WEBVTT 00:00:00.000 --> 00:00:12.308 align:middle line:90% 00:00:12.308 --> 00:00:14.850 align:middle line:84% CASEY RODRIGUEZ: We'll continue our discussion of the Riemann 00:00:14.850 --> 00:00:15.400 align:middle line:90% integral. 00:00:15.400 --> 00:00:21.490 align:middle line:84% So let me just recall a few bits of notation 00:00:21.490 --> 00:00:24.310 align:middle line:84% from last time and then also, the main result that we proved 00:00:24.310 --> 00:00:26.620 align:middle line:90% at the end of last class. 00:00:26.620 --> 00:00:37.180 align:middle line:84% So we had partitions, which are just finite sets. 00:00:37.180 --> 00:00:40.350 align:middle line:90% 00:00:40.350 --> 00:00:44.400 align:middle line:84% So here, a is less than b, and we're 00:00:44.400 --> 00:00:47.400 align:middle line:90% looking at an interval ab. 00:00:47.400 --> 00:00:52.320 align:middle line:84% Partitions, they're just finite sets of ab. 00:00:52.320 --> 00:00:59.520 align:middle line:84% And then we also had tags, which were finite sets 00:00:59.520 --> 00:01:13.490 align:middle line:84% as well with c one between x0 and x1, which is less than 00:01:13.490 --> 00:01:19.100 align:middle line:84% or equal to xi 2, which is less than or equal to x2 and so on. 00:01:19.100 --> 00:01:22.290 align:middle line:90% 00:01:22.290 --> 00:01:30.180 align:middle line:84% And we call these tag partitions. 00:01:30.180 --> 00:01:34.620 align:middle line:90% 00:01:34.620 --> 00:01:41.310 align:middle line:84% And we also had the norm of a partition. 00:01:41.310 --> 00:01:43.980 align:middle line:90% This is the max of-- 00:01:43.980 --> 00:01:47.340 align:middle line:84% so we think of partitions as being-- 00:01:47.340 --> 00:01:53.460 align:middle line:84% as breaking up the interval ab up into smaller subintervals, 00:01:53.460 --> 00:01:54.510 align:middle line:90% x0 up to x1. 00:01:54.510 --> 00:02:01.853 align:middle line:90% 00:02:01.853 --> 00:02:03.520 align:middle line:84% I got to get a different piece of chalk. 00:02:03.520 --> 00:02:05.620 align:middle line:84% This one is going psychedelic on me. 00:02:05.620 --> 00:02:13.670 align:middle line:90% 00:02:13.670 --> 00:02:18.470 align:middle line:84% And then the xi ones are just points chosen in these smaller 00:02:18.470 --> 00:02:19.430 align:middle line:90% subintervals. 00:02:19.430 --> 00:02:22.190 align:middle line:84% The norm of a partition is the length 00:02:22.190 --> 00:02:24.830 align:middle line:90% of the largest subinterval. 00:02:24.830 --> 00:02:28.700 align:middle line:90% 00:02:28.700 --> 00:02:34.490 align:middle line:84% And then we had the Riemann sum associated to a partition, tag 00:02:34.490 --> 00:02:40.760 align:middle line:84% partition xi, which is the sum from j 00:02:40.760 --> 00:02:52.490 align:middle line:84% equals 1 to n, f xi j xj minus xj minus 1. 00:02:52.490 --> 00:02:54.040 align:middle line:90% OK? 00:02:54.040 --> 00:02:58.810 align:middle line:84% This just some notation from last time. 00:02:58.810 --> 00:03:03.250 align:middle line:90% And then what did we prove? 00:03:03.250 --> 00:03:06.100 align:middle line:84% We proved the following theorem, which 00:03:06.100 --> 00:03:09.280 align:middle line:84% is the existence for the Riemann integral, which 00:03:09.280 --> 00:03:18.220 align:middle line:84% is the following that for all continuous functions on ab, 00:03:18.220 --> 00:03:29.590 align:middle line:84% they exist a unique number, which we denote integral abf 00:03:29.590 --> 00:03:30.715 align:middle line:90% with the property that-- 00:03:30.715 --> 00:03:36.400 align:middle line:90% 00:03:36.400 --> 00:03:37.930 align:middle line:90% with the following property. 00:03:37.930 --> 00:03:41.110 align:middle line:84% That if I take any sequence of tag partitions 00:03:41.110 --> 00:03:44.080 align:middle line:84% with norm going to 0-- so this is the partitions 00:03:44.080 --> 00:03:45.970 align:middle line:90% are getting finer and finer. 00:03:45.970 --> 00:03:51.220 align:middle line:90% 00:03:51.220 --> 00:04:03.550 align:middle line:84% They're all a sequence of tag partitions with norms going 00:04:03.550 --> 00:04:17.320 align:middle line:84% to zero, we have the associated Riemann sums, this sequence 00:04:17.320 --> 00:04:17.990 align:middle line:90% converges. 00:04:17.990 --> 00:04:23.500 align:middle line:84% So this is now a sequence of real numbers, 00:04:23.500 --> 00:04:29.960 align:middle line:84% and equals this number, integral abf. 00:04:29.960 --> 00:04:32.730 align:middle line:84% No matter what sequence of tag partitions 00:04:32.730 --> 00:04:36.620 align:middle line:84% I take with norms converging to 0, 00:04:36.620 --> 00:04:39.920 align:middle line:84% the Riemann sums converge to this number, which we 00:04:39.920 --> 00:04:43.220 align:middle line:90% call the Riemann integral of f. 00:04:43.220 --> 00:04:46.040 align:middle line:84% Net, I think the very last thing we proved 00:04:46.040 --> 00:04:49.430 align:middle line:84% was that the Riemann integral is linear. 00:04:49.430 --> 00:04:52.890 align:middle line:84% The integral of the sum is the sum of the integrals. 00:04:52.890 --> 00:04:58.610 align:middle line:84% And if I multiply a continuous function by a scalar, 00:04:58.610 --> 00:05:00.470 align:middle line:90% then the scalar pulls out. 00:05:00.470 --> 00:05:06.280 align:middle line:90% 00:05:06.280 --> 00:05:06.780 align:middle line:90% OK? 00:05:06.780 --> 00:05:19.370 align:middle line:90% 00:05:19.370 --> 00:05:22.100 align:middle line:84% So now we're going to discuss some properties of the Riemann 00:05:22.100 --> 00:05:23.040 align:middle line:90% integral. 00:05:23.040 --> 00:05:34.570 align:middle line:84% So the next is that in some sense, the area of the union, 00:05:34.570 --> 00:05:37.180 align:middle line:84% if you like, is the sum of the areas 00:05:37.180 --> 00:05:41.840 align:middle line:84% if we think of the integral as being 00:05:41.840 --> 00:05:44.130 align:middle line:90% the area underneath the curve. 00:05:44.130 --> 00:05:51.120 align:middle line:84% So this next property is the additivity 00:05:51.120 --> 00:05:55.200 align:middle line:84% of the Riemann integral, which states the following-- 00:05:55.200 --> 00:06:06.340 align:middle line:84% if f is in cab, and I take a point between c and b, 00:06:06.340 --> 00:06:11.090 align:middle line:84% then the integral from a to b of f 00:06:11.090 --> 00:06:15.290 align:middle line:84% is equal to the integral from a to c of f plus the integral 00:06:15.290 --> 00:06:19.340 align:middle line:90% from c to b of f. 00:06:19.340 --> 00:06:25.860 align:middle line:90% 00:06:25.860 --> 00:06:30.570 align:middle line:84% So to prove it, again, all we know 00:06:30.570 --> 00:06:34.180 align:middle line:84% about these numbers is that they satisfy this property. 00:06:34.180 --> 00:06:39.960 align:middle line:84% So what we'll do is, we'll take a sequence of tag partitions 00:06:39.960 --> 00:06:43.680 align:middle line:84% of ac converging to this integral in a sequence of tag 00:06:43.680 --> 00:06:48.600 align:middle line:84% partitions from c to b converging to this integral 00:06:48.600 --> 00:06:51.030 align:middle line:84% when I stick them into the Riemann sums. 00:06:51.030 --> 00:06:53.710 align:middle line:84% But then when I take the union of those tag partitions, 00:06:53.710 --> 00:06:58.840 align:middle line:84% I get a partition of ab, which will 00:06:58.840 --> 00:07:00.490 align:middle line:90% approach the integral of ab. 00:07:00.490 --> 00:07:04.990 align:middle line:90% That's the basic intuition. 00:07:04.990 --> 00:07:13.480 align:middle line:84% And of course, what was I saying a minute ago about area 00:07:13.480 --> 00:07:17.680 align:middle line:84% of the union being sum of the areas? 00:07:17.680 --> 00:07:22.270 align:middle line:84% Well again, like I said, if we think of the Riemann integral 00:07:22.270 --> 00:07:24.590 align:middle line:84% as being the area underneath the curve, 00:07:24.590 --> 00:07:28.210 align:middle line:84% then this says that the area from a 00:07:28.210 --> 00:07:30.670 align:middle line:84% to b underneath the curve is equal to the area 00:07:30.670 --> 00:07:32.080 align:middle line:84% underneath the curve from a to c, 00:07:32.080 --> 00:07:34.480 align:middle line:84% plus the area underneath the curve from c to b. 00:07:34.480 --> 00:07:38.430 align:middle line:90% 00:07:38.430 --> 00:07:45.060 align:middle line:90% OK, so let's take-- 00:07:45.060 --> 00:07:48.890 align:middle line:90% 00:07:48.890 --> 00:07:52.680 align:middle line:84% someone needs to write notation here. 00:07:52.680 --> 00:08:00.550 align:middle line:90% Let's take two partitions. 00:08:00.550 --> 00:08:07.370 align:middle line:84% Let be a sequence of partitions-- 00:08:07.370 --> 00:08:11.460 align:middle line:90% 00:08:11.460 --> 00:08:14.060 align:middle line:84% so a sequence of tag partitions, I should say-- 00:08:14.060 --> 00:08:29.380 align:middle line:84% of ac such that norm of this partition goes to 0. 00:08:29.380 --> 00:08:36.409 align:middle line:84% So this is now a partition of ac. 00:08:36.409 --> 00:08:38.090 align:middle line:84% And then we'll take another sequence 00:08:38.090 --> 00:08:41.584 align:middle line:90% of tag partitions of cb. 00:08:41.584 --> 00:08:56.080 align:middle line:90% 00:08:56.080 --> 00:08:59.470 align:middle line:84% For those watching at home wondering what these letters 00:08:59.470 --> 00:09:00.938 align:middle line:90% are, that's the Greek eta. 00:09:00.938 --> 00:09:01.855 align:middle line:90% That's the Greek zeta. 00:09:01.855 --> 00:09:05.290 align:middle line:90% 00:09:05.290 --> 00:09:12.370 align:middle line:84% So we have a sequence of partitions of ac. 00:09:12.370 --> 00:09:13.870 align:middle line:90% These are the y and eta. 00:09:13.870 --> 00:09:15.910 align:middle line:84% And then we also have a sequence of partitions 00:09:15.910 --> 00:09:27.050 align:middle line:84% of cb such that the norm of these guys go to 0 00:09:27.050 --> 00:09:31.220 align:middle line:90% as well as r goes to infinity. 00:09:31.220 --> 00:09:35.660 align:middle line:84% So now if I have a partition of ac and a partition of cb, 00:09:35.660 --> 00:09:40.310 align:middle line:84% if I take their union, then I will get a partition of ab. 00:09:40.310 --> 00:09:47.390 align:middle line:90% 00:09:47.390 --> 00:09:49.505 align:middle line:84% This will be the union of these guys. 00:09:49.505 --> 00:10:06.030 align:middle line:90% 00:10:06.030 --> 00:10:10.050 align:middle line:84% So it's easy to see that this is now 00:10:10.050 --> 00:10:15.410 align:middle line:84% a sequence of tag partitions of ab. 00:10:15.410 --> 00:10:32.720 align:middle line:90% 00:10:32.720 --> 00:10:37.880 align:middle line:84% And moreover, so what is the norm of this new tag partition? 00:10:37.880 --> 00:10:40.520 align:middle line:84% What is the maximum length of the subintervals? 00:10:40.520 --> 00:10:42.320 align:middle line:84% Well, it's just going to be the maximum 00:10:42.320 --> 00:10:45.005 align:middle line:90% of the norms of y and z. 00:10:45.005 --> 00:10:52.070 align:middle line:90% 00:10:52.070 --> 00:10:58.310 align:middle line:84% And note that the norm of x sub r, this 00:10:58.310 --> 00:11:13.510 align:middle line:84% is equal to the maximum of the norm of yr and the norm of zr. 00:11:13.510 --> 00:11:16.090 align:middle line:84% And since both of these converges to zero, 00:11:16.090 --> 00:11:18.340 align:middle line:84% the maximum of them also converges to zero. 00:11:18.340 --> 00:11:24.978 align:middle line:90% 00:11:24.978 --> 00:11:26.520 align:middle line:84% So now we just put this all together. 00:11:26.520 --> 00:11:29.460 align:middle line:90% 00:11:29.460 --> 00:11:40.000 align:middle line:84% So by this theorem, what do we know? 00:11:40.000 --> 00:11:47.730 align:middle line:84% We know that the Riemann sum associated to this converges 00:11:47.730 --> 00:11:50.940 align:middle line:90% to the integral ab of f. 00:11:50.940 --> 00:12:00.310 align:middle line:84% The Riemann sum associated to this sequence of tag partitions 00:12:00.310 --> 00:12:07.250 align:middle line:90% converges to acf. 00:12:07.250 --> 00:12:13.570 align:middle line:84% And then, this sequence of Riemann sums 00:12:13.570 --> 00:12:18.990 align:middle line:84% associated to these tag partitions converge to cbf. 00:12:18.990 --> 00:12:21.880 align:middle line:90% 00:12:21.880 --> 00:12:23.200 align:middle line:90% OK? 00:12:23.200 --> 00:12:28.700 align:middle line:84% And now we just use the fact that taking a Riemann sum 00:12:28.700 --> 00:12:33.040 align:middle line:90% is additive, right? 00:12:33.040 --> 00:12:42.080 align:middle line:84% Since the Riemann sum associated to this partition, which 00:12:42.080 --> 00:12:48.490 align:middle line:84% is just a union of these partitions, is equal to-- 00:12:48.490 --> 00:13:05.960 align:middle line:90% 00:13:05.960 --> 00:13:09.920 align:middle line:84% this implies taking the limit as r goes to infinity. 00:13:09.920 --> 00:13:15.650 align:middle line:84% The left hand side, as I wrote down here, converges to abf. 00:13:15.650 --> 00:13:18.830 align:middle line:84% And then the right hand side, the limit of the sum 00:13:18.830 --> 00:13:19.920 align:middle line:90% is the sum of the limits. 00:13:19.920 --> 00:13:23.660 align:middle line:90% So this converges to acf. 00:13:23.660 --> 00:13:27.530 align:middle line:84% And then the right thing here converges to-- 00:13:27.530 --> 00:13:30.110 align:middle line:84% the second thing here converges to the integral 00:13:30.110 --> 00:13:32.370 align:middle line:90% from c to b of f. 00:13:32.370 --> 00:13:34.080 align:middle line:90% And that's it. 00:13:34.080 --> 00:13:38.930 align:middle line:84% So really, the fact that the integral is additive 00:13:38.930 --> 00:13:51.320 align:middle line:84% follows from the simple fact that Riemann sums are additive, 00:13:51.320 --> 00:13:54.620 align:middle line:84% and the fact that limits respect algebraic operations. 00:13:54.620 --> 00:13:57.590 align:middle line:84% The limit of the sum is the sum of the limits. 00:13:57.590 --> 00:14:15.910 align:middle line:90% 00:14:15.910 --> 00:14:18.790 align:middle line:84% All right, so the Riemann integral 00:14:18.790 --> 00:14:22.030 align:middle line:90% is additive and also linear. 00:14:22.030 --> 00:14:24.910 align:middle line:84% So we see that the Riemann integral 00:14:24.910 --> 00:14:32.170 align:middle line:84% is something that respects algebraic operations 00:14:32.170 --> 00:14:39.680 align:middle line:84% with respect to the real numbers namely, adding and scalar 00:14:39.680 --> 00:14:40.640 align:middle line:90% multiplication. 00:14:40.640 --> 00:14:43.170 align:middle line:90% 00:14:43.170 --> 00:14:48.930 align:middle line:84% And it respects I guess, what you could 00:14:48.930 --> 00:14:52.890 align:middle line:90% say topological considerations. 00:14:52.890 --> 00:14:59.020 align:middle line:84% Or, not really topological, but this additive property 00:14:59.020 --> 00:15:02.070 align:middle line:90% is a very natural property. 00:15:02.070 --> 00:15:03.460 align:middle line:90% What am I getting at here? 00:15:03.460 --> 00:15:07.470 align:middle line:84% So now one could ask, how does the integral 00:15:07.470 --> 00:15:09.070 align:middle line:90% interact with inequality? 00:15:09.070 --> 00:15:14.187 align:middle line:84% So this is something we ask about all limits, right? 00:15:14.187 --> 00:15:16.020 align:middle line:84% Or at least limits of sequences that we saw. 00:15:16.020 --> 00:15:19.770 align:middle line:84% If one sequence lies below another one, 00:15:19.770 --> 00:15:21.270 align:middle line:90% taking limits respects that. 00:15:21.270 --> 00:15:24.220 align:middle line:90% 00:15:24.220 --> 00:15:26.860 align:middle line:84% Of course, that's not true for the derivative. 00:15:26.860 --> 00:15:29.330 align:middle line:84% I can have two functions, one bigger than the other, 00:15:29.330 --> 00:15:33.460 align:middle line:84% and the derivative of the smaller one 00:15:33.460 --> 00:15:37.870 align:middle line:84% be bigger than the derivative of the first one. 00:15:37.870 --> 00:15:41.080 align:middle line:84% But so one can ask the question let's suppose I have 00:15:41.080 --> 00:15:42.895 align:middle line:84% two functions, one smaller than the other. 00:15:42.895 --> 00:15:45.550 align:middle line:90% 00:15:45.550 --> 00:15:53.350 align:middle line:84% What is the-- what's the relationship 00:15:53.350 --> 00:15:54.490 align:middle line:90% between the integrals? 00:15:54.490 --> 00:16:03.180 align:middle line:90% 00:16:03.180 --> 00:16:03.680 align:middle line:90% OK. 00:16:03.680 --> 00:16:07.010 align:middle line:90% 00:16:07.010 --> 00:16:20.600 align:middle line:84% So suppose f and g are now two continuous functions on ab. 00:16:20.600 --> 00:16:23.550 align:middle line:90% 00:16:23.550 --> 00:16:28.500 align:middle line:84% So then the first part of this theorem 00:16:28.500 --> 00:16:36.540 align:middle line:84% is that if for all x and ab f of x 00:16:36.540 --> 00:16:44.260 align:middle line:84% is less than or equal to g of x, then the integral from ab of f 00:16:44.260 --> 00:16:47.670 align:middle line:84% is less than or equal to the integral from ab of g. 00:16:47.670 --> 00:16:48.170 align:middle line:90% All right? 00:16:48.170 --> 00:16:49.780 align:middle line:84% And this is kind of understandable. 00:16:49.780 --> 00:16:52.210 align:middle line:84% I mean again, if we think of the integral 00:16:52.210 --> 00:16:55.000 align:middle line:84% as being a theory of the area underneath the curve. 00:16:55.000 --> 00:16:57.950 align:middle line:84% If I have one function sitting above another function, 00:16:57.950 --> 00:17:01.900 align:middle line:84% then the area should be bigger than the area of the smaller 00:17:01.900 --> 00:17:07.670 align:middle line:84% one underneath the curve, which sits below the first one. 00:17:07.670 --> 00:17:17.750 align:middle line:84% And now the Riemann integral is a limit of sums in some sense, 00:17:17.750 --> 00:17:22.960 align:middle line:84% and a limit of certain sums involving f. 00:17:22.960 --> 00:17:25.390 align:middle line:90% And what? 00:17:25.390 --> 00:17:31.640 align:middle line:84% So we do have this relationship involving 00:17:31.640 --> 00:17:34.932 align:middle line:84% sums and absolute values with the triangle inequality, which 00:17:34.932 --> 00:17:36.890 align:middle line:84% says the absolute value of the sum is less than 00:17:36.890 --> 00:17:39.680 align:middle line:84% or equal to the sum of the absolute values. 00:17:39.680 --> 00:17:41.750 align:middle line:84% And if you think of the integral as 00:17:41.750 --> 00:17:46.760 align:middle line:84% being essentially in a limit of sums, or if you like, 00:17:46.760 --> 00:17:51.830 align:middle line:84% a continuous sum, then you should 00:17:51.830 --> 00:17:55.212 align:middle line:84% expect that the absolute value of the integral is less than 00:17:55.212 --> 00:17:57.170 align:middle line:84% or equal to the integral of the absolute value. 00:17:57.170 --> 00:18:00.450 align:middle line:90% And that's indeed what we have. 00:18:00.450 --> 00:18:04.770 align:middle line:84% Second is the absolute value of the integral 00:18:04.770 --> 00:18:08.850 align:middle line:84% from ab of f is less than or equal to the integral from ab 00:18:08.850 --> 00:18:11.970 align:middle line:90% of the absolute value of f. 00:18:11.970 --> 00:18:18.420 align:middle line:84% Like this is the triangle inequality for integrals. 00:18:18.420 --> 00:18:24.670 align:middle line:90% 00:18:24.670 --> 00:18:31.130 align:middle line:84% All right, so we'll prove one, using again, 00:18:31.130 --> 00:18:33.970 align:middle line:84% the main property we have about Riemann 00:18:33.970 --> 00:18:38.990 align:middle line:84% integrals that they satisfy this property. 00:18:38.990 --> 00:18:41.710 align:middle line:84% And then we'll deduce two directly from number one. 00:18:41.710 --> 00:18:48.300 align:middle line:90% 00:18:48.300 --> 00:18:59.180 align:middle line:84% So let's take a sequence of tag partitions 00:18:59.180 --> 00:19:00.665 align:middle line:90% with norms converging to zero. 00:19:00.665 --> 00:19:09.400 align:middle line:90% 00:19:09.400 --> 00:19:18.250 align:middle line:84% Then for all r, if we take the Riemann sum with respect 00:19:18.250 --> 00:19:28.910 align:middle line:84% to f of this tag partition, this is equal to j equals 1 to n. 00:19:28.910 --> 00:19:30.380 align:middle line:90% This is just the definition. 00:19:30.380 --> 00:19:35.520 align:middle line:90% 00:19:35.520 --> 00:19:40.170 align:middle line:84% r here is just indexing the sequence of tag partitions. 00:19:40.170 --> 00:19:42.420 align:middle line:84% And I should say this doesn't have 00:19:42.420 --> 00:19:46.110 align:middle line:90% to end at some single number-- 00:19:46.110 --> 00:19:47.580 align:middle line:90% in fact, it can't-- 00:19:47.580 --> 00:19:49.920 align:middle line:84% in common with all of these tag partitions. 00:19:49.920 --> 00:19:58.240 align:middle line:90% 00:19:58.240 --> 00:20:03.560 align:middle line:84% So here n of r is just the number of partition points 00:20:03.560 --> 00:20:06.860 align:middle line:84% that we have in our partition x superscript r. 00:20:06.860 --> 00:20:09.710 align:middle line:84% And since we're assuming f of x is less than or equal to g 00:20:09.710 --> 00:20:11.900 align:middle line:84% of x for all x and ab, this is less 00:20:11.900 --> 00:20:15.980 align:middle line:84% than or equal to sum from j equals 1 to n 00:20:15.980 --> 00:20:20.615 align:middle line:90% of r, g of cj of r. 00:20:20.615 --> 00:20:28.510 align:middle line:90% 00:20:28.510 --> 00:20:31.300 align:middle line:90% OK, and what is this last thing? 00:20:31.300 --> 00:20:37.870 align:middle line:84% This last thing is just equal to the Riemann sum of g, 00:20:37.870 --> 00:20:45.150 align:middle line:84% with respect to this tag partition x and xi. 00:20:45.150 --> 00:20:56.800 align:middle line:84% So we started off with the Riemann sum associated to f, 00:20:56.800 --> 00:20:58.975 align:middle line:84% and we showed that's less than or equal to-- 00:20:58.975 --> 00:21:03.610 align:middle line:90% 00:21:03.610 --> 00:21:05.950 align:middle line:90% just kind of summarizing here. 00:21:05.950 --> 00:21:09.760 align:middle line:90% 00:21:09.760 --> 00:21:12.250 align:middle line:84% That this Riemann sum is less than 00:21:12.250 --> 00:21:13.730 align:middle line:90% or equal to this Riemann sum. 00:21:13.730 --> 00:21:19.580 align:middle line:90% 00:21:19.580 --> 00:21:26.460 align:middle line:84% So now if I take the limit as r goes to infinity, 00:21:26.460 --> 00:21:29.190 align:middle line:84% this on the left hand side approaches the integral 00:21:29.190 --> 00:21:30.870 align:middle line:90% from ab. 00:21:30.870 --> 00:21:39.610 align:middle line:84% And since limits respect inequalities, 00:21:39.610 --> 00:21:44.195 align:middle line:84% I get this, which is what we wanted to prove. 00:21:44.195 --> 00:21:52.160 align:middle line:90% 00:21:52.160 --> 00:21:53.690 align:middle line:90% All right, and from number two-- 00:21:53.690 --> 00:21:57.070 align:middle line:84% I mean, for number two, we get from number one-- 00:21:57.070 --> 00:21:58.820 align:middle line:84% I mean, we get number two from number one, 00:21:58.820 --> 00:22:00.540 align:middle line:90% essentially quite quickly. 00:22:00.540 --> 00:22:06.290 align:middle line:84% So since plus or minus f is less than 00:22:06.290 --> 00:22:09.800 align:middle line:84% or equal to the absolute value of f, 00:22:09.800 --> 00:22:12.410 align:middle line:90% we get that plus or minus-- 00:22:12.410 --> 00:22:15.260 align:middle line:84% so integral of plus or minus f is 00:22:15.260 --> 00:22:19.700 align:middle line:84% less than or equal to the integral 00:22:19.700 --> 00:22:22.530 align:middle line:90% from ab of the absolute value. 00:22:22.530 --> 00:22:29.150 align:middle line:84% Now scalar multiplication we proved last time pulls out. 00:22:29.150 --> 00:22:35.120 align:middle line:84% So that means plus or minus the integral of f 00:22:35.120 --> 00:22:39.140 align:middle line:84% is less than or equal to the integral of the absolute value 00:22:39.140 --> 00:22:42.560 align:middle line:90% of f, which is the same as-- 00:22:42.560 --> 00:22:51.210 align:middle line:84% so for the minus sign this tells me this inequality. 00:22:51.210 --> 00:22:58.040 align:middle line:84% And for the plus sign, I get that inequality. 00:22:58.040 --> 00:23:00.350 align:middle line:84% But that's equivalent to saying that the absolute value 00:23:00.350 --> 00:23:13.303 align:middle line:84% of the integral of f is less than or equal to the integral 00:23:13.303 --> 00:23:14.220 align:middle line:90% of the absolute value. 00:23:14.220 --> 00:23:25.100 align:middle line:90% 00:23:25.100 --> 00:23:26.600 align:middle line:84% And that's the proof of the theorem. 00:23:26.600 --> 00:23:31.000 align:middle line:84% Let me make a small remark here-- 00:23:31.000 --> 00:23:37.510 align:middle line:84% and this will be an exercise on the homework to prove this-- 00:23:37.510 --> 00:23:38.650 align:middle line:90% is the following. 00:23:38.650 --> 00:23:42.000 align:middle line:90% 00:23:42.000 --> 00:23:44.070 align:middle line:84% We know that taking limits respects 00:23:44.070 --> 00:23:47.400 align:middle line:84% inequalities but not necessarily strict inequalities. 00:23:47.400 --> 00:23:53.180 align:middle line:84% So I can have two sequences, one strictly less 00:23:53.180 --> 00:23:54.860 align:middle line:90% than the other for all n. 00:23:54.860 --> 00:23:56.900 align:middle line:84% And then as n goes to infinity, their limits 00:23:56.900 --> 00:23:59.300 align:middle line:90% could in fact equal each other. 00:23:59.300 --> 00:24:03.170 align:middle line:84% So for example, if I take the first sequence 00:24:03.170 --> 00:24:06.320 align:middle line:84% to be 0, the second sequence to be 1 over n, 1 over n 00:24:06.320 --> 00:24:07.880 align:middle line:90% is always bigger than zero. 00:24:07.880 --> 00:24:11.600 align:middle line:90% Yet, they both converge to 0. 00:24:11.600 --> 00:24:18.350 align:middle line:84% So taking limits does not respect strict inequality. 00:24:18.350 --> 00:24:24.900 align:middle line:84% However, in this setting, which is very nice-- 00:24:24.900 --> 00:24:30.800 align:middle line:84% I mean, it's actually an extremely important property 00:24:30.800 --> 00:24:34.310 align:middle line:84% of the integral is that this limiting process 00:24:34.310 --> 00:24:38.630 align:middle line:84% that we do to take integrals in fact does-- 00:24:38.630 --> 00:24:40.730 align:middle line:84% it's not just any old limiting process. 00:24:40.730 --> 00:24:44.850 align:middle line:84% It actually does respect strict inequality. 00:24:44.850 --> 00:24:47.360 align:middle line:90% So what am I saying? 00:24:47.360 --> 00:24:54.010 align:middle line:84% Is that we can prove something a little bit stronger. 00:24:54.010 --> 00:24:57.350 align:middle line:84% And that's-- which I'll say it in its kind of reduced form 00:24:57.350 --> 00:24:57.850 align:middle line:90% here. 00:24:57.850 --> 00:25:10.280 align:middle line:84% If I take a continuous function and it's positive, 00:25:10.280 --> 00:25:14.880 align:middle line:84% then we already know that the integral of f 00:25:14.880 --> 00:25:17.760 align:middle line:84% is going to be non-negative by what we just proved. 00:25:17.760 --> 00:25:20.520 align:middle line:84% But in fact, the integral of f is positive. 00:25:20.520 --> 00:25:23.730 align:middle line:90% 00:25:23.730 --> 00:25:26.130 align:middle line:84% And again, this is also kind of important from 00:25:26.130 --> 00:25:28.920 align:middle line:90% a psychological standpoint. 00:25:28.920 --> 00:25:30.390 align:middle line:84% If we're thinking of the integral 00:25:30.390 --> 00:25:34.510 align:middle line:84% as being a theory of the area underneath the curve. 00:25:34.510 --> 00:25:37.600 align:middle line:90% So what this states is that-- 00:25:37.600 --> 00:25:39.960 align:middle line:84% and again, I'm not going to prove this. 00:25:39.960 --> 00:25:41.670 align:middle line:90% This will be on the assignment. 00:25:41.670 --> 00:25:53.810 align:middle line:90% 00:25:53.810 --> 00:25:59.560 align:middle line:84% Again, this coincides with our intuition 00:25:59.560 --> 00:26:02.590 align:middle line:84% of what a theory of the area underneath the curve should be. 00:26:02.590 --> 00:26:04.930 align:middle line:84% So this says if a continuous function 00:26:04.930 --> 00:26:06.760 align:middle line:84% is positive-- like the picture I drew you-- 00:26:06.760 --> 00:26:09.760 align:middle line:84% then the integral, which we interpret again, 00:26:09.760 --> 00:26:11.920 align:middle line:84% as the area underneath the curve, is positive. 00:26:11.920 --> 00:26:16.900 align:middle line:84% So that matches what our desire for this 00:26:16.900 --> 00:26:20.180 align:middle line:84% to be a theory of the area underneath the curve should be. 00:26:20.180 --> 00:26:20.680 align:middle line:90% All right? 00:26:20.680 --> 00:26:31.990 align:middle line:90% 00:26:31.990 --> 00:26:38.140 align:middle line:90% And so we have- if you like-- 00:26:38.140 --> 00:26:43.920 align:middle line:84% I mean, we could approve this from basic considerations 00:26:43.920 --> 00:26:46.560 align:middle line:84% in what the definition of the integral is. 00:26:46.560 --> 00:26:51.960 align:middle line:84% But we have the following theorem 00:26:51.960 --> 00:26:58.080 align:middle line:90% that for all alpha and r-- 00:26:58.080 --> 00:27:02.522 align:middle line:90% 00:27:02.522 --> 00:27:04.637 align:middle line:84% or I could just say the following. 00:27:04.637 --> 00:27:13.240 align:middle line:90% 00:27:13.240 --> 00:27:17.910 align:middle line:90% Well, maybe not stated this way. 00:27:17.910 --> 00:27:36.940 align:middle line:90% 00:27:36.940 --> 00:27:39.640 align:middle line:84% Going a little off script here, but it's OK. 00:27:39.640 --> 00:27:45.780 align:middle line:84% So we actually haven't computed a single integral yet. 00:27:45.780 --> 00:27:48.450 align:middle line:84% Really, it's tough to with this definition. 00:27:48.450 --> 00:27:50.970 align:middle line:84% In a minute, we'll prove the fundamental theorem of calculus 00:27:50.970 --> 00:27:53.650 align:middle line:84% and be able to do it very easily. 00:27:53.650 --> 00:27:58.810 align:middle line:84% But let me first prove at least the most basic integral. 00:27:58.810 --> 00:28:06.090 align:middle line:84% So integral from ab of the function 1 equals b minus a. 00:28:06.090 --> 00:28:14.880 align:middle line:90% 00:28:14.880 --> 00:28:19.370 align:middle line:90% So what's the proof? 00:28:19.370 --> 00:28:23.450 align:middle line:84% Take a sequence of partitions with the norms converging to 0. 00:28:23.450 --> 00:28:24.980 align:middle line:84% And we just look at the Riemann sum. 00:28:24.980 --> 00:28:30.540 align:middle line:90% 00:28:30.540 --> 00:28:33.990 align:middle line:84% And for one this will be quite easy to compute 00:28:33.990 --> 00:28:38.970 align:middle line:90% what the Riemann sums are. 00:28:38.970 --> 00:28:40.800 align:middle line:84% So this is a sequence of partitions 00:28:40.800 --> 00:28:46.410 align:middle line:90% with norm converging to 0. 00:28:46.410 --> 00:28:56.120 align:middle line:84% Then if I look at associated Riemann sums, 00:28:56.120 --> 00:29:01.930 align:middle line:84% this is equal to j equals 1 to n of r. 00:29:01.930 --> 00:29:04.360 align:middle line:84% No matter what I-- no matter what xi is, 00:29:04.360 --> 00:29:08.050 align:middle line:84% I just get 1, and then times the length of the interval, 00:29:08.050 --> 00:29:09.280 align:middle line:90% the subinterval. 00:29:09.280 --> 00:29:11.970 align:middle line:90% 00:29:11.970 --> 00:29:13.425 align:middle line:90% Now this is a telescoping sum. 00:29:13.425 --> 00:29:18.680 align:middle line:90% 00:29:18.680 --> 00:29:24.830 align:middle line:84% Meaning this is equal to x1 minus x0, plus x2, minus x1, 00:29:24.830 --> 00:29:28.850 align:middle line:84% plus up until xn minus xn minus 1. 00:29:28.850 --> 00:29:30.620 align:middle line:90% So all I pick up is-- 00:29:30.620 --> 00:29:33.890 align:middle line:90% 00:29:33.890 --> 00:29:37.490 align:middle line:90% not x1-- but x0 and xn. 00:29:37.490 --> 00:29:49.070 align:middle line:84% So this is equal to xj of r xn, sorry, of r minus x0 of r. 00:29:49.070 --> 00:29:51.500 align:middle line:84% And this is the last point in the partition. 00:29:51.500 --> 00:29:53.540 align:middle line:84% This is the first point in the partition. 00:29:53.540 --> 00:29:56.780 align:middle line:84% And as always for partitions, the last point is b. 00:29:56.780 --> 00:29:59.510 align:middle line:90% The first point is a. 00:29:59.510 --> 00:30:02.210 align:middle line:90% So this is equal to b minus a. 00:30:02.210 --> 00:30:04.760 align:middle line:90% 00:30:04.760 --> 00:30:11.030 align:middle line:84% Now this thing on the left hand side is equal to b minus a. 00:30:11.030 --> 00:30:21.490 align:middle line:84% So this tells me that the integral, which 00:30:21.490 --> 00:30:24.490 align:middle line:84% is equal to this limit of Riemann sums 00:30:24.490 --> 00:30:26.840 align:middle line:90% is equal to b minus a. 00:30:26.840 --> 00:30:29.035 align:middle line:84% All right, and then the assignment for this-- 00:30:29.035 --> 00:30:32.050 align:middle line:84% or I guess, when you see this last week, 00:30:32.050 --> 00:30:34.960 align:middle line:84% was to essentially, compute the integral for-- 00:30:34.960 --> 00:30:37.170 align:middle line:90% the integral from ab of x, dx. 00:30:37.170 --> 00:30:49.800 align:middle line:90% 00:30:49.800 --> 00:30:56.130 align:middle line:84% And so we get from this theorem and this theorem following 00:30:56.130 --> 00:31:01.060 align:middle line:90% bound for the area. 00:31:01.060 --> 00:31:05.020 align:middle line:84% So let me draw a picture so that this is not so surprising. 00:31:05.020 --> 00:31:08.730 align:middle line:84% Let's say I have a function from a b. 00:31:08.730 --> 00:31:13.080 align:middle line:90% 00:31:13.080 --> 00:31:14.700 align:middle line:90% There's the function f. 00:31:14.700 --> 00:31:19.213 align:middle line:90% 00:31:19.213 --> 00:31:21.130 align:middle line:84% So that was supposed to go through this point. 00:31:21.130 --> 00:31:22.505 align:middle line:84% So let's make that a little bit-- 00:31:22.505 --> 00:31:32.590 align:middle line:90% 00:31:32.590 --> 00:31:34.100 align:middle line:84% I suppose I don't need these parts. 00:31:34.100 --> 00:31:44.940 align:middle line:90% 00:31:44.940 --> 00:31:47.200 align:middle line:84% Let's say I have-- so far a continuous function, 00:31:47.200 --> 00:31:49.618 align:middle line:84% it always achieves a minimum and maximum. 00:31:49.618 --> 00:31:51.910 align:middle line:84% And at least for this picture, the minimum occurs here. 00:31:51.910 --> 00:31:55.010 align:middle line:90% The maximum occurs here. 00:31:55.010 --> 00:32:00.070 align:middle line:84% Now, what's the comparison about areas from this picture? 00:32:00.070 --> 00:32:03.670 align:middle line:84% Well, the area underneath the curve of f 00:32:03.670 --> 00:32:05.590 align:middle line:84% will be bigger than or equal to the area 00:32:05.590 --> 00:32:09.430 align:middle line:84% underneath the curve of the constant function 00:32:09.430 --> 00:32:11.860 align:middle line:90% equal to little m sub f. 00:32:11.860 --> 00:32:17.710 align:middle line:84% And that's just b times a times m sub f, again, by intuition. 00:32:17.710 --> 00:32:20.878 align:middle line:84% But this is also seen from the theorem we just proved along 00:32:20.878 --> 00:32:22.670 align:middle line:84% with the linearity of the Riemann integral, 00:32:22.670 --> 00:32:24.700 align:middle line:90% which we've already proved. 00:32:24.700 --> 00:32:28.510 align:middle line:84% And the integral of f, the area underneath this curve 00:32:28.510 --> 00:32:31.540 align:middle line:84% is also less than or equal to the area underneath the curve 00:32:31.540 --> 00:32:33.445 align:middle line:90% where f achieves a maximum. 00:32:33.445 --> 00:32:36.240 align:middle line:90% 00:32:36.240 --> 00:32:39.030 align:middle line:84% It's the 12th week of class, and I still 00:32:39.030 --> 00:32:43.110 align:middle line:84% haven't brought colored chalk, so hopefully, that 00:32:43.110 --> 00:32:46.050 align:middle line:84% was clear enough without me having to color stuff in. 00:32:46.050 --> 00:32:50.590 align:middle line:90% So the theorem is the following. 00:32:50.590 --> 00:32:54.210 align:middle line:84% If f-- again, we're working with continuous functions 00:32:54.210 --> 00:32:57.150 align:middle line:84% because that's all we can integrate or are going 00:32:57.150 --> 00:33:01.050 align:middle line:90% to integrate in this section-- 00:33:01.050 --> 00:33:13.410 align:middle line:84% and I have these two numbers, the inf of fx, which we know 00:33:13.410 --> 00:33:18.280 align:middle line:84% is in fact, a min, meaning there exist a point in here-- 00:33:18.280 --> 00:33:21.090 align:middle line:84% so in fact, I'm going to write "min." we know that by the min, 00:33:21.090 --> 00:33:23.100 align:middle line:84% max theorem that for every continuous function 00:33:23.100 --> 00:33:24.540 align:middle line:90% it achieves its minimum-- 00:33:24.540 --> 00:33:29.720 align:middle line:90% 00:33:29.720 --> 00:33:30.890 align:middle line:90% and max. 00:33:30.890 --> 00:33:39.040 align:middle line:90% 00:33:39.040 --> 00:33:41.630 align:middle line:84% Then, as I was saying a minute ago, 00:33:41.630 --> 00:33:44.620 align:middle line:84% the area underneath the curve little m sub 00:33:44.620 --> 00:33:46.990 align:middle line:84% f, which is this smaller square, is 00:33:46.990 --> 00:33:51.850 align:middle line:84% less than or equal to the area underneath the curve of f. 00:33:51.850 --> 00:33:55.130 align:middle line:90% 00:33:55.130 --> 00:33:58.700 align:middle line:84% So little m sub f times b minus a, 00:33:58.700 --> 00:34:01.100 align:middle line:84% the area underneath the minimum, is 00:34:01.100 --> 00:34:06.950 align:middle line:84% less than or equal to the area underneath the curve of f, 00:34:06.950 --> 00:34:09.679 align:middle line:84% which is less than or equal to the area 00:34:09.679 --> 00:34:16.770 align:middle line:84% underneath the curve of the taller line, capital M sub f. 00:34:16.770 --> 00:34:24.179 align:middle line:90% 00:34:24.179 --> 00:34:25.150 align:middle line:90% How do we prove this? 00:34:25.150 --> 00:34:27.690 align:middle line:84% Well, we just use these previous two theorems, right? 00:34:27.690 --> 00:34:31.580 align:middle line:90% 00:34:31.580 --> 00:34:38.889 align:middle line:84% Since little m sub f is less than or equal to capital 00:34:38.889 --> 00:34:41.560 align:middle line:90% M sub f-- 00:34:41.560 --> 00:34:44.300 align:middle line:84% so since little m sub f is less than or equal to f of x, 00:34:44.300 --> 00:34:47.500 align:middle line:84% which is less than or equal to capital M sub f, 00:34:47.500 --> 00:34:53.800 align:middle line:84% for all x and ab we can apply that theorem up there to get 00:34:53.800 --> 00:34:58.480 align:middle line:84% that the integral of little m sub f-- again, 00:34:58.480 --> 00:35:00.580 align:middle line:90% this is just a fixed number-- 00:35:00.580 --> 00:35:04.165 align:middle line:84% is less than or equal to the integral from ab of f. 00:35:04.165 --> 00:35:08.390 align:middle line:90% 00:35:08.390 --> 00:35:11.280 align:middle line:84% And now again, these are just fixed numbers. 00:35:11.280 --> 00:35:15.980 align:middle line:84% So by what we proved here that the integral from a to b of 1 00:35:15.980 --> 00:35:18.920 align:middle line:84% is b minus a, the fact that scalers 00:35:18.920 --> 00:35:21.590 align:middle line:84% pull out-- so this number here can come out-- 00:35:21.590 --> 00:35:27.190 align:middle line:84% and this is just m sub f times the integral from ab of one. 00:35:27.190 --> 00:35:30.850 align:middle line:84% So this is little m sub f times b minus a. 00:35:30.850 --> 00:35:38.665 align:middle line:84% Over here, this is capital M sub f times b minus a. 00:35:38.665 --> 00:35:39.540 align:middle line:90% And that's the proof. 00:35:39.540 --> 00:35:50.730 align:middle line:90% 00:35:50.730 --> 00:35:52.860 align:middle line:84% So now let me make a few comments 00:35:52.860 --> 00:35:58.675 align:middle line:90% about some conventions, really. 00:35:58.675 --> 00:36:01.840 align:middle line:90% 00:36:01.840 --> 00:36:04.520 align:middle line:90% So we've been talking about-- 00:36:04.520 --> 00:36:06.770 align:middle line:84% or at least, I've been talking about, you're not here. 00:36:06.770 --> 00:36:09.110 align:middle line:84% So we haven't been talking at all. 00:36:09.110 --> 00:36:14.880 align:middle line:84% So we've been talking about the integral from ab 00:36:14.880 --> 00:36:21.550 align:middle line:84% to f when a is less than b, But I'm going to use-- 00:36:21.550 --> 00:36:27.390 align:middle line:84% I'm just going to set down some convention, 00:36:27.390 --> 00:36:29.520 align:middle line:84% or you could say this is really notation. 00:36:29.520 --> 00:36:33.830 align:middle line:90% 00:36:33.830 --> 00:36:35.195 align:middle line:90% When I have this-- 00:36:35.195 --> 00:36:37.720 align:middle line:90% 00:36:37.720 --> 00:36:43.130 align:middle line:84% when I write down this symbol, integral a to a of f, this 00:36:43.130 --> 00:36:45.830 align:middle line:84% is really just another way of writing 0. 00:36:45.830 --> 00:36:54.250 align:middle line:84% And another is that if, in fact, b is less than a, 00:36:54.250 --> 00:37:01.190 align:middle line:90% then the integral from ab to f-- 00:37:01.190 --> 00:37:03.650 align:middle line:84% so remember, we've been talking about the integrals from-- 00:37:03.650 --> 00:37:09.170 align:middle line:84% over intervals-- or where the number on the bottom 00:37:09.170 --> 00:37:11.720 align:middle line:84% is less than the number on the top, 00:37:11.720 --> 00:37:15.530 align:middle line:84% so whenever I write this symbol with the bigger 00:37:15.530 --> 00:37:18.530 align:middle line:84% number on the bottom and the smaller number on the top, 00:37:18.530 --> 00:37:26.630 align:middle line:84% you should read this as minus the integral of, 00:37:26.630 --> 00:37:31.010 align:middle line:84% with this being in the right place, 00:37:31.010 --> 00:37:34.790 align:middle line:84% meaning this number here is less than this number here. 00:37:34.790 --> 00:37:41.480 align:middle line:84% OK and so again, these are really just notations, 00:37:41.480 --> 00:37:44.420 align:middle line:84% that if I write this, this is a fancy way of writing 0. 00:37:44.420 --> 00:37:49.610 align:middle line:84% When I write this, this means minus the integral 00:37:49.610 --> 00:37:52.760 align:middle line:84% with the right numbers on top and bottom, 00:37:52.760 --> 00:37:56.120 align:middle line:84% meaning the smaller numbers on the bottom, the bigger 00:37:56.120 --> 00:37:56.915 align:middle line:90% numbers on the top. 00:37:56.915 --> 00:38:02.300 align:middle line:90% 00:38:02.300 --> 00:38:05.240 align:middle line:90% So maybe you ask-- so this is-- 00:38:05.240 --> 00:38:09.730 align:middle line:90% so why do I define that to be 0. 00:38:09.730 --> 00:38:20.080 align:middle line:84% Well, if you like, 1 is consistent with the fact 00:38:20.080 --> 00:38:25.060 align:middle line:84% that for all continuous functions, 00:38:25.060 --> 00:38:31.225 align:middle line:84% if I take the integral from a to b with b bigger than a, 00:38:31.225 --> 00:38:34.840 align:middle line:84% and I take the limit as b goes to a, this equals 0. 00:38:34.840 --> 00:38:38.660 align:middle line:90% 00:38:38.660 --> 00:38:42.073 align:middle line:84% And two is then consistent with number one and then additivity. 00:38:42.073 --> 00:38:59.600 align:middle line:90% 00:38:59.600 --> 00:39:02.990 align:middle line:84% With additivity in number one, because then by number one, 00:39:02.990 --> 00:39:04.790 align:middle line:90% I get that the-- 00:39:04.790 --> 00:39:08.120 align:middle line:84% so additivity will tell me that the integral from a to a of f 00:39:08.120 --> 00:39:10.640 align:middle line:84% is equal to the integral from ab to f, 00:39:10.640 --> 00:39:16.580 align:middle line:84% plus the integral from baf, here assuming b is less than a. 00:39:16.580 --> 00:39:23.610 align:middle line:84% So if by 1 this is 0, then to be consistent with additivity, 00:39:23.610 --> 00:39:26.970 align:middle line:84% this would force the integral from ab 00:39:26.970 --> 00:39:30.780 align:middle line:84% to f to be minus the integral from ba of f. 00:39:30.780 --> 00:39:36.360 align:middle line:84% So both of these conventions, if you like, 00:39:36.360 --> 00:39:47.010 align:middle line:84% are consistent with the properties of the Riemann 00:39:47.010 --> 00:39:50.040 align:middle line:84% integral, namely that the limit as b 00:39:50.040 --> 00:39:54.240 align:middle line:84% goes to a of the area underneath this curve, 00:39:54.240 --> 00:39:57.180 align:middle line:84% as the base gets smaller and smaller is 0. 00:39:57.180 --> 00:39:59.880 align:middle line:84% And then from-- if I assume this, 00:39:59.880 --> 00:40:03.060 align:middle line:84% then additivity tells me that the integral from ab 00:40:03.060 --> 00:40:05.430 align:middle line:84% has to be minus the integral from b to a of f. 00:40:05.430 --> 00:40:19.790 align:middle line:90% 00:40:19.790 --> 00:40:23.945 align:middle line:84% All right, so now we computed successfully one integral. 00:40:23.945 --> 00:40:26.560 align:middle line:90% 00:40:26.560 --> 00:40:29.000 align:middle line:84% Of course, when you took calculus, that was not all 00:40:29.000 --> 00:40:29.750 align:middle line:90% you could do. 00:40:29.750 --> 00:40:32.720 align:middle line:90% 00:40:32.720 --> 00:40:39.050 align:middle line:84% Calculus is what it is because of the hero of the story, which 00:40:39.050 --> 00:40:41.580 align:middle line:84% is Fundamental Theorem of Calculus, 00:40:41.580 --> 00:40:42.830 align:middle line:90% which is what we'll prove now. 00:40:42.830 --> 00:40:49.090 align:middle line:90% 00:40:49.090 --> 00:40:55.940 align:middle line:84% Fundamental Theorem of Calculus, which states following. 00:40:55.940 --> 00:41:01.150 align:middle line:90% If I have a function big F-- 00:41:01.150 --> 00:41:07.075 align:middle line:84% so first off, let little f be continuous function. 00:41:07.075 --> 00:41:11.660 align:middle line:90% 00:41:11.660 --> 00:41:15.370 align:middle line:84% So the first statement is basically 00:41:15.370 --> 00:41:18.190 align:middle line:90% how to compute integrals. 00:41:18.190 --> 00:41:27.670 align:middle line:84% If capital F integral from ab to r 00:41:27.670 --> 00:41:40.960 align:middle line:84% is differentiable everywhere on ab and f prime equals little f, 00:41:40.960 --> 00:41:44.110 align:middle line:84% then the integral from ab of little f 00:41:44.110 --> 00:41:49.450 align:middle line:90% equals f of b minus f of a. 00:41:49.450 --> 00:41:52.350 align:middle line:90% 00:41:52.350 --> 00:41:54.840 align:middle line:84% Another way of saying this is that the integral 00:41:54.840 --> 00:42:01.500 align:middle line:84% from ab of capital F prime is equal to f of b minus f of a. 00:42:01.500 --> 00:42:05.430 align:middle line:90% 00:42:05.430 --> 00:42:08.010 align:middle line:84% And then the second part-- so the first part 00:42:08.010 --> 00:42:09.840 align:middle line:90% is about computing integrals. 00:42:09.840 --> 00:42:13.980 align:middle line:84% The second part is about solving differential equations, 00:42:13.980 --> 00:42:15.540 align:middle line:90% basically. 00:42:15.540 --> 00:42:34.680 align:middle line:84% So the function g of x equals the integral from a to x of f. 00:42:34.680 --> 00:42:45.590 align:middle line:84% So for each x and ab, I stick it in as the upper limit 00:42:45.590 --> 00:42:48.880 align:middle line:90% on this integral. 00:42:48.880 --> 00:42:54.430 align:middle line:84% This function is differentiable on ab. 00:42:54.430 --> 00:43:01.380 align:middle line:90% 00:43:01.380 --> 00:43:04.020 align:middle line:84% And it satisfies the simplest differential 00:43:04.020 --> 00:43:12.065 align:middle line:84% equation, which is g prime equals f, and g of a equals 0. 00:43:12.065 --> 00:43:18.040 align:middle line:90% 00:43:18.040 --> 00:43:19.630 align:middle line:84% So this is the Fundamental Theorem 00:43:19.630 --> 00:43:21.760 align:middle line:84% of Calculus, the first part being about how do you 00:43:21.760 --> 00:43:25.000 align:middle line:84% compute integrals, the second, how do you compute solutions 00:43:25.000 --> 00:43:26.680 align:middle line:90% to a differential equation. 00:43:26.680 --> 00:43:30.280 align:middle line:84% Namely, how do I find the solution to the problem 00:43:30.280 --> 00:43:32.110 align:middle line:90% g prime equals little f. 00:43:32.110 --> 00:43:33.430 align:middle line:90% So little f is the given. 00:43:33.430 --> 00:43:37.330 align:middle line:84% I want to find a function g that satisfies g prime equals little 00:43:37.330 --> 00:43:40.840 align:middle line:84% f with initial condition capital G of a equals 0. 00:43:40.840 --> 00:43:44.080 align:middle line:84% That's given by this function here where I take little left 00:43:44.080 --> 00:43:46.630 align:middle line:90% and I integrate it from a to x. 00:43:46.630 --> 00:43:50.440 align:middle line:84% So that's how I interpret the Fundamental Theorem 00:43:50.440 --> 00:43:53.350 align:middle line:84% of Calculus, the first about solving integrals, 00:43:53.350 --> 00:43:56.890 align:middle line:84% the second about solving differential equations. 00:43:56.890 --> 00:44:03.500 align:middle line:90% 00:44:03.500 --> 00:44:07.810 align:middle line:84% All right so this is the hero of calculus, 00:44:07.810 --> 00:44:10.900 align:middle line:90% the Batman of the story. 00:44:10.900 --> 00:44:13.630 align:middle line:84% And I said at one point that the Mean Value Theorem 00:44:13.630 --> 00:44:17.690 align:middle line:90% is the Alfred of this story. 00:44:17.690 --> 00:44:19.970 align:middle line:84% And therefore, it should play a decisive role 00:44:19.970 --> 00:44:27.555 align:middle line:90% in proving this theorem. 00:44:27.555 --> 00:44:34.410 align:middle line:90% 00:44:34.410 --> 00:44:34.910 align:middle line:90% All right. 00:44:34.910 --> 00:44:48.280 align:middle line:90% 00:44:48.280 --> 00:44:54.160 align:middle line:84% So we're going to connect these two things. 00:44:54.160 --> 00:44:56.730 align:middle line:84% So in order to even get at this guy, 00:44:56.730 --> 00:44:58.950 align:middle line:84% we have to take a partition, a sequence of partitions 00:44:58.950 --> 00:45:01.260 align:middle line:84% converging with norm converging to 0. 00:45:01.260 --> 00:45:04.750 align:middle line:90% So and then see what we get. 00:45:04.750 --> 00:45:07.920 align:middle line:90% So this is the proof part one. 00:45:07.920 --> 00:45:16.520 align:middle line:90% 00:45:16.520 --> 00:45:24.880 align:middle line:84% So let's take a sequence of tag partitions-- 00:45:24.880 --> 00:45:27.060 align:middle line:84% I keep forgetting to write tag and say the word tag, 00:45:27.060 --> 00:45:32.290 align:middle line:84% but you should hear that even though I don't say it-- 00:45:32.290 --> 00:45:33.445 align:middle line:90% with norm. 00:45:33.445 --> 00:45:36.690 align:middle line:90% 00:45:36.690 --> 00:45:40.380 align:middle line:84% Actually, we're not even going to-- 00:45:40.380 --> 00:45:41.210 align:middle line:90% this is correct. 00:45:41.210 --> 00:45:46.297 align:middle line:90% 00:45:46.297 --> 00:45:48.630 align:middle line:84% I'm actually going to come up with the tags in a minute. 00:45:48.630 --> 00:45:53.090 align:middle line:84% So let's first take a sequence of partition. 00:45:53.090 --> 00:45:55.820 align:middle line:84% So this is actually completely correct. 00:45:55.820 --> 00:45:59.570 align:middle line:84% So no tags yet, no points chosen in between, 00:45:59.570 --> 00:46:02.540 align:middle line:90% with Norm converging to 0. 00:46:02.540 --> 00:46:08.120 align:middle line:84% And then I'm going to choose a special tag for each r 00:46:08.120 --> 00:46:12.680 align:middle line:84% so that in the limit I get this equality between the integral 00:46:12.680 --> 00:46:16.040 align:middle line:84% of f and capital F of b minus f, capital F of a. 00:46:16.040 --> 00:46:20.210 align:middle line:90% 00:46:20.210 --> 00:46:27.210 align:middle line:84% So here comes Alfred by the Mean Value Theorem. 00:46:27.210 --> 00:46:42.040 align:middle line:84% So we have this subinterval x of j minus 1 and x sub j. 00:46:42.040 --> 00:46:46.780 align:middle line:84% So by the Mean Value Theorem, there exist-- 00:46:46.780 --> 00:47:05.740 align:middle line:84% for each j there exists a point in between these two guys 00:47:05.740 --> 00:47:17.660 align:middle line:84% such that if I take f of x sub j r minus f-- 00:47:17.660 --> 00:47:19.640 align:middle line:84% so capital F, remember, is this function 00:47:19.640 --> 00:47:23.360 align:middle line:84% that's differential and whose derivative gives me little f-- 00:47:23.360 --> 00:47:28.482 align:middle line:90% 00:47:28.482 --> 00:47:32.170 align:middle line:84% this is equal to capital F prime of xi j 00:47:32.170 --> 00:47:37.795 align:middle line:90% of r times xj minus xj minus 1. 00:47:37.795 --> 00:47:40.520 align:middle line:90% 00:47:40.520 --> 00:47:43.250 align:middle line:84% And by assumption, the derivative of capital F 00:47:43.250 --> 00:47:45.560 align:middle line:90% is equal to little f, right? 00:47:45.560 --> 00:47:50.510 align:middle line:84% So I'm going to replace this f prime with little f. 00:47:50.510 --> 00:47:56.010 align:middle line:90% 00:47:56.010 --> 00:48:01.005 align:middle line:84% So now these xi's I will take as my tag, my sequence of tags. 00:48:01.005 --> 00:48:05.040 align:middle line:90% 00:48:05.040 --> 00:48:07.090 align:middle line:84% Put a star by this relation here. 00:48:07.090 --> 00:48:14.830 align:middle line:90% 00:48:14.830 --> 00:48:18.080 align:middle line:90% What do we conclude? 00:48:18.080 --> 00:48:24.920 align:middle line:84% If I look at the Riemann sum for f associated 00:48:24.920 --> 00:48:36.560 align:middle line:84% to now this sequence of tags, where now the xi's are 00:48:36.560 --> 00:48:41.340 align:middle line:84% exactly these xi's that satisfy this relation. 00:48:41.340 --> 00:48:44.460 align:middle line:84% This is equal to as before, sum from j 00:48:44.460 --> 00:48:51.090 align:middle line:90% equals 1 to nr f of xg xi sub j. 00:48:51.090 --> 00:48:54.810 align:middle line:90% 00:48:54.810 --> 00:48:57.900 align:middle line:84% Now by how these xi's were chosen-- remember, 00:48:57.900 --> 00:49:02.340 align:middle line:84% they were chosen to satisfy this relation here, star 00:49:02.340 --> 00:49:03.520 align:middle line:90% by the Mean Value Theorem. 00:49:03.520 --> 00:49:07.830 align:middle line:84% So this is equal to the sum from j equals 1 two n sub 00:49:07.830 --> 00:49:15.600 align:middle line:84% r, f of x sub j minus f of x sub j minus 1. 00:49:15.600 --> 00:49:19.700 align:middle line:90% 00:49:19.700 --> 00:49:23.490 align:middle line:84% And again, this is a telescoping sum. 00:49:23.490 --> 00:49:29.010 align:middle line:84% And so all I pick up is when j equals n of r, the last point. 00:49:29.010 --> 00:49:38.560 align:middle line:84% And when j equals 0-- so this is equal to f of the last point 00:49:38.560 --> 00:49:40.525 align:middle line:90% minus f of the first point. 00:49:40.525 --> 00:49:43.730 align:middle line:90% 00:49:43.730 --> 00:49:48.010 align:middle line:84% Now for partition, this is always b. 00:49:48.010 --> 00:49:49.150 align:middle line:90% And this is always a. 00:49:49.150 --> 00:49:53.470 align:middle line:90% So this is f of b minus f of a. 00:49:53.470 --> 00:50:00.280 align:middle line:84% So every one of these Riemann sums for this sequence of tag 00:50:00.280 --> 00:50:02.650 align:middle line:84% partitions gives me f of b minus f of a 00:50:02.650 --> 00:50:05.890 align:middle line:84% So then I get to take the limit, and I get the integral. 00:50:05.890 --> 00:50:10.580 align:middle line:84% And therefore, integral from ab to f, 00:50:10.580 --> 00:50:21.640 align:middle line:84% which is limited as r goes to infinity of s of f equals-- 00:50:21.640 --> 00:50:23.140 align:middle line:84% and we just computed this is always 00:50:23.140 --> 00:50:24.910 align:middle line:90% equal to f of b minus f of a. 00:50:24.910 --> 00:50:34.140 align:middle line:90% 00:50:34.140 --> 00:50:37.360 align:middle line:84% We took a sequence of-- just a recap-- 00:50:37.360 --> 00:50:40.650 align:middle line:84% we took a sequence of partitions with norms converging to 0. 00:50:40.650 --> 00:50:46.480 align:middle line:84% And we chose special tags using the Mean Value Theorem. 00:50:46.480 --> 00:50:48.300 align:middle line:84% So by the Mean Value Theorem, on each one 00:50:48.300 --> 00:50:54.580 align:middle line:84% of these subintervals capital F evaluated at the right endpoint 00:50:54.580 --> 00:50:57.310 align:middle line:84% minus f evaluated at the left endpoint 00:50:57.310 --> 00:51:01.180 align:middle line:84% is equal to f, which is the derivative of capital 00:51:01.180 --> 00:51:03.550 align:middle line:84% F evaluated at some point in between times 00:51:03.550 --> 00:51:07.000 align:middle line:90% the length of the interval. 00:51:07.000 --> 00:51:09.190 align:middle line:84% So now if we take the Riemann sum associated 00:51:09.190 --> 00:51:11.620 align:middle line:84% to this sequence of tag partitions 00:51:11.620 --> 00:51:16.670 align:middle line:84% where these xi's are defined by this condition, 00:51:16.670 --> 00:51:19.670 align:middle line:84% then we can actually compute this Riemann sum, 00:51:19.670 --> 00:51:22.280 align:middle line:84% and it just gives me f of b minus f of a no matter 00:51:22.280 --> 00:51:23.870 align:middle line:90% what r is. 00:51:23.870 --> 00:51:27.320 align:middle line:84% So then when I take the limit as r goes to infinity, 00:51:27.320 --> 00:51:30.530 align:middle line:84% the left side goes to the integral, 00:51:30.530 --> 00:51:34.670 align:middle line:84% but it's always equal to capital F of b minus f of a. 00:51:34.670 --> 00:51:35.930 align:middle line:90% So that proves number one. 00:51:35.930 --> 00:51:53.310 align:middle line:90% 00:51:53.310 --> 00:51:59.190 align:middle line:84% So for number two, what do we want to show? 00:51:59.190 --> 00:52:00.460 align:middle line:90% We want to show this function. 00:52:00.460 --> 00:52:06.270 align:middle line:84% So first off, it's by convention g 00:52:06.270 --> 00:52:09.570 align:middle line:84% of a, which is the integral from a to a of f, this is zero. 00:52:09.570 --> 00:52:11.790 align:middle line:84% So I don't have to check the second condition. 00:52:11.790 --> 00:52:14.670 align:middle line:84% I just need to check that this function is differentiable, 00:52:14.670 --> 00:52:17.730 align:middle line:84% and the derivative of that function is little f. 00:52:17.730 --> 00:52:22.420 align:middle line:90% 00:52:22.420 --> 00:52:27.780 align:middle line:90% Let c be in ab. 00:52:27.780 --> 00:52:29.820 align:middle line:90% So what do we want to show? 00:52:29.820 --> 00:52:33.090 align:middle line:84% We want to show the derivative of capital 00:52:33.090 --> 00:52:35.130 align:middle line:90% G is equal to little f. 00:52:35.130 --> 00:52:38.550 align:middle line:84% So we would like to show that the limit as x goes 00:52:38.550 --> 00:52:47.150 align:middle line:84% to c of the integral from a to x of f minus integral from a 00:52:47.150 --> 00:52:52.430 align:middle line:84% to c of f, over x minus c, which is just g of x minus g of c, 00:52:52.430 --> 00:52:54.380 align:middle line:90% equals f of c. 00:52:54.380 --> 00:53:00.280 align:middle line:90% 00:53:00.280 --> 00:53:04.270 align:middle line:84% so we're going to do this by the books-- 00:53:04.270 --> 00:53:05.770 align:middle line:90% by the book, I guess. 00:53:05.770 --> 00:53:08.280 align:middle line:90% 00:53:08.280 --> 00:53:11.758 align:middle line:84% When I say-- so that's kind of a bad pun, I guess. 00:53:11.758 --> 00:53:12.300 align:middle line:90% I don't know. 00:53:12.300 --> 00:53:17.290 align:middle line:84% But meaning, an epsilon delta proof for this limit. 00:53:17.290 --> 00:53:21.730 align:middle line:90% So let epsilon be positive. 00:53:21.730 --> 00:53:26.320 align:middle line:84% The Delta that we pick in the end depends on-- 00:53:26.320 --> 00:53:30.580 align:middle line:84% and we'll use crucially the fact that f is continuous at c. 00:53:30.580 --> 00:53:37.180 align:middle line:90% 00:53:37.180 --> 00:53:46.740 align:middle line:84% So since f is continuous at c, there 00:53:46.740 --> 00:53:53.850 align:middle line:84% exists a delta 0 positive such that what? 00:53:53.850 --> 00:53:57.780 align:middle line:84% t minus c less than delta 0 implies 00:53:57.780 --> 00:54:03.900 align:middle line:84% that f of t minus f of c is less than epsilon-- 00:54:03.900 --> 00:54:06.600 align:middle line:84% let's give ourselves a little room-- over 2. 00:54:06.600 --> 00:54:09.670 align:middle line:90% 00:54:09.670 --> 00:54:14.870 align:middle line:90% Choose delta to be this delta 0. 00:54:14.870 --> 00:54:17.220 align:middle line:84% Remember, we have to show for every epsilon, 00:54:17.220 --> 00:54:20.730 align:middle line:84% there exists a delta so that if x minus c is less than delta, 00:54:20.730 --> 00:54:24.200 align:middle line:84% then this minus this in absolute value is less than epsilon. 00:54:24.200 --> 00:54:26.930 align:middle line:84% I'm now saying, choose delta to be this delta 0 00:54:26.930 --> 00:54:34.610 align:middle line:84% where this delta 0 is ensuring this right here. 00:54:34.610 --> 00:54:37.700 align:middle line:84% OK as long as t minus C is less than delta 0. 00:54:37.700 --> 00:54:41.550 align:middle line:90% 00:54:41.550 --> 00:54:48.680 align:middle line:84% Now because of sign conventions, we'll do two cases. 00:54:48.680 --> 00:54:51.710 align:middle line:84% So not really, I'm just going to do one case. 00:54:51.710 --> 00:55:03.960 align:middle line:84% So suppose that x is between c and c plus delta. 00:55:03.960 --> 00:55:07.500 align:middle line:90% So we want to also show-- 00:55:07.500 --> 00:55:09.870 align:middle line:84% we'll also need to be able to do the case that x 00:55:09.870 --> 00:55:12.330 align:middle line:90% is between c minus delta and c. 00:55:12.330 --> 00:55:16.170 align:middle line:84% That would cover the whole range of the absolute value 00:55:16.170 --> 00:55:19.993 align:middle line:84% of x minus c is less than delta but bigger than zero. 00:55:19.993 --> 00:55:21.285 align:middle line:90% I'm just going to do this case. 00:55:21.285 --> 00:55:28.000 align:middle line:90% 00:55:28.000 --> 00:55:30.360 align:middle line:84% So now we want to show that if x is in here, 00:55:30.360 --> 00:55:33.550 align:middle line:84% then this thing minus this thing in absolute value 00:55:33.550 --> 00:55:34.425 align:middle line:90% is less than epsilon. 00:55:34.425 --> 00:55:37.750 align:middle line:90% 00:55:37.750 --> 00:55:41.310 align:middle line:84% So first off, I would like to note 00:55:41.310 --> 00:55:53.400 align:middle line:84% that if t is in this interval c to x, then that tells me 00:55:53.400 --> 00:55:59.070 align:middle line:84% that the absolute value of t minus c, which is equal 00:55:59.070 --> 00:56:05.140 align:middle line:84% just t minus c, this is less than or equal to x minus c, 00:56:05.140 --> 00:56:07.240 align:middle line:90% which is less than delta. 00:56:07.240 --> 00:56:10.390 align:middle line:84% And remember, this equals delta 0, all right? 00:56:10.390 --> 00:56:17.980 align:middle line:84% So as long as I'm in this interval here, 00:56:17.980 --> 00:56:20.560 align:middle line:84% the absolute value of t minus c is less than delta 0. 00:56:20.560 --> 00:56:23.650 align:middle line:84% So let me just draw a picture so that this is pretty clear. 00:56:23.650 --> 00:56:25.360 align:middle line:90% This length is delta. 00:56:25.360 --> 00:56:31.210 align:middle line:84% If I take any t in between, then this distance from t to c 00:56:31.210 --> 00:56:33.297 align:middle line:84% is also going to be less than delta. 00:56:33.297 --> 00:56:34.630 align:middle line:90% So that's all I've written down. 00:56:34.630 --> 00:56:41.440 align:middle line:90% 00:56:41.440 --> 00:56:52.980 align:middle line:84% Thus, by now compute the difference quotient 00:56:52.980 --> 00:56:55.530 align:middle line:90% minus the proposed limit. 00:56:55.530 --> 00:57:03.300 align:middle line:84% So looking at 1 over x minus c times the integral minus 00:57:03.300 --> 00:57:10.600 align:middle line:84% the integral from ac of f minus f of c. 00:57:10.600 --> 00:57:13.340 align:middle line:84% This is the thing I want to show is less than epsilon. 00:57:13.340 --> 00:57:18.190 align:middle line:84% Now by additivity, the integral from a to x since x is-- 00:57:18.190 --> 00:57:19.810 align:middle line:84% well, the integral from a to x of f 00:57:19.810 --> 00:57:23.560 align:middle line:84% is equal to the integral from a to c plus the integral 00:57:23.560 --> 00:57:25.060 align:middle line:90% from c to x. 00:57:25.060 --> 00:57:29.980 align:middle line:84% So this is equal to 1 over x minus c integral 00:57:29.980 --> 00:57:34.930 align:middle line:90% from c to x of f minus fc. 00:57:34.930 --> 00:57:40.130 align:middle line:90% 00:57:40.130 --> 00:57:43.175 align:middle line:84% Now, this is just a fixed number. 00:57:43.175 --> 00:57:45.730 align:middle line:90% 00:57:45.730 --> 00:57:52.800 align:middle line:84% And therefore, I'm going to do a little trick. 00:57:52.800 --> 00:57:55.080 align:middle line:90% Minus the integral from-- 00:57:55.080 --> 00:57:58.680 align:middle line:90% so in fact, let me do this. 00:57:58.680 --> 00:58:01.690 align:middle line:84% Let me throw in some integration variables 00:58:01.690 --> 00:58:05.420 align:middle line:90% so that this becomes clearer. 00:58:05.420 --> 00:58:13.310 align:middle line:84% So this is f of t, dt minus integral of f of t, dt. 00:58:13.310 --> 00:58:14.990 align:middle line:84% I've been using the notation where 00:58:14.990 --> 00:58:17.300 align:middle line:84% I just don't write down the integration variable, 00:58:17.300 --> 00:58:19.880 align:middle line:90% but let me do that here. 00:58:19.880 --> 00:58:22.910 align:middle line:90% F of t, dt minus f of c. 00:58:22.910 --> 00:58:30.820 align:middle line:90% 00:58:30.820 --> 00:58:37.300 align:middle line:84% Now this is equal to minus f of c over x minus c integral 00:58:37.300 --> 00:58:43.400 align:middle line:90% from c to x of 1 dt. 00:58:43.400 --> 00:58:46.550 align:middle line:84% Because the integral from c to x of the constant 1 00:58:46.550 --> 00:58:47.630 align:middle line:90% gives me x minus c. 00:58:47.630 --> 00:58:49.170 align:middle line:90% That cancels with this guy. 00:58:49.170 --> 00:58:50.930 align:middle line:90% Now f of c is just a number. 00:58:50.930 --> 00:58:52.320 align:middle line:90% It's just a fixed number. 00:58:52.320 --> 00:58:55.110 align:middle line:84% So I can bring this inside the integral. 00:58:55.110 --> 00:59:00.530 align:middle line:84% And then use the linearity of the integral to in fact, 00:59:00.530 --> 00:59:03.800 align:middle line:84% rewrite all of this as 1 over the absolute value 00:59:03.800 --> 00:59:12.845 align:middle line:84% of x minus c integral from c to x of f of t minus f of c, 00:59:12.845 --> 00:59:16.235 align:middle line:90% dt in absolute value. 00:59:16.235 --> 00:59:20.220 align:middle line:90% 00:59:20.220 --> 00:59:22.940 align:middle line:84% Now 1 over x minus c, this is positive 00:59:22.940 --> 00:59:24.380 align:middle line:90% because x is bigger than c. 00:59:24.380 --> 00:59:27.890 align:middle line:84% So I can pull this out just like that. 00:59:27.890 --> 00:59:31.110 align:middle line:90% 00:59:31.110 --> 00:59:32.090 align:middle line:90% And now what do I know? 00:59:32.090 --> 00:59:39.860 align:middle line:84% For t in between this c and x, the absolute value of t minus c 00:59:39.860 --> 00:59:42.290 align:middle line:90% is less than delta 0, right? 00:59:42.290 --> 00:59:52.150 align:middle line:84% So by what I have here, as long as I'm in this interval cx, 00:59:52.150 --> 00:59:54.625 align:middle line:84% I'm going to have this inequality hold. 00:59:54.625 --> 00:59:57.950 align:middle line:90% 00:59:57.950 --> 01:00:02.090 align:middle line:84% So before I do that, let me first apply the triangle 01:00:02.090 --> 01:00:03.410 align:middle line:90% inequality for integrals. 01:00:03.410 --> 01:00:05.480 align:middle line:84% This is less than or equal to the integral from c 01:00:05.480 --> 01:00:09.050 align:middle line:90% to x of f of t minus f of c. 01:00:09.050 --> 01:00:13.340 align:middle line:90% 01:00:13.340 --> 01:00:16.550 align:middle line:84% So by the second thing that I've underlined, 01:00:16.550 --> 01:00:19.400 align:middle line:84% for t in this interval c to x, this thing 01:00:19.400 --> 01:00:23.720 align:middle line:84% here is always less than epsilon over 2. 01:00:23.720 --> 01:00:27.380 align:middle line:84% So by what we know about integrals respecting 01:00:27.380 --> 01:00:30.610 align:middle line:84% inequalities, this is going to be less than 01:00:30.610 --> 01:00:34.300 align:middle line:84% or equal to 1 over integral x minus c, integral c 01:00:34.300 --> 01:00:37.000 align:middle line:90% to x, epsilon over 2 dt. 01:00:37.000 --> 01:00:39.570 align:middle line:90% 01:00:39.570 --> 01:00:41.430 align:middle line:90% And now this is just equal to-- 01:00:41.430 --> 01:00:42.600 align:middle line:90% so this is a number. 01:00:42.600 --> 01:00:43.650 align:middle line:90% It comes out. 01:00:43.650 --> 01:00:46.830 align:middle line:84% So I get 1 integral from c to x of 1. 01:00:46.830 --> 01:00:54.320 align:middle line:84% So I get x minus c times the integral of 1. 01:00:54.320 --> 01:01:01.760 align:middle line:84% So x minus c, and that cancels, less than epsilon. 01:01:01.760 --> 01:01:04.750 align:middle line:90% 01:01:04.750 --> 01:01:09.700 align:middle line:84% And this was for x between c and c plus delta. 01:01:09.700 --> 01:01:12.880 align:middle line:84% The argument for c between c and c minus delta 01:01:12.880 --> 01:01:17.260 align:middle line:84% is essentially the same, just with minor sign changes. 01:01:17.260 --> 01:01:24.540 align:middle line:90% 01:01:24.540 --> 01:01:30.500 align:middle line:84% So similarly, c minus delta less than c 01:01:30.500 --> 01:01:35.420 align:middle line:84% implies that the integral from a to x 01:01:35.420 --> 01:01:42.110 align:middle line:84% of f minus integral a to c of f over x minus c minus f of c 01:01:42.110 --> 01:01:48.390 align:middle line:90% is less than epsilon. 01:01:48.390 --> 01:01:53.270 align:middle line:84% So we've done x between c minus delta and c, x between c and c 01:01:53.270 --> 01:01:59.140 align:middle line:84% plus delta, which implies that if-- 01:01:59.140 --> 01:02:04.770 align:middle line:90% 01:02:04.770 --> 01:02:20.310 align:middle line:84% then, which is what we wanted to prove. 01:02:20.310 --> 01:02:31.990 align:middle line:90% 01:02:31.990 --> 01:02:41.450 align:middle line:84% OK, so we've concluded the proof of the Batman of the story. 01:02:41.450 --> 01:02:44.390 align:middle line:84% And this gives us not only a way of computing integrals, 01:02:44.390 --> 01:02:53.920 align:middle line:84% but useful way of, if you like, shifting the burden 01:02:53.920 --> 01:02:56.140 align:middle line:84% or shifting the blame whenever we 01:02:56.140 --> 01:02:59.540 align:middle line:84% do compute the integrals of certain products. 01:02:59.540 --> 01:03:02.230 align:middle line:84% So what I'm talking about is the integration 01:03:02.230 --> 01:03:05.830 align:middle line:84% by parts, which I can't remember if I 01:03:05.830 --> 01:03:07.030 align:middle line:90% said it was first or second. 01:03:07.030 --> 01:03:08.740 align:middle line:90% I think he was second. 01:03:08.740 --> 01:03:10.900 align:middle line:84% First most useful thing from analysis 01:03:10.900 --> 01:03:14.290 align:middle line:84% is the triangle inequality, the second being integration 01:03:14.290 --> 01:03:17.250 align:middle line:90% by parts. 01:03:17.250 --> 01:03:21.810 align:middle line:84% And a close third being the Cauchy-Schwartz inequality, 01:03:21.810 --> 01:03:24.460 align:middle line:84% which in conjunction with integration by parts is quite 01:03:24.460 --> 01:03:24.960 align:middle line:90% a-- 01:03:24.960 --> 01:03:29.800 align:middle line:90% 01:03:29.800 --> 01:03:34.060 align:middle line:84% most current research papers are based on these two things 01:03:34.060 --> 01:03:37.130 align:middle line:84% and using them in a very clever way. 01:03:37.130 --> 01:03:42.190 align:middle line:84% So integration by parts is suppose f 01:03:42.190 --> 01:03:57.050 align:middle line:84% and g are continuously differentiable, 01:03:57.050 --> 01:04:02.630 align:middle line:84% what that means is that the derivatives exists on ab, 01:04:02.630 --> 01:04:06.050 align:middle line:90% and they're also continuous. 01:04:06.050 --> 01:04:07.815 align:middle line:84% So if you hear me say continuously 01:04:07.815 --> 01:04:09.190 align:middle line:84% differentiable in the future that 01:04:09.190 --> 01:04:12.310 align:middle line:84% means that the function has a derivative that's continuous. 01:04:12.310 --> 01:04:15.920 align:middle line:90% 01:04:15.920 --> 01:04:22.550 align:middle line:84% Then the integral from ab of f prime times 01:04:22.550 --> 01:04:24.920 align:middle line:84% g-- so the one with the burden, because remember, 01:04:24.920 --> 01:04:26.570 align:middle line:84% differentiablity is kind of a miracle. 01:04:26.570 --> 01:04:31.460 align:middle line:84% So miracles always come with some sort of burden. 01:04:31.460 --> 01:04:35.780 align:middle line:84% This is equal to f of b times g of b 01:04:35.780 --> 01:04:42.410 align:middle line:84% minus f of a times g of a minus the integral from a to b. 01:04:42.410 --> 01:04:45.260 align:middle line:84% And now we shift that burden to g times g prime. 01:04:45.260 --> 01:04:50.880 align:middle line:90% 01:04:50.880 --> 01:04:54.210 align:middle line:84% So let me put a parentheses around that so you can-- 01:04:54.210 --> 01:04:58.620 align:middle line:84% I don't-- let's do it this way, g prime f. 01:04:58.620 --> 01:05:01.680 align:middle line:84% All right, so I can take that derivative and put it on g. 01:05:01.680 --> 01:05:02.680 align:middle line:90% And what's the proof? 01:05:02.680 --> 01:05:05.490 align:middle line:84% The proof is just the Fundamental Theorem of Calculus 01:05:05.490 --> 01:05:06.810 align:middle line:90% and the product rule. 01:05:06.810 --> 01:05:11.820 align:middle line:90% 01:05:11.820 --> 01:05:15.150 align:middle line:84% Since the derivative of f times g 01:05:15.150 --> 01:05:21.900 align:middle line:84% is equal to f prime times g plus g prime times f, 01:05:21.900 --> 01:05:23.850 align:middle line:84% we get by the Fundamental Theorem of Calculus 01:05:23.850 --> 01:05:29.530 align:middle line:84% the integral from ab of, let's say the right hand side, 01:05:29.530 --> 01:05:34.200 align:middle line:84% so this is f prime g plus g prime f 01:05:34.200 --> 01:05:40.990 align:middle line:84% is equal to the integral from a to b of f times g prime, 01:05:40.990 --> 01:05:45.840 align:middle line:84% so by Fundamental Theorem of Calculus. 01:05:45.840 --> 01:05:47.610 align:middle line:84% And now the integral of the derivative 01:05:47.610 --> 01:05:51.290 align:middle line:84% is the function evaluated at the endpoint. 01:05:51.290 --> 01:05:58.940 align:middle line:84% So that's f of b times g of b minus f of a times g of a. 01:05:58.940 --> 01:06:14.040 align:middle line:90% 01:06:14.040 --> 01:06:17.910 align:middle line:84% So that's integration by parts, which 01:06:17.910 --> 01:06:24.390 align:middle line:84% is a consequence of the product rule. 01:06:24.390 --> 01:06:29.520 align:middle line:90% 01:06:29.520 --> 01:06:32.460 align:middle line:84% the quotient rule is something-- is just really the product 01:06:32.460 --> 01:06:33.750 align:middle line:90% rule in disguise kind of. 01:06:33.750 --> 01:06:36.150 align:middle line:84% So we don't get really anything new from there. 01:06:36.150 --> 01:06:38.910 align:middle line:90% 01:06:38.910 --> 01:06:40.650 align:middle line:84% But we also remember, have the chain 01:06:40.650 --> 01:06:47.670 align:middle line:84% rule, which then results in the change of variables formula. 01:06:47.670 --> 01:07:13.240 align:middle line:84% So theorem-- let b from ab to cd be continuously differentiable. 01:07:13.240 --> 01:07:16.570 align:middle line:84% So this means the function is continuous, 01:07:16.570 --> 01:07:18.550 align:middle line:84% and its derivative is also continuous. 01:07:18.550 --> 01:07:24.040 align:middle line:90% 01:07:24.040 --> 01:07:34.120 align:middle line:84% With the property that the derivative is positive on ab. 01:07:34.120 --> 01:07:36.922 align:middle line:90% b of a equals c. 01:07:36.922 --> 01:07:41.110 align:middle line:90% b of b equals d. 01:07:41.110 --> 01:07:42.160 align:middle line:90% It's hard to see that a. 01:07:42.160 --> 01:07:48.070 align:middle line:90% 01:07:48.070 --> 01:07:51.910 align:middle line:84% So you can think of phi as being a change of variables from cd 01:07:51.910 --> 01:07:52.450 align:middle line:90% to ab. 01:07:52.450 --> 01:07:55.140 align:middle line:90% 01:07:55.140 --> 01:08:03.800 align:middle line:84% Then the integral from c to d of f of u, 01:08:03.800 --> 01:08:07.250 align:middle line:84% du is equal to the integral from a 01:08:07.250 --> 01:08:16.340 align:middle line:84% to b of f of phi of x, phi prime of x, dx. 01:08:16.340 --> 01:08:22.490 align:middle line:84% So maybe instead of change of variables-- 01:08:22.490 --> 01:08:28.115 align:middle line:84% so this is change of variables, a.k.a. 01:08:28.115 --> 01:08:32.270 align:middle line:90% 01:08:32.270 --> 01:08:37.069 align:middle line:84% u substitution, where you said u equal to phi of x, 01:08:37.069 --> 01:08:42.029 align:middle line:84% then du is equal to phi prime of x dx. 01:08:42.029 --> 01:08:46.080 align:middle line:84% And again, the proof of this just 01:08:46.080 --> 01:08:47.580 align:middle line:84% follows from the Fundamental Theorem 01:08:47.580 --> 01:08:55.850 align:middle line:84% of Calculus and the chain rule that we know. 01:08:55.850 --> 01:09:04.640 align:middle line:84% So let f be a function from ab to r such that 01:09:04.640 --> 01:09:07.640 align:middle line:90% f prime equals little f. 01:09:07.640 --> 01:09:09.890 align:middle line:84% We can always find one by the second part 01:09:09.890 --> 01:09:11.990 align:middle line:84% of the Fundamental Theorem of Calculus. 01:09:11.990 --> 01:09:15.439 align:middle line:84% We can always solve this differential equation up 01:09:15.439 --> 01:09:18.290 align:middle line:90% to a constant. 01:09:18.290 --> 01:09:25.479 align:middle line:84% Then if I look at the function f of phi of x, 01:09:25.479 --> 01:09:28.600 align:middle line:84% and I take its derivative with respect to x by the chain rule, 01:09:28.600 --> 01:09:33.609 align:middle line:84% this is equal to f prime of phi of x 01:09:33.609 --> 01:09:36.910 align:middle line:84% times phi prime of x, which equals 01:09:36.910 --> 01:09:41.350 align:middle line:90% f of phi of x, phi prime of x. 01:09:41.350 --> 01:09:44.010 align:middle line:90% 01:09:44.010 --> 01:09:55.700 align:middle line:84% So then when I integrate that, integral from ab, phi of x-- 01:09:55.700 --> 01:10:05.000 align:middle line:84% f of phi of x phi prime of x dx, this 01:10:05.000 --> 01:10:15.150 align:middle line:84% is equal to the integral from ab of f of phi of x prime dx. 01:10:15.150 --> 01:10:17.900 align:middle line:84% Now by part one of the Fundamental Theorem 01:10:17.900 --> 01:10:21.620 align:middle line:90% of Calculus, this is equal to-- 01:10:21.620 --> 01:10:23.540 align:middle line:84% the integral of the derivative is this thing 01:10:23.540 --> 01:10:24.900 align:middle line:90% evaluated at the endpoint. 01:10:24.900 --> 01:10:31.280 align:middle line:84% So f of phi of b minus f of phi of a. 01:10:31.280 --> 01:10:37.400 align:middle line:84% This is equal to f of d minus f of c because phi of b 01:10:37.400 --> 01:10:43.430 align:middle line:84% equals d, phi of a equals c, and phi of b equals d. 01:10:43.430 --> 01:10:46.010 align:middle line:84% But again, by the Fundamental Theorem of Calculus 01:10:46.010 --> 01:10:47.810 align:middle line:90% since capital F is-- 01:10:47.810 --> 01:10:51.760 align:middle line:84% when you differentiate, it gives me little f, 01:10:51.760 --> 01:10:59.020 align:middle line:84% this is equal to the integral from c to d of f prime of u du, 01:10:59.020 --> 01:11:06.790 align:middle line:90% which is equal to f of u du. 01:11:06.790 --> 01:11:09.460 align:middle line:84% Again, so we just applied the Fundamental Theorem of Calculus 01:11:09.460 --> 01:11:11.800 align:middle line:90% really, three times, OK? 01:11:11.800 --> 01:11:15.370 align:middle line:84% First to find a function whose derivative is little f. 01:11:15.370 --> 01:11:17.470 align:middle line:84% Then we used the chain rule to conclude 01:11:17.470 --> 01:11:19.360 align:middle line:84% that the derivative of capital F of phi 01:11:19.360 --> 01:11:22.270 align:middle line:84% equals this thing, which is the thing we're integrating 01:11:22.270 --> 01:11:23.957 align:middle line:90% on the right hand side. 01:11:23.957 --> 01:11:25.540 align:middle line:84% So then when we integrate that, that's 01:11:25.540 --> 01:11:27.740 align:middle line:84% equal to the integral of a derivative. 01:11:27.740 --> 01:11:31.460 align:middle line:84% And therefore, we just pick up the endpoints, 01:11:31.460 --> 01:11:32.690 align:middle line:90% f of d minus f of c. 01:11:32.690 --> 01:11:36.350 align:middle line:84% But because capital F is equal to-- 01:11:36.350 --> 01:11:38.090 align:middle line:84% is an anti-derivative of little f, 01:11:38.090 --> 01:11:41.210 align:middle line:84% meaning it's derivative is little f, 01:11:41.210 --> 01:11:44.700 align:middle line:84% this number here is also equal to this integral here, 01:11:44.700 --> 01:11:47.110 align:middle line:90% which is this guy. 01:11:47.110 --> 01:11:52.050 align:middle line:84% And so then we get the change of variables formula. 01:11:52.050 --> 01:11:53.318 align:middle line:90% So I think we'll stop there. 01:11:53.318 --> 01:11:55.110 align:middle line:84% And next time, we'll do a quick application 01:11:55.110 --> 01:11:59.910 align:middle line:84% of the second most useful thing on Earth, 01:11:59.910 --> 01:12:03.180 align:middle line:90% which is integration by parts. 01:12:03.180 --> 01:12:06.800 align:middle line:84% And then we'll move on to sequences of functions. 01:12:06.800 --> 01:12:13.000 align:middle line:90%