90% No. 00:51:13.610 --> 00:51:15.670 align:middle line:84% We are giving-- we are constructing 00:51:15.670 --> 00:51:21.880 align:middle line:84% a theory of the area underneath the curve, which, 00:51:21.880 --> 00:51:27.580 align:middle line:84% for example, for a half-circle, or square, or ellipse, 00:51:27.580 --> 00:51:29.560 align:middle line:84% do, in fact, coincide with stuff you 00:51:29.560 --> 00:51:32.530 align:middle line:90% know from ordinary geometry. 00:51:32.530 --> 00:51:35.890 align:middle line:84% And therefore, it gives a good theory 00:51:35.890 --> 00:51:38.470 align:middle line:90% of area underneath the curve. 00:51:38.470 --> 00:51:43.820 align:middle line:84% But in order for us to construct that theory, 00:51:43.820 --> 00:51:47.180 align:middle line:84% or how we're constructing that theory, 00:51:47.180 --> 00:51:50.750 align:middle line:84% is for a continuous function, we define a number 00:51:50.750 --> 00:51:52.790 align:middle line:84% associated to a partition, which we 00:51:52.790 --> 00:51:54.740 align:middle line:90% interpret as approximate areas. 00:51:54.740 --> 00:51:57.620 align:middle line:84% We would like to say these approximate areas converge 00:51:57.620 --> 00:52:03.380 align:middle line:84% to some number as the partitions get finer and finer, 00:52:03.380 --> 00:52:07.280 align:middle line:84% or as the norm gets smaller and smaller. 00:52:07.280 --> 00:52:09.680 align:middle line:84% And that limiting number we interpret 00:52:09.680 --> 00:52:11.840 align:middle line:84% as the area underneath the curve. 00:52:11.840 --> 00:52:17.510 align:middle line:84% That will be what we do next time, is actually show 00:52:17.510 --> 00:52:20.160 align:middle line:84% the existence of this limiting number, 00:52:20.160 --> 00:52:23.270 align:middle line:84% which is the Riemann integral of f. 00:52:23.270 --> 00:52:25.390 align:middle line:90% And we'll stop there. 00:52:25.390 --> 00:52:32.000 align:middle line:90%