WEBVTT 00:00:00.000 --> 00:00:00.980 align:middle line:90% 00:00:00.980 --> 00:00:02.450 align:middle line:90% [SQUEAKING] 00:00:02.450 --> 00:00:03.430 align:middle line:90% [RUSTLING] 00:00:03.430 --> 00:00:06.860 align:middle line:90% [CLICKING] 00:00:06.860 --> 00:00:10.527 align:middle line:90% 00:00:10.527 --> 00:00:12.360 align:middle line:84% TOBIAS COLDING: Last time, one of the things 00:00:12.360 --> 00:00:17.820 align:middle line:84% we did were we proved true mean value inequalities. 00:00:17.820 --> 00:00:22.360 align:middle line:90% 00:00:22.360 --> 00:00:26.080 align:middle line:84% So we proved true mean value inequality. 00:00:26.080 --> 00:00:27.610 align:middle line:90% And let me recall those. 00:00:27.610 --> 00:00:35.760 align:middle line:90% 00:00:35.760 --> 00:00:38.980 align:middle line:90% And the first one was that-- 00:00:38.980 --> 00:00:45.720 align:middle line:84% so if you had a function f on some interval from a to b into R 00:00:45.720 --> 00:00:54.520 align:middle line:84% and this function f is differentiable 00:00:54.520 --> 00:01:02.480 align:middle line:84% then it said that there exists some x0 with-- 00:01:02.480 --> 00:01:07.440 align:middle line:84% x0 between strictly between a and b so that 00:01:07.440 --> 00:01:14.900 align:middle line:84% the derivative at x0 is equal to the average of the function-- 00:01:14.900 --> 00:01:18.600 align:middle line:84% or the average of the function at the endpoints. 00:01:18.600 --> 00:01:22.620 align:middle line:84% So this is the average on the endpoint divided-- 00:01:22.620 --> 00:01:26.000 align:middle line:84% so it's average over the length of the interval. 00:01:26.000 --> 00:01:33.000 align:middle line:84% So this is what the mean value theorem said. 00:01:33.000 --> 00:01:35.920 align:middle line:84% And then there was a second mean value theorem, 00:01:35.920 --> 00:01:39.640 align:middle line:84% and that was the Cauchy mean value theorem. 00:01:39.640 --> 00:01:41.330 align:middle line:84% So the Cauchy mean value theorem. 00:01:41.330 --> 00:01:52.160 align:middle line:90% 00:01:52.160 --> 00:01:57.650 align:middle line:84% And that was where you had two functions, f and g, 00:01:57.650 --> 00:02:02.250 align:middle line:84% on this interval into R. And they are both 00:02:02.250 --> 00:02:04.030 align:middle line:90% assumed to be differentiable. 00:02:04.030 --> 00:02:09.330 align:middle line:90% 00:02:09.330 --> 00:02:15.490 align:middle line:84% And then again, it was that there exists some x0 strictly 00:02:15.490 --> 00:02:18.270 align:middle line:90% between a and b. 00:02:18.270 --> 00:02:20.970 align:middle line:90% 00:02:20.970 --> 00:02:33.650 align:middle line:84% And if you take the derivative at x0 of f times the value-- 00:02:33.650 --> 00:02:38.410 align:middle line:84% the difference in value of g at the endpoints, 00:02:38.410 --> 00:02:44.970 align:middle line:84% then this is equal to the derivative of g at x0 times 00:02:44.970 --> 00:02:48.390 align:middle line:84% the difference in value of f at the endpoints. 00:02:48.390 --> 00:02:51.799 align:middle line:90% 00:02:51.799 --> 00:02:54.225 align:middle line:84% All right, so that was the Cauchy mean value theorem. 00:02:54.225 --> 00:02:55.850 align:middle line:84% So it was a little bit more complicated 00:02:55.850 --> 00:02:57.930 align:middle line:84% than the actual mean value theorem 00:02:57.930 --> 00:03:01.890 align:middle line:84% because you had two functions at the same time, 00:03:01.890 --> 00:03:05.490 align:middle line:84% and the point here was that you can arrange this exterior 00:03:05.490 --> 00:03:08.310 align:middle line:84% to work for both points-- for both functions. 00:03:08.310 --> 00:03:16.130 align:middle line:84% Let me just make an observation, that if-- 00:03:16.130 --> 00:03:17.810 align:middle line:90% so if we're looking at-- 00:03:17.810 --> 00:03:26.227 align:middle line:90% 00:03:26.227 --> 00:03:27.810 align:middle line:84% so the Cauchy mean value theorem seems 00:03:27.810 --> 00:03:31.030 align:middle line:84% like a more elaborate version of the mean value theorem, 00:03:31.030 --> 00:03:37.770 align:middle line:84% and it really is because if you suppose that g here was 00:03:37.770 --> 00:03:41.130 align:middle line:90% a function that is equal to x-- 00:03:41.130 --> 00:03:48.010 align:middle line:84% so g of x is equal to x, I have f here on the integral 00:03:48.010 --> 00:03:51.250 align:middle line:90% from a to b-- 00:03:51.250 --> 00:03:53.470 align:middle line:90% this is, again, differentiable. 00:03:53.470 --> 00:04:00.860 align:middle line:90% 00:04:00.860 --> 00:04:03.380 align:middle line:84% Well, if the other function was just 00:04:03.380 --> 00:04:06.680 align:middle line:84% this rather trivial function g, of course, 00:04:06.680 --> 00:04:09.420 align:middle line:84% g is also differentiable, and we even 00:04:09.420 --> 00:04:15.260 align:middle line:84% know that the derivative of g at all points is 1. 00:04:15.260 --> 00:04:19.100 align:middle line:84% So if you are now applying the Cauchy mean value theorem, 00:04:19.100 --> 00:04:34.460 align:middle line:84% then you say that by the Cauchy mean value theorem, 00:04:34.460 --> 00:04:42.940 align:middle line:84% there exists some x0 with x0 strictly between a and b. 00:04:42.940 --> 00:04:45.940 align:middle line:84% And so that the derivative of f-- 00:04:45.940 --> 00:04:48.300 align:middle line:90% so that's whatever that is. 00:04:48.300 --> 00:04:51.460 align:middle line:84% That's the derivative of f at x0. 00:04:51.460 --> 00:04:54.760 align:middle line:84% The value of g at these two endpoints 00:04:54.760 --> 00:04:56.760 align:middle line:84% where g is just a function of x-- 00:04:56.760 --> 00:04:59.900 align:middle line:90% so this is just b minus a. 00:04:59.900 --> 00:05:04.940 align:middle line:90% And the derivative of g is 1. 00:05:04.940 --> 00:05:11.400 align:middle line:84% And then it's the difference in value of f at the endpoints. 00:05:11.400 --> 00:05:16.040 align:middle line:84% And so now, if you just divide by b minus a-- so this, 00:05:16.040 --> 00:05:19.660 align:middle line:84% of course, implies that f prime of x0 00:05:19.660 --> 00:05:26.660 align:middle line:84% is equal to f of b minus f of a divided by b minus a. 00:05:26.660 --> 00:05:28.660 align:middle line:90% So in other words-- 00:05:28.660 --> 00:05:31.580 align:middle line:84% and you see, this here gives you-- 00:05:31.580 --> 00:05:35.780 align:middle line:84% this gives you the more standard mean value inequality. 00:05:35.780 --> 00:05:38.500 align:middle line:84% So in other words, the Cauchy mean value theorem 00:05:38.500 --> 00:05:44.140 align:middle line:84% is more general than this mean value inequality 00:05:44.140 --> 00:05:46.180 align:middle line:90% because you recover this-- 00:05:46.180 --> 00:05:48.040 align:middle line:84% if you just let one of the two functions, 00:05:48.040 --> 00:05:50.220 align:middle line:84% it'd be just the trivial function x. 00:05:50.220 --> 00:05:57.150 align:middle line:90% 00:05:57.150 --> 00:05:59.830 align:middle line:84% OK, so that was just an observation. 00:05:59.830 --> 00:06:03.110 align:middle line:84% The second thing I want to talk about 00:06:03.110 --> 00:06:06.810 align:middle line:84% is something we didn't get to last time. 00:06:06.810 --> 00:06:11.350 align:middle line:90% And these are these-- 00:06:11.350 --> 00:06:14.220 align:middle line:90% L'Hopital's rule. 00:06:14.220 --> 00:06:20.030 align:middle line:90% 00:06:20.030 --> 00:06:22.770 align:middle line:84% And I'm just going to state one version of it. 00:06:22.770 --> 00:06:27.010 align:middle line:84% There are two versions, and they're both in the notes, 00:06:27.010 --> 00:06:29.750 align:middle line:84% but I will just state one of them. 00:06:29.750 --> 00:06:31.510 align:middle line:90% And so this is-- 00:06:31.510 --> 00:06:33.790 align:middle line:84% so you have-- so the situation here 00:06:33.790 --> 00:06:38.910 align:middle line:84% is that you have two functions, f and g. 00:06:38.910 --> 00:06:42.030 align:middle line:84% And they are defined on some interval, 00:06:42.030 --> 00:06:50.030 align:middle line:84% say from a to b into R. The interval does not 00:06:50.030 --> 00:06:51.710 align:middle line:90% include the endpoint. 00:06:51.710 --> 00:06:54.490 align:middle line:90% So these are differentiable. 00:06:54.490 --> 00:07:03.790 align:middle line:90% 00:07:03.790 --> 00:07:08.150 align:middle line:84% And then we assume that the limit here-- 00:07:08.150 --> 00:07:13.670 align:middle line:90% we assume that g of x is-- 00:07:13.670 --> 00:07:19.310 align:middle line:84% and not only g of x, but also g prime of x, are not 00:07:19.310 --> 00:07:22.030 align:middle line:90% equal to 0 for all x. 00:07:22.030 --> 00:07:27.550 align:middle line:90% 00:07:27.550 --> 00:07:34.390 align:middle line:84% And then we assume that the limit here of g of x as x 00:07:34.390 --> 00:07:39.190 align:middle line:84% goes to a, that this here is 0, and the same 00:07:39.190 --> 00:07:42.610 align:middle line:90% holds for the function f. 00:07:42.610 --> 00:07:46.030 align:middle line:90% 00:07:46.030 --> 00:07:49.080 align:middle line:90% So this here. 00:07:49.080 --> 00:07:51.680 align:middle line:84% And then what you're interested in is-- 00:07:51.680 --> 00:07:54.840 align:middle line:84% what you're interested in is that you're 00:07:54.840 --> 00:08:01.160 align:middle line:84% interested in looking at f of x divided by g of x. 00:08:01.160 --> 00:08:03.960 align:middle line:84% So this here makes sense, g doesn't vanish. 00:08:03.960 --> 00:08:05.620 align:middle line:90% So this here makes sense. 00:08:05.620 --> 00:08:10.760 align:middle line:84% And you are interested in looking at this fraction 00:08:10.760 --> 00:08:16.600 align:middle line:90% here as x goes to a. 00:08:16.600 --> 00:08:20.600 align:middle line:84% If x goes to a, then the denominator goes to 0, 00:08:20.600 --> 00:08:23.760 align:middle line:84% and the denominator also goes to 0. 00:08:23.760 --> 00:08:27.060 align:middle line:84% And the question is, does it have a limit or not? 00:08:27.060 --> 00:08:31.480 align:middle line:84% And so there's one more assumption here that you need, 00:08:31.480 --> 00:08:37.679 align:middle line:84% and that is that if you're looking at the-- 00:08:37.679 --> 00:08:55.280 align:middle line:84% so assume also that the limit here as x goes to a of f 00:08:55.280 --> 00:09:02.360 align:middle line:84% prime of x divided by g prime, assume that this exists 00:09:02.360 --> 00:09:08.760 align:middle line:90% and is equal to some L. 00:09:08.760 --> 00:09:11.480 align:middle line:84% And then the claim is that then-- 00:09:11.480 --> 00:09:21.520 align:middle line:84% so the claim here is that this fraction here, 00:09:21.520 --> 00:09:26.720 align:middle line:84% f of x divided by g of x, that this does 00:09:26.720 --> 00:09:36.800 align:middle line:84% have a limit as x goes to a, and that limit is actually L. 00:09:36.800 --> 00:09:42.980 align:middle line:84% So this is one version of this rule. 00:09:42.980 --> 00:09:45.560 align:middle line:84% And there's another one, very similar. 00:09:45.560 --> 00:09:49.130 align:middle line:90% 00:09:49.130 --> 00:09:51.405 align:middle line:90% OK. 00:09:51.405 --> 00:09:54.290 align:middle line:90% And so we want to prove this. 00:09:54.290 --> 00:09:56.690 align:middle line:90% So we are looking at-- 00:09:56.690 --> 00:10:01.410 align:middle line:90% 00:10:01.410 --> 00:10:05.690 align:middle line:84% so we want to apply the Cauchy mean value 00:10:05.690 --> 00:10:08.090 align:middle line:90% theorem to prove this theorem. 00:10:08.090 --> 00:10:12.610 align:middle line:84% And so we're looking at an interval here. 00:10:12.610 --> 00:10:16.690 align:middle line:90% So you're looking at, say, y. 00:10:16.690 --> 00:10:22.630 align:middle line:84% So here, you have a; here, you have x; 00:10:22.630 --> 00:10:27.970 align:middle line:84% and then here, you have y; and here, you have b. 00:10:27.970 --> 00:10:30.050 align:middle line:90% And I'm now going to apply-- 00:10:30.050 --> 00:10:31.560 align:middle line:90% so we will apply the-- 00:10:31.560 --> 00:10:34.490 align:middle line:90% 00:10:34.490 --> 00:10:39.830 align:middle line:84% so apply the Cauchy mean value theorem. 00:10:39.830 --> 00:10:50.250 align:middle line:90% 00:10:50.250 --> 00:10:55.890 align:middle line:84% But we are applying it on this interval here. 00:10:55.890 --> 00:11:02.330 align:middle line:84% Between-- so y here is less than x, 00:11:02.330 --> 00:11:04.970 align:middle line:90% but is strictly bigger than a. 00:11:04.970 --> 00:11:08.970 align:middle line:84% And this is x is strictly less than x. 00:11:08.970 --> 00:11:12.210 align:middle line:84% So, the function was differentiable 00:11:12.210 --> 00:11:14.170 align:middle line:84% on this open interval-- so of course, 00:11:14.170 --> 00:11:15.650 align:middle line:84% differentiable on both functions, 00:11:15.650 --> 00:11:18.450 align:middle line:84% differentiable on this closed interval. 00:11:18.450 --> 00:11:25.330 align:middle line:84% So we can apply it-- and so we get that f prime at some 00:11:25.330 --> 00:11:44.220 align:middle line:84% intermediate point x0, times g of x minus g of y, 00:11:44.220 --> 00:11:47.580 align:middle line:90% this is equal to g prime-- 00:11:47.580 --> 00:11:50.580 align:middle line:84% again, x0 here is some intermediate point, 00:11:50.580 --> 00:11:53.620 align:middle line:90% it lie between these two. 00:11:53.620 --> 00:12:01.280 align:middle line:84% g prime at x0 times f of x minus g of x-- g of y, 00:12:01.280 --> 00:12:05.500 align:middle line:90% sorry, like that. 00:12:05.500 --> 00:12:10.070 align:middle line:84% This is just applying the Cauchy mean value theorem. 00:12:10.070 --> 00:12:10.570 align:middle line:90% Sorry? 00:12:10.570 --> 00:12:14.900 align:middle line:90% 00:12:14.900 --> 00:12:16.160 align:middle line:90% What did you say? 00:12:16.160 --> 00:12:18.360 align:middle line:84% Oh, yeah, yeah, yeah, sorry about that. 00:12:18.360 --> 00:12:19.740 align:middle line:90% Thanks. 00:12:19.740 --> 00:12:22.800 align:middle line:90% Yeah, thank you. 00:12:22.800 --> 00:12:23.300 align:middle line:90% Right. 00:12:23.300 --> 00:12:25.520 align:middle line:84% And so now, I just divide-- of course, 00:12:25.520 --> 00:12:32.260 align:middle line:84% this here is supposed to be a non-zero, so I can just divide. 00:12:32.260 --> 00:12:48.540 align:middle line:84% So we have that f prime of x0 divided by g prime of x0. 00:12:48.540 --> 00:12:49.880 align:middle line:90% And then I will-- 00:12:49.880 --> 00:12:52.460 align:middle line:84% I want to divide by this, but I just 00:12:52.460 --> 00:12:55.546 align:middle line:84% need to make sure that this here is-- 00:12:55.546 --> 00:13:01.340 align:middle line:90% 00:13:01.340 --> 00:13:05.860 align:middle line:84% that I can divide by it, that is not 0, this thing here. 00:13:05.860 --> 00:13:06.960 align:middle line:90% So I have this. 00:13:06.960 --> 00:13:15.340 align:middle line:90% 00:13:15.340 --> 00:13:22.140 align:middle line:84% Now, think about x as being fixed and do this for all y. 00:13:22.140 --> 00:13:25.080 align:middle line:90% If I do this-- 00:13:25.080 --> 00:13:31.460 align:middle line:84% if I do this for y where y is very, very close to a, 00:13:31.460 --> 00:13:35.210 align:middle line:84% this here becomes really, really close to 0. 00:13:35.210 --> 00:13:40.230 align:middle line:84% This here is fixed and it's not equal to 0. 00:13:40.230 --> 00:13:44.050 align:middle line:84% So this here is very close to 0, this is not equal to 0. 00:13:44.050 --> 00:13:47.510 align:middle line:84% So this here, if y is really close, if x is fixed, 00:13:47.510 --> 00:13:51.810 align:middle line:84% y is really close to a, this is definitely not zero, 00:13:51.810 --> 00:13:53.410 align:middle line:90% so I can divide by it. 00:13:53.410 --> 00:14:16.310 align:middle line:84% So, for x fixed and y sufficiently close to 0, 00:14:16.310 --> 00:14:27.830 align:middle line:84% then you have, then, g of x minus g of y is not equal to 0. 00:14:27.830 --> 00:14:34.470 align:middle line:84% And so I get that f prime at this intermediate point divided 00:14:34.470 --> 00:14:38.030 align:middle line:84% by g prime at the same intermediate point 00:14:38.030 --> 00:14:48.510 align:middle line:84% is equal to f of x minus f of y divided by g of x minus g of y. 00:14:48.510 --> 00:14:52.730 align:middle line:84% And now you see that's y here goes to 0-- 00:14:52.730 --> 00:14:58.390 align:middle line:84% so x is fixed still as y goes to-- sorry, not to 0, but to a, 00:14:58.390 --> 00:15:01.250 align:middle line:84% this here goes to 0 by assumption, 00:15:01.250 --> 00:15:04.110 align:middle line:84% this here goes to 0, so the whole thing goes 00:15:04.110 --> 00:15:07.590 align:middle line:90% to f of x divided by g of x. 00:15:07.590 --> 00:15:11.430 align:middle line:90% 00:15:11.430 --> 00:15:14.560 align:middle line:90% Now the thing is that if you-- 00:15:14.560 --> 00:15:18.510 align:middle line:90% 00:15:18.510 --> 00:15:21.640 align:middle line:90% so this is for all x fixed. 00:15:21.640 --> 00:15:27.010 align:middle line:90% 00:15:27.010 --> 00:15:32.470 align:middle line:90% All right, so now if you-- 00:15:32.470 --> 00:15:34.250 align:middle line:90% so now, we want to use-- 00:15:34.250 --> 00:15:47.320 align:middle line:84% so now, since f prime at, say, z divided by g prime at z, 00:15:47.320 --> 00:15:51.640 align:middle line:84% since this fraction here, as z goes to a, 00:15:51.640 --> 00:15:55.400 align:middle line:90% this is assumed to go to L. 00:15:55.400 --> 00:16:02.760 align:middle line:84% So this means that given epsilon greater than 0, 00:16:02.760 --> 00:16:06.460 align:middle line:84% there exists a delta greater than 0 00:16:06.460 --> 00:16:19.280 align:middle line:84% so that if z here is less than a plus delta, then 00:16:19.280 --> 00:16:20.740 align:middle line:90% this fraction here-- 00:16:20.740 --> 00:16:27.720 align:middle line:84% then f prime of z minus g prime of z is-- 00:16:27.720 --> 00:16:32.100 align:middle line:90% minus L is less than epsilon. 00:16:32.100 --> 00:16:36.120 align:middle line:90% 00:16:36.120 --> 00:16:42.080 align:middle line:84% So now coming back here, if you were starting with-- 00:16:42.080 --> 00:16:45.540 align:middle line:84% so this here, again, was for all fixed x. 00:16:45.540 --> 00:16:50.240 align:middle line:90% 00:16:50.240 --> 00:16:52.140 align:middle line:90% You had this statement. 00:16:52.140 --> 00:16:57.880 align:middle line:84% And again, x0 was between y and x. 00:16:57.880 --> 00:17:03.960 align:middle line:84% So now if you-- so if this x here-- 00:17:03.960 --> 00:17:17.720 align:middle line:84% if x is-- is delta close to the left endpoint of the interval, 00:17:17.720 --> 00:17:22.119 align:middle line:90% then you would have that x0. 00:17:22.119 --> 00:17:26.819 align:middle line:84% But x0-- remember that x0 is lying here. 00:17:26.819 --> 00:17:35.290 align:middle line:84% So since x0 would lie here, this means that also, x0 here 00:17:35.290 --> 00:17:40.670 align:middle line:90% is in this interval here. 00:17:40.670 --> 00:17:44.000 align:middle line:90% 00:17:44.000 --> 00:17:46.370 align:middle line:84% And so this means that this thing-- 00:17:46.370 --> 00:17:54.410 align:middle line:84% this number here you get is L up to some-- 00:17:54.410 --> 00:17:59.470 align:middle line:84% it's L up to some error of epsilon. 00:17:59.470 --> 00:18:02.250 align:middle line:90% 00:18:02.250 --> 00:18:06.290 align:middle line:84% So this here is almost equal to L. 00:18:06.290 --> 00:18:09.770 align:middle line:84% And so this means that this limit here 00:18:09.770 --> 00:18:15.550 align:middle line:84% is almost equal to L within an error of epsilon. 00:18:15.550 --> 00:18:20.690 align:middle line:84% So this proof-- this version of L'Hopital's rule. 00:18:20.690 --> 00:18:24.250 align:middle line:84% And that's the second version, and that's in the notes, 00:18:24.250 --> 00:18:28.370 align:middle line:84% but I'll skip it, actually, for now just because there 00:18:28.370 --> 00:18:31.550 align:middle line:84% was-- from the last lecture, and it's very similar to this. 00:18:31.550 --> 00:18:32.388 align:middle line:90% Yeah? 00:18:32.388 --> 00:18:37.453 align:middle line:84% STUDENT: Why did you deduce the g x minus g y is not equal to 0? 00:18:37.453 --> 00:18:38.370 align:middle line:90% TOBIAS COLDING: Sorry? 00:18:38.370 --> 00:18:39.010 align:middle line:90% Why-- 00:18:39.010 --> 00:18:41.510 align:middle line:84% STUDENT: Why would you deduce x that g x minus g 00:18:41.510 --> 00:18:43.090 align:middle line:90% y is not equal to 0? 00:18:43.090 --> 00:18:44.250 align:middle line:90% TOBIAS COLDING: Yeah, OK. 00:18:44.250 --> 00:18:52.370 align:middle line:84% So this is because g of x is supposed 00:18:52.370 --> 00:18:54.870 align:middle line:90% to be not equal to x for all x. 00:18:54.870 --> 00:18:58.630 align:middle line:90% 00:18:58.630 --> 00:19:01.650 align:middle line:90% And now I'm fixing x. 00:19:01.650 --> 00:19:03.410 align:middle line:90% So this here is not 0. 00:19:03.410 --> 00:19:07.330 align:middle line:84% Then I'm letting y go to a, but then this 00:19:07.330 --> 00:19:11.270 align:middle line:90% here goes to 0 by assumption. 00:19:11.270 --> 00:19:13.770 align:middle line:84% So you see that if you're fixing x 00:19:13.770 --> 00:19:18.010 align:middle line:84% and you just take out y sufficiently close to a, then 00:19:18.010 --> 00:19:20.730 align:middle line:84% this denominator is definitely not 0. 00:19:20.730 --> 00:19:23.290 align:middle line:84% You really want to let this-- you 00:19:23.290 --> 00:19:28.830 align:middle line:84% want to let y goes to a because then this here is eliminated. 00:19:28.830 --> 00:19:31.160 align:middle line:84% This is not part of the statement, that g of y. 00:19:31.160 --> 00:19:33.860 align:middle line:90% 00:19:33.860 --> 00:19:36.540 align:middle line:84% And so, again, as long as we started 00:19:36.540 --> 00:19:41.600 align:middle line:84% with an x that was sufficiently close to y-- 00:19:41.600 --> 00:19:46.920 align:middle line:84% if x was in this interval, then x0 is even further to the left, 00:19:46.920 --> 00:19:48.880 align:middle line:84% so this is also in this interval. 00:19:48.880 --> 00:19:53.160 align:middle line:84% We don't know exactly what x0 is, but it is in that interval, 00:19:53.160 --> 00:19:56.730 align:middle line:84% and so this means that this is strictly smaller than epsilon. 00:19:56.730 --> 00:19:59.780 align:middle line:90% 00:19:59.780 --> 00:20:01.570 align:middle line:90% OK. 00:20:01.570 --> 00:20:05.580 align:middle line:84% So another thing we talked about last time 00:20:05.580 --> 00:20:07.640 align:middle line:84% was this with the Taylor expansion. 00:20:07.640 --> 00:20:13.140 align:middle line:90% 00:20:13.140 --> 00:20:15.940 align:middle line:84% In that all of these things really 00:20:15.940 --> 00:20:20.540 align:middle line:84% came out of this broader theorem. 00:20:20.540 --> 00:20:26.940 align:middle line:84% And that was also the case for the Taylor expansion. 00:20:26.940 --> 00:20:29.220 align:middle line:90% So you have a function f-- 00:20:29.220 --> 00:20:32.840 align:middle line:84% so this is-- just to remind you about the Taylor expansion. 00:20:32.840 --> 00:20:40.940 align:middle line:90% 00:20:40.940 --> 00:20:45.500 align:middle line:90% So you have a function f. 00:20:45.500 --> 00:20:48.460 align:middle line:90% 00:20:48.460 --> 00:20:53.740 align:middle line:84% And let's say that it's defined on some interval here into R. 00:20:53.740 --> 00:20:56.100 align:middle line:90% And the function is-- 00:20:56.100 --> 00:21:02.780 align:middle line:84% so f here is k times differentiable. 00:21:02.780 --> 00:21:08.660 align:middle line:90% 00:21:08.660 --> 00:21:12.660 align:middle line:84% So this means that f prime exists, 00:21:12.660 --> 00:21:18.500 align:middle line:84% f double prime up to the k derivative. 00:21:18.500 --> 00:21:19.450 align:middle line:90% So they all exist. 00:21:19.450 --> 00:21:24.660 align:middle line:90% 00:21:24.660 --> 00:21:28.820 align:middle line:84% And then the statement was-- so then 00:21:28.820 --> 00:21:32.420 align:middle line:90% you looked at the k minus 1. 00:21:32.420 --> 00:21:39.040 align:middle line:84% This is the k minus 1 Taylor polynomial. 00:21:39.040 --> 00:21:47.100 align:middle line:90% 00:21:47.100 --> 00:21:55.660 align:middle line:84% And it was given by that k minus 1 of x is equal to-- 00:21:55.660 --> 00:22:01.620 align:middle line:84% and now you're summing here from i equal to 0 to k minus 1. 00:22:01.620 --> 00:22:04.420 align:middle line:90% And you take the i derivative-- 00:22:04.420 --> 00:22:10.500 align:middle line:84% this is-- sorry, this is the i derivative of f, 00:22:10.500 --> 00:22:15.820 align:middle line:84% and you do this in a, and you divide by a factorial, 00:22:15.820 --> 00:22:23.020 align:middle line:84% and then you're looking at x minus a to the power i. 00:22:23.020 --> 00:22:31.190 align:middle line:84% So this was the k minus 1 Taylor polynomial. 00:22:31.190 --> 00:22:37.870 align:middle line:84% And then the Taylor expansion just 00:22:37.870 --> 00:22:51.510 align:middle line:84% told you that f here at some x was 00:22:51.510 --> 00:23:02.470 align:middle line:84% equal to P k minus 1, this Taylor polynomial plus. 00:23:02.470 --> 00:23:06.740 align:middle line:84% And then this is what is usually referred to as the remainder 00:23:06.740 --> 00:23:07.240 align:middle line:90% term. 00:23:07.240 --> 00:23:13.390 align:middle line:90% 00:23:13.390 --> 00:23:19.530 align:middle line:84% And this looked very similar to one of these terms. 00:23:19.530 --> 00:23:26.350 align:middle line:84% It was the k derivative of f, but then it 00:23:26.350 --> 00:23:38.350 align:middle line:84% was evaluated at a point c, divided by k factorial times 00:23:38.350 --> 00:23:45.470 align:middle line:84% x minus a to the power k And c here 00:23:45.470 --> 00:23:50.730 align:middle line:84% was supposed to lie between a and x. 00:23:50.730 --> 00:23:54.190 align:middle line:90% 00:23:54.190 --> 00:23:58.810 align:middle line:84% So that was-- this was the Taylor expansion theorem. 00:23:58.810 --> 00:24:04.570 align:middle line:84% And so the thing is that-- so why is this so useful? 00:24:04.570 --> 00:24:08.190 align:middle line:84% And it is extremely useful in many different settings, 00:24:08.190 --> 00:24:10.070 align:middle line:84% also in much more general settings 00:24:10.070 --> 00:24:12.230 align:middle line:90% than what I'm describing here. 00:24:12.230 --> 00:24:19.470 align:middle line:84% And that is because that if you take a function here, then-- 00:24:19.470 --> 00:24:22.160 align:middle line:84% and it may be a very complicated function, 00:24:22.160 --> 00:24:25.160 align:middle line:84% but this Taylor expansion theorem tells you 00:24:25.160 --> 00:24:32.640 align:middle line:84% that it is close to-- if you want to evaluate it at some x, 00:24:32.640 --> 00:24:35.900 align:middle line:84% you can evaluate the Taylor polynomial-- 00:24:35.900 --> 00:24:37.740 align:middle line:84% that's, after all, just a polynomial, 00:24:37.740 --> 00:24:40.200 align:middle line:90% so that's easy to evaluate. 00:24:40.200 --> 00:24:44.600 align:middle line:84% And then you get within this remainder term 00:24:44.600 --> 00:24:46.000 align:middle line:90% of the right answer. 00:24:46.000 --> 00:24:49.920 align:middle line:90% So that's how it always is used. 00:24:49.920 --> 00:24:52.320 align:middle line:90% And so the question is now-- 00:24:52.320 --> 00:24:58.320 align:middle line:84% the natural question is, how well does the Taylor polynomial 00:24:58.320 --> 00:25:01.930 align:middle line:84% approximate the value of the given function? 00:25:01.930 --> 00:25:05.940 align:middle line:90% 00:25:05.940 --> 00:25:23.360 align:middle line:84% Right, so how well does the Taylor polynomial 00:25:23.360 --> 00:25:30.790 align:middle line:90% approximate the function? 00:25:30.790 --> 00:25:36.304 align:middle line:90% 00:25:36.304 --> 00:25:42.000 align:middle line:84% And of course, to understand that, 00:25:42.000 --> 00:25:45.200 align:middle line:84% you need to estimate the remainder term. 00:25:45.200 --> 00:25:48.040 align:middle line:84% Let me write the remainder term here. 00:25:48.040 --> 00:25:49.790 align:middle line:90% So the remainder term-- 00:25:49.790 --> 00:25:54.840 align:middle line:90% 00:25:54.840 --> 00:25:58.680 align:middle line:84% right, so the remainder term is-- 00:25:58.680 --> 00:26:01.360 align:middle line:90% so I write this here as-- 00:26:01.360 --> 00:26:06.560 align:middle line:84% so it really depends on x also, but I will just 00:26:06.560 --> 00:26:08.480 align:middle line:90% write it like this in here. 00:26:08.480 --> 00:26:12.880 align:middle line:84% So by this, I mean the k derivative 00:26:12.880 --> 00:26:21.890 align:middle line:84% of f in c divided by k factorial, and then x minus a 00:26:21.890 --> 00:26:24.050 align:middle line:90% to the power k. 00:26:24.050 --> 00:26:27.410 align:middle line:84% So again, it depends on really also-- 00:26:27.410 --> 00:26:30.730 align:middle line:84% of course, it depends on a, but it also depends on x, 00:26:30.730 --> 00:26:32.490 align:middle line:90% and it's this thing here. 00:26:32.490 --> 00:26:38.230 align:middle line:84% This thing-- the catch is, that you don't really know what c is. 00:26:38.230 --> 00:26:41.290 align:middle line:84% The only thing you know about c is that c 00:26:41.290 --> 00:26:46.170 align:middle line:90% is lying between a and x. 00:26:46.170 --> 00:26:49.450 align:middle line:84% So this is essentially all you need-- 00:26:49.450 --> 00:27:00.870 align:middle line:90% all you know about c. 00:27:00.870 --> 00:27:05.050 align:middle line:90% 00:27:05.050 --> 00:27:08.110 align:middle line:84% So now, let's look at some examples. 00:27:08.110 --> 00:27:09.910 align:middle line:90% Let's look at two examples. 00:27:09.910 --> 00:27:13.510 align:middle line:84% So the first one is like a promising example, 00:27:13.510 --> 00:27:16.570 align:middle line:84% and the second one is much less promising. 00:27:16.570 --> 00:27:23.570 align:middle line:84% And so the thing is that how well it approximates really 00:27:23.570 --> 00:27:25.130 align:middle line:90% depends on the function. 00:27:25.130 --> 00:27:32.130 align:middle line:84% But the first one is quite typical, though, example. 00:27:32.130 --> 00:27:35.570 align:middle line:84% So suppose that the function f of x 00:27:35.570 --> 00:27:37.400 align:middle line:90% is the exponential function. 00:27:37.400 --> 00:27:40.730 align:middle line:90% 00:27:40.730 --> 00:27:46.890 align:middle line:84% And then we have that the derivative is just 00:27:46.890 --> 00:27:48.970 align:middle line:90% the exponential function itself. 00:27:48.970 --> 00:27:55.850 align:middle line:84% So this means that if you take the derivative k times, 00:27:55.850 --> 00:27:58.040 align:middle line:84% then you just get the function itself. 00:27:58.040 --> 00:28:01.130 align:middle line:90% 00:28:01.130 --> 00:28:05.170 align:middle line:84% So this means that the k minus 1-- 00:28:05.170 --> 00:28:10.810 align:middle line:84% which means that the k minus 1 Taylor polynomial, this 00:28:10.810 --> 00:28:12.630 align:middle line:84% is just the value of the function-- 00:28:12.630 --> 00:28:14.330 align:middle line:90% I'll just write it out here. 00:28:14.330 --> 00:28:17.340 align:middle line:84% So this is the value, and now I do it-- 00:28:17.340 --> 00:28:21.910 align:middle line:90% I do it for a equal to 0. 00:28:21.910 --> 00:28:24.460 align:middle line:90% 00:28:24.460 --> 00:28:27.780 align:middle line:84% And I'm assuming that b here-- so I'm thinking about-- 00:28:27.780 --> 00:28:35.378 align:middle line:84% I will think about this function here f of x on this function, 00:28:35.378 --> 00:28:36.420 align:middle line:90% the exponential function. 00:28:36.420 --> 00:28:39.060 align:middle line:84% I'm going to think about it on the interval, 00:28:39.060 --> 00:28:54.340 align:middle line:84% say, from minus 1 to 1 into R. Maybe I'll just 00:28:54.340 --> 00:28:59.534 align:middle line:90% take it from 0 to 1 into R. 00:28:59.534 --> 00:29:00.752 align:middle line:90% OK. 00:29:00.752 --> 00:29:03.940 align:middle line:84% And the Taylor polynomial-- so I'm going to-- 00:29:03.940 --> 00:29:10.180 align:middle line:84% so the Taylor polynomial, one, also talk about that this-- 00:29:10.180 --> 00:29:15.180 align:middle line:84% the Taylor polynomial is expanded around a. 00:29:15.180 --> 00:29:18.020 align:middle line:84% So we talk about the Taylor polynomial. 00:29:18.020 --> 00:29:29.900 align:middle line:84% So we talk about Taylor expansion around a. 00:29:29.900 --> 00:29:34.340 align:middle line:84% And this just refers to that when 00:29:34.340 --> 00:29:36.920 align:middle line:84% you plug in a here, well then, of course, 00:29:36.920 --> 00:29:38.660 align:middle line:90% you have exact equality. 00:29:38.660 --> 00:29:44.160 align:middle line:84% And that all of the derivatives are evaluated at a. 00:29:44.160 --> 00:29:46.800 align:middle line:84% So it's really-- like the Taylor polynomial, 00:29:46.800 --> 00:29:51.180 align:middle line:84% it's only a good approximation near a. 00:29:51.180 --> 00:30:00.260 align:middle line:84% So I want now to Taylor expand near the theorem. 00:30:00.260 --> 00:30:06.460 align:middle line:84% And so the value here of the function, this 1 at 0, 00:30:06.460 --> 00:30:09.040 align:middle line:90% the derivative is-- 00:30:09.040 --> 00:30:13.190 align:middle line:84% so the derivative is also 1 because it's just a function 00:30:13.190 --> 00:30:14.510 align:middle line:90% itself. 00:30:14.510 --> 00:30:16.030 align:middle line:90% And so this is-- 00:30:16.030 --> 00:30:20.350 align:middle line:90% so you get x, and then you get-- 00:30:20.350 --> 00:30:23.190 align:middle line:84% the next element in the Taylor expansion 00:30:23.190 --> 00:30:31.630 align:middle line:84% is x squared over 2 plus x squared over 3 factorial. 00:30:31.630 --> 00:30:36.470 align:middle line:84% And then x to the fourth over 4 factorial, 00:30:36.470 --> 00:30:39.630 align:middle line:84% et cetera, up to x to the power k 00:30:39.630 --> 00:30:47.430 align:middle line:84% minus 1 over k minus 1 factorial. 00:30:47.430 --> 00:30:54.590 align:middle line:84% So this is the k Taylor expansion, or k minus 1 Taylor 00:30:54.590 --> 00:30:59.870 align:middle line:90% polynomial, around 0 00:30:59.870 --> 00:31:05.310 align:middle line:84% And so now, if you are interested in evaluating P, 00:31:05.310 --> 00:31:08.670 align:middle line:84% you want to say, how well does this here 00:31:08.670 --> 00:31:13.050 align:middle line:84% approximate the original function, 00:31:13.050 --> 00:31:17.050 align:middle line:84% the exponential function, so you want to bound this thing here. 00:31:17.050 --> 00:31:19.470 align:middle line:84% So you want to have a bound for that. 00:31:19.470 --> 00:31:24.190 align:middle line:84% Well, the thing, again, is that f of x, you know that f of x 00:31:24.190 --> 00:31:29.510 align:middle line:84% is equal to this Taylor polynomial, and then 00:31:29.510 --> 00:31:34.630 align:middle line:84% plus this remainder term in the c. 00:31:34.630 --> 00:31:37.190 align:middle line:90% 00:31:37.190 --> 00:31:40.330 align:middle line:90% Now c, again, has to lie-- 00:31:40.330 --> 00:31:48.510 align:middle line:84% c will see will lie between a and x, and in this case, a is 0. 00:31:48.510 --> 00:31:50.530 align:middle line:90% So it means that c here-- 00:31:50.530 --> 00:31:53.870 align:middle line:90% 00:31:53.870 --> 00:32:02.350 align:middle line:84% so this means that c here is between 0 and x. 00:32:02.350 --> 00:32:11.320 align:middle line:84% And the remainder term here, this here was equal to-- 00:32:11.320 --> 00:32:16.360 align:middle line:84% that you took the k derivative of the function. 00:32:16.360 --> 00:32:18.720 align:middle line:84% So in this case, the k-th derivative of the function 00:32:18.720 --> 00:32:20.560 align:middle line:90% is the function itself. 00:32:20.560 --> 00:32:27.400 align:middle line:84% So this here is the value of the s derivative, and this is at c. 00:32:27.400 --> 00:32:31.840 align:middle line:84% This is the k-th derivative of the original function. 00:32:31.840 --> 00:32:35.160 align:middle line:84% And then you have to divide by k factorial, 00:32:35.160 --> 00:32:39.880 align:middle line:84% and then it's x to the power k like that. 00:32:39.880 --> 00:32:43.240 align:middle line:90% 00:32:43.240 --> 00:32:50.080 align:middle line:84% And so you see that if x here is now between 0 and 1, 00:32:50.080 --> 00:32:56.640 align:middle line:84% this means that this thing here, this remainder term here, 00:32:56.640 --> 00:32:59.240 align:middle line:90% the absolute value is-- 00:32:59.240 --> 00:33:04.320 align:middle line:84% so c here is less than 1, so this means that this thing here 00:33:04.320 --> 00:33:06.760 align:middle line:90% is less than e. 00:33:06.760 --> 00:33:12.760 align:middle line:84% So it's less than or equal to e divided by k factorial. 00:33:12.760 --> 00:33:15.080 align:middle line:90% This here is bounded by-- 00:33:15.080 --> 00:33:18.180 align:middle line:84% since x is bounded by 1, this is bounded by 1, 00:33:18.180 --> 00:33:21.080 align:middle line:84% so you're getting this inequality here. 00:33:21.080 --> 00:33:27.600 align:middle line:84% And so you see that if k is very large, this is a tiny number. 00:33:27.600 --> 00:33:29.780 align:middle line:84% So it's a really good approximation. 00:33:29.780 --> 00:33:33.040 align:middle line:84% I mean, you don't even need to make k very large because it's 00:33:33.040 --> 00:33:34.840 align:middle line:90% exponential. 00:33:34.840 --> 00:33:36.260 align:middle line:90% Not exponential, it's factorial. 00:33:36.260 --> 00:33:36.760 align:middle line:90% Yeah. 00:33:36.760 --> 00:33:39.640 align:middle line:90% Right. 00:33:39.640 --> 00:33:40.412 align:middle line:90% OK. 00:33:40.412 --> 00:33:42.940 align:middle line:84% So that's a kind of encouraging example. 00:33:42.940 --> 00:33:47.960 align:middle line:84% Let's say that the Taylor expansion is a very-- in this 00:33:47.960 --> 00:33:50.650 align:middle line:84% case, it's a very good approximation to the function. 00:33:50.650 --> 00:33:53.760 align:middle line:90% 00:33:53.760 --> 00:33:56.960 align:middle line:84% Another example that is less encouraging-- 00:33:56.960 --> 00:34:00.440 align:middle line:84% and so again, the moral is that how well 00:34:00.440 --> 00:34:02.960 align:middle line:84% the Taylor expansion approximate the function 00:34:02.960 --> 00:34:05.450 align:middle line:90% depends on the function. 00:34:05.450 --> 00:34:07.970 align:middle line:84% So suppose that the function here 00:34:07.970 --> 00:34:16.570 align:middle line:84% was a function that was 0 here when x is less than 0-- 00:34:16.570 --> 00:34:18.810 align:middle line:90% less than or equal to 0. 00:34:18.810 --> 00:34:26.290 align:middle line:84% And when x is bigger than 0, then it is e to the minus 1 00:34:26.290 --> 00:34:27.229 align:middle line:90% over x squared. 00:34:27.229 --> 00:34:30.090 align:middle line:90% 00:34:30.090 --> 00:34:36.210 align:middle line:84% So you see that when x becomes very, very small, 00:34:36.210 --> 00:34:39.710 align:middle line:84% you are looking at 1 over the square. 00:34:39.710 --> 00:34:43.510 align:middle line:90% So this here is huge. 00:34:43.510 --> 00:34:45.690 align:middle line:84% But now it's with a minus sign, so this 00:34:45.690 --> 00:34:49.250 align:middle line:84% means that this thing here is incredibly small. 00:34:49.250 --> 00:34:53.010 align:middle line:84% And so it's clear that this function here 00:34:53.010 --> 00:34:55.050 align:middle line:90% is continuous at 0. 00:34:55.050 --> 00:34:57.730 align:middle line:84% It's clearly continuous everywhere else. 00:34:57.730 --> 00:35:02.570 align:middle line:84% But actually, using in this argument, 00:35:02.570 --> 00:35:06.050 align:middle line:84% it's easy to see that this function f here 00:35:06.050 --> 00:35:09.850 align:middle line:90% is infinite differentiable. 00:35:09.850 --> 00:35:12.930 align:middle line:84% So you can differentiate it infinitely many times, 00:35:12.930 --> 00:35:14.010 align:middle line:90% and even f0. 00:35:14.010 --> 00:35:19.950 align:middle line:90% 00:35:19.950 --> 00:35:25.770 align:middle line:84% And by its end, of course, the derivative is-- 00:35:25.770 --> 00:35:28.510 align:middle line:84% so it's an example of a function that looks like this. 00:35:28.510 --> 00:35:34.530 align:middle line:84% It's that 0 until 0, and then it goes up like this. 00:35:34.530 --> 00:35:38.950 align:middle line:90% And so the derivative at 0 is 0. 00:35:38.950 --> 00:35:41.130 align:middle line:84% Any derivative, however many times 00:35:41.130 --> 00:35:44.450 align:middle line:84% you take the derivative of this function-- 00:35:44.450 --> 00:35:50.610 align:middle line:84% so this thing here, if you take any derivative 00:35:50.610 --> 00:35:54.090 align:middle line:84% and you evaluate it at 0, you always get 0. 00:35:54.090 --> 00:35:58.970 align:middle line:90% 00:35:58.970 --> 00:36:09.660 align:middle line:84% So this means that if you're looking at the Taylor expansion 00:36:09.660 --> 00:36:10.580 align:middle line:90% at 0-- 00:36:10.580 --> 00:36:16.940 align:middle line:84% so you're looking at this k minus 1 Taylor polynomial, 00:36:16.940 --> 00:36:19.380 align:middle line:84% this polynomial is-- all of the coefficients 00:36:19.380 --> 00:36:21.700 align:middle line:90% in the polynomial are 0. 00:36:21.700 --> 00:36:24.420 align:middle line:90% So this polynomial is just 0. 00:36:24.420 --> 00:36:27.940 align:middle line:84% So this means that if you're looking 00:36:27.940 --> 00:36:34.660 align:middle line:84% at f of x where f of x is actually equal to this remainder 00:36:34.660 --> 00:36:38.540 align:middle line:90% term, so it's somehow-- 00:36:38.540 --> 00:36:40.720 align:middle line:84% the Taylor polynomial is too trivial. 00:36:40.720 --> 00:36:43.520 align:middle line:84% It doesn't really help you all that much. 00:36:43.520 --> 00:36:48.300 align:middle line:84% So, it's equal to this remainder term. 00:36:48.300 --> 00:36:50.020 align:middle line:90% OK. 00:36:50.020 --> 00:36:50.740 align:middle line:90% Yeah? 00:36:50.740 --> 00:36:53.220 align:middle line:90% STUDENT: Is there a way of-- 00:36:53.220 --> 00:36:55.820 align:middle line:84% so the remainder-- like, boundedness 00:36:55.820 --> 00:36:59.860 align:middle line:84% helps us figure out how well the expansion approximates 00:36:59.860 --> 00:37:01.080 align:middle line:90% the actual function. 00:37:01.080 --> 00:37:05.300 align:middle line:84% Is there a way to determine how far from the point 00:37:05.300 --> 00:37:06.240 align:middle line:90% you're approximating? 00:37:06.240 --> 00:37:11.060 align:middle line:84% Like a or 0 in this case, you can go to a certain endpoint? 00:37:11.060 --> 00:37:13.180 align:middle line:84% TOBIAS COLDING: Yeah, right, that's right. 00:37:13.180 --> 00:37:15.080 align:middle line:90% I mean, so you are asking-- 00:37:15.080 --> 00:37:16.700 align:middle line:90% so you're asking-- suppose-- 00:37:16.700 --> 00:37:18.880 align:middle line:84% and you definitely could do this. 00:37:18.880 --> 00:37:25.644 align:middle line:84% You're asking-- you're thinking that you have to have a-- 00:37:25.644 --> 00:37:28.560 align:middle line:84% you see a slightly different situation than here, 00:37:28.560 --> 00:37:30.300 align:middle line:90% but closely related. 00:37:30.300 --> 00:37:34.300 align:middle line:90% That you have some a here. 00:37:34.300 --> 00:37:37.300 align:middle line:90% And you have a function f. 00:37:37.300 --> 00:37:39.500 align:middle line:90% And you want to-- 00:37:39.500 --> 00:37:43.820 align:middle line:84% and you are willing to maybe look 00:37:43.820 --> 00:37:49.580 align:middle line:84% at-- you're willing to take some fixed large k. 00:37:49.580 --> 00:37:52.540 align:middle line:90% And then you're asking-- 00:37:52.540 --> 00:37:56.700 align:middle line:84% what I'm interested in, how well does this-- 00:37:56.700 --> 00:38:00.070 align:middle line:84% how well does this function, this polynomial 00:38:00.070 --> 00:38:01.170 align:middle line:90% approximate this? 00:38:01.170 --> 00:38:04.310 align:middle line:84% But I want it within an error, certain error. 00:38:04.310 --> 00:38:08.390 align:middle line:84% So there's a given epsilon, and you are interested 00:38:08.390 --> 00:38:12.030 align:middle line:84% in how far away can you go from a so 00:38:12.030 --> 00:38:14.630 align:middle line:84% that this here is less than epsilon? 00:38:14.630 --> 00:38:16.790 align:middle line:90% And so this is-- 00:38:16.790 --> 00:38:21.210 align:middle line:84% so this becomes a question of this remainder term. 00:38:21.210 --> 00:38:23.570 align:middle line:84% So this becomes a question of this remainder term. 00:38:23.570 --> 00:38:25.830 align:middle line:84% When can you always say that this remainder 00:38:25.830 --> 00:38:28.710 align:middle line:90% term is less than epsilon? 00:38:28.710 --> 00:38:30.830 align:middle line:84% And so in this particular case, you 00:38:30.830 --> 00:38:34.830 align:middle line:84% would have to do it on a case-by-case basis. 00:38:34.830 --> 00:38:39.530 align:middle line:84% But in this particular case, the remainder term is-- 00:38:39.530 --> 00:38:44.430 align:middle line:84% If this interval here, in that example over there, 00:38:44.430 --> 00:38:49.630 align:middle line:84% if you have here 0, this was a in our example, 00:38:49.630 --> 00:38:54.670 align:middle line:84% and then you want to go out here to some x, 00:38:54.670 --> 00:38:59.150 align:middle line:84% and now you know that C here is between 0 and x, 00:38:59.150 --> 00:39:02.110 align:middle line:84% but you don't really know anything else, 00:39:02.110 --> 00:39:05.570 align:middle line:84% so then it's like this remainder term, 00:39:05.570 --> 00:39:10.510 align:middle line:84% you will have this remainder term in this example adds 00:39:10.510 --> 00:39:12.070 align:middle line:90% this c here. 00:39:12.070 --> 00:39:14.870 align:middle line:84% This, again, was equal to this thing here. 00:39:14.870 --> 00:39:16.290 align:middle line:90% Over k factorial. 00:39:16.290 --> 00:39:19.110 align:middle line:90% 00:39:19.110 --> 00:39:23.550 align:middle line:84% And so you would have-- that you would want this thing here, 00:39:23.550 --> 00:39:26.470 align:middle line:90% Rk c, You. 00:39:26.470 --> 00:39:29.430 align:middle line:84% Would want this to be less than epsilon. 00:39:29.430 --> 00:39:33.762 align:middle line:84% And so you would have that this thing here-- 00:39:33.762 --> 00:39:37.190 align:middle line:84% the only thing you can really do is 00:39:37.190 --> 00:39:40.960 align:middle line:84% that you can say that c is less than or equal to x. 00:39:40.960 --> 00:39:42.430 align:middle line:90% STUDENT: And we solved for c-- 00:39:42.430 --> 00:39:43.305 align:middle line:90% TOBIAS COLDING: Yeah. 00:39:43.305 --> 00:39:45.290 align:middle line:84% That whenever-- so you would do-- 00:39:45.290 --> 00:39:50.310 align:middle line:84% so what you would do here is you would say Rk c, 00:39:50.310 --> 00:39:53.070 align:middle line:90% this here has to be bounded by-- 00:39:53.070 --> 00:39:55.840 align:middle line:90% because x c is less than x. 00:39:55.840 --> 00:39:59.880 align:middle line:84% So this is e to the x-- you have to eliminate c. 00:39:59.880 --> 00:40:04.920 align:middle line:84% So you would do like this, k factorial times x 00:40:04.920 --> 00:40:05.980 align:middle line:90% to the power k. 00:40:05.980 --> 00:40:10.400 align:middle line:84% You have to eliminate c because you don't know anything really 00:40:10.400 --> 00:40:11.600 align:middle line:90% about c. 00:40:11.600 --> 00:40:15.920 align:middle line:84% And so this here now has to be less than epsilon. 00:40:15.920 --> 00:40:19.960 align:middle line:84% So now you only have-- you have fixed your k, 00:40:19.960 --> 00:40:23.400 align:middle line:84% and so now you only have one variable and you can solve that. 00:40:23.400 --> 00:40:26.220 align:middle line:84% But you really have to eliminate c, yeah. 00:40:26.220 --> 00:40:29.476 align:middle line:90% 00:40:29.476 --> 00:40:29.976 align:middle line:90% OK. 00:40:29.976 --> 00:40:32.664 align:middle line:90% 00:40:32.664 --> 00:40:37.600 align:middle line:84% So again, so how well the Taylor expansion approximate 00:40:37.600 --> 00:40:39.960 align:middle line:90% depends on the function. 00:40:39.960 --> 00:40:43.020 align:middle line:84% OK, the next topic is another important topic. 00:40:43.020 --> 00:40:47.280 align:middle line:90% And so this is integrals. 00:40:47.280 --> 00:40:58.520 align:middle line:84% And there are different ways of defining integrals. 00:40:58.520 --> 00:41:07.280 align:middle line:84% This here is-- this is certainly one of the-- 00:41:07.280 --> 00:41:10.720 align:middle line:90% maybe the most natural ones. 00:41:10.720 --> 00:41:15.040 align:middle line:84% And so this is what's called Riemannese. 00:41:15.040 --> 00:41:19.160 align:middle line:84% So it's after German mathematician Riemann, 00:41:19.160 --> 00:41:19.980 align:middle line:90% integrals. 00:41:19.980 --> 00:41:26.880 align:middle line:90% 00:41:26.880 --> 00:41:31.600 align:middle line:84% And so the thing is that here now, you look at a function. 00:41:31.600 --> 00:41:37.800 align:middle line:84% You have a function on some interval from a to b into R. 00:41:37.800 --> 00:41:42.710 align:middle line:84% And this function f here is bounded. 00:41:42.710 --> 00:41:47.920 align:middle line:90% 00:41:47.920 --> 00:41:49.720 align:middle line:90% And the idea is-- 00:41:49.720 --> 00:41:52.940 align:middle line:84% so the idea is for integrals is-- 00:41:52.940 --> 00:41:55.560 align:middle line:90% 00:41:55.560 --> 00:41:58.680 align:middle line:90% so you have a function. 00:41:58.680 --> 00:42:01.920 align:middle line:84% And you want to be able to define-- 00:42:01.920 --> 00:42:15.990 align:middle line:90% so define an area below graph. 00:42:15.990 --> 00:42:18.880 align:middle line:90% 00:42:18.880 --> 00:42:21.880 align:middle line:84% So if this has a function, you want 00:42:21.880 --> 00:42:26.680 align:middle line:84% to be able to calculate what is this area. 00:42:26.680 --> 00:42:29.840 align:middle line:90% Now, if the function was to-- 00:42:29.840 --> 00:42:35.560 align:middle line:84% if the function was to have crossed the x-axis, 00:42:35.560 --> 00:42:39.080 align:middle line:84% then this area here would count negatively 00:42:39.080 --> 00:42:41.800 align:middle line:84% and this area here would count positively. 00:42:41.800 --> 00:42:42.720 align:middle line:90% Anyway. 00:42:42.720 --> 00:42:47.120 align:middle line:90% So this is what you want to do. 00:42:47.120 --> 00:42:51.890 align:middle line:84% And you're always starting with a bounded function. 00:42:51.890 --> 00:42:54.740 align:middle line:90% Now, to do that-- 00:42:54.740 --> 00:42:58.050 align:middle line:84% right, and so you want to define that. 00:42:58.050 --> 00:43:01.680 align:middle line:84% The only thing that you know for sure is-- 00:43:01.680 --> 00:43:04.810 align:middle line:90% 00:43:04.810 --> 00:43:07.350 align:middle line:84% so there's only one ingredient you have. 00:43:07.350 --> 00:43:11.050 align:middle line:90% 00:43:11.050 --> 00:43:26.440 align:middle line:84% So only thing you know at the outset. 00:43:26.440 --> 00:43:30.050 align:middle line:90% 00:43:30.050 --> 00:43:35.290 align:middle line:84% So the only thing is that if you take a right angle-- 00:43:35.290 --> 00:43:37.160 align:middle line:90% so if this length here is-- 00:43:37.160 --> 00:43:41.050 align:middle line:90% 00:43:41.050 --> 00:43:44.210 align:middle line:84% let's call it d, and this here is e, 00:43:44.210 --> 00:43:48.490 align:middle line:84% then you know that the area here-- 00:43:48.490 --> 00:43:55.730 align:middle line:90% so area-- is d times e. 00:43:55.730 --> 00:43:59.250 align:middle line:90% That's the only thing we have. 00:43:59.250 --> 00:44:05.690 align:middle line:84% So now, how do we define the area below the graph? 00:44:05.690 --> 00:44:10.382 align:middle line:84% And to do that, we first need to talk about partitions. 00:44:10.382 --> 00:44:18.170 align:middle line:90% 00:44:18.170 --> 00:44:19.310 align:middle line:90% So partition. 00:44:19.310 --> 00:44:23.410 align:middle line:90% 00:44:23.410 --> 00:44:28.370 align:middle line:84% So suppose I have an integral here. 00:44:28.370 --> 00:44:29.950 align:middle line:90% a and b are real numbers. 00:44:29.950 --> 00:44:32.130 align:middle line:90% So I have integral here. 00:44:32.130 --> 00:44:34.190 align:middle line:90% Then I talk about a partition. 00:44:34.190 --> 00:44:39.330 align:middle line:90% 00:44:39.330 --> 00:44:43.150 align:middle line:84% And it's useful to have a notation for partition. 00:44:43.150 --> 00:44:50.340 align:middle line:84% So I call it P of this interval here. 00:44:50.340 --> 00:44:59.510 align:middle line:84% This is finitely many dividing points. 00:44:59.510 --> 00:45:05.740 align:middle line:90% 00:45:05.740 --> 00:45:10.460 align:middle line:84% And the dividing points always have to start at a and end at b. 00:45:10.460 --> 00:45:14.140 align:middle line:84% So you have finitely many-- let's say k many. 00:45:14.140 --> 00:45:19.300 align:middle line:84% And you're looking at k plus 1 many. 00:45:19.300 --> 00:45:25.340 align:middle line:90% So the first one has to be a. 00:45:25.340 --> 00:45:34.864 align:middle line:84% And then you have x1 up to the last one, and that has to be b. 00:45:34.864 --> 00:45:37.940 align:middle line:90% 00:45:37.940 --> 00:45:42.260 align:middle line:84% And so what you are thinking about 00:45:42.260 --> 00:45:46.900 align:middle line:84% is-- so you have this interval here from a to b, 00:45:46.900 --> 00:45:51.100 align:middle line:84% and you have these dividing points. 00:45:51.100 --> 00:45:56.580 align:middle line:84% And you are thinking about dividing the original interval 00:45:56.580 --> 00:45:58.780 align:middle line:90% into these subintervals. 00:45:58.780 --> 00:46:01.580 align:middle line:90% So these are the subintervals. 00:46:01.580 --> 00:46:19.040 align:middle line:84% So you divide the original interval into subintervals. 00:46:19.040 --> 00:46:23.660 align:middle line:90% 00:46:23.660 --> 00:46:26.420 align:middle line:90% And the subintervals are-- 00:46:26.420 --> 00:46:27.760 align:middle line:90% I'll do it like this. 00:46:27.760 --> 00:46:32.100 align:middle line:84% I will do it where I take from-- it's always 00:46:32.100 --> 00:46:35.100 align:middle line:90% from one to the next. 00:46:35.100 --> 00:46:38.020 align:middle line:84% And I'll write it typically where I'm starting-- 00:46:38.020 --> 00:46:43.340 align:middle line:84% or I'm ending at i and beginning at i minus 1. 00:46:43.340 --> 00:46:49.030 align:middle line:84% So I'm dividing it into these subintervals. 00:46:49.030 --> 00:46:54.270 align:middle line:84% So in this case here, in this case here where this is x0, 00:46:54.270 --> 00:47:00.590 align:middle line:84% there's going to be k subintervals. 00:47:00.590 --> 00:47:05.450 align:middle line:84% And you only always go from one to the next. 00:47:05.450 --> 00:47:10.870 align:middle line:90% So that's a partition. 00:47:10.870 --> 00:47:14.370 align:middle line:84% And now we need a second notion related to this, 00:47:14.370 --> 00:47:16.210 align:middle line:90% and this is a subpartition. 00:47:16.210 --> 00:47:23.950 align:middle line:90% 00:47:23.950 --> 00:47:25.330 align:middle line:90% So subpartition. 00:47:25.330 --> 00:47:29.390 align:middle line:90% 00:47:29.390 --> 00:47:30.350 align:middle line:90% Subpartition. 00:47:30.350 --> 00:47:39.890 align:middle line:84% This is also sometimes called a refinement. 00:47:39.890 --> 00:47:44.190 align:middle line:90% 00:47:44.190 --> 00:47:47.470 align:middle line:84% So a subpartition is-- so if you have a partition-- 00:47:47.470 --> 00:48:03.390 align:middle line:84% P1 is a partition, and then P2 is a subpartition of refinement 00:48:03.390 --> 00:48:21.910 align:middle line:84% if P2 has all the dividing point of P1 and then some-- 00:48:21.910 --> 00:48:23.150 align:middle line:90% and then some. 00:48:23.150 --> 00:48:26.270 align:middle line:90% I mean-- right. 00:48:26.270 --> 00:48:30.070 align:middle line:84% I mean, of course, it could be a trivial subpartition where 00:48:30.070 --> 00:48:32.430 align:middle line:90% P1 and P2 are the same. 00:48:32.430 --> 00:48:33.870 align:middle line:90% That's OK. 00:48:33.870 --> 00:48:36.310 align:middle line:84% But in general, it will have some more points-- also 00:48:36.310 --> 00:48:37.510 align:middle line:90% some more dividing points. 00:48:37.510 --> 00:48:42.400 align:middle line:90% 00:48:42.400 --> 00:48:46.820 align:middle line:90% Now, right. 00:48:46.820 --> 00:48:49.820 align:middle line:90% So now, let me look at-- 00:48:49.820 --> 00:48:52.840 align:middle line:90% so what is, then-- 00:48:52.840 --> 00:48:57.560 align:middle line:84% so now, let's try to get closer to try to define this 00:48:57.560 --> 00:49:00.210 align:middle line:90% with the area below the graph. 00:49:00.210 --> 00:49:09.112 align:middle line:90% 00:49:09.112 --> 00:49:19.000 align:middle line:84% So again, we have a function f from a to b into R. 00:49:19.000 --> 00:49:22.530 align:middle line:84% We assume that this is-- so that is bounded. 00:49:22.530 --> 00:49:30.240 align:middle line:90% 00:49:30.240 --> 00:49:40.100 align:middle line:84% We have P here is a partition of this interval here. 00:49:40.100 --> 00:49:42.680 align:middle line:90% 00:49:42.680 --> 00:49:47.240 align:middle line:84% Then we defined a upper and a lower sum. 00:49:47.240 --> 00:50:06.660 align:middle line:84% Then we define upper and lower sums as follows. 00:50:06.660 --> 00:50:09.640 align:middle line:84% So again, you have a given function that is bounded 00:50:09.640 --> 00:50:11.360 align:middle line:90% and you have a partition. 00:50:11.360 --> 00:50:13.760 align:middle line:84% And so we define upper and lower sums, 00:50:13.760 --> 00:50:18.600 align:middle line:84% and the upper and lower sums I will denote like this. 00:50:18.600 --> 00:50:21.040 align:middle line:84% It depends on the function, of course, 00:50:21.040 --> 00:50:22.920 align:middle line:90% and then on the partition. 00:50:22.920 --> 00:50:24.880 align:middle line:90% That's an upper sum. 00:50:24.880 --> 00:50:28.460 align:middle line:84% And the lower sum, similarly, I will denote like this. 00:50:28.460 --> 00:50:33.695 align:middle line:90% 00:50:33.695 --> 00:50:35.310 align:middle line:90% And so how are these defined? 00:50:35.310 --> 00:50:39.610 align:middle line:90% 00:50:39.610 --> 00:50:41.320 align:middle line:90% So they are defined as follows. 00:50:41.320 --> 00:50:44.410 align:middle line:90% 00:50:44.410 --> 00:50:50.770 align:middle line:84% That you have-- here, you have your interval from a to b. 00:50:50.770 --> 00:50:56.170 align:middle line:84% Then you have-- this here is x0; here, I have x1; 00:50:56.170 --> 00:50:58.370 align:middle line:90% and here, I have maybe x2. 00:50:58.370 --> 00:51:00.840 align:middle line:90% And then this thing here is x3. 00:51:00.840 --> 00:51:04.930 align:middle line:90% 00:51:04.930 --> 00:51:06.850 align:middle line:90% So I take-- 00:51:06.850 --> 00:51:09.930 align:middle line:84% I look at one of these intervals. 00:51:09.930 --> 00:51:12.310 align:middle line:90% Well, it could be this one here. 00:51:12.310 --> 00:51:19.690 align:middle line:84% It's between-- in this case, it would be between x1 and x2. 00:51:19.690 --> 00:51:24.290 align:middle line:84% And I would look at the function in this interval here, 00:51:24.290 --> 00:51:25.550 align:middle line:90% and I would-- 00:51:25.550 --> 00:51:28.450 align:middle line:84% so the contribution from this here 00:51:28.450 --> 00:51:36.730 align:middle line:84% would be that I would take-- so for the upper sum with respect 00:51:36.730 --> 00:51:40.210 align:middle line:84% to this partition, the one that comes from this subinterval 00:51:40.210 --> 00:51:51.150 align:middle line:84% here would be the sup of f on this subinterval here. 00:51:51.150 --> 00:51:53.690 align:middle line:90% 00:51:53.690 --> 00:52:02.130 align:middle line:84% So the sup of the function times the length of the interval 00:52:02.130 --> 00:52:03.090 align:middle line:90% like that. 00:52:03.090 --> 00:52:05.770 align:middle line:84% So this would be the value that-- 00:52:05.770 --> 00:52:08.990 align:middle line:84% in the upper that would come-- in a minute 00:52:08.990 --> 00:52:11.130 align:middle line:90% I'll write it out more clearly. 00:52:11.130 --> 00:52:13.970 align:middle line:84% But this would be what is coming from here. 00:52:13.970 --> 00:52:21.090 align:middle line:84% And likewise, for the lower one, then the contribution 00:52:21.090 --> 00:52:23.730 align:middle line:84% from this little interval here would be 00:52:23.730 --> 00:52:32.410 align:middle line:90% the inf of f, and then times. 00:52:32.410 --> 00:52:34.860 align:middle line:90% And the idea is-- 00:52:34.860 --> 00:52:39.100 align:middle line:84% OK, so that would be the contribution. 00:52:39.100 --> 00:52:41.970 align:middle line:84% The idea, of course, is that you have-- 00:52:41.970 --> 00:52:48.460 align:middle line:90% 00:52:48.460 --> 00:52:52.910 align:middle line:84% the only thing we knew was this was the area of a right angle. 00:52:52.910 --> 00:52:55.540 align:middle line:90% 00:52:55.540 --> 00:52:59.180 align:middle line:84% And so you have this function here, 00:52:59.180 --> 00:53:02.580 align:middle line:84% and you have some little interval here 00:53:02.580 --> 00:53:06.060 align:middle line:90% between xi and xi minus-- 00:53:06.060 --> 00:53:08.140 align:middle line:90% sorry, xi minus 1 and xi. 00:53:08.140 --> 00:53:10.900 align:middle line:90% 00:53:10.900 --> 00:53:15.342 align:middle line:84% And you're focusing on this little interval here. 00:53:15.342 --> 00:53:16.800 align:middle line:84% And you're looking at the function. 00:53:16.800 --> 00:53:18.160 align:middle line:90% The function was bounded. 00:53:18.160 --> 00:53:20.500 align:middle line:90% So there is a sup on this. 00:53:20.500 --> 00:53:24.800 align:middle line:84% And you try to look at the smallest right 00:53:24.800 --> 00:53:30.180 align:middle line:84% angle that kind of have the graph that fits-- 00:53:30.180 --> 00:53:32.580 align:middle line:84% where the graph fits inside that. 00:53:32.580 --> 00:53:36.220 align:middle line:90% And then you're looking at-- 00:53:36.220 --> 00:53:43.380 align:middle line:84% so this is the contribution from this little interval to p-- 00:53:43.380 --> 00:53:46.020 align:middle line:90% to the upper sum. 00:53:46.020 --> 00:53:51.580 align:middle line:84% And then you have that you're looking at the largest right 00:53:51.580 --> 00:53:55.900 align:middle line:84% angle like this that fits under the graph, 00:53:55.900 --> 00:54:00.940 align:middle line:84% and that's in contribution to the lower one. 00:54:00.940 --> 00:54:05.740 align:middle line:84% And the idea is that if the function is reasonable, 00:54:05.740 --> 00:54:08.460 align:middle line:84% if you're looking at finite partition, then 00:54:08.460 --> 00:54:12.860 align:middle line:84% you should certainly get a better approximation to the area 00:54:12.860 --> 00:54:13.960 align:middle line:90% below the graph. 00:54:13.960 --> 00:54:18.820 align:middle line:84% I mean-- and if the function is reasonable, these two-- 00:54:18.820 --> 00:54:21.860 align:middle line:90% OK, so now what is, then, the-- 00:54:21.860 --> 00:54:24.220 align:middle line:90% let me just define this. 00:54:24.220 --> 00:54:31.340 align:middle line:84% So this thing here, it would be convenient to the length 00:54:31.340 --> 00:54:35.530 align:middle line:84% of this interval to denote it by this. 00:54:35.530 --> 00:54:39.030 align:middle line:90% So this is xi, xi minus 1. 00:54:39.030 --> 00:54:42.430 align:middle line:84% This is just the length of that interval. 00:54:42.430 --> 00:54:42.930 align:middle line:90% OK. 00:54:42.930 --> 00:54:46.670 align:middle line:84% And so now written out, this thing-- and-- 00:54:46.670 --> 00:54:50.110 align:middle line:84% OK, let me make one other notation. 00:54:50.110 --> 00:54:59.110 align:middle line:84% Mi is the sup of f on xi minus 1 xi. 00:54:59.110 --> 00:55:01.310 align:middle line:90% That's capital Mi. 00:55:01.310 --> 00:55:13.110 align:middle line:84% And little mi is then the inf on the same interval like that. 00:55:13.110 --> 00:55:18.310 align:middle line:84% And so the upper sum for this partition, this 00:55:18.310 --> 00:55:30.350 align:middle line:84% is the sum from i equal to 1 to k of Mi times delta xi. 00:55:30.350 --> 00:55:32.510 align:middle line:90% That's the upper sum. 00:55:32.510 --> 00:55:44.710 align:middle line:84% And the lower sum is, then, the sum from i equal to 1 to k of Mi 00:55:44.710 --> 00:55:46.990 align:middle line:90% delta xi. 00:55:46.990 --> 00:55:50.710 align:middle line:84% So that's the upper and the lower sum. 00:55:50.710 --> 00:55:54.550 align:middle line:84% The first thing you observe is-- the first thing 00:55:54.550 --> 00:55:57.800 align:middle line:90% you observe is a trivial fact. 00:55:57.800 --> 00:56:00.390 align:middle line:90% 00:56:00.390 --> 00:56:04.830 align:middle line:84% And it really should be the case because one is like what 00:56:04.830 --> 00:56:07.430 align:middle line:90% fits inside the graph-- 00:56:07.430 --> 00:56:10.590 align:middle line:84% below the graph, the other is where the graph 00:56:10.590 --> 00:56:12.510 align:middle line:90% fits below the right angle. 00:56:12.510 --> 00:56:14.230 align:middle line:90% So you clearly have just-- 00:56:14.230 --> 00:56:17.190 align:middle line:84% and this is just-- because this number here is the inf, 00:56:17.190 --> 00:56:19.470 align:middle line:84% this is the sup on the same interval. 00:56:19.470 --> 00:56:28.430 align:middle line:84% So you clearly have that L f, P is less than or equal to U f, 00:56:28.430 --> 00:56:30.340 align:middle line:90% P. So this is-- 00:56:30.340 --> 00:56:33.040 align:middle line:90% 00:56:33.040 --> 00:56:34.180 align:middle line:90% so this is clear. 00:56:34.180 --> 00:56:37.520 align:middle line:90% 00:56:37.520 --> 00:56:40.220 align:middle line:90% OK. 00:56:40.220 --> 00:56:46.010 align:middle line:84% Let's look at-- let's look at a couple of examples. 00:56:46.010 --> 00:56:52.080 align:middle line:90% 00:56:52.080 --> 00:56:56.600 align:middle line:84% So just to make sure that this is totally clear. 00:56:56.600 --> 00:56:58.270 align:middle line:84% Let me look at a couple of examples. 00:56:58.270 --> 00:57:01.040 align:middle line:90% 00:57:01.040 --> 00:57:10.920 align:middle line:90% So the first example is-- 00:57:10.920 --> 00:57:19.230 align:middle line:84% so I take a function f of x equal to x squared plus 1. 00:57:19.230 --> 00:57:22.040 align:middle line:90% 00:57:22.040 --> 00:57:24.720 align:middle line:84% And my partition and the interval 00:57:24.720 --> 00:57:29.530 align:middle line:84% here, I'm going to think about this as a function on minus 2 00:57:29.530 --> 00:57:41.920 align:middle line:84% to 2 into R. And my partition here is going to be minus 2 00:57:41.920 --> 00:57:45.980 align:middle line:90% minus 1, 0, 1, and 2. 00:57:45.980 --> 00:57:48.880 align:middle line:90% 00:57:48.880 --> 00:57:50.640 align:middle line:90% And, right. 00:57:50.640 --> 00:57:52.480 align:middle line:90% And so you have-- 00:57:52.480 --> 00:57:53.220 align:middle line:90% so let's see. 00:57:53.220 --> 00:57:59.360 align:middle line:84% So this here-- so if I draw the function, then this looks like-- 00:57:59.360 --> 00:58:02.360 align:middle line:90% if this is 1-- 00:58:02.360 --> 00:58:05.720 align:middle line:90% here, you have 1. 00:58:05.720 --> 00:58:08.160 align:middle line:90% Then it looks sort of like this. 00:58:08.160 --> 00:58:10.200 align:middle line:84% It doesn't-- more or less like this. 00:58:10.200 --> 00:58:12.880 align:middle line:90% 00:58:12.880 --> 00:58:15.980 align:middle line:84% And so this means that here and here, you have 2. 00:58:15.980 --> 00:58:18.600 align:middle line:90% 00:58:18.600 --> 00:58:20.580 align:middle line:90% And this here is minus 1. 00:58:20.580 --> 00:58:23.320 align:middle line:90% 00:58:23.320 --> 00:58:34.050 align:middle line:90% And here, you have minus 2. 00:58:34.050 --> 00:58:38.690 align:middle line:84% So now, if you're looking at-- so, right. 00:58:38.690 --> 00:58:40.770 align:middle line:90% So this means here-- 00:58:40.770 --> 00:58:42.970 align:middle line:90% right, so this is my partition. 00:58:42.970 --> 00:58:48.690 align:middle line:84% So this means that x0 is equal to minus 2. 00:58:48.690 --> 00:58:52.850 align:middle line:84% x1, that's the next one, minus 1. 00:58:52.850 --> 00:58:56.650 align:middle line:90% x3 is the next one. 00:58:56.650 --> 00:58:57.970 align:middle line:90% That's 0. 00:58:57.970 --> 00:59:05.570 align:middle line:90% x4 is 1, and x5 is 2. 00:59:05.570 --> 00:59:09.370 align:middle line:84% And so, of course, all of these intervals here, 00:59:09.370 --> 00:59:12.170 align:middle line:84% they all have the same length in this case. 00:59:12.170 --> 00:59:20.210 align:middle line:84% So xi minus xi minus 1, this was what we call delta xi. 00:59:20.210 --> 00:59:23.190 align:middle line:84% This here is, for each of these, is equal to 1. 00:59:23.190 --> 00:59:32.130 align:middle line:90% 00:59:32.130 --> 00:59:35.480 align:middle line:90% And now, we have that the-- 00:59:35.480 --> 00:59:41.410 align:middle line:90% 00:59:41.410 --> 00:59:44.650 align:middle line:90% and we have that the-- 00:59:44.650 --> 00:59:48.570 align:middle line:84% when you're looking between minus 2 and 2-- 00:59:48.570 --> 00:59:50.730 align:middle line:90% sorry, minus 2 and minus 1-- 00:59:50.730 --> 00:59:55.170 align:middle line:84% so in this interval here, the function 00:59:55.170 --> 00:59:58.490 align:middle line:84% is-- the function on this interval here is decreasing, 00:59:58.490 --> 01:00:04.570 align:middle line:84% so the largest value is achieved at this point here. 01:00:04.570 --> 01:00:19.650 align:middle line:84% So M1, this is the same as the sup of f between x0 and x1. 01:00:19.650 --> 01:00:26.980 align:middle line:84% And so this is the same as the sup of minus 2 to negative 1 01:00:26.980 --> 01:00:28.540 align:middle line:90% right of f. 01:00:28.540 --> 01:00:32.580 align:middle line:84% And the function was decreasing, so they lie on this interval. 01:00:32.580 --> 01:00:37.160 align:middle line:84% So the largest value is achieved here, and that's 5. 01:00:37.160 --> 01:00:40.820 align:middle line:90% So this would be 5. 01:00:40.820 --> 01:00:45.740 align:middle line:84% And on the second interval here, so this 01:00:45.740 --> 01:00:48.540 align:middle line:90% is between negative 1 and 0. 01:00:48.540 --> 01:00:51.820 align:middle line:84% Still decreasing, so the value is-- the largest value 01:00:51.820 --> 01:00:54.140 align:middle line:90% is here where it's 2. 01:00:54.140 --> 01:00:56.940 align:middle line:90% So this here is 2. 01:00:56.940 --> 01:01:03.000 align:middle line:84% And M3, this is now between 0 and 1. 01:01:03.000 --> 01:01:06.080 align:middle line:84% So now it's increasing, so the largest value is over here. 01:01:06.080 --> 01:01:09.660 align:middle line:90% So that's, again, 2. 01:01:09.660 --> 01:01:13.810 align:middle line:84% And then the last one is, again, 5. 01:01:13.810 --> 01:01:16.400 align:middle line:90% 01:01:16.400 --> 01:01:22.900 align:middle line:84% And then likewise, for the ones that are the infimum-- 01:01:22.900 --> 01:01:27.340 align:middle line:84% so the first one, M1, well, the function 01:01:27.340 --> 01:01:31.980 align:middle line:84% was decreasing, so the smallest value is over at that endpoint. 01:01:31.980 --> 01:01:34.620 align:middle line:90% There, the value is 2. 01:01:34.620 --> 01:01:39.020 align:middle line:84% So you're getting-- this here is 2. 01:01:39.020 --> 01:01:46.900 align:middle line:84% And then between negative 1 and 0, it's still decreasing, 01:01:46.900 --> 01:01:50.540 align:middle line:84% so there, the smallest is at this point where it's 1. 01:01:50.540 --> 01:01:51.590 align:middle line:90% So this is M2. 01:01:51.590 --> 01:01:56.420 align:middle line:90% 01:01:56.420 --> 01:02:00.900 align:middle line:90% And then M3, this is now-- 01:02:00.900 --> 01:02:03.480 align:middle line:84% here, the function is increasing, 01:02:03.480 --> 01:02:05.820 align:middle line:90% so it's still the value here. 01:02:05.820 --> 01:02:07.700 align:middle line:90% So it's still 1. 01:02:07.700 --> 01:02:14.180 align:middle line:84% And then the last one is between 1 and 2, 01:02:14.180 --> 01:02:17.940 align:middle line:84% and the infimum is the value here, so that's 2. 01:02:17.940 --> 01:02:22.100 align:middle line:90% 01:02:22.100 --> 01:02:24.720 align:middle line:84% And so you see that in this case, 01:02:24.720 --> 01:02:29.500 align:middle line:90% the upper for this function-- 01:02:29.500 --> 01:02:33.180 align:middle line:90% for this partition, it's this-- 01:02:33.180 --> 01:02:36.080 align:middle line:84% all of the intervals have lengths. 01:02:36.080 --> 01:02:38.140 align:middle line:84% The small interval have length 1, 01:02:38.140 --> 01:02:41.700 align:middle line:84% so it's just the sum of these guys. 01:02:41.700 --> 01:02:44.420 align:middle line:90% So this is just equal to 5. 01:02:44.420 --> 01:02:46.220 align:middle line:84% If you will, it's 5 times 1, that 01:02:46.220 --> 01:02:52.860 align:middle line:84% was the length of the interval, plus 2 times 1 plus 2 times 01:02:52.860 --> 01:02:55.340 align:middle line:90% 1 plus 5 times 1. 01:02:55.340 --> 01:02:58.260 align:middle line:84% So in other words, you get here 14. 01:02:58.260 --> 01:03:04.180 align:middle line:84% And for the lower ones, it's, again-- 01:03:04.180 --> 01:03:06.600 align:middle line:90% this time I won't write times 1. 01:03:06.600 --> 01:03:09.360 align:middle line:84% It's the sum of those guys, and that's actually 6. 01:03:09.360 --> 01:03:12.340 align:middle line:90% 01:03:12.340 --> 01:03:17.240 align:middle line:90% So that's upper and lower sum. 01:03:17.240 --> 01:03:25.070 align:middle line:84% Now-- and we had one observation that 01:03:25.070 --> 01:03:32.390 align:middle line:84% was an immediate observation, and this was this thing here. 01:03:32.390 --> 01:03:39.310 align:middle line:84% Now, what is useful is that you're 01:03:39.310 --> 01:03:44.670 align:middle line:84% starting with a partition, and then you have a subpartition. 01:03:44.670 --> 01:03:46.470 align:middle line:90% So the next one is-- 01:03:46.470 --> 01:03:52.150 align:middle line:84% so I take a function f on some integral from a to b, 01:03:52.150 --> 01:03:53.820 align:middle line:84% and this function here is bounded. 01:03:53.820 --> 01:03:57.310 align:middle line:90% 01:03:57.310 --> 01:03:59.910 align:middle line:84% And then I have P is a partition. 01:03:59.910 --> 01:04:01.890 align:middle line:90% P1, say, is a partition. 01:04:01.890 --> 01:04:05.110 align:middle line:90% 01:04:05.110 --> 01:04:16.670 align:middle line:84% And then P2 is a subpartition of P1. 01:04:16.670 --> 01:04:21.650 align:middle line:84% And now, the thing that is useful is the following. 01:04:21.650 --> 01:04:38.110 align:middle line:90% 01:04:38.110 --> 01:04:50.030 align:middle line:84% And so the following claim, the following lemma here, 01:04:50.030 --> 01:04:54.030 align:middle line:84% that if you take a subpartition, then I 01:04:54.030 --> 01:04:59.550 align:middle line:84% claim that the value of the upper sum go down 01:04:59.550 --> 01:05:02.130 align:middle line:84% and the value of the lower go up. 01:05:02.130 --> 01:05:03.910 align:middle line:90% So this is the following. 01:05:03.910 --> 01:05:12.870 align:middle line:84% So, lower of f times P1, this is less than 01:05:12.870 --> 01:05:17.960 align:middle line:84% or equal to the lower of f of P2. 01:05:17.960 --> 01:05:21.000 align:middle line:84% We already proved that this thing here, this one here-- 01:05:21.000 --> 01:05:23.740 align:middle line:84% this inequality we haven't proved yet, 01:05:23.740 --> 01:05:29.680 align:middle line:84% but this one here we have, that this is less than the upper one 01:05:29.680 --> 01:05:31.600 align:middle line:90% for the same partition. 01:05:31.600 --> 01:05:34.040 align:middle line:84% And the second claim is that this here 01:05:34.040 --> 01:05:37.850 align:middle line:84% is then less than the upper one for the first. 01:05:37.850 --> 01:05:40.880 align:middle line:90% 01:05:40.880 --> 01:05:43.220 align:middle line:90% Now, why is this the case? 01:05:43.220 --> 01:05:46.540 align:middle line:84% This is also actually a trivial statement, 01:05:46.540 --> 01:05:49.920 align:middle line:90% but it's very useful statement. 01:05:49.920 --> 01:05:54.760 align:middle line:84% So imagine that you have your partition P1 here, 01:05:54.760 --> 01:06:00.920 align:middle line:84% and then you have the dividing points i minus 1, 01:06:00.920 --> 01:06:06.800 align:middle line:84% and here, you have i P2 is a subpartition. 01:06:06.800 --> 01:06:10.400 align:middle line:84% So there may be integrals like this where there 01:06:10.400 --> 01:06:12.640 align:middle line:90% are no further dividing points. 01:06:12.640 --> 01:06:17.160 align:middle line:84% But in that case, your the contribution for the lower 01:06:17.160 --> 01:06:20.200 align:middle line:84% and upper is the same for the two partitions. 01:06:20.200 --> 01:06:24.840 align:middle line:90% But should there be one or more? 01:06:24.840 --> 01:06:26.560 align:middle line:90% Let's just say there's one. 01:06:26.560 --> 01:06:31.080 align:middle line:84% Then for the upper one, you would 01:06:31.080 --> 01:06:33.400 align:middle line:84% look at-- you would then get-- so you 01:06:33.400 --> 01:06:39.960 align:middle line:84% would get a new dividing point, let's call it y. 01:06:39.960 --> 01:06:41.640 align:middle line:90% And then you would get-- 01:06:41.640 --> 01:06:46.760 align:middle line:84% for the new one, you would look at the sup here on this interval 01:06:46.760 --> 01:06:47.520 align:middle line:90% here. 01:06:47.520 --> 01:06:51.400 align:middle line:84% But the sup on this interval here of the function 01:06:51.400 --> 01:06:54.760 align:middle line:84% is less than the sup on the larger interval. 01:06:54.760 --> 01:07:04.120 align:middle line:84% So you would get a sup of f on the interval from i minus 1 to y 01:07:04.120 --> 01:07:06.200 align:middle line:90% of f. 01:07:06.200 --> 01:07:07.840 align:middle line:90% You get that. 01:07:07.840 --> 01:07:11.210 align:middle line:84% And then you would get times the length of the interval that's 01:07:11.210 --> 01:07:15.210 align:middle line:90% y minus xi minus 1. 01:07:15.210 --> 01:07:18.570 align:middle line:84% And then you would get a second sup, 01:07:18.570 --> 01:07:25.730 align:middle line:84% but this is now from y to xi of f. 01:07:25.730 --> 01:07:28.090 align:middle line:84% And then you would have multiplied 01:07:28.090 --> 01:07:30.110 align:middle line:90% by the length of this interval. 01:07:30.110 --> 01:07:34.730 align:middle line:84% So this here is then from xi to y. 01:07:34.730 --> 01:07:38.170 align:middle line:90% 01:07:38.170 --> 01:07:43.450 align:middle line:84% So this is what you get from the new partition, the contribution 01:07:43.450 --> 01:07:47.090 align:middle line:90% from the second-- 01:07:47.090 --> 01:07:49.050 align:middle line:90% from the-- this is what? 01:07:49.050 --> 01:07:50.970 align:middle line:84% This is the contribution that you 01:07:50.970 --> 01:08:01.050 align:middle line:84% get from the upper sum for this partition P2 from this part. 01:08:01.050 --> 01:08:04.010 align:middle line:84% But now you see that this thing here, 01:08:04.010 --> 01:08:08.150 align:middle line:84% the sup over this little interval, well, that, of course, 01:08:08.150 --> 01:08:11.290 align:middle line:84% is less than the sup over the larger interval. 01:08:11.290 --> 01:08:17.770 align:middle line:84% So this is less than Mi, and I'll just keep this one here. 01:08:17.770 --> 01:08:21.130 align:middle line:84% And this one here is also a subinterval 01:08:21.130 --> 01:08:22.770 align:middle line:90% of this larger interval. 01:08:22.770 --> 01:08:30.609 align:middle line:84% So this is less than Mi again times xi minus x, y. 01:08:30.609 --> 01:08:33.970 align:middle line:84% But now you can factor out the Mi. 01:08:33.970 --> 01:08:36.450 align:middle line:84% So you're getting Mi, and you're getting 01:08:36.450 --> 01:08:45.490 align:middle line:84% the sum of these two, y minus xi minus 1 plus xi minus y. 01:08:45.490 --> 01:08:48.010 align:middle line:84% And so you see that this is exactly 01:08:48.010 --> 01:08:55.130 align:middle line:90% Mi times xi minus xi minus 1. 01:08:55.130 --> 01:08:59.450 align:middle line:84% And so you see that the contribution 01:08:59.450 --> 01:09:06.810 align:middle line:84% that we get here to the upper sum from this part here-- 01:09:06.810 --> 01:09:11.100 align:middle line:84% that was this and this, those two things, 01:09:11.100 --> 01:09:13.380 align:middle line:84% this is less than the contribution 01:09:13.380 --> 01:09:17.140 align:middle line:84% we get from the uppers sum, from the first partition. 01:09:17.140 --> 01:09:22.660 align:middle line:84% And so this proves this inequality, 01:09:22.660 --> 01:09:26.260 align:middle line:84% and this inequality is similar with just 01:09:26.260 --> 01:09:29.939 align:middle line:84% with the infs instead of the sups. 01:09:29.939 --> 01:09:34.265 align:middle line:90% OK, so this is like a key point. 01:09:34.265 --> 01:09:38.340 align:middle line:84% Each of these steps is completely trivial, 01:09:38.340 --> 01:09:41.979 align:middle line:84% but together, it actually means that you can really 01:09:41.979 --> 01:09:45.420 align:middle line:90% define something non-trivial. 01:09:45.420 --> 01:09:45.920 align:middle line:90% OK. 01:09:45.920 --> 01:09:50.120 align:middle line:84% So now-- and this may seem trivial. 01:09:50.120 --> 01:09:53.700 align:middle line:84% Now we-- well, first of all, we define the upper and lower 01:09:53.700 --> 01:09:56.700 align:middle line:90% integrals. 01:09:56.700 --> 01:09:59.540 align:middle line:84% So we have upper and lower sum, and now 01:09:59.540 --> 01:10:02.080 align:middle line:84% we define the upper and lower integrals. 01:10:02.080 --> 01:10:12.860 align:middle line:90% 01:10:12.860 --> 01:10:19.480 align:middle line:90% So upper and lower integrals. 01:10:19.480 --> 01:10:29.700 align:middle line:90% 01:10:29.700 --> 01:10:36.060 align:middle line:84% So the upper-lower integral-- so the upper 01:10:36.060 --> 01:10:39.540 align:middle line:90% here is denoted like this. 01:10:39.540 --> 01:10:42.020 align:middle line:90% We take a bar over this. 01:10:42.020 --> 01:10:44.700 align:middle line:84% And this is the upper-lower integral. 01:10:44.700 --> 01:10:48.620 align:middle line:84% So this is the integral, this is the function, 01:10:48.620 --> 01:10:51.300 align:middle line:90% and then you write this. 01:10:51.300 --> 01:10:54.140 align:middle line:84% And so this is simply defined to be 01:10:54.140 --> 01:11:02.700 align:middle line:84% the infimum over all partitions of the upper integral. 01:11:02.700 --> 01:11:05.870 align:middle line:90% Upper sum, sorry. 01:11:05.870 --> 01:11:10.350 align:middle line:84% And so remember again that if you take a partition 01:11:10.350 --> 01:11:12.950 align:middle line:84% and you refined it, if you're looking 01:11:12.950 --> 01:11:15.950 align:middle line:84% at a subpartition of the given partition, 01:11:15.950 --> 01:11:18.110 align:middle line:90% then this number here goes down. 01:11:18.110 --> 01:11:22.030 align:middle line:84% I mean, it may stay the same, but it's certainly not bigger. 01:11:22.030 --> 01:11:28.190 align:middle line:84% So, this is-- so the idea here is that essentially, this number 01:11:28.190 --> 01:11:32.090 align:middle line:84% here you get by looking at finer and finer partition. 01:11:32.090 --> 01:11:43.590 align:middle line:84% So, idea is that the infimum that this 01:11:43.590 --> 01:11:50.050 align:middle line:84% is a limit over finer and finer partition. 01:11:50.050 --> 01:11:59.230 align:middle line:90% 01:11:59.230 --> 01:12:02.030 align:middle line:90% So that's the idea, is the-- 01:12:02.030 --> 01:12:07.030 align:middle line:84% but the precise statement is that it's just the infimum. 01:12:07.030 --> 01:12:09.430 align:middle line:84% So this is the upper integral, and then 01:12:09.430 --> 01:12:12.010 align:middle line:84% likewise, we define the lower integrals. 01:12:12.010 --> 01:12:15.815 align:middle line:90% 01:12:15.815 --> 01:12:17.190 align:middle line:84% STUDENT: Is there any assumptions 01:12:17.190 --> 01:12:20.110 align:middle line:84% that you have finitely many partitions? 01:12:20.110 --> 01:12:20.693 align:middle line:90% Or obviously-- 01:12:20.693 --> 01:12:21.610 align:middle line:90% TOBIAS COLDING: Right. 01:12:21.610 --> 01:12:24.838 align:middle line:90% 01:12:24.838 --> 01:12:29.950 align:middle line:84% But there is a key assumption is, that for each partitions, 01:12:29.950 --> 01:12:33.270 align:middle line:84% there are only finitely many divisions. 01:12:33.270 --> 01:12:36.350 align:middle line:90% But there's not a-- 01:12:36.350 --> 01:12:38.650 align:middle line:84% but there are infinitely many partitions. 01:12:38.650 --> 01:12:41.670 align:middle line:84% So it's really-- and so when I say 01:12:41.670 --> 01:12:45.870 align:middle line:84% that it's a limit over finer and finer partitions, 01:12:45.870 --> 01:12:48.850 align:middle line:84% you should really think about that in quotation marks. 01:12:48.850 --> 01:12:51.950 align:middle line:84% It's not very precise statement, but the precise statement 01:12:51.950 --> 01:12:56.030 align:middle line:84% is just that it's infimum over all partitions. 01:12:56.030 --> 01:12:59.310 align:middle line:90% And so-- so-- 01:12:59.310 --> 01:13:02.320 align:middle line:90% 01:13:02.320 --> 01:13:04.540 align:middle line:84% I mean, you can make that precise, 01:13:04.540 --> 01:13:07.880 align:middle line:84% but then that requires a lot of stuff 01:13:07.880 --> 01:13:10.408 align:middle line:90% that we don't want to do here. 01:13:10.408 --> 01:13:12.200 align:middle line:84% And there's no reason to it because there's 01:13:12.200 --> 01:13:16.400 align:middle line:84% a perfectly reasonable way of defining it. 01:13:16.400 --> 01:13:18.640 align:middle line:84% And so that was the upper integral-- 01:13:18.640 --> 01:13:19.890 align:middle line:90% so this is the upper integral. 01:13:19.890 --> 01:13:25.560 align:middle line:90% 01:13:25.560 --> 01:13:35.520 align:middle line:84% And then similarly, the lower integral, the lower integral-- 01:13:35.520 --> 01:13:38.610 align:middle line:84% and there, I'm just putting the bar below. 01:13:38.610 --> 01:13:42.360 align:middle line:90% 01:13:42.360 --> 01:13:45.120 align:middle line:90% And you define it like this. 01:13:45.120 --> 01:13:48.520 align:middle line:84% And this is now the sup because remember, 01:13:48.520 --> 01:13:52.600 align:middle line:84% that if you took a partition and then you take a subpartition, 01:13:52.600 --> 01:13:55.760 align:middle line:84% then this lower sum actually went up. 01:13:55.760 --> 01:14:00.240 align:middle line:84% So it's a sup of all partitions of this lower. 01:14:00.240 --> 01:14:07.520 align:middle line:90% 01:14:07.520 --> 01:14:10.080 align:middle line:90% OK. 01:14:10.080 --> 01:14:10.580 align:middle line:90% Right. 01:14:10.580 --> 01:14:13.300 align:middle line:90% And so now we say that-- 01:14:13.300 --> 01:14:14.390 align:middle line:90% and then we say that-- 01:14:14.390 --> 01:14:17.000 align:middle line:90% 01:14:17.000 --> 01:14:19.980 align:middle line:90% so now, here's the definition. 01:14:19.980 --> 01:14:28.800 align:middle line:84% And again, this may initially seem we are wimping out, 01:14:28.800 --> 01:14:34.320 align:middle line:84% but in a minute, we will then determine when this is the case. 01:14:34.320 --> 01:14:38.120 align:middle line:90% So we define-- so f here-- 01:14:38.120 --> 01:14:45.920 align:middle line:84% so if you have a bounded function like this, 01:14:45.920 --> 01:14:58.330 align:middle line:84% then we say that f is Riemann integral. 01:14:58.330 --> 01:15:19.390 align:middle line:84% So if the lower sum is equal to the upper sum-- 01:15:19.390 --> 01:15:25.650 align:middle line:90% 01:15:25.650 --> 01:15:27.000 align:middle line:90% so if this is the case. 01:15:27.000 --> 01:15:33.330 align:middle line:90% 01:15:33.330 --> 01:15:39.850 align:middle line:84% And in that case, and then the Riemann 01:15:39.850 --> 01:15:50.690 align:middle line:90% integral is this common-- 01:15:50.690 --> 01:15:56.010 align:middle line:84% so the Riemann integral is written like this. 01:15:56.010 --> 01:16:00.060 align:middle line:84% And this is just this common value. 01:16:00.060 --> 01:16:06.370 align:middle line:90% 01:16:06.370 --> 01:16:07.795 align:middle line:90% All right. 01:16:07.795 --> 01:16:09.650 align:middle line:90% OK. 01:16:09.650 --> 01:16:21.810 align:middle line:84% Now, one thing I just want to remark is that we have-- 01:16:21.810 --> 01:16:25.550 align:middle line:90% you always have that the-- 01:16:25.550 --> 01:16:28.930 align:middle line:90% 01:16:28.930 --> 01:16:33.650 align:middle line:90% so one remark is that-- 01:16:33.650 --> 01:16:41.970 align:middle line:84% so note that-- so first of all, we had this trivial statement, 01:16:41.970 --> 01:16:45.530 align:middle line:84% that if you take a partition, then 01:16:45.530 --> 01:16:52.183 align:middle line:84% the lower sum is less or equal to the upper sum. 01:16:52.183 --> 01:16:56.860 align:middle line:84% Then we had that if you took a subpartition, if this here 01:16:56.860 --> 01:17:10.820 align:middle line:84% was a subpartition of P1, then this lower sum here 01:17:10.820 --> 01:17:16.140 align:middle line:84% under a subpartition, the lower sum went up, 01:17:16.140 --> 01:17:17.750 align:middle line:90% but it's still staying below. 01:17:17.750 --> 01:17:25.860 align:middle line:90% 01:17:25.860 --> 01:17:29.340 align:middle line:84% So you have this inequality here. 01:17:29.340 --> 01:17:33.220 align:middle line:90% OK, now what is this useful for? 01:17:33.220 --> 01:17:37.240 align:middle line:84% So as a consequence of this, we get that-- 01:17:37.240 --> 01:17:40.000 align:middle line:84% v or not the function is Riemann integral, 01:17:40.000 --> 01:17:46.620 align:middle line:84% you always have that the lower integral here of f dx 01:17:46.620 --> 01:17:49.090 align:middle line:84% is less than or equal to the upper integral. 01:17:49.090 --> 01:17:52.340 align:middle line:90% 01:17:52.340 --> 01:17:54.360 align:middle line:84% So this here is for all functions. 01:17:54.360 --> 01:17:56.580 align:middle line:90% And why is this the case? 01:17:56.580 --> 01:17:58.620 align:middle line:90% This is because if you take-- 01:17:58.620 --> 01:18:05.540 align:middle line:84% this here is the Soup of these lower partitions, 01:18:05.540 --> 01:18:10.660 align:middle line:84% but if you take any given partition, then 01:18:10.660 --> 01:18:16.730 align:middle line:84% this lower thing here is less than the upper one. 01:18:16.730 --> 01:18:21.620 align:middle line:90% 01:18:21.620 --> 01:18:27.660 align:middle line:84% And so now, imagine that you take-- 01:18:27.660 --> 01:18:33.000 align:middle line:84% so if you take a lower partition-- 01:18:33.000 --> 01:18:34.040 align:middle line:90% suppose-- sorry. 01:18:34.040 --> 01:18:39.620 align:middle line:84% Suppose now that you take the sup over all partitions, 01:18:39.620 --> 01:18:43.820 align:middle line:84% then it means that there is one partition so that this value 01:18:43.820 --> 01:18:45.570 align:middle line:90% here is close to the sup. 01:18:45.570 --> 01:18:48.996 align:middle line:90% 01:18:48.996 --> 01:18:54.350 align:middle line:84% But for this one here, you have this inequality here. 01:18:54.350 --> 01:18:59.910 align:middle line:84% And so in fact-- but in fact, if you take any other partition, 01:18:59.910 --> 01:19:02.830 align:middle line:84% you can think about-- maybe I'll explain this in more detail 01:19:02.830 --> 01:19:04.790 align:middle line:84% next time because I only have a minute. 01:19:04.790 --> 01:19:09.150 align:middle line:84% But it basically comes from-- this inequality here is coming 01:19:09.150 --> 01:19:14.830 align:middle line:90% from this inequality here. 01:19:14.830 --> 01:19:17.710 align:middle line:84% It will take just a few minutes to explain it clearly. 01:19:17.710 --> 01:19:20.136 align:middle line:84% So maybe I'll just leave it here. 01:19:20.136 --> 01:19:25.590 align:middle line:84% But a key point here is that you have this inequality here. 01:19:25.590 --> 01:19:28.842 align:middle line:90% And this comes from that. 01:19:28.842 --> 01:19:33.310 align:middle line:84% But we return to that next time because to make it really clear, 01:19:33.310 --> 01:19:36.550 align:middle line:90% it will take a few minutes. 01:19:36.550 --> 01:19:41.250 align:middle line:84% Now, the last thing I want to say is that-- 01:19:41.250 --> 01:19:46.590 align:middle line:84% so you may think that we wimped out of saying when this-- 01:19:46.590 --> 01:19:48.110 align:middle line:84% because we just define it, we just 01:19:48.110 --> 01:19:55.435 align:middle line:84% say that it's Riemann-integrable if those two things are equal. 01:19:55.435 --> 01:19:56.810 align:middle line:84% But then the question, of course, 01:19:56.810 --> 01:19:59.670 align:middle line:84% becomes, when are those two things equal? 01:19:59.670 --> 01:20:01.950 align:middle line:90% So that's the key question. 01:20:01.950 --> 01:20:09.260 align:middle line:84% So the key question is, when are they equal? 01:20:09.260 --> 01:20:15.910 align:middle line:90% 01:20:15.910 --> 01:20:22.270 align:middle line:84% And we'll answer this next time where 01:20:22.270 --> 01:20:29.260 align:middle line:84% we will show that if f is continuous, then they're equal. 01:20:29.260 --> 01:20:38.070 align:middle line:90% 01:20:38.070 --> 01:20:40.030 align:middle line:84% And so anyways, I'll return to this. 01:20:40.030 --> 01:20:46.190 align:middle line:84% I will first-- we need, anyway, some more information 01:20:46.190 --> 01:20:47.860 align:middle line:84% about these upper and lower integrals, 01:20:47.860 --> 01:20:50.920 align:middle line:84% and I will first show why the lower 01:20:50.920 --> 01:20:53.820 align:middle line:84% one is less than or equal to-- the lower one is less than 01:20:53.820 --> 01:20:55.680 align:middle line:84% or equal to the upper one always. 01:20:55.680 --> 01:20:58.960 align:middle line:84% And then we will talk about some criteria 01:20:58.960 --> 01:21:00.420 align:middle line:90% for when they are equal. 01:21:00.420 --> 01:21:03.560 align:middle line:84% And then we will prove this for continuous functions. 01:21:03.560 --> 01:21:05.852 align:middle line:84% STUDENT: Is that an if and only if or are there other-- 01:21:05.852 --> 01:21:06.643 align:middle line:90% TOBIAS COLDING: No. 01:21:06.643 --> 01:21:07.160 align:middle line:90% No, no, no. 01:21:07.160 --> 01:21:07.660 align:middle line:90% No. 01:21:07.660 --> 01:21:09.160 align:middle line:90% It's not. 01:21:09.160 --> 01:21:12.800 align:middle line:90% It's not-- it's-- 01:21:12.800 --> 01:21:15.480 align:middle line:90% yeah, it's not a total answer. 01:21:15.480 --> 01:21:20.200 align:middle line:84% It's a good initial answer, but there 01:21:20.200 --> 01:21:24.720 align:middle line:84% are many functions that are not continuous that are very worth-- 01:21:24.720 --> 01:21:31.440 align:middle line:84% that plays key roles, and they're still-- 01:21:31.440 --> 01:21:33.390 align:middle line:84% and we will also look at an example-- 01:21:33.390 --> 01:21:35.140 align:middle line:84% it's in the notes, but I didn't get to it, 01:21:35.140 --> 01:21:40.720 align:middle line:84% we look at an example of function that is not-- 01:21:40.720 --> 01:21:43.520 align:middle line:84% where you don't have this equality. 01:21:43.520 --> 01:21:45.990 align:middle line:90% And this is like an old. 01:21:45.990 --> 01:21:48.000 align:middle line:90%