0:00:00.690,0:00:04.360
In the last video, we started to explore[br]the notion of an error function.

0:00:04.360,0:00:06.120
Not to be confused with the expected value

0:00:06.120,0:00:08.000
because it really does reflect the same[br]notation.

0:00:08.000,0:00:09.810
But here E is for error.

0:00:09.810,0:00:10.840
And we could also thought it will

0:00:10.840,0:00:13.180
some times here referred to as Reminder[br]function.

0:00:13.180,0:00:16.750
And we saw it's really just the difference[br]as we,

0:00:16.750,0:00:20.440
the difference between the function and[br]our approximation of the function.

0:00:20.440,0:00:25.980
So for example, this, this distance right[br]over here, that is our error.

0:00:25.980,0:00:29.680
That is our error at the x is equal to b.

0:00:29.680,0:00:32.340
And what we really care about is the[br]absolute value of it.

0:00:32.340,0:00:35.290
Because at some points f of x might be[br]larger than the polynomial.

0:00:35.290,0:00:37.500
Sometimes the polynomial might be larger[br]than f of x.

0:00:37.500,0:00:40.860
What we care is the absolute distance[br]between them.

0:00:40.860,0:00:42.500
And so what I want to do in this video is

0:00:42.500,0:00:48.430
try to bound, try to bound our error at[br]some b.

0:00:48.430,0:00:49.560
Try to bound our error.

0:00:49.560,0:00:52.640
So say it's less than or equal to some[br]constant value.

0:00:52.640,0:00:55.840
Try to bound it at b for some b is greater[br]than a.

0:00:55.840,0:00:58.070
We're just going to assume that b is[br]greater than a.

0:00:58.070,0:01:01.620
And we saw some tantalizing, we, we got to[br]a bit of a tantalizing

0:01:01.620,0:01:04.519
result that seems like we might be able to[br]bound it in the last video.

0:01:04.519,0:01:07.660
We saw that the n plus 1th derivative of[br]our error

0:01:07.660,0:01:12.060
function is equal to the n plus 1th[br]derivative of our function.

0:01:12.060,0:01:14.760
Or their absolute values would also be[br]equal to.

0:01:14.760,0:01:18.330
So if we could somehow bound the n plus[br]1th derivative

0:01:18.330,0:01:22.240
of our function over some interval, an[br]interval that matters to us.

0:01:22.240,0:01:24.770
An interval that maybe has b in it.

0:01:24.770,0:01:29.980
Then, we can, at least bound the n plus[br]1th derivative our error function.

0:01:29.980,0:01:31.390
And then, maybe we can do a little bit of

0:01:31.390,0:01:36.120
integration to bound the error itself at[br]some value b.

0:01:36.120,0:01:37.160
So, let's see if we can do that.

0:01:37.160,0:01:40.060
Well, let's just assume, let's just assume[br]that we're in a reality where

0:01:40.060,0:01:44.300
we do know something about the n plus1[br]derivative of f of x.

0:01:44.300,0:01:46.420
Let's say we do know that this.

0:01:46.420,0:01:49.150
We do it in a color I that haven't used[br]yet.

0:01:49.150,0:01:50.580
Well, I'll do it in white.

0:01:50.580,0:01:55.400
So let's say that that thing over there[br]looks something like that.

0:01:55.400,0:01:59.420
So that is f the n plus 1th derivative.

0:01:59.420,0:02:00.500
The n plus 1th derivative.

0:02:00.500,0:02:03.740
And I only care about it over this[br]interval right over here.

0:02:03.740,0:02:06.140
Who cares what it does later, I just gotta[br]bound it over the interval

0:02:06.140,0:02:09.759
cuz at the end of the day I just wanna[br]balance b right over there.

0:02:09.759,0:02:12.750
So let's say that the absolute value of[br]this.

0:02:12.750,0:02:13.740
Let's say that we know.

0:02:13.740,0:02:17.350
Let me write it over here, let's say that[br]we know.

0:02:19.170,0:02:23.800
We know that the absolute value of the n[br]plus 1th derivative, the n plus 1th.

0:02:23.800,0:02:26.520
And, I apologize I actually switch between[br]the capital N and

0:02:26.520,0:02:28.120
the lower-case n and I did that in the[br]last video.

0:02:28.120,0:02:29.690
I shouldn't have, but now that you know

0:02:29.690,0:02:32.078
that I did that hopefully it doesn't[br]confuse it.

0:02:32.078,0:02:35.090
N plus 1th, so let's say we know that the[br]n plus 1th

0:02:35.090,0:02:40.110
derivative of f of x, the absolute value[br]of it, let's say it's bounded.

0:02:40.110,0:02:43.010
Let's say it's less than or equal to some[br]m

0:02:43.010,0:02:45.160
over the interval, cuz we only care about[br]the interval.

0:02:45.160,0:02:47.540
It might not be bounded in general, but[br]all we

0:02:47.540,0:02:50.168
care is it takes some maximum value over[br]this interval.

0:02:50.168,0:02:57.190
So over, over, over the interval x, I[br]could write it this way,

0:02:57.190,0:03:04.190
over the interval x is a member between a[br]and b, so this includes both of them.

0:03:04.190,0:03:06.330
It's a closed interval, x could be a, x

0:03:06.330,0:03:09.940
could be b, or x could be anything in[br]between.

0:03:09.940,0:03:11.760
And we can say this generally that,

0:03:11.760,0:03:15.230
that this derivative will have some[br]maximum value.

0:03:15.230,0:03:20.060
So this is its, the absolute value,[br]maximum value, max value, m for max.

0:03:20.060,0:03:23.980
We know that it will have a maximum value,[br]if this thing is continuous.

0:03:23.980,0:03:26.620
So once again we're going to assume that[br]it is continuous,

0:03:26.620,0:03:30.710
that it has some maximum value over this[br]interval right over here.

0:03:30.710,0:03:34.796
Well this thing, this thing right over[br]here, we know is the

0:03:34.796,0:03:38.978
same thing as the n plus 1 derivative of[br]the error function.

0:03:38.978,0:03:46.220
So then we know, so then that, that[br]implies, that implies that,

0:03:46.220,0:03:51.980
that implies that the, that's a new color,[br]let me do that in blue, or that green.

0:03:51.980,0:03:58.720
That implies that the, the, the end plus[br]one derivative of the error function.

0:03:58.720,0:04:00.270
The absolute value of it because these are[br]the

0:04:00.270,0:04:04.570
same thing is also, is also bounded by m.

0:04:04.570,0:04:07.500
So that's a little bit of an interesting[br]result but it gets us no where near there.

0:04:07.500,0:04:11.450
It might look similar but this is the n[br]plus 1 derivative of the error function.

0:04:11.450,0:04:14.000
And, and we'll have to think about how we[br]can get an m in the future.

0:04:14.000,0:04:16.140
We're assuming that we some how know it[br]and maybe

0:04:16.140,0:04:18.589
we'll do some example problems where we[br]figure that out.

0:04:18.589,0:04:20.160
But this is the m plus 1th derivative.

0:04:20.160,0:04:21.750
We bounded it's absolute value but we

0:04:21.750,0:04:24.210
really want to bound the actual error[br]function.

0:04:24.210,0:04:27.710
The 0 is the derivative, you could say,[br]the actual function itself.

0:04:27.710,0:04:31.380
What we could try to integrate both sides[br]of this and see

0:04:31.380,0:04:34.960
if we can eventually get to e, to get to e[br]of x.

0:04:34.960,0:04:38.095
To get our, to our error function or our[br]remainder function so let's do that.

0:04:38.095,0:04:44.050
Let's take the integral, let's take the[br]integral of both sides of this.

0:04:44.050,0:04:46.290
Now the integral on this left hand side,[br]it's a little interesting.

0:04:46.290,0:04:47.930
We take the integral of the absolute[br]value.

0:04:47.930,0:04:51.570
It would be easier if we were taking the[br]absolute value of the integral.

0:04:51.570,0:04:54.220
And lucky for us, the way it's set up.

0:04:54.220,0:04:56.480
So let me just write a little aside here.

0:04:56.480,0:04:59.369
We know generally that if I take, and it's[br]something for you to think about.

0:04:59.369,0:05:03.029
If I take, so if I have two options, if I[br]have two

0:05:03.029,0:05:09.090
options, this option versus and I don't[br]know, they look the same right now.

0:05:10.530,0:05:12.870
I know they look the same right now.

0:05:12.870,0:05:15.810
So over here, I'm gonna have the integral[br]of the absolute value

0:05:15.810,0:05:19.690
and over here I'm going to have the[br]absolute value of the interval.

0:05:19.690,0:05:24.310
Which of these is going to be, which of[br]these can be larger?

0:05:24.310,0:05:26.790
Well, you just have to think about the[br]scenarios.

0:05:26.790,0:05:30.170
So, if f of x is always positive over the[br]interval that

0:05:30.170,0:05:33.470
you're taking the integration, then[br]they're going to be the same thing.

0:05:33.470,0:05:34.990
They're, you're gonna get positive values.

0:05:34.990,0:05:36.760
Take the absolute of a value of a positive[br]value.

0:05:36.760,0:05:38.260
It doesn't make a difference.

0:05:38.260,0:05:40.990
What matters is if f of x is negative.

0:05:40.990,0:05:43.780
If f of x, if f of x is negative the

0:05:43.780,0:05:48.170
entire time, so if this our x-axis, that[br]is our y-axis.

0:05:48.170,0:05:51.070
If f of x is, well we saw if it's positive[br]the entire

0:05:51.070,0:05:55.310
time, you're taking the absolute value of[br]a positive, absolute value of positive.

0:05:55.310,0:05:56.130
It's not going to matter.

0:05:56.130,0:05:57.860
These two things are going to be equal.

0:05:57.860,0:06:00.800
If f of x is negative the whole time, then[br]you're going

0:06:00.800,0:06:04.920
to get, then this integral going to[br]evaluate to a negative value.

0:06:04.920,0:06:07.440
But then, you would take the absolute[br]value of it.

0:06:07.440,0:06:10.090
And then over here, you're just going to,[br]this is, the integral going to

0:06:10.090,0:06:12.820
value to a positive value and it's still[br]going to be the same thing.

0:06:12.820,0:06:15.300
The interesting case is when f of x is[br]both

0:06:15.300,0:06:18.970
positive and negative, so you can imagine[br]a situation like this.

0:06:18.970,0:06:22.580
If f of x looks something like that, then

0:06:22.580,0:06:25.580
this right over here, the integral, you'd[br]have positive.

0:06:25.580,0:06:28.560
This would be positive and then this would[br]be negative right over here.

0:06:28.560,0:06:30.810
And so they would cancel each other out.

0:06:30.810,0:06:32.230
So this would be a smaller value than

0:06:32.230,0:06:35.580
if you took the integral of the absolute[br]value.

0:06:35.580,0:06:39.470
So the integral, the absolute value of f[br]would look something like this.

0:06:39.470,0:06:42.260
So all of the areas are going to be, if[br]you view the

0:06:42.260,0:06:43.120
integral, if you view this it is

0:06:43.120,0:06:44.730
definitely going to be a definite[br]integral.

0:06:44.730,0:06:48.380
All of the areas, all of the areas would[br]be positive.

0:06:48.380,0:06:49.750
So when you it, you are going to get a

0:06:49.750,0:06:53.210
bigger value when you take the integral of[br]an absolute value.

0:06:53.210,0:06:54.791
Then you will, especially when f of x

0:06:54.791,0:06:57.038
goes both positive and negative over the[br]interval.

0:06:57.038,0:07:02.005
Then you would if you took the integral[br]first and then the absolute value.

0:07:02.005,0:07:04.090
Cuz once again, if you took the integral[br]first, for something like

0:07:04.090,0:07:07.020
this, you'd get a low value cause this[br]stuff would cancel out.

0:07:07.020,0:07:09.500
Would cancel out with this stuff right[br]over here then you'd

0:07:09.500,0:07:13.470
take the absolute value of just a lower, a[br]lower magnitude number.

0:07:13.470,0:07:15.880
And so in general, the integral, the

0:07:15.880,0:07:18.260
integral, sorry the absolute value of the[br]integral

0:07:18.260,0:07:22.870
is going to be less than or equal to the[br]integral of the absolute value.

0:07:22.870,0:07:24.670
So we can say, so this right here is the[br]integral of

0:07:24.670,0:07:27.740
the absolute value which is going to be[br]greater than or equal.

0:07:27.740,0:07:29.840
What we have written over here is just[br]this.

0:07:29.840,0:07:31.910
That's going to be greater than or equal[br]to, and I

0:07:31.910,0:07:34.550
think you'll see why I'm why I'm doing[br]this in a second.

0:07:34.550,0:07:39.670
Greater than or equal to the absolute[br]value, the absolute

0:07:39.670,0:07:45.920
value of the integral of, of the n plus[br]1th derivative.

0:07:45.920,0:07:48.960
The n plus 1th derivative of, x, dx.

0:07:48.960,0:07:51.490
And the reason why this is useful, is that[br]we can still

0:07:51.490,0:07:55.090
keep the inequality that, this is less[br]than, or equal to this.

0:07:55.090,0:07:58.700
But now, this is a pretty straight forward[br]integral to evaluate.

0:07:58.700,0:08:00.932
The indo, the anti-derivative of the n[br]plus

0:08:00.932,0:08:04.240
1th derivative, is going to be the nth[br]derivative.

0:08:04.240,0:08:06.510
So this business, right over here.

0:08:06.510,0:08:09.960
Is just going to the absolute value of the[br]nth derivative.

0:08:11.150,0:08:16.310
The absolute value of the nth derivative[br]of our error function.

0:08:16.310,0:08:17.330
Did I say expected value?

0:08:17.330,0:08:17.730
I shouldn't.

0:08:17.730,0:08:18.820
See, it even confuses me.

0:08:18.820,0:08:19.710
This is the error function.

0:08:19.710,0:08:21.900
I should've used r, r for remainder.

0:08:21.900,0:08:22.660
But this all error.

0:08:22.660,0:08:25.170
The, noth, nothing about probability or[br]expected value in this video.

0:08:25.170,0:08:25.850
This is.

0:08:25.850,0:08:27.250
E for error.

0:08:27.250,0:08:30.030
So anyway, this is going to be the nth[br]derivative of our

0:08:30.030,0:08:32.880
error function, which is going to be less[br]than or equal to this.

0:08:32.880,0:08:37.230
Which is less than or equal to the[br]anti-derivative of M.

0:08:37.230,0:08:38.760
Well, that's a constant.

0:08:38.760,0:08:42.630
So that's going to be mx, mx.

0:08:42.630,0:08:44.179
And since we're just taking indefinite[br]integrals.

0:08:44.179,0:08:48.220
We can't forget the idea that we have a[br]constant over here.

0:08:48.220,0:08:49.840
And in general, when you're trying to[br]create an upper

0:08:49.840,0:08:52.220
bound you want as low of an upper bound as[br]possible.

0:08:52.220,0:08:56.640
So we wanna minimize, we wanna minimize[br]what this constant is.

0:08:56.640,0:09:00.180
And lucky for us, we do have, we do know[br]what this,

0:09:00.180,0:09:04.410
what this function, what value this[br]function takes on at a point.

0:09:04.410,0:09:08.430
We know that the nth derivative of our[br]error function at a is equal to 0.

0:09:08.430,0:09:09.940
I think we wrote it over here.

0:09:09.940,0:09:12.480
The nth derivative at a is equal to 0.

0:09:12.480,0:09:15.370
And that's because the nth derivative of[br]the function and the

0:09:15.370,0:09:19.550
approximation at a are going to be the[br]same exact thing.

0:09:19.550,0:09:22.860
And so, if we evaluate both sides of this[br]at a, I'll

0:09:22.860,0:09:27.010
do that over here on the side, we know[br]that the absolute value.

0:09:27.010,0:09:31.560
We know the absolute value of the nth[br]derivative at a, we know

0:09:31.560,0:09:34.670
that this thing is going to be equal to[br]the absolute value of 0.

0:09:34.670,0:09:35.400
Which is 0.

0:09:35.400,0:09:37.820
Which needs to be less than or equal to[br]when you evaluate this

0:09:37.820,0:09:43.420
thing at a, which is less than or equal to[br]m a plus c.

0:09:43.420,0:09:45.260
And so you can, if you look at this part

0:09:45.260,0:09:47.710
of the inequality, you subtract m a from[br]both sides.

0:09:47.710,0:09:51.460
You get negative m a is less than or equal[br]to c.

0:09:51.460,0:09:53.590
So our constant here, based on that little[br]condition

0:09:53.590,0:09:56.310
that we were able to get in the last[br]video.

0:09:56.310,0:10:00.820
Our constant is going to be greater than[br]or equal to negative ma.

0:10:00.820,0:10:03.880
So if we want to minimize the constant, if[br]we wanna get this as low

0:10:03.880,0:10:08.090
of a bound as possible, we would wanna[br]pick c is equal to negative Ma.

0:10:08.090,0:10:10.250
That is the lowest possible c that will

0:10:10.250,0:10:13.170
meet these constraints that we know are[br]true.

0:10:13.170,0:10:16.969
So, we will actually pick c to be negative[br]Ma.

0:10:16.969,0:10:19.364
And then we can rewrite this whole thing[br]as the

0:10:19.364,0:10:22.590
absolute value of the nth derivative of[br]the error function.

0:10:22.590,0:10:24.640
The nth derivative of the error function.

0:10:24.640,0:10:25.970
Not the expected value.

0:10:25.970,0:10:28.010
I have a strange suspicion I might have[br]said expected value.

0:10:28.010,0:10:29.790
But, this is the error function.

0:10:29.790,0:10:30.440
The nth der.

0:10:30.440,0:10:33.230
The absolute value of the nth derivative[br]of the error function

0:10:33.230,0:10:38.600
is less than or equal to M times x minus[br]a.

0:10:38.600,0:10:40.820
And once again all of the constraints[br]hold.

0:10:40.820,0:10:43.880
This is for, this is for x as part of the[br]interval.

0:10:43.880,0:10:48.910
The closed interval between, the closed[br]interval between a and b.

0:10:48.910,0:10:50.220
But looks like we're making progress.

0:10:50.220,0:10:52.910
We at least went from the m plus 1[br]derivative to the n derivative.

0:10:52.910,0:10:55.170
Lets see if we can keep going.

0:10:55.170,0:10:57.750
So same general idea.

0:10:57.750,0:11:00.090
This if we know this then we know that

0:11:00.090,0:11:00.740
we can take the integral of both sides of[br]this.

0:11:00.740,0:11:02.850
So we can take the integral of both sides[br]of this

0:11:06.280,0:11:08.360
the anti derivative of both sides.

0:11:08.360,0:11:10.740
And we know from what we figured out up[br]here

0:11:10.740,0:11:14.780
that something's that's even smaller than[br]this right over here.

0:11:14.780,0:11:19.820
Is, is the absolute value of the integral[br]of the expected value.

0:11:19.820,0:11:21.070
Now [LAUGH] see, I said it.

0:11:21.070,0:11:22.900
Of the error function, not the expected[br]value.

0:11:22.900,0:11:23.900
Of the error function.

0:11:23.900,0:11:27.170
The nth derivative of the error function[br]of x.

0:11:27.170,0:11:29.940
The nth derivative of the error function[br]of x dx.

0:11:29.940,0:11:33.510
So we know that this is less than or equal[br]to based on the exact same logic there.

0:11:33.510,0:11:37.450
And this is useful because this is just[br]going to be, this is just

0:11:37.450,0:11:42.640
going to be the nth minus 1 derivative of[br]our error function of x.

0:11:42.640,0:11:45.160
And of course we have the absolute value[br]outside of it.

0:11:45.160,0:11:46.650
And now this is going to be less than or[br]equal to.

0:11:46.650,0:11:48.390
It's less than or equal to this, which is[br]less than or equal

0:11:48.390,0:11:50.940
to this, which is less than or equal to[br]this right over here.

0:11:50.940,0:11:53.340
The anti-derivative of this right over[br]here is going

0:11:53.340,0:11:58.060
to be M times x minus a squared over 2.

0:11:58.060,0:12:01.410
You could do U substitution if you want or[br]you could just say hey look.

0:12:01.410,0:12:03.820
I have a little expression here, it's[br]derivative is 1.

0:12:03.820,0:12:06.480
So it's implicitly there so I can just[br]treat it as kind of a U.

0:12:06.480,0:12:09.320
So raise it to an exponent and then divide[br]that exponent.

0:12:09.320,0:12:11.460
But once again I'm taking indefinite[br]integrals.

0:12:11.460,0:12:14.350
So I'm going to say a plus C over here.

0:12:14.350,0:12:16.600
But let's use that same exact logic.

0:12:16.600,0:12:19.130
If we evaluate this at A, you're going to[br]have it.

0:12:19.130,0:12:22.250
If you evaluate this while, let's evaluate[br]both sides of this at A.

0:12:22.250,0:12:25.990
the left side, evaluated at A, we know, is[br]going to be zero.

0:12:25.990,0:12:29.250
We figured that out, all, up here in the[br]last video.

0:12:29.250,0:12:31.630
So you get, I'm gonna do it on the right[br]over here.

0:12:31.630,0:12:34.130
You get zero, when you valued the left[br]side of a.

0:12:34.130,0:12:36.820
The right side of a, if you, the right[br]side of the

0:12:36.820,0:12:39.850
value of a you get m times a menus a[br]square over 2.

0:12:39.850,0:12:45.220
So you are gonna get 0 plus c, so you are[br]gonna get, 0 is less or equal to c.

0:12:45.220,0:12:47.620
Once again we want to minimize our constant,

0:12:47.620,0:12:49.800
we wanna minimize our upper boundary up[br]here.

0:12:49.800,0:12:52.930
So we wanna pick the lowest possible c[br]that we talk constrains.

0:12:52.930,0:12:57.440
So the lowest possible c that meets our[br]constraint is zero.

0:12:57.440,0:13:01.070
And so the general idea here is that we[br]can keep doing this, we can

0:13:01.070,0:13:07.270
keep doing exactly what we're doing all[br]the way, all the way, all the way until.

0:13:07.270,0:13:10.440
And so we keep integrating it at the exact[br]same, same way that I've

0:13:10.440,0:13:14.040
done it all the way that we get and using[br]this exact same property here.

0:13:14.040,0:13:19.180
All the way until we get, the bound on the[br]error function of x.

0:13:19.180,0:13:21.550
So you could view this as the 0th[br]derivative.

0:13:21.550,0:13:22.740
You know, we're going all the way to the

0:13:22.740,0:13:25.360
0th derivative, which is really just the[br]error function.

0:13:25.360,0:13:27.620
The bound on the error function of x is[br]going to

0:13:27.620,0:13:29.660
be less than or equal to, and what's it[br]going to be?

0:13:29.660,0:13:31.940
And you can already see the pattern here.

0:13:31.940,0:13:36.270
Is that it's going to be m times x, minus[br]a.

0:13:36.270,0:13:39.490
And the exponent, the one way to think[br]about it, this exponent

0:13:39.490,0:13:42.950
plus this derivative is going to be equal[br]to n plus 1.

0:13:42.950,0:13:46.980
Now this derivative is zero so this[br]exponent is going to be n plus 1.

0:13:46.980,0:13:50.210
And whatever the exponent is, you're going[br]to have,a nd maybe I should

0:13:50.210,0:13:54.280
have done it, you're going to have n plus[br]one factorial over here.

0:13:54.280,0:13:56.950
And if say wait why, where does this n[br]plus 1 factorial come from?

0:13:56.950,0:13:58.370
I just had a two here.

0:13:58.370,0:14:01.120
Well think about what happens when we[br]integrate this again.

0:14:01.120,0:14:04.700
You're going to raise this to the third[br]power and then divide by three.

0:14:04.700,0:14:07.050
So your denominator is going to have two[br]times three.

0:14:07.050,0:14:08.540
Then when you integrate it again, you're[br]going to raise

0:14:08.540,0:14:10.800
it to the fourth power and then divide by[br]four.

0:14:10.800,0:14:12.960
So then your denominator is going to be[br]two times three times four.

0:14:12.960,0:14:14.140
Four factorial.

0:14:14.140,0:14:15.530
So whatever power you're raising to, the

0:14:15.530,0:14:18.500
denominator is going to be that power[br]factorial.

0:14:18.500,0:14:21.240
But what's really interesting now is if we[br]are

0:14:21.240,0:14:24.360
able to figure out that maximum value of[br]our function.

0:14:24.360,0:14:28.510
If we're able to figure out that maximum[br]value of our function right there.

0:14:28.510,0:14:31.800
We now have a way of bounding our error[br]function

0:14:31.800,0:14:36.500
over that interval, over that interval[br]between a and b.

0:14:36.500,0:14:39.530
So for example, the error function at b.

0:14:39.530,0:14:42.040
We can now bound it if we know what an m[br]is.

0:14:42.040,0:14:49.190
We can say the error function at b is[br]going to be less than or equal to m times

0:14:49.190,0:14:57.190
b minus a to the n plus 1th power over n[br]plus 1 factorial.

0:14:57.190,0:15:00.030
So that gets us a really powerful, I guess[br]you

0:15:00.030,0:15:03.720
could call it, result, kinda the, the math[br]behind it.

0:15:03.720,0:15:06.849
And now we can show some examples where[br]this could actually be applied.