Proof for the meaning of Lagrange multipliers

WEBVTT
Kind: captions
Language: en

00:00:00.370 --> 00:00:01.540
- [Grant] All right, so last video

00:00:01.540 --> 00:00:03.960
I showed you guys this really crazy fact.

00:00:03.960 --> 00:00:05.440
Now, we have our usual setup here

00:00:05.440 --> 00:00:07.850
for this constrained
optimization situation.

00:00:07.850 --> 00:00:09.510
We have a function we wanna maximize,

00:00:09.510 --> 00:00:12.370
which I'm thinking of as
revenues for some company,

00:00:12.370 --> 00:00:14.210
a constraint, which I'm thinking of

00:00:14.210 --> 00:00:16.204
as some kind of budget for that company,

00:00:16.204 --> 00:00:18.500
and as you know if you've
gotten to this video,

00:00:18.500 --> 00:00:21.700
one way to solve this
constrained optimization problem

00:00:21.700 --> 00:00:24.730
is to define this function
here, the Lagrangian,

00:00:24.730 --> 00:00:26.180
which involves taking this function

00:00:26.180 --> 00:00:27.230
that you're trying to maximize,

00:00:27.230 --> 00:00:28.380
in this case the revenue,

00:00:28.380 --> 00:00:31.000
and subtracting a new variable, lambda,

00:00:31.000 --> 00:00:32.850
what's called the Lagrange multiplier,

00:00:32.850 --> 00:00:37.000
times this quantity, which
is the budget function,

00:00:37.000 --> 00:00:37.940
however much you spend

00:00:37.940 --> 00:00:39.840
as a function of your input parameters,

00:00:39.840 --> 00:00:41.150
minus the budget itself,

00:00:41.150 --> 00:00:44.040
which you might think of
as $10,000 in our example.

00:00:44.040 --> 00:00:45.440
So that's all the usual setup,

00:00:45.440 --> 00:00:47.880
and the crazy fact,
which is just declared,

00:00:47.880 --> 00:00:51.130
is that when you set this
gradient equal to zero,

00:00:51.130 --> 00:00:52.830
and you find some solution,

00:00:52.830 --> 00:00:54.840
and there will be three
variables in this solution:

00:00:54.840 --> 00:00:57.560
h*, s*, and lambda*,

00:00:57.560 --> 00:00:59.500
that this lambda* is not meaningless.

00:00:59.500 --> 00:01:01.850
It's not just a proportionality constant

00:01:01.850 --> 00:01:03.760
between these gradient vectors,

00:01:03.760 --> 00:01:05.570
but it will actually tell you

00:01:05.570 --> 00:01:08.510
how much the maximum
possible revenue changes

00:01:08.510 --> 00:01:10.640
as a function of your budget,

00:01:10.640 --> 00:01:13.200
and the way to start writing
all of that in formulas

00:01:13.200 --> 00:01:15.700
would be to make explicit the fact that,

00:01:15.700 --> 00:01:19.730
if you consider this value, the
$10,000 that is your budget,

00:01:19.730 --> 00:01:22.970
which I'm calling b, a
variable and not a constant,

00:01:22.970 --> 00:01:25.120
then you have to
acknowledge that h* and s*

00:01:26.100 --> 00:01:28.120
are dependent on b, right?

00:01:28.120 --> 00:01:29.720
It's a very implicit relationship,

00:01:29.720 --> 00:01:31.890
something that's kind of
hard to think about at first,

00:01:31.890 --> 00:01:35.270
because as you change b, it
changes what the Lagrangian is,

00:01:35.270 --> 00:01:37.730
which is gonna change where
its gradient equals zero,

00:01:37.730 --> 00:01:40.700
which changes what h*,
s*, and lambda* are,

00:01:40.700 --> 00:01:41.770
but in principle,

00:01:41.770 --> 00:01:44.723
they are some function
of that budget, of b,

00:01:46.300 --> 00:01:48.060
and the maximum possible revenue

00:01:48.060 --> 00:01:50.750
is whatever you get when you
just plug in that solution

00:01:50.750 --> 00:01:52.220
to your function r,

00:01:52.220 --> 00:01:54.830
and the claim I made, that I
just pulled out of the hat,

00:01:54.830 --> 00:01:56.250
is that lambda*,

00:01:56.250 --> 00:01:58.680
the lambda value that comes
packaged in with these two

00:01:58.680 --> 00:02:01.890
when you set the gradient of
the Lagrangian equal to zero,

00:02:01.890 --> 00:02:05.350
equals the derivative
of this maximum value,

00:02:05.350 --> 00:02:06.500
thought of as a function of b.

00:02:06.500 --> 00:02:08.070
Maybe I should emphasize that.

00:02:08.070 --> 00:02:09.330
We're thinking of this maximum value

00:02:09.330 --> 00:02:12.680
as a function of b, with respect to b.

00:02:12.680 --> 00:02:13.820
So that's kind of a mouthful.

00:02:13.820 --> 00:02:17.380
It takes a lot just to even
phrase what's going on,

00:02:17.380 --> 00:02:19.250
but in the context of an economic example,

00:02:19.250 --> 00:02:21.360
it has a very clear, precise meaning,

00:02:21.360 --> 00:02:25.280
which is, if you increase
your budget by a dollar,

00:02:25.280 --> 00:02:29.550
if you increase it from
$10,000 to $10,001,

00:02:29.550 --> 00:02:30.383
you're wondering,

00:02:30.383 --> 00:02:33.020
for that tiny change in
budget, that tiny db,

00:02:33.020 --> 00:02:36.150
what is the ratio of the
resulting change in revenue?

00:02:36.150 --> 00:02:38.960
So in a sense, this lambda* tells you,

00:02:38.960 --> 00:02:41.010
for every dollar that
you increase the budget,

00:02:41.010 --> 00:02:42.910
how much can your revenue increase

00:02:42.910 --> 00:02:44.740
if you're always maximizing it?

00:02:44.740 --> 00:02:47.070
So why on earth is this true, right?

00:02:47.070 --> 00:02:49.320
This just seems like it
comes out of nowhere.

00:02:50.700 --> 00:02:52.390
Well, there are a couple
clever observations

00:02:52.390 --> 00:02:53.920
that go into proving this.

00:02:53.920 --> 00:02:56.890
The first is to notice what happens

00:02:56.890 --> 00:03:00.770
if we evaluate this
Lagrangian function itself

00:03:00.770 --> 00:03:04.130
at this critical point, when you input

00:03:04.130 --> 00:03:07.857
h*, s*, and lambda*,

00:03:08.860 --> 00:03:11.530
and remember, the way that
these guys are defined

00:03:11.530 --> 00:03:15.040
is that you look at all of the values

00:03:15.040 --> 00:03:18.490
where the gradient of the
Lagrangian equals the zero vector,

00:03:18.490 --> 00:03:20.130
and then, if you get
multiple options, you know,

00:03:20.130 --> 00:03:21.930
sometimes when you set
the gradient equal to zero

00:03:21.930 --> 00:03:23.540
you get multiple solutions,

00:03:23.540 --> 00:03:27.947
and whichever one maximizes
R, that is h*, s*, lambda*.

00:03:28.880 --> 00:03:30.430
So now I'm just asking, if you plug that,

00:03:30.430 --> 00:03:32.360
not into the gradient of the Lagrangian,

00:03:32.360 --> 00:03:35.440
but to the Lagrangian
itself, what do you get?

00:03:35.440 --> 00:03:37.790
Well, you're going to get

00:03:37.790 --> 00:03:40.280
we just look at its definition up here,

00:03:40.280 --> 00:03:44.857
R evaluated at h* and s*,

00:03:46.070 --> 00:03:46.903
right?

00:03:47.810 --> 00:03:50.253
And we subtract off lambda*

00:03:52.020 --> 00:03:53.160
times

00:03:56.030 --> 00:04:00.637
B of h*, s*,

00:04:02.210 --> 00:04:04.490
minus the constant that
is your budget, you know,

00:04:04.490 --> 00:04:07.010
something you might think of as $10,000,

00:04:07.010 --> 00:04:08.760
whatever you set little b equal to.

00:04:09.930 --> 00:04:13.210
Okay, Grant, you might say,
why does this tell us anything?

00:04:13.210 --> 00:04:14.430
You're just plugging in stars

00:04:14.430 --> 00:04:16.150
instead of the usual variables,

00:04:16.150 --> 00:04:21.030
but the key is that if
you plug in h* and s*,

00:04:21.030 --> 00:04:23.400
this value has to equal zero

00:04:24.310 --> 00:04:28.020
because h* and s* have to
satisfy the constraint.

00:04:28.020 --> 00:04:29.360
Remember, one of the cool parts

00:04:29.360 --> 00:04:31.810
about this Lagrangian
function as a whole is that

00:04:31.810 --> 00:04:34.890
when you take its partial
derivative with respect to lambda,

00:04:34.890 --> 00:04:37.260
all that's left is this
constraint function

00:04:37.260 --> 00:04:38.980
minus the constraint portion.

00:04:38.980 --> 00:04:40.730
When you set the gradient
of the Lagrangian

00:04:40.730 --> 00:04:42.280
equal to the zero vector,

00:04:42.280 --> 00:04:43.540
one component of that

00:04:43.540 --> 00:04:46.620
is to set the partial derivative
with respect to lambda

00:04:46.620 --> 00:04:47.613
equal to zero,

00:04:49.987 --> 00:04:52.120
and if you remember from
the Lagrangian video,

00:04:52.120 --> 00:04:53.590
all that really boils down to

00:04:53.590 --> 00:04:56.100
is the fact that the
constraint holds, right?

00:04:56.100 --> 00:04:58.520
Which would be your
budget achieves $10,000.

00:04:58.520 --> 00:05:01.230
When you plug in the appropriate
h* and s* to this value,

00:05:01.230 --> 00:05:03.050
you are hitting this
constrained amount of money

00:05:03.050 --> 00:05:04.033
that you can spend.

00:05:04.950 --> 00:05:07.650
So by virtue of how h* and s* are defined,

00:05:07.650 --> 00:05:09.090
the fact that they are solutions

00:05:09.090 --> 00:05:11.150
to the constrained optimization problem

00:05:11.150 --> 00:05:13.303
means this whole portion goes to zero.

00:05:14.500 --> 00:05:18.340
So we can just kind of cancel
all that out, and what's left

00:05:18.340 --> 00:05:23.030
what's left here is the maximum
possible revenue, right?

00:05:23.030 --> 00:05:23.863
So evidently,

00:05:23.863 --> 00:05:26.760
when you evaluate the Lagrangian
at this critical point,

00:05:26.760 --> 00:05:30.980
at h*, s*, and lambda*, it equals M*.

00:05:30.980 --> 00:05:33.170
It equals the maximum possible value

00:05:33.170 --> 00:05:35.320
for the function you're
trying to maximize.

00:05:36.290 --> 00:05:37.550
So ultimately, what we want

00:05:37.550 --> 00:05:40.630
is to understand how that
maximum value changes

00:05:40.630 --> 00:05:43.060
when you consider it a
function of the budget.

00:05:43.060 --> 00:05:45.630
So evidently, what we can look for

00:05:45.630 --> 00:05:48.860
is to just ask how the Lagrangian changes

00:05:48.860 --> 00:05:51.333
as you consider it a
function of the budget.

00:05:52.620 --> 00:05:55.200
Now, this is an interesting
thing to observe,

00:05:55.200 --> 00:05:56.420
because if we just look up

00:05:56.420 --> 00:05:58.210
at the definition of the Lagrangian,

00:05:58.210 --> 00:06:00.100
if you just look at this formula,

00:06:00.100 --> 00:06:02.320
if I told you to take
the derivative of this

00:06:02.320 --> 00:06:04.330
with respect to little b,

00:06:04.330 --> 00:06:06.840
how much does this change
with respect to little b,

00:06:06.840 --> 00:06:08.410
you would notice that this goes to zero.

00:06:08.410 --> 00:06:09.870
It doesn't have a little b.

00:06:09.870 --> 00:06:11.533
This would also go to zero,

00:06:12.520 --> 00:06:13.690
and all you'd be left with

00:06:13.690 --> 00:06:16.940
would be negative lambda times negative b,

00:06:16.940 --> 00:06:18.970
and the derivative of
that with respect to b

00:06:18.970 --> 00:06:20.513
would be lambda.

00:06:21.400 --> 00:06:23.570
So you might say, oh yeah,
of course, of course,

00:06:23.570 --> 00:06:26.270
the derivative of that
Lagrangian with respect to b,

00:06:26.270 --> 00:06:27.200
once we work it all out,

00:06:27.200 --> 00:06:29.813
the only term that was
left there was the lambda,

00:06:30.920 --> 00:06:32.260
and that's compelling,

00:06:32.260 --> 00:06:35.560
but ultimately it's not entirely right.

00:06:35.560 --> 00:06:36.530
That overlooks the fact

00:06:36.530 --> 00:06:39.810
that L is not actually
defined as a function of b.

00:06:39.810 --> 00:06:41.600
When we defined the Lagrangian,

00:06:41.600 --> 00:06:44.090
we were considering b to be a constant.

00:06:44.090 --> 00:06:45.900
So if you really wanna consider this

00:06:45.900 --> 00:06:48.403
to be a function that involves b,

00:06:49.340 --> 00:06:51.460
the way we should write it,

00:06:51.460 --> 00:06:53.700
and I'll go ahead and erase this guy,

00:06:53.700 --> 00:06:55.870
the way we should write this Lagrangian

00:06:55.870 --> 00:07:00.010
is to say, you are a function of h*,

00:07:00.010 --> 00:07:01.913
which itself is dependent on b,

00:07:06.160 --> 00:07:10.070
and s*, which is also
a function of b, right?

00:07:10.070 --> 00:07:12.290
As soon as we start
considering b a variable

00:07:12.290 --> 00:07:13.320
and not a constant,

00:07:13.320 --> 00:07:15.540
we have to acknowledge
that this critical point,

00:07:15.540 --> 00:07:19.360
h*, s*, and lambda*,
depends on the value of b.

00:07:19.360 --> 00:07:21.267
So likewise, that lambda*,

00:07:22.850 --> 00:07:26.453
lambda* is also gonna be a function of b,

00:07:27.700 --> 00:07:30.440
and then we can consider,
as a fourth variable, right,

00:07:30.440 --> 00:07:33.323
so we're adding on yet another
variable to this function,

00:07:35.443 --> 00:07:36.833
the value of b itself,

00:07:38.760 --> 00:07:40.303
the value of b itself here.

00:07:41.490 --> 00:07:42.860
So now, when we wanna know,

00:07:42.860 --> 00:07:44.920
what is the value of the Lagrangian

00:07:44.920 --> 00:07:47.730
at the critical point, h*, s*, lambda*,

00:07:47.730 --> 00:07:49.380
as a function of b, all right?

00:07:49.380 --> 00:07:51.850
So that can be kind of confusing.

00:07:51.850 --> 00:07:55.060
What you basically have is this function

00:07:55.060 --> 00:07:57.090
that only really depends
on one value, right?

00:07:57.090 --> 00:07:58.570
It only depends on b,

00:07:58.570 --> 00:08:01.603
but it kinda goes through
a four-variable function,

00:08:02.960 --> 00:08:04.290
and so, just to make it explicit,

00:08:04.290 --> 00:08:08.530
this would equal the value of R

00:08:08.530 --> 00:08:10.227
as a function of h*, and s*,

00:08:13.690 --> 00:08:16.683
and each one of those is a
function of little b, right?

00:08:20.290 --> 00:08:22.230
So this term is saying,
what's your revenue?

00:08:22.230 --> 00:08:26.133
Evaluate it on the maximizing
h and s for the given budget,

00:08:27.090 --> 00:08:30.037
and then you subtract off lambda*.

00:08:31.780 --> 00:08:32.730
Oh here, I should probably

00:08:32.730 --> 00:08:34.990
I'm not gonna have room here, am I?

00:08:34.990 --> 00:08:36.793
So what you subtract off,

00:08:38.550 --> 00:08:40.083
minus lambda*

00:08:42.230 --> 00:08:45.230
at B

00:08:45.230 --> 00:08:46.947
of h*,

00:08:51.046 --> 00:08:51.879
and s*,

00:08:52.910 --> 00:08:55.610
but each of these guys is
also a function of little b,

00:08:56.670 --> 00:08:57.503
little b,

00:08:59.330 --> 00:09:01.853
minus little b, right?

00:09:06.150 --> 00:09:07.390
So you have this large

00:09:07.390 --> 00:09:09.470
kind of complicated
multivariable function.

00:09:09.470 --> 00:09:12.210
It's defined in terms of h*'s and s*'s,

00:09:12.210 --> 00:09:14.510
which are themselves very implicit, right?

00:09:14.510 --> 00:09:15.930
We just say, by definition

00:09:15.930 --> 00:09:20.330
these are whatever values make
the gradient of L equal zero.

00:09:20.330 --> 00:09:23.820
So very hard to think about
what that means concretely,

00:09:23.820 --> 00:09:25.700
but all of this is really just dependent

00:09:25.700 --> 00:09:27.563
on the single value, little b,

00:09:29.700 --> 00:09:34.050
and from here, if we wanna
evaluate the derivative of L,

00:09:34.050 --> 00:09:38.340
we wanna evaluate the
derivative of this Lagrangian

00:09:38.340 --> 00:09:40.193
with respect to little b,

00:09:41.190 --> 00:09:42.880
which is really the only
thing it depends on.

00:09:42.880 --> 00:09:46.950
It's just via all of
these other variables,

00:09:46.950 --> 00:09:48.773
we use the multivariable chain rule,

00:09:50.560 --> 00:09:51.530
and at this point,

00:09:51.530 --> 00:09:53.690
if you don't know the
multivariable chain rule,

00:09:53.690 --> 00:09:54.730
I have a video on that.

00:09:54.730 --> 00:09:55.860
Definitely pause.

00:09:55.860 --> 00:09:56.693
Go take a look.

00:09:56.693 --> 00:09:58.260
Make sure that it all makes sense.

00:09:58.260 --> 00:09:59.740
But right here, I'm just gonna be assuming

00:09:59.740 --> 00:10:02.560
that you know what the
multivariable chain rule is.

00:10:02.560 --> 00:10:05.050
So what it is, is we take the partial

00:10:05.050 --> 00:10:06.660
we're gonna look at
the partial derivatives

00:10:06.660 --> 00:10:09.020
with respect to all four of these inputs.

00:10:09.020 --> 00:10:09.853
So we'll start with

00:10:09.853 --> 00:10:13.257
the partial derivative
of L with respect to h*,

00:10:14.170 --> 00:10:15.387
with respect to h*,

00:10:16.550 --> 00:10:18.000
and we're gonna multiply that

00:10:18.000 --> 00:10:22.763
by the derivative of h* with respect to b,

00:10:24.500 --> 00:10:26.450
and this might seem like a
very hard thing to think about,

00:10:26.450 --> 00:10:28.970
like, how do we know how
h* changes as b changes?

00:10:28.970 --> 00:10:30.520
But don't worry about it.

00:10:30.520 --> 00:10:33.090
You'll see something magic
happen in just a moment.

00:10:33.090 --> 00:10:36.220
And then we add in partial derivative of L

00:10:36.220 --> 00:10:38.377
with respect to that second variable, s*,

00:10:39.410 --> 00:10:42.280
with respect to whatever
the second variable is,

00:10:42.280 --> 00:10:45.143
multiplied by the derivative of s*

00:10:46.930 --> 00:10:48.133
with respect to b.

00:10:49.550 --> 00:10:51.090
You can see how you really need to know

00:10:51.090 --> 00:10:52.620
what the multivariable
chain rule is, right?

00:10:52.620 --> 00:10:55.520
This would all seem
kind of out of the blue.

00:10:55.520 --> 00:10:56.460
So what we now add in

00:10:56.460 --> 00:11:00.687
is the partial derivative of L
with respect to that lambda*,

00:11:01.950 --> 00:11:03.150
with respect to lambda*,

00:11:06.490 --> 00:11:09.533
multiplied by the derivative of lambda*

00:11:11.670 --> 00:11:13.033
with respect to little b,

00:11:15.060 --> 00:11:16.063
and then finally,

00:11:16.980 --> 00:11:21.520
finally, we take the partial
derivative of this Lagrangian

00:11:21.520 --> 00:11:23.253
with respect to that little b,

00:11:24.950 --> 00:11:26.960
which we're now considering
a variable in there, right?

00:11:26.960 --> 00:11:29.050
We're no longer considering
that b a constant.

00:11:29.050 --> 00:11:31.420
Multiplied by, well,
something kinda silly:

00:11:31.420 --> 00:11:34.493
the derivative of b
with respect to itself.

00:11:37.720 --> 00:11:38.760
So now, if you're thinking

00:11:38.760 --> 00:11:41.040
that this is gonna be
horrifying to compute,

00:11:41.040 --> 00:11:42.650
I can understand where you're coming from.

00:11:42.650 --> 00:11:44.940
You have to know the derivative of lambda*

00:11:44.940 --> 00:11:46.230
with respect to b.

00:11:46.230 --> 00:11:48.230
You have to somehow
intimately be familiar with

00:11:48.230 --> 00:11:51.470
how this lambda* changes as
you change b, and like I said,

00:11:51.470 --> 00:11:53.740
that's such an implicit
relationship, right?

00:11:53.740 --> 00:11:56.090
We just said that lambda*
is, by definition,

00:11:56.090 --> 00:11:58.980
whatever the solution to
this gradient equation is.

00:11:58.980 --> 00:12:00.250
So somehow you're supposed to know

00:12:00.250 --> 00:12:04.460
how that changes when you
slightly alter b over here?

00:12:04.460 --> 00:12:07.070
Well, you don't really
have to worry about things,

00:12:07.070 --> 00:12:10.510
because by definition, h*, s*, and lambda*

00:12:10.510 --> 00:12:14.050
are whatever values make the
gradient of L equal to zero,

00:12:14.050 --> 00:12:15.610
but if you think about
that, what does it mean

00:12:15.610 --> 00:12:18.640
for the gradient of L to
equal the zero vector.

00:12:18.640 --> 00:12:21.530
Well, what it means is that
when you take its derivative

00:12:21.530 --> 00:12:25.600
with respect to that first
variable, h*, it equals zero.

00:12:25.600 --> 00:12:26.570
When you take its derivative

00:12:26.570 --> 00:12:28.450
with respect to the second variable,

00:12:28.450 --> 00:12:30.100
that equals zero as well,

00:12:30.100 --> 00:12:32.300
and with respect to this third variable,

00:12:32.300 --> 00:12:33.513
that's gonna equal zero.

00:12:35.270 --> 00:12:38.530
By definition, h*, s*, and lambda*,

00:12:38.530 --> 00:12:40.310
are whatever values make it the case,

00:12:40.310 --> 00:12:41.920
so that when you plug them in,

00:12:41.920 --> 00:12:44.420
the partial derivative of the Lagrangian,

00:12:44.420 --> 00:12:48.110
with respect to any one of
those variables, equal zero.

00:12:48.110 --> 00:12:51.100
So we don't even have to worry
about most of this equation.

00:12:51.100 --> 00:12:52.992
The only part that matters here

00:12:52.992 --> 00:12:56.290
is the partial derivative
of L with respect to b,

00:12:56.290 --> 00:12:58.200
that we're now considering a variable,

00:12:58.200 --> 00:13:00.923
multiplied by, well, what's db/db?

00:13:01.770 --> 00:13:03.150
What is the rate of change

00:13:03.150 --> 00:13:05.470
of a variable with respect to itself?

00:13:05.470 --> 00:13:06.303
It's one.

00:13:07.590 --> 00:13:08.423
It is one.

00:13:09.530 --> 00:13:12.830
So all of this stuff, this
entire multivariable chain rule,

00:13:12.830 --> 00:13:16.363
boils down to a single
innocent-looking factor,

00:13:17.380 --> 00:13:20.000
which is the partial derivative of L,

00:13:20.000 --> 00:13:23.520
partial derivative of L
with respect to little b,

00:13:23.520 --> 00:13:26.490
and now there's something
very subtle here, right?

00:13:26.490 --> 00:13:27.450
Because this might seem obvious.

00:13:27.450 --> 00:13:29.430
I'm saying the derivative
of L with respect to b

00:13:29.430 --> 00:13:31.823
equals the derivative
of L with respect to b,

00:13:33.520 --> 00:13:35.620
but maybe I should give a
different notation here, right?

00:13:35.620 --> 00:13:38.040
Because here, when I'm taking
the derivative, really,

00:13:38.040 --> 00:13:41.180
I'm considering L as a
single-variable function, right?

00:13:41.180 --> 00:13:42.760
I'm considering, not what happens

00:13:42.760 --> 00:13:45.700
as you can freely change
all four of these variables.

00:13:45.700 --> 00:13:48.140
Three of them are locked into place by b.

00:13:48.140 --> 00:13:50.010
So maybe I should really
give that a different name.

00:13:50.010 --> 00:13:52.070
I should call that L*.

00:13:52.070 --> 00:13:54.400
L* is a single-variable function,

00:13:54.400 --> 00:13:57.840
whereas this L is a
multivariable function.

00:13:57.840 --> 00:14:00.690
This is the function where you
can freely change the values

00:14:00.690 --> 00:14:04.530
of h, and s, and lambda,
and b, as you put them in.

00:14:04.530 --> 00:14:06.840
So if we kind of scroll up
to look at its definition,

00:14:06.840 --> 00:14:08.200
which I've written all over, I guess.

00:14:08.200 --> 00:14:10.130
Here, let me actually rewrite
its definition, right?

00:14:10.130 --> 00:14:11.450
I think that'll be useful.

00:14:11.450 --> 00:14:13.180
I'm gonna rewrite that L,

00:14:13.180 --> 00:14:17.233
if I consider it as a
four-variable function of h, s,

00:14:18.140 --> 00:14:21.333
lambda, and b,

00:14:23.550 --> 00:14:26.990
that what that equals is R

00:14:28.010 --> 00:14:30.180
evaluated at h and s,

00:14:30.180 --> 00:14:32.523
minus lambda,

00:14:36.380 --> 00:14:39.230
multiplied by this constraint function B

00:14:39.230 --> 00:14:42.783
evaluated at h and s, minus little b,

00:14:44.290 --> 00:14:45.123
and this is now

00:14:45.123 --> 00:14:47.200
when I'm considering
little b to be a variable.

00:14:47.200 --> 00:14:48.700
So this is the Lagrangian

00:14:48.700 --> 00:14:50.120
when you consider all four of these

00:14:50.120 --> 00:14:52.240
to be freely changing as you want,

00:14:52.240 --> 00:14:54.810
whereas the thing up here

00:14:54.810 --> 00:14:56.870
that I'm considering a
single-variable function,

00:14:56.870 --> 00:14:58.930
has three of its inputs locked into place.

00:14:58.930 --> 00:14:59.763
So effectively,

00:14:59.763 --> 00:15:02.620
it's just a single-variable
function with respect to b.

00:15:02.620 --> 00:15:04.090
So it's actually quite miraculous

00:15:04.090 --> 00:15:07.440
that the single-variable
derivative of that L, here,

00:15:07.440 --> 00:15:10.950
I should L*, with respect to b,

00:15:10.950 --> 00:15:14.080
ends up being the same as
the partial derivative of L,

00:15:14.080 --> 00:15:16.570
this L, where you're free
to change all the variables,

00:15:16.570 --> 00:15:18.090
that these should be the same.

00:15:18.090 --> 00:15:20.390
Usually, in any usual circumstance,

00:15:20.390 --> 00:15:23.340
all of these other terms would
have come into play somehow,

00:15:24.540 --> 00:15:26.220
but what's special here is that

00:15:26.220 --> 00:15:28.570
by the definition of this L*,

00:15:28.570 --> 00:15:30.100
the specific way

00:15:30.100 --> 00:15:33.940
in which these h*, s*, and
lambda*'s are locked into place

00:15:33.940 --> 00:15:34.920
happens to be one

00:15:34.920 --> 00:15:38.530
in which all of these partial
derivatives go to zero.

00:15:38.530 --> 00:15:42.010
So that's pretty subtle, and
I think it's quite clever,

00:15:42.010 --> 00:15:44.000
and what it leaves us with is that

00:15:44.000 --> 00:15:46.640
we just have to evaluate
this partial derivative,

00:15:46.640 --> 00:15:47.473
which is quite simple,

00:15:47.473 --> 00:15:49.060
'cause we look down here, and you say,

00:15:49.060 --> 00:15:52.140
what's the partial derivative
of L with respect to b?

00:15:52.140 --> 00:15:54.300
Well, this R has no b's in it,

00:15:54.300 --> 00:15:56.310
so don't need to care about that.

00:15:56.310 --> 00:16:00.880
This term over here, its partial
derivative is negative one

00:16:00.880 --> 00:16:02.180
just 'cause there's a b here,

00:16:02.180 --> 00:16:04.430
and that's multiplied
by the constant lambda,

00:16:05.380 --> 00:16:07.643
so that all just equals lambda,

00:16:10.020 --> 00:16:11.320
but if we're in the situation

00:16:11.320 --> 00:16:14.480
where lambda is locked into
place as a function of little b,

00:16:14.480 --> 00:16:19.123
then we'd write lambda* as a
function of little b, right?

00:16:20.900 --> 00:16:23.960
So if that feels a little
notationally confusing,

00:16:23.960 --> 00:16:25.110
I'm right there with you,

00:16:25.110 --> 00:16:28.540
but the important part here,
the important thing to remember

00:16:28.540 --> 00:16:32.183
is that we just started
considering b as a variable,

00:16:33.100 --> 00:16:36.310
and we were looking at
the h*, s*, and lambda*,

00:16:36.310 --> 00:16:38.850
as they depended on that variable.

00:16:38.850 --> 00:16:41.610
We made the observation
that the Lagrangian,

00:16:41.610 --> 00:16:43.720
evaluated at that critical point,

00:16:43.720 --> 00:16:46.330
equals the revenue evaluated
at that critical point.

00:16:46.330 --> 00:16:48.380
The rest of the stuff just cancels out.

00:16:48.380 --> 00:16:51.700
So if you wanna know the derivative of M*,

00:16:51.700 --> 00:16:55.190
the maximizing revenue,
with respect to the budget,

00:16:55.190 --> 00:16:56.820
how much does your maximum revenue change

00:16:56.820 --> 00:16:58.570
for tiny changes in your budget?

00:16:58.570 --> 00:17:02.010
That's the same as
looking at the derivative

00:17:02.010 --> 00:17:04.570
of the Lagrangian with
respect to the budget,

00:17:04.570 --> 00:17:06.000
so long as you're considering it

00:17:06.000 --> 00:17:09.020
only on values h*, s*, lambda*,

00:17:09.020 --> 00:17:11.640
that are critical points
of the Lagrangian,

00:17:11.640 --> 00:17:13.850
and all of that really nicely boils down

00:17:13.850 --> 00:17:16.200
to just taking a simple partial derivative

00:17:16.200 --> 00:17:17.950
that gives us the relation we want.


Creating scale drawings

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:02.470
- [Instructor] Sue is a software engineer.

00:00:02.470 --> 00:00:04.470
She wants to create a large scale drawing

00:00:04.470 --> 00:00:06.860
of a processor inside a cell phone.

00:00:06.860 --> 00:00:09.060
The processor is a square chip,

00:00:09.060 --> 00:00:11.540
nine millimeters on each side.

00:00:11.540 --> 00:00:14.530
Draw the processor such that one unit

00:00:14.530 --> 00:00:18.460
on the grid below represents
one half of a millimeter.

00:00:18.460 --> 00:00:21.670
All right, so let's just
think about these two worlds.

00:00:21.670 --> 00:00:23.180
You have the drawing,

00:00:23.180 --> 00:00:26.160
and you have the chip in the real world.

00:00:26.160 --> 00:00:27.920
The drawing, we're thinking in terms

00:00:27.920 --> 00:00:30.180
of these units here on this graph paper,

00:00:30.180 --> 00:00:31.460
so units.

00:00:31.460 --> 00:00:35.150
And the chip, we're thinking
in terms of millimeters.

00:00:35.150 --> 00:00:37.403
And so we can set up a table here.

00:00:38.270 --> 00:00:41.470
And they tell us draw
it such that one unit

00:00:41.470 --> 00:00:43.320
represents half a millimeter.

00:00:43.320 --> 00:00:47.760
So one unit would represent
one half of a millimeter.

00:00:47.760 --> 00:00:49.660
They tell us that the processor

00:00:49.660 --> 00:00:52.580
is nine millimeters on each side.

00:00:52.580 --> 00:00:55.670
So that is how many times
as many millimeters?

00:00:55.670 --> 00:00:57.130
Well, to go from one half to nine,

00:00:57.130 --> 00:00:59.460
you have to multiply by 18.

00:00:59.460 --> 00:01:04.150
So that's going to be 18
times as many units as well.

00:01:04.150 --> 00:01:07.410
So if I want to do a scale
drawing right over here,

00:01:07.410 --> 00:01:09.110
it's going to be a square.

00:01:09.110 --> 00:01:10.600
It's a square chip.

00:01:10.600 --> 00:01:13.680
And since the scale is one unit,

00:01:13.680 --> 00:01:16.030
it represents half a millimeter.

00:01:16.030 --> 00:01:18.930
18 units would represent nine millimeters,

00:01:18.930 --> 00:01:22.860
and so I would want to
do 18 units on a side.

00:01:22.860 --> 00:01:24.170
So one,

00:01:24.170 --> 00:01:25.550
two, three,

00:01:25.550 --> 00:01:27.570
four, five, six,

00:01:27.570 --> 00:01:29.600
seven, eight, nine,

00:01:29.600 --> 00:01:31.250
10, 11,

00:01:31.250 --> 00:01:32.973
12, 13,

00:01:32.973 --> 00:01:34.339
14, 15,

00:01:34.339 --> 00:01:35.256
16, 17, 18.

00:01:36.520 --> 00:01:38.490
So that would be the
left side of the chip.

00:01:38.490 --> 00:01:40.410
Let me just see if I can
draw a straight line there,

00:01:40.410 --> 00:01:41.470
just like that.

00:01:41.470 --> 00:01:43.900
And then it would also be 18 wide,

00:01:43.900 --> 00:01:44.960
18 units wide,

00:01:44.960 --> 00:01:47.450
representing nine millimeters
in the real world.

00:01:47.450 --> 00:01:52.450
One, two, three, four, five,
six, seven, eight, nine, 10,

00:01:52.640 --> 00:01:53.847
11, 12,

00:01:53.847 --> 00:01:54.778
13, 14,

00:01:54.778 --> 00:01:55.880
15, 16,

00:01:55.880 --> 00:01:56.713
17, 18.

00:01:56.713 --> 00:01:58.430
This is making my eyes hurt.

00:01:58.430 --> 00:02:01.080
All right, so just like that,

00:02:01.080 --> 00:02:03.780
and then I could draw
the other parts of it.

00:02:03.780 --> 00:02:05.830
It would be like that,

00:02:05.830 --> 00:02:09.110
and then I would just go
straight down like that.

00:02:09.110 --> 00:02:10.520
And there you have it,

00:02:10.520 --> 00:02:15.140
I have drawn a scale
drawing of our processor,

00:02:15.140 --> 00:02:16.473
and we are all done.


Scale drawings

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:03.250
- [Instructor] We're told a
scale on a blue print drawing

00:00:03.250 --> 00:00:08.250
of a house shows 10 centimeters
represents 2 meters.

00:00:08.870 --> 00:00:12.330
What number of actual
meters are represented

00:00:12.330 --> 00:00:15.620
by 18 centimeters on the blue print?

00:00:15.620 --> 00:00:18.860
So pause this video and see
if you can figure it out.

00:00:18.860 --> 00:00:22.360
So the main thing to realize
is that a blue print drawing

00:00:22.360 --> 00:00:25.400
is a scale drawing of
something in the real world.

00:00:25.400 --> 00:00:27.400
In this case of a house.

00:00:27.400 --> 00:00:29.360
And so what we can do
is set-up a little bit

00:00:29.360 --> 00:00:32.150
of a table here, so let's put the drawing

00:00:32.150 --> 00:00:36.080
on the left, so this is
the blue print drawing.

00:00:36.080 --> 00:00:38.322
And then this is the real world.

00:00:38.322 --> 00:00:43.322
Real world on the right.

00:00:43.530 --> 00:00:46.400
And the unit that we're using,
the units that we're using

00:00:46.400 --> 00:00:48.490
in our drawing are centimeters.

00:00:48.490 --> 00:00:51.100
So we have centimeters over here.

00:00:51.100 --> 00:00:55.520
And then the real world, we're
thinking in terms of meters.

00:00:55.520 --> 00:00:58.030
And so let me make a
little bit of a table.

00:00:58.030 --> 00:01:01.540
And so we see that 10
centimeters on the drawing

00:01:01.540 --> 00:01:03.940
corresponds to 2 meters in the real world.

00:01:03.940 --> 00:01:08.420
So 10 centimeters in the
drawing, corresponds to 2 meters

00:01:08.420 --> 00:01:10.060
in the real world.

00:01:10.060 --> 00:01:12.450
Then they ask us what
number of actual meters

00:01:12.450 --> 00:01:16.070
are represented by 18
centimeters on the blue print?

00:01:16.070 --> 00:01:19.250
So 18 centimeters on the
blue print would correspond

00:01:19.250 --> 00:01:21.120
to what in the real world?

00:01:21.120 --> 00:01:23.020
Well there's several ways to approach it.

00:01:23.020 --> 00:01:24.440
One way to think about it is,

00:01:24.440 --> 00:01:29.313
look to go from 10 to 18, we
are going to multiply by 1.8.

00:01:30.860 --> 00:01:34.270
So to go from what 10
centimeters represents

00:01:34.270 --> 00:01:36.960
in the real world to what
18 centimeters represents

00:01:36.960 --> 00:01:41.790
in the real world, you would
similarly multiply by 1.8.

00:01:41.790 --> 00:01:45.053
So times 1.8, which would give us 3.6.

00:01:45.955 --> 00:01:47.370
3.6 what?

00:01:47.370 --> 00:01:50.059
3.6 meters in the real world.

00:01:50.059 --> 00:01:53.680
So 3.6 meters in the real world would be

00:01:53.680 --> 00:01:57.160
what 18 centimeters on
the blue print represents.

00:01:57.160 --> 00:01:59.005
Let's do another example.

00:01:59.005 --> 00:02:03.250
Jaylynn draws a hen
with a scale of 2 units

00:02:03.250 --> 00:02:06.690
on her graph paper represents 6 meters,

00:02:06.690 --> 00:02:09.850
6 centimeters in the real world.

00:02:09.850 --> 00:02:13.810
The hen is 16 units tall in the drawing.

00:02:13.810 --> 00:02:16.880
What is the height in
centimeters of the actual hen?

00:02:16.880 --> 00:02:18.230
So once again, pause this video

00:02:18.230 --> 00:02:20.600
and see if you can figure that out.

00:02:20.600 --> 00:02:23.140
Alright, so let's just
set-up our tables again.

00:02:23.140 --> 00:02:26.710
So for our table, so we have our drawing,

00:02:26.710 --> 00:02:31.590
and then we have the real, real world.

00:02:31.590 --> 00:02:34.510
And in our drawing, it's
just these little units

00:02:34.510 --> 00:02:37.600
on our graph paper, so I'll
call it just that, units.

00:02:37.600 --> 00:02:39.450
And the real world,
we're thinking in terms

00:02:39.450 --> 00:02:41.830
of centimeters, centimeters.

00:02:41.830 --> 00:02:44.920
And so let me set-up a
little bit of a table here,

00:02:44.920 --> 00:02:48.080
and so we know, and we
see that right over here

00:02:48.080 --> 00:02:49.350
from this scale.

00:02:49.350 --> 00:02:53.020
We can see two units
represent 6 centimeters

00:02:53.020 --> 00:02:54.190
in the real world.

00:02:54.190 --> 00:02:57.630
So two units in our drawing
represent six centimeters

00:02:57.630 --> 00:03:00.320
in the real world, and
then they say the hen

00:03:00.320 --> 00:03:03.380
is 16 units tall in the drawing.

00:03:03.380 --> 00:03:05.800
So it's 16 units tall in the drawing.

00:03:05.800 --> 00:03:08.350
What would that represent
in the real world

00:03:08.350 --> 00:03:11.320
which would be the
actual height of the hen?

00:03:11.320 --> 00:03:13.830
If two units represent six centimeters,

00:03:13.830 --> 00:03:17.730
and now we have eight times as many units,

00:03:17.730 --> 00:03:21.980
well that's gonna represent
eight times as many centimeters.

00:03:21.980 --> 00:03:25.030
So six times eight is 48 centimeters.

00:03:25.030 --> 00:03:27.870
So what is the height in
centimeters of the actual hen?

00:03:27.870 --> 00:03:32.403
48 centimeters, 48 and we're done.


Making scaled copies

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:04.830
- [Instructor] Figure A is
a scaled copy of figure B.

00:00:04.830 --> 00:00:07.980
And then they say what is the value of x?

00:00:07.980 --> 00:00:11.400
Pause this video and see
if you can figure that out.

00:00:11.400 --> 00:00:16.400
All right, given that figure A
is a scaled copy of figure B,

00:00:16.500 --> 00:00:18.270
that would also mean that figure B is

00:00:18.270 --> 00:00:20.920
a scaled copy of figure A.

00:00:20.920 --> 00:00:23.970
And so we would have a common scale factor

00:00:23.970 --> 00:00:25.910
between corresponding sides.

00:00:25.910 --> 00:00:28.920
So, for example, this
side right over here,

00:00:28.920 --> 00:00:30.940
it looks like it's playing the same role

00:00:30.940 --> 00:00:33.140
as this side right over here.

00:00:33.140 --> 00:00:35.090
And so what would be the scale factor

00:00:35.090 --> 00:00:37.230
to go from 10 to 12.5?

00:00:37.230 --> 00:00:41.790
What do I have to multiply
10 by to get 12.5?

00:00:41.790 --> 00:00:45.260
Well, I'm multiplying
it by a scale factor.

00:00:45.260 --> 00:00:49.580
I'm using a scale factor of 1.25.

00:00:49.580 --> 00:00:52.620
I'm multiplying times 1.25,

00:00:52.620 --> 00:00:54.030
or you could also think of it

00:00:54.030 --> 00:00:57.580
as I'm multiplying times 1 1/4,

00:00:57.580 --> 00:01:00.230
either way of expressing the same number.

00:01:00.230 --> 00:01:02.450
Well, if I'm thinking about x,

00:01:02.450 --> 00:01:05.540
if I want to know this
length right over here,

00:01:05.540 --> 00:01:08.440
corresponding to this side on figure B

00:01:08.440 --> 00:01:10.680
would be this side on figure A,

00:01:10.680 --> 00:01:12.450
which has length 16.

00:01:12.450 --> 00:01:15.930
And so I would use the same scale factor.

00:01:15.930 --> 00:01:19.093
So my scale factor here,
once again, would be 1 1/4.

00:01:20.210 --> 00:01:22.903
So I'll multiply times 1 1/4.

00:01:23.770 --> 00:01:26.820
And 1/4 of 16 is four.

00:01:26.820 --> 00:01:30.000
So it's going to be 16
and then another four

00:01:30.000 --> 00:01:32.833
is going to be equal to 20.

00:01:32.833 --> 00:01:37.550
1.25 times 16 is equal
to 20, and we're done.

00:01:37.550 --> 00:01:39.483
Let's do another example.

00:01:40.570 --> 00:01:45.570
Here we're told figure A is
a scaled copy of figure B.

00:01:45.770 --> 00:01:46.780
All right.

00:01:46.780 --> 00:01:49.560
Figure A is made from figure B

00:01:49.560 --> 00:01:53.072
using a scale factor of 5/2.

00:01:53.072 --> 00:01:55.260
What is the value of x?

00:01:55.260 --> 00:01:58.340
Pause this video and see
if you can figure that out.

00:01:58.340 --> 00:02:00.030
Well, when they tell us that figure A

00:02:00.030 --> 00:02:04.550
is made from figure B using
a scale factor of 5/2,

00:02:04.550 --> 00:02:07.790
that means when we're making figure A,

00:02:07.790 --> 00:02:09.990
if we look at the
corresponding side on figure B,

00:02:09.990 --> 00:02:12.640
so we have this side on figure B.

00:02:12.640 --> 00:02:16.260
It looks like it corresponds
to this side on figure A.

00:02:16.260 --> 00:02:19.960
And so to go from five to x,

00:02:19.960 --> 00:02:22.190
we would use a scale factor of 5/2.

00:02:22.190 --> 00:02:25.430
We're gonna multiply by 5/2.

00:02:25.430 --> 00:02:27.920
Well, what is five times 5/2?

00:02:27.920 --> 00:02:31.170
Let me just write here, x
is going to be equal to five

00:02:31.170 --> 00:02:33.960
times our scale factor, times 5/2,

00:02:33.960 --> 00:02:37.540
which is going to be equal to 25.

00:02:37.540 --> 00:02:38.690
Let me write it this way.

00:02:38.690 --> 00:02:42.000
It's going to be equal to 25 over two,

00:02:42.000 --> 00:02:46.750
which, if we want, we
could write it as 12.5.

00:02:46.750 --> 00:02:50.430
So x right over here, the
length right over here, is 12.5.


Identifying corresponding parts of scaled copies

WEBVTT
Kind: captions
Language: en

00:00:00.200 --> 00:00:01.290
- [Instructor] We are told

00:00:01.290 --> 00:00:05.840
that figure two is a
scaled copy of figure one.

00:00:05.840 --> 00:00:10.220
And we can verify that by
comparing corresponding sides.

00:00:10.220 --> 00:00:11.870
Corresponding sides are sides

00:00:11.870 --> 00:00:14.790
that have the same relative position,

00:00:14.790 --> 00:00:16.760
the same, they're playing
the same role in each

00:00:16.760 --> 00:00:20.330
of the diagrams even if the
diagrams are scaled versions

00:00:20.330 --> 00:00:22.690
of each other, even if
they are different sizes.

00:00:22.690 --> 00:00:24.880
So for example, if we were

00:00:24.880 --> 00:00:28.190
to compare segment ea right over here,

00:00:28.190 --> 00:00:30.753
it looks like it
corresponds to segment op.

00:00:31.960 --> 00:00:34.260
And the length of ea is three,

00:00:34.260 --> 00:00:38.940
the length of op is one,
two, three, four, five, six.

00:00:38.940 --> 00:00:41.710
And so for this to be a scaled copy,

00:00:41.710 --> 00:00:43.760
the scaling factor from,

00:00:43.760 --> 00:00:45.830
what from the corresponding
side in figure one

00:00:45.830 --> 00:00:46.860
to the corresponding side

00:00:46.860 --> 00:00:49.920
in figure two, should be a factor of two.

00:00:49.920 --> 00:00:52.570
So it's times two right over there.

00:00:52.570 --> 00:00:55.020
So let's just answer their
questions that they're asking us,

00:00:55.020 --> 00:00:58.110
and then we can also verify
that it is a scaled copy.

00:00:58.110 --> 00:01:00.920
What point on figure one corresponds

00:01:00.920 --> 00:01:03.990
to the point q on figure two?

00:01:03.990 --> 00:01:05.070
All right, pause this video

00:01:05.070 --> 00:01:07.570
and see if you can figure that out.

00:01:07.570 --> 00:01:11.450
All right, so point q on
figure two is right over there,

00:01:11.450 --> 00:01:15.240
so what point on figure
one corresponds to that?

00:01:15.240 --> 00:01:18.120
Well, it would be playing the same role.

00:01:18.120 --> 00:01:20.550
It would be in the same relative position.

00:01:20.550 --> 00:01:22.730
And so it looks like this
point right over here,

00:01:22.730 --> 00:01:26.400
point b, is in that
same relative position.

00:01:26.400 --> 00:01:30.730
So point b corresponds
to point q on figure two.

00:01:30.730 --> 00:01:33.320
Identify the side of figure two

00:01:33.320 --> 00:01:38.030
that corresponds to
segment dc in figure one.

00:01:38.030 --> 00:01:40.930
Pause this video again and see
if you can figure that out.

00:01:41.870 --> 00:01:45.310
All right, so segment
dc in figure one, that

00:01:45.310 --> 00:01:47.750
is that right over there,

00:01:47.750 --> 00:01:50.560
and your eye might've
immediately catched that hey,

00:01:50.560 --> 00:01:53.550
the segment that's playing
the same role in figure two

00:01:53.550 --> 00:01:56.860
is this one right over here.

00:01:56.860 --> 00:02:00.100
And so that is segment nm.

00:02:00.100 --> 00:02:01.470
Put the line over it to make sure

00:02:01.470 --> 00:02:02.930
that I'm specifying the segment.

00:02:02.930 --> 00:02:06.870
And we can once again verify
the scale factor to ensure

00:02:06.870 --> 00:02:08.580
that this is a scaled copy.

00:02:08.580 --> 00:02:10.630
For these two to correspond to each other,

00:02:10.630 --> 00:02:14.010
and for these to be scaled
copies of each other,

00:02:14.010 --> 00:02:18.570
dc has a length of one, two, three, four.

00:02:18.570 --> 00:02:20.870
And nm has a length of one,

00:02:20.870 --> 00:02:24.780
two, three, four, five, six, seven, eight.

00:02:24.780 --> 00:02:26.940
So once again, we are verifying

00:02:26.940 --> 00:02:30.113
that our scale factor is two.


Identifying scale factor in drawings

WEBVTT
Kind: captions
Language: en

00:00:00.920 --> 00:00:04.410
- [Instructor] Right over
here, figure b is a scaled copy

00:00:04.410 --> 00:00:05.960
of figure a.

00:00:05.960 --> 00:00:07.550
What we want to do is figure out

00:00:07.550 --> 00:00:11.570
what is the scale factor to
go from figure a to figure b?

00:00:11.570 --> 00:00:14.323
Pause the video and see if
you can figure that out.

00:00:15.380 --> 00:00:18.050
All we have to do is look
at corresponding sides

00:00:18.050 --> 00:00:21.280
and think about how much
they have been scaled by.

00:00:21.280 --> 00:00:24.340
For example, this side right over here

00:00:24.340 --> 00:00:28.950
would correspond to this side
right over here on figure b.

00:00:28.950 --> 00:00:31.020
And over here, it had length two

00:00:31.020 --> 00:00:33.830
and over here it has
length one, two three,

00:00:33.830 --> 00:00:36.020
four, five, six.

00:00:36.020 --> 00:00:39.320
So, it look like that
side has been scaled up

00:00:39.320 --> 00:00:41.673
by a factor of three.

00:00:42.507 --> 00:00:45.310
If figure b truly is a scaled copy,

00:00:45.310 --> 00:00:48.940
then every side should be
scaled up by a factor of three.

00:00:48.940 --> 00:00:49.773
We could verify that.

00:00:49.773 --> 00:00:51.050
We don't have to do it with every side.

00:00:51.050 --> 00:00:53.500
We're being told that
these are scaled copies

00:00:53.500 --> 00:00:55.260
but we can see that this is the case.

00:00:55.260 --> 00:00:56.960
For example, this side right over here,

00:00:56.960 --> 00:01:00.350
corresponds to this base right over here.

00:01:00.350 --> 00:01:02.390
This has length three.

00:01:02.390 --> 00:01:05.760
So, if we're scaling up
by a factor of three,

00:01:05.760 --> 00:01:07.460
we should multiply that by three

00:01:07.460 --> 00:01:08.940
and this should be of length nine.

00:01:08.940 --> 00:01:10.260
Let's see if that's the case.

00:01:10.260 --> 00:01:13.990
One, two, three, four, five, six,

00:01:13.990 --> 00:01:17.350
seven, eight, and nine.

00:01:17.350 --> 00:01:19.870
So, you can see, we can feel pretty good

00:01:19.870 --> 00:01:22.060
that figure b is a scaled copy of figure a

00:01:22.060 --> 00:01:24.893
and that scaling factor is three.


Interpreting scale factors in drawings

WEBVTT
Kind: captions
Language: en

00:00:00.130 --> 00:00:03.030
- [Instructor] We are told
Ismael made a scaled copy

00:00:03.030 --> 00:00:04.920
of the following quadrilateral.

00:00:04.920 --> 00:00:09.920
He used a scale factor
less than one, all right.

00:00:10.090 --> 00:00:12.160
And then they say what could be the length

00:00:12.160 --> 00:00:14.880
of the side that corresponds to AD?

00:00:14.880 --> 00:00:17.290
So AD is right over here.

00:00:17.290 --> 00:00:22.240
AD has length 16 units in
our original quadrilateral.

00:00:22.240 --> 00:00:23.700
What could be the length of the side

00:00:23.700 --> 00:00:26.270
that corresponds with
AD on the scaled copy

00:00:26.270 --> 00:00:28.090
of the quadrilateral?

00:00:28.090 --> 00:00:30.000
So it's a scale factor less than one.

00:00:30.000 --> 00:00:33.870
So we're gonna get something
that is less than 16

00:00:33.870 --> 00:00:35.090
for that side.

00:00:35.090 --> 00:00:39.180
And the rest of it will all
be scaled by the same factor.

00:00:39.180 --> 00:00:42.960
So the resulting quadrilateral
might look something,

00:00:42.960 --> 00:00:44.220
might look something like this.

00:00:44.220 --> 00:00:46.450
This is just my hand-drawn version.

00:00:46.450 --> 00:00:47.900
So the key realization is

00:00:47.900 --> 00:00:50.290
is if our scale factor is less than one,

00:00:50.290 --> 00:00:52.100
this thing right over here is going to be

00:00:52.100 --> 00:00:55.000
less than 16 units.

00:00:55.000 --> 00:00:56.310
So let's look at the choices.

00:00:56.310 --> 00:00:58.343
And it says choose three answers.

00:00:59.290 --> 00:01:00.500
So pause the video.

00:01:00.500 --> 00:01:01.900
Which of these would match

00:01:01.900 --> 00:01:05.370
if we're scaling by a
factor of less than one?

00:01:05.370 --> 00:01:06.420
Well, we just have to see which of these

00:01:06.420 --> 00:01:08.000
are less than 16 units.

00:01:08.000 --> 00:01:09.620
This is less than 16.

00:01:09.620 --> 00:01:11.070
This is less than 16.

00:01:11.070 --> 00:01:12.330
This is less than 16.

00:01:12.330 --> 00:01:15.040
And those are the only
three that are less than 16.

00:01:15.040 --> 00:01:18.580
32 units, this would be
a scale factor of two.

00:01:18.580 --> 00:01:20.940
64 units, this would be
a scale factor of four,

00:01:20.940 --> 00:01:24.603
clearly a scale factor
that is not less than one.


Chain rule

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:01.810
- [Instructor] What we're going to go over

00:00:01.810 --> 00:00:05.270
in this video is one of the
core principles in calculus,

00:00:05.270 --> 00:00:07.210
and you're going to use it any
time you take the derivative,

00:00:07.210 --> 00:00:09.380
anything even reasonably complex.

00:00:09.380 --> 00:00:11.280
And it's called the chain rule.

00:00:11.280 --> 00:00:12.610
And when you're first exposed to it,

00:00:12.610 --> 00:00:14.900
it can seem a little daunting
and a little bit convoluted.

00:00:14.900 --> 00:00:16.550
But as you see more and more examples,

00:00:16.550 --> 00:00:18.520
it'll start to make sense,
and hopefully it'd even start

00:00:18.520 --> 00:00:21.930
to seem a little bit simple
and intuitive over time.

00:00:21.930 --> 00:00:24.060
So let's say that I had a function.

00:00:24.060 --> 00:00:28.990
Let's say I have a function
h of x, and it is equal to,

00:00:28.990 --> 00:00:33.280
just for example, let's say
it's equal to sine of x,

00:00:33.280 --> 00:00:36.740
let's say it's equal to sine of x squared.

00:00:36.740 --> 00:00:37.970
Now, I could've written that,

00:00:37.970 --> 00:00:39.260
I could've written it like this,

00:00:39.260 --> 00:00:41.770
sine squared of x, but it'll
be a little bit clearer

00:00:41.770 --> 00:00:44.010
using that type of notation.

00:00:44.010 --> 00:00:46.300
So let me make it so I have h of x.

00:00:46.300 --> 00:00:49.850
And what I'm curious about
is what is h prime of x?

00:00:49.850 --> 00:00:52.400
So I want to know h prime of x,

00:00:52.400 --> 00:00:54.070
which another way of writing it is

00:00:54.070 --> 00:00:56.890
the derivative of h with respect to x.

00:00:56.890 --> 00:00:58.610
These are just different notations.

00:00:58.610 --> 00:01:01.310
And to do this, I'm going
to use the chain rule.

00:01:01.310 --> 00:01:03.040
I'm going to use the chain rule,

00:01:03.040 --> 00:01:06.070
and the chain rule comes
into play every time,

00:01:06.070 --> 00:01:08.950
any time your function can
be used as a composition

00:01:08.950 --> 00:01:10.090
of more than one function.

00:01:10.090 --> 00:01:12.610
And as that might not
seem obvious right now,

00:01:12.610 --> 00:01:13.810
but it will hopefully,

00:01:13.810 --> 00:01:16.660
maybe by the end of this
video or the next one.

00:01:16.660 --> 00:01:17.493
Now, what I want to do

00:01:17.493 --> 00:01:19.440
is a little bit of a thought experiment,

00:01:19.440 --> 00:01:21.470
a little bit of a thought experiment.

00:01:21.470 --> 00:01:22.640
If I were to ask you

00:01:22.640 --> 00:01:24.770
what is the derivative with respect to x,

00:01:24.770 --> 00:01:26.900
if I were to just apply
the derivative operator

00:01:26.900 --> 00:01:30.670
to x squared with respect
to x, what do I get?

00:01:30.670 --> 00:01:32.030
Well, this gives me two x.

00:01:32.030 --> 00:01:34.530
We've seen that many,
many, many, many times.

00:01:34.530 --> 00:01:37.570
Now, what if I were to take the
derivative with respect to a

00:01:37.570 --> 00:01:39.210
of a squared?

00:01:39.210 --> 00:01:40.350
Well, it's the exact same thing.

00:01:40.350 --> 00:01:42.850
I just swapped an a for the x's.

00:01:42.850 --> 00:01:45.720
This is still going to be equal to two a.

00:01:45.720 --> 00:01:46.840
Now I will do something

00:01:46.840 --> 00:01:48.810
that might be a little bit more bizarre.

00:01:48.810 --> 00:01:52.140
What if I were to take the
derivative with respect to

00:01:52.140 --> 00:01:53.800
sine of x,

00:01:53.800 --> 00:01:56.970
with respect to sine of x of,

00:01:56.970 --> 00:01:59.120
of sine of x,

00:01:59.120 --> 00:02:01.730
sine of x squared?

00:02:01.730 --> 00:02:04.860
Well, wherever I had the x's
up here, the a's over here,

00:02:04.860 --> 00:02:06.850
I just replace it with a sine of x.

00:02:06.850 --> 00:02:10.320
So this is just going to be
two times the thing that I had,

00:02:10.320 --> 00:02:12.300
so whatever I'm taking the
derivative with respect to.

00:02:12.300 --> 00:02:13.700
Here it was with respect to x.

00:02:13.700 --> 00:02:14.670
Here with respect to a.

00:02:14.670 --> 00:02:16.790
Here's with respect to sine of x.

00:02:16.790 --> 00:02:19.550
So it's going to be two times

00:02:19.550 --> 00:02:21.880
sine of x.

00:02:21.880 --> 00:02:23.850
Now, so the chain rule tells us

00:02:23.850 --> 00:02:27.200
that this derivative is
going to be the derivative

00:02:27.200 --> 00:02:29.950
of our whole function with respect,

00:02:29.950 --> 00:02:33.520
or the derivative of this
outer function, x squared,

00:02:33.520 --> 00:02:36.760
the derivative of x squared,

00:02:36.760 --> 00:02:39.570
the derivative of this outer function

00:02:39.570 --> 00:02:41.920
with respect to sine of x.

00:02:41.920 --> 00:02:45.050
So that's going to be two sine of x,

00:02:45.050 --> 00:02:46.110
two

00:02:46.110 --> 00:02:47.180
sine of x.

00:02:47.180 --> 00:02:49.830
So we could view it as the
derivative of the outer function

00:02:49.830 --> 00:02:52.420
with respect to the inner, two sine of x.

00:02:52.420 --> 00:02:54.820
We could just treat sine of
x like it's kind of an x.

00:02:54.820 --> 00:02:55.980
And it would've been just two x,

00:02:55.980 --> 00:02:57.060
but instead it's a sine of x.

00:02:57.060 --> 00:02:59.980
We say two sine of x times,

00:02:59.980 --> 00:03:02.580
times the derivative, do this is green,

00:03:02.580 --> 00:03:06.760
times the derivative of
sine of x with respect to x.

00:03:06.760 --> 00:03:10.450
Times the derivative of
sine of x with respect to x,

00:03:10.450 --> 00:03:11.640
well, that's more straightforward,

00:03:11.640 --> 00:03:13.140
a little bit more intuitive.

00:03:13.140 --> 00:03:14.610
The derivative of sine of x

00:03:14.610 --> 00:03:17.520
with respect to x, we've
seen multiple times,

00:03:17.520 --> 00:03:19.520
is cosine of x,

00:03:19.520 --> 00:03:21.450
so times cosine of x.

00:03:21.450 --> 00:03:23.420
And so there we've applied the chain rule.

00:03:23.420 --> 00:03:25.180
It was the derivative
of the outer function

00:03:25.180 --> 00:03:27.060
with respect to the inner.

00:03:27.060 --> 00:03:28.660
So derivative of sine of x squared

00:03:28.660 --> 00:03:31.190
with respect to sine
of x is two sine of x,

00:03:31.190 --> 00:03:34.590
and then we multiply that times
the derivative of sine of x

00:03:34.590 --> 00:03:37.160
with respect to x.

00:03:37.160 --> 00:03:38.200
So let me make it clear.

00:03:38.200 --> 00:03:41.520
This right over here is the derivative.

00:03:41.520 --> 00:03:43.080
We're taking the derivative of,

00:03:43.080 --> 00:03:48.080
we're taking the derivative
of sine of x squared.

00:03:48.210 --> 00:03:49.043
So let me make it clear.

00:03:49.043 --> 00:03:50.800
That's what we were
taking the derivative of

00:03:50.800 --> 00:03:53.370
with respect to sine of x,

00:03:53.370 --> 00:03:55.630
with respect to sine of x.

00:03:55.630 --> 00:03:57.960
And then we're multiplying that times

00:03:57.960 --> 00:03:59.940
the derivative of sine of x,

00:03:59.940 --> 00:04:02.540
the derivative of sine

00:04:02.540 --> 00:04:03.680
of x

00:04:03.680 --> 00:04:05.250
with respect to,

00:04:05.250 --> 00:04:07.100
with respect to x.

00:04:07.100 --> 00:04:09.010
And this is where it might start making

00:04:09.010 --> 00:04:11.160
a little bit of intuition.

00:04:11.160 --> 00:04:13.330
You can't really treat
these differentials,

00:04:13.330 --> 00:04:18.180
this d whatever, this
dx, this d sine of x,

00:04:18.180 --> 00:04:19.013
as a number.

00:04:19.013 --> 00:04:19.900
And you really can't,

00:04:19.900 --> 00:04:21.610
this notation makes it
look like a fraction

00:04:21.610 --> 00:04:23.280
because intuitively
that's what we're doing.

00:04:23.280 --> 00:04:25.280
But if you were to treat
'em like fractions,

00:04:25.280 --> 00:04:28.050
then you could think about
canceling that and that.

00:04:28.050 --> 00:04:29.610
And once again, this isn't
a rigorous thing to do,

00:04:29.610 --> 00:04:30.970
but it can help with the intuition.

00:04:30.970 --> 00:04:33.470
And then what you're left
with is the derivative

00:04:33.470 --> 00:04:37.200
of this whole sine of x
squared with respect to x.

00:04:37.200 --> 00:04:38.760
So you're left with,

00:04:38.760 --> 00:04:41.550
you're left with the derivative of

00:04:41.550 --> 00:04:45.600
essentially our original
function, sine of x squared

00:04:45.600 --> 00:04:47.400
with respect to x,

00:04:47.400 --> 00:04:50.420
with respect to x, which
is exactly what dh/dx is.

00:04:50.420 --> 00:04:51.950
This right over here,

00:04:51.950 --> 00:04:56.180
this right over here is
our original function h.

00:04:56.180 --> 00:04:58.410
That's our original function h.

00:04:58.410 --> 00:05:00.740
So it might seem a
little bit daunting now.

00:05:00.740 --> 00:05:03.700
What I'll do in the next video
is another several examples,

00:05:03.700 --> 00:05:06.413
and then we'll try to
abstract this a little bit.


Introduction to Ratios

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:02.090
- [Instructor] We've got some apples here

00:00:02.090 --> 00:00:03.027
and we've got some oranges.

00:00:03.027 --> 00:00:07.410
And what I wanna think
about is what is the ratio?

00:00:07.410 --> 00:00:11.740
What is the ratio of apples,

00:00:11.740 --> 00:00:14.030
of apples,

00:00:14.030 --> 00:00:17.563
to oranges, to oranges?

00:00:18.660 --> 00:00:21.870
And to clarify what
we're even talking about,

00:00:21.870 --> 00:00:24.350
a ratio is giving us the relationship

00:00:24.350 --> 00:00:27.380
between quantities of
two different things.

00:00:27.380 --> 00:00:30.040
So there's a couple of ways
that we can specify this.

00:00:30.040 --> 00:00:32.100
We can literally count
the number of apples,

00:00:32.100 --> 00:00:34.720
one, two, three, four, five, six.

00:00:34.720 --> 00:00:37.260
So we have six apples.

00:00:37.260 --> 00:00:41.430
And we could say the ratio
is going to be six to,

00:00:41.430 --> 00:00:45.070
six to, and then how
many oranges do we have?

00:00:45.070 --> 00:00:49.050
One, two, three, four, five,
six, seven, eight, nine.

00:00:49.050 --> 00:00:51.060
It is six to nine.

00:00:51.060 --> 00:00:55.700
Ratio of apples to oranges is six to nine.

00:00:55.700 --> 00:00:57.540
And you could use different notation.

00:00:57.540 --> 00:00:59.260
You could also write it this way.

00:00:59.260 --> 00:01:02.150
Six to, you would still read it

00:01:02.150 --> 00:01:05.970
as the ratio as being six to nine.

00:01:05.970 --> 00:01:08.010
But we don't have to just
satisfy us with this,

00:01:08.010 --> 00:01:10.089
because one way to think about ratios,

00:01:10.089 --> 00:01:13.130
especially if we're thinking
about apples to oranges,

00:01:13.130 --> 00:01:14.910
is how many apples do we have

00:01:14.910 --> 00:01:16.800
for a certain number of oranges?

00:01:16.800 --> 00:01:18.130
And when you think about it that way,

00:01:18.130 --> 00:01:22.240
both six and nine are divisible by three.

00:01:22.240 --> 00:01:25.570
So if you divide six
and nine both by three,

00:01:25.570 --> 00:01:28.200
six divided by three is two,

00:01:28.200 --> 00:01:30.630
six divided by three is two,

00:01:30.630 --> 00:01:33.720
and nine divided by three is three.

00:01:33.720 --> 00:01:37.040
So we could also say that the
ratio of apples to oranges

00:01:37.040 --> 00:01:39.070
is two to three.

00:01:39.070 --> 00:01:41.260
Or if we want to use this notation,

00:01:41.260 --> 00:01:45.400
two to three.

00:01:45.400 --> 00:01:46.480
Two to three.

00:01:46.480 --> 00:01:48.400
Now does that make sense?

00:01:48.400 --> 00:01:51.700
Well, look, we divided each
of these groups into three.

00:01:51.700 --> 00:01:53.280
So one way to think about it,

00:01:53.280 --> 00:01:54.700
so if you divide,

00:01:54.700 --> 00:01:57.480
if you divide this whole
total into three groups,

00:01:57.480 --> 00:01:58.633
so one group,

00:01:59.780 --> 00:02:01.190
one group,

00:02:01.190 --> 00:02:04.053
two groups, two groups,

00:02:04.940 --> 00:02:07.490
and three equal groups,

00:02:07.490 --> 00:02:10.230
we see that in each of those groups,

00:02:10.230 --> 00:02:11.870
for every two apples,

00:02:11.870 --> 00:02:14.650
for every two apples
we have three oranges.

00:02:14.650 --> 00:02:17.000
For two apples, we have three oranges.

00:02:17.000 --> 00:02:20.380
For two apples, we have three oranges.

00:02:20.380 --> 00:02:21.440
So once again,

00:02:21.440 --> 00:02:23.080
the ratio of apples to oranges,

00:02:23.080 --> 00:02:24.180
for every two apples,

00:02:24.180 --> 00:02:26.000
we have three oranges.

00:02:26.000 --> 00:02:27.090
But we could think about things

00:02:27.090 --> 00:02:28.670
the other way around as well.

00:02:28.670 --> 00:02:31.563
We could also think
about what is the ratio,

00:02:32.920 --> 00:02:36.300
ratio of oranges to apples,

00:02:36.300 --> 00:02:39.980
orange to apples.

00:02:39.980 --> 00:02:43.210
And here we would essentially
switch the numbers.

00:02:43.210 --> 00:02:47.690
The ratio of oranges to apples.

00:02:47.690 --> 00:02:50.910
Notice up here we said apples to oranges,

00:02:50.910 --> 00:02:53.310
which is six to nine or two to three,

00:02:53.310 --> 00:02:54.660
which is an equivalent ratio,

00:02:54.660 --> 00:02:56.820
and here we're going to say
the ratio of oranges to apples.

00:02:56.820 --> 00:02:58.320
So we swapped these two,

00:02:58.320 --> 00:03:00.260
and so we're going to swap the numbers.

00:03:00.260 --> 00:03:02.820
Here we have nine oranges
for every six apples,

00:03:02.820 --> 00:03:07.820
so we could say the ratio
is going to be nine to six.

00:03:09.230 --> 00:03:11.180
The ratio is nine to six,

00:03:11.180 --> 00:03:12.430
or an equivalent ratio,

00:03:12.430 --> 00:03:14.120
for every three oranges,

00:03:14.120 --> 00:03:15.660
so we're gonna divide this by three,

00:03:15.660 --> 00:03:17.700
so for every three oranges,

00:03:17.700 --> 00:03:21.320
we are going to have two apples.

00:03:21.320 --> 00:03:23.680
We are going to have two apples.

00:03:23.680 --> 00:03:26.740
So notice this is just
exactly what we had up here,

00:03:26.740 --> 00:03:28.660
but where we had apples to oranges,

00:03:28.660 --> 00:03:29.840
it was six to nine,

00:03:29.840 --> 00:03:32.070
six apples for every nine oranges,

00:03:32.070 --> 00:03:34.150
and now when it's oranges to apples,

00:03:34.150 --> 00:03:35.930
we say it's nine to six.

00:03:35.930 --> 00:03:38.590
Nine oranges for every six apples.

00:03:38.590 --> 00:03:40.870
Or we could say for every three oranges,

00:03:40.870 --> 00:03:44.163
we have exactly two apples.


Senate confirmation as a check on the judicial branch

WEBVTT
Kind: captions
Language: en

00:00:00.090 --> 00:00:01.660
- [Instructor] When we think
about how the executive

00:00:01.660 --> 00:00:04.890
or the legislative branch
have some form of check

00:00:04.890 --> 00:00:06.990
or power over the judicial branch,

00:00:06.990 --> 00:00:10.750
a key element of that is
the executive's ability

00:00:10.750 --> 00:00:13.500
to appoint judges to federal courts,

00:00:13.500 --> 00:00:15.510
including the US Supreme Court.

00:00:15.510 --> 00:00:17.580
But it's not just that
the president can decide

00:00:17.580 --> 00:00:18.920
who gets to be a justice;

00:00:18.920 --> 00:00:21.940
they have to be confirmed by the Senate.

00:00:21.940 --> 00:00:24.890
And what you see over here,
what I'm about to play,

00:00:24.890 --> 00:00:26.900
are the confirmation hearings

00:00:26.900 --> 00:00:29.623
from Justice Sonia Sotomayor in 2009,

00:00:31.629 --> 00:00:35.160
and what I hope you get from
it is it's not an easy process.

00:00:35.160 --> 00:00:37.670
This is going to be a question
from Senator Chuck Grassley,

00:00:37.670 --> 00:00:39.220
who's a Republican from Iowa,

00:00:39.220 --> 00:00:42.150
and he's going to be asking
about whether marriage

00:00:42.150 --> 00:00:45.130
is a state or a federal question.

00:00:45.130 --> 00:00:49.500
- So, do you agree that
marriage is a question

00:00:49.500 --> 00:00:54.500
reserved for the states to
decide, based on Baker v. Nelson?

00:00:55.130 --> 00:00:59.770
- That also is a question
that's pending and impending

00:00:59.770 --> 00:01:01.203
in many courts.

00:01:03.270 --> 00:01:08.270
As you know, the issue of
marriage and what constitutes it

00:01:09.280 --> 00:01:12.180
is a subject of much public discussion,

00:01:12.180 --> 00:01:16.170
and there's a number of
cases in state courts

00:01:16.170 --> 00:01:21.170
addressing the issue of who
regulates it, under what terms--

00:01:23.130 --> 00:01:24.580
- Can I please interrupt you?

00:01:26.210 --> 00:01:28.820
I thought I was asking
a very simple question

00:01:28.820 --> 00:01:33.820
based upon a precedent
that Baker v. Nelson is,

00:01:34.450 --> 00:01:37.130
based on the proposition that yesterday,

00:01:37.130 --> 00:01:39.560
in so many cases, whether it was Griswold,

00:01:39.560 --> 00:01:42.410
whether it was Roe v. Wade,
whether it was Chevron,

00:01:42.410 --> 00:01:44.280
whether it's a whole bunch of other cases

00:01:44.280 --> 00:01:47.410
that you made reference
to, the Casey case,

00:01:47.410 --> 00:01:49.240
the Gonzales case,

00:01:49.240 --> 00:01:54.230
the Leegin Creative Leather
Products case, the Kelo case.

00:01:54.230 --> 00:01:56.480
You made that case to me.

00:01:56.480 --> 00:01:58.700
You said these are precedents.

00:01:58.700 --> 00:01:59.860
Now, are you saying to me

00:01:59.860 --> 00:02:04.370
that Baker v. Nelson is not a precedent?

00:02:04.370 --> 00:02:09.370
- No sir, I just haven't
reviewed Baker in a while.

00:02:10.670 --> 00:02:13.930
And so, I actually don't
know what the status is.

00:02:13.930 --> 00:02:15.550
If it is the court's precedent,

00:02:15.550 --> 00:02:18.510
as I've indicated in all of my answers,

00:02:18.510 --> 00:02:21.440
I will apply that precedent

00:02:21.440 --> 00:02:26.440
to the facts of any new
situation that implicates it.

00:02:26.560 --> 00:02:27.930
Always, the first question for a judge--

00:02:27.930 --> 00:02:30.330
- Well then, tell me
what sort of a process

00:02:30.330 --> 00:02:31.970
you might go through--

00:02:31.970 --> 00:02:33.100
- [Instructor] So anyway, you can see

00:02:33.100 --> 00:02:35.390
that this is not an easy process.

00:02:35.390 --> 00:02:36.350
And in this situation,

00:02:36.350 --> 00:02:38.690
you have an appointment
by a Democratic president

00:02:38.690 --> 00:02:40.780
and you have questions
from a Republican senator,

00:02:40.780 --> 00:02:42.090
and this is fairly typical.

00:02:42.090 --> 00:02:44.150
They're likely to ask
more pointed questions

00:02:44.150 --> 00:02:46.980
and try to get the appointee to trip up.

00:02:46.980 --> 00:02:49.640
And if you were to see questions
from a Democratic senator,

00:02:49.640 --> 00:02:51.700
they'd be more likely to ask questions

00:02:51.700 --> 00:02:54.560
that would make the nominee
look a little bit better,

00:02:54.560 --> 00:02:56.340
and that tends to always be the case.

00:02:56.340 --> 00:02:57.750
If it's from the same party,

00:02:57.750 --> 00:03:00.130
they tend to try to smooth the process,

00:03:00.130 --> 00:03:01.270
while the opposition party

00:03:01.270 --> 00:03:03.860
tries to make it a little
bit more difficult.

00:03:03.860 --> 00:03:06.930
And an interesting thing to
think about is Justice Sotomayor

00:03:06.930 --> 00:03:09.820
eventually does get appointed
to the Supreme Court,

00:03:09.820 --> 00:03:12.850
but what was the goal
of Senator Grassley here

00:03:12.850 --> 00:03:15.210
in asking these questions, even if he knew

00:03:15.210 --> 00:03:17.620
that she was eventually
going to be appointed?

00:03:17.620 --> 00:03:19.640
There's other themes
that we've talked about

00:03:19.640 --> 00:03:21.820
in government that he might be playing to.

00:03:21.820 --> 00:03:25.100
He might be asking these questions
more for his constituents

00:03:25.100 --> 00:03:27.210
or there might be an element of,

00:03:27.210 --> 00:03:30.560
even if a Supreme Court justice
is going to be appointed,

00:03:30.560 --> 00:03:31.600
at the end of the day,

00:03:31.600 --> 00:03:34.930
there's also the court of
public opinion, so to speak,

00:03:34.930 --> 00:03:38.510
and these might be opportunities
to sow some seeds of doubt

00:03:38.510 --> 00:03:40.550
or to make it a little bit more difficult

00:03:40.550 --> 00:03:42.230
for a Supreme Court justice to,

00:03:42.230 --> 00:03:44.630
in the future, vote one way or another

00:03:44.630 --> 00:03:47.350
based on what they say
during these hearings.

00:03:47.350 --> 00:03:49.000
Interesting things to talk about.


State checks on the judicial branch

WEBVTT
Kind: captions
Language: en

00:00:00.350 --> 00:00:01.500
- [Instructor] In previous videos,

00:00:01.500 --> 00:00:06.040
we had talked about the
1896 Supreme Court case,

00:00:06.040 --> 00:00:08.610
Plessy versus Ferguson,

00:00:08.610 --> 00:00:10.330
which is a good one to know in general

00:00:10.330 --> 00:00:12.350
if you're studying United States history

00:00:12.350 --> 00:00:14.380
and/or United States government.

00:00:14.380 --> 00:00:17.090
But this is where we got the principle

00:00:17.090 --> 00:00:22.090
of separate but equal

00:00:22.400 --> 00:00:25.280
where the Supreme Court said in 1896

00:00:25.280 --> 00:00:27.460
that it was okay to have segregation.

00:00:27.460 --> 00:00:30.950
It was okay to separate
folks on a racial basis

00:00:30.950 --> 00:00:34.120
as long as what they got was equal.

00:00:34.120 --> 00:00:36.670
This was a case around whether
it was unconstitutional

00:00:36.670 --> 00:00:39.540
for an African-American in Louisiana

00:00:39.540 --> 00:00:43.710
to have to sit in a separate
rail car from white passengers.

00:00:43.710 --> 00:00:44.750
Now, as you can imagine,

00:00:44.750 --> 00:00:47.450
this idea of segregation
and separate but equal,

00:00:47.450 --> 00:00:50.740
it was taken to justify racial segregation

00:00:50.740 --> 00:00:54.720
in all walks of life,
including in schools.

00:00:54.720 --> 00:00:58.950
And what you see here is a
map of the United States,

00:00:58.950 --> 00:01:03.300
and it's colored in by the
degree of school segregation,

00:01:03.300 --> 00:01:08.300
educational segregation,
as we enter into the 1950s.

00:01:08.460 --> 00:01:11.350
And what you could see is
in these red states here,

00:01:11.350 --> 00:01:13.620
which is for the most
part the Southern states,

00:01:13.620 --> 00:01:16.540
it required the separation of races.

00:01:16.540 --> 00:01:18.600
In the green states you see here

00:01:18.600 --> 00:01:21.010
segregation was forbidden.

00:01:21.010 --> 00:01:23.960
And in the yellow and blue
states it was kind of in between.

00:01:23.960 --> 00:01:25.930
The yellow states had no legislation,

00:01:25.930 --> 00:01:28.410
and in the blue states it was limited.

00:01:28.410 --> 00:01:29.910
It wasn't required.

00:01:29.910 --> 00:01:31.590
And all of this comes to a head as we get

00:01:31.590 --> 00:01:36.070
into the 1950s in Topeka, Kansas.

00:01:36.070 --> 00:01:39.280
You have a situation where the NAACP

00:01:39.280 --> 00:01:41.990
in coordination with 13 parents

00:01:41.990 --> 00:01:45.310
tried to register
African-American students

00:01:45.310 --> 00:01:48.720
into white schools in Topeka, Kansas.

00:01:48.720 --> 00:01:50.640
The schools tried to prevent them,

00:01:50.640 --> 00:01:53.510
and it eventually goes to the
United States Supreme Court

00:01:53.510 --> 00:01:57.250
in the case of Brown
versus Board of Education

00:01:57.250 --> 00:02:00.180
of Topeka 1954.

00:02:00.180 --> 00:02:02.930
Oliver Brown was one of those 13 parents

00:02:02.930 --> 00:02:04.160
trying to get his daughter

00:02:04.160 --> 00:02:06.950
into an otherwise segregated school.

00:02:06.950 --> 00:02:09.970
And this is a seminal Supreme Court case

00:02:09.970 --> 00:02:13.900
because here the Supreme
Court revisits segregation

00:02:13.900 --> 00:02:17.680
and essentially overturns
Plessy versus Ferguson.

00:02:17.680 --> 00:02:20.300
The Supreme Court, in
a unanimous decision,

00:02:20.300 --> 00:02:25.300
ruled, We conclude that, in
the field of public education,

00:02:25.440 --> 00:02:29.780
the doctrine of separate
but equal has no place.

00:02:29.780 --> 00:02:33.690
Separate educational facilities
are inherently unequal.

00:02:33.690 --> 00:02:37.510
Therefore, we hold that
the plaintiffs and others

00:02:37.510 --> 00:02:40.780
similarly situated for whom
the actions have been brought

00:02:40.780 --> 00:02:44.270
are, by reason of the
segregation complained of,

00:02:44.270 --> 00:02:48.050
deprived of the equal
protection of the laws

00:02:48.050 --> 00:02:50.730
guaranteed by the 14th Amendment.

00:02:50.730 --> 00:02:52.560
And we cite the 14th Amendment

00:02:52.560 --> 00:02:54.200
a lot in these Supreme Court cases.

00:02:54.200 --> 00:02:56.300
And it never hurts to take another look

00:02:56.300 --> 00:02:58.560
at at least an excerpt
of the 14th Amendment.

00:02:58.560 --> 00:03:01.050
And this is a focus on the
equal protection clause.

00:03:01.050 --> 00:03:03.890
This is part of Section
1 of the 14th Amendment.

00:03:03.890 --> 00:03:06.970
Nor shall any State
deprive any person of life,

00:03:06.970 --> 00:03:10.230
liberty, or property,
without due process of law.

00:03:10.230 --> 00:03:12.750
And then this is the
equal protection clause.

00:03:12.750 --> 00:03:15.788
Nor deny to any person
within its jurisdiction

00:03:15.788 --> 00:03:19.430
the equal protection of the laws.

00:03:19.430 --> 00:03:21.500
And this is a big deal
because not only does it say

00:03:21.500 --> 00:03:25.610
that hey, segregation is
fundamentally unconstitutional.

00:03:25.610 --> 00:03:27.500
What's also interesting about this

00:03:27.500 --> 00:03:28.970
is that this unanimous ruling

00:03:28.970 --> 00:03:30.770
by the United States Supreme Court

00:03:30.770 --> 00:03:34.150
then sets off a whole series of actions

00:03:34.150 --> 00:03:37.020
by politicians from
those states that you saw

00:03:37.020 --> 00:03:39.400
where segregation was mandatory

00:03:39.400 --> 00:03:42.790
to try to go around the
Supreme Court's decision.

00:03:42.790 --> 00:03:46.330
And so it's a case of how
other branches of government

00:03:46.330 --> 00:03:49.480
might try to limit Supreme Court powers

00:03:49.480 --> 00:03:53.440
or try to ignore, in some
cases, Supreme Court powers.

00:03:53.440 --> 00:03:57.910
A good example of this was
Virginia Senator Harry Byrd.

00:03:57.910 --> 00:04:01.250
In 1956, shortly after the Brown versus

00:04:01.250 --> 00:04:03.250
Board of Education decision,

00:04:03.250 --> 00:04:05.770
he issues the Southern Manifesto

00:04:05.770 --> 00:04:08.920
where he gets 100 politicians in the South

00:04:08.920 --> 00:04:10.630
to essentially sign on saying, hey,

00:04:10.630 --> 00:04:14.150
we are going to do what
we can, legislatively,

00:04:14.150 --> 00:04:17.300
to try to get around the Brown versus

00:04:17.300 --> 00:04:19.320
Board of Education decision.

00:04:19.320 --> 00:04:20.910
We are going to try to ensure

00:04:20.910 --> 00:04:22.700
that that does not go into effect

00:04:22.700 --> 00:04:25.460
in segregated states like Virginia.

00:04:25.460 --> 00:04:28.240
And later that year, in 1956,

00:04:28.240 --> 00:04:31.120
it took the form of this
massive resistance movement

00:04:31.120 --> 00:04:32.870
where they had the Stanley Plan,

00:04:32.870 --> 00:04:35.370
named after the Virginia governor

00:04:35.370 --> 00:04:37.300
in which they started to pass laws

00:04:37.300 --> 00:04:40.180
to try to make Brown
versus Board of Education

00:04:40.180 --> 00:04:41.550
not take effect.

00:04:41.550 --> 00:04:42.630
One of them, for example,

00:04:42.630 --> 00:04:45.410
said that any school that
actually desegregated

00:04:45.410 --> 00:04:47.680
would not receive state funds,

00:04:47.680 --> 00:04:52.340
and it authorized the governor
to close those schools down.

00:04:52.340 --> 00:04:54.730
And this massive resistance continued,

00:04:54.730 --> 00:04:57.670
and it was challenged in the
courts all the way to 1959

00:04:57.670 --> 00:05:00.030
where you actually have the
first integrated schools

00:05:00.030 --> 00:05:01.280
in the state of Virginia.

00:05:02.420 --> 00:05:05.350
Another state-level
challenge to the ruling

00:05:05.350 --> 00:05:08.160
by the Supreme Court happened in 1957

00:05:08.160 --> 00:05:11.960
where the governor of
Arkansas, Orval Faubus,

00:05:11.960 --> 00:05:15.680
tried to actively prevent nine students,

00:05:15.680 --> 00:05:17.670
nine African-American students,

00:05:17.670 --> 00:05:20.940
from attending Little
Rock Central High School.

00:05:20.940 --> 00:05:23.550
And so what you see in
this picture over here,

00:05:23.550 --> 00:05:28.550
these are US Army troops
sent by President Eisenhower

00:05:28.940 --> 00:05:31.390
to escort these nine students

00:05:31.390 --> 00:05:33.890
into Little Rock Central High School.

00:05:33.890 --> 00:05:35.680
These nine students are famously known

00:05:35.680 --> 00:05:37.140
as the Little Rock Nine.

00:05:37.140 --> 00:05:40.420
And this is actually a picture
of Governor Orval Faubus

00:05:40.420 --> 00:05:43.500
later complaining about this intervention.

00:05:43.500 --> 00:05:46.350
And this continues on for some time.

00:05:46.350 --> 00:05:49.290
You fast forward to 1963.

00:05:49.290 --> 00:05:52.820
This is Alabama Governor George Wallace

00:05:52.820 --> 00:05:55.760
physically using himself to block

00:05:55.760 --> 00:05:58.460
two African-American
students from registering

00:05:58.460 --> 00:06:01.210
at the University of Alabama.

00:06:01.210 --> 00:06:02.870
Once again, in this situation,

00:06:02.870 --> 00:06:04.460
the president had to take action.

00:06:04.460 --> 00:06:08.860
This time John F. Kennedy, he
federalized the National Guard

00:06:08.860 --> 00:06:13.020
and he made them force
Governor Wallace to step aside

00:06:13.020 --> 00:06:14.870
so that those students could register

00:06:14.870 --> 00:06:17.200
at the University of Alabama.

00:06:17.200 --> 00:06:18.530
So the big picture here is

00:06:18.530 --> 00:06:20.570
that Brown versus Board of Education

00:06:20.570 --> 00:06:24.110
is a seminal case in American history,

00:06:24.110 --> 00:06:26.520
but it also tells us how
the legislative branch

00:06:26.520 --> 00:06:30.810
and state government might
try to go around decisions

00:06:30.810 --> 00:06:32.100
that they don't like.

00:06:32.100 --> 00:06:34.120
Sometimes they might
just ignore the decision

00:06:34.120 --> 00:06:36.270
until they're forced
to pay attention to it.

00:06:36.270 --> 00:06:38.850
Or they might try to
legislate their way around it

00:06:38.850 --> 00:06:40.610
and have a fight in the court system

00:06:40.610 --> 00:06:43.453
until they're forced to
go one way or the other.


Introduction to average rate of change

WEBVTT
Kind: captions
Language: en

00:00:00.290 --> 00:00:02.170
- So we have different definitions

00:00:02.170 --> 00:00:04.600
for d of t on the left and the right

00:00:04.600 --> 00:00:07.210
and let's say that d is
distance and t is time,

00:00:07.210 --> 00:00:10.700
so this is giving us our
distance as a function of time,

00:00:10.700 --> 00:00:12.620
on the left, it's equal to 3t plus one

00:00:12.620 --> 00:00:16.180
and you can see the graph
of how distance is changing

00:00:16.180 --> 00:00:18.760
as a function of time here is a line

00:00:18.760 --> 00:00:22.230
and just as a review from algebra,

00:00:22.230 --> 00:00:24.120
the rate of change of a line,

00:00:24.120 --> 00:00:28.550
we refer to as the slope of a
line and we can figure it out,

00:00:28.550 --> 00:00:32.500
we can figure out, well,
for any change in time,

00:00:32.500 --> 00:00:36.450
what is our change in distance?

00:00:36.450 --> 00:00:37.367
And so in this situation,

00:00:37.367 --> 00:00:40.890
if we're going from time
equal one to time equal two,

00:00:40.890 --> 00:00:44.786
our change in time,
delta t is equal to one

00:00:44.786 --> 00:00:47.030
and what is our change in distance?

00:00:47.030 --> 00:00:49.260
We go from distance is
equal to four meters,

00:00:49.260 --> 00:00:50.760
at time equals one,

00:00:50.760 --> 00:00:54.110
to distance in seven
meters at time equal two

00:00:54.110 --> 00:00:58.836
and so our change in distance
here is equal to three

00:00:58.836 --> 00:01:00.410
and if we wanna put our units,

00:01:00.410 --> 00:01:03.560
it's three meters for
every one second in time

00:01:03.560 --> 00:01:07.310
and so our slope would be
our change in our vertical

00:01:07.310 --> 00:01:09.297
divided by our change in our horizontal,

00:01:09.297 --> 00:01:13.471
which would be change in
d, delta d over delta t,

00:01:13.471 --> 00:01:15.507
which is equal to three over one

00:01:15.507 --> 00:01:19.679
or we could just write that
as three meters per second

00:01:19.679 --> 00:01:22.590
and you might recognize this as a rate,

00:01:22.590 --> 00:01:24.970
if you're thinking about
your change in distance

00:01:24.970 --> 00:01:27.620
over change in time,
this rate right over here

00:01:27.620 --> 00:01:29.460
is going to be your speed.

00:01:29.460 --> 00:01:31.760
This is all a review of
what you've seen before

00:01:31.760 --> 00:01:32.890
and what's interesting about a line,

00:01:32.890 --> 00:01:35.180
or if we're talking
about a linear function,

00:01:35.180 --> 00:01:38.120
is that your rate does
not change at any point,

00:01:38.120 --> 00:01:40.810
the slope of this line
between any two points

00:01:40.810 --> 00:01:43.500
is always going to be three,

00:01:43.500 --> 00:01:46.670
but what's interesting about
this function on the right

00:01:46.670 --> 00:01:48.260
is that is not true,

00:01:48.260 --> 00:01:50.690
our rate of change is constantly changing

00:01:50.690 --> 00:01:52.852
and we're going to study
that in a lot more depth,

00:01:52.852 --> 00:01:55.160
when we get to differential calculus

00:01:55.160 --> 00:01:56.840
and really this video's a little bit

00:01:56.840 --> 00:01:59.350
of a foundational primer
for that future state,

00:01:59.350 --> 00:02:01.870
where we learn about differential calculus

00:02:01.870 --> 00:02:03.390
and the thing to appreciate here

00:02:03.390 --> 00:02:07.010
is think about the instantaneous
rate of change someplace,

00:02:07.010 --> 00:02:08.490
so let's say right over there,

00:02:08.490 --> 00:02:10.260
if you ever think about
the slope of a line,

00:02:10.260 --> 00:02:12.640
that just barely touches this graph,

00:02:12.640 --> 00:02:14.420
it might look something like that,

00:02:14.420 --> 00:02:15.760
the slope of a tangent line

00:02:15.760 --> 00:02:16.840
and then right over here,

00:02:16.840 --> 00:02:18.360
it looks like it's a little bit steeper

00:02:18.360 --> 00:02:20.800
and then over here, it looks
like it's a little bit steeper,

00:02:20.800 --> 00:02:22.350
so it looks like your rate of change

00:02:22.350 --> 00:02:24.790
is increasing as t increases.

00:02:24.790 --> 00:02:26.580
As I mentioned, we will build the tools

00:02:26.580 --> 00:02:28.950
to later think about
instantaneous rate of change,

00:02:28.950 --> 00:02:31.050
but what we can start to think about

00:02:31.050 --> 00:02:33.800
is an average rate of change,

00:02:33.800 --> 00:02:36.960
average rate

00:02:36.960 --> 00:02:38.740
of change,

00:02:38.740 --> 00:02:41.024
and the way that we think about
our average rate of change

00:02:41.024 --> 00:02:44.700
is we use the same tools, that
we first learned in algebra,

00:02:44.700 --> 00:02:47.480
we think about slopes of secant lines,

00:02:47.480 --> 00:02:49.010
what is a secant line?

00:02:49.010 --> 00:02:51.080
Well, we talk about this in geometry,

00:02:51.080 --> 00:02:53.810
that a secant is something
that intersects a curve

00:02:53.810 --> 00:02:57.160
in two points, so let's
say that there's a line,

00:02:57.160 --> 00:03:01.030
that intersects at t equals
zero and t equals one

00:03:01.030 --> 00:03:04.590
and so let me draw that
line, I'll draw it in orange,

00:03:04.590 --> 00:03:08.830
so this right over here is a secant line

00:03:09.720 --> 00:03:12.100
and you could do the
slope of the secant line

00:03:12.100 --> 00:03:14.020
as the average rate of change

00:03:14.020 --> 00:03:16.068
from t equals zero to t equals one,

00:03:16.068 --> 00:03:19.740
well, what is that average
rate of change going to be?

00:03:19.740 --> 00:03:21.442
Well, the slope of our
secant line is going to be

00:03:21.442 --> 00:03:25.660
our change in distance
divided by our change in time,

00:03:25.660 --> 00:03:26.962
which is going to be equal to,

00:03:26.962 --> 00:03:31.310
well, our change in time is one second,

00:03:31.310 --> 00:03:33.238
one, I'll put the units here, one second

00:03:33.238 --> 00:03:35.420
and what is our change in distance?

00:03:35.420 --> 00:03:40.098
At t equals zero or d of zero is one

00:03:40.098 --> 00:03:41.600
and d of one is two,

00:03:41.600 --> 00:03:44.050
so our distance has
increased by one meter,

00:03:44.050 --> 00:03:46.375
so we've gone one meter in one second

00:03:46.375 --> 00:03:49.170
or we could say that our
average rate of change

00:03:49.170 --> 00:03:51.340
over that first second from t equals zero,

00:03:51.340 --> 00:03:54.532
t equals one is one meter per second,

00:03:54.532 --> 00:03:56.430
but let's think about what it is,

00:03:56.430 --> 00:03:59.920
if we're going from t equals two

00:03:59.920 --> 00:04:02.100
to t equals three.

00:04:02.100 --> 00:04:05.440
Well, once again, we can
look at this secant line

00:04:05.440 --> 00:04:08.200
and we can figure out its slope,

00:04:08.200 --> 00:04:11.150
so the slope here,
which you could also use

00:04:11.150 --> 00:04:12.620
the average rate of change

00:04:12.620 --> 00:04:15.180
from t equals two to t equals three,

00:04:15.180 --> 00:04:16.030
as I already mentioned,

00:04:16.030 --> 00:04:17.315
the rate of change seems
to be constantly changing,

00:04:17.315 --> 00:04:20.000
but we can think about
the average rate of change

00:04:20.000 --> 00:04:22.170
and so that's going to
be our change in distance

00:04:22.170 --> 00:04:24.170
over our change in time,

00:04:24.170 --> 00:04:28.730
which is going to be equal
to when t is equal to two,

00:04:28.730 --> 00:04:32.430
our distance is equal to five,

00:04:32.430 --> 00:04:34.910
so one, two, three, four, five,

00:04:34.910 --> 00:04:36.840
so that's five right over there

00:04:36.840 --> 00:04:40.066
and when t is equal to three,
our distance is equal to 10,

00:04:40.066 --> 00:04:44.750
six, seven, eight, nine, 10,
so that is 10 right over there,

00:04:44.750 --> 00:04:47.185
so our change in time, that's
pretty straightforward,

00:04:47.185 --> 00:04:50.230
we've just gone forward one
second, so that's one second

00:04:50.230 --> 00:04:51.787
and then our change in
distance right over here,

00:04:51.787 --> 00:04:55.560
we go from five meters to
10 meters is five meters,

00:04:55.560 --> 00:05:00.070
so this is equal to five meters per second

00:05:00.070 --> 00:05:01.660
and so this makes it very clear,

00:05:01.660 --> 00:05:04.030
that our average rate
of change has changed

00:05:04.030 --> 00:05:05.880
from t equals zero, t equals one

00:05:05.880 --> 00:05:08.320
to t equals two to t equals three,

00:05:08.320 --> 00:05:10.920
our average rate of change is higher

00:05:10.920 --> 00:05:13.690
on this second interval,
than on this first one

00:05:13.690 --> 00:05:14.825
and as you can imagine,

00:05:14.825 --> 00:05:16.790
something very interesting to think about

00:05:16.790 --> 00:05:18.940
is what if you were to take the slope

00:05:18.940 --> 00:05:21.746
of the secant line of
closer and closer points?

00:05:21.746 --> 00:05:23.880
Well, then you would get closer and closer

00:05:23.880 --> 00:05:26.820
to approximating that
slope of the tangent line

00:05:26.820 --> 00:05:30.653
and that's actually what we
will do when we get to calculus.