2011 Calculus BC free response #6d

WEBVTT
Kind: captions
Language: en

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Part D. Let p sub 4 of x
be the fourth degree Taylor

00:00:04.770 --> 00:00:07.560
polynomial for f
about x equals 0.

00:00:07.560 --> 00:00:09.940
Using information
from the graph of y

00:00:09.940 --> 00:00:13.700
is equal to the absolute value
of the fifth derivative of f

00:00:13.700 --> 00:00:17.750
of x, shown above-- and that's
the graph right over here,

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I put it on the side
to save space-- show

00:00:19.870 --> 00:00:22.530
that the absolute
value of the difference

00:00:22.530 --> 00:00:25.370
between the polynomial--
the fourth degree polynomial

00:00:25.370 --> 00:00:28.510
evaluated at 1/4 and the
function evaluated at 1/4

00:00:28.510 --> 00:00:32.060
is less than one over 3,000.

00:00:32.060 --> 00:00:35.430
So this is, it seems like quite
an interesting problem here.

00:00:35.430 --> 00:00:37.676
So first, let's just think
about what they're asking.

00:00:37.676 --> 00:00:39.050
They're asking
for the difference

00:00:39.050 --> 00:00:42.120
between the polynomial and
the function evaluated 1/4.

00:00:42.120 --> 00:00:43.780
In other words,
they're asking us

00:00:43.780 --> 00:00:48.660
to-- they're asking us to
bound the error of the function

00:00:48.660 --> 00:00:53.494
approximation at
x is equal to 1/4.

00:00:53.494 --> 00:00:54.910
So let me draw a
little graph here

00:00:54.910 --> 00:00:56.743
just so we can visualize
it a little better.

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So if that's our y-axis and--
let me draw it like this.

00:01:00.560 --> 00:01:05.230
So this is our y-axis
and this is our x-axis.

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And our function, let's
say our function looks

00:01:07.440 --> 00:01:08.610
something like this.

00:01:08.610 --> 00:01:11.310
Our polynomial is centered at 0.

00:01:11.310 --> 00:01:14.320
Our fourth degree
polynomial is centered at 0.

00:01:14.320 --> 00:01:17.140
And so it'll be equal
to the function at 0.

00:01:17.140 --> 00:01:20.570
And then it'll become
probably a worse approximation

00:01:20.570 --> 00:01:23.436
as we get further and
further away from 0.

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And what they're
saying is, let's

00:01:24.810 --> 00:01:26.950
bound how bad of
an approximation

00:01:26.950 --> 00:01:29.540
that is when x is equal to 1/4.

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So if we say that this is x is
equal to 1/4 right over here,

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they're saying, look, take the
value of the polynomial at 1/4,

00:01:38.030 --> 00:01:41.192
subtract from that the value
of the function at 1/4,

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and then take the
absolute value and then

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that'll essentially give us the
absolute value of the error.

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Or that will give us the
absolute value of the remainder

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or give us the distance between
the two functions there.

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And they want us to
bound that distance.

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Now, to do this problem
there is something

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that you just kind
of have to know.

00:02:00.930 --> 00:02:02.940
And I've proven it
in another series

00:02:02.940 --> 00:02:04.960
of videos-- two videos
I think it took me--

00:02:04.960 --> 00:02:07.210
in the Khan Academy
Calculus playlist.

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If you do a search for bounded
or error or approximation

00:02:11.120 --> 00:02:13.830
or Taylor, you should
probably find it.

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And that there's a general idea
here which we've proved that

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if you have some function-- so
I'm going to show the general--

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so if you have some function
f of x and that you have some

00:02:28.650 --> 00:02:32.230
polynomial approximation
of it of degree n---

00:02:32.230 --> 00:02:35.230
some polynomial
approximation of degree n.

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And let's say that this
polynomial approximation is

00:02:37.620 --> 00:02:38.515
centered at a.

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In our particular case, a is 0.

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So that you're centered at a.

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So we could write-- we
could say this is about a.

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It's an approximation
about x equals a.

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We can bound the error.

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So let me first define
this error function here.

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Sometimes it's
called a remainder.

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So we could say, the error
function for the nth degree

00:03:01.710 --> 00:03:05.520
polynomial about a-- the
n-th degree Taylor polynomial

00:03:05.520 --> 00:03:08.750
approximation about a-- it
will be a function of x.

00:03:08.750 --> 00:03:12.490
So for any other x that you
pick, what will this error be?

00:03:12.490 --> 00:03:15.600
Is equal to the difference
between these two things.

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You could say it's f of x minus
that or that minus f of x.

00:03:19.070 --> 00:03:20.790
But let me just
write it this way.

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This is equal to f of x minus
the polynomial approximation

00:03:27.060 --> 00:03:27.896
of x.

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And if you want to take
the absolute value of this,

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that's equal to
absolute value of that.

00:03:31.430 --> 00:03:32.860
And if you're taking
absolute values,

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then you can switch the order.

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So this is the same thing
as the absolute value

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of the polynomial approximation
at x minus f of x.

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So the thing that you just need
to know is, we can bound this.

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We can bound this if
we know some properties

00:03:51.050 --> 00:03:52.740
about the n plus 1.

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So this is the nth
degree approximation.

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If we know some properties about
the n plus 1th derivative of f.

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So if we know that
the n plus 1th is

00:04:07.830 --> 00:04:15.060
the derivative of f is less
than or equal to some maximum

00:04:15.060 --> 00:04:16.990
value-- and in particular,
the absolute value

00:04:16.990 --> 00:04:19.570
of the n plus 1th
derivative of f--

00:04:19.570 --> 00:04:22.470
is less than or equal to
some-- we could call this

00:04:22.470 --> 00:04:29.200
some maximum value--
over the interval, over x

00:04:29.200 --> 00:04:33.910
is in the interval
between a and some b,

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where b is greater than a.

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Then we can make the statement--
and this is, once again,

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something that I
prove in that video.

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I think I call it the
boundedness of the error

00:04:49.376 --> 00:04:51.020
remainder for Taylor
approximation.

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Something like that.

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Then we know that the
error function at any given

00:04:59.460 --> 00:05:04.180
x where x is greater than or--
let me say this in particular.

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Is any given x that's part
of this interval-- where

00:05:08.070 --> 00:05:10.110
x is part of this
interval between a

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and b-- is going to be less
than or equal to M times

00:05:17.310 --> 00:05:23.950
x minus a to the n plus
1th power over n plus 1.

00:05:23.950 --> 00:05:25.450
So this is the thing
that you really

00:05:25.450 --> 00:05:29.630
have to know ahead of time to
be able to do this problem.

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It would be nearly impossible in
the time constraints of the AP

00:05:32.610 --> 00:05:35.440
exam to prove this
from first principles.

00:05:35.440 --> 00:05:36.990
I encourage you to
watch that video

00:05:36.990 --> 00:05:39.370
so you can understand
how this proof happens

00:05:39.370 --> 00:05:40.520
from first principles.

00:05:40.520 --> 00:05:44.020
But going into the AP
exam, based on the facts

00:05:44.020 --> 00:05:46.610
that this did show
up in the 2011 exam,

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it's probably good to
know this property,

00:05:48.640 --> 00:05:49.772
to know how to bound it.

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And this is clearly what they're
asking for because they're

00:05:52.230 --> 00:05:56.400
giving us-- so we know
what the fourth-- we

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know what p sub four is.

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p sub 4 of x-- it's up
to the fourth degree

00:06:00.480 --> 00:06:03.480
term of the polynomial
approximation of f about 0.

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In part B, we got all the
way the sixth degree term.

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So if you want to go just
to the fourth degree term,

00:06:07.930 --> 00:06:09.840
this is what p sub 4 is.

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So this is 1 plus x squared over
2 plus x to the fourth over 4.

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So we know that.

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And we want to bound this
thing right over here.

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So if we can find an M-- so
this is the fourth degree

00:06:28.350 --> 00:06:31.100
polynomial.

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This is the fourth
degree error that we're

00:06:32.850 --> 00:06:34.860
talking about in this case.

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Let me write it this way.

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This is the nth error
in the general case.

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But we want to bound
the fourth degree error.

00:06:39.970 --> 00:06:42.490
So if we want bound the
fourth degree error--

00:06:42.490 --> 00:06:47.794
so we can say the fourth
degree error-- at b.

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In this case, we're
going to say--

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or at 1/4 I should say, not b.

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I never mentioned a b before.

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I mentioned the b
when I prove it.

00:06:54.450 --> 00:06:56.990
The fourth degree
error at 1/4 is

00:06:56.990 --> 00:06:59.850
going to be less than or equal
to-- I did mention a b here

00:06:59.850 --> 00:07:03.020
so if you take this
to be 1/4, and 1/4

00:07:03.020 --> 00:07:05.550
is part of that interval
so you can apply it here.

00:07:05.550 --> 00:07:08.720
So the fourth degree error,
I guess you could say,

00:07:08.720 --> 00:07:13.620
at 1/4 is going to be less
than or equal to some M times

00:07:13.620 --> 00:07:18.730
b-- I should say, 1/4 minus a.

00:07:18.730 --> 00:07:22.430
a in our particular case
for this problem is a 0.

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So 1/4 minus is just 0
to the n plus 1th power.

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So this is 4-- this is the
fourth degree polynomial. n is

00:07:31.770 --> 00:07:34.740
4 so the n plus 1 power
is the fifth power.

00:07:34.740 --> 00:07:37.430
All of that over--
and sorry, this

00:07:37.430 --> 00:07:39.510
was supposed to be n
plus 1-- all of that

00:07:39.510 --> 00:07:41.090
over n plus 1 factorial.

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So this is going to
be over 5 factorial.

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Don't want to confuse you.

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This was-- all of that is
over n plus 1 factorial.

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So all of this is
over 5 factorial.

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So we can make this statement
if we know that the fifth

00:08:05.140 --> 00:08:07.200
derivative-- the n
plus 1th derivative--

00:08:07.200 --> 00:08:11.150
if we know that the fifth
derivative of f is less than

00:08:11.150 --> 00:08:14.750
or equal to some value M-- the
absolute value of the fifth

00:08:14.750 --> 00:08:17.720
derivative of f, I should say---
is less than or equal to some

00:08:17.720 --> 00:08:29.810
value at M over the interval
x is between 0 and 1/4.

00:08:29.810 --> 00:08:32.419
Those are our a's and b's
in this particular case.

00:08:32.419 --> 00:08:34.169
So when you look at
the graph-- and that's

00:08:34.169 --> 00:08:35.252
exactly what they gave us.

00:08:35.252 --> 00:08:37.590
They gave us the graph
of the absolute value

00:08:37.590 --> 00:08:39.409
of the fifth derivative of x.

00:08:39.409 --> 00:08:42.270
And 1/4 is sitting
right over here.

00:08:42.270 --> 00:08:43.669
So this is 1/4.

00:08:43.669 --> 00:08:46.510
We don't know exactly
what we hit over here.

00:08:46.510 --> 00:08:50.070
This looks like it's roughly,
I don't know, 31 or 32.

00:08:50.070 --> 00:08:54.330
But we know it's
definitely less than 40

00:08:54.330 --> 00:08:57.200
over the interval
between 0 and 1/4.

00:08:57.200 --> 00:08:59.125
So we can pick M to be 40.

00:09:04.100 --> 00:09:06.950
We know that this is
less than or equal to 40

00:09:06.950 --> 00:09:08.280
over that interval.

00:09:08.280 --> 00:09:10.140
So we can make this statement.

00:09:10.140 --> 00:09:12.340
And so we know that this
thing right over here

00:09:12.340 --> 00:09:14.400
is going to be less
because this over here

00:09:14.400 --> 00:09:16.080
is the same as this over here.

00:09:16.080 --> 00:09:20.000
We know this is less than--
let me write it all out.

00:09:20.000 --> 00:09:23.421
This is-- and I know of people
who got fives on this example

00:09:23.421 --> 00:09:24.670
without being able to do this.

00:09:24.670 --> 00:09:26.920
So don't stress out if this
seems really bizarre too.

00:09:26.920 --> 00:09:28.970
This is really so
they can separate out

00:09:28.970 --> 00:09:31.110
who happened to
know this, really.

00:09:31.110 --> 00:09:38.930
So this we can say is
based on this property,

00:09:38.930 --> 00:09:43.850
we can say is less than
or equal to 40 times 1/4

00:09:43.850 --> 00:09:45.560
to the fifth power.

00:09:45.560 --> 00:09:48.320
So this is 40 times
1/4 to the fifth power.

00:09:48.320 --> 00:09:51.360
So I could just say divided
by 4 to the fifth power

00:09:51.360 --> 00:09:53.400
times-- and then the
denominator over here--

00:09:53.400 --> 00:09:54.250
we have 5 factorial.

00:09:54.250 --> 00:09:55.208
I'll just write it out.

00:09:55.208 --> 00:09:57.416
5 times 4 times 3 times 2.

00:09:57.416 --> 00:09:58.790
And we don't have
to write the 1.

00:09:58.790 --> 00:10:00.430
That doesn't change the value.

00:10:00.430 --> 00:10:02.140
And now we can simplify this.

00:10:02.140 --> 00:10:04.980
We can divide the numerator
and the denominator by 4.

00:10:04.980 --> 00:10:07.710
So this becomes 10,
that just becomes a 1.

00:10:07.710 --> 00:10:09.120
We can divide it by 5.

00:10:09.120 --> 00:10:11.560
This becomes a 2,
that becomes a 1.

00:10:11.560 --> 00:10:13.180
We can divide by 1.

00:10:13.180 --> 00:10:15.840
That becomes a-- oh,
sorry, we can divide by 2.

00:10:15.840 --> 00:10:18.090
That becomes a 1,
that becomes a 1.

00:10:18.090 --> 00:10:20.400
And you say that this
quantity over here

00:10:20.400 --> 00:10:23.610
is going to be equal to 1.

00:10:23.610 --> 00:10:25.200
So let me write it.

00:10:25.200 --> 00:10:30.130
So all of this stuff is less
than or equal to 1 over 4

00:10:30.130 --> 00:10:35.080
to the fifth times 3.

00:10:38.390 --> 00:10:40.680
That's all we have left
in the denominator.

00:10:40.680 --> 00:10:42.131
And so what is 4 to the fifth?

00:10:42.131 --> 00:10:43.130
You may or may not know.

00:10:43.130 --> 00:10:45.660
You can work it out by hand if
you like, but 4 to the fifth

00:10:45.660 --> 00:10:47.840
is the same thing as
2 to the 10th power

00:10:47.840 --> 00:10:49.457
because 4 is 2 squared.

00:10:49.457 --> 00:10:52.040
2 to the 10th power, if you're
familiar with computer science,

00:10:52.040 --> 00:10:52.900
is 1,024.

00:10:52.900 --> 00:10:54.660
But you can work
it out yourself.

00:10:54.660 --> 00:10:57.050
So this is going to
be-- this is less than

00:10:57.050 --> 00:11:05.770
or equal to 1,024 times 3.

00:11:05.770 --> 00:11:06.640
So what is that?

00:11:06.640 --> 00:11:13.020
That is less than or
equal to 1 over 3.

00:11:13.020 --> 00:11:14.113
Oh, what is it?

00:11:14.113 --> 00:11:15.350
72?

00:11:15.350 --> 00:11:16.691
3,000.

00:11:16.691 --> 00:11:19.570
1 over 3,072.

00:11:19.570 --> 00:11:24.050
So this quantity right over
here is less than 1 over 3,072.

00:11:24.050 --> 00:11:24.930
We know.

00:11:24.930 --> 00:11:26.540
We know that for a fact.

00:11:26.540 --> 00:11:29.830
Well, if something is
less than 1 over 3,072,

00:11:29.830 --> 00:11:32.300
it's definitely going to
be less than 1 over 3,000.

00:11:32.300 --> 00:11:35.102
1 over 3,000 is a larger number.

00:11:35.102 --> 00:11:36.310
It has a smaller denominator.

00:11:36.310 --> 00:11:41.380
That's definitely less than
or equal to 1 over 3,000.

00:11:41.380 --> 00:11:43.500
And then we are done.

00:11:43.500 --> 00:11:46.400
We've proven that this is
less than 1 over 3,000.

00:11:46.400 --> 00:11:51.860
We've proven that the error at
1/4 is less than 1 over 3,000.


Taylor polynomial remainder (part 2)

WEBVTT
Kind: captions
Language: en

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.


Taylor polynomial remainder (part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:02.658
- [Voiceover] Let's say
that we have some function

00:00:02.658 --> 00:00:04.135
f of x right over here.

00:00:04.135 --> 00:00:07.500
And let me graph an arbitrary f of x.

00:00:07.500 --> 00:00:11.280
So, that's my y-axis, that is my x-axis

00:00:11.280 --> 00:00:14.360
and maybe f of x looks
something like that.

00:00:14.360 --> 00:00:17.322
And what I wanna do is I
wanna approximate f of x

00:00:17.322 --> 00:00:18.970
with a Taylor polynomial

00:00:20.490 --> 00:00:22.830
centered around x is equal to a.

00:00:22.830 --> 00:00:25.590
So this is the x-axis, this is the y-axis.

00:00:25.590 --> 00:00:28.326
So I want a Taylor polynomial
centered around there.

00:00:28.326 --> 00:00:29.926
And we've seen how this works.

00:00:29.926 --> 00:00:32.254
The Taylor polynomial
comes out of the idea

00:00:32.254 --> 00:00:34.820
that for all of the derivatives up to

00:00:34.820 --> 00:00:37.300
and including the degree
of the polynomial,

00:00:37.300 --> 00:00:40.573
those derivatives of that
polynomial evaluated at a

00:00:40.573 --> 00:00:42.760
should be equal to the derivatives

00:00:42.760 --> 00:00:44.390
of our function evaluated at a.

00:00:44.390 --> 00:00:46.090
And that polynomial evaluated at a

00:00:46.090 --> 00:00:49.522
should also be equal to that
function evaluated at a.

00:00:49.522 --> 00:00:51.420
So our polynomial, our Taylor polynomial

00:00:51.420 --> 00:00:54.760
approximation would look
something like this.

00:00:54.760 --> 00:00:57.316
So, I'll call it P of x.

00:00:57.316 --> 00:00:59.795
And sometimes you might see
a subscript, a big N there

00:00:59.795 --> 00:01:02.030
to say it's an Nth degree approximation

00:01:02.030 --> 00:01:04.381
and sometimes you'll
see something like this.

00:01:04.381 --> 00:01:06.923
Sometimes you'll see
something like N comma a

00:01:06.923 --> 00:01:10.096
to say it's an Nth degree
approximation centered at a.

00:01:10.096 --> 00:01:11.690
Actually, I'll write that right now.

00:01:11.690 --> 00:01:13.350
Maybe we might lose it if
we have to keep writing it

00:01:13.350 --> 00:01:15.090
over and over but you
should assume that it is

00:01:15.090 --> 00:01:17.630
an Nth degree polynomial centered at a.

00:01:17.630 --> 00:01:18.960
And it's going to look like this.

00:01:18.960 --> 00:01:21.280
It is going to be f of a,

00:01:22.090 --> 00:01:25.071
plus f prime of a,

00:01:27.118 --> 00:01:29.500
times x minus a,

00:01:29.500 --> 00:01:32.900
plus f prime prime of a,

00:01:32.900 --> 00:01:36.391
times x minus a squared over--

00:01:36.391 --> 00:01:38.550
Either you could write
two or two factorial,

00:01:38.550 --> 00:01:39.750
they're the same value.

00:01:39.750 --> 00:01:40.760
I'll write two factorial.

00:01:40.760 --> 00:01:41.890
You could write a divided by

00:01:41.890 --> 00:01:44.119
one factorial over here, if you like.

00:01:44.119 --> 00:01:48.990
And then plus, you go to the
third derivative of f at a

00:01:48.990 --> 00:01:51.710
times x minus a to the third power,

00:01:51.710 --> 00:01:53.295
I think you see where this is going,

00:01:53.295 --> 00:01:54.820
over three factorial.

00:01:54.820 --> 00:01:57.020
And you keep going, I'll
go to this line right here,

00:01:57.020 --> 00:02:00.220
all the way to your Nth degree term

00:02:00.220 --> 00:02:03.220
which is the Nth derivative of f

00:02:03.220 --> 00:02:07.651
evaluated at a times x minus a

00:02:07.651 --> 00:02:11.700
to the N over N factorial.

00:02:11.700 --> 00:02:13.226
And this polynomial right over here,

00:02:13.226 --> 00:02:15.936
this Nth degree polynomial centered at a,

00:02:18.756 --> 00:02:21.379
f or P of a is going to be
the same thing as f of a.

00:02:21.379 --> 00:02:23.160
And you can verify that
because all of these

00:02:23.160 --> 00:02:25.142
other terms have an x minus a here.

00:02:25.142 --> 00:02:27.129
So if you put an a in the polynomial,

00:02:27.129 --> 00:02:28.956
all of these other terms
are going to be zero.

00:02:28.956 --> 00:02:31.330
And you'll have P of a is equal to f of a.

00:02:31.330 --> 00:02:32.373
Let me write that down.

00:02:34.512 --> 00:02:36.717
P of a is equal to f of a.

00:02:36.717 --> 00:02:38.513
And so it might look something like this.

00:02:38.513 --> 00:02:40.626
And it's going to fit the curve better

00:02:40.626 --> 00:02:43.226
the more of these terms
that we actually have.

00:02:43.226 --> 00:02:44.856
So it might look something like this.

00:02:44.856 --> 00:02:47.836
I'll try my best

00:02:49.636 --> 00:02:51.793
to show what it might look like.

00:02:53.393 --> 00:02:55.943
So this is all review,
I have this polynomial

00:02:55.943 --> 00:02:57.496
that's approximating this function.

00:02:57.496 --> 00:03:00.310
The more terms I have, the
higher degree of this polynomial,

00:03:00.310 --> 00:03:02.096
the better that it will fit this curve

00:03:02.096 --> 00:03:04.852
the further that I get away from a.

00:03:04.852 --> 00:03:07.450
But what I wanna do in
this video is think about

00:03:07.450 --> 00:03:10.026
if we can bound how good it's fitting

00:03:11.756 --> 00:03:15.099
this function as we move away from a.

00:03:15.099 --> 00:03:17.659
So what I wanna do is
define a remainder function.

00:03:17.659 --> 00:03:19.258
Or sometimes, I've seen some text books

00:03:19.258 --> 00:03:21.053
call it an error function.

00:03:21.053 --> 00:03:22.686
And I'm going to call this--

00:03:24.931 --> 00:03:26.666
I'll just call it an error--

00:03:27.526 --> 00:03:28.826
Just so you're consistent with all the

00:03:28.826 --> 00:03:30.311
different notations you
might see in a book,

00:03:30.311 --> 00:03:32.201
some people will call
this a remainder function

00:03:32.201 --> 00:03:33.691
and sometimes they'll
write a remainder function

00:03:33.691 --> 00:03:37.256
for an Nth degree
polynomial centered at a.

00:03:37.256 --> 00:03:40.022
Sometimes you'll see this
as an error function.

00:03:41.350 --> 00:03:42.993
The error function is sometimes avoided

00:03:42.993 --> 00:03:45.767
because it looks like expected
value from probability.

00:03:45.767 --> 00:03:49.191
But you'll see this often,
this is E for error.

00:03:49.191 --> 00:03:50.886
E for error, R for remainder.

00:03:50.886 --> 00:03:52.026
And sometimes they'll also have

00:03:52.026 --> 00:03:54.496
the subscripts over there like that.

00:03:54.496 --> 00:03:56.286
And what we'll do is, we'll just define

00:03:56.286 --> 00:03:58.676
this function to be the difference between

00:04:00.346 --> 00:04:05.252
f of x and our approximation
of f of x for any given x.

00:04:05.252 --> 00:04:07.566
So it's really just going to be,

00:04:07.566 --> 00:04:10.469
I'll do it in the same colors,
it's going to be f of x

00:04:11.911 --> 00:04:13.722
minus P of x.

00:04:15.520 --> 00:04:17.886
Where this is an Nth degree polynomial

00:04:19.148 --> 00:04:20.391
centered at a.

00:04:20.391 --> 00:04:23.486
So for example, if
someone were to ask you,

00:04:23.486 --> 00:04:24.918
or if you wanted to visualize.

00:04:24.918 --> 00:04:27.390
What are they talking
about if they're saying

00:04:27.390 --> 00:04:30.657
the error of this Nth degree
polynomial centered at a

00:04:30.657 --> 00:04:33.124
when we are at x is equal to b.

00:04:33.124 --> 00:04:36.456
What is thing equal to or how
should you think about this.

00:04:36.456 --> 00:04:38.723
Well, if b is right over here.

00:04:39.850 --> 00:04:41.911
So the error of b is going to be f of b

00:04:41.911 --> 00:04:43.943
minus the polynomial at b.

00:04:43.943 --> 00:04:46.566
So f of b there, the
polynomial's right over there.

00:04:46.566 --> 00:04:48.966
So it'll be this distance right over here.

00:04:48.966 --> 00:04:51.194
So if you measure the error at a,

00:04:52.426 --> 00:04:54.177
it would actually be zero.

00:04:54.177 --> 00:04:56.556
Because the polynomial and the
function are the same there.

00:04:56.556 --> 00:04:58.329
F of a is equal to P of a,

00:04:58.329 --> 00:05:00.064
so the error at a is equal to zero.

00:05:00.064 --> 00:05:01.202
And let me actually write that down

00:05:01.202 --> 00:05:02.477
because that's an interesting property.

00:05:02.477 --> 00:05:05.780
It'll help us bound it
eventually so let me write that.

00:05:07.327 --> 00:05:09.050
The error function at a.

00:05:09.050 --> 00:05:11.296
And for the rest of this
video you can assume

00:05:11.296 --> 00:05:13.675
that I could write a subscript.

00:05:13.675 --> 00:05:16.849
This is for the Nth degree
polynomial centered at a.

00:05:16.849 --> 00:05:18.426
I'm just gonna not write that everytime

00:05:18.426 --> 00:05:21.030
just to save ourselves a
little bit of time in writing,

00:05:21.030 --> 00:05:22.777
to keep my hand fresh.

00:05:22.777 --> 00:05:25.118
So the error at a is equal to

00:05:25.118 --> 00:05:27.696
f of a minus P of a.

00:05:27.696 --> 00:05:29.356
And once again, I won't
write the sub-N, sub-a.

00:05:29.356 --> 00:05:30.686
You can assume it, this is an

00:05:30.686 --> 00:05:32.953
Nth degree polynomial centered at a.

00:05:32.953 --> 00:05:35.696
And these two things
are equal to each other.

00:05:35.696 --> 00:05:37.333
So this is going to be equal to zero.

00:05:37.333 --> 00:05:38.426
And we see that right over here.

00:05:38.426 --> 00:05:41.426
The distance between the
two functions is zero there.

00:05:41.426 --> 00:05:43.248
Now let's think about something else.

00:05:43.248 --> 00:05:47.023
Let's think about what the derivative

00:05:49.783 --> 00:05:52.187
of the error function evaluated at a is.

00:05:52.187 --> 00:05:55.051
Well that's going to be the
derivative of our function at a

00:05:55.051 --> 00:05:58.096
minus the first derivative

00:05:58.096 --> 00:06:00.270
of our polynomial at a.

00:06:00.270 --> 00:06:03.514
And if we assume that this is

00:06:03.514 --> 00:06:06.595
higher than degree one, we
know that these derivates

00:06:06.595 --> 00:06:08.496
are going to be the same at a.

00:06:08.496 --> 00:06:10.112
You can try to take the
first derivative here.

00:06:10.112 --> 00:06:12.790
If you take the first
derivative of this whole mess--

00:06:12.790 --> 00:06:16.331
And this is actually why Taylor
polynomials are so useful,

00:06:16.331 --> 00:06:19.350
is that up to and including
the degree of the polynomial

00:06:19.350 --> 00:06:21.988
when you evaluate the derivatives
of your polynomial at a

00:06:21.988 --> 00:06:22.956
they're going to be the same

00:06:22.956 --> 00:06:25.341
as the derivatives of the function at a.

00:06:25.341 --> 00:06:27.426
And that's what starts to
make it a good approximation.

00:06:27.426 --> 00:06:29.156
But if you took a derivative here,

00:06:29.156 --> 00:06:31.760
this term right here will
disappear, it'll go to zero.

00:06:31.760 --> 00:06:33.086
I'll cross it out for now.

00:06:33.086 --> 00:06:35.970
This term right over here
will just be f prime of a

00:06:35.970 --> 00:06:38.725
and then all of these other
terms are going to be left with

00:06:38.725 --> 00:06:41.108
some type of an x minus a in them.

00:06:41.108 --> 00:06:42.519
And so when you evaluate it at a,

00:06:42.519 --> 00:06:44.626
all the terms with an x minus a disappear,

00:06:44.626 --> 00:06:46.419
because you have an a minus a on them.

00:06:46.419 --> 00:06:47.766
This one already disappeared

00:06:47.766 --> 00:06:50.750
and you're literally just left with

00:06:50.750 --> 00:06:53.426
P prime of a will equal f prime of a.

00:06:53.426 --> 00:06:56.356
And we've seen that before.
So let me write that.

00:06:56.356 --> 00:06:58.954
So because we know that P prime of a

00:06:58.954 --> 00:07:01.626
is equal to f prime of a,

00:07:01.626 --> 00:07:04.743
when you evaluate the error function,

00:07:04.743 --> 00:07:06.556
the derivative of the error function at a,

00:07:06.556 --> 00:07:09.756
that also is going to be equal to zero.

00:07:09.756 --> 00:07:12.294
And this general property right over here,

00:07:12.294 --> 00:07:14.696
is true up to an including N.

00:07:14.696 --> 00:07:16.676
So let me write this down.

00:07:16.676 --> 00:07:21.096
So we already know that P
of a is equal to f of a.

00:07:21.096 --> 00:07:25.086
We already know that P prime
of a is equal to f prime of a.

00:07:25.086 --> 00:07:26.696
This really comes straight
out of the definition

00:07:26.696 --> 00:07:28.286
of the Taylor polynomials.

00:07:28.286 --> 00:07:30.426
And this is going to be true all the way

00:07:30.426 --> 00:07:34.664
until the Nth derivative
of our polynomial is going,

00:07:34.664 --> 00:07:37.756
evaluated at a, not everywhere,
just evaluated at a,

00:07:37.756 --> 00:07:40.390
is going to be equal to the Nth derivative

00:07:40.390 --> 00:07:44.156
of our function evaluated at a.

00:07:44.156 --> 00:07:46.486
So what that tells us is
that we can keep doing this

00:07:46.486 --> 00:07:50.084
with the error function all
the way to the Nth derivative

00:07:50.084 --> 00:07:53.391
of the error function evaluated at a

00:07:53.391 --> 00:07:55.076
is going to be equal to,

00:07:55.076 --> 00:07:57.055
well that's just going
to be the Nth derivative

00:07:57.055 --> 00:08:00.886
of f evaluated at a,
minus the Nth derivative

00:08:00.886 --> 00:08:03.151
of our polynomial evaluated at a.

00:08:03.151 --> 00:08:04.672
And we already said that
these are going to be

00:08:04.672 --> 00:08:06.714
equal to each other up
to the Nth derivative

00:08:06.714 --> 00:08:08.226
when we evaluate them at a.

00:08:08.226 --> 00:08:10.793
So these are all going
to be equal to zero.

00:08:10.793 --> 00:08:12.156
So this is an interesting property

00:08:12.156 --> 00:08:13.992
and it's also going to
be useful when we start

00:08:13.992 --> 00:08:16.686
to try to bound this error function.

00:08:16.686 --> 00:08:18.166
And that's the whole
point of where I'm going

00:08:18.166 --> 00:08:20.179
with this video and
probably the next video,

00:08:20.179 --> 00:08:22.084
is we're gonna try to bound it so we know

00:08:22.084 --> 00:08:24.540
how good of an estimate we have.

00:08:24.540 --> 00:08:26.659
Especially as we go further and further

00:08:26.659 --> 00:08:27.991
from where we are centered.

00:08:27.991 --> 00:08:30.426
>From where are approximation is centered.

00:08:30.426 --> 00:08:33.877
Now let's think about when we
take a derivative beyond that.

00:08:33.877 --> 00:08:35.178
So let's think about what
happens when we take the

00:08:35.178 --> 00:08:36.878
N plus oneth derivative.

00:08:38.498 --> 00:08:39.316
What's a good place to write?

00:08:39.316 --> 00:08:41.923
Well I have some screen
real estate right over here.

00:08:43.261 --> 00:08:46.143
What is the N plus oneth derivative

00:08:46.143 --> 00:08:48.086
of our error function?

00:08:48.086 --> 00:08:49.836
And not even if I'm just evaluating at a.

00:08:49.836 --> 00:08:52.226
If I just say generally,
the error function E of x,

00:08:52.226 --> 00:08:54.306
what's the N plus oneth derivative of it?

00:08:54.306 --> 00:08:56.034
Well it's going to be the
N plus oneth derivative

00:08:56.034 --> 00:08:57.669
of our function

00:09:02.954 --> 00:09:06.665
minus the N plus oneth derivative of our--

00:09:06.665 --> 00:09:08.284
We're not just evaluating
at a here either.

00:09:08.284 --> 00:09:10.036
Let me write a x there.

00:09:11.850 --> 00:09:16.713
I'm literally just taking
the N plus oneth derivative

00:09:16.713 --> 00:09:20.246
of both sides of this
equation right over here.

00:09:20.246 --> 00:09:21.756
So it's literally the
N plus oneth derivative

00:09:21.756 --> 00:09:24.414
of our function minus the
N plus oneth derivative

00:09:24.414 --> 00:09:26.476
of our Nth degree polynomial.

00:09:28.896 --> 00:09:32.544
The N plus oneth derivative
of our Nth degree polynomial.

00:09:32.544 --> 00:09:34.826
I could write a N here,
I could write an a here

00:09:34.826 --> 00:09:38.258
to show it's an Nth degree centered at a.

00:09:38.258 --> 00:09:42.151
Now, what is the N plus onethe derivative

00:09:42.151 --> 00:09:44.858
of an Nth degree polynomial?

00:09:44.858 --> 00:09:47.166
And if you want some hints,

00:09:49.691 --> 00:09:52.686
take the second derivative
of y is equal to x.

00:09:52.686 --> 00:09:56.154
It's a first degree polynomial,
take the second derivative,

00:09:56.154 --> 00:09:57.994
you're gonna get zero.

00:09:57.994 --> 00:10:02.366
Take the third derivative
of y is equal to x squared.

00:10:02.366 --> 00:10:05.226
The first derivative is 2x,
the second derivative is 2,

00:10:05.226 --> 00:10:07.026
the third derivative is zero.

00:10:07.026 --> 00:10:10.486
In general, if you take
an N plus oneth derivative

00:10:10.486 --> 00:10:11.911
of an Nth degree polynomial,

00:10:11.911 --> 00:10:13.062
and you could prove it for yourself,

00:10:13.062 --> 00:10:14.226
you could even prove it generally

00:10:14.226 --> 00:10:16.860
but I think it might make
a little sense to you,

00:10:16.860 --> 00:10:19.255
it's going to be equal to zero.

00:10:19.255 --> 00:10:21.336
It is going to be equal to zero.

00:10:21.336 --> 00:10:24.021
So this thing right here, this
is an N plus oneth derivative

00:10:24.021 --> 00:10:26.086
of an Nth degree polynomial.

00:10:26.086 --> 00:10:29.593
This is going to be equal to zero.

00:10:33.826 --> 00:10:34.726
Let me write this over here.

00:10:34.726 --> 00:10:38.397
The N plus oneth derivative
of our error function

00:10:38.397 --> 00:10:40.326
or our remainder function,
we could call it,

00:10:40.326 --> 00:10:43.290
is equal to the N plus oneth

00:10:45.250 --> 00:10:47.591
derivative of our function.

00:10:47.591 --> 00:10:49.459
And so, what we could do now

00:10:49.459 --> 00:10:51.923
and we'll probably have to
continue this in the next video,

00:10:51.923 --> 00:10:54.742
is figure out, at least can we bound this?

00:10:54.742 --> 00:10:57.426
Can we bound this and if
we are able to bound this,

00:10:57.426 --> 00:10:58.509
if we're able to figure out

00:10:58.509 --> 00:11:00.086
an upper bound on its magnitude--

00:11:00.086 --> 00:11:02.044
So actually, what we want to do is,

00:11:02.044 --> 00:11:03.941
we wanna bound its overall magnitude.

00:11:03.941 --> 00:11:06.036
We wanna bound its absolute value.

00:11:06.036 --> 00:11:07.680
If we can determine that it is

00:11:07.680 --> 00:11:11.198
less than or equal to some value M,

00:11:11.198 --> 00:11:12.400
so if we can actually bound it,

00:11:12.400 --> 00:11:14.486
maybe we can do a little bit of calculus,

00:11:14.486 --> 00:11:15.896
we could keep integrating it

00:11:15.896 --> 00:11:18.529
and maybe we can go back
to the original function

00:11:18.529 --> 00:11:20.156
and bound that in some way.

00:11:20.156 --> 00:11:23.756
If we do know some type of
bound like this over here.

00:11:23.756 --> 00:11:26.580
So I'll take that up in the next video.


Brainwave Sep 2011

WEBVTT
Kind: captions
Language: en

00:00:00.541 --> 00:00:02.139
Voiceover: Here's some
questions that I got from

00:00:02.139 --> 00:00:07.032
Brainwave, a children's
science magazine in India.

00:00:07.032 --> 00:00:09.540
The first question right here
is last month, September,

00:00:09.540 --> 00:00:11.669
the students of the world
celebrated Teacher's Day,

00:00:11.669 --> 00:00:13.400
an overwhelming number of whom would

00:00:13.400 --> 00:00:14.874
undoubtedly have sent their wishes and

00:00:14.874 --> 00:00:15.809
gratitude your way.

00:00:15.809 --> 00:00:18.284
Hopefully and I thank
everyone for all of the

00:00:18.284 --> 00:00:20.473
thank you notes we do get on our YouTube

00:00:20.473 --> 00:00:22.139
channel and all the rest.

00:00:22.139 --> 00:00:25.934
Who was your favorite teacher while

00:00:25.934 --> 00:00:28.011
studying and why?

00:00:28.011 --> 00:00:32.062
I'd have to say I had a
bunch of favorite teachers

00:00:32.062 --> 00:00:34.410
for, I guess, multiple
reasons, but I guess there is

00:00:34.410 --> 00:00:35.873
a common thread.

00:00:35.873 --> 00:00:39.116
I would say in high
school the teachers who,

00:00:39.116 --> 00:00:40.703
and I don't want to just pick favorites,

00:00:40.703 --> 00:00:42.428
I'm sure I'm forgetting some

00:00:42.428 --> 00:00:44.139
very important people here,

00:00:44.139 --> 00:00:47.205
but Mr. Hernandez.

00:00:47.205 --> 00:00:48.810
The first time I had a
class with him was for

00:00:48.810 --> 00:00:51.472
Algebra II, but he was
also the adviser for,

00:00:51.472 --> 00:00:52.870
it was called Mu Alpha
Theta, which was the

00:00:52.870 --> 00:00:54.872
Math Honor Society and also

00:00:54.872 --> 00:00:56.871
the math competition team.

00:00:56.871 --> 00:01:01.805
He was our adviser for
that and Miss. Kennedy.

00:01:01.805 --> 00:01:02.935
She was an English teacher

00:01:02.935 --> 00:01:04.283
and also a journalism instructor.

00:01:04.283 --> 00:01:05.472
When I was on the school newspaper,

00:01:05.472 --> 00:01:09.200
she was our teacher and our adviser.

00:01:09.200 --> 00:01:14.472
I think what made both
of them memorable to me

00:01:14.472 --> 00:01:16.615
and I think were important
in my own development

00:01:16.615 --> 00:01:20.958
is that they really did
treat me like a peer.

00:01:20.958 --> 00:01:22.739
They never talked down to me.

00:01:22.739 --> 00:01:24.953
It never felt like,

00:01:24.953 --> 00:01:27.410
"I am the teacher, you are the student."

00:01:27.410 --> 00:01:29.804
It was kind of a two-way
flow of information.

00:01:29.804 --> 00:01:33.009
It almost felt like we
were working together

00:01:33.009 --> 00:01:34.444
towards a common goal.

00:01:34.444 --> 00:01:35.761
Mr. Hernandez with some of the stuff

00:01:35.761 --> 00:01:37.615
we did with the math team.

00:01:37.615 --> 00:01:38.742
Miss Kennedy with some of the stuff

00:01:38.742 --> 00:01:40.538
we would do with the newspaper.

00:01:40.538 --> 00:01:42.072
She was just a person who was

00:01:42.072 --> 00:01:43.790
more knowledgeable and more experienced,

00:01:43.790 --> 00:01:45.372
but she really respected everything that

00:01:45.372 --> 00:01:46.957
we were doing.

00:01:46.957 --> 00:01:48.956
I think probably all of the teachers

00:01:48.956 --> 00:01:51.367
that I had that I think
really impacted me,

00:01:51.367 --> 00:01:55.866
that's what did it for me is that.

00:01:55.866 --> 00:01:57.412
I remember even in 5th
grade I had a teacher,

00:01:57.412 --> 00:02:00.740
Ms. Ellis and she would run her 5th grade

00:02:00.740 --> 00:02:03.372
social studies classes kind
of like a college seminar.

00:02:03.372 --> 00:02:07.033
I distinctly remember her being
at the front of the class,

00:02:07.033 --> 00:02:08.117
peeling an orange,

00:02:08.117 --> 00:02:10.539
and just asking us tough questions.

00:02:10.539 --> 00:02:12.473
The type of questions where you'd have to

00:02:12.473 --> 00:02:13.737
ponder it for a few seconds

00:02:13.737 --> 00:02:15.289
before you'd even have an

00:02:15.289 --> 00:02:16.455
interesting thought or you would

00:02:16.455 --> 00:02:18.631
continue pondering it when you went home.

00:02:18.631 --> 00:02:22.234
I remember there was Miss Krause and

00:02:22.234 --> 00:02:24.139
Miss Rucell in elementary school.

00:02:24.139 --> 00:02:26.136
They were gifted teachers
and once again, they had

00:02:26.136 --> 00:02:30.202
the luxury of defining what
their own curriculum was.

00:02:30.202 --> 00:02:33.476
They really used to
challenge us and give us

00:02:33.476 --> 00:02:37.205
truly creative projects to work on.

00:02:37.205 --> 00:02:39.787
When you take art class outside of that,

00:02:39.787 --> 00:02:42.367
oftentimes it's very
formulaic, it's kind of

00:02:42.367 --> 00:02:45.808
paint by numbers for
lack of a better word,

00:02:45.808 --> 00:02:47.468
but I remember when we did art projects

00:02:47.468 --> 00:02:49.656
in the context of the gifted class,

00:02:49.656 --> 00:02:51.468
it was really very open-ended.

00:02:51.468 --> 00:02:54.701
It was like here's some
tools; make something

00:02:54.701 --> 00:02:57.368
and I think that was a
really strong message and

00:02:57.368 --> 00:02:59.869
allowed students to really
develop their creativity

00:02:59.869 --> 00:03:02.616
as opposed to just following a formula.

00:03:02.616 --> 00:03:04.983
Across the board, I think
it's the teachers who

00:03:04.983 --> 00:03:09.534
really expected something
from the students

00:03:09.534 --> 00:03:12.038
and treated them with
respect, even though these

00:03:12.038 --> 00:03:14.668
students were clearly
early in their learning.

00:03:14.668 --> 00:03:17.040
Also, assumed that the students were

00:03:17.040 --> 00:03:18.613
capable of true creativity.

00:03:18.613 --> 00:03:20.953
I'll throw another one
in there, Dr. Santania.

00:03:20.953 --> 00:03:24.000
My junior and senior year in
high school, I took courses

00:03:24.000 --> 00:03:26.700
at the University of New
Orleans and Dr. Santania

00:03:26.700 --> 00:03:29.400
was the professor for partial
differential equations.

00:03:29.400 --> 00:03:33.701
He brought me under his
wing and for a summer I was

00:03:33.701 --> 00:03:36.881
essentially one of his researchers.

00:03:36.881 --> 00:03:40.787
He would come into the little
computer lab where I was

00:03:40.787 --> 00:03:43.073
working and he would
talk about his research.

00:03:43.073 --> 00:03:48.954
We would bounce ideas off of each other.

00:03:48.954 --> 00:03:51.535
Clearly, here you are a tenured
professor at a university,

00:03:51.535 --> 00:03:54.067
but you're willing to hang
out with a 16- or 17-year-old.

00:03:54.067 --> 00:03:56.944
It really sent a very strong
message to me and really

00:03:56.944 --> 00:03:59.202
I think built my own self-esteem and

00:03:59.202 --> 00:04:02.538
made me think that I could do something.

00:04:02.538 --> 00:04:05.470
Question number two;
what sort of a student

00:04:05.470 --> 00:04:07.818
were you at school?

00:04:07.818 --> 00:04:17.303
I was a kid who loved,
so I loved - this is kind

00:04:17.303 --> 00:04:20.284
of an interesting answer
I think - I loved a lot of

00:04:20.284 --> 00:04:27.020
the subject matter and I
hated a lot of the classes.

00:04:27.020 --> 00:04:28.616
I found a lot of the
subject matter really,

00:04:28.616 --> 00:04:29.694
really, really interesting;

00:04:29.694 --> 00:04:33.003
I just didn't like how it was sometimes

00:04:33.003 --> 00:04:34.805
presented and it wasn't
the fault of the teacher.

00:04:34.805 --> 00:04:37.522
It was often more usually
the fault I would say of

00:04:37.522 --> 00:04:40.803
the textbook or of just
the curriculum generally.

00:04:40.803 --> 00:04:43.867
The general idea, everything
was devoid of connections.

00:04:43.867 --> 00:04:47.484
You would learn how to
borrow numbers without anyone

00:04:47.484 --> 00:04:49.538
telling you why you were borrowing that

00:04:49.538 --> 00:04:51.426
or even without anyone
ever asking the students

00:04:51.426 --> 00:04:53.006
why does this work,

00:04:53.006 --> 00:04:54.710
without ever having the chance to ponder

00:04:54.710 --> 00:04:56.961
why does it work.

00:04:57.884 --> 00:05:01.539
You'd see peers memorize
one formula for one exam

00:05:01.539 --> 00:05:03.955
and then on the next exam
or by the next month,

00:05:03.955 --> 00:05:07.408
they'd forgotten it and
the next year when there's

00:05:07.408 --> 00:05:09.994
a topic that just builds
off of that same formula,

00:05:09.994 --> 00:05:12.284
if you just intuitively
understood the original formula

00:05:12.284 --> 00:05:13.740
the second one would be intuitive,

00:05:13.740 --> 00:05:15.469
they're rememorizing an ex-formula.

00:05:15.469 --> 00:05:20.805
I really didn't like that
aspect of what was going on

00:05:20.805 --> 00:05:23.951
and I sometimes, I think
for lack of a better word,

00:05:23.951 --> 00:05:27.551
I think I sometimes
acted up because of that.

00:05:30.476 --> 00:05:32.384
I straightened up a
little bit near the end of

00:05:32.384 --> 00:05:34.977
high school, but earlier
on I wasn't like the

00:05:34.977 --> 00:05:38.034
discipline problem kid, but
I definitely got a lot of

00:05:38.034 --> 00:05:42.114
teachers talking to my
mom about Sal not being

00:05:42.114 --> 00:05:45.611
fully engaged in class.

00:05:45.611 --> 00:05:52.194
I think once I got to
college, I was lucky enough

00:05:52.194 --> 00:05:55.030
to go to a place where there
was a lot of intellectual

00:05:55.030 --> 00:05:57.037
curiosity going on.

00:05:57.037 --> 00:06:02.703
There was a culture of
people wanting to learn

00:06:02.703 --> 00:06:05.076
and I frankly think MIT is
somewhat unique in that.

00:06:05.076 --> 00:06:07.449
The culture really is,
there's other things going on

00:06:07.449 --> 00:06:09.947
on campus, but most of
the people there genuinely

00:06:09.947 --> 00:06:12.027
want to learn and there's
no embarrassment of saying

00:06:12.027 --> 00:06:14.842
you're actually fascinated
by something academic.

00:06:14.842 --> 00:06:18.157
While at a lot of schools
in the world, you almost

00:06:18.157 --> 00:06:21.335
have to hide that you're
interested in something.

00:06:21.335 --> 00:06:24.137
I know that was definitely
the case in my high school

00:06:24.137 --> 00:06:28.926
experience is that I would
not, even if I was fascinated

00:06:28.926 --> 00:06:32.737
by something, I would keep
it to myself and sometimes

00:06:32.737 --> 00:06:35.869
even if I did enjoy talking
to the teacher, I would

00:06:35.869 --> 00:06:37.604
actually keep that to myself.

00:06:37.604 --> 00:06:41.116
I wouldn't want my peers
seeing that I was even or

00:06:41.116 --> 00:06:43.952
sometimes I wouldn't talk
to the teacher for fear of

00:06:43.952 --> 00:06:46.667
looking like you're
trying to be teacher's pet

00:06:46.667 --> 00:06:48.237
or something and I think sometimes I would

00:06:48.237 --> 00:06:51.965
maybe overcompensate
in the other direction.

00:06:53.700 --> 00:06:56.535
I would say - I'm giving
a long meandering answer -

00:06:56.535 --> 00:06:58.338
I would say as a student, I was

00:06:58.338 --> 00:07:00.471
definitely intellectually curious.

00:07:00.471 --> 00:07:02.678
I definitely liked to ponder things.

00:07:02.678 --> 00:07:05.699
I did not like to be
given; I didn't like to

00:07:05.699 --> 00:07:08.488
paint by numbers I guess is
the simple way to do it and

00:07:08.488 --> 00:07:13.807
I oftentimes butted heads
with parts of the school

00:07:13.807 --> 00:07:16.806
system that wanted you to paint by numbers

00:07:16.806 --> 00:07:18.539
and wanted you to move lock-step

00:07:18.539 --> 00:07:21.698
with everyone else and wanted you to

00:07:21.698 --> 00:07:24.448
just accept things
without questioning them.

00:07:24.448 --> 00:07:27.667
Yes, sometimes that led to some issues.

00:07:27.667 --> 00:07:31.950
Number three; could you
tell us about any one

00:07:31.950 --> 00:07:34.700
experience or epiphany
that you may have had

00:07:34.700 --> 00:07:36.452
while growing up that shaped your attitude

00:07:36.452 --> 00:07:38.270
towards learning and teaching?

00:07:39.270 --> 00:07:41.648
There's a bunch.

00:07:41.648 --> 00:07:49.407
I think one story is that
we tell often now because

00:07:49.407 --> 00:07:51.535
Shantanu now works at the Khan Academy,

00:07:51.535 --> 00:07:54.200
he's the president here.

00:07:54.200 --> 00:07:57.666
I first met him in 10th
grade at a math competition

00:07:57.666 --> 00:07:59.359
and we met in the finals
at the math competition.

00:07:59.359 --> 00:08:00.940
It was a district wide math competition.

00:08:00.940 --> 00:08:02.865
In the last round it was
me, Shantanu and there was

00:08:02.865 --> 00:08:06.604
another boy and Shantuna
did beat me and so that was

00:08:06.604 --> 00:08:08.749
a good experience, a
humbling experience, but

00:08:08.749 --> 00:08:11.269
above and beyond that
while we were competing,

00:08:11.269 --> 00:08:13.176
we actually had a little
bit of a conversation.

00:08:13.176 --> 00:08:15.658
It actually turned out
that even though we were

00:08:15.658 --> 00:08:16.748
both in the same grade, he was

00:08:16.748 --> 00:08:18.695
a math class ahead of me.

00:08:18.695 --> 00:08:21.069
I asked him, I was like
how does that happen?

00:08:21.069 --> 00:08:22.426
He said, "Well, I just
tested out of the math class

00:08:22.426 --> 00:08:24.272
you're taking right now."

00:08:24.272 --> 00:08:25.604
For me, that was an
epiphany that one, that was

00:08:25.604 --> 00:08:26.872
even a possibility.

00:08:26.872 --> 00:08:29.966
I then went back to my high school and

00:08:29.966 --> 00:08:31.874
went back to the administration.

00:08:31.874 --> 00:08:32.840
I said, "Hey, can I test out of stuff?"

00:08:32.840 --> 00:08:35.113
They're like, "Oh, we're not
aware of anything like that."

00:08:35.113 --> 00:08:38.436
What Shantanu did was he
awakened the possibility

00:08:38.436 --> 00:08:39.734
that you can do that.

00:08:39.734 --> 00:08:41.874
It ended up being a lot
harder, not harder in terms

00:08:41.874 --> 00:08:43.810
of the subject matter
mastery, but a lot harder in

00:08:43.810 --> 00:08:46.434
terms of convincing the
system that you should be

00:08:46.434 --> 00:08:47.693
allowed to test out of something and

00:08:47.693 --> 00:08:49.970
once you prove that you know something,

00:08:49.970 --> 00:08:52.472
you should be allowed to
go on to the next thing.

00:08:52.472 --> 00:08:55.001
That gave me the energy
to do it and that really

00:08:55.001 --> 00:08:57.767
did accelerate my own learning
and frankly that allowed

00:08:57.767 --> 00:08:59.402
me to have a lot of
those experiences that I

00:08:59.402 --> 00:09:01.275
just mentioned; someone like Dr. Santania.

00:09:01.275 --> 00:09:02.604
I would have never ended up taking partial

00:09:02.604 --> 00:09:04.146
differential equations at
the University of New Orleans

00:09:04.146 --> 00:09:07.937
while in high school had I
not had that conversation

00:09:07.937 --> 00:09:11.208
with Shantanu that opened
my mind to the idea

00:09:11.208 --> 00:09:14.904
that you really can, for
lack of a better word,

00:09:14.904 --> 00:09:17.519
fight the system and go at your own pace.

00:09:17.519 --> 00:09:21.352
I think other learnings,
once I started taking classes

00:09:21.352 --> 00:09:24.805
at the university even in
high school, university

00:09:24.805 --> 00:09:28.139
class attendance was
completely optional and I

00:09:28.139 --> 00:09:30.608
sometimes I took advantage
of that and I realized

00:09:30.608 --> 00:09:34.267
that I did learn a lot
better from a book than

00:09:34.267 --> 00:09:35.545
oftentimes in a lecture.

00:09:35.545 --> 00:09:37.215
It wasn't that lectures
are fundamentally bad.

00:09:37.215 --> 00:09:40.208
I mean here I am making
all these YouTube Videos.

00:09:40.208 --> 00:09:41.463
It's just that the book
allowed me to learn

00:09:41.463 --> 00:09:42.717
at my own pace.

00:09:42.717 --> 00:09:43.957
If I was confused by
something, I could read the

00:09:43.957 --> 00:09:45.975
same passage ten times.

00:09:45.975 --> 00:09:49.471
If I found something easy,
I could just skim something.

00:09:49.471 --> 00:09:52.601
In lecture, it was hard to stay engaged.

00:09:52.601 --> 00:09:55.339
I always found that I was
either lost or I was bored.

00:09:55.339 --> 00:09:57.411
Very few times in a lecture that I was

00:09:57.411 --> 00:09:58.460
exactly at the right place,

00:09:58.460 --> 00:10:01.957
exactly connecting with
the pace of the class.

00:10:01.957 --> 00:10:05.116
At the Univeristy of New
Orleans, where I spent

00:10:05.116 --> 00:10:09.123
essentially my 11th and
12th grades in high school,

00:10:09.123 --> 00:10:13.059
I, to a large degree,
showed up for the exams

00:10:13.059 --> 00:10:15.484
but did a lot of self-paced study and that

00:10:15.484 --> 00:10:18.676
carried with me when I
went to MIT is that I

00:10:18.676 --> 00:10:22.128
realized that class time
maybe it works for some,

00:10:22.128 --> 00:10:23.540
but didn't work a lot for me.

00:10:23.540 --> 00:10:25.294
I would have loved class
time that was interactive,

00:10:25.294 --> 00:10:26.631
that was hands on.

00:10:26.631 --> 00:10:27.958
I loved lab time.

00:10:27.958 --> 00:10:29.740
In all my computer science
classes, I would skip

00:10:29.740 --> 00:10:31.077
class and go to the
computer lab and work on my

00:10:31.077 --> 00:10:33.426
projects and I felt I was
learning the material much

00:10:33.426 --> 00:10:37.915
better than sitting in a
300 person lecture hall

00:10:37.915 --> 00:10:39.466
and listening to a
lecture about something.

00:10:39.466 --> 00:10:41.793
It was much better to
fiddle with the thing.

00:10:41.793 --> 00:10:43.869
I think the ideal would
have been having videos

00:10:43.869 --> 00:10:46.997
on demand so that if I
was curious about a topic,

00:10:46.997 --> 00:10:48.495
I could get some videos, pause, repeat,

00:10:48.495 --> 00:10:51.266
watch them as much as I
need to, but then have all

00:10:51.266 --> 00:10:53.612
this extra time and
preferably with the rest of my

00:10:53.612 --> 00:10:56.412
peers and professor around so that I could

00:10:56.412 --> 00:10:58.803
actually fiddle with
stuff and build stuff.

00:10:58.803 --> 00:11:01.740
That's where I feel the
best learning happens.

00:11:01.740 --> 00:11:04.136
That's what we're trying to
do with the Khan Academy.

00:11:04.136 --> 00:11:08.030
We're trying to make class
time about interaction

00:11:08.030 --> 00:11:11.204
and about fiddling and
about helping each other,

00:11:11.204 --> 00:11:13.853
letting humans be humans and allow

00:11:13.853 --> 00:11:16.822
a lot of the core skills to
happen at someone's own pace

00:11:16.822 --> 00:11:20.140
through on demand video
and through exercise.

00:11:20.140 --> 00:11:20.872
I want to be clear.

00:11:20.872 --> 00:11:23.005
I said that I liked textbooks
a lot and I still like

00:11:23.005 --> 00:11:26.842
textbooks; I'm reading
many right now, but I think

00:11:26.842 --> 00:11:30.739
a combination of on demand
videos and exercises that

00:11:30.739 --> 00:11:33.646
give immediate feedback can be better

00:11:33.646 --> 00:11:36.945
for the purposes of self-paced learning.

00:11:36.945 --> 00:11:41.372
When I was 16 and I was
trying to chug through

00:11:41.372 --> 00:11:45.736
some type of an analysis book
or a partial differential

00:11:45.736 --> 00:11:49.346
equations book, it's dense
reading and you had to be

00:11:49.346 --> 00:11:51.548
very persistent to read through that.

00:11:51.548 --> 00:11:54.868
I would have loved a ten
minute video on the exact

00:11:54.868 --> 00:11:56.947
topic that I was ready
for and I would have loved

00:11:56.947 --> 00:11:59.615
practice problems where
you don't just have every

00:11:59.615 --> 00:12:01.403
other answer at the back of the book.

00:12:01.403 --> 00:12:02.952
You have immediate
feedback; you know how to do

00:12:02.952 --> 00:12:04.365
the steps of that problem.

00:12:04.365 --> 00:12:06.202
You know what videos to
watch if you don't know

00:12:06.202 --> 00:12:08.200
how to do that problem and
the videos are not just

00:12:08.200 --> 00:12:10.808
dry text; it's actually
someone trying to communicate

00:12:10.808 --> 00:12:13.228
intuition to you, trying to communicate

00:12:13.228 --> 00:12:15.017
not just the methodology but really

00:12:15.017 --> 00:12:18.069
the art of the mathematics or science.

00:12:18.069 --> 00:12:22.536
Question number four; what
excites you most about

00:12:22.536 --> 00:12:25.002
learning science?

00:12:25.002 --> 00:12:28.475
It really is that science
is all about really

00:12:28.475 --> 00:12:31.369
trying to find the truth in the universe.

00:12:31.369 --> 00:12:33.898
We'll never probably
know all the truth in the

00:12:33.898 --> 00:12:36.202
universe, but I think
that's what separates humans

00:12:36.202 --> 00:12:38.953
from every other species
that we know is that

00:12:38.953 --> 00:12:42.203
we are just naturally
curious and we are naturally

00:12:42.203 --> 00:12:45.699
in awe of the mysteries of this universe.

00:12:45.699 --> 00:12:51.075
Science is our best tool at trying to

00:12:51.075 --> 00:12:53.807
demystify the mysterious.

00:12:53.807 --> 00:12:55.623
Frankly what is does is it
just introduces even more

00:12:55.623 --> 00:12:57.206
and even more profound mysteries

00:12:57.206 --> 00:12:59.474
the deeper and deeper you go into science.

00:12:59.474 --> 00:13:02.951
Just this journey of
working on the Khan Academy

00:13:02.951 --> 00:13:05.288
videos, I've gone so
much deeper in so many

00:13:05.288 --> 00:13:07.201
areas of science, but
it's given me so much more

00:13:07.201 --> 00:13:10.136
appreciation for how much
deeper the open questions

00:13:10.136 --> 00:13:16.953
are and so I think science,
when someone is really in

00:13:16.953 --> 00:13:24.695
science, it caters to that
part of the human instinct

00:13:24.695 --> 00:13:30.075
that gives you awe, that
you start to appreciate

00:13:30.075 --> 00:13:31.953
where you are in the universe
and what the universe

00:13:31.953 --> 00:13:36.754
might be about or how
intertwined and elegant the

00:13:36.754 --> 00:13:39.792
universe that we're in actually is.

00:13:44.207 --> 00:13:46.575
It's just exciting.

00:13:46.575 --> 00:13:50.806
Question number five; as
a teacher who has impacted

00:13:50.806 --> 00:13:53.472
so many lives, do you
have any advice for our

00:13:53.472 --> 00:13:55.136
young readers who find
it hard to concentrate on

00:13:55.136 --> 00:13:59.333
their studies or perform well in exams?

00:13:59.333 --> 00:14:02.841
Yes, my main advice is as much as you can,

00:14:02.841 --> 00:14:05.364
try to ponder what you are learning.

00:14:05.364 --> 00:14:08.614
Don't just try to cram for an exam.

00:14:08.614 --> 00:14:10.806
Don't try to memorize formulas.

00:14:10.806 --> 00:14:13.202
If you feel that the formulas
you're looking at for

00:14:13.202 --> 00:14:16.139
your current class are just
beyond your understanding,

00:14:16.139 --> 00:14:17.953
then back up, look at what
you learned last year,

00:14:17.953 --> 00:14:19.203
the year before.

00:14:19.203 --> 00:14:20.619
Get to a point where
you can start to see why

00:14:20.619 --> 00:14:22.536
those things are intuitive
and hopefully you watch

00:14:22.536 --> 00:14:25.073
Khan Academy videos and
that's what we try and do.

00:14:25.073 --> 00:14:27.011
Show you why these things are intuitive,

00:14:27.011 --> 00:14:30.261
why they aren't formulas to memorize.

00:14:30.261 --> 00:14:32.958
Once you start to ponder
things, you'll just start

00:14:32.958 --> 00:14:34.955
to ponder things all the time.

00:14:34.955 --> 00:14:37.669
My wife jokes that it often
takes me a half an hour

00:14:37.669 --> 00:14:40.334
to throw out the trash
because I take out the trash

00:14:40.334 --> 00:14:42.248
and while I'm holding the
trash can, I sometimes am

00:14:42.248 --> 00:14:44.866
just staring out at the
sky for like ten minutes

00:14:44.866 --> 00:14:47.119
because I'm pondering something.

00:14:47.119 --> 00:14:48.608
This may be the most
fun thing to be doing is

00:14:48.608 --> 00:14:50.664
pondering something
and the more you ponder

00:14:50.664 --> 00:14:54.954
something, it gets your brain
just building connections.

00:14:54.954 --> 00:14:57.335
Then as you progress through more and more

00:14:57.335 --> 00:14:59.672
advanced courses, they'll
just start to seem

00:14:59.672 --> 00:15:01.000
more and more obvious.

00:15:01.000 --> 00:15:02.751
You'll almost say why
is this a new course.

00:15:02.751 --> 00:15:06.485
This is just a natural
by-product of what I had already

00:15:06.485 --> 00:15:10.100
pondered and frankly it will help you

00:15:10.100 --> 00:15:11.612
not just in your academics.

00:15:11.612 --> 00:15:14.206
I think it will help you
be a more creative person.

00:15:14.206 --> 00:15:16.583
That's frankly what really,
really, really matters

00:15:16.583 --> 00:15:19.649
is the ability to see
things where maybe other

00:15:19.649 --> 00:15:21.211
people aren't seeing things,

00:15:21.211 --> 00:15:22.528
see better ways of doing things,

00:15:22.528 --> 00:15:23.875
seeing innovations, seeing connections

00:15:23.875 --> 00:15:25.539
that other people aren't seeing.

00:15:25.539 --> 00:15:27.542
I would say ponder as much as you can,

00:15:27.542 --> 00:15:29.660
learn as deeply as you can,

00:15:29.660 --> 00:15:31.415
make sure that your foundations are as

00:15:31.415 --> 00:15:37.669
strong as you can and after
that, if there is the mundane

00:15:37.669 --> 00:15:40.337
aspects and there are mundane
aspects of our current

00:15:40.337 --> 00:15:43.827
school systems and our
current exams, just view

00:15:43.827 --> 00:15:46.436
that as a game, as a
game that lets you get

00:15:46.436 --> 00:15:48.299
to the good stuff and
that's how I viewed it.

00:15:48.299 --> 00:15:50.622
There are definitely
moments in my own education

00:15:50.622 --> 00:15:51.729
where I was like this is
silly, they're making me

00:15:51.729 --> 00:15:55.259
do this or that, but I
said I have to do this if I

00:15:55.259 --> 00:15:58.068
want to get to a point
where I get to do more

00:15:58.068 --> 00:16:00.663
pondering and I get to look
at the really cool stuff.

00:16:00.663 --> 00:16:06.252
Yes, I would say ponder
as much as you can,

00:16:06.252 --> 00:16:08.207
so that you can really
distill things to what's

00:16:08.207 --> 00:16:10.935
most important and then
the stuff that's still not

00:16:10.935 --> 00:16:15.677
fun after that, just
remind yourself it's a bit

00:16:15.677 --> 00:16:19.758
of a marathon and it's
going to get you to where

00:16:19.758 --> 00:16:21.754
you're going so that your
life will be more fun at

00:16:21.754 --> 00:16:24.322
some point and hopefully it's already

00:16:24.322 --> 00:16:26.902
a little fun right now.


2011 Calculus BC free response #6c

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:04.870
Part C. Find the value
of the sixth derivative

00:00:04.870 --> 00:00:07.880
of f evaluated at 0.

00:00:07.880 --> 00:00:10.320
So you could imagine
if you just tried

00:00:10.320 --> 00:00:13.470
to find the sixth derivative of
f, that would take you forever.

00:00:13.470 --> 00:00:16.512
And then to evaluate it at 0,
because this is x squared here.

00:00:16.512 --> 00:00:18.970
And you'd have to keep doing
the product rule over and over

00:00:18.970 --> 00:00:20.761
again, and the chain
rule and all the rest.

00:00:20.761 --> 00:00:22.540
It would become very,
very, very messy.

00:00:22.540 --> 00:00:24.000
But we have a big clue here.

00:00:24.000 --> 00:00:26.420
The fact that they made
us find the first four

00:00:26.420 --> 00:00:29.890
terms of the Taylor series
of f about x equals 0

00:00:29.890 --> 00:00:31.600
tells us that there
might be a simpler

00:00:31.600 --> 00:00:33.870
way to do this, as
opposed to just taking

00:00:33.870 --> 00:00:36.800
the sixth derivative of
this and evaluating at 0.

00:00:36.800 --> 00:00:39.380
The simplest way to do
this is to just go back.

00:00:39.380 --> 00:00:41.590
In the last problem,
we were able to come up

00:00:41.590 --> 00:00:44.860
with the first four
non-zero terms of the Taylor

00:00:44.860 --> 00:00:46.670
series of f.

00:00:46.670 --> 00:00:51.960
And if you look at your
definition of the Taylor series

00:00:51.960 --> 00:00:54.510
right here-- and we go into
depth on this in another Khan

00:00:54.510 --> 00:00:58.030
Academy video where we talk
about why this makes sense--

00:00:58.030 --> 00:01:00.360
you see that each degree
term of the Taylor

00:01:00.360 --> 00:01:04.637
series, its coefficient
is that derivative.

00:01:04.637 --> 00:01:06.470
And this Taylor Series
is centered around 0,

00:01:06.470 --> 00:01:09.770
and that's what we care about
in terms of this problem.

00:01:09.770 --> 00:01:14.270
We see the coefficient is
that derivative divided

00:01:14.270 --> 00:01:17.860
by that degrees, that
derivative evaluated at 0

00:01:17.860 --> 00:01:21.080
divided by that
degrees factorials.

00:01:21.080 --> 00:01:24.080
So the second degree term,
it's the second derivative

00:01:24.080 --> 00:01:27.250
of f evaluated at 0
divided by 2 factorial.

00:01:27.250 --> 00:01:32.330
The fourth degree term
is the fourth derivative

00:01:32.330 --> 00:01:35.950
of f evaluated at 0
divided by 4 factorial.

00:01:35.950 --> 00:01:42.490
So the sixth degree term--
let's remind ourselves

00:01:42.490 --> 00:01:44.270
what we're even
trying to figure out--

00:01:44.270 --> 00:01:47.080
so they want us to figure
out the sixth derivative of f

00:01:47.080 --> 00:01:48.440
evaluated at 0.

00:01:48.440 --> 00:01:50.540
That's what they want
us to figure out.

00:01:50.540 --> 00:01:54.320
Well, if you think about
the Taylor series centered

00:01:54.320 --> 00:01:59.290
at 0, or at 0, or
approximated around 0,

00:01:59.290 --> 00:02:02.220
the sixth degree term in the
Taylor series approximation

00:02:02.220 --> 00:02:09.310
of f is going to be f prime
of the sixth derivative of f

00:02:09.310 --> 00:02:16.640
evaluated at 0 times x to
the sixth over 6 factorial.

00:02:16.640 --> 00:02:18.910
This is going to
be the sixth degree

00:02:18.910 --> 00:02:25.875
term in Taylor approximation,
in Taylor series.

00:02:28.454 --> 00:02:30.370
And we have that term
sitting right over here.

00:02:30.370 --> 00:02:31.620
This is the sixth degree term.

00:02:31.620 --> 00:02:33.670
We figured it out
in the last problem.

00:02:33.670 --> 00:02:36.790
This right here is
the sixth degree term.

00:02:36.790 --> 00:02:39.500
So you have x to the sixth over
here, x to the sixth over here,

00:02:39.500 --> 00:02:43.020
you have 6 factorial over
here, 6 factorial over here.

00:02:43.020 --> 00:02:49.610
So this negative 121 must
be the sixth derivative

00:02:49.610 --> 00:02:51.890
of f evaluated at 0.

00:02:51.890 --> 00:02:52.800
So that's our answer.

00:02:52.800 --> 00:02:56.120
This is equal to negative 121.

00:02:56.120 --> 00:02:57.850
And we're done.