WEBVTT 00:00:00.690 --> 00:00:04.360 In the last video, we started to explore the notion of an error function. 00:00:04.360 --> 00:00:06.120 Not to be confused with the expected value 00:00:06.120 --> 00:00:08.000 because it really does reflect the same notation. 00:00:08.000 --> 00:00:09.810 But here E is for error. 00:00:09.810 --> 00:00:10.840 And we could also thought it will 00:00:10.840 --> 00:00:13.180 some times here referred to as Reminder function. 00:00:13.180 --> 00:00:16.750 And we saw it's really just the difference as we, 00:00:16.750 --> 00:00:20.440 the difference between the function and our approximation of the function. 00:00:20.440 --> 00:00:25.980 So for example, this, this distance right over here, that is our error. 00:00:25.980 --> 00:00:29.680 That is our error at the x is equal to b. 00:00:29.680 --> 00:00:32.340 And what we really care about is the absolute value of it. 00:00:32.340 --> 00:00:35.290 Because at some points f of x might be larger than the polynomial. 00:00:35.290 --> 00:00:37.500 Sometimes the polynomial might be larger than f of x. 00:00:37.500 --> 00:00:40.860 What we care is the absolute distance between them. 00:00:40.860 --> 00:00:42.500 And so what I want to do in this video is 00:00:42.500 --> 00:00:48.430 try to bound, try to bound our error at some b. 00:00:48.430 --> 00:00:49.560 Try to bound our error. 00:00:49.560 --> 00:00:52.640 So say it's less than or equal to some constant value. 00:00:52.640 --> 00:00:55.840 Try to bound it at b for some b is greater than a. 00:00:55.840 --> 00:00:58.070 We're just going to assume that b is greater than a. 00:00:58.070 --> 00:01:01.620 And we saw some tantalizing, we, we got to a bit of a tantalizing 00:01:01.620 --> 00:01:04.519 result that seems like we might be able to bound it in the last video. 00:01:04.519 --> 00:01:07.660 We saw that the n plus 1th derivative of our error 00:01:07.660 --> 00:01:12.060 function is equal to the n plus 1th derivative of our function. 00:01:12.060 --> 00:01:14.760 Or their absolute values would also be equal to. 00:01:14.760 --> 00:01:18.330 So if we could somehow bound the n plus 1th derivative 00:01:18.330 --> 00:01:22.240 of our function over some interval, an interval that matters to us. 00:01:22.240 --> 00:01:24.770 An interval that maybe has b in it. 00:01:24.770 --> 00:01:29.980 Then, we can, at least bound the n plus 1th derivative our error function. 00:01:29.980 --> 00:01:31.390 And then, maybe we can do a little bit of 00:01:31.390 --> 00:01:36.120 integration to bound the error itself at some value b. 00:01:36.120 --> 00:01:37.160 So, let's see if we can do that. 00:01:37.160 --> 00:01:40.060 Well, let's just assume, let's just assume that we're in a reality where 00:01:40.060 --> 00:01:44.300 we do know something about the n plus1 derivative of f of x. 00:01:44.300 --> 00:01:46.420 Let's say we do know that this. 00:01:46.420 --> 00:01:49.150 We do it in a color I that haven't used yet. 00:01:49.150 --> 00:01:50.580 Well, I'll do it in white. 00:01:50.580 --> 00:01:55.400 So let's say that that thing over there looks something like that. 00:01:55.400 --> 00:01:59.420 So that is f the n plus 1th derivative. 00:01:59.420 --> 00:02:00.500 The n plus 1th derivative. 00:02:00.500 --> 00:02:03.740 And I only care about it over this interval right over here. 00:02:03.740 --> 00:02:06.140 Who cares what it does later, I just gotta bound it over the interval 00:02:06.140 --> 00:02:09.759 cuz at the end of the day I just wanna balance b right over there. 00:02:09.759 --> 00:02:12.750 So let's say that the absolute value of this. 00:02:12.750 --> 00:02:13.740 Let's say that we know. 00:02:13.740 --> 00:02:17.350 Let me write it over here, let's say that we know. 00:02:19.170 --> 00:02:23.800 We know that the absolute value of the n plus 1th derivative, the n plus 1th. 00:02:23.800 --> 00:02:26.520 And, I apologize I actually switch between the capital N and 00:02:26.520 --> 00:02:28.120 the lower-case n and I did that in the last video. 00:02:28.120 --> 00:02:29.690 I shouldn't have, but now that you know 00:02:29.690 --> 00:02:32.078 that I did that hopefully it doesn't confuse it. 00:02:32.078 --> 00:02:35.090 N plus 1th, so let's say we know that the n plus 1th 00:02:35.090 --> 00:02:40.110 derivative of f of x, the absolute value of it, let's say it's bounded. 00:02:40.110 --> 00:02:43.010 Let's say it's less than or equal to some m 00:02:43.010 --> 00:02:45.160 over the interval, cuz we only care about the interval. 00:02:45.160 --> 00:02:47.540 It might not be bounded in general, but all we 00:02:47.540 --> 00:02:50.168 care is it takes some maximum value over this interval. 00:02:50.168 --> 00:02:57.190 So over, over, over the interval x, I could write it this way, 00:02:57.190 --> 00:03:04.190 over the interval x is a member between a and b, so this includes both of them. 00:03:04.190 --> 00:03:06.330 It's a closed interval, x could be a, x 00:03:06.330 --> 00:03:09.940 could be b, or x could be anything in between. 00:03:09.940 --> 00:03:11.760 And we can say this generally that, 00:03:11.760 --> 00:03:15.230 that this derivative will have some maximum value. 00:03:15.230 --> 00:03:20.060 So this is its, the absolute value, maximum value, max value, m for max. 00:03:20.060 --> 00:03:23.980 We know that it will have a maximum value, if this thing is continuous. 00:03:23.980 --> 00:03:26.620 So once again we're going to assume that it is continuous, 00:03:26.620 --> 00:03:30.710 that it has some maximum value over this interval right over here. 00:03:30.710 --> 00:03:34.796 Well this thing, this thing right over here, we know is the 00:03:34.796 --> 00:03:38.978 same thing as the n plus 1 derivative of the error function. 00:03:38.978 --> 00:03:46.220 So then we know, so then that, that implies, that implies that, 00:03:46.220 --> 00:03:51.980 that implies that the, that's a new color, let me do that in blue, or that green. 00:03:51.980 --> 00:03:58.720 That implies that the, the, the end plus one derivative of the error function. 00:03:58.720 --> 00:04:00.270 The absolute value of it because these are the 00:04:00.270 --> 00:04:04.570 same thing is also, is also bounded by m. 00:04:04.570 --> 00:04:07.500 So that's a little bit of an interesting result but it gets us no where near there. 00:04:07.500 --> 00:04:11.450 It might look similar but this is the n plus 1 derivative of the error function. 00:04:11.450 --> 00:04:14.000 And, and we'll have to think about how we can get an m in the future. 00:04:14.000 --> 00:04:16.140 We're assuming that we some how know it and maybe 00:04:16.140 --> 00:04:18.589 we'll do some example problems where we figure that out. 00:04:18.589 --> 00:04:20.160 But this is the m plus 1th derivative. 00:04:20.160 --> 00:04:21.750 We bounded it's absolute value but we 00:04:21.750 --> 00:04:24.210 really want to bound the actual error function. 00:04:24.210 --> 00:04:27.710 The 0 is the derivative, you could say, the actual function itself. 00:04:27.710 --> 00:04:31.380 What we could try to integrate both sides of this and see 00:04:31.380 --> 00:04:34.960 if we can eventually get to e, to get to e of x. 00:04:34.960 --> 00:04:38.095 To get our, to our error function or our remainder function so let's do that. 00:04:38.095 --> 00:04:44.050 Let's take the integral, let's take the integral of both sides of this. 00:04:44.050 --> 00:04:46.290 Now the integral on this left hand side, it's a little interesting. 00:04:46.290 --> 00:04:47.930 We take the integral of the absolute value. 00:04:47.930 --> 00:04:51.570 It would be easier if we were taking the absolute value of the integral. 00:04:51.570 --> 00:04:54.220 And lucky for us, the way it's set up. 00:04:54.220 --> 00:04:56.480 So let me just write a little aside here. 00:04:56.480 --> 00:04:59.369 We know generally that if I take, and it's something for you to think about. 00:04:59.369 --> 00:05:03.029 If I take, so if I have two options, if I have two 00:05:03.029 --> 00:05:09.090 options, this option versus and I don't know, they look the same right now. 00:05:10.530 --> 00:05:12.870 I know they look the same right now. 00:05:12.870 --> 00:05:15.810 So over here, I'm gonna have the integral of the absolute value 00:05:15.810 --> 00:05:19.690 and over here I'm going to have the absolute value of the interval. 00:05:19.690 --> 00:05:24.310 Which of these is going to be, which of these can be larger? 00:05:24.310 --> 00:05:26.790 Well, you just have to think about the scenarios. 00:05:26.790 --> 00:05:30.170 So, if f of x is always positive over the interval that 00:05:30.170 --> 00:05:33.470 you're taking the integration, then they're going to be the same thing. 00:05:33.470 --> 00:05:34.990 They're, you're gonna get positive values. 00:05:34.990 --> 00:05:36.760 Take the absolute of a value of a positive value. 00:05:36.760 --> 00:05:38.260 It doesn't make a difference. 00:05:38.260 --> 00:05:40.990 What matters is if f of x is negative. 00:05:40.990 --> 00:05:43.780 If f of x, if f of x is negative the 00:05:43.780 --> 00:05:48.170 entire time, so if this our x-axis, that is our y-axis. 00:05:48.170 --> 00:05:51.070 If f of x is, well we saw if it's positive the entire 00:05:51.070 --> 00:05:55.310 time, you're taking the absolute value of a positive, absolute value of positive. 00:05:55.310 --> 00:05:56.130 It's not going to matter. 00:05:56.130 --> 00:05:57.860 These two things are going to be equal. 00:05:57.860 --> 00:06:00.800 If f of x is negative the whole time, then you're going 00:06:00.800 --> 00:06:04.920 to get, then this integral going to evaluate to a negative value. 00:06:04.920 --> 00:06:07.440 But then, you would take the absolute value of it. 00:06:07.440 --> 00:06:10.090 And then over here, you're just going to, this is, the integral going to 00:06:10.090 --> 00:06:12.820 value to a positive value and it's still going to be the same thing. 00:06:12.820 --> 00:06:15.300 The interesting case is when f of x is both 00:06:15.300 --> 00:06:18.970 positive and negative, so you can imagine a situation like this. 00:06:18.970 --> 00:06:22.580 If f of x looks something like that, then 00:06:22.580 --> 00:06:25.580 this right over here, the integral, you'd have positive. 00:06:25.580 --> 00:06:28.560 This would be positive and then this would be negative right over here. 00:06:28.560 --> 00:06:30.810 And so they would cancel each other out. 00:06:30.810 --> 00:06:32.230 So this would be a smaller value than 00:06:32.230 --> 00:06:35.580 if you took the integral of the absolute value. 00:06:35.580 --> 00:06:39.470 So the integral, the absolute value of f would look something like this. 00:06:39.470 --> 00:06:42.260 So all of the areas are going to be, if you view the 00:06:42.260 --> 00:06:43.120 integral, if you view this it is 00:06:43.120 --> 00:06:44.730 definitely going to be a definite integral. 00:06:44.730 --> 00:06:48.380 All of the areas, all of the areas would be positive. 00:06:48.380 --> 00:06:49.750 So when you it, you are going to get a 00:06:49.750 --> 00:06:53.210 bigger value when you take the integral of an absolute value. 00:06:53.210 --> 00:06:54.791 Then you will, especially when f of x 00:06:54.791 --> 00:06:57.038 goes both positive and negative over the interval. 00:06:57.038 --> 00:07:02.005 Then you would if you took the integral first and then the absolute value. 00:07:02.005 --> 00:07:04.090 Cuz once again, if you took the integral first, for something like 00:07:04.090 --> 00:07:07.020 this, you'd get a low value cause this stuff would cancel out. 00:07:07.020 --> 00:07:09.500 Would cancel out with this stuff right over here then you'd 00:07:09.500 --> 00:07:13.470 take the absolute value of just a lower, a lower magnitude number. 00:07:13.470 --> 00:07:15.880 And so in general, the integral, the 00:07:15.880 --> 00:07:18.260 integral, sorry the absolute value of the integral 00:07:18.260 --> 00:07:22.870 is going to be less than or equal to the integral of the absolute value. 00:07:22.870 --> 00:07:24.670 So we can say, so this right here is the integral of 00:07:24.670 --> 00:07:27.740 the absolute value which is going to be greater than or equal. 00:07:27.740 --> 00:07:29.840 What we have written over here is just this. 00:07:29.840 --> 00:07:31.910 That's going to be greater than or equal to, and I 00:07:31.910 --> 00:07:34.550 think you'll see why I'm why I'm doing this in a second. 00:07:34.550 --> 00:07:39.670 Greater than or equal to the absolute value, the absolute 00:07:39.670 --> 00:07:45.920 value of the integral of, of the n plus 1th derivative. 00:07:45.920 --> 00:07:48.960 The n plus 1th derivative of, x, dx. 00:07:48.960 --> 00:07:51.490 And the reason why this is useful, is that we can still 00:07:51.490 --> 00:07:55.090 keep the inequality that, this is less than, or equal to this. 00:07:55.090 --> 00:07:58.700 But now, this is a pretty straight forward integral to evaluate. 00:07:58.700 --> 00:08:00.932 The indo, the anti-derivative of the n plus 00:08:00.932 --> 00:08:04.240 1th derivative, is going to be the nth derivative. 00:08:04.240 --> 00:08:06.510 So this business, right over here. 00:08:06.510 --> 00:08:09.960 Is just going to the absolute value of the nth derivative. 00:08:11.150 --> 00:08:16.310 The absolute value of the nth derivative of our error function. 00:08:16.310 --> 00:08:17.330 Did I say expected value? 00:08:17.330 --> 00:08:17.730 I shouldn't. 00:08:17.730 --> 00:08:18.820 See, it even confuses me. 00:08:18.820 --> 00:08:19.710 This is the error function. 00:08:19.710 --> 00:08:21.900 I should've used r, r for remainder. 00:08:21.900 --> 00:08:22.660 But this all error. 00:08:22.660 --> 00:08:25.170 The, noth, nothing about probability or expected value in this video. 00:08:25.170 --> 00:08:25.850 This is. 00:08:25.850 --> 00:08:27.250 E for error. 00:08:27.250 --> 00:08:30.030 So anyway, this is going to be the nth derivative of our 00:08:30.030 --> 00:08:32.880 error function, which is going to be less than or equal to this. 00:08:32.880 --> 00:08:37.230 Which is less than or equal to the anti-derivative of M. 00:08:37.230 --> 00:08:38.760 Well, that's a constant. 00:08:38.760 --> 00:08:42.630 So that's going to be mx, mx. 00:08:42.630 --> 00:08:44.179 And since we're just taking indefinite integrals. 00:08:44.179 --> 00:08:48.220 We can't forget the idea that we have a constant over here. 00:08:48.220 --> 00:08:49.840 And in general, when you're trying to create an upper 00:08:49.840 --> 00:08:52.220 bound you want as low of an upper bound as possible. 00:08:52.220 --> 00:08:56.640 So we wanna minimize, we wanna minimize what this constant is. 00:08:56.640 --> 00:09:00.180 And lucky for us, we do have, we do know what this, 00:09:00.180 --> 00:09:04.410 what this function, what value this function takes on at a point. 00:09:04.410 --> 00:09:08.430 We know that the nth derivative of our error function at a is equal to 0. 00:09:08.430 --> 00:09:09.940 I think we wrote it over here. 00:09:09.940 --> 00:09:12.480 The nth derivative at a is equal to 0. 00:09:12.480 --> 00:09:15.370 And that's because the nth derivative of the function and the 00:09:15.370 --> 00:09:19.550 approximation at a are going to be the same exact thing. 00:09:19.550 --> 00:09:22.860 And so, if we evaluate both sides of this at a, I'll 00:09:22.860 --> 00:09:27.010 do that over here on the side, we know that the absolute value. 00:09:27.010 --> 00:09:31.560 We know the absolute value of the nth derivative at a, we know 00:09:31.560 --> 00:09:34.670 that this thing is going to be equal to the absolute value of 0. 00:09:34.670 --> 00:09:35.400 Which is 0. 00:09:35.400 --> 00:09:37.820 Which needs to be less than or equal to when you evaluate this 00:09:37.820 --> 00:09:43.420 thing at a, which is less than or equal to m a plus c. 00:09:43.420 --> 00:09:45.260 And so you can, if you look at this part 00:09:45.260 --> 00:09:47.710 of the inequality, you subtract m a from both sides. 00:09:47.710 --> 00:09:51.460 You get negative m a is less than or equal to c. 00:09:51.460 --> 00:09:53.590 So our constant here, based on that little condition 00:09:53.590 --> 00:09:56.310 that we were able to get in the last video. 00:09:56.310 --> 00:10:00.820 Our constant is going to be greater than or equal to negative ma. 00:10:00.820 --> 00:10:03.880 So if we want to minimize the constant, if we wanna get this as low 00:10:03.880 --> 00:10:08.090 of a bound as possible, we would wanna pick c is equal to negative Ma. 00:10:08.090 --> 00:10:10.250 That is the lowest possible c that will 00:10:10.250 --> 00:10:13.170 meet these constraints that we know are true. 00:10:13.170 --> 00:10:16.969 So, we will actually pick c to be negative Ma. 00:10:16.969 --> 00:10:19.364 And then we can rewrite this whole thing as the 00:10:19.364 --> 00:10:22.590 absolute value of the nth derivative of the error function. 00:10:22.590 --> 00:10:24.640 The nth derivative of the error function. 00:10:24.640 --> 00:10:25.970 Not the expected value. 00:10:25.970 --> 00:10:28.010 I have a strange suspicion I might have said expected value. 00:10:28.010 --> 00:10:29.790 But, this is the error function. 00:10:29.790 --> 00:10:30.440 The nth der. 00:10:30.440 --> 00:10:33.230 The absolute value of the nth derivative of the error function 00:10:33.230 --> 00:10:38.600 is less than or equal to M times x minus a. 00:10:38.600 --> 00:10:40.820 And once again all of the constraints hold. 00:10:40.820 --> 00:10:43.880 This is for, this is for x as part of the interval. 00:10:43.880 --> 00:10:48.910 The closed interval between, the closed interval between a and b. 00:10:48.910 --> 00:10:50.220 But looks like we're making progress. 00:10:50.220 --> 00:10:52.910 We at least went from the m plus 1 derivative to the n derivative. 00:10:52.910 --> 00:10:55.170 Lets see if we can keep going. 00:10:55.170 --> 00:10:57.750 So same general idea. 00:10:57.750 --> 00:11:00.090 This if we know this then we know that 00:11:00.090 --> 00:11:00.740 we can take the integral of both sides of this. 00:11:00.740 --> 00:11:02.850 So we can take the integral of both sides of this 00:11:06.280 --> 00:11:08.360 the anti derivative of both sides. 00:11:08.360 --> 00:11:10.740 And we know from what we figured out up here 00:11:10.740 --> 00:11:14.780 that something's that's even smaller than this right over here. 00:11:14.780 --> 00:11:19.820 Is, is the absolute value of the integral of the expected value. 00:11:19.820 --> 00:11:21.070 Now [LAUGH] see, I said it. 00:11:21.070 --> 00:11:22.900 Of the error function, not the expected value. 00:11:22.900 --> 00:11:23.900 Of the error function. 00:11:23.900 --> 00:11:27.170 The nth derivative of the error function of x. 00:11:27.170 --> 00:11:29.940 The nth derivative of the error function of x dx. 00:11:29.940 --> 00:11:33.510 So we know that this is less than or equal to based on the exact same logic there. 00:11:33.510 --> 00:11:37.450 And this is useful because this is just going to be, this is just 00:11:37.450 --> 00:11:42.640 going to be the nth minus 1 derivative of our error function of x. 00:11:42.640 --> 00:11:45.160 And of course we have the absolute value outside of it. 00:11:45.160 --> 00:11:46.650 And now this is going to be less than or equal to. 00:11:46.650 --> 00:11:48.390 It's less than or equal to this, which is less than or equal 00:11:48.390 --> 00:11:50.940 to this, which is less than or equal to this right over here. 00:11:50.940 --> 00:11:53.340 The anti-derivative of this right over here is going 00:11:53.340 --> 00:11:58.060 to be M times x minus a squared over 2. 00:11:58.060 --> 00:12:01.410 You could do U substitution if you want or you could just say hey look. 00:12:01.410 --> 00:12:03.820 I have a little expression here, it's derivative is 1. 00:12:03.820 --> 00:12:06.480 So it's implicitly there so I can just treat it as kind of a U. 00:12:06.480 --> 00:12:09.320 So raise it to an exponent and then divide that exponent. 00:12:09.320 --> 00:12:11.460 But once again I'm taking indefinite integrals. 00:12:11.460 --> 00:12:14.350 So I'm going to say a plus C over here. 00:12:14.350 --> 00:12:16.600 But let's use that same exact logic. 00:12:16.600 --> 00:12:19.130 If we evaluate this at A, you're going to have it. 00:12:19.130 --> 00:12:22.250 If you evaluate this while, let's evaluate both sides of this at A. 00:12:22.250 --> 00:12:25.990 the left side, evaluated at A, we know, is going to be zero. 00:12:25.990 --> 00:12:29.250 We figured that out, all, up here in the last video. 00:12:29.250 --> 00:12:31.630 So you get, I'm gonna do it on the right over here. 00:12:31.630 --> 00:12:34.130 You get zero, when you valued the left side of a. 00:12:34.130 --> 00:12:36.820 The right side of a, if you, the right side of the 00:12:36.820 --> 00:12:39.850 value of a you get m times a menus a square over 2. 00:12:39.850 --> 00:12:45.220 So you are gonna get 0 plus c, so you are gonna get, 0 is less or equal to c. 00:12:45.220 --> 00:12:47.620 Once again we want to minimize our constant, 00:12:47.620 --> 00:12:49.800 we wanna minimize our upper boundary up here. 00:12:49.800 --> 00:12:52.930 So we wanna pick the lowest possible c that we talk constrains. 00:12:52.930 --> 00:12:57.440 So the lowest possible c that meets our constraint is zero. 00:12:57.440 --> 00:13:01.070 And so the general idea here is that we can keep doing this, we can 00:13:01.070 --> 00:13:07.270 keep doing exactly what we're doing all the way, all the way, all the way until. 00:13:07.270 --> 00:13:10.440 And so we keep integrating it at the exact same, same way that I've 00:13:10.440 --> 00:13:14.040 done it all the way that we get and using this exact same property here. 00:13:14.040 --> 00:13:19.180 All the way until we get, the bound on the error function of x. 00:13:19.180 --> 00:13:21.550 So you could view this as the 0th derivative. 00:13:21.550 --> 00:13:22.740 You know, we're going all the way to the 00:13:22.740 --> 00:13:25.360 0th derivative, which is really just the error function. 00:13:25.360 --> 00:13:27.620 The bound on the error function of x is going to 00:13:27.620 --> 00:13:29.660 be less than or equal to, and what's it going to be? 00:13:29.660 --> 00:13:31.940 And you can already see the pattern here. 00:13:31.940 --> 00:13:36.270 Is that it's going to be m times x, minus a. 00:13:36.270 --> 00:13:39.490 And the exponent, the one way to think about it, this exponent 00:13:39.490 --> 00:13:42.950 plus this derivative is going to be equal to n plus 1. 00:13:42.950 --> 00:13:46.980 Now this derivative is zero so this exponent is going to be n plus 1. 00:13:46.980 --> 00:13:50.210 And whatever the exponent is, you're going to have,a nd maybe I should 00:13:50.210 --> 00:13:54.280 have done it, you're going to have n plus one factorial over here. 00:13:54.280 --> 00:13:56.950 And if say wait why, where does this n plus 1 factorial come from? 00:13:56.950 --> 00:13:58.370 I just had a two here. 00:13:58.370 --> 00:14:01.120 Well think about what happens when we integrate this again. 00:14:01.120 --> 00:14:04.700 You're going to raise this to the third power and then divide by three. 00:14:04.700 --> 00:14:07.050 So your denominator is going to have two times three. 00:14:07.050 --> 00:14:08.540 Then when you integrate it again, you're going to raise 00:14:08.540 --> 00:14:10.800 it to the fourth power and then divide by four. 00:14:10.800 --> 00:14:12.960 So then your denominator is going to be two times three times four. 00:14:12.960 --> 00:14:14.140 Four factorial. 00:14:14.140 --> 00:14:15.530 So whatever power you're raising to, the 00:14:15.530 --> 00:14:18.500 denominator is going to be that power factorial. 00:14:18.500 --> 00:14:21.240 But what's really interesting now is if we are 00:14:21.240 --> 00:14:24.360 able to figure out that maximum value of our function. 00:14:24.360 --> 00:14:28.510 If we're able to figure out that maximum value of our function right there. 00:14:28.510 --> 00:14:31.800 We now have a way of bounding our error function 00:14:31.800 --> 00:14:36.500 over that interval, over that interval between a and b. 00:14:36.500 --> 00:14:39.530 So for example, the error function at b. 00:14:39.530 --> 00:14:42.040 We can now bound it if we know what an m is. 00:14:42.040 --> 00:14:49.190 We can say the error function at b is going to be less than or equal to m times 00:14:49.190 --> 00:14:57.190 b minus a to the n plus 1th power over n plus 1 factorial. 00:14:57.190 --> 00:15:00.030 So that gets us a really powerful, I guess you 00:15:00.030 --> 00:15:03.720 could call it, result, kinda the, the math behind it. 00:15:03.720 --> 00:15:06.849 And now we can show some examples where this could actually be applied.