2011 Calculus BC free response #6b

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:01.770
We're on part b.

00:00:01.770 --> 00:00:05.940
Write the first four non-zero
terms of the Taylor series

00:00:05.940 --> 00:00:09.200
for cosine of x
about x equals 0.

00:00:09.200 --> 00:00:12.420
Use this series and the
series for sine of x squared,

00:00:12.420 --> 00:00:15.440
found in part A, to write
the first four non-zero terms

00:00:15.440 --> 00:00:17.990
of the Taylor series
for f-- so, for this f

00:00:17.990 --> 00:00:20.340
right over here--
about x equals 0.

00:00:20.340 --> 00:00:22.260
So let's just do the first part.

00:00:22.260 --> 00:00:25.450
Let's find the first four
non-zero terms of the Taylor

00:00:25.450 --> 00:00:29.240
series for cosine of
x about x equals 0.

00:00:29.240 --> 00:00:33.820
So let's just say-- well, we'll
put our formula for the Taylor

00:00:33.820 --> 00:00:36.780
series up here around
x is equal to 0.

00:00:36.780 --> 00:00:39.320
So we're centering it around
x equals 0 right over here.

00:00:39.320 --> 00:00:46.520
And so if we say that h of x
is equal to cosine of x, then

00:00:46.520 --> 00:00:51.550
h of 0, is going to
be equal to cosine

00:00:51.550 --> 00:00:53.860
of 0, which is equal to 1.

00:00:53.860 --> 00:00:56.460
And then, if you
have h prime of x,

00:00:56.460 --> 00:00:58.842
that's going to be equal
to negative sine of x.

00:00:58.842 --> 00:01:00.050
And you could look over here.

00:01:00.050 --> 00:01:01.270
This is essentially h of x.

00:01:01.270 --> 00:01:02.870
And this is going
to be h prime of x.

00:01:02.870 --> 00:01:04.459
So it's essentially
one step ahead

00:01:04.459 --> 00:01:06.090
of where we were with g of x.

00:01:06.090 --> 00:01:09.570
Well, we'll just redo it
over here just so it's clear.

00:01:09.570 --> 00:01:12.900
So the first derivative
at 0 is negative sine

00:01:12.900 --> 00:01:14.830
of 0, which is just 0.

00:01:14.830 --> 00:01:21.344
And then you have the
second derivative of h of x.

00:01:21.344 --> 00:01:23.260
So the derivative of
sine of x is cosine of x.

00:01:23.260 --> 00:01:24.760
But we have that
negative out front,

00:01:24.760 --> 00:01:26.760
so it's negative cosine of x.

00:01:26.760 --> 00:01:31.210
So h prime prime of 0 is going
to be equal to cosine of 0

00:01:31.210 --> 00:01:31.796
is 1.

00:01:31.796 --> 00:01:33.170
You have that
negative out front.

00:01:33.170 --> 00:01:34.650
So it's negative 1.

00:01:34.650 --> 00:01:39.740
And then you have the
third derivative of h

00:01:39.740 --> 00:01:42.680
is equal to derivative
of negative cosine of x

00:01:42.680 --> 00:01:46.080
is positive sine of x.

00:01:46.080 --> 00:01:52.630
So the third derivative at
0 is going to be equal to 0.

00:01:52.630 --> 00:01:54.410
And then you take the
derivative of this,

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you're going to get
cosine of x again.

00:01:56.290 --> 00:01:57.650
So then the cycle starts again.

00:01:57.650 --> 00:02:05.920
So I could write, this is also
equal to the fourth derivative

00:02:05.920 --> 00:02:06.460
at 0.

00:02:06.460 --> 00:02:08.960
Right, you take-- let me just
make it clear what I'm saying.

00:02:08.960 --> 00:02:11.430
If I take the derivative
of this, I get cosine of x.

00:02:11.430 --> 00:02:14.037
The fourth derivative of
h is cosine of x again.

00:02:14.037 --> 00:02:15.620
So the fourth
derivative at 0 is going

00:02:15.620 --> 00:02:17.786
to be the same thing as the
function evaluated at 0,

00:02:17.786 --> 00:02:18.720
which is 1.

00:02:18.720 --> 00:02:23.860
And so the fifth derivative
at 0 is going to be 0.

00:02:23.860 --> 00:02:27.780
The sixth derivative at 0
is going to be negative 1.

00:02:27.780 --> 00:02:31.400
And then the seventh
the derivative at 0

00:02:31.400 --> 00:02:33.070
is going to be 0.

00:02:33.070 --> 00:02:34.240
So that's enough, now.

00:02:34.240 --> 00:02:37.050
I think we definitely
have four non-zero terms.

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We have this term-- let me
pick a nice color here--

00:02:40.450 --> 00:02:42.870
we have this term, because
we have a 1 coefficient.

00:02:42.870 --> 00:02:45.390
This is going to be a 0 because
we have a 0 coefficient.

00:02:45.390 --> 00:02:46.870
This is a non-zero term.

00:02:46.870 --> 00:02:49.470
Then this is a non-zero term
because we have the 1 again.

00:02:49.470 --> 00:02:53.040
And then this is a non-zero term
because we have the negative 1.

00:02:53.040 --> 00:02:59.430
So we could say that cosine
of x is approximately

00:02:59.430 --> 00:03:01.600
equal to-- because we're
just approximating it

00:03:01.600 --> 00:03:04.360
with the first four
non-zero terms.

00:03:04.360 --> 00:03:08.130
So f of 0 is 1.

00:03:08.130 --> 00:03:10.010
So that's this term
right over here.

00:03:10.010 --> 00:03:12.121
Instead of f of x,
we're calling it

00:03:12.121 --> 00:03:14.120
h of x right here, so
that we don't get confused

00:03:14.120 --> 00:03:16.117
with that original f
defined in the problem,

00:03:16.117 --> 00:03:18.200
and that we're going to
work on in the second part

00:03:18.200 --> 00:03:20.630
of this section of the problem.

00:03:20.630 --> 00:03:23.620
So cosine of x is equal to 1.

00:03:23.620 --> 00:03:26.850
h prime of 0 is 0, so
there's no term there.

00:03:26.850 --> 00:03:30.240
Then f prime prime of 0, second
derivative, it's negative 1.

00:03:30.240 --> 00:03:32.760
So it's negative x
squared over 2 factorial.

00:03:35.920 --> 00:03:38.740
And then you get another 0
for the third degree term.

00:03:38.740 --> 00:03:40.860
Then you get a 1 for
the fourth degree term.

00:03:40.860 --> 00:03:42.030
So plus 1.

00:03:42.030 --> 00:03:44.760
But we don't have to write
the 1, it's implicitly there.

00:03:44.760 --> 00:03:48.930
Plus 1 times x to the
4th over 4 factorial.

00:03:48.930 --> 00:03:52.720
And then the next non-zero term
is this negative 1 over here.

00:03:52.720 --> 00:03:55.360
So minus 1-- we can
just write minus--

00:03:55.360 --> 00:03:57.080
and that's the
sixth degree term.

00:03:57.080 --> 00:04:00.216
So x to the sixth
over 6 factorial.

00:04:00.216 --> 00:04:00.840
And we're done.

00:04:00.840 --> 00:04:04.660
Those are the first four
non-zero terms of cosine of x.

00:04:04.660 --> 00:04:07.787
Now let's do the second
part of the problem.

00:04:07.787 --> 00:04:08.870
So we did this first part.

00:04:08.870 --> 00:04:11.930
Use this series and the
series for sine of x squared,

00:04:11.930 --> 00:04:15.190
which we found right over
here, sine of x squared--

00:04:15.190 --> 00:04:17.820
actually, I think
we wrote it, this

00:04:17.820 --> 00:04:21.470
is the series for sine of x
squared when we simplified it.

00:04:21.470 --> 00:04:25.600
Use those-- that was
found in part A--

00:04:25.600 --> 00:04:28.220
to write the first four non-zero
terms of the Taylor series

00:04:28.220 --> 00:04:30.510
for f about x equals 0.

00:04:30.510 --> 00:04:31.830
So where's our function again?

00:04:31.830 --> 00:04:34.140
So our function is sine of
x squared plus cosine of x.

00:04:34.140 --> 00:04:35.670
Let me rewrite that.

00:04:35.670 --> 00:04:47.230
So f of x is equal to sine of
x squared plus cosine of x.

00:04:47.230 --> 00:04:49.806
So let's write the first
four non-zero terms.

00:04:49.806 --> 00:04:51.680
And you want to start
with the lowest degree.

00:04:51.680 --> 00:04:53.749
So you could say
that f of x is going

00:04:53.749 --> 00:04:55.790
to be approximately equal
to-- because we're just

00:04:55.790 --> 00:04:58.970
approximating it with this
polynomial right over here.

00:04:58.970 --> 00:05:04.930
So, from cosine of x-- so
you have this right over here

00:05:04.930 --> 00:05:06.540
is cosine of x.

00:05:06.540 --> 00:05:07.950
And sine of x is over there.

00:05:07.950 --> 00:05:10.750
So between both of these,
the lowest degree term

00:05:10.750 --> 00:05:12.900
right over here is this 1.

00:05:12.900 --> 00:05:15.450
So I'll write this 1 over here.

00:05:15.450 --> 00:05:20.360
And then after that, the next
lowest-- in either of these,

00:05:20.360 --> 00:05:21.700
we have no first degree terms.

00:05:21.700 --> 00:05:23.340
We have no x terms.

00:05:23.340 --> 00:05:26.340
But then, in both of these,
we have an x squared term.

00:05:26.340 --> 00:05:29.375
Actually, let me write it this
way to simplify what I'm doing.

00:05:29.375 --> 00:05:30.500
Let me just write them out.

00:05:30.500 --> 00:05:34.280
So sine of x squared-- let
me do this in a new color.

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Sine of x squared is this thing.

00:05:36.390 --> 00:05:38.020
This is what we
figured out in part A.

00:05:38.020 --> 00:05:41.454
So that is x squared minus
x to the sixth over 3

00:05:41.454 --> 00:05:43.620
factorial-- I'm going to
just write the whole thing,

00:05:43.620 --> 00:05:46.000
and then later we'll pick
out the first four terms--

00:05:46.000 --> 00:05:52.000
plus x to the 10th over 5
factorial minus x to the 14th

00:05:52.000 --> 00:05:53.580
over 7 factorial.

00:05:53.580 --> 00:05:56.186
So that's the first
four terms of that.

00:05:56.186 --> 00:05:57.310
So this isn't exactly that.

00:05:57.310 --> 00:05:58.768
That's approximately
equal to that.

00:05:58.768 --> 00:06:00.060
So approximation.

00:06:00.060 --> 00:06:03.250
And then cosine of x,
we just figured out,

00:06:03.250 --> 00:06:09.600
is plus 1 minus x squared
over 2 factorial plus x

00:06:09.600 --> 00:06:15.210
to the 4th over 4 factorial
minus x to the 6th over 6

00:06:15.210 --> 00:06:16.400
factorial.

00:06:16.400 --> 00:06:19.130
Now we can pick out the
first four non-zero terms.

00:06:19.130 --> 00:06:21.172
And since we're assuming
this is a Taylor series,

00:06:21.172 --> 00:06:23.171
the first term is going
to be the lowest degree,

00:06:23.171 --> 00:06:25.750
then we're going to keep
going at higher degree terms.

00:06:25.750 --> 00:06:28.070
So the lowest degree
term right over here.

00:06:28.070 --> 00:06:32.230
So once again, f of x is going
to be approximately equal to.

00:06:32.230 --> 00:06:35.090
The lowest degree term
over here is this 1.

00:06:35.090 --> 00:06:36.782
So let's write the 1 over here.

00:06:36.782 --> 00:06:38.740
And then, we don't have
any first degree terms.

00:06:38.740 --> 00:06:40.781
We don't how many just x
to the firsts over here.

00:06:40.781 --> 00:06:42.570
But we have a couple
of x squareds.

00:06:42.570 --> 00:06:44.390
So that will actually
be the next term,

00:06:44.390 --> 00:06:46.140
because we can add
those x squareds.

00:06:46.140 --> 00:06:49.167
So this right over here,
you have x squared-- so let

00:06:49.167 --> 00:06:50.250
me just write it this way.

00:06:50.250 --> 00:06:57.160
So plus this x squared minus
x squared over 2 factorial.

00:06:57.160 --> 00:07:00.270
And x squared over 2 factorial
is just x squared over 2.

00:07:00.270 --> 00:07:03.670
So minus, I could
write 1/2 x squared.

00:07:03.670 --> 00:07:05.620
And I'll simplify
this in the next step.

00:07:05.620 --> 00:07:07.536
That's just going to be
one term, because this

00:07:07.536 --> 00:07:10.310
is going to end up
being 1/2 x squared.

00:07:10.310 --> 00:07:10.810
All right.

00:07:10.810 --> 00:07:12.720
Do we have any third
degree terms here?

00:07:12.720 --> 00:07:14.940
We do not have any
third degree terms.

00:07:14.940 --> 00:07:17.130
Do we have any fourth
degree terms here?

00:07:17.130 --> 00:07:17.860
We do.

00:07:17.860 --> 00:07:20.970
We have this character
right over here.

00:07:20.970 --> 00:07:22.470
It's a fourth degree term.

00:07:22.470 --> 00:07:26.025
So plus x to the 4th
over 4 factorial.

00:07:29.040 --> 00:07:33.270
And then finally, do we
have any 5th degree terms?

00:07:33.270 --> 00:07:35.430
We don't have any 5th
degree terms, no x

00:07:35.430 --> 00:07:36.450
to the 5ths over here.

00:07:36.450 --> 00:07:38.070
Do we have any 6th degree terms?

00:07:38.070 --> 00:07:38.800
We do.

00:07:38.800 --> 00:07:43.610
We have this one and this one.

00:07:43.610 --> 00:07:49.089
So this is going to be minus--
there's a negative sign out

00:07:49.089 --> 00:07:50.130
in front of both of them.

00:07:50.130 --> 00:07:53.900
So it's going to be minus x
to the 6th over 3 factorial

00:07:53.900 --> 00:07:57.400
plus x to the 6th
over 6 factorial.

00:07:57.400 --> 00:07:59.770
You could say it's minus x
to the 6th over 3 factorial,

00:07:59.770 --> 00:08:02.269
and then you could distribute
the negative sign if you want.

00:08:02.269 --> 00:08:03.530
I just factored it out.

00:08:03.530 --> 00:08:06.320
And now we can just simplify
this thing right over here.

00:08:06.320 --> 00:08:10.860
So this is going to be equal
to-- so f of x is approximately

00:08:10.860 --> 00:08:16.330
equal to-- 1 plus-- and then,
if you have an x squared

00:08:16.330 --> 00:08:18.560
minus half of an x squared,
you're going to have 1/2

00:08:18.560 --> 00:08:19.210
x squared.

00:08:19.210 --> 00:08:23.390
Or I could just write x squared
over 2, plus x squared over 2.

00:08:23.390 --> 00:08:28.860
And then you have, in that blue
color, plus x to the 4th over 4

00:08:28.860 --> 00:08:30.150
factorial.

00:08:30.150 --> 00:08:36.130
And then, to simplify this,
3 factorial 6 factorial.

00:08:36.130 --> 00:08:39.059
So this 6 factorial
is the same thing,

00:08:39.059 --> 00:08:41.450
just to make
everything clear to us.

00:08:41.450 --> 00:08:46.940
6 factorial is the same thing
as 3 factorial times 4 times

00:08:46.940 --> 00:08:53.410
5 times 6, or 3
factorial times 120.

00:08:53.410 --> 00:08:55.390
And so what we can do is
multiply the numerator

00:08:55.390 --> 00:09:00.370
and the denominator here by
4 times 5 times 6, or by 120.

00:09:00.370 --> 00:09:01.710
So let me just rewrite this.

00:09:01.710 --> 00:09:03.580
So I'll do this
over here because I

00:09:03.580 --> 00:09:05.540
want this to be my final answer.

00:09:05.540 --> 00:09:10.080
So if I have x to the
sixth over 3 factorial,

00:09:10.080 --> 00:09:13.930
I can multiply the
denominator by 4 times 5 times

00:09:13.930 --> 00:09:16.750
6, and then multiply the
numerator times 4 times 5 times

00:09:16.750 --> 00:09:20.580
6, which is the same thing
as multiplying it times 120.

00:09:20.580 --> 00:09:22.440
So you have 120 up there.

00:09:22.440 --> 00:09:24.729
4 times 5 times 6 is also 120.

00:09:24.729 --> 00:09:26.270
And now we have the
same denominator,

00:09:26.270 --> 00:09:28.620
because this is the same
thing as 6 factorial.

00:09:28.620 --> 00:09:31.320
So you have 120 x to
the 6 over 6 factorial

00:09:31.320 --> 00:09:37.290
plus x to the 6th over 6
factorial gives you 121 x

00:09:37.290 --> 00:09:40.570
to the sixth over 6 factorial.

00:09:40.570 --> 00:09:42.470
And we have this
negative sign out here.

00:09:42.470 --> 00:09:50.550
So it becomes negative 121 x
to the sixth over 6 factorial.

00:09:50.550 --> 00:09:51.740
And we're done.

00:09:51.740 --> 00:09:58.630
We've figured out the first four
non-zero terms of our f of x.


2011 Calculus BC free response #6a

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:01.550
Problem number six.

00:00:01.550 --> 00:00:05.850
Let f of x is equal to sine
of x squared plus cosine of x.

00:00:05.850 --> 00:00:08.010
The graph of y is equal
to the absolute value

00:00:08.010 --> 00:00:11.479
of the fifth derivative
of f at x is shown above.

00:00:11.479 --> 00:00:13.770
And I haven't shown it here
just so we have some space.

00:00:13.770 --> 00:00:15.311
I'll show it when
we need to show it.

00:00:15.311 --> 00:00:16.920
I think we have to
show it in part D.

00:00:16.920 --> 00:00:19.440
So first, let's do
part A right over here.

00:00:19.440 --> 00:00:23.130
Write the first non-zero
terms for the Taylor series

00:00:23.130 --> 00:00:26.000
for sine of x about
x is equal to 0.

00:00:26.000 --> 00:00:29.070
And write the first for non-zero
terms of the Taylor series

00:00:29.070 --> 00:00:31.270
for sine of x squared
about x equals 0.

00:00:31.270 --> 00:00:32.740
So let's do this first part.

00:00:32.740 --> 00:00:34.460
And just a reminder,
a Taylor Series

00:00:34.460 --> 00:00:36.840
is a polynomial
approximation of a function.

00:00:36.840 --> 00:00:39.560
So just to give us
a quick reminder.

00:00:39.560 --> 00:00:42.380
We go into much
more detail in this

00:00:42.380 --> 00:00:44.530
on the videos about
Taylor series.

00:00:44.530 --> 00:00:46.887
But if you have a function
that looks like this,

00:00:46.887 --> 00:00:49.220
and if you wanted to approximate
it with a Taylor series

00:00:49.220 --> 00:00:53.480
around 0, if you only have one
term in your Taylor series,

00:00:53.480 --> 00:00:55.330
you would just
literally have it--

00:00:55.330 --> 00:00:57.450
it would just be a
constant, just like that.

00:00:57.450 --> 00:00:59.520
If you have two terms
in your Taylor series

00:00:59.520 --> 00:01:01.660
it would be a line
that looks like that.

00:01:01.660 --> 00:01:03.660
If you have three terms
in your Taylor series,

00:01:03.660 --> 00:01:05.140
you'd get to a
second degree term.

00:01:05.140 --> 00:01:08.580
And you'd start approximating
it something like that.

00:01:08.580 --> 00:01:10.555
If you get to the
third degree term

00:01:10.555 --> 00:01:11.930
it might start
looking like that.

00:01:11.930 --> 00:01:13.388
And as you add more
and more terms,

00:01:13.388 --> 00:01:15.740
you get a better, better
approximation of your function.

00:01:15.740 --> 00:01:17.156
And if you add an
infinite number,

00:01:17.156 --> 00:01:19.080
it might actually
converge to your function.

00:01:19.080 --> 00:01:20.920
So let's just remind
ourselves, if I

00:01:20.920 --> 00:01:24.700
have a function-- if I have f
of x, this Taylor series-- I can

00:01:24.700 --> 00:01:26.410
approximate it with
this Taylor series.

00:01:26.410 --> 00:01:29.600
And if we want to center
that approximation around 0,

00:01:29.600 --> 00:01:34.440
it'll be equal to f of
0 plus f prime of 0.

00:01:34.440 --> 00:01:38.145
The derivative at 0 times x
plus the second derivative

00:01:38.145 --> 00:01:42.289
at 0 times x squared
over 2 factorial.

00:01:42.289 --> 00:01:44.080
You could have divided
this term right here

00:01:44.080 --> 00:01:45.700
by one factorial,
which is just 1.

00:01:45.700 --> 00:01:47.580
Two factorials is just 2.

00:01:47.580 --> 00:01:51.673
And then plus the third
derivative of f at 0

00:01:51.673 --> 00:01:55.290
times x to the third
over 3 factorial.

00:01:55.290 --> 00:01:58.480
Plus the fourth derivative-- I
think you get the idea here--

00:01:58.480 --> 00:02:00.720
at the fourth
derivative at 0 times

00:02:00.720 --> 00:02:03.890
x to the fourth
divided by 4 factorial,

00:02:03.890 --> 00:02:05.900
and so on and so forth.

00:02:05.900 --> 00:02:08.060
So what they want us to
do is find the first four

00:02:08.060 --> 00:02:11.524
non-zero terms of the
Taylor series for sine of x.

00:02:11.524 --> 00:02:13.190
And some of you might
already know this.

00:02:13.190 --> 00:02:14.970
We actually cover
this in the video

00:02:14.970 --> 00:02:17.600
where we show Euler's identity.

00:02:17.600 --> 00:02:20.030
But we'll do it again
right over here.

00:02:20.030 --> 00:02:23.400
So if we just
take-- I'll call it

00:02:23.400 --> 00:02:26.270
g of x, because they already
defined f of x up here.

00:02:26.270 --> 00:02:28.990
So let's say that g of x--
let me do this in a new color

00:02:28.990 --> 00:02:31.020
just to ease the monotony.

00:02:31.020 --> 00:02:36.400
Let's say that g of x
is equal to sine of x.

00:02:36.400 --> 00:02:41.010
Then we know that g of 0
is going to be equal to 0.

00:02:41.010 --> 00:02:43.020
And then if we take
the derivative, g

00:02:43.020 --> 00:02:46.200
prime of x is going to be
equal to negative-- no, it's

00:02:46.200 --> 00:02:49.570
going to be positive
cosine of x.

00:02:49.570 --> 00:02:54.350
g prime of 0 is now
going to be equal to 1.

00:02:54.350 --> 00:02:56.230
Cosine of 0 is 1.

00:02:56.230 --> 00:02:58.520
Then if you take the
second derivative,

00:02:58.520 --> 00:03:01.700
the derivative of cosine
of x is negative sine of x.

00:03:01.700 --> 00:03:05.250
Negative sine of x and
the second derivative at 0

00:03:05.250 --> 00:03:08.600
once again is going to
equal zero, sine of 0, 0.

00:03:08.600 --> 00:03:10.920
And now let's take
the third derivative.

00:03:10.920 --> 00:03:15.350
Third derivative of
our function of g.

00:03:15.350 --> 00:03:18.950
The derivative of negative sine
of x is negative cosine of x.

00:03:18.950 --> 00:03:21.840
And the third
derivative at 0 is now

00:03:21.840 --> 00:03:24.260
going to be equal to negative 1.

00:03:24.260 --> 00:03:25.720
And we could keep going.

00:03:25.720 --> 00:03:28.330
You can already guess
where this might lead to,

00:03:28.330 --> 00:03:30.720
but I'll just keep
going just in case.

00:03:30.720 --> 00:03:33.720
The fourth derivative
is once again

00:03:33.720 --> 00:03:36.970
going to be equal to sine of x.

00:03:36.970 --> 00:03:39.040
It is now going to be
equal to sine of x.

00:03:39.040 --> 00:03:42.550
And then the fourth
derivative at 0,

00:03:42.550 --> 00:03:44.710
because it's the same
thing as the function,

00:03:44.710 --> 00:03:46.820
is now going to be 0 as well.

00:03:46.820 --> 00:03:51.522
So this is going to be equal
to g to the fourth at 0.

00:03:51.522 --> 00:03:52.730
And then we could keep going.

00:03:52.730 --> 00:03:55.230
This is going to be the same
thing as g to the fifth,

00:03:55.230 --> 00:03:58.100
because we start cycling as we
take more and more derivatives.

00:03:58.100 --> 00:03:59.641
So this is going to
be the same thing

00:03:59.641 --> 00:04:01.194
as the fifth derivative at 0.

00:04:01.194 --> 00:04:02.610
This is going to
be the same thing

00:04:02.610 --> 00:04:07.700
as the sixth derivative at 0.

00:04:07.700 --> 00:04:11.020
And this is going
to be the same.

00:04:11.020 --> 00:04:12.887
This is an equal sign
right here, equals 1.

00:04:12.887 --> 00:04:14.470
And this is going
to be the same thing

00:04:14.470 --> 00:04:17.630
as the seventh derivative at 0.

00:04:17.630 --> 00:04:21.416
So if you want the first
four non-zero terms,

00:04:21.416 --> 00:04:22.540
let's just work through it.

00:04:22.540 --> 00:04:26.740
So first of all, f of 0--
I'll do this in a new color.

00:04:26.740 --> 00:04:28.510
If we want to
approximate g of x,

00:04:28.510 --> 00:04:30.640
if we want to
approximate sine of x.

00:04:30.640 --> 00:04:34.380
So we'll say sine of x is going
to be approximately equal to.

00:04:34.380 --> 00:04:37.320
So this first term right
over here, that is sine of 0.

00:04:37.320 --> 00:04:38.220
That's g of 0.

00:04:38.220 --> 00:04:40.620
That's going to be 0, so we
don't even have to write it.

00:04:40.620 --> 00:04:42.650
Then we go to this
term right over here.

00:04:42.650 --> 00:04:45.730
Well, the first derivative, f
prime of 0-- or in this case,

00:04:45.730 --> 00:04:49.320
g prime of 0-- is
going to be equal to 1.

00:04:49.320 --> 00:04:50.660
So it's going to be 1 times x.

00:04:50.660 --> 00:04:52.510
So you're going to
have an x there.

00:04:52.510 --> 00:04:55.900
Then the next term
is going to be 0.

00:04:55.900 --> 00:04:57.490
We see that right over there.

00:04:57.490 --> 00:05:01.620
Because the second derivative
of our function evaluated at 0

00:05:01.620 --> 00:05:02.920
is 0.

00:05:02.920 --> 00:05:06.580
Our third derivative of our
function evaluated at 0 is 1.

00:05:06.580 --> 00:05:09.030
So that term is going
to show up again.

00:05:09.030 --> 00:05:10.875
Actually, that
term is negative 1.

00:05:10.875 --> 00:05:12.570
I don't want to
make a mistake here.

00:05:12.570 --> 00:05:14.710
That is negative 1, negative 1.

00:05:14.710 --> 00:05:16.120
It was negative cosine of x.

00:05:16.120 --> 00:05:19.320
You evaluate negative cosine
of x and 0, you get negative 1

00:05:19.320 --> 00:05:20.620
right over here.

00:05:20.620 --> 00:05:22.800
So the third derivative,
this thing right over here,

00:05:22.800 --> 00:05:24.250
is negative 1.

00:05:24.250 --> 00:05:29.050
So then we have minus x to
the third over 3 factorial.

00:05:29.050 --> 00:05:31.660
The fourth derivative,
once again, is 0.

00:05:31.660 --> 00:05:35.410
The fifth derivative
evaluated at 0 is 1.

00:05:35.410 --> 00:05:38.330
So then you're going
to have plus 1 times x

00:05:38.330 --> 00:05:40.940
to the fifth over 5 factorial.

00:05:40.940 --> 00:05:42.522
Then the six derivative is 0.

00:05:42.522 --> 00:05:43.980
So that term is
going to disappear.

00:05:43.980 --> 00:05:45.420
I didn't even write it up here.

00:05:45.420 --> 00:05:48.425
And then the seventh term,
the coefficient is negative 1.

00:05:48.425 --> 00:05:51.590
Or the seventh derivative
evaluated at 0 is negative 1.

00:05:51.590 --> 00:05:55.880
So you have negative 1 times
x to the seventh over 7

00:05:55.880 --> 00:05:57.030
factorial.

00:05:57.030 --> 00:05:59.990
And we have to go all the
way to the seven degree term

00:05:59.990 --> 00:06:04.550
to find the first four non-zero
terms for our Taylor Series.

00:06:04.550 --> 00:06:07.120
And so we're done with
at least the first part.

00:06:07.120 --> 00:06:11.650
We found the first four
non-zero terms of sine of x Now,

00:06:11.650 --> 00:06:16.216
what about sine of x squared?

00:06:16.216 --> 00:06:17.340
We have to be careful here.

00:06:17.340 --> 00:06:19.750
Because you might just say, OK,
let me just apply this formula.

00:06:19.750 --> 00:06:20.940
And what you're going
to find very quickly

00:06:20.940 --> 00:06:23.410
is when you start taking the
second and third derivatives

00:06:23.410 --> 00:06:26.170
of this thing right over here,
it's going to get really messy.

00:06:26.170 --> 00:06:28.570
But what you can say
is, look, sine of x

00:06:28.570 --> 00:06:30.610
is approximately this
thing right over here.

00:06:30.610 --> 00:06:33.195
What happens if I just replace
the x with the x squared?

00:06:36.480 --> 00:06:38.740
Then I get sine of x
squared is approximately

00:06:38.740 --> 00:06:41.700
equal to-- instead of an x here,
I'm going to put an x squared.

00:06:41.700 --> 00:06:43.570
Instead of an x
to the third here,

00:06:43.570 --> 00:06:46.120
I'm going to put an x
squared to the third

00:06:46.120 --> 00:06:48.380
over here, over 3 factorial.

00:06:48.380 --> 00:06:49.690
Instead of an x to the fifth.

00:06:49.690 --> 00:06:54.204
I can put an x squared to the
fifth power over 5 factorial.

00:06:54.204 --> 00:06:55.620
And instead of an
x to the seventh

00:06:55.620 --> 00:07:00.880
I could write x squared to the
seventh power over 7 factorial.

00:07:00.880 --> 00:07:02.770
So this is a very
important thing to realize.

00:07:02.770 --> 00:07:05.210
Because if you had
started directly

00:07:05.210 --> 00:07:08.309
take the Taylor Series around
0 of this thing right here,

00:07:08.309 --> 00:07:09.850
you would have taken
up all your time

00:07:09.850 --> 00:07:11.110
trying to take the derivatives.

00:07:11.110 --> 00:07:12.600
And you probably wouldn't have
been able to do it anyway,

00:07:12.600 --> 00:07:14.350
because it would have
gotten really messy.

00:07:14.350 --> 00:07:17.660
And the key here is just to
realize that you substitute

00:07:17.660 --> 00:07:19.420
x squared for x,
you're then going

00:07:19.420 --> 00:07:23.550
to get the approximation
for sine of x squared.

00:07:23.550 --> 00:07:25.105
We could simplify
this a little bit.

00:07:25.105 --> 00:07:26.480
This is going to
be approximately

00:07:26.480 --> 00:07:30.510
equal to x squared minus
x to the sixth over 3

00:07:30.510 --> 00:07:33.640
factorial plus x
to the 10th over 5

00:07:33.640 --> 00:07:39.030
factorial minus x to the
14th over 7 factorial.

00:07:39.030 --> 00:07:42.030
So that's our second
part of the problem.


2011 Calculus BC free response #3 (b & c)

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.070
We're on part b.

00:00:02.070 --> 00:00:05.960
The region R is rotated about
the x-axis to form a solid.

00:00:05.960 --> 00:00:09.120
Find the volume, V, of
the solid in terms of k.

00:00:09.120 --> 00:00:11.060
So this is our
region R, and it's

00:00:11.060 --> 00:00:13.680
going to be rotated
around the x-axis.

00:00:13.680 --> 00:00:17.120
It's going to be rotated around
the x-axis to form a solid.

00:00:17.120 --> 00:00:19.860
So that solid is going to
look something like this.

00:00:19.860 --> 00:00:21.110
It's going to look like this.

00:00:21.110 --> 00:00:22.818
This is going to be
this end of the solid

00:00:22.818 --> 00:00:25.350
when it's rotated
around, and it's

00:00:25.350 --> 00:00:27.760
going to look
something like this.

00:00:27.760 --> 00:00:32.540
Let me do my best attempt
to draw our solid.

00:00:32.540 --> 00:00:35.340
And then that side is going
to look something like that.

00:00:35.340 --> 00:00:37.830
It'll look a little
bit like a loudspeaker.

00:00:37.830 --> 00:00:39.996
And so let me shade
it in a little bit.

00:00:39.996 --> 00:00:41.620
So that's going to
be our solid and you

00:00:41.620 --> 00:00:48.630
can imagine that maybe this
right over here is our y-axis.

00:00:48.630 --> 00:00:50.000
That is our y-axis.

00:00:50.000 --> 00:00:56.310
And then going straight through
the center of the solid,

00:00:56.310 --> 00:00:59.230
you have your x-axis.

00:00:59.230 --> 00:01:01.120
If the solid was
slightly transparent,

00:01:01.120 --> 00:01:04.220
you would see the other side of
this opening right over here,

00:01:04.220 --> 00:01:06.990
and the x-axis goes
straight through the center.

00:01:06.990 --> 00:01:09.460
So how do we find the
volume of this solid?

00:01:09.460 --> 00:01:11.810
Well, we could think about
each of the individual disks

00:01:11.810 --> 00:01:14.105
of this solid right over here.

00:01:14.105 --> 00:01:15.230
So think about it this way.

00:01:15.230 --> 00:01:16.770
Let's take a disk
right over here.

00:01:16.770 --> 00:01:19.670
Let me take this
disk right over here

00:01:19.670 --> 00:01:22.300
so that you can imagine this
little bit of a cross section.

00:01:22.300 --> 00:01:25.280
It's a little bit of a
cross section of this solid.

00:01:25.280 --> 00:01:27.250
So this disk right
over here, it's

00:01:27.250 --> 00:01:30.010
going to have some area,
some surface area, and then

00:01:30.010 --> 00:01:31.589
it's going to have some depth.

00:01:31.589 --> 00:01:32.380
I'll draw the edge.

00:01:32.380 --> 00:01:33.970
You can imagine
the depth is kind

00:01:33.970 --> 00:01:36.820
of the edge of the quarter,
while the surface area is kind

00:01:36.820 --> 00:01:38.700
of the face of the
quarter, the area

00:01:38.700 --> 00:01:40.010
of the face of the quarter.

00:01:40.010 --> 00:01:42.710
So what's going to be the
volume of this white disk

00:01:42.710 --> 00:01:43.524
right over here?

00:01:43.524 --> 00:01:44.940
Well, it's going
to be the surface

00:01:44.940 --> 00:01:49.010
area of the disk
times the depth.

00:01:49.010 --> 00:01:51.110
So what is the surface area?

00:01:51.110 --> 00:01:55.165
Well, surface area of a circle
is pi times the radius squared.

00:01:57.820 --> 00:02:02.320
So it's going to be pi times--
and what's this radius?

00:02:02.320 --> 00:02:04.470
Well, the radius is the
height between the x-axis

00:02:04.470 --> 00:02:05.630
and the function.

00:02:05.630 --> 00:02:08.930
So this radius is
e to the 2x power.

00:02:08.930 --> 00:02:12.650
So the area of each
of these disks,

00:02:12.650 --> 00:02:15.520
for this x right over
here, for this value of x,

00:02:15.520 --> 00:02:21.315
is going to be pi times
e to the 2x squared.

00:02:21.315 --> 00:02:23.440
And then if we want the
volume of this entire disk,

00:02:23.440 --> 00:02:26.580
we then multiply the
area times the depth.

00:02:26.580 --> 00:02:29.070
So times dx.

00:02:29.070 --> 00:02:31.550
So this would be the volume
of this little disk over here

00:02:31.550 --> 00:02:35.540
that has a very small
infinitesimally wide depth.

00:02:35.540 --> 00:02:39.050
What we want to do is
sum all of these disks.

00:02:39.050 --> 00:02:40.810
We want to have a
bunch of these disks,

00:02:40.810 --> 00:02:42.274
that we want to
sum up all of them.

00:02:42.274 --> 00:02:43.690
So we want to take
an infinite sum

00:02:43.690 --> 00:02:46.010
of these infinitely thin disks.

00:02:46.010 --> 00:02:48.750
So essentially we're just going
to take the sum of all of these

00:02:48.750 --> 00:02:54.010
from x is equal to 0, all
the way to x is equal to k.

00:02:54.010 --> 00:02:56.080
Right, because this
right over here,

00:02:56.080 --> 00:02:58.780
this is x is equal to k,
this is x is equal to 0,

00:02:58.780 --> 00:03:00.250
we take the sum
of all the disks.

00:03:00.250 --> 00:03:02.069
We will get our actual volume.

00:03:02.069 --> 00:03:04.360
So we can actually evaluate
this integral analytically,

00:03:04.360 --> 00:03:05.780
it's not too bad.

00:03:05.780 --> 00:03:09.970
Because either--
let me rewrite it.

00:03:09.970 --> 00:03:16.190
Pi times-- e to the 2x squared
is e to the 4x, just 2x times

00:03:16.190 --> 00:03:17.040
2.

00:03:17.040 --> 00:03:18.990
And then, if we want
to take the integral,

00:03:18.990 --> 00:03:21.035
you might be able to do
this just by inspection,

00:03:21.035 --> 00:03:22.410
but if you want
to say, hey look,

00:03:22.410 --> 00:03:23.670
I have a little
function here, I want

00:03:23.670 --> 00:03:25.490
its derivative sitting
here, what we can do

00:03:25.490 --> 00:03:28.410
is we can multiply
this expression by 4,

00:03:28.410 --> 00:03:30.310
and we can also divide it by 4.

00:03:30.310 --> 00:03:32.320
If you multiply something
by 4 and divide by 4,

00:03:32.320 --> 00:03:33.569
you haven't changed its value.

00:03:33.569 --> 00:03:35.190
You've multiplied it by 1.

00:03:35.190 --> 00:03:37.210
But what this does
is it gives us--

00:03:37.210 --> 00:03:39.070
let me rewrite it a little bit.

00:03:39.070 --> 00:03:49.030
We can rewrite this as pi
over 4 times 4e to the 4x.

00:03:49.030 --> 00:03:50.100
I just rewrote this.

00:03:50.100 --> 00:03:53.529
I multiplied and divided by 4,
multiplied and divided by 4.

00:03:53.529 --> 00:03:55.320
And the reason why I
did that is so that we

00:03:55.320 --> 00:03:57.730
have the derivative of 4x
sitting right over here,

00:03:57.730 --> 00:03:58.860
we have this 4.

00:03:58.860 --> 00:04:01.320
Pi over 4 is a constant, so
we can actually take it out

00:04:01.320 --> 00:04:02.600
of the integral.

00:04:02.600 --> 00:04:05.160
And so we can say that this
expression right over here

00:04:05.160 --> 00:04:09.740
is the same as this
expression right over here,

00:04:09.740 --> 00:04:13.670
from pi over 4 times the
integral from 0 to k of 4e

00:04:13.670 --> 00:04:17.130
to the 4x dx.

00:04:17.130 --> 00:04:19.579
And so the whole reason why
I multiplied and divided by 4

00:04:19.579 --> 00:04:23.220
is I have something here,
I have its derivative.

00:04:23.220 --> 00:04:27.050
So I can really just pretend
like this is an e to the x

00:04:27.050 --> 00:04:32.700
and you can take the integral,
or the anti-derivative,

00:04:32.700 --> 00:04:34.610
with respect to 4x.

00:04:34.610 --> 00:04:38.360
So the inside is e to the 4x.

00:04:38.360 --> 00:04:39.250
And you can verify.

00:04:39.250 --> 00:04:41.000
If you take the
derivative of e to the 4x,

00:04:41.000 --> 00:04:44.820
it is the derivative of 4x,
which is 4, times e to the 4x.

00:04:44.820 --> 00:04:48.340
So this right here is the
anti-derivative of that.

00:04:48.340 --> 00:04:50.310
And we're going to evaluate it.

00:04:50.310 --> 00:04:52.930
We are going to
evaluate it from 0 to k.

00:04:55.500 --> 00:04:58.210
All of that times pi over 4.

00:04:58.210 --> 00:05:03.590
And so when you evaluate
it at k, it's e to the 4k.

00:05:03.590 --> 00:05:09.700
So this is going to be pi
over 4 times e to the 4k minus

00:05:09.700 --> 00:05:13.880
e to the 4 times 0, which is
e to the 0, which is minus 1.

00:05:13.880 --> 00:05:17.850
So this is our volume, pi over
4 times e to the 4k minus 1.

00:05:17.850 --> 00:05:19.970
So that is part b.

00:05:19.970 --> 00:05:21.180
We did that.

00:05:21.180 --> 00:05:21.890
Now part c.

00:05:21.890 --> 00:05:23.480
I think we have time for this.

00:05:23.480 --> 00:05:27.850
The volume, V, found in part
b, changes as k changes.

00:05:27.850 --> 00:05:31.730
If dk, the derivative of k
with respect to t is 1/3,

00:05:31.730 --> 00:05:33.760
determine the derivative
of V with respect

00:05:33.760 --> 00:05:35.830
to t when k is equal to 1/2.

00:05:35.830 --> 00:05:38.530
So this is straight up
from the chain rule.

00:05:38.530 --> 00:05:41.700
So the chain rule will tell us--
and if you view differentials

00:05:41.700 --> 00:05:43.150
as just very, very
small numbers,

00:05:43.150 --> 00:05:45.670
it actually seems like a
little bit of common sense--

00:05:45.670 --> 00:05:48.790
the rate of change of V,
so a very small change

00:05:48.790 --> 00:05:51.400
in V with respect to a
very small change in t

00:05:51.400 --> 00:05:55.910
is equal to a very small
change in V with respect

00:05:55.910 --> 00:06:02.840
to a very small change in k
times a very small change in k

00:06:02.840 --> 00:06:05.440
divided by a very
small change in t.

00:06:05.440 --> 00:06:07.770
Or the derivative of
V with respect to t

00:06:07.770 --> 00:06:10.270
is equal to the derivative
of V with respect to k times

00:06:10.270 --> 00:06:12.091
the derivative of k
with respect to t.

00:06:12.091 --> 00:06:13.840
And the reason why I
said this makes sense

00:06:13.840 --> 00:06:17.640
if you view a differential as
a small number is you would--

00:06:17.640 --> 00:06:20.206
and that's not a very
rigorous way of doing it,

00:06:20.206 --> 00:06:22.580
but it's a good kind of common
sense way of just thinking

00:06:22.580 --> 00:06:25.220
about things-- is that
these guys would cancel out

00:06:25.220 --> 00:06:26.820
if these were
numbers, and then you

00:06:26.820 --> 00:06:29.040
would have dV dt on both sides.

00:06:29.040 --> 00:06:31.610
So that's why I said that
this is slightly common sense.

00:06:31.610 --> 00:06:33.145
You can kind of
view it that way.

00:06:33.145 --> 00:06:35.520
But this essentially gives us
everything we need to know.

00:06:35.520 --> 00:06:38.230
We need to solve for dV dt.

00:06:38.230 --> 00:06:45.340
They already gave us dk dt at
1/3 when k is equal to 1/2.

00:06:45.340 --> 00:06:48.140
So they're already
telling us the rate

00:06:48.140 --> 00:06:50.602
of change of k
with respect to t.

00:06:50.602 --> 00:06:52.560
We already know that this
thing right over here

00:06:52.560 --> 00:06:54.790
is going to be 1/3.

00:06:54.790 --> 00:06:57.700
And we can figure out the
derivative of V with respect

00:06:57.700 --> 00:07:00.460
to k very easily because we
have V as a function of k right

00:07:00.460 --> 00:07:01.010
over here.

00:07:01.010 --> 00:07:02.600
So let's figure that out.

00:07:02.600 --> 00:07:07.850
So the derivative of
V with respect to k

00:07:07.850 --> 00:07:13.830
is equal to pi over 4 times
the derivative in here.

00:07:13.830 --> 00:07:16.160
And the derivative of
this is just 4e to the 4k.

00:07:16.160 --> 00:07:19.560
The derivative of
negative 1 is just 0.

00:07:19.560 --> 00:07:24.790
So this is times 4e to
the 4k, is the derivative.

00:07:24.790 --> 00:07:26.120
These guys cancel out.

00:07:26.120 --> 00:07:31.540
So it's equal to pi
e to the 4k, that's

00:07:31.540 --> 00:07:33.700
the derivative of V
with respect to k.

00:07:33.700 --> 00:07:37.270
We want to find it
when k is equal to 1/2,

00:07:37.270 --> 00:07:40.820
so we could write
V prime at 1/2,

00:07:40.820 --> 00:07:43.960
or when k is equal to 1/2 dV dk.

00:07:43.960 --> 00:07:50.020
dV dk is going to be equal to
pi times e to the 4 times 1/2

00:07:50.020 --> 00:07:53.430
which is equal to pi e squared.

00:07:53.430 --> 00:07:58.610
So this part right over
here is pi e squared.

00:07:58.610 --> 00:08:00.970
And so the derivative
of V with respect to t

00:08:00.970 --> 00:08:06.800
when k is equal to
1/2 and the derivative

00:08:06.800 --> 00:08:09.720
of k with respect to t is
1/3 is going to be just

00:08:09.720 --> 00:08:11.340
pi e squared times 1/3.

00:08:11.340 --> 00:08:13.280
So let me just
rewrite it over here.

00:08:13.280 --> 00:08:19.030
So dV dt is going to be
equal to 1/3 times this,

00:08:19.030 --> 00:08:23.490
or we could write
pi e squared over 3.

00:08:23.490 --> 00:08:25.440
And we're done.


2011 Calculus BC free response #3a

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:01.900
Problem three.

00:00:01.900 --> 00:00:04.930
Let f of x is equal
to e to the 2x.

00:00:04.930 --> 00:00:07.750
Let R be the region in
the first quadrant bounded

00:00:07.750 --> 00:00:10.600
by the graph of f,
the coordinate axes,

00:00:10.600 --> 00:00:12.780
and the vertical
line, x is equal to k.

00:00:12.780 --> 00:00:14.290
So they drew that
right over here.

00:00:14.290 --> 00:00:17.160
And it's in the figure
above in the actual AP exam.

00:00:17.160 --> 00:00:19.860
But I put it below
here so I didn't

00:00:19.860 --> 00:00:22.080
have to waste the screen
real estate above that.

00:00:22.080 --> 00:00:25.090
And you see right here,
this is our f of x,

00:00:25.090 --> 00:00:27.110
this is the line
x is equal to k,

00:00:27.110 --> 00:00:29.590
and this region R
is between those,

00:00:29.590 --> 00:00:31.960
and it's in the first
quadrant right over here.

00:00:31.960 --> 00:00:36.010
So this right over
here is our region R.

00:00:36.010 --> 00:00:37.960
So what are they
asking us to do?

00:00:37.960 --> 00:00:39.610
So k is greater
than 0, the region R

00:00:39.610 --> 00:00:40.790
shown the figure above.

00:00:40.790 --> 00:00:42.030
And this is the figure.

00:00:42.030 --> 00:00:43.260
Part a.

00:00:43.260 --> 00:00:46.420
Write, but do not evaluate
an expression involving

00:00:46.420 --> 00:00:51.620
an integral that gives the
perimeter of R in terms of k.

00:00:51.620 --> 00:00:53.208
So for the perimeter
of R, we're going

00:00:53.208 --> 00:00:55.083
to have to find the
length of the sides of R,

00:00:55.083 --> 00:00:57.000
and the hardest of
these is to find

00:00:57.000 --> 00:01:02.610
the length of the actual
curve between this point

00:01:02.610 --> 00:01:05.200
and this point right over here.

00:01:05.200 --> 00:01:08.505
And I like to always rederive
the arc length formula,

00:01:08.505 --> 00:01:10.380
although if you're about
to take the AP exam,

00:01:10.380 --> 00:01:12.050
it probably doesn't hurt
to have it memorized just

00:01:12.050 --> 00:01:13.210
so you could save some time.

00:01:13.210 --> 00:01:14.668
But you should
always know how it's

00:01:14.668 --> 00:01:16.557
derived so that when
you're 35 years old

00:01:16.557 --> 00:01:18.640
and you don't have a
calculus book as a reference,

00:01:18.640 --> 00:01:20.015
you can come up
with it yourself.

00:01:20.015 --> 00:01:22.570
And so you know where
the formula comes from.

00:01:22.570 --> 00:01:23.690
So let's rederive it.

00:01:23.690 --> 00:01:28.090
So let's say we were to zoom
in on a little bit of the curve

00:01:28.090 --> 00:01:28.860
right over there.

00:01:28.860 --> 00:01:31.510
So let's say let's zoom in
on that part of the curve

00:01:31.510 --> 00:01:32.900
right over there.

00:01:32.900 --> 00:01:36.770
And what we could do is,
same way we derived an arc

00:01:36.770 --> 00:01:42.260
length when the function
was defined parametrically,

00:01:42.260 --> 00:01:47.061
which we did, I think, in
part one of this AP exam.

00:01:47.061 --> 00:01:48.810
This time it's not
defined parametrically,

00:01:48.810 --> 00:01:51.120
so we can do it in
terms of just x and y.

00:01:51.120 --> 00:01:54.340
So if you viewed this
distance right over here,

00:01:54.340 --> 00:01:57.140
this is going to be a
very small change in x.

00:01:57.140 --> 00:01:58.960
Let's call that dx.

00:01:58.960 --> 00:02:02.740
And then this right
over here is going

00:02:02.740 --> 00:02:05.840
to be a very small
change in y, dy.

00:02:05.840 --> 00:02:09.330
And we know that dy, the
derivative of y with respect

00:02:09.330 --> 00:02:14.260
to x, or dy/dx, this is
equal to f prime of x.

00:02:14.260 --> 00:02:17.290
Or we could say that if we
multiply both sides by dx,

00:02:17.290 --> 00:02:22.050
we know that dy is equal
to f prime of x dx.

00:02:22.050 --> 00:02:26.170
So this is equal
to f prime of x dx.

00:02:26.170 --> 00:02:29.080
And then for a very small change
in x and very small change

00:02:29.080 --> 00:02:32.100
in y, we can approximate
this curve length

00:02:32.100 --> 00:02:33.400
with the Pythagorean theorem.

00:02:33.400 --> 00:02:36.320
And if we get small enough,
it'll probably directly

00:02:36.320 --> 00:02:38.060
kind of measure
that curve length.

00:02:38.060 --> 00:02:40.170
So we use the Pythagorean
theorem right here.

00:02:40.170 --> 00:02:45.110
This length is going to be
the square root of dx squared

00:02:45.110 --> 00:02:47.310
plus dy squared.

00:02:47.310 --> 00:02:50.364
And if we write dy this
way, that arc length

00:02:50.364 --> 00:02:52.280
right over here, this
little small arc length,

00:02:52.280 --> 00:02:54.710
is going to be equal
to, right over here,

00:02:54.710 --> 00:03:00.890
the square root of dx squared
plus-- this is dy-- so

00:03:00.890 --> 00:03:08.090
plus f prime of x
squared times dx squared.

00:03:08.090 --> 00:03:10.200
All I did is I squared
this. dy squared

00:03:10.200 --> 00:03:12.090
is this thing right over here.

00:03:12.090 --> 00:03:15.330
And let me expand the
radical sign a little bit.

00:03:15.330 --> 00:03:17.030
And just the same
exact way we did it

00:03:17.030 --> 00:03:20.050
when we had the function
defined parametrically,

00:03:20.050 --> 00:03:23.150
we can factor out
a dx squared here.

00:03:23.150 --> 00:03:28.850
So this is going to be equal
to the square root of-- if you

00:03:28.850 --> 00:03:31.030
factor out a dx squared
of this, you get a 1.

00:03:31.030 --> 00:03:33.030
You factor out a dx
squared out of this part,

00:03:33.030 --> 00:03:37.535
plus f prime of x squared.

00:03:37.535 --> 00:03:40.820
And we factor a dx squared
out of the radical sign,

00:03:40.820 --> 00:03:43.640
it just becomes a dx.

00:03:43.640 --> 00:03:46.224
And so that would be the length
of this little small arc here,

00:03:46.224 --> 00:03:47.598
and it wouldn't
even be this big.

00:03:47.598 --> 00:03:49.630
It would be a super
infinitesimally small piece

00:03:49.630 --> 00:03:50.130
of arc.

00:03:50.130 --> 00:03:52.530
We could call it d
arc right over there.

00:03:52.530 --> 00:03:55.440
And essentially want to sum up
all of these little segments.

00:03:55.440 --> 00:04:02.350
So we want to sum them up from x
is equal to 0, all the way to x

00:04:02.350 --> 00:04:07.070
is equal to k, because we're
integrating with respect to x

00:04:07.070 --> 00:04:08.230
right over here.

00:04:08.230 --> 00:04:10.660
So this arc length
right over here-- so

00:04:10.660 --> 00:04:14.130
let me write what they
want us to figure out;

00:04:14.130 --> 00:04:18.850
we're trying to figure
out the perimeter of R--

00:04:18.850 --> 00:04:21.870
is equal to first, this part.

00:04:21.870 --> 00:04:25.670
So this is the integral
from 0 to k. x is equal to 0

00:04:25.670 --> 00:04:30.880
to x is equal to k of the
square root of 1 plus--

00:04:30.880 --> 00:04:32.405
and f prime of x.

00:04:32.405 --> 00:04:34.500
f of x is e to the 2x.

00:04:34.500 --> 00:04:38.660
So f prime of x is equal to the
derivative of this with respect

00:04:38.660 --> 00:04:42.360
to x, which is 2e to the 2x.

00:04:42.360 --> 00:04:44.990
The derivative of 2x is 2.

00:04:44.990 --> 00:04:48.870
Derivative of e to the 2x with
respect to 2x is e to the 2x.

00:04:48.870 --> 00:04:51.800
So 2e to the 2x, we
want to square that,

00:04:51.800 --> 00:04:53.400
because this is f
prime of x, but we

00:04:53.400 --> 00:04:54.600
want to square f prime of x.

00:04:54.600 --> 00:04:59.650
So that's 4e to the 4x.

00:04:59.650 --> 00:05:01.140
And of course, we have a dx.

00:05:01.140 --> 00:05:05.400
So this is the length of the
arc, and that's the hard part.

00:05:05.400 --> 00:05:07.820
And now we just have to get
the rest of the perimeter.

00:05:07.820 --> 00:05:09.990
So you have this little
segment right over here.

00:05:09.990 --> 00:05:10.890
That's length 1.

00:05:10.890 --> 00:05:13.370
We're going from
1 to 0 or 0 to 1.

00:05:13.370 --> 00:05:14.720
So plus 1.

00:05:14.720 --> 00:05:18.750
Then you have this part
right along the x-axis.

00:05:18.750 --> 00:05:21.360
Well, that's going
to be of length k.

00:05:21.360 --> 00:05:22.480
Plus k.

00:05:22.480 --> 00:05:26.840
And then finally, you have
this height right over here.

00:05:26.840 --> 00:05:31.420
And that's going to be of length
of f of k, or e to the 2k.

00:05:31.420 --> 00:05:34.790
Plus e to the 2k.

00:05:34.790 --> 00:05:35.510
And we're done.

00:05:35.510 --> 00:05:38.700
We found the perimeter of R, and
we don't have to evaluate it.

00:05:38.700 --> 00:05:41.300
So we're done with this part.


School Leadership Briefing Questions (Sep 12 2011) 2

WEBVTT
Kind: captions
Language: en

00:00:00.437 --> 00:00:02.612
Voiceover: All right, we're on question,

00:00:02.612 --> 00:00:04.195
on this one right over here.

00:00:04.195 --> 00:00:06.535
We often hear people say
if you're teaching math

00:00:06.535 --> 00:00:08.189
in this country, there's
no way you haven't

00:00:08.189 --> 00:00:09.443
heard of Salman Khan.

00:00:09.443 --> 00:00:11.068
That's a little bit surreal
to me if that's true.

00:00:11.068 --> 00:00:13.362
How does your Khan Academy model work

00:00:13.362 --> 00:00:18.012
for less concrete subjects,
say creative writing?

00:00:18.012 --> 00:00:20.329
It depends where you want
to go on the spectrum

00:00:20.329 --> 00:00:21.786
into kind of subjective subjects.

00:00:21.786 --> 00:00:26.527
I think if we go into
something like history,

00:00:26.527 --> 00:00:27.874
if there are lectures,

00:00:27.874 --> 00:00:29.536
and clearly there are history lectures,

00:00:29.536 --> 00:00:31.417
I've experienced many of them,

00:00:31.417 --> 00:00:34.132
we think, once again, that those shouldn't

00:00:34.132 --> 00:00:36.463
necessarily happen during class time;

00:00:36.463 --> 00:00:37.936
that those are better
at the student's pace,

00:00:37.936 --> 00:00:39.079
at the student's time.

00:00:39.079 --> 00:00:41.133
Then if you get them out of the way,

00:00:41.133 --> 00:00:42.998
using something like the Khan Academy,

00:00:42.998 --> 00:00:47.002
then class time can have
a richer discussion.

00:00:47.002 --> 00:00:50.000
On top of that, right
now our exercise platform

00:00:50.000 --> 00:00:52.255
for the math, it generates math problems,

00:00:52.255 --> 00:00:55.331
it's harder for a
computer to automatically

00:00:55.331 --> 00:00:58.271
generate a history
question, or even, frankly,

00:00:58.271 --> 00:01:00.168
some types of math questions.

00:01:00.168 --> 00:01:02.269
So what we think we can
do is probably create

00:01:02.269 --> 00:01:05.671
a repository of questions
that have appropriate

00:01:05.671 --> 00:01:07.751
data with them: How hard are they?

00:01:07.751 --> 00:01:09.267
How long does it take someone to do?

00:01:09.267 --> 00:01:11.465
How relevant are they to a specific topic?

00:01:11.465 --> 00:01:16.251
We can quiz people on
topics like history so that,

00:01:16.251 --> 00:01:18.601
we don't have that yet
but that's something

00:01:18.601 --> 00:01:20.900
that we would like to do
in the not too far future,

00:01:20.900 --> 00:01:23.233
so that someone can, with the lectures,

00:01:23.233 --> 00:01:25.333
get a basic scaffold
of their understanding

00:01:25.333 --> 00:01:27.004
of history through the videos,

00:01:27.004 --> 00:01:29.875
and they can get practice
so that they make sure

00:01:29.875 --> 00:01:31.665
that they at least have the basic,

00:01:31.665 --> 00:01:33.575
they understand when World War I was,

00:01:33.575 --> 00:01:34.738
when World War II is,

00:01:34.738 --> 00:01:36.208
what's the difference
between the League of Nations

00:01:36.208 --> 00:01:37.895
and the UN, and all of
these type of things.

00:01:37.895 --> 00:01:39.673
So when they go to class time,

00:01:39.673 --> 00:01:41.403
if they have access to a classroom,

00:01:41.403 --> 00:01:43.970
the classroom can be there
for really rich discussion

00:01:43.970 --> 00:01:45.810
and the teacher or the professor knows,

00:01:45.810 --> 00:01:47.398
because they're going to have data

00:01:47.398 --> 00:01:49.667
based on how people perform
on the Khan Academy,

00:01:49.667 --> 00:01:52.864
what videos they watch and how
they perform on the questions,

00:01:52.864 --> 00:01:55.136
the teacher will know
how prepared the class is

00:01:55.136 --> 00:01:57.218
for a richer discussion.

00:01:57.218 --> 00:02:00.007
For something like creative writing,

00:02:00.007 --> 00:02:03.091
and creative writing, I'd
put that probably with

00:02:03.091 --> 00:02:09.629
music composition and art,
painting art, or studio art,

00:02:09.629 --> 00:02:10.931
things that are purely creative,

00:02:10.931 --> 00:02:14.932
I would say that if
there are any lectures,

00:02:14.932 --> 00:02:16.927
so if right now in a
creative writing class,

00:02:16.927 --> 00:02:20.464
there are teachers
giving basic lectures on

00:02:20.464 --> 00:02:21.998
just things to think about,

00:02:21.998 --> 00:02:24.334
we think that those things can be taken

00:02:24.334 --> 00:02:26.894
out of class time, that
those can be done in a video.

00:02:26.894 --> 00:02:29.259
I'm probably not the
right person to do it,

00:02:29.259 --> 00:02:30.400
but we'll be able to find people

00:02:30.400 --> 00:02:33.611
who can do those creative
writing lectures.

00:02:33.611 --> 00:02:35.130
Also, even in creative writing,

00:02:35.130 --> 00:02:38.019
there is a basic skill
set that you want to have

00:02:38.019 --> 00:02:40.330
in order to be a
respectable creative writer;

00:02:40.330 --> 00:02:45.800
basic grammar, vocabulary,

00:02:45.800 --> 00:02:49.347
doing a critical analysis
of other writing.

00:02:49.347 --> 00:02:52.373
Some of that can be done
in a video form factor.

00:02:52.373 --> 00:02:54.802
Everything we want to do is to enable

00:02:54.802 --> 00:02:59.052
the class time to be all about
writing and getting feedback

00:02:59.052 --> 00:03:01.659
and getting assessment
and critically analyzing

00:03:01.659 --> 00:03:04.051
other peoples' work so it's actively ...

00:03:04.051 --> 00:03:07.693
It's activity-based
work as opposed to just

00:03:07.693 --> 00:03:09.702
listening to a lecture.

00:03:09.702 --> 00:03:11.162
Creative writing, we don't have any,

00:03:11.162 --> 00:03:15.072
I can imagine there will probably
be some videos eventually

00:03:15.072 --> 00:03:17.593
for maybe writing generally;
grammar, vocabulary,

00:03:17.593 --> 00:03:21.328
some forms of critical analysis.

00:03:21.328 --> 00:03:24.454
What I think we could do
eventually for critical writing,

00:03:24.454 --> 00:03:26.186
and once again, this is
completely on the drawing board

00:03:26.186 --> 00:03:28.683
and something we won't be able
to get to for the near term,

00:03:28.683 --> 00:03:32.257
but I think you can do a
model where Khan Academy

00:03:32.257 --> 00:03:39.056
has creative writing projects
and students write creative,

00:03:39.056 --> 00:03:41.259
they write essays and
they write short stories

00:03:41.259 --> 00:03:44.487
and they write poetry and members

00:03:44.487 --> 00:03:47.409
of the Khan Academy community
can give them feedback

00:03:47.409 --> 00:03:48.797
on how they did and they can assess

00:03:48.797 --> 00:03:53.409
how good the writing was,
and the students themselves

00:03:53.409 --> 00:03:55.447
will assess other people's writing.

00:03:55.447 --> 00:03:57.134
I think if we can do something like that,

00:03:57.134 --> 00:03:59.537
once again, it will become
a tool where we can automate

00:03:59.537 --> 00:04:03.356
a lot of the overhead
of trying to coordinate

00:04:03.356 --> 00:04:08.531
who's going to assess whom's
work and all of the rest,

00:04:08.531 --> 00:04:10.271
and then you can, once
again, use class time

00:04:10.271 --> 00:04:13.352
with as much human
interaction as possible.

00:04:13.352 --> 00:04:15.272
I don't want to claim that
we've solved that problem,

00:04:15.272 --> 00:04:17.275
but I suspect that there are ways.

00:04:17.275 --> 00:04:20.130
One thing I want to make
clear in this interview

00:04:20.130 --> 00:04:29.089
is that we aren't married to a
particular type of technology.

00:04:29.089 --> 00:04:31.537
We aren't saying it has
to be a black background

00:04:31.537 --> 00:04:35.052
with Sal scrawling in pastel colors

00:04:35.052 --> 00:04:36.984
and just talking about a subject.

00:04:36.984 --> 00:04:39.251
We aren't married to the
idea that you have to have

00:04:39.251 --> 00:04:40.987
a knowledge map that's
structured that way.

00:04:40.987 --> 00:04:45.871
Our number one goal is how
can we make a student's

00:04:45.871 --> 00:04:48.210
learning experience as
productive as possible?

00:04:48.210 --> 00:04:52.030
And how can we make that
scarce time with the teacher,

00:04:52.030 --> 00:04:54.114
with the professor, and
with the student's peers

00:04:54.114 --> 00:04:56.535
in that classroom, how
can we make that as rich

00:04:56.535 --> 00:04:58.049
and productive as possible?

00:04:58.049 --> 00:04:59.614
That's what drives all of our thinking.

00:04:59.614 --> 00:05:04.864
Over time, I suspect in 10
years I might not recognize

00:05:04.864 --> 00:05:06.653
a lot of what Khan Academy
is, but I'll be very

00:05:06.653 --> 00:05:08.196
excited because it's
going to be empowering

00:05:08.196 --> 00:05:10.368
a lot of those basic core principles.

00:05:10.368 --> 00:05:13.201
We're always on the lookout
for things that can do that.

00:05:13.201 --> 00:05:15.537
In something like creative
writing, it's true,

00:05:15.537 --> 00:05:17.283
we don't offer something right now.

00:05:17.283 --> 00:05:19.321
History we do offer a few videos

00:05:19.321 --> 00:05:21.657
and we'll probably have a question bank

00:05:21.657 --> 00:05:22.982
in the not too far off future.

00:05:22.982 --> 00:05:25.280
But for creative writing, we
want to explore that space.

00:05:25.280 --> 00:05:27.458
We do eventually want to empower

00:05:27.458 --> 00:05:29.790
even the creative writing classrooms.

00:05:29.790 --> 00:05:31.495
Next question.

00:05:31.495 --> 00:05:33.595
You said that most educational software

00:05:33.595 --> 00:05:35.088
loses sight of the end user.

00:05:35.088 --> 00:05:36.789
What should the people
in charge of purchasing

00:05:36.789 --> 00:05:38.753
these materials understand about them?

00:05:38.753 --> 00:05:40.857
It's a very simple idea.

00:05:40.857 --> 00:05:45.789
I would say if you are in
the decision-making role

00:05:45.789 --> 00:05:47.922
of purchasing materials, and this is true

00:05:47.922 --> 00:05:49.503
whether you're in education or anything,

00:05:49.503 --> 00:05:53.666
don't just be impressed by the polish

00:05:53.666 --> 00:05:55.924
of what the sales person tells you,

00:05:55.924 --> 00:05:58.527
or the fact that other
districts are doing it,

00:05:58.527 --> 00:06:00.124
or when you flip through the book

00:06:00.124 --> 00:06:02.527
it seems to have fancy graphics
and it seems to be expensive

00:06:02.527 --> 00:06:05.258
and it seems to meet all
of the State standards.

00:06:05.258 --> 00:06:07.794
I think the best thing
that anyone can do is say,

00:06:07.794 --> 00:06:08.517
"You know what?

00:06:08.517 --> 00:06:09.784
"Let me take out this book,

00:06:09.784 --> 00:06:11.868
"or let me take out this piece of software

00:06:11.868 --> 00:06:13.265
"and let me try to learn from it."

00:06:13.265 --> 00:06:17.883
Pick a subject that,
if you're the school's

00:06:17.883 --> 00:06:19.738
principal or the district superintendant

00:06:19.738 --> 00:06:20.964
or whoever you are, say,

00:06:20.964 --> 00:06:23.638
"You know what, I'm a little
rusty on systems of equations.

00:06:23.638 --> 00:06:25.308
"Let me see if I can
learn systems of equations

00:06:25.308 --> 00:06:26.866
"from this book, and let
me see if it's really

00:06:26.866 --> 00:06:28.634
"that dramatically different
than the older textbook,

00:06:28.634 --> 00:06:31.276
"or if the software's
dramatically different

00:06:31.276 --> 00:06:32.923
"than the textbook."

00:06:32.923 --> 00:06:36.532
That's really the best way
to see if it's effective.

00:06:36.532 --> 00:06:39.004
In fact, I think that's true
about assessing anything.

00:06:39.004 --> 00:06:42.414
I think all of us have
experienced that sometimes,

00:06:42.414 --> 00:06:46.059
even amongst teachers, there could be

00:06:46.059 --> 00:06:49.004
a very flashy presentation.

00:06:49.004 --> 00:06:50.201
And not just teachers, videos even.

00:06:50.201 --> 00:06:51.657
There could be a very flashy presentation.

00:06:51.657 --> 00:06:53.922
If you look at it superficially, you say,

00:06:53.922 --> 00:06:56.195
"Oh my god, this is going
to be good instruction,"

00:06:56.195 --> 00:06:58.861
but you're not actually
trying to learn from,

00:06:58.861 --> 00:07:00.672
or that thing might
not actually teach you.

00:07:00.672 --> 00:07:04.864
The more basic lecture,
or the more basic video,

00:07:04.864 --> 00:07:08.339
oftentimes might do a
better job at teaching you.

00:07:08.339 --> 00:07:12.467
So the real way to assess something

00:07:12.467 --> 00:07:14.169
is to actually try to learn from it.

00:07:14.169 --> 00:07:15.929
I'm not saying that
flashy is necessarily bad,

00:07:15.929 --> 00:07:18.331
I'm just saying that flashy
is not necessarily good.

00:07:18.331 --> 00:07:21.597
A lot of us, it's just
easier to assess something

00:07:21.597 --> 00:07:23.592
very quickly saying, "Oh,
look, that looks polished;

00:07:23.592 --> 00:07:25.458
"that looks like there's a
lot of money spent into it,

00:07:25.458 --> 00:07:27.333
it must be good," but the reality is

00:07:27.333 --> 00:07:28.929
try to actually learn from it

00:07:28.929 --> 00:07:30.660
and you'll find that some
of the more basic things

00:07:30.660 --> 00:07:32.071
can often do better.

00:07:32.071 --> 00:07:37.066
I encourage anyone who's
looking at anything like that

00:07:37.066 --> 00:07:40.402
is to question their assumptions.

00:07:40.402 --> 00:07:46.335
Does a calculus textbook have
to have 1,000 pages in it?

00:07:46.335 --> 00:07:49.117
Does it have to even be a hardcover book?

00:07:49.117 --> 00:07:51.619
Why don't you separate it
into multiple chapters?

00:07:51.619 --> 00:07:55.696
All of these questions I think people ...

00:07:55.696 --> 00:07:58.202
They all boil down to the
basic principle that you

00:07:58.202 --> 00:08:00.525
should just try to learn
from that content yourselves

00:08:00.525 --> 00:08:02.390
and if you learn better from that content

00:08:02.390 --> 00:08:04.727
and you find that content engaging

00:08:04.727 --> 00:08:07.255
and you find it's something
that you can engage with,

00:08:07.255 --> 00:08:09.818
it's likely that the
students will be able to.

00:08:09.818 --> 00:08:11.196
If you can't and if it's just flashy

00:08:11.196 --> 00:08:13.129
and looks polished, it's
likely that the students

00:08:13.129 --> 00:08:15.799
won't be able to either.

00:08:15.799 --> 00:08:18.266
Next question.

00:08:18.266 --> 00:08:20.145
At the heart of the Khan
Academy's philosophy

00:08:20.145 --> 00:08:21.677
is the importance of enabling students

00:08:21.677 --> 00:08:22.979
to learn at their own pace.

00:08:22.979 --> 00:08:24.891
What happens when these
students enter the work force?

00:08:24.891 --> 00:08:27.737
Do they work, perform surgery, try a case,

00:08:27.737 --> 00:08:30.583
analyze data, or even
teach at their own pace?

00:08:30.583 --> 00:08:32.662
Yes.

00:08:32.662 --> 00:08:34.677
I think the learning at your own pace

00:08:34.677 --> 00:08:37.263
is exactly what happens in the real world.

00:08:37.263 --> 00:08:41.466
Every job I've been at,
there was no 30-person

00:08:41.466 --> 00:08:45.600
classroom where everyone was
moved ahead at the same time.

00:08:45.600 --> 00:08:48.598
First of all, it was a lot
of learning on your own.

00:08:48.598 --> 00:08:51.595
You had to figure out
what needs to be learned.

00:08:51.595 --> 00:08:54.101
There wasn't anyone holding your hand.

00:08:54.101 --> 00:08:56.012
You had to learn at your own pace.

00:08:56.012 --> 00:08:58.933
You work at a firm, you have
to figure out what's important,

00:08:58.933 --> 00:09:01.527
then you have to go,
usually at most firms,

00:09:01.527 --> 00:09:03.769
ask the right questions
from the right people

00:09:03.769 --> 00:09:05.529
and then proceed at your own pace.

00:09:05.529 --> 00:09:08.860
You actually see that; that
is actually the reality

00:09:08.860 --> 00:09:12.195
in most fields.

00:09:12.195 --> 00:09:17.262
You learn from experience
and from your own inquiry,

00:09:17.262 --> 00:09:20.685
and that tends to happen at your own pace.

00:09:20.685 --> 00:09:24.859
No one forces you along until you have ...

00:09:24.859 --> 00:09:26.512
And you don't force yourself along

00:09:26.512 --> 00:09:28.096
until you've mastered something.

00:09:28.096 --> 00:09:31.462
So, yes, I actually
think that's why so many

00:09:31.462 --> 00:09:33.343
students, right now,
suffer when they go into

00:09:33.343 --> 00:09:36.767
the work force is that
they're so used to being

00:09:36.767 --> 00:09:39.608
kind of shepherded along,
kind of as a cohort,

00:09:39.608 --> 00:09:41.486
that all of a sudden,
when they're thrown into

00:09:41.486 --> 00:09:43.068
the work environment and all of a sudden

00:09:43.068 --> 00:09:44.662
you have to decide when it's time

00:09:44.662 --> 00:09:46.478
to ask the boss for a promotion,

00:09:46.478 --> 00:09:48.148
you have to decide when you think

00:09:48.148 --> 00:09:50.004
you're ready to go back to grad school,

00:09:50.004 --> 00:09:51.402
you have to decide when you think

00:09:51.402 --> 00:09:53.737
you're ready to start your own business.

00:09:53.737 --> 00:09:57.581
Some people struggle when
they're not being told

00:09:57.581 --> 00:09:59.149
now you're going to go to 12th grade,

00:09:59.149 --> 00:10:00.403
now you're going to go to college,

00:10:00.403 --> 00:10:02.605
so I think that's an added benefit.

00:10:02.605 --> 00:10:04.468
When you give people at a young age,

00:10:04.468 --> 00:10:06.065
if you give them control
of their learning,

00:10:06.065 --> 00:10:07.867
if you make them self directed,

00:10:07.867 --> 00:10:10.275
that's a skill that's going to translate

00:10:10.275 --> 00:10:12.801
really well and really empower them

00:10:12.801 --> 00:10:15.867
once they enter the workforce.


School Leadership Briefing (Sep 12 2011)

WEBVTT
Kind: captions
Language: en

00:00:00.679 --> 00:00:05.678
Voiceover: I have a set of questions
here from school leadership briefing.

00:00:05.678 --> 00:00:09.342
The first question is, "The Khan Academy
uses a unique system of teaching,

00:00:09.342 --> 00:00:13.189
"where most of the taking in of
information, concrete learning,

00:00:13.189 --> 00:00:15.855
"occurs outside the classroom.
And inside the classroom,

00:00:15.855 --> 00:00:17.997
"time is spent solving
problems in other pursuits.

00:00:17.997 --> 00:00:20.129
"Can you briefly explain how this works?"

00:00:20.129 --> 00:00:23.389
So, on a very high level
this is exactly right.

00:00:23.389 --> 00:00:26.335
We believe, or the overarching idea here,

00:00:26.335 --> 00:00:28.590
is that when people
are in a room together,

00:00:28.590 --> 00:00:30.998
students with each other
and students with teachers,

00:00:30.998 --> 00:00:34.256
we think the best use of that
time and that scarce time,

00:00:34.256 --> 00:00:36.995
is for them to interact with
each other as much as possible.

00:00:36.995 --> 00:00:39.335
Leverage the fact that you
have other humans in the room.

00:00:39.335 --> 00:00:43.599
We want that time to be spent with
the students teaching each other,

00:00:43.599 --> 00:00:48.989
the teachers doing much smaller
group interactions with the students.

00:00:48.989 --> 00:00:52.393
As opposed to a traditional kind
of lecture for all of the students,

00:00:52.393 --> 00:00:53.564
at the same pace.

00:00:53.564 --> 00:00:56.933
And doing things that
there's often not time for.

00:00:58.486 --> 00:01:01.319
In a traditional classroom there's
a lot of time spent on lecture.

00:01:01.319 --> 00:01:03.521
We like to see teachers
doing more projects,

00:01:03.521 --> 00:01:08.855
doing more peer to peer tutoring,
leading to deeper simulations,

00:01:08.855 --> 00:01:10.725
and types of experiments.

00:01:10.725 --> 00:01:13.724
In terms of how it works,
the simple answer is,

00:01:16.123 --> 00:01:20.392
the details we're still learning
what the best way to do it is.

00:01:20.392 --> 00:01:23.259
We've done pilots and even in the
pilots that we've been running,

00:01:25.212 --> 00:01:26.927
every teacher does it a
different way and we're trying,

00:01:26.927 --> 00:01:28.267
to share best practices.

00:01:28.267 --> 00:01:32.391
The general idea is, exactly
as this question phrased,

00:01:32.391 --> 00:01:35.939
students spent a good amount of
their time so I guess their homework,

00:01:35.939 --> 00:01:38.864
is they're watching videos and they're
doing exercises at their own pace.

00:01:38.864 --> 00:01:40.864
So they are doing exercises as well.

00:01:41.802 --> 00:01:43.262
The teacher gets data on it.

00:01:43.262 --> 00:01:44.247
They get a dashboard on it.

00:01:44.247 --> 00:01:47.590
They know when and where and what
the students have been working on.

00:01:47.590 --> 00:01:50.500
They also get flagged on
what students are stuck on.

00:01:50.500 --> 00:01:53.500
When students go into the
classroom they are able,

00:01:53.500 --> 00:01:57.202
to continue working at their own
pace on the exercise and the videos.

00:01:57.202 --> 00:02:00.453
For maybe 15 to 20 minutes of class time.

00:02:00.453 --> 00:02:03.846
While that's occurring, since all the
students are working at their own pace,

00:02:03.846 --> 00:02:06.717
the teacher can look at
the dashboard and say,

00:02:06.717 --> 00:02:08.697
"Hey look. Billy is stuck
on negative numbers.

00:02:08.697 --> 00:02:11.454
"Let me get Susie who's already
mastered that concept to help Billy.

00:02:11.454 --> 00:02:15.451
"Or let me go sit next to Billy and see
if I can tutor him on negative numbers."

00:02:15.451 --> 00:02:17.515
Or maybe it's something different
because the teacher will also see,

00:02:17.515 --> 00:02:22.447
in his or her dashboard that hey
look, Billy has been working,

00:02:22.447 --> 00:02:25.029
on Khan Academy at 2 in the morning
maybe he's just not sleeping right.

00:02:25.029 --> 00:02:26.941
Maybe something else is going on at home.

00:02:26.941 --> 00:02:30.618
To do that very, kind of,
close human interaction,

00:02:30.618 --> 00:02:34.402
I think is the most valuable
thing we've been seeing.

00:02:34.402 --> 00:02:37.667
The rest of class time, outside
of that Khan Academy experience,

00:02:37.667 --> 00:02:40.611
where the students are working with
each other and teaching with each other,

00:02:40.611 --> 00:02:45.866
we like to see teachers do as many
projects as they can and simulations.

00:02:45.866 --> 00:02:48.949
One thing that we hope to add to Khan
Academy is to actually tell teachers,

00:02:48.949 --> 00:02:53.529
"Look these 5 students have
mastered these 5 concepts,

00:02:53.529 --> 00:02:57.204
"or these 20 concepts. We think
here's a good example of a project,

00:02:57.204 --> 00:03:00.360
"that will stimulate them. That
will let them learn that concept,

00:03:00.360 --> 00:03:02.282
"even a little bit deeper."

00:03:02.282 --> 00:03:03.738
Other things we've been seeing in classes.

00:03:03.738 --> 00:03:06.276
We've seen almost automatic
self-organization take place,

00:03:06.276 --> 00:03:09.108
with the students where students
start writing on a wipe board,

00:03:09.108 --> 00:03:11.614
what problems or what concepts
they're having trouble with,

00:03:11.614 --> 00:03:14.863
and other students are saying what
concepts they're comfortable teaching.

00:03:14.863 --> 00:03:17.613
Then they start to kind of
sort it out between them.

00:03:17.613 --> 00:03:21.615
With that said, we've been
doing it at Los Altos,

00:03:21.615 --> 00:03:26.203
which is an upper middle class to
affluent public school district.

00:03:26.203 --> 00:03:31.181
Although one of the classes that
we've been doing a pilot with,

00:03:31.181 --> 00:03:31.856
is for their algebra readiness course.

00:03:31.856 --> 00:03:36.418
Which is less affluent students,
many English as a second language,

00:03:36.956 --> 00:03:38.407
who are having trouble in mathematics.

00:03:38.407 --> 00:03:40.148
The results have been
really tremendous there.

00:03:40.148 --> 00:03:45.253
We're also doing pilots in East Palo
Alto which is an underserved community.

00:03:45.253 --> 00:03:48.261
We're doing one at a charter
school in Redwood City.

00:03:48.261 --> 00:03:52.328
We're talking to some other
pretty prominent charter networks,

00:03:52.328 --> 00:03:53.518
of experiments with them.

00:03:53.518 --> 00:03:55.450
Also in Oakland and in other places.

00:03:55.450 --> 00:03:59.724
In all of these pilots,
our goal isn't to say,

00:03:59.724 --> 00:04:02.450
"This is how you use Khan Academy, go
use it and let's see what happens."

00:04:02.450 --> 00:04:04.926
Our goal is to say we want to
collaborate with these teachers.

00:04:04.926 --> 00:04:08.118
The main reason we want to do these
pilots is because we want to explore,

00:04:08.118 --> 00:04:09.984
what is the best way to use this tool?

00:04:09.984 --> 00:04:15.753
How can we best make class time as
rich of an experience as possible?

00:04:15.753 --> 00:04:18.426
And how can we, with the
teachers, the physical experience,

00:04:18.426 --> 00:04:21.851
and the Khan Academy experience make
the overall learning experience,

00:04:21.851 --> 00:04:25.721
as deep and as long lasting as possible?

00:04:26.477 --> 00:04:28.214
Question number two.

00:04:28.214 --> 00:04:31.544
"You often talk about using
technology to humanize the classroom.

00:04:31.544 --> 00:04:33.807
"How can it do that? How do
you build community around,

00:04:33.807 --> 00:04:36.201
"the use of technology in
a classroom or school?"

00:04:37.431 --> 00:04:40.264
This kind of goes right
off of the first answer.

00:04:42.617 --> 00:04:46.060
I think the mistake with technology
in the classroom in the past is that,

00:04:46.060 --> 00:04:49.196
they've been using technology,
frankly, to dehumanize the classroom.

00:04:49.196 --> 00:04:52.585
Before, in a traditional classroom
where there's no computer,

00:04:52.585 --> 00:04:54.262
you have at least some interaction.

00:04:54.262 --> 00:04:57.528
The teacher might lecture for some
time but you do have some opportunity,

00:04:57.528 --> 00:05:00.191
to ask the teacher a question or
you might have some opportunity,

00:05:00.191 --> 00:05:01.858
to work on some projects together.

00:05:01.858 --> 00:05:03.864
Then when you introduce
the computer in that,

00:05:03.864 --> 00:05:06.595
where students are spending all of
class time working on computers,

00:05:06.595 --> 00:05:08.920
by themselves then all of
a sudden you crowd out,

00:05:08.920 --> 00:05:10.659
whatever interaction was there before.

00:05:10.659 --> 00:05:11.757
So you're dehumanizing it.

00:05:11.757 --> 00:05:13.193
We're seeing it do the opposite.

00:05:13.193 --> 00:05:15.336
Let's take to computer,
let's use the computer,

00:05:15.336 --> 00:05:19.536
so that the teacher no longer has to
give that one pace fits all lecture.

00:05:19.536 --> 00:05:23.331
So that kids don't have to sit
passively for an hour in math class,

00:05:23.331 --> 00:05:24.334
or whatever class.

00:05:24.334 --> 00:05:25.674
We're going to have physics content.

00:05:25.674 --> 00:05:28.671
Physics, chemistry,
biology, and other topics.

00:05:29.717 --> 00:05:31.590
When you have every student
working at their own pace,

00:05:31.590 --> 00:05:34.261
one it's catering to them so
that's one level of humanity.

00:05:34.261 --> 00:05:36.587
Then also, since you freed
up all of this class time,

00:05:36.587 --> 00:05:38.254
and you have students
interacting with each other,

00:05:38.254 --> 00:05:41.497
and actually interacting with
the teacher in a deeper way,

00:05:41.497 --> 00:05:45.160
that's kind of the second order
of humanizing the classroom.

00:05:45.160 --> 00:05:47.589
When some people think about
something like Khan Academy,

00:05:47.589 --> 00:05:50.591
the knee-jerk reaction is, "Well
how can someone ask a question?

00:05:50.591 --> 00:05:54.464
"These are videos. You can't
ask a question on a video."

00:05:54.464 --> 00:05:56.450
What we say is that they're exactly right.

00:05:56.450 --> 00:05:58.870
We also think that's true
of a lot of lectures.

00:05:58.870 --> 00:06:03.521
If you're in a room of 30 kids,
some kids might have the guts,

00:06:03.521 --> 00:06:04.928
to raise their hand and ask a question.

00:06:04.928 --> 00:06:07.521
It's still not even clear
if that's a good thing,

00:06:07.521 --> 00:06:10.118
because it might not be appropriate
question for all of the students.

00:06:10.118 --> 00:06:12.858
It might be a question
that's better served offline.

00:06:12.858 --> 00:06:14.584
We think a lot of students
aren't asking questions,

00:06:14.584 --> 00:06:15.918
in the current form factor.

00:06:15.918 --> 00:06:17.117
Maybe just because they're a little lost,

00:06:17.117 --> 00:06:19.401
they're not even engaged
in the content enough,

00:06:19.401 --> 00:06:21.492
that they can even think of questions.

00:06:21.492 --> 00:06:24.491
What we say is, let the kids
do exercises, watch videos,

00:06:24.491 --> 00:06:26.653
at their own time, at their own pace.

00:06:26.653 --> 00:06:30.593
Let them watch the same video 30
times if that's what it takes.

00:06:30.593 --> 00:06:33.929
We've seen examples of that is
what students sometimes need,

00:06:33.929 --> 00:06:36.196
to really understand something.

00:06:36.196 --> 00:06:38.417
Let the teacher have as much
information as they need,

00:06:38.417 --> 00:06:41.820
to make really powerful
judgements about what's going on,

00:06:41.820 --> 00:06:44.184
in a student's mind.

00:06:44.464 --> 00:06:46.787
Then free up not just a
small fraction of class time,

00:06:46.787 --> 00:06:49.071
to ask a teacher question.

00:06:49.071 --> 00:06:50.661
Free up all of class time.

00:06:50.661 --> 00:06:52.866
So all of class time is
nothing but interactions.

00:06:52.866 --> 00:06:55.604
And nothing but questions.

00:06:55.604 --> 00:06:57.722
In terms of how do you
build a community around,

00:06:57.722 --> 00:06:59.995
the use of technology in
a classroom or school?

00:06:59.995 --> 00:07:02.998
Once again, what we're doing in
our pilots is we're observing.

00:07:02.998 --> 00:07:04.394
We have ideas.

00:07:04.394 --> 00:07:06.988
Some of the amazing teachers
and principals and students,

00:07:06.988 --> 00:07:08.933
we've been working with have ideas.

00:07:08.933 --> 00:07:10.197
We're essentially trying them out,

00:07:10.197 --> 00:07:11.699
and we're seeing what
works and what doesn't.

00:07:11.699 --> 00:07:14.456
What you can expect from us over
the course of the next year,

00:07:14.456 --> 00:07:17.058
is us sharing what we think
are the best practices,

00:07:17.058 --> 00:07:19.732
or us encouraging some of these
awesome teachers and principals,

00:07:19.732 --> 00:07:22.037
to share what are the best practices.

00:07:22.775 --> 00:07:24.854
They've already had a
ton of really good ideas,

00:07:24.854 --> 00:07:26.649
for building that
community in the classroom.

00:07:26.649 --> 00:07:29.130
What we're going to do on our end, is
try to build as many technology tools,

00:07:29.130 --> 00:07:30.654
that also facilitate it.

00:07:30.654 --> 00:07:34.138
Right now kids are writing on the
wipe board what they need help with,

00:07:34.138 --> 00:07:35.131
or what they can help.

00:07:35.131 --> 00:07:37.060
Well, we could probably build a
little tool to facilitate that,

00:07:37.060 --> 00:07:38.133
within the classroom.

00:07:38.133 --> 00:07:39.715
It's all on the table though.

00:07:39.715 --> 00:07:42.532
We're always exploring
and always open to ideas.

00:07:43.742 --> 00:07:44.935
The third question.

00:07:44.935 --> 00:07:48.371
"One of your most talked about ideas
concerns flipping the classroom.

00:07:48.371 --> 00:07:50.683
"Can you describe how and why that works?"

00:07:50.683 --> 00:07:54.887
This is interesting because
I gave a Ted Talk in April,

00:07:54.887 --> 00:07:56.780
and I talk about flipping
the class room where,

00:07:56.780 --> 00:08:01.679
early on in Khan Academy I had made these
videos and I got letters from teachers,

00:08:01.679 --> 00:08:05.543
saying, "Hey look you already gave a
video on finding a limit of a function,

00:08:05.543 --> 00:08:09.338
"or adding and subtracting
negative numbers."

00:08:09.338 --> 00:08:11.602
So those teachers said, hey they don't
have to give that lecture anymore,

00:08:11.602 --> 00:08:14.150
so they're having those students
watch those lectures at home.

00:08:14.150 --> 00:08:17.404
Then they were freeing up
class time to do more problems,

00:08:17.404 --> 00:08:19.145
essentially what used to be homework.

00:08:19.145 --> 00:08:22.390
That is kind of a traditional pure flip.

00:08:23.805 --> 00:08:26.422
That by itself, I think,
is a very powerful thing.

00:08:26.422 --> 00:08:29.465
You're using class time,
once again, to interact.

00:08:29.465 --> 00:08:33.879
Just even that flip, by itself, even
before we talk about any other technology,

00:08:33.879 --> 00:08:36.830
some people say, "Well,
unmotivated students right now,

00:08:36.830 --> 00:08:38.733
"aren't doing their homework.
What makes you think,

00:08:38.733 --> 00:08:40.875
"that they're going to watch videos?"

00:08:40.875 --> 00:08:43.936
A partial answer is,
well we have technology,

00:08:43.936 --> 00:08:45.827
that the teachers can track
that and all the rest.

00:08:45.827 --> 00:08:48.827
But the more important
answer is, they might not.

00:08:48.827 --> 00:08:52.878
Although, I suspect they will
because it's easier to watch a video,

00:08:52.878 --> 00:08:55.502
than to do a problem that
you don't know how to do.

00:08:55.502 --> 00:08:58.673
That's most of the reason why I think
a lot of kids don't do their homework.

00:08:58.673 --> 00:09:00.824
Because they start and they
get frustrated and they stop.

00:09:00.824 --> 00:09:05.002
Even if you assume that the same
kid who didn't do homework before,

00:09:05.002 --> 00:09:07.862
isn't going to watch a
video now, what I say is,

00:09:07.862 --> 00:09:10.344
"Well, if you're going to skip out
on anything, it's better to skip out,

00:09:10.344 --> 00:09:13.868
"on the lecture than to
skip out on the exercises."

00:09:13.868 --> 00:09:15.204
What happens is with
the flipped classroom,

00:09:15.204 --> 00:09:17.080
at least when you're in
the room with the teacher,

00:09:17.080 --> 00:09:18.545
you're able to do exercises.

00:09:18.545 --> 00:09:20.344
You're able to do it with
the help of your friends,

00:09:20.344 --> 00:09:21.831
and whoever else is there.

00:09:21.831 --> 00:09:23.402
The real way that you're learning things,

00:09:23.402 --> 00:09:25.497
the exercises are what really
get you to the next level.

00:09:25.497 --> 00:09:28.661
The opposite, where you're using class
time to passively listen to a lecture,

00:09:28.661 --> 00:09:30.267
and then you don't do homework.

00:09:30.267 --> 00:09:31.935
That's the worst of everything.

00:09:31.935 --> 00:09:34.873
You're probably not paying attention to
the lecture so you're zoned out there.

00:09:34.873 --> 00:09:37.002
And you're also not doing the exercises.

00:09:37.002 --> 00:09:38.877
The ideal is obviously,
students watch the videos.

00:09:38.877 --> 00:09:40.907
We think they're much more
likely to watch videos,

00:09:40.907 --> 00:09:43.753
and everything we've observed
seems to verify that,

00:09:43.753 --> 00:09:47.442
that students want to learn it's
just that traditional homework,

00:09:47.442 --> 00:09:49.476
they often get frustrated because
they don't know how to do it,

00:09:49.476 --> 00:09:51.630
and there's no one there to help them.

00:09:51.630 --> 00:09:52.441
So you can flip it that way.

00:09:52.441 --> 00:09:54.065
With all of that said we want to be clear,

00:09:54.065 --> 00:09:56.803
that we're not just about
flipping the classroom anymore.

00:09:56.803 --> 00:09:58.938
I kind of mentioned the word
halfway into my Ted Talk,

00:09:58.938 --> 00:10:02.690
and I think the media and the
press love a very simple word,

00:10:02.690 --> 00:10:04.739
like flip the classroom
and homework at school,

00:10:04.739 --> 00:10:07.332
and schoolwork at home and
all of that type of thing.

00:10:07.332 --> 00:10:11.290
We're not actually saying that
this is the 100 percent ideal.

00:10:11.290 --> 00:10:13.738
What we're saying is, 100 percent
ideal is doing everything,

00:10:13.738 --> 00:10:16.522
to allow every student to work at
their own pace, at their own time.

00:10:16.522 --> 00:10:18.361
For every student to master concepts,

00:10:18.361 --> 00:10:20.196
so that they don't have
gaps in their knowledge,

00:10:20.196 --> 00:10:22.363
so they don't move on with
just C level of understanding,

00:10:22.363 --> 00:10:24.102
or B level of understanding.

00:10:24.102 --> 00:10:27.690
And to humanize the classroom
as much as possible.

00:10:27.690 --> 00:10:32.129
The flip is the start to do this,
but we think you can go even further,

00:10:32.129 --> 00:10:34.782
if you have self-paced
exercises and you have feedback,

00:10:34.782 --> 00:10:37.661
and you have dashboards, and you
have dashboards for teachers.

00:10:37.661 --> 00:10:39.078
I want to make it very clear.

00:10:39.078 --> 00:10:41.861
This is something that the
media has grabbed on to.

00:10:41.861 --> 00:10:46.659
That we kind of view as
a intermediary solution.

00:10:46.659 --> 00:10:49.457
But we think the ideal solution,
and we're still exploring,

00:10:49.457 --> 00:10:51.058
what that ideal solution is.

00:10:51.058 --> 00:10:52.938
But we think that the right answer is,

00:10:52.938 --> 00:10:55.396
don't just flip for the sake of flipping,

00:10:55.396 --> 00:10:57.615
or just because the word
flip is a nice buzz word,

00:10:57.615 --> 00:10:59.127
that people are excited about.

00:10:59.127 --> 00:11:01.664
Do everything in your power to
make class time as interactive,

00:11:01.664 --> 00:11:04.334
and as deep, and as
exploratory as possible.

00:11:04.334 --> 00:11:08.274
And use technology in ways to empower
kids to work at their own pace,

00:11:08.274 --> 00:11:10.521
at their own time, and
get as much data possible,

00:11:10.521 --> 00:11:14.521
to teachers and parents, and
frankly to the students themselves.