[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.69,0:00:04.36,Default,,0000,0000,0000,,In the last video, we started to explore\Nthe notion of an error function.
Dialogue: 0,0:00:04.36,0:00:06.12,Default,,0000,0000,0000,,Not to be confused with the expected value
Dialogue: 0,0:00:06.12,0:00:08.00,Default,,0000,0000,0000,,because it really does reflect the same\Nnotation.
Dialogue: 0,0:00:08.00,0:00:09.81,Default,,0000,0000,0000,,But here E is for error.
Dialogue: 0,0:00:09.81,0:00:10.84,Default,,0000,0000,0000,,And we could also thought it will
Dialogue: 0,0:00:10.84,0:00:13.18,Default,,0000,0000,0000,,some times here referred to as Reminder\Nfunction.
Dialogue: 0,0:00:13.18,0:00:16.75,Default,,0000,0000,0000,,And we saw it's really just the difference\Nas we,
Dialogue: 0,0:00:16.75,0:00:20.44,Default,,0000,0000,0000,,the difference between the function and\Nour approximation of the function.
Dialogue: 0,0:00:20.44,0:00:25.98,Default,,0000,0000,0000,,So for example, this, this distance right\Nover here, that is our error.
Dialogue: 0,0:00:25.98,0:00:29.68,Default,,0000,0000,0000,,That is our error at the x is equal to b.
Dialogue: 0,0:00:29.68,0:00:32.34,Default,,0000,0000,0000,,And what we really care about is the\Nabsolute value of it.
Dialogue: 0,0:00:32.34,0:00:35.29,Default,,0000,0000,0000,,Because at some points f of x might be\Nlarger than the polynomial.
Dialogue: 0,0:00:35.29,0:00:37.50,Default,,0000,0000,0000,,Sometimes the polynomial might be larger\Nthan f of x.
Dialogue: 0,0:00:37.50,0:00:40.86,Default,,0000,0000,0000,,What we care is the absolute distance\Nbetween them.
Dialogue: 0,0:00:40.86,0:00:42.50,Default,,0000,0000,0000,,And so what I want to do in this video is
Dialogue: 0,0:00:42.50,0:00:48.43,Default,,0000,0000,0000,,try to bound, try to bound our error at\Nsome b.
Dialogue: 0,0:00:48.43,0:00:49.56,Default,,0000,0000,0000,,Try to bound our error.
Dialogue: 0,0:00:49.56,0:00:52.64,Default,,0000,0000,0000,,So say it's less than or equal to some\Nconstant value.
Dialogue: 0,0:00:52.64,0:00:55.84,Default,,0000,0000,0000,,Try to bound it at b for some b is greater\Nthan a.
Dialogue: 0,0:00:55.84,0:00:58.07,Default,,0000,0000,0000,,We're just going to assume that b is\Ngreater than a.
Dialogue: 0,0:00:58.07,0:01:01.62,Default,,0000,0000,0000,,And we saw some tantalizing, we, we got to\Na bit of a tantalizing
Dialogue: 0,0:01:01.62,0:01:04.52,Default,,0000,0000,0000,,result that seems like we might be able to\Nbound it in the last video.
Dialogue: 0,0:01:04.52,0:01:07.66,Default,,0000,0000,0000,,We saw that the n plus 1th derivative of\Nour error
Dialogue: 0,0:01:07.66,0:01:12.06,Default,,0000,0000,0000,,function is equal to the n plus 1th\Nderivative of our function.
Dialogue: 0,0:01:12.06,0:01:14.76,Default,,0000,0000,0000,,Or their absolute values would also be\Nequal to.
Dialogue: 0,0:01:14.76,0:01:18.33,Default,,0000,0000,0000,,So if we could somehow bound the n plus\N1th derivative
Dialogue: 0,0:01:18.33,0:01:22.24,Default,,0000,0000,0000,,of our function over some interval, an\Ninterval that matters to us.
Dialogue: 0,0:01:22.24,0:01:24.77,Default,,0000,0000,0000,,An interval that maybe has b in it.
Dialogue: 0,0:01:24.77,0:01:29.98,Default,,0000,0000,0000,,Then, we can, at least bound the n plus\N1th derivative our error function.
Dialogue: 0,0:01:29.98,0:01:31.39,Default,,0000,0000,0000,,And then, maybe we can do a little bit of
Dialogue: 0,0:01:31.39,0:01:36.12,Default,,0000,0000,0000,,integration to bound the error itself at\Nsome value b.
Dialogue: 0,0:01:36.12,0:01:37.16,Default,,0000,0000,0000,,So, let's see if we can do that.
Dialogue: 0,0:01:37.16,0:01:40.06,Default,,0000,0000,0000,,Well, let's just assume, let's just assume\Nthat we're in a reality where
Dialogue: 0,0:01:40.06,0:01:44.30,Default,,0000,0000,0000,,we do know something about the n plus1\Nderivative of f of x.
Dialogue: 0,0:01:44.30,0:01:46.42,Default,,0000,0000,0000,,Let's say we do know that this.
Dialogue: 0,0:01:46.42,0:01:49.15,Default,,0000,0000,0000,,We do it in a color I that haven't used\Nyet.
Dialogue: 0,0:01:49.15,0:01:50.58,Default,,0000,0000,0000,,Well, I'll do it in white.
Dialogue: 0,0:01:50.58,0:01:55.40,Default,,0000,0000,0000,,So let's say that that thing over there\Nlooks something like that.
Dialogue: 0,0:01:55.40,0:01:59.42,Default,,0000,0000,0000,,So that is f the n plus 1th derivative.
Dialogue: 0,0:01:59.42,0:02:00.50,Default,,0000,0000,0000,,The n plus 1th derivative.
Dialogue: 0,0:02:00.50,0:02:03.74,Default,,0000,0000,0000,,And I only care about it over this\Ninterval right over here.
Dialogue: 0,0:02:03.74,0:02:06.14,Default,,0000,0000,0000,,Who cares what it does later, I just gotta\Nbound it over the interval
Dialogue: 0,0:02:06.14,0:02:09.76,Default,,0000,0000,0000,,cuz at the end of the day I just wanna\Nbalance b right over there.
Dialogue: 0,0:02:09.76,0:02:12.75,Default,,0000,0000,0000,,So let's say that the absolute value of\Nthis.
Dialogue: 0,0:02:12.75,0:02:13.74,Default,,0000,0000,0000,,Let's say that we know.
Dialogue: 0,0:02:13.74,0:02:17.35,Default,,0000,0000,0000,,Let me write it over here, let's say that\Nwe know.
Dialogue: 0,0:02:19.17,0:02:23.80,Default,,0000,0000,0000,,We know that the absolute value of the n\Nplus 1th derivative, the n plus 1th.
Dialogue: 0,0:02:23.80,0:02:26.52,Default,,0000,0000,0000,,And, I apologize I actually switch between\Nthe capital N and
Dialogue: 0,0:02:26.52,0:02:28.12,Default,,0000,0000,0000,,the lower-case n and I did that in the\Nlast video.
Dialogue: 0,0:02:28.12,0:02:29.69,Default,,0000,0000,0000,,I shouldn't have, but now that you know
Dialogue: 0,0:02:29.69,0:02:32.08,Default,,0000,0000,0000,,that I did that hopefully it doesn't\Nconfuse it.
Dialogue: 0,0:02:32.08,0:02:35.09,Default,,0000,0000,0000,,N plus 1th, so let's say we know that the\Nn plus 1th
Dialogue: 0,0:02:35.09,0:02:40.11,Default,,0000,0000,0000,,derivative of f of x, the absolute value\Nof it, let's say it's bounded.
Dialogue: 0,0:02:40.11,0:02:43.01,Default,,0000,0000,0000,,Let's say it's less than or equal to some\Nm
Dialogue: 0,0:02:43.01,0:02:45.16,Default,,0000,0000,0000,,over the interval, cuz we only care about\Nthe interval.
Dialogue: 0,0:02:45.16,0:02:47.54,Default,,0000,0000,0000,,It might not be bounded in general, but\Nall we
Dialogue: 0,0:02:47.54,0:02:50.17,Default,,0000,0000,0000,,care is it takes some maximum value over\Nthis interval.
Dialogue: 0,0:02:50.17,0:02:57.19,Default,,0000,0000,0000,,So over, over, over the interval x, I\Ncould write it this way,
Dialogue: 0,0:02:57.19,0:03:04.19,Default,,0000,0000,0000,,over the interval x is a member between a\Nand b, so this includes both of them.
Dialogue: 0,0:03:04.19,0:03:06.33,Default,,0000,0000,0000,,It's a closed interval, x could be a, x
Dialogue: 0,0:03:06.33,0:03:09.94,Default,,0000,0000,0000,,could be b, or x could be anything in\Nbetween.
Dialogue: 0,0:03:09.94,0:03:11.76,Default,,0000,0000,0000,,And we can say this generally that,
Dialogue: 0,0:03:11.76,0:03:15.23,Default,,0000,0000,0000,,that this derivative will have some\Nmaximum value.
Dialogue: 0,0:03:15.23,0:03:20.06,Default,,0000,0000,0000,,So this is its, the absolute value,\Nmaximum value, max value, m for max.
Dialogue: 0,0:03:20.06,0:03:23.98,Default,,0000,0000,0000,,We know that it will have a maximum value,\Nif this thing is continuous.
Dialogue: 0,0:03:23.98,0:03:26.62,Default,,0000,0000,0000,,So once again we're going to assume that\Nit is continuous,
Dialogue: 0,0:03:26.62,0:03:30.71,Default,,0000,0000,0000,,that it has some maximum value over this\Ninterval right over here.
Dialogue: 0,0:03:30.71,0:03:34.80,Default,,0000,0000,0000,,Well this thing, this thing right over\Nhere, we know is the
Dialogue: 0,0:03:34.80,0:03:38.98,Default,,0000,0000,0000,,same thing as the n plus 1 derivative of\Nthe error function.
Dialogue: 0,0:03:38.98,0:03:46.22,Default,,0000,0000,0000,,So then we know, so then that, that\Nimplies, that implies that,
Dialogue: 0,0:03:46.22,0:03:51.98,Default,,0000,0000,0000,,that implies that the, that's a new color,\Nlet me do that in blue, or that green.
Dialogue: 0,0:03:51.98,0:03:58.72,Default,,0000,0000,0000,,That implies that the, the, the end plus\None derivative of the error function.
Dialogue: 0,0:03:58.72,0:04:00.27,Default,,0000,0000,0000,,The absolute value of it because these are\Nthe
Dialogue: 0,0:04:00.27,0:04:04.57,Default,,0000,0000,0000,,same thing is also, is also bounded by m.
Dialogue: 0,0:04:04.57,0:04:07.50,Default,,0000,0000,0000,,So that's a little bit of an interesting\Nresult but it gets us no where near there.
Dialogue: 0,0:04:07.50,0:04:11.45,Default,,0000,0000,0000,,It might look similar but this is the n\Nplus 1 derivative of the error function.
Dialogue: 0,0:04:11.45,0:04:14.00,Default,,0000,0000,0000,,And, and we'll have to think about how we\Ncan get an m in the future.
Dialogue: 0,0:04:14.00,0:04:16.14,Default,,0000,0000,0000,,We're assuming that we some how know it\Nand maybe
Dialogue: 0,0:04:16.14,0:04:18.59,Default,,0000,0000,0000,,we'll do some example problems where we\Nfigure that out.
Dialogue: 0,0:04:18.59,0:04:20.16,Default,,0000,0000,0000,,But this is the m plus 1th derivative.
Dialogue: 0,0:04:20.16,0:04:21.75,Default,,0000,0000,0000,,We bounded it's absolute value but we
Dialogue: 0,0:04:21.75,0:04:24.21,Default,,0000,0000,0000,,really want to bound the actual error\Nfunction.
Dialogue: 0,0:04:24.21,0:04:27.71,Default,,0000,0000,0000,,The 0 is the derivative, you could say,\Nthe actual function itself.
Dialogue: 0,0:04:27.71,0:04:31.38,Default,,0000,0000,0000,,What we could try to integrate both sides\Nof this and see
Dialogue: 0,0:04:31.38,0:04:34.96,Default,,0000,0000,0000,,if we can eventually get to e, to get to e\Nof x.
Dialogue: 0,0:04:34.96,0:04:38.10,Default,,0000,0000,0000,,To get our, to our error function or our\Nremainder function so let's do that.
Dialogue: 0,0:04:38.10,0:04:44.05,Default,,0000,0000,0000,,Let's take the integral, let's take the\Nintegral of both sides of this.
Dialogue: 0,0:04:44.05,0:04:46.29,Default,,0000,0000,0000,,Now the integral on this left hand side,\Nit's a little interesting.
Dialogue: 0,0:04:46.29,0:04:47.93,Default,,0000,0000,0000,,We take the integral of the absolute\Nvalue.
Dialogue: 0,0:04:47.93,0:04:51.57,Default,,0000,0000,0000,,It would be easier if we were taking the\Nabsolute value of the integral.
Dialogue: 0,0:04:51.57,0:04:54.22,Default,,0000,0000,0000,,And lucky for us, the way it's set up.
Dialogue: 0,0:04:54.22,0:04:56.48,Default,,0000,0000,0000,,So let me just write a little aside here.
Dialogue: 0,0:04:56.48,0:04:59.37,Default,,0000,0000,0000,,We know generally that if I take, and it's\Nsomething for you to think about.
Dialogue: 0,0:04:59.37,0:05:03.03,Default,,0000,0000,0000,,If I take, so if I have two options, if I\Nhave two
Dialogue: 0,0:05:03.03,0:05:09.09,Default,,0000,0000,0000,,options, this option versus and I don't\Nknow, they look the same right now.
Dialogue: 0,0:05:10.53,0:05:12.87,Default,,0000,0000,0000,,I know they look the same right now.
Dialogue: 0,0:05:12.87,0:05:15.81,Default,,0000,0000,0000,,So over here, I'm gonna have the integral\Nof the absolute value
Dialogue: 0,0:05:15.81,0:05:19.69,Default,,0000,0000,0000,,and over here I'm going to have the\Nabsolute value of the interval.
Dialogue: 0,0:05:19.69,0:05:24.31,Default,,0000,0000,0000,,Which of these is going to be, which of\Nthese can be larger?
Dialogue: 0,0:05:24.31,0:05:26.79,Default,,0000,0000,0000,,Well, you just have to think about the\Nscenarios.
Dialogue: 0,0:05:26.79,0:05:30.17,Default,,0000,0000,0000,,So, if f of x is always positive over the\Ninterval that
Dialogue: 0,0:05:30.17,0:05:33.47,Default,,0000,0000,0000,,you're taking the integration, then\Nthey're going to be the same thing.
Dialogue: 0,0:05:33.47,0:05:34.99,Default,,0000,0000,0000,,They're, you're gonna get positive values.
Dialogue: 0,0:05:34.99,0:05:36.76,Default,,0000,0000,0000,,Take the absolute of a value of a positive\Nvalue.
Dialogue: 0,0:05:36.76,0:05:38.26,Default,,0000,0000,0000,,It doesn't make a difference.
Dialogue: 0,0:05:38.26,0:05:40.99,Default,,0000,0000,0000,,What matters is if f of x is negative.
Dialogue: 0,0:05:40.99,0:05:43.78,Default,,0000,0000,0000,,If f of x, if f of x is negative the
Dialogue: 0,0:05:43.78,0:05:48.17,Default,,0000,0000,0000,,entire time, so if this our x-axis, that\Nis our y-axis.
Dialogue: 0,0:05:48.17,0:05:51.07,Default,,0000,0000,0000,,If f of x is, well we saw if it's positive\Nthe entire
Dialogue: 0,0:05:51.07,0:05:55.31,Default,,0000,0000,0000,,time, you're taking the absolute value of\Na positive, absolute value of positive.
Dialogue: 0,0:05:55.31,0:05:56.13,Default,,0000,0000,0000,,It's not going to matter.
Dialogue: 0,0:05:56.13,0:05:57.86,Default,,0000,0000,0000,,These two things are going to be equal.
Dialogue: 0,0:05:57.86,0:06:00.80,Default,,0000,0000,0000,,If f of x is negative the whole time, then\Nyou're going
Dialogue: 0,0:06:00.80,0:06:04.92,Default,,0000,0000,0000,,to get, then this integral going to\Nevaluate to a negative value.
Dialogue: 0,0:06:04.92,0:06:07.44,Default,,0000,0000,0000,,But then, you would take the absolute\Nvalue of it.
Dialogue: 0,0:06:07.44,0:06:10.09,Default,,0000,0000,0000,,And then over here, you're just going to,\Nthis is, the integral going to
Dialogue: 0,0:06:10.09,0:06:12.82,Default,,0000,0000,0000,,value to a positive value and it's still\Ngoing to be the same thing.
Dialogue: 0,0:06:12.82,0:06:15.30,Default,,0000,0000,0000,,The interesting case is when f of x is\Nboth
Dialogue: 0,0:06:15.30,0:06:18.97,Default,,0000,0000,0000,,positive and negative, so you can imagine\Na situation like this.
Dialogue: 0,0:06:18.97,0:06:22.58,Default,,0000,0000,0000,,If f of x looks something like that, then
Dialogue: 0,0:06:22.58,0:06:25.58,Default,,0000,0000,0000,,this right over here, the integral, you'd\Nhave positive.
Dialogue: 0,0:06:25.58,0:06:28.56,Default,,0000,0000,0000,,This would be positive and then this would\Nbe negative right over here.
Dialogue: 0,0:06:28.56,0:06:30.81,Default,,0000,0000,0000,,And so they would cancel each other out.
Dialogue: 0,0:06:30.81,0:06:32.23,Default,,0000,0000,0000,,So this would be a smaller value than
Dialogue: 0,0:06:32.23,0:06:35.58,Default,,0000,0000,0000,,if you took the integral of the absolute\Nvalue.
Dialogue: 0,0:06:35.58,0:06:39.47,Default,,0000,0000,0000,,So the integral, the absolute value of f\Nwould look something like this.
Dialogue: 0,0:06:39.47,0:06:42.26,Default,,0000,0000,0000,,So all of the areas are going to be, if\Nyou view the
Dialogue: 0,0:06:42.26,0:06:43.12,Default,,0000,0000,0000,,integral, if you view this it is
Dialogue: 0,0:06:43.12,0:06:44.73,Default,,0000,0000,0000,,definitely going to be a definite\Nintegral.
Dialogue: 0,0:06:44.73,0:06:48.38,Default,,0000,0000,0000,,All of the areas, all of the areas would\Nbe positive.
Dialogue: 0,0:06:48.38,0:06:49.75,Default,,0000,0000,0000,,So when you it, you are going to get a
Dialogue: 0,0:06:49.75,0:06:53.21,Default,,0000,0000,0000,,bigger value when you take the integral of\Nan absolute value.
Dialogue: 0,0:06:53.21,0:06:54.79,Default,,0000,0000,0000,,Then you will, especially when f of x
Dialogue: 0,0:06:54.79,0:06:57.04,Default,,0000,0000,0000,,goes both positive and negative over the\Ninterval.
Dialogue: 0,0:06:57.04,0:07:02.00,Default,,0000,0000,0000,,Then you would if you took the integral\Nfirst and then the absolute value.
Dialogue: 0,0:07:02.00,0:07:04.09,Default,,0000,0000,0000,,Cuz once again, if you took the integral\Nfirst, for something like
Dialogue: 0,0:07:04.09,0:07:07.02,Default,,0000,0000,0000,,this, you'd get a low value cause this\Nstuff would cancel out.
Dialogue: 0,0:07:07.02,0:07:09.50,Default,,0000,0000,0000,,Would cancel out with this stuff right\Nover here then you'd
Dialogue: 0,0:07:09.50,0:07:13.47,Default,,0000,0000,0000,,take the absolute value of just a lower, a\Nlower magnitude number.
Dialogue: 0,0:07:13.47,0:07:15.88,Default,,0000,0000,0000,,And so in general, the integral, the
Dialogue: 0,0:07:15.88,0:07:18.26,Default,,0000,0000,0000,,integral, sorry the absolute value of the\Nintegral
Dialogue: 0,0:07:18.26,0:07:22.87,Default,,0000,0000,0000,,is going to be less than or equal to the\Nintegral of the absolute value.
Dialogue: 0,0:07:22.87,0:07:24.67,Default,,0000,0000,0000,,So we can say, so this right here is the\Nintegral of
Dialogue: 0,0:07:24.67,0:07:27.74,Default,,0000,0000,0000,,the absolute value which is going to be\Ngreater than or equal.
Dialogue: 0,0:07:27.74,0:07:29.84,Default,,0000,0000,0000,,What we have written over here is just\Nthis.
Dialogue: 0,0:07:29.84,0:07:31.91,Default,,0000,0000,0000,,That's going to be greater than or equal\Nto, and I
Dialogue: 0,0:07:31.91,0:07:34.55,Default,,0000,0000,0000,,think you'll see why I'm why I'm doing\Nthis in a second.
Dialogue: 0,0:07:34.55,0:07:39.67,Default,,0000,0000,0000,,Greater than or equal to the absolute\Nvalue, the absolute
Dialogue: 0,0:07:39.67,0:07:45.92,Default,,0000,0000,0000,,value of the integral of, of the n plus\N1th derivative.
Dialogue: 0,0:07:45.92,0:07:48.96,Default,,0000,0000,0000,,The n plus 1th derivative of, x, dx.
Dialogue: 0,0:07:48.96,0:07:51.49,Default,,0000,0000,0000,,And the reason why this is useful, is that\Nwe can still
Dialogue: 0,0:07:51.49,0:07:55.09,Default,,0000,0000,0000,,keep the inequality that, this is less\Nthan, or equal to this.
Dialogue: 0,0:07:55.09,0:07:58.70,Default,,0000,0000,0000,,But now, this is a pretty straight forward\Nintegral to evaluate.
Dialogue: 0,0:07:58.70,0:08:00.93,Default,,0000,0000,0000,,The indo, the anti-derivative of the n\Nplus
Dialogue: 0,0:08:00.93,0:08:04.24,Default,,0000,0000,0000,,1th derivative, is going to be the nth\Nderivative.
Dialogue: 0,0:08:04.24,0:08:06.51,Default,,0000,0000,0000,,So this business, right over here.
Dialogue: 0,0:08:06.51,0:08:09.96,Default,,0000,0000,0000,,Is just going to the absolute value of the\Nnth derivative.
Dialogue: 0,0:08:11.15,0:08:16.31,Default,,0000,0000,0000,,The absolute value of the nth derivative\Nof our error function.
Dialogue: 0,0:08:16.31,0:08:17.33,Default,,0000,0000,0000,,Did I say expected value?
Dialogue: 0,0:08:17.33,0:08:17.73,Default,,0000,0000,0000,,I shouldn't.
Dialogue: 0,0:08:17.73,0:08:18.82,Default,,0000,0000,0000,,See, it even confuses me.
Dialogue: 0,0:08:18.82,0:08:19.71,Default,,0000,0000,0000,,This is the error function.
Dialogue: 0,0:08:19.71,0:08:21.90,Default,,0000,0000,0000,,I should've used r, r for remainder.
Dialogue: 0,0:08:21.90,0:08:22.66,Default,,0000,0000,0000,,But this all error.
Dialogue: 0,0:08:22.66,0:08:25.17,Default,,0000,0000,0000,,The, noth, nothing about probability or\Nexpected value in this video.
Dialogue: 0,0:08:25.17,0:08:25.85,Default,,0000,0000,0000,,This is.
Dialogue: 0,0:08:25.85,0:08:27.25,Default,,0000,0000,0000,,E for error.
Dialogue: 0,0:08:27.25,0:08:30.03,Default,,0000,0000,0000,,So anyway, this is going to be the nth\Nderivative of our
Dialogue: 0,0:08:30.03,0:08:32.88,Default,,0000,0000,0000,,error function, which is going to be less\Nthan or equal to this.
Dialogue: 0,0:08:32.88,0:08:37.23,Default,,0000,0000,0000,,Which is less than or equal to the\Nanti-derivative of M.
Dialogue: 0,0:08:37.23,0:08:38.76,Default,,0000,0000,0000,,Well, that's a constant.
Dialogue: 0,0:08:38.76,0:08:42.63,Default,,0000,0000,0000,,So that's going to be mx, mx.
Dialogue: 0,0:08:42.63,0:08:44.18,Default,,0000,0000,0000,,And since we're just taking indefinite\Nintegrals.
Dialogue: 0,0:08:44.18,0:08:48.22,Default,,0000,0000,0000,,We can't forget the idea that we have a\Nconstant over here.
Dialogue: 0,0:08:48.22,0:08:49.84,Default,,0000,0000,0000,,And in general, when you're trying to\Ncreate an upper
Dialogue: 0,0:08:49.84,0:08:52.22,Default,,0000,0000,0000,,bound you want as low of an upper bound as\Npossible.
Dialogue: 0,0:08:52.22,0:08:56.64,Default,,0000,0000,0000,,So we wanna minimize, we wanna minimize\Nwhat this constant is.
Dialogue: 0,0:08:56.64,0:09:00.18,Default,,0000,0000,0000,,And lucky for us, we do have, we do know\Nwhat this,
Dialogue: 0,0:09:00.18,0:09:04.41,Default,,0000,0000,0000,,what this function, what value this\Nfunction takes on at a point.
Dialogue: 0,0:09:04.41,0:09:08.43,Default,,0000,0000,0000,,We know that the nth derivative of our\Nerror function at a is equal to 0.
Dialogue: 0,0:09:08.43,0:09:09.94,Default,,0000,0000,0000,,I think we wrote it over here.
Dialogue: 0,0:09:09.94,0:09:12.48,Default,,0000,0000,0000,,The nth derivative at a is equal to 0.
Dialogue: 0,0:09:12.48,0:09:15.37,Default,,0000,0000,0000,,And that's because the nth derivative of\Nthe function and the
Dialogue: 0,0:09:15.37,0:09:19.55,Default,,0000,0000,0000,,approximation at a are going to be the\Nsame exact thing.
Dialogue: 0,0:09:19.55,0:09:22.86,Default,,0000,0000,0000,,And so, if we evaluate both sides of this\Nat a, I'll
Dialogue: 0,0:09:22.86,0:09:27.01,Default,,0000,0000,0000,,do that over here on the side, we know\Nthat the absolute value.
Dialogue: 0,0:09:27.01,0:09:31.56,Default,,0000,0000,0000,,We know the absolute value of the nth\Nderivative at a, we know
Dialogue: 0,0:09:31.56,0:09:34.67,Default,,0000,0000,0000,,that this thing is going to be equal to\Nthe absolute value of 0.
Dialogue: 0,0:09:34.67,0:09:35.40,Default,,0000,0000,0000,,Which is 0.
Dialogue: 0,0:09:35.40,0:09:37.82,Default,,0000,0000,0000,,Which needs to be less than or equal to\Nwhen you evaluate this
Dialogue: 0,0:09:37.82,0:09:43.42,Default,,0000,0000,0000,,thing at a, which is less than or equal to\Nm a plus c.
Dialogue: 0,0:09:43.42,0:09:45.26,Default,,0000,0000,0000,,And so you can, if you look at this part
Dialogue: 0,0:09:45.26,0:09:47.71,Default,,0000,0000,0000,,of the inequality, you subtract m a from\Nboth sides.
Dialogue: 0,0:09:47.71,0:09:51.46,Default,,0000,0000,0000,,You get negative m a is less than or equal\Nto c.
Dialogue: 0,0:09:51.46,0:09:53.59,Default,,0000,0000,0000,,So our constant here, based on that little\Ncondition
Dialogue: 0,0:09:53.59,0:09:56.31,Default,,0000,0000,0000,,that we were able to get in the last\Nvideo.
Dialogue: 0,0:09:56.31,0:10:00.82,Default,,0000,0000,0000,,Our constant is going to be greater than\Nor equal to negative ma.
Dialogue: 0,0:10:00.82,0:10:03.88,Default,,0000,0000,0000,,So if we want to minimize the constant, if\Nwe wanna get this as low
Dialogue: 0,0:10:03.88,0:10:08.09,Default,,0000,0000,0000,,of a bound as possible, we would wanna\Npick c is equal to negative Ma.
Dialogue: 0,0:10:08.09,0:10:10.25,Default,,0000,0000,0000,,That is the lowest possible c that will
Dialogue: 0,0:10:10.25,0:10:13.17,Default,,0000,0000,0000,,meet these constraints that we know are\Ntrue.
Dialogue: 0,0:10:13.17,0:10:16.97,Default,,0000,0000,0000,,So, we will actually pick c to be negative\NMa.
Dialogue: 0,0:10:16.97,0:10:19.36,Default,,0000,0000,0000,,And then we can rewrite this whole thing\Nas the
Dialogue: 0,0:10:19.36,0:10:22.59,Default,,0000,0000,0000,,absolute value of the nth derivative of\Nthe error function.
Dialogue: 0,0:10:22.59,0:10:24.64,Default,,0000,0000,0000,,The nth derivative of the error function.
Dialogue: 0,0:10:24.64,0:10:25.97,Default,,0000,0000,0000,,Not the expected value.
Dialogue: 0,0:10:25.97,0:10:28.01,Default,,0000,0000,0000,,I have a strange suspicion I might have\Nsaid expected value.
Dialogue: 0,0:10:28.01,0:10:29.79,Default,,0000,0000,0000,,But, this is the error function.
Dialogue: 0,0:10:29.79,0:10:30.44,Default,,0000,0000,0000,,The nth der.
Dialogue: 0,0:10:30.44,0:10:33.23,Default,,0000,0000,0000,,The absolute value of the nth derivative\Nof the error function
Dialogue: 0,0:10:33.23,0:10:38.60,Default,,0000,0000,0000,,is less than or equal to M times x minus\Na.
Dialogue: 0,0:10:38.60,0:10:40.82,Default,,0000,0000,0000,,And once again all of the constraints\Nhold.
Dialogue: 0,0:10:40.82,0:10:43.88,Default,,0000,0000,0000,,This is for, this is for x as part of the\Ninterval.
Dialogue: 0,0:10:43.88,0:10:48.91,Default,,0000,0000,0000,,The closed interval between, the closed\Ninterval between a and b.
Dialogue: 0,0:10:48.91,0:10:50.22,Default,,0000,0000,0000,,But looks like we're making progress.
Dialogue: 0,0:10:50.22,0:10:52.91,Default,,0000,0000,0000,,We at least went from the m plus 1\Nderivative to the n derivative.
Dialogue: 0,0:10:52.91,0:10:55.17,Default,,0000,0000,0000,,Lets see if we can keep going.
Dialogue: 0,0:10:55.17,0:10:57.75,Default,,0000,0000,0000,,So same general idea.
Dialogue: 0,0:10:57.75,0:11:00.09,Default,,0000,0000,0000,,This if we know this then we know that
Dialogue: 0,0:11:00.09,0:11:00.74,Default,,0000,0000,0000,,we can take the integral of both sides of\Nthis.
Dialogue: 0,0:11:00.74,0:11:02.85,Default,,0000,0000,0000,,So we can take the integral of both sides\Nof this
Dialogue: 0,0:11:06.28,0:11:08.36,Default,,0000,0000,0000,,the anti derivative of both sides.
Dialogue: 0,0:11:08.36,0:11:10.74,Default,,0000,0000,0000,,And we know from what we figured out up\Nhere
Dialogue: 0,0:11:10.74,0:11:14.78,Default,,0000,0000,0000,,that something's that's even smaller than\Nthis right over here.
Dialogue: 0,0:11:14.78,0:11:19.82,Default,,0000,0000,0000,,Is, is the absolute value of the integral\Nof the expected value.
Dialogue: 0,0:11:19.82,0:11:21.07,Default,,0000,0000,0000,,Now [LAUGH] see, I said it.
Dialogue: 0,0:11:21.07,0:11:22.90,Default,,0000,0000,0000,,Of the error function, not the expected\Nvalue.
Dialogue: 0,0:11:22.90,0:11:23.90,Default,,0000,0000,0000,,Of the error function.
Dialogue: 0,0:11:23.90,0:11:27.17,Default,,0000,0000,0000,,The nth derivative of the error function\Nof x.
Dialogue: 0,0:11:27.17,0:11:29.94,Default,,0000,0000,0000,,The nth derivative of the error function\Nof x dx.
Dialogue: 0,0:11:29.94,0:11:33.51,Default,,0000,0000,0000,,So we know that this is less than or equal\Nto based on the exact same logic there.
Dialogue: 0,0:11:33.51,0:11:37.45,Default,,0000,0000,0000,,And this is useful because this is just\Ngoing to be, this is just
Dialogue: 0,0:11:37.45,0:11:42.64,Default,,0000,0000,0000,,going to be the nth minus 1 derivative of\Nour error function of x.
Dialogue: 0,0:11:42.64,0:11:45.16,Default,,0000,0000,0000,,And of course we have the absolute value\Noutside of it.
Dialogue: 0,0:11:45.16,0:11:46.65,Default,,0000,0000,0000,,And now this is going to be less than or\Nequal to.
Dialogue: 0,0:11:46.65,0:11:48.39,Default,,0000,0000,0000,,It's less than or equal to this, which is\Nless than or equal
Dialogue: 0,0:11:48.39,0:11:50.94,Default,,0000,0000,0000,,to this, which is less than or equal to\Nthis right over here.
Dialogue: 0,0:11:50.94,0:11:53.34,Default,,0000,0000,0000,,The anti-derivative of this right over\Nhere is going
Dialogue: 0,0:11:53.34,0:11:58.06,Default,,0000,0000,0000,,to be M times x minus a squared over 2.
Dialogue: 0,0:11:58.06,0:12:01.41,Default,,0000,0000,0000,,You could do U substitution if you want or\Nyou could just say hey look.
Dialogue: 0,0:12:01.41,0:12:03.82,Default,,0000,0000,0000,,I have a little expression here, it's\Nderivative is 1.
Dialogue: 0,0:12:03.82,0:12:06.48,Default,,0000,0000,0000,,So it's implicitly there so I can just\Ntreat it as kind of a U.
Dialogue: 0,0:12:06.48,0:12:09.32,Default,,0000,0000,0000,,So raise it to an exponent and then divide\Nthat exponent.
Dialogue: 0,0:12:09.32,0:12:11.46,Default,,0000,0000,0000,,But once again I'm taking indefinite\Nintegrals.
Dialogue: 0,0:12:11.46,0:12:14.35,Default,,0000,0000,0000,,So I'm going to say a plus C over here.
Dialogue: 0,0:12:14.35,0:12:16.60,Default,,0000,0000,0000,,But let's use that same exact logic.
Dialogue: 0,0:12:16.60,0:12:19.13,Default,,0000,0000,0000,,If we evaluate this at A, you're going to\Nhave it.
Dialogue: 0,0:12:19.13,0:12:22.25,Default,,0000,0000,0000,,If you evaluate this while, let's evaluate\Nboth sides of this at A.
Dialogue: 0,0:12:22.25,0:12:25.99,Default,,0000,0000,0000,,the left side, evaluated at A, we know, is\Ngoing to be zero.
Dialogue: 0,0:12:25.99,0:12:29.25,Default,,0000,0000,0000,,We figured that out, all, up here in the\Nlast video.
Dialogue: 0,0:12:29.25,0:12:31.63,Default,,0000,0000,0000,,So you get, I'm gonna do it on the right\Nover here.
Dialogue: 0,0:12:31.63,0:12:34.13,Default,,0000,0000,0000,,You get zero, when you valued the left\Nside of a.
Dialogue: 0,0:12:34.13,0:12:36.82,Default,,0000,0000,0000,,The right side of a, if you, the right\Nside of the
Dialogue: 0,0:12:36.82,0:12:39.85,Default,,0000,0000,0000,,value of a you get m times a menus a\Nsquare over 2.
Dialogue: 0,0:12:39.85,0:12:45.22,Default,,0000,0000,0000,,So you are gonna get 0 plus c, so you are\Ngonna get, 0 is less or equal to c.
Dialogue: 0,0:12:45.22,0:12:47.62,Default,,0000,0000,0000,,Once again we want to minimize our constant,
Dialogue: 0,0:12:47.62,0:12:49.80,Default,,0000,0000,0000,,we wanna minimize our upper boundary up\Nhere.
Dialogue: 0,0:12:49.80,0:12:52.93,Default,,0000,0000,0000,,So we wanna pick the lowest possible c\Nthat we talk constrains.
Dialogue: 0,0:12:52.93,0:12:57.44,Default,,0000,0000,0000,,So the lowest possible c that meets our\Nconstraint is zero.
Dialogue: 0,0:12:57.44,0:13:01.07,Default,,0000,0000,0000,,And so the general idea here is that we\Ncan keep doing this, we can
Dialogue: 0,0:13:01.07,0:13:07.27,Default,,0000,0000,0000,,keep doing exactly what we're doing all\Nthe way, all the way, all the way until.
Dialogue: 0,0:13:07.27,0:13:10.44,Default,,0000,0000,0000,,And so we keep integrating it at the exact\Nsame, same way that I've
Dialogue: 0,0:13:10.44,0:13:14.04,Default,,0000,0000,0000,,done it all the way that we get and using\Nthis exact same property here.
Dialogue: 0,0:13:14.04,0:13:19.18,Default,,0000,0000,0000,,All the way until we get, the bound on the\Nerror function of x.
Dialogue: 0,0:13:19.18,0:13:21.55,Default,,0000,0000,0000,,So you could view this as the 0th\Nderivative.
Dialogue: 0,0:13:21.55,0:13:22.74,Default,,0000,0000,0000,,You know, we're going all the way to the
Dialogue: 0,0:13:22.74,0:13:25.36,Default,,0000,0000,0000,,0th derivative, which is really just the\Nerror function.
Dialogue: 0,0:13:25.36,0:13:27.62,Default,,0000,0000,0000,,The bound on the error function of x is\Ngoing to
Dialogue: 0,0:13:27.62,0:13:29.66,Default,,0000,0000,0000,,be less than or equal to, and what's it\Ngoing to be?
Dialogue: 0,0:13:29.66,0:13:31.94,Default,,0000,0000,0000,,And you can already see the pattern here.
Dialogue: 0,0:13:31.94,0:13:36.27,Default,,0000,0000,0000,,Is that it's going to be m times x, minus\Na.
Dialogue: 0,0:13:36.27,0:13:39.49,Default,,0000,0000,0000,,And the exponent, the one way to think\Nabout it, this exponent
Dialogue: 0,0:13:39.49,0:13:42.95,Default,,0000,0000,0000,,plus this derivative is going to be equal\Nto n plus 1.
Dialogue: 0,0:13:42.95,0:13:46.98,Default,,0000,0000,0000,,Now this derivative is zero so this\Nexponent is going to be n plus 1.
Dialogue: 0,0:13:46.98,0:13:50.21,Default,,0000,0000,0000,,And whatever the exponent is, you're going\Nto have,a nd maybe I should
Dialogue: 0,0:13:50.21,0:13:54.28,Default,,0000,0000,0000,,have done it, you're going to have n plus\None factorial over here.
Dialogue: 0,0:13:54.28,0:13:56.95,Default,,0000,0000,0000,,And if say wait why, where does this n\Nplus 1 factorial come from?
Dialogue: 0,0:13:56.95,0:13:58.37,Default,,0000,0000,0000,,I just had a two here.
Dialogue: 0,0:13:58.37,0:14:01.12,Default,,0000,0000,0000,,Well think about what happens when we\Nintegrate this again.
Dialogue: 0,0:14:01.12,0:14:04.70,Default,,0000,0000,0000,,You're going to raise this to the third\Npower and then divide by three.
Dialogue: 0,0:14:04.70,0:14:07.05,Default,,0000,0000,0000,,So your denominator is going to have two\Ntimes three.
Dialogue: 0,0:14:07.05,0:14:08.54,Default,,0000,0000,0000,,Then when you integrate it again, you're\Ngoing to raise
Dialogue: 0,0:14:08.54,0:14:10.80,Default,,0000,0000,0000,,it to the fourth power and then divide by\Nfour.
Dialogue: 0,0:14:10.80,0:14:12.96,Default,,0000,0000,0000,,So then your denominator is going to be\Ntwo times three times four.
Dialogue: 0,0:14:12.96,0:14:14.14,Default,,0000,0000,0000,,Four factorial.
Dialogue: 0,0:14:14.14,0:14:15.53,Default,,0000,0000,0000,,So whatever power you're raising to, the
Dialogue: 0,0:14:15.53,0:14:18.50,Default,,0000,0000,0000,,denominator is going to be that power\Nfactorial.
Dialogue: 0,0:14:18.50,0:14:21.24,Default,,0000,0000,0000,,But what's really interesting now is if we\Nare
Dialogue: 0,0:14:21.24,0:14:24.36,Default,,0000,0000,0000,,able to figure out that maximum value of\Nour function.
Dialogue: 0,0:14:24.36,0:14:28.51,Default,,0000,0000,0000,,If we're able to figure out that maximum\Nvalue of our function right there.
Dialogue: 0,0:14:28.51,0:14:31.80,Default,,0000,0000,0000,,We now have a way of bounding our error\Nfunction
Dialogue: 0,0:14:31.80,0:14:36.50,Default,,0000,0000,0000,,over that interval, over that interval\Nbetween a and b.
Dialogue: 0,0:14:36.50,0:14:39.53,Default,,0000,0000,0000,,So for example, the error function at b.
Dialogue: 0,0:14:39.53,0:14:42.04,Default,,0000,0000,0000,,We can now bound it if we know what an m\Nis.
Dialogue: 0,0:14:42.04,0:14:49.19,Default,,0000,0000,0000,,We can say the error function at b is\Ngoing to be less than or equal to m times
Dialogue: 0,0:14:49.19,0:14:57.19,Default,,0000,0000,0000,,b minus a to the n plus 1th power over n\Nplus 1 factorial.
Dialogue: 0,0:14:57.19,0:15:00.03,Default,,0000,0000,0000,,So that gets us a really powerful, I guess\Nyou
Dialogue: 0,0:15:00.03,0:15:03.72,Default,,0000,0000,0000,,could call it, result, kinda the, the math\Nbehind it.
Dialogue: 0,0:15:03.72,0:15:06.85,Default,,0000,0000,0000,,And now we can show some examples where\Nthis could actually be applied.