Subtracting 1 vs. subtracting 10

WEBVTT
Kind: captions
Language: en

00:00:00.266 --> 00:00:02.565
- What I want you to do is pause the video

00:00:02.565 --> 00:00:05.480
and think about what 27 minus one is,

00:00:05.480 --> 00:00:09.322
and then think about what 27 minus 10 is.

00:00:09.322 --> 00:00:11.579
Alright, you might have found it pretty straightforward,

00:00:11.579 --> 00:00:14.828
but I want to think about it in terms of place value.

00:00:14.828 --> 00:00:18.669
So let's focus on 27, so 27,

00:00:18.669 --> 00:00:23.062
I have two in the 10's place so that two means two 10s

00:00:23.062 --> 00:00:25.332
and we seen them here, two groups of 10,

00:00:25.332 --> 00:00:27.612
and then we have seven in the one's place

00:00:27.612 --> 00:00:30.445
and we see those seven ones right over here.

00:00:30.445 --> 00:00:32.254
Now I'm going to subtract one,

00:00:32.254 --> 00:00:35.075
I'm subtracting one, one is another way to think about it

00:00:35.075 --> 00:00:37.260
because this is in the one's place here,

00:00:37.260 --> 00:00:38.568
so I'm going to take away one.

00:00:38.568 --> 00:00:40.626
So let's just take away one like that,

00:00:40.626 --> 00:00:42.223
and then what am I left with?

00:00:42.223 --> 00:00:44.266
Well I still have my two 10s

00:00:44.266 --> 00:00:46.901
so I could put a two in the 10's place,

00:00:46.901 --> 00:00:49.591
so I still have a two in the 10's place.

00:00:49.591 --> 00:00:51.814
And now how many ones do I have?

00:00:51.814 --> 00:00:53.989
Well I used to have seven, I took one away

00:00:53.989 --> 00:00:58.665
now I have six ones,

00:00:58.665 --> 00:01:03.421
so two 10s and six ones, well we call that 26.

00:01:03.421 --> 00:01:05.785
Now let's think about 27 minus 10.

00:01:05.785 --> 00:01:10.595
So once again this is 27, two 10s and seven ones

00:01:10.595 --> 00:01:13.241
and now we're not going to take away any ones,

00:01:13.241 --> 00:01:15.663
but we're going to take away one 10.

00:01:15.663 --> 00:01:20.498
Let's just remind ourselves, a 10 is literally one 10.

00:01:20.498 --> 00:01:22.243
I know that sounds like I'm saying the same thing

00:01:22.243 --> 00:01:23.437
over and over again,

00:01:23.437 --> 00:01:26.093
but I really want you to appreciate the place value.

00:01:26.093 --> 00:01:28.217
This is the 10's place and I have a one there

00:01:28.217 --> 00:01:32.014
so I am going to take away one 10 and zero ones.

00:01:32.014 --> 00:01:34.022
So let's take away that 10.

00:01:34.022 --> 00:01:36.623
So let's take away, maybe I'll take this one away.

00:01:36.623 --> 00:01:40.316
I'll do it in a color that's easier to see.

00:01:40.316 --> 00:01:43.053
So I could take that away, and then what am I left with?

00:01:43.053 --> 00:01:45.069
How many 10s am I left with?

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Well now I only have one full group of 10

00:01:48.121 --> 00:01:52.493
so I have one 10, so that's the 10's place right over there.

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And how many ones do I have?

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Well I still have seven ones,

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so I still have seven in the one's place.

00:02:00.892 --> 00:02:05.975
So 27 minus 10 is one 10 and seven ones, or 17.

00:02:06.645 --> 00:02:08.298
And another way that you could think about it,

00:02:08.298 --> 00:02:09.772
and this might be the way that you did it

00:02:09.772 --> 00:02:11.287
when I first asked you to do it.

00:02:11.287 --> 00:02:13.041
Hey, if I'm taking away a one

00:02:13.041 --> 00:02:15.567
I'm going to reduce my one's place by one,

00:02:15.567 --> 00:02:17.888
so I'm going to go from 27 to 26.

00:02:17.888 --> 00:02:19.732
But if I'm taking away a 10,

00:02:19.732 --> 00:02:21.992
then my ten's place is going to go down by one

00:02:21.992 --> 00:02:23.850
so I go from a two in the ten's place

00:02:23.850 --> 00:02:25.309
to a one in the ten's place,

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I go from 27 to 17.

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The one's place didn't change.


Example of vector magnitude from initial and terminal points

WEBVTT
Kind: captions
Language: en

00:00:00.516 --> 00:00:02.456
- [Instructor] What we have depicted here

00:00:02.456 --> 00:00:04.373
we could call vector w,

00:00:05.336 --> 00:00:07.548
and you can see from this diagram

00:00:07.548 --> 00:00:09.967
that its initial point is right over here.

00:00:09.967 --> 00:00:13.956
It's the point negative
seven comma positive three,

00:00:13.956 --> 00:00:18.383
and its terminal point is
this point right over here,

00:00:18.383 --> 00:00:22.079
which is the point two comma negative one.

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What I want to do in
this video is think about

00:00:24.040 --> 00:00:27.040
what is the magnitude of our vector?

00:00:28.232 --> 00:00:30.821
And if you're saying what
do I mean by magnitude,

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well, one way to think about it is,

00:00:32.321 --> 00:00:34.399
what is the length of this vector?

00:00:34.399 --> 00:00:36.287
How long is it?

00:00:36.287 --> 00:00:38.409
Pause this video, and see
if you can figure it out

00:00:38.409 --> 00:00:41.265
based on the information that's given.

00:00:41.265 --> 00:00:43.090
Well, one thing that
might jump out at you is

00:00:43.090 --> 00:00:44.929
that the magnitude of this vector,

00:00:44.929 --> 00:00:47.640
the length of this vector
is really just the distance

00:00:47.640 --> 00:00:50.095
between these two points.

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And so if you want the magnitude,

00:00:52.213 --> 00:00:54.371
you just have to apply essentially

00:00:54.371 --> 00:00:56.124
the distance formula here,

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which is essentially just
the Pythagorean theorem.

00:00:58.980 --> 00:01:03.856
So what we could do is
construct a right triangle.

00:01:03.856 --> 00:01:06.333
I will do that like this.

00:01:06.333 --> 00:01:10.768
So this height in red, that
would be our change in y.

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That would be our change in y.

00:01:12.459 --> 00:01:15.979
And then what I am doing
in this light blue color,

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this would be our change in x,

00:01:20.726 --> 00:01:21.726
change in x.

00:01:23.007 --> 00:01:25.010
And we know from the Pythagorean theorem

00:01:25.010 --> 00:01:27.512
that the length of the hypotenuse,

00:01:27.512 --> 00:01:30.095
which would be the
magnitude of our vector,

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that that is going to be equal to,

00:01:32.586 --> 00:01:35.703
that's going to be equal
to the square root of

00:01:35.703 --> 00:01:38.126
our change in x squared,

00:01:38.126 --> 00:01:40.043
change in x squared,

00:01:41.386 --> 00:01:43.875
plus change in y squared,

00:01:43.875 --> 00:01:46.772
plus change in y squared.

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And so, what will this be?

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Well, what is our change in x?

00:01:50.727 --> 00:01:52.805
Our change in x,

00:01:52.805 --> 00:01:53.888
our change in x,

00:01:53.888 --> 00:01:57.843
you could view it as your
x final minus x initial.

00:01:57.843 --> 00:02:01.302
So this would be two minus negative seven.

00:02:01.302 --> 00:02:04.113
So this is two minus negative seven,

00:02:04.113 --> 00:02:06.930
which is equal to positive nine.

00:02:06.930 --> 00:02:09.763
And so this would be nine squared.

00:02:09.763 --> 00:02:11.793
And then what is our change in y?

00:02:11.793 --> 00:02:13.126
Our change in y,

00:02:14.318 --> 00:02:18.210
you could view this as your y
final, which is negative one,

00:02:18.210 --> 00:02:21.689
minus your y initial, which is three,

00:02:21.689 --> 00:02:23.590
minus three, which is
equal to negative four.

00:02:23.590 --> 00:02:26.365
And you did indeed go down by four,

00:02:26.365 --> 00:02:28.467
so this is going to be negative four.

00:02:28.467 --> 00:02:32.140
And so our magnitude is going
to be equal to the square root

00:02:32.140 --> 00:02:33.390
of nine squared

00:02:34.297 --> 00:02:35.130
is 81,

00:02:36.500 --> 00:02:39.250
plus negative four squared is 16.

00:02:40.209 --> 00:02:41.914
And so what is that going to be?

00:02:41.914 --> 00:02:45.893
Let's see, if you add six
to it, that's gets to 87,

00:02:45.893 --> 00:02:48.810
and you add another 10,
the square root of 97.

00:02:48.810 --> 00:02:52.926
So this is going to be equal
to the square root of 97,

00:02:52.926 --> 00:02:55.943
which I don't think can
be simplified anymore.

00:02:55.943 --> 00:02:57.902
But if you wanted to
estimate what that is,

00:02:57.902 --> 00:03:00.005
that's almost the square root of 100.

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So this number is going to
be a little bit less than 10

00:03:03.921 --> 00:03:07.380
is the magnitude of this vector.

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And in this case, we were able to do that

00:03:09.440 --> 00:03:12.549
from its initial points
and its ending point.

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Now, another way that a
vector might be specified,

00:03:15.105 --> 00:03:18.886
they might just be given an
x-component and a y-component.

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And so, for example,

00:03:21.526 --> 00:03:23.249
in this situation,

00:03:23.249 --> 00:03:26.388
you could actually define our vector w

00:03:26.388 --> 00:03:28.555
by the sum of two vectors,

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one of which is,

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let me do this in the blue color,

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one of which is the x-component.

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So you could view this
as the x-component of w,

00:03:38.937 --> 00:03:41.103
and then the other is the y-component.

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You could view this as,

00:03:42.743 --> 00:03:47.116
you could view this as
the y-component of w.

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And you could immediately see that

00:03:48.475 --> 00:03:51.251
that y-component is the
same as our change in y,

00:03:51.251 --> 00:03:54.599
and the x-component is the
same thing as our change in x.

00:03:54.599 --> 00:03:57.194
And so sometimes you will
see something like this.

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The vector,

00:03:59.388 --> 00:04:00.388
the vector w

00:04:01.973 --> 00:04:03.666
is equal to,

00:04:03.666 --> 00:04:04.912
and it might look like coordinates,

00:04:04.912 --> 00:04:07.581
but they're really giving
you the components.

00:04:07.581 --> 00:04:09.887
So the x-component

00:04:09.887 --> 00:04:11.304
is positive nine.

00:04:12.172 --> 00:04:14.800
The x-component is positive nine,

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and then the y-component is negative four.

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It is negative four.

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Now, you might say, hey, well,
with something like this,

00:04:23.540 --> 00:04:25.229
all I know is the x- and y-component,

00:04:25.229 --> 00:04:27.771
I don't know where it
exactly starts and ends.

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And that's actually on
purpose because, a vector,

00:04:30.767 --> 00:04:34.524
you only care about the
magnitude and the direction,

00:04:34.524 --> 00:04:36.496
and this is actually specifying both.

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If you wanted the magnitude here,

00:04:38.196 --> 00:04:39.681
you'd just take the square root

00:04:39.681 --> 00:04:42.554
of the sum of the squares
of the magnitudes.

00:04:42.554 --> 00:04:44.794
So once again, the square
root of nine squared

00:04:44.794 --> 00:04:46.218
plus negative four squared

00:04:46.218 --> 00:04:49.749
is going to be the square root of 97.

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So you want the magnitude
and the direction,

00:04:51.852 --> 00:04:53.245
which this will specify,

00:04:53.245 --> 00:04:56.094
but you can shift it
around all that you want.

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This vector w, you could
also have it starting,

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you could also have it
starting right over here

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and going nine in the positive x-direction

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and then negative four in
the positive y-direction

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and, or negative four down.

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And so it might look something like this.

00:05:12.841 --> 00:05:14.635
And so once again, you
can shift vectors around.

00:05:14.635 --> 00:05:16.531
You care about magnitude and direction,

00:05:16.531 --> 00:05:18.307
but hopefully this gives
you sense of how to

00:05:18.307 --> 00:05:20.018
find magnitude given the components

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or given the starting and ending points.


Worked example: separable differential equation (with taking log of both sides)

WEBVTT
Kind: captions
Language: en

00:00:00.250 --> 00:00:02.475
- [Instructor] Let's say
we need to find a solution

00:00:02.475 --> 00:00:04.280
to the differential equation

00:00:04.280 --> 00:00:07.379
that the derivative of y
with respect to x is equal

00:00:07.379 --> 00:00:10.046
to x squared over e to the y.

00:00:12.183 --> 00:00:15.015
Pause this video and see
if you can have a go at it,

00:00:15.015 --> 00:00:16.742
and I will give you a clue.

00:00:16.742 --> 00:00:20.766
It is a separable differential equation.

00:00:20.766 --> 00:00:22.316
All right, now let's do this together.

00:00:22.316 --> 00:00:24.827
So, whenever you do any
differential equation,

00:00:24.827 --> 00:00:27.829
the first thing you should try
to see is, is it separable?

00:00:27.829 --> 00:00:29.016
And when I say separable,

00:00:29.016 --> 00:00:33.083
I mean I can get all the
expressions that deal with y

00:00:33.083 --> 00:00:36.768
on the same side as dy,
and I can separate those

00:00:36.768 --> 00:00:38.508
from the expressions that deal with x,

00:00:38.508 --> 00:00:42.708
and they need to be on the same
side as my differential dx.

00:00:42.708 --> 00:00:44.274
So how can we do that?

00:00:44.274 --> 00:00:48.357
Well, if we multiply both
sides times e to the y,

00:00:49.305 --> 00:00:53.055
then e to the y will go
away right over here.

00:00:53.985 --> 00:00:56.332
So we will get rid of this y expression

00:00:56.332 --> 00:00:57.823
from the right hand side,

00:00:57.823 --> 00:01:00.948
and then we can multiply both sides by dx.

00:01:00.948 --> 00:01:04.038
So, if we did that, just
let me move my screen

00:01:04.038 --> 00:01:05.632
over a bit to the left.

00:01:05.632 --> 00:01:08.477
So I'm gonna multiply
both sides by e to the y,

00:01:08.477 --> 00:01:12.644
and I'm also gonna multiply
both sides by dx, times dx.

00:01:14.759 --> 00:01:16.357
And multiplying by dx gets rid

00:01:16.357 --> 00:01:17.949
of the dx on the left hand side,

00:01:17.949 --> 00:01:20.815
and it sits on the right
hand side with the x squared.

00:01:20.815 --> 00:01:23.815
And so all of this is now e to the y

00:01:24.741 --> 00:01:27.074
dy is equal to x squared,

00:01:30.560 --> 00:01:31.643
x squared dx.

00:01:33.217 --> 00:01:35.629
And just the fact that we
were able to do this shows

00:01:35.629 --> 00:01:37.138
that it is separable.

00:01:37.138 --> 00:01:39.078
Now, what we can do now is integrate

00:01:39.078 --> 00:01:43.571
both sides of this
equation, so let's do that.

00:01:43.571 --> 00:01:46.988
So what is the integral of e to the y dy?

00:01:48.019 --> 00:01:51.447
Well, one of the amazing
things about the expression,

00:01:51.447 --> 00:01:53.320
or you could say the function,

00:01:53.320 --> 00:01:55.525
something is equal to the e to the,

00:01:55.525 --> 00:01:56.871
and normally we say e to the x

00:01:56.871 --> 00:01:58.477
but in this case it's e to the y,

00:01:58.477 --> 00:02:01.591
is that the antiderivative
of this is just e to the y.

00:02:01.591 --> 00:02:02.973
We've learned that in multiple videos.

00:02:02.973 --> 00:02:05.893
I always express my fascination with it.

00:02:05.893 --> 00:02:07.825
So this is just e to the y.

00:02:07.825 --> 00:02:09.302
And likewise, if you took the derivative

00:02:09.302 --> 00:02:13.055
of e to the y with respect
to y, it would be e to the y.

00:02:13.055 --> 00:02:15.134
Remember, this works
because we are integrating

00:02:15.134 --> 00:02:16.619
with respect to y here.

00:02:16.619 --> 00:02:20.376
So the integral of e to
the y with respect to y,

00:02:20.376 --> 00:02:22.000
well, that's e to the y.

00:02:22.000 --> 00:02:24.219
And so that is going to be equal to

00:02:24.219 --> 00:02:25.882
the antiderivative of x squared.

00:02:25.882 --> 00:02:28.037
Well, that is we increment the exponent,

00:02:28.037 --> 00:02:29.581
so that gets us to x to the third power,

00:02:29.581 --> 00:02:32.320
and we divide by that
incremented exponent.

00:02:32.320 --> 00:02:34.789
And since I took the indefinite
integral of both sides,

00:02:34.789 --> 00:02:37.854
I have to put a constant on
at least one of these sides,

00:02:37.854 --> 00:02:41.236
so let me throw it over here, so plus c.

00:02:41.236 --> 00:02:42.601
And just to finish up,

00:02:42.601 --> 00:02:45.828
especially on a lot of
examination, like the AP exam,

00:02:45.828 --> 00:02:47.458
they might want you to write it in a form

00:02:47.458 --> 00:02:50.541
where y is an explicit function of x.

00:02:51.786 --> 00:02:55.107
So to do that, we can take
the natural log of both sides.

00:02:55.107 --> 00:02:57.018
So we take the natural log of that side,

00:02:57.018 --> 00:03:00.074
and we take the natural log of that side.

00:03:00.074 --> 00:03:02.214
Well, the natural log of e to the y,

00:03:02.214 --> 00:03:05.510
what power do we have to raise
e to to get to e to the y?

00:03:05.510 --> 00:03:07.187
Well, that's why we took the natural log.

00:03:07.187 --> 00:03:10.056
This just simplifies as y.

00:03:10.056 --> 00:03:12.139
And we get y is equal to,

00:03:13.621 --> 00:03:17.788
is equal to the natural log of
what we have right over here,

00:03:19.461 --> 00:03:22.211
x to the third over three plus c.

00:03:24.637 --> 00:03:25.970
And we are done.


Normative and positive statements

WEBVTT
Kind: captions
Language: en

00:00:00.808 --> 00:00:02.517
- [Instructor] What we're
going to do in this video

00:00:02.517 --> 00:00:05.485
is discuss the difference
between normative statements

00:00:05.485 --> 00:00:09.664
and positive statements, and
you'll see these words used

00:00:09.664 --> 00:00:11.894
usually in an economic context,

00:00:11.894 --> 00:00:13.921
sometimes a philosophical one.

00:00:13.921 --> 00:00:16.629
A normative statement is
one that really is a matter

00:00:16.629 --> 00:00:19.312
of opinion, maybe a matter of ethics,

00:00:19.312 --> 00:00:23.415
something that someone thinks
is how the world should be.

00:00:23.415 --> 00:00:25.834
While a positive statement
is something that,

00:00:25.834 --> 00:00:27.920
it doesn't necessarily have to be true

00:00:27.920 --> 00:00:30.616
but it's something that can be tested.

00:00:30.616 --> 00:00:32.329
So what we're going to do in this video

00:00:32.329 --> 00:00:35.598
is look at a bunch of
statements around economics

00:00:35.598 --> 00:00:37.633
and think about whether
they would be classified

00:00:37.633 --> 00:00:40.494
as normative statements,
things that are opinions,

00:00:40.494 --> 00:00:42.588
that are a matter of ethics or morals,

00:00:42.588 --> 00:00:44.689
or whether they are positive statements,

00:00:44.689 --> 00:00:47.005
things that can be tested.

00:00:47.005 --> 00:00:50.088
So let's look at our first statement.

00:00:50.088 --> 00:00:53.006
This says, "Paying people
who aren't working,

00:00:53.006 --> 00:00:57.389
"even though they could
work, is wrong and unfair."

00:00:57.389 --> 00:00:59.454
So regardless of whether or not you agree

00:00:59.454 --> 00:01:01.681
with this statement, is
it a normative statement

00:01:01.681 --> 00:01:04.120
or a positive statement?

00:01:04.120 --> 00:01:06.152
Well the fact that
someone's saying it's wrong

00:01:06.152 --> 00:01:10.362
and it's unfair, this is pretty
clearly a matter of opinion

00:01:10.362 --> 00:01:13.904
so this would be a normative statement.

00:01:13.904 --> 00:01:17.708
You can't test whether
this is wrong or unfair,

00:01:17.708 --> 00:01:21.871
you would just have to believe
that it is wrong and unfair.

00:01:21.871 --> 00:01:24.871
Now let's look at another statement.

00:01:26.787 --> 00:01:30.407
Programs like welfare reduce the incentive

00:01:30.407 --> 00:01:31.990
for people to work.

00:01:32.850 --> 00:01:37.348
Is this a normative statement
or a positive statement?

00:01:37.348 --> 00:01:40.354
Well it might feel a little normative,

00:01:40.354 --> 00:01:42.029
it might feel like this is an opinion,

00:01:42.029 --> 00:01:44.209
but it actually can be tested.

00:01:44.209 --> 00:01:47.864
You could institute some welfare
program on some small scale

00:01:47.864 --> 00:01:49.755
and compare it to a comparable place

00:01:49.755 --> 00:01:52.590
where there isn't a welfare
program and see what it does

00:01:52.590 --> 00:01:55.262
for incentives to work, you survey people,

00:01:55.262 --> 00:01:58.388
you see how many people work
in one situation or another.

00:01:58.388 --> 00:02:01.336
It might be a false statement,
it might be a true statement,

00:02:01.336 --> 00:02:04.499
but either way, it actually can be tested,

00:02:04.499 --> 00:02:07.582
so this would be a positive statement

00:02:08.681 --> 00:02:12.785
so I'll put it in this
category right over here,

00:02:12.785 --> 00:02:16.328
this is a positive, positive statement.

00:02:16.328 --> 00:02:18.258
Alright, let's look at another one.

00:02:18.258 --> 00:02:22.091
This say, "Raising taxes
on the wealthy to pay

00:02:22.967 --> 00:02:26.474
"for government programs
grows the economy."

00:02:26.474 --> 00:02:31.235
Is that a normative statement
or a positive statement?

00:02:31.235 --> 00:02:34.459
Well once again, this can be tested.

00:02:34.459 --> 00:02:37.427
It might be true, it might
be false, maybe your test

00:02:37.427 --> 00:02:40.061
is even inconclusive,
but it can be tested,

00:02:40.061 --> 00:02:42.156
you could try to run a
simulation, you could look

00:02:42.156 --> 00:02:44.627
at case studies of
countries that did do this

00:02:44.627 --> 00:02:46.991
and see what happens to their economy

00:02:46.991 --> 00:02:48.573
versus ones that didn't do it.

00:02:48.573 --> 00:02:50.600
And so this is, even though
it looks like something

00:02:50.600 --> 00:02:53.966
that someone who favors
raising taxes on the wealthy

00:02:53.966 --> 00:02:56.513
maybe out of fairness arguments,
something that they would

00:02:56.513 --> 00:03:00.298
say, this statement
itself is not normative,

00:03:00.298 --> 00:03:02.460
the statement can be tested,

00:03:02.460 --> 00:03:05.127
so this is a positive statement.

00:03:07.158 --> 00:03:10.317
A good giveaway for normative statement,

00:03:10.317 --> 00:03:13.019
if it said something like
it is fair to raise taxes

00:03:13.019 --> 00:03:15.395
on the wealthy to pay
for government programs,

00:03:15.395 --> 00:03:17.050
that would have been a normative statement

00:03:17.050 --> 00:03:19.406
or we should do this, that
would have been a normative

00:03:19.406 --> 00:03:22.622
statement but here, this is
something that's testable.

00:03:22.622 --> 00:03:27.467
Now the next statement,
"Raising taxes on the wealthy

00:03:27.467 --> 00:03:30.252
"slows economic growth."

00:03:30.252 --> 00:03:34.419
Is that a normative statement
or a positive statement?

00:03:35.899 --> 00:03:38.334
Well, once again, this
might feel like someone

00:03:38.334 --> 00:03:41.384
who is against raising
taxes, who think it's unfair

00:03:41.384 --> 00:03:44.185
to raise taxes on the wealthy,
something that they would say

00:03:44.185 --> 00:03:47.133
but the statement itself
can actually be tested.

00:03:47.133 --> 00:03:49.206
So this is also a positive statement,

00:03:49.206 --> 00:03:51.394
even though in some ways
it's the opposite statement

00:03:51.394 --> 00:03:53.419
as the one that we just did.

00:03:53.419 --> 00:03:57.527
Because once again, we could
look at countries that did this

00:03:57.527 --> 00:03:58.796
and countries that didn't do this,

00:03:58.796 --> 00:04:01.964
we could run a computer
simulation to try to understand

00:04:01.964 --> 00:04:04.359
whether this statement is true.

00:04:04.359 --> 00:04:07.122
Let's do one last statement.

00:04:07.122 --> 00:04:11.083
This says, "The government
should raise taxes

00:04:11.083 --> 00:04:13.872
"on the wealthy to pay
for helping the poor."

00:04:13.872 --> 00:04:16.621
Is this normative or positive?

00:04:16.621 --> 00:04:17.972
Well in this situation,

00:04:17.972 --> 00:04:20.319
the word should is a pretty big giveaway,

00:04:20.319 --> 00:04:23.946
should, or it's fair or unfair,
this is someone's opinion,

00:04:23.946 --> 00:04:25.648
it's not something that's testable,

00:04:25.648 --> 00:04:27.738
you can't test whether this
statement is right or wrong,

00:04:27.738 --> 00:04:30.815
it's based on, do you
believe ethically, morally,

00:04:30.815 --> 00:04:32.315
that this is true?

00:04:33.279 --> 00:04:36.817
And so this is a normative statement,

00:04:36.817 --> 00:04:40.067
so I'll put it in the normative column.

00:04:41.482 --> 00:04:44.868
So big picture, these words
normative and positive,

00:04:44.868 --> 00:04:48.531
these are fancy words but
all they mean is normative

00:04:48.531 --> 00:04:51.617
is a matter of morals or opinion
and can't really be tested

00:04:51.617 --> 00:04:54.752
while a positive statement,
whether they're right or wrong

00:04:54.752 --> 00:04:56.766
or whether you agree
or disagree with them,

00:04:56.766 --> 00:04:59.859
these are things that in
theory could be tested.


Solar and lunar eclipses

WEBVTT
Kind: captions
Language: en

00:00:00.607 --> 00:00:02.262
- [Instructor] In a previous
video, we asked ourselves

00:00:02.262 --> 00:00:04.112
a very important question.

00:00:04.112 --> 00:00:08.340
As the moon has its 28-day
cycle around the Earth,

00:00:08.340 --> 00:00:11.151
we talked about how a
new moon is when the moon

00:00:11.151 --> 00:00:14.878
is between the earth and the
sun, and so from the Earth's

00:00:14.878 --> 00:00:17.845
point of view, or from the
point of view of someone

00:00:17.845 --> 00:00:20.421
standing on Earth, you're
seeing the side of the moon

00:00:20.421 --> 00:00:21.892
that is not lit up.

00:00:21.892 --> 00:00:24.106
But an obvious question is
why doesn't that block out

00:00:24.106 --> 00:00:27.728
the sun every time we are
in a new moon position?

00:00:27.728 --> 00:00:32.621
After all, the sun would be out
here, 93 million miles away,

00:00:32.621 --> 00:00:37.047
and so wouldn't the moon block
out the sun in that scenario?

00:00:37.047 --> 00:00:38.873
And we also talked about
when the moon is on the other

00:00:38.873 --> 00:00:42.196
side of the Earth, when we
would typically see a full moon,

00:00:42.196 --> 00:00:45.859
the Earth is between the moon and the sun.

00:00:45.859 --> 00:00:48.952
Why doesn't the Earth block
the light from the sun

00:00:48.952 --> 00:00:51.291
that's being reflected on the moon?

00:00:51.291 --> 00:00:54.055
So the big question is, why
don't we see a solar eclipse

00:00:54.055 --> 00:00:57.177
every new moon and why
don't we see a lunar eclipse

00:00:57.177 --> 00:00:58.975
every full moon?

00:00:58.975 --> 00:01:01.520
If we wanna ask the same
question, looking at the scale

00:01:01.520 --> 00:01:04.767
of the Earth and the moon,
we can see it right over here

00:01:04.767 --> 00:01:08.114
and so this would be the new moon position

00:01:08.114 --> 00:01:10.850
where the moon is between
the Earth and the sun.

00:01:10.850 --> 00:01:14.869
The key explanation for why
we do not have a solar eclipse

00:01:14.869 --> 00:01:17.630
every time we are in the new moon position

00:01:17.630 --> 00:01:20.707
is that the rotation of
the moon around the Earth

00:01:20.707 --> 00:01:23.874
is not in the exact same
plane as the rotation

00:01:23.874 --> 00:01:26.207
of the Earth around the sun.

00:01:27.827 --> 00:01:30.759
It actually turns out that the
plane of the moon's rotation

00:01:30.759 --> 00:01:33.423
is at a five-degree angle with the plane

00:01:33.423 --> 00:01:35.506
of the Earth and the sun.

00:01:36.430 --> 00:01:39.141
If this is the sun here, and
this is not drawn to scale

00:01:39.141 --> 00:01:42.047
by any means, this is the
Earth, and so the Earth

00:01:42.047 --> 00:01:45.130
is in its orbit, something like that,

00:01:46.563 --> 00:01:50.462
the moon's orbit does not
sit exactly in this plane,

00:01:50.462 --> 00:01:52.654
it does not sit exactly in this plane,

00:01:52.654 --> 00:01:56.018
the moon's orbit is at
a five-degree angle,

00:01:56.018 --> 00:01:58.409
is at a five-degree angle to this plane,

00:01:58.409 --> 00:02:00.746
so depending what time of year you are at

00:02:00.746 --> 00:02:02.604
and where the moon is in its cycle,

00:02:02.604 --> 00:02:06.779
the moon will often sit
above or below the plane

00:02:06.779 --> 00:02:10.946
of the Earth's orbit, so for
example, from the vantage point

00:02:11.888 --> 00:02:16.055
of the Earth, the moon will
vary up to five degrees

00:02:18.893 --> 00:02:21.883
above the plane of Earth's
rotation around the sun,

00:02:21.883 --> 00:02:26.584
often known as the ecliptic,
and five degrees below that.

00:02:26.584 --> 00:02:30.347
And so you can see here,
the shadow of the moon

00:02:30.347 --> 00:02:34.188
will only fall on the Earth
when the moon is crossing

00:02:34.188 --> 00:02:37.911
through the plane of the
rotation of Earth around the sun.

00:02:37.911 --> 00:02:42.332
Many times, the moon might
be here or here or here

00:02:42.332 --> 00:02:47.299
and in those times, the
shadow will not hit the Earth.

00:02:47.299 --> 00:02:49.780
Now what you do see here,
depicted in this picture,

00:02:49.780 --> 00:02:52.069
would be a solar eclipse.

00:02:52.069 --> 00:02:54.269
Now there's two things
that I've drawn here

00:02:54.269 --> 00:02:57.856
and this is important for
understanding a solar eclipse,

00:02:57.856 --> 00:02:59.682
what you see in those yellow lines,

00:02:59.682 --> 00:03:02.907
that's the umbra of the
moon, so if you are in,

00:03:02.907 --> 00:03:05.559
if you're at that point right
over here in this picture,

00:03:05.559 --> 00:03:08.718
then the moon will
completely block out the sun

00:03:08.718 --> 00:03:11.513
and it turns out depending
on the solar eclipse,

00:03:11.513 --> 00:03:14.275
where that umbra is hitting the Earth,

00:03:14.275 --> 00:03:16.962
it might only be a few
hundred miles where the moon

00:03:16.962 --> 00:03:19.696
completely blocks out the sun.

00:03:19.696 --> 00:03:23.315
What you see in this dark blue
color, that is the penumbra,

00:03:23.315 --> 00:03:26.057
that is the outer shadow and
if you're in one of those

00:03:26.057 --> 00:03:30.727
areas, you will see the moon
partially block out the sun.

00:03:30.727 --> 00:03:32.852
Now this is the exact same reason,

00:03:32.852 --> 00:03:36.088
the fact that the plane of its
rotation is at a five-degree

00:03:36.088 --> 00:03:38.231
angle with the plane of
rotation of the Earth

00:03:38.231 --> 00:03:41.258
around the sun, that's also
the reason why we don't see

00:03:41.258 --> 00:03:43.758
a lunar eclipse every 28 days.

00:03:45.599 --> 00:03:48.380
Here we see different
scenarios where the moon

00:03:48.380 --> 00:03:51.594
is on the other side
of Earth from the sun,

00:03:51.594 --> 00:03:54.798
when the moon is what we would
call a full moon position,

00:03:54.798 --> 00:03:57.567
and as we can see, it can
vary five degrees above

00:03:57.567 --> 00:04:00.219
the plane of the Earth and
the sun or five degrees

00:04:00.219 --> 00:04:04.377
below that and you're only
going to get a scenario

00:04:04.377 --> 00:04:08.377
of a lunar eclipse when
the moon happens to fall

00:04:09.943 --> 00:04:12.875
in the shadow of the
Earth and here once again,

00:04:12.875 --> 00:04:16.555
between the yellow
lines, you see the umbra,

00:04:16.555 --> 00:04:20.616
and between the blue lines,
you see the penumbra,

00:04:20.616 --> 00:04:23.042
which would be kind of a partial shadow.

00:04:23.042 --> 00:04:25.226
So next time you look up at the moon,

00:04:25.226 --> 00:04:27.145
hopefully you are armed
with a lot more information

00:04:27.145 --> 00:04:30.704
to think about how it's oriented
with respect to the Earth.

00:04:30.704 --> 00:04:32.590
In fact, now when you
look at the moon at night,

00:04:32.590 --> 00:04:35.503
you can usually look at the
moon and tell which direction

00:04:35.503 --> 00:04:37.940
the sun is in and based on
that, you can even think about

00:04:37.940 --> 00:04:40.845
what time of day it is
and you could figure out

00:04:40.845 --> 00:04:43.550
east, west, north, and south.

00:04:43.550 --> 00:04:45.619
And we've talked a lot
about the sun, in fact,

00:04:45.619 --> 00:04:47.500
this entire video and the one before it

00:04:47.500 --> 00:04:49.431
is all about light from the sun.

00:04:49.431 --> 00:04:51.769
And just to finish off
with a little bit of awe,

00:04:51.769 --> 00:04:55.491
let's just appreciate how
large and how far the sun is.

00:04:55.491 --> 00:04:58.315
As we mentioned before, this
is a scaled representation

00:04:58.315 --> 00:05:01.271
of the Earth and the moon,
the moon being roughly

00:05:01.271 --> 00:05:04.111
240,000 miles away from Earth.

00:05:04.111 --> 00:05:06.969
If we were to try to model the sun here,

00:05:06.969 --> 00:05:10.425
it is 400 times as far,
depending on your screen size,

00:05:10.425 --> 00:05:12.966
it would be a football field to the right

00:05:12.966 --> 00:05:15.441
of your screen and you
would have something

00:05:15.441 --> 00:05:19.103
roughly the size of a beach
ball to represent the sun.

00:05:19.103 --> 00:05:21.624
If you wanted to just
compare the sizes of the sun

00:05:21.624 --> 00:05:24.874
to the Earth and the moon, here you go.

00:05:25.945 --> 00:05:27.362
The sun is huge.


The Moon

WEBVTT
Kind: captions
Language: en

00:00:00.438 --> 00:00:01.362
- [Narrator] When you look at

00:00:01.362 --> 00:00:02.405
the moon in the night sky,

00:00:02.405 --> 00:00:04.718
it might seem reasonably close.

00:00:04.718 --> 00:00:08.665
But it's actually incredibly far away.

00:00:08.665 --> 00:00:10.685
Right over here is a scaled picture

00:00:10.685 --> 00:00:12.136
of the distance between

00:00:12.136 --> 00:00:14.763
the earth and the moon.

00:00:14.763 --> 00:00:16.271
Earth has a diameter

00:00:16.271 --> 00:00:18.688
of approximately 8,000 miles.

00:00:20.754 --> 00:00:22.653
While the moon has a diameter

00:00:22.653 --> 00:00:25.070
of approximately 2,200 miles.

00:00:26.533 --> 00:00:28.241
So, a little bit more

00:00:28.241 --> 00:00:30.663
than one fourth the diameter of earth.

00:00:30.663 --> 00:00:33.057
Now the distance between the two

00:00:33.057 --> 00:00:34.474
is 239,000 miles.

00:00:39.120 --> 00:00:41.939
Which you can imagine is incredibly far.

00:00:41.939 --> 00:00:44.189
Now one thing that is very interesting,

00:00:44.189 --> 00:00:45.628
this is why the moon looks like

00:00:45.628 --> 00:00:48.641
it's the same size as the sun in the sky,

00:00:48.641 --> 00:00:50.569
even though it is one four hundredth

00:00:50.569 --> 00:00:52.150
of the diameter,

00:00:52.150 --> 00:00:53.916
is the ratio between

00:00:53.916 --> 00:00:55.069
the distance to the moon

00:00:55.069 --> 00:00:56.915
and the diameter of the moon

00:00:56.915 --> 00:00:58.370
is roughly the same as

00:00:58.370 --> 00:01:00.196
the ratio of the distance

00:01:00.196 --> 00:01:02.316
of the sun to the diameter of the sun.

00:01:02.316 --> 00:01:03.570
So, for the moon,

00:01:03.570 --> 00:01:05.492
that ratio, the distance to the moon,

00:01:05.492 --> 00:01:07.909
239,000 miles over the radius

00:01:11.132 --> 00:01:13.449
of the moon, 2,200.

00:01:13.449 --> 00:01:14.662
It's approximately the same

00:01:14.662 --> 00:01:15.998
as the ratio between

00:01:15.998 --> 00:01:17.499
the distance from earth

00:01:17.499 --> 00:01:18.979
to the sun which would be

00:01:18.979 --> 00:01:20.729
93 million miles over

00:01:24.731 --> 00:01:26.898
865,000 miles in diameter.

00:01:29.954 --> 00:01:31.041
So this number is roughly

00:01:31.041 --> 00:01:33.068
400 times larger than that number.

00:01:33.068 --> 00:01:34.196
And this number is roughly

00:01:34.196 --> 00:01:36.117
400 times larger than that number.

00:01:36.117 --> 00:01:37.296
But what's neat is,

00:01:37.296 --> 00:01:39.025
and this depends on where earth is

00:01:39.025 --> 00:01:39.960
in its orbit,

00:01:39.960 --> 00:01:43.380
but this is approximately equal to 108.

00:01:43.380 --> 00:01:44.514
This gives me goosebumps

00:01:44.514 --> 00:01:45.856
whenever I think about it

00:01:45.856 --> 00:01:49.021
because it didn't have to be this way.

00:01:49.021 --> 00:01:50.543
That the moon and the sun,

00:01:50.543 --> 00:01:51.863
even though they have these vastly

00:01:51.863 --> 00:01:53.509
different diameters,

00:01:53.509 --> 00:01:55.658
that they look roughly the same size

00:01:55.658 --> 00:01:57.850
in the sky because the ratio

00:01:57.850 --> 00:01:59.834
of the distances is comparable

00:01:59.834 --> 00:02:02.188
to the ratios of their diameters.

00:02:02.188 --> 00:02:03.421
And this is why the number 108

00:02:03.421 --> 00:02:05.029
is actually an auspicious number

00:02:05.029 --> 00:02:05.927
in some cultures,

00:02:05.927 --> 00:02:07.519
especially in Hinduism.

00:02:07.519 --> 00:02:08.437
Now, when you're discussing

00:02:08.437 --> 00:02:10.383
the moon and especially the moon

00:02:10.383 --> 00:02:11.656
as viewed from earth,

00:02:11.656 --> 00:02:12.931
one of the obvious questions

00:02:12.931 --> 00:02:16.304
is why does the moon go through phases?

00:02:16.304 --> 00:02:18.188
Why do we see different portions

00:02:18.188 --> 00:02:19.021
of the moon lit up

00:02:19.021 --> 00:02:20.354
from day to day?

00:02:22.009 --> 00:02:23.328
This right over here is a diagram

00:02:23.328 --> 00:02:25.727
from NASA's website.

00:02:25.727 --> 00:02:27.019
And what you see here,

00:02:27.019 --> 00:02:28.317
clearly this is not at scale.

00:02:28.317 --> 00:02:29.850
This picture over here,

00:02:29.850 --> 00:02:31.133
this inner picture,

00:02:31.133 --> 00:02:32.197
the size of the moon

00:02:32.197 --> 00:02:33.960
and the earth is roughly at scale,

00:02:33.960 --> 00:02:35.080
but clearly the distances

00:02:35.080 --> 00:02:36.338
between them are not.

00:02:36.338 --> 00:02:38.271
You don't see that 239,000 miles

00:02:38.271 --> 00:02:39.203
between them.

00:02:39.203 --> 00:02:40.362
In this inner circle,

00:02:40.362 --> 00:02:41.748
what you see is that the moon

00:02:41.748 --> 00:02:43.653
and the earth is always lit

00:02:43.653 --> 00:02:45.310
up from the right.

00:02:45.310 --> 00:02:46.146
So, it's assuming that the sun

00:02:46.146 --> 00:02:47.999
is 93 million miles

00:02:47.999 --> 00:02:50.063
in that direction.

00:02:50.063 --> 00:02:51.299
And it is lighting up

00:02:51.299 --> 00:02:53.057
both the earth and the moon

00:02:53.057 --> 00:02:54.954
from the right.

00:02:54.954 --> 00:02:58.371
Now, as the moon rotates around the earth

00:02:59.305 --> 00:03:01.196
in its 28 day cycle,

00:03:01.196 --> 00:03:02.946
and if you're wondering,

00:03:02.946 --> 00:03:04.403
gee a 28 day cycle

00:03:04.403 --> 00:03:06.941
seems awfully close to a month,

00:03:06.941 --> 00:03:08.474
that is not a coincidence.

00:03:08.474 --> 00:03:09.980
The notion of a month comes

00:03:09.980 --> 00:03:12.812
from the cycles of the moon.

00:03:12.812 --> 00:03:13.645
In fact,

00:03:13.645 --> 00:03:14.916
even the word month

00:03:14.916 --> 00:03:16.936
and the word moon

00:03:16.936 --> 00:03:18.416
have the exact same root

00:03:18.416 --> 00:03:20.471
in old English.

00:03:20.471 --> 00:03:21.767
And in proto Germanic.

00:03:21.767 --> 00:03:24.755
They are essentially the same word.

00:03:24.755 --> 00:03:26.253
But what you see is the moon goes

00:03:26.253 --> 00:03:28.439
in this 28 day cycle,

00:03:28.439 --> 00:03:30.744
when the moon is roughly between

00:03:30.744 --> 00:03:32.857
the earth and the sun,

00:03:32.857 --> 00:03:34.577
the lit up half of the moon

00:03:34.577 --> 00:03:36.061
is away from what

00:03:36.061 --> 00:03:37.508
we can see here on earth.

00:03:37.508 --> 00:03:39.099
And so, we see the non lit up side

00:03:39.099 --> 00:03:41.327
which would be a new moon.

00:03:41.327 --> 00:03:42.985
Now, as the moon goes,

00:03:42.985 --> 00:03:44.985
this is viewing from above the earth,

00:03:44.985 --> 00:03:46.923
this would be the North
Pole right over here,

00:03:46.923 --> 00:03:48.633
as it goes counter clockwise

00:03:48.633 --> 00:03:50.278
as viewed from above the earth,

00:03:50.278 --> 00:03:52.058
we start to be able to see

00:03:52.058 --> 00:03:53.779
a little bit of that half of the moon

00:03:53.779 --> 00:03:55.051
that gets lit up.

00:03:55.051 --> 00:03:56.202
So, when the moon is in this position,

00:03:56.202 --> 00:03:58.064
we see, us from earth,

00:03:58.064 --> 00:03:59.360
from this vantage point,

00:03:59.360 --> 00:04:01.049
we're able to see a little bit

00:04:01.049 --> 00:04:02.372
of the lit up side.

00:04:02.372 --> 00:04:03.708
When the moon is over here,

00:04:03.708 --> 00:04:05.342
we're able to see half

00:04:05.342 --> 00:04:06.419
the lit up side

00:04:06.419 --> 00:04:08.189
and half the non lit up side.

00:04:08.189 --> 00:04:09.849
And that's called a first quarter moon.

00:04:09.849 --> 00:04:11.019
And that keeps on going.

00:04:11.019 --> 00:04:12.402
When we're halfway through

00:04:12.402 --> 00:04:13.273
our cycle here,

00:04:13.273 --> 00:04:14.445
at a full moon,

00:04:14.445 --> 00:04:16.355
the earth is between the moon

00:04:16.355 --> 00:04:19.337
and the sun and so from our vantage point,

00:04:19.337 --> 00:04:20.968
we are able to see

00:04:20.968 --> 00:04:22.885
the entire lit up side.

00:04:23.836 --> 00:04:25.931
And that's why it is a full moon.

00:04:25.931 --> 00:04:27.035
And then we keep going

00:04:27.035 --> 00:04:28.670
all the way until we get back

00:04:28.670 --> 00:04:29.837
to a new moon.

00:04:30.725 --> 00:04:33.291
Now, one question that
might be bothering you,

00:04:33.291 --> 00:04:35.386
it bothered me for many years,

00:04:35.386 --> 00:04:37.462
is as soon as I understood this,

00:04:37.462 --> 00:04:38.586
the cycle of the moon,

00:04:38.586 --> 00:04:40.524
how the moon has this 28 day cycle

00:04:40.524 --> 00:04:43.464
as earth rotates around the sun,

00:04:43.464 --> 00:04:45.881
I always wondered well the new moon,

00:04:45.881 --> 00:04:48.048
it looks like the moon is between

00:04:48.048 --> 00:04:50.374
the earth and the sun,

00:04:50.374 --> 00:04:52.550
why doesn't it block out the sun

00:04:52.550 --> 00:04:54.473
every time we have a new moon?

00:04:54.473 --> 00:04:55.865
Why don't we have a solar eclipse

00:04:55.865 --> 00:04:57.032
every 28 days?

00:04:58.051 --> 00:04:58.884
Similarly,

00:04:58.884 --> 00:05:00.268
for the full moons,

00:05:00.268 --> 00:05:01.883
when we're in this position,

00:05:01.883 --> 00:05:03.111
it looks like the earth

00:05:03.111 --> 00:05:05.688
is between the moon and the sun.

00:05:05.688 --> 00:05:07.274
Why doesn't earth's shadow

00:05:07.274 --> 00:05:08.999
block out the sun

00:05:08.999 --> 00:05:10.641
so that we have a lunar eclipse

00:05:10.641 --> 00:05:12.110
instead of a full moon

00:05:12.110 --> 00:05:13.029
and we would have one of those

00:05:13.029 --> 00:05:14.362
every 28 days?

00:05:14.362 --> 00:05:16.034
Why don't we see that?

00:05:16.034 --> 00:05:17.685
Think about it yourself.

00:05:17.685 --> 00:05:19.667
What's a plausible explanation?

00:05:19.667 --> 00:05:20.670
In the next video,

00:05:20.670 --> 00:05:23.021
we'll actually talk about why eclipses,

00:05:23.021 --> 00:05:25.115
both solar eclipses when the moon

00:05:25.115 --> 00:05:26.569
blocks out the sun,

00:05:26.569 --> 00:05:28.018
and lunar eclipses when

00:05:28.018 --> 00:05:29.497
the earth's shadow

00:05:29.497 --> 00:05:30.624
falls on the moon,

00:05:30.624 --> 00:05:33.760
why they're not happening every 28 days.


Justification using first derivative

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:02.611
- [Instructor] "The
differentiable function f

00:00:02.611 --> 00:00:06.088
"and its derivative f prime are graphed."

00:00:06.088 --> 00:00:07.138
So, let's see.

00:00:07.138 --> 00:00:11.594
We see the graph of y is equal
to f of x here in the blue.

00:00:11.594 --> 00:00:15.818
And then f prime we see in
this brownish-orangish color

00:00:15.818 --> 00:00:17.900
right over here.

00:00:17.900 --> 00:00:22.781
"What is an appropriate
calculus-based justification

00:00:22.781 --> 00:00:25.172
"for the fact that f is decreasing

00:00:25.172 --> 00:00:28.447
"when x is greater than three?"

00:00:28.447 --> 00:00:31.054
So, we can see that that
actually is indeed the case.

00:00:31.054 --> 00:00:33.511
When x is greater than three,

00:00:33.511 --> 00:00:35.914
we see that our function
is indeed decreasing.

00:00:35.914 --> 00:00:38.497
As x increases, the y value,

00:00:39.440 --> 00:00:42.429
the value of our function, decreases.

00:00:42.429 --> 00:00:44.092
So, a calculus-based justification,

00:00:44.092 --> 00:00:45.985
without even looking at the choices,

00:00:45.985 --> 00:00:47.538
well, I could look at the derivative.

00:00:47.538 --> 00:00:49.288
And we're going to be decreasing

00:00:49.288 --> 00:00:52.145
if the slope of the
tangent line is negative,

00:00:52.145 --> 00:00:54.297
which means that the
derivative is negative.

00:00:54.297 --> 00:00:57.465
And we can see that for
x is greater than three,

00:00:57.465 --> 00:00:59.731
the derivative is less than zero.

00:00:59.731 --> 00:01:00.985
So, my justification,
I haven't even looked

00:01:00.985 --> 00:01:02.174
at these choices yet.

00:01:02.174 --> 00:01:06.453
I would say for x is greater than three,

00:01:06.453 --> 00:01:10.199
f prime of x is less than a zero.

00:01:10.199 --> 00:01:11.488
That would be my justification,

00:01:11.488 --> 00:01:13.416
not even looking at these choices.

00:01:13.416 --> 00:01:15.215
Now let's look at the choices.

00:01:15.215 --> 00:01:19.382
"f prime is decreasing when
x is greater than three."

00:01:20.404 --> 00:01:21.334
Now, this isn't right.

00:01:21.334 --> 00:01:22.249
What we care about is

00:01:22.249 --> 00:01:25.105
whether f prime is positive or negative.

00:01:25.105 --> 00:01:28.280
If f prime is negative,
if it's less than zero,

00:01:28.280 --> 00:01:30.247
than the function itself is decreasing.

00:01:30.247 --> 00:01:33.723
The slope of the tangent
line will be negative.

00:01:33.723 --> 00:01:36.037
f prime could be positive
while decreasing.

00:01:36.037 --> 00:01:39.153
For example, f prime could
be doing something like this.

00:01:39.153 --> 00:01:41.750
And even though f prime
would be decreasing

00:01:41.750 --> 00:01:44.769
in this situation, the actual value

00:01:44.769 --> 00:01:46.283
of the derivative would be positive,

00:01:46.283 --> 00:01:47.862
which means the function
would be increasing

00:01:47.862 --> 00:01:51.305
in that scenario, so I
would rule this one out.

00:01:51.305 --> 00:01:55.778
"For values of x larger than
three, as x values increase,

00:01:55.778 --> 00:01:58.005
"the values of f of x decrease."

00:01:58.005 --> 00:01:59.510
Now, that is actually true.

00:01:59.510 --> 00:02:02.736
This is actually the definition
that f is decreasing.

00:02:02.736 --> 00:02:06.356
As x values increase, the
values of f of x decrease.

00:02:06.356 --> 00:02:09.039
But this is not a
calculus-based justification,

00:02:09.039 --> 00:02:12.980
so I am going to rule
this one out as well.

00:02:12.980 --> 00:02:17.077
"f prime is negative when
x is greater than three."

00:02:17.077 --> 00:02:19.015
Well, that's exactly what I wrote up here.

00:02:19.015 --> 00:02:22.494
If f prime is negative, then that means

00:02:22.494 --> 00:02:24.101
that our slope of the tangent line

00:02:24.101 --> 00:02:27.301
of our original function f is
going to be downward sloping,

00:02:27.301 --> 00:02:29.861
or that our function is decreasing,

00:02:29.861 --> 00:02:32.429
so this one is looking good.

00:02:32.429 --> 00:02:33.702
And this one right over here says,

00:02:33.702 --> 00:02:38.038
"f prime of zero is
equal to negative three,"

00:02:38.038 --> 00:02:39.743
so they're just pointing out this point.

00:02:39.743 --> 00:02:42.431
This isn't relative to the
interval that we care about,

00:02:42.431 --> 00:02:46.345
or this isn't even relevant
when x is greater than three,

00:02:46.345 --> 00:02:48.597
so we definitely wanna rule that one out.

00:02:48.597 --> 00:02:50.847
Let's do one more of these.

00:02:52.018 --> 00:02:55.228
So, here we're told, "The
differentiable function g

00:02:55.228 --> 00:02:58.055
"and its derivative g prime are graphed."

00:02:58.055 --> 00:03:00.664
So, once again, g is in this bluish color,

00:03:00.664 --> 00:03:04.306
and then g prime, its derivative,
is in this orange color.

00:03:04.306 --> 00:03:07.710
"What is an appropriate
calculus-based justification

00:03:07.710 --> 00:03:11.511
"for the fact that g has
a relative minimum point

00:03:11.511 --> 00:03:13.546
"at x is equal to negative three?"

00:03:13.546 --> 00:03:16.012
And we could see here, when
x is equal to negative three,

00:03:16.012 --> 00:03:19.348
it looks like g is equal to negative six,

00:03:19.348 --> 00:03:21.672
and it looks like a relative
minimum point there.

00:03:21.672 --> 00:03:23.945
So, what's the best justification?

00:03:23.945 --> 00:03:26.292
So, once again, without
even looking at the choices,

00:03:26.292 --> 00:03:29.392
I would say a good justification is

00:03:29.392 --> 00:03:32.286
before we get to x equals negative three,

00:03:32.286 --> 00:03:35.052
before we get to x equals negative three,

00:03:35.052 --> 00:03:38.173
our derivative, and this is a
calculus-based justification,

00:03:38.173 --> 00:03:40.728
before we go to x equals negative three,

00:03:40.728 --> 00:03:43.508
our derivative is negative.

00:03:43.508 --> 00:03:45.530
And after x equals negative three,

00:03:45.530 --> 00:03:47.274
our derivative is positive.

00:03:47.274 --> 00:03:49.187
That would be my justification.

00:03:49.187 --> 00:03:53.354
Because if our derivative is
negative before that value,

00:03:54.671 --> 00:03:59.153
that means that we are downward
sloping before that value.

00:03:59.153 --> 00:04:00.722
And if it's positive after that value,

00:04:00.722 --> 00:04:02.809
that means we're upward
sloping after that,

00:04:02.809 --> 00:04:04.457
which is a good justification

00:04:04.457 --> 00:04:08.033
that we are at relative
minimum point right over there.

00:04:08.033 --> 00:04:10.189
So, let's see, "The point
where x equals negative three

00:04:10.189 --> 00:04:12.101
"is the lowest point on the graph of g

00:04:12.101 --> 00:04:14.086
"in its surrounding interval."

00:04:14.086 --> 00:04:15.557
That is true,

00:04:15.557 --> 00:04:17.594
but that's not a
calculus-based justification.

00:04:17.594 --> 00:04:18.992
You wouldn't even have
to look at the derivative

00:04:18.992 --> 00:04:22.749
to make that statement, so
let's rule that one out.

00:04:22.749 --> 00:04:26.462
"g prime has a relative
maximum at zero comma three."

00:04:26.462 --> 00:04:29.962
At zero comma three, it actually does not.

00:04:31.747 --> 00:04:33.279
Oh, g prime, yes,

00:04:33.279 --> 00:04:36.261
g prime actually does
have a relative maximum

00:04:36.261 --> 00:04:39.848
at zero comma three, but
that doesn't tell us anything

00:04:39.848 --> 00:04:43.399
about whether we're at
a relative minimum point

00:04:43.399 --> 00:04:47.566
at x equals negative three,
so I would rule that out.

00:04:49.399 --> 00:04:53.754
"g prime of negative
three is equal to zero."

00:04:53.754 --> 00:04:56.469
So, g prime of negative
three is equal to zero,

00:04:56.469 --> 00:04:58.484
so that tells us that the
slope of the tangent line

00:04:58.484 --> 00:05:01.221
of our function is going to
be zero right over there,

00:05:01.221 --> 00:05:03.231
but that by itself is not enough

00:05:03.231 --> 00:05:05.398
to say that we are at
relative minimum point.

00:05:05.398 --> 00:05:07.912
For example, I could be at
a point that does something

00:05:07.912 --> 00:05:09.586
like this where the slope
of the tangent line is zero,

00:05:09.586 --> 00:05:11.595
and then it keeps increasing again,

00:05:11.595 --> 00:05:14.575
or it does something like
this and it keeps decreasing.

00:05:14.575 --> 00:05:16.219
So, even though you're at a point

00:05:16.219 --> 00:05:19.479
where the slope of your
tangent line is zero,

00:05:19.479 --> 00:05:21.754
it doesn't mean you're at
a relative minimum point,

00:05:21.754 --> 00:05:23.293
so I would rule that out.

00:05:23.293 --> 00:05:26.739
"g prime crosses the x-axis
from below it to above it

00:05:26.739 --> 00:05:28.371
"at x equals negative three."

00:05:28.371 --> 00:05:31.968
g prime crosses the x-axis
from below it to above it.

00:05:31.968 --> 00:05:33.782
Yep, and that's the argument that I made,

00:05:33.782 --> 00:05:35.725
that we're going from below the x-axis,

00:05:35.725 --> 00:05:37.916
so g prime goes from being
negative to positive,

00:05:37.916 --> 00:05:40.617
which means the slope of the tangent lines

00:05:40.617 --> 00:05:43.006
of our points as we approach
x equals negative three go

00:05:43.006 --> 00:05:46.379
from being downward
sloping to upward sloping,

00:05:46.379 --> 00:05:47.743
which is an indication

00:05:47.743 --> 00:05:51.076
that we are at a relative minimum point.


Examples establishing conditions for MVT

WEBVTT
Kind: captions
Language: en

00:00:01.309 --> 00:00:03.190
- [Instructor] This table
gives us a few values

00:00:03.190 --> 00:00:04.966
of the function g.

00:00:04.966 --> 00:00:07.753
So we know what g of x is
equal to with these values

00:00:07.753 --> 00:00:08.751
right over here.

00:00:08.751 --> 00:00:11.143
X is equal to negative two,
negative one, zero, and one.

00:00:11.143 --> 00:00:15.310
It says Rafael said that
since g of one minus g of zero

00:00:16.646 --> 00:00:20.905
over one minus zero is equal
to negative five, there must be

00:00:20.905 --> 00:00:24.745
a number c in the interval in
the closed interval from zero

00:00:24.745 --> 00:00:28.912
to one for which g prime of
c is equal to negative five.

00:00:31.017 --> 00:00:35.174
Which condition makes Rafael's
claim true and they give us

00:00:35.174 --> 00:00:36.674
four choices here.

00:00:37.660 --> 00:00:40.992
So I encourage you to pause
this video and try to work

00:00:40.992 --> 00:00:43.380
through it on your own.

00:00:43.380 --> 00:00:45.311
Alright before we even look
at these choices let's just

00:00:45.311 --> 00:00:48.243
revisit what's going on over here.

00:00:48.243 --> 00:00:53.076
So Rafael is saying since this,
so he's looking at g of one

00:00:53.076 --> 00:00:57.243
minus g of zero, so this is
g of one, minus g of zero

00:01:00.276 --> 00:01:03.421
that's this right over here,
over one minus zero, over one

00:01:03.421 --> 00:01:04.338
minus zero.

00:01:05.231 --> 00:01:06.406
So what is this?

00:01:06.406 --> 00:01:10.573
This right over here is
the average rate of change,

00:01:12.938 --> 00:01:17.105
rate of change between x equals
zero and x is equal to one.

00:01:26.358 --> 00:01:28.332
Another way to think about it
this is the slope of the line

00:01:28.332 --> 00:01:32.082
that connects the point
0,8 to the point 1,3.

00:01:33.626 --> 00:01:37.292
And that average rate of
change is negative five.

00:01:37.292 --> 00:01:39.708
And that he's saying
there must be a number c

00:01:39.708 --> 00:01:43.630
in the interval, in the closed
interval from zero to one

00:01:43.630 --> 00:01:47.797
for which the derivative of
that x value what x is equal

00:01:49.228 --> 00:01:52.940
to c is the same thing as
this average rate of change.

00:01:52.940 --> 00:01:55.436
So this is what the mean
value theorem is all about,

00:01:55.436 --> 00:01:57.833
it's this notion that...

00:01:57.833 --> 00:02:01.548
We plot it a little bit
just as a bit of a review,

00:02:01.548 --> 00:02:04.811
so we're going from 0,8 which
is maybe right over here

00:02:04.811 --> 00:02:08.978
to 1,3 so that's one, 1,3
might look something like this.

00:02:10.720 --> 00:02:13.040
So we're seeing the average
rate of change, the slope

00:02:13.040 --> 00:02:15.886
of the line that connects
those two points is like this.

00:02:15.886 --> 00:02:18.092
What the mean value
theorem tells us if we have

00:02:18.092 --> 00:02:22.386
the conditions for the mean
value theorem, that let's say

00:02:22.386 --> 00:02:25.697
that our function does
something like this, as long

00:02:25.697 --> 00:02:27.984
as our functions meet the
conditions of the mean value

00:02:27.984 --> 00:02:30.709
theorem, which essentially
has to be differentiable

00:02:30.709 --> 00:02:33.588
over the open interval and
continuous over the closed

00:02:33.588 --> 00:02:37.569
interval, then there has to
be a point c where the slope

00:02:37.569 --> 00:02:41.088
of the tangent line at point
c is equal to the average

00:02:41.088 --> 00:02:42.097
rate of change.

00:02:42.097 --> 00:02:46.769
So over here the slope of the
tangent line looks the same

00:02:46.769 --> 00:02:49.505
as our average rate of
change, right of there just

00:02:49.505 --> 00:02:50.642
like that.

00:02:50.642 --> 00:02:52.896
So this is the mean value
theorem, but in order to make

00:02:52.896 --> 00:02:57.479
this claim that in order to
apply the mean value theorem

00:02:57.479 --> 00:03:00.024
to say there must be a c in
the interval where the slope

00:03:00.024 --> 00:03:02.794
of the tangent line at x equals
c is the same as the average

00:03:02.794 --> 00:03:05.748
rate of change, we have to
feel very confident that we are

00:03:05.748 --> 00:03:08.471
meeting the conditions for
the mean value theorem, namely

00:03:08.471 --> 00:03:11.495
that the function is continuous
over the closed interval

00:03:11.495 --> 00:03:14.920
and differentiable over the
open interval from zero to one.

00:03:14.920 --> 00:03:18.328
So let's look at the choices now.

00:03:18.328 --> 00:03:22.648
G is continuous at x equals
zero and x equals one.

00:03:22.648 --> 00:03:24.727
Well that by itself isn't
sufficient, we also have

00:03:24.727 --> 00:03:26.807
to be continuous at all the
points in between, so that alone

00:03:26.807 --> 00:03:29.965
does not make this claim true.

00:03:29.965 --> 00:03:33.647
G is continuous over the closed
interval from zero to one.

00:03:33.647 --> 00:03:36.305
So that's nice, but that
doesn't ensure that we're

00:03:36.305 --> 00:03:39.835
differentiable over the open
interval from zero to one,

00:03:39.835 --> 00:03:43.353
we have to be continuous
over the closed interval

00:03:43.353 --> 00:03:46.879
and differentiable over the
open interval, so we can rule

00:03:46.879 --> 00:03:48.642
that one out, that's not enough.

00:03:48.642 --> 00:03:51.662
G is differentiable over the
open interval from negative one

00:03:51.662 --> 00:03:55.829
to one and so that includes
this right over here includes

00:03:58.457 --> 00:04:01.636
the open interval that we care about.

00:04:01.636 --> 00:04:04.541
This includes the open interval
from zero to one, that's

00:04:04.541 --> 00:04:07.664
what we care about, so
that is a check that we're

00:04:07.664 --> 00:04:10.206
differentiable over the
open interval and continuous

00:04:10.206 --> 00:04:13.731
over the closed interval
from negative one to one.

00:04:13.731 --> 00:04:17.137
Well once again this includes
the closed interval that we

00:04:17.137 --> 00:04:18.159
care about.

00:04:18.159 --> 00:04:22.326
This includes the closed
interval from zero to one

00:04:23.953 --> 00:04:25.613
and so this is looking good.

00:04:25.613 --> 00:04:27.761
What we care about is that
we're differentiable over this

00:04:27.761 --> 00:04:29.665
interval and if we're
differentiable here we're definitely

00:04:29.665 --> 00:04:31.568
differentiable here and
then we're continuous over

00:04:31.568 --> 00:04:32.567
this interval.

00:04:32.567 --> 00:04:34.400
So I like this choice.

00:04:35.551 --> 00:04:37.697
This right over here says
g is differentiable over

00:04:37.697 --> 00:04:41.425
the open interval and at x
is equal to zero, so this

00:04:41.425 --> 00:04:44.160
is getting us close because
if you are differentiable

00:04:44.160 --> 00:04:45.919
you are continuous.

00:04:45.919 --> 00:04:48.159
So one way to interpret this
is that we are differentiable

00:04:48.159 --> 00:04:52.385
over the interval that is
from zero to one that includes

00:04:52.385 --> 00:04:53.635
the point zero.

00:04:54.753 --> 00:04:56.722
Now this is close if they said
that we are differentiable

00:04:56.722 --> 00:05:00.033
for over the closed interval
from zero to one, that actually

00:05:00.033 --> 00:05:02.113
would have been sufficient
because if you are differentiable

00:05:02.113 --> 00:05:04.517
then you are also continuous
but they did not say that.

00:05:04.517 --> 00:05:07.314
So we can rule this one out
as well, we would have to know

00:05:07.314 --> 00:05:10.561
this last choice still does
not make us confident that

00:05:10.561 --> 00:05:13.539
we are continuous at the
point one, that's what being

00:05:13.539 --> 00:05:16.836
continuous over the closed
interval would have told us.

00:05:16.836 --> 00:05:18.798
Let's do one more example
that's a little bit different

00:05:18.798 --> 00:05:19.965
than this one.

00:05:22.351 --> 00:05:26.657
So here we are told that g
is a differentiable function

00:05:26.657 --> 00:05:30.002
once again they've given us
the function sample that some

00:05:30.002 --> 00:05:31.264
values of x.

00:05:31.264 --> 00:05:34.971
Max was asked whether
there's a solution to g prime

00:05:34.971 --> 00:05:38.860
of x is equal to two on the
closed interval from three

00:05:38.860 --> 00:05:39.693
to six.

00:05:40.751 --> 00:05:44.084
So we see three here, we see three here.

00:05:46.488 --> 00:05:49.402
So we want to care about the
closed interval from three

00:05:49.402 --> 00:05:50.235
to six.

00:05:51.917 --> 00:05:53.767
So the first thing he did,
he says okay let's find

00:05:53.767 --> 00:05:56.971
the average rate of change
from x equals three to x equals

00:05:56.971 --> 00:06:01.138
six, so this is our change
in our function, so g of six

00:06:02.156 --> 00:06:06.515
minus g of three over six
minus three, let's see g of six

00:06:06.515 --> 00:06:10.682
is equal to one, g of three
is equal to negative five,

00:06:11.864 --> 00:06:13.562
this is three.

00:06:13.562 --> 00:06:16.103
One minus negative five over
three, that's six over three,

00:06:16.103 --> 00:06:18.270
it is indeed equal to two.

00:06:19.192 --> 00:06:20.792
So that step makes sense.

00:06:20.792 --> 00:06:22.711
The reason I'm going through
they say is Max's work correct?

00:06:22.711 --> 00:06:24.648
If not, what is his mistake?

00:06:24.648 --> 00:06:26.229
So let's look at step two.

00:06:26.229 --> 00:06:28.899
The mean value theorem so Max
is now saying alright I found

00:06:28.899 --> 00:06:32.359
the average rate of change
between x equals three and x

00:06:32.359 --> 00:06:35.176
equals six and then he
says the mean value theorem

00:06:35.176 --> 00:06:38.930
guarantees a solution where
g prime of x equals two

00:06:38.930 --> 00:06:40.347
on this interval.

00:06:41.450 --> 00:06:44.491
So we have to be careful here,
because he just immediately

00:06:44.491 --> 00:06:47.052
applies the mean value
theorem without establishing

00:06:47.052 --> 00:06:49.878
the conditions for the mean value theorem.

00:06:49.878 --> 00:06:52.502
So I would say even though
it does say that we're

00:06:52.502 --> 00:06:55.475
differentiable over it, I would
make it, I think his mistake

00:06:55.475 --> 00:06:59.000
is that he did not make this clear.

00:06:59.000 --> 00:07:01.651
He should have said step two
should have been something

00:07:01.651 --> 00:07:04.984
like because g is differentiable over...

00:07:12.404 --> 00:07:14.807
It sounds like it's a
differentiable function so we could

00:07:14.807 --> 00:07:18.557
say for all x is differentiable for all x.

00:07:20.007 --> 00:07:24.174
We can say the mean value theorem
applies over this interval

00:07:32.109 --> 00:07:34.349
if you wanted to be really
careful you could've said

00:07:34.349 --> 00:07:37.055
because g is differentiable
for all x is definitely

00:07:37.055 --> 00:07:40.122
differentiable over this
closed interval and if it's

00:07:40.122 --> 00:07:42.235
differentiable over the
closed interval it's also

00:07:42.235 --> 00:07:44.973
continuous over the closed
interval and so the mean value

00:07:44.973 --> 00:07:47.227
theorem definitely applies
and then say the mean value

00:07:47.227 --> 00:07:49.294
theorem guarantees a solution.

00:07:49.294 --> 00:07:51.634
So the reason why it's
important to stress is if you're

00:07:51.634 --> 00:07:54.364
taking a test especially
things like the AP exam,

00:07:54.364 --> 00:07:57.746
if you're going to apply one
of these, really any theorems,

00:07:57.746 --> 00:08:01.275
they're going to want you to
put down all the conditions

00:08:01.275 --> 00:08:03.644
for the theorem to ensure
that you can apply it as part

00:08:03.644 --> 00:08:05.061
of your argument.


Mean of sum and difference of random variables

WEBVTT
Kind: captions
Language: en

00:00:00.752 --> 00:00:02.724
- [Instructor] Let's say that
I have a random variable X

00:00:02.724 --> 00:00:06.891
which is equal to the number
of dogs that I see in a day.

00:00:07.810 --> 00:00:12.606
And random variable Y is
equal to the number of cats

00:00:12.606 --> 00:00:14.273
that I see in a day.

00:00:15.300 --> 00:00:18.593
Let's say I also know what
the mean of each of these

00:00:18.593 --> 00:00:20.849
random variables are, the expected value.

00:00:20.849 --> 00:00:24.258
So the expected value of X
which I could also denote

00:00:24.258 --> 00:00:28.754
as the mean of our random
variable X let's say I expect

00:00:28.754 --> 00:00:32.921
to see three dogs a day
and similarly for the cats,

00:00:33.922 --> 00:00:38.879
the expected value of Y is
equal to I could also denote

00:00:38.879 --> 00:00:43.046
that as the mean of Y is going
to be equal to and this is

00:00:43.988 --> 00:00:45.857
just for the sake of (mumbles)
let's say I expect to see

00:00:45.857 --> 00:00:47.201
four cats a day.

00:00:47.201 --> 00:00:51.649
And pretty much we define how
you take the mean of a random

00:00:51.649 --> 00:00:54.785
variable or the expected
value for a random variable.

00:00:54.785 --> 00:00:57.171
What we're going to think
about now is what would be

00:00:57.171 --> 00:01:01.338
the expected value of X plus
Y or other way of saying

00:01:02.479 --> 00:01:06.646
that the mean of the sum of
these two random variables.

00:01:08.078 --> 00:01:11.058
Well it turns out, and I'm
not proving it just yet,

00:01:11.058 --> 00:01:15.180
that the mean of the sum of
random variables is equal to

00:01:15.180 --> 00:01:17.099
the sum of the means.

00:01:17.099 --> 00:01:20.972
So this is going to be equal
to the mean of random variable

00:01:20.972 --> 00:01:24.055
X plus the mean of random variable Y.

00:01:26.557 --> 00:01:29.180
And so in this particular case,
if I were to say well what's

00:01:29.180 --> 00:01:32.297
the expected number of dogs
and cats that I would see

00:01:32.297 --> 00:01:34.119
in a given day.

00:01:34.119 --> 00:01:37.439
Well I would add these two
means, it would be three plus

00:01:37.439 --> 00:01:39.979
four it would be equal to seven,
so in this particular case

00:01:39.979 --> 00:01:44.146
it is equal to three plus
four which is equal to seven.

00:01:48.504 --> 00:01:51.012
And similarly if I were to ask
you the difference if I were

00:01:51.012 --> 00:01:55.179
to say how many more cats in
a given day would I expect

00:01:56.376 --> 00:02:01.062
to see than dogs, so the
expected value of Y minus X.

00:02:01.062 --> 00:02:03.364
What would that be?

00:02:03.364 --> 00:02:05.961
Well intuitively you might
say well hey if we can add

00:02:05.961 --> 00:02:06.997
random...

00:02:06.997 --> 00:02:09.367
If the expected value of the
sum is the sum of the expected

00:02:09.367 --> 00:02:12.049
values, then the expected
value or the mean of

00:02:12.049 --> 00:02:14.485
the difference will be the
differences of the means

00:02:14.485 --> 00:02:16.322
and that is absolutely true.

00:02:16.322 --> 00:02:20.453
So this is the same thing as
the mean of Y minus X which is

00:02:20.453 --> 00:02:24.620
equal to the mean of Y is going
to be equal to the mean of Y

00:02:28.984 --> 00:02:32.401
minus the mean of X, minus the mean of X.

00:02:33.446 --> 00:02:37.065
And in this particular case,
it would be equal to four

00:02:37.065 --> 00:02:40.482
minus three, minus three is equal to one.

00:02:44.100 --> 00:02:47.397
So another way of thinking about
this intuitively is I would

00:02:47.397 --> 00:02:51.564
expect to see on a given
day one more cat than dog.

00:02:52.597 --> 00:02:54.995
The example that I just
used this is discrete random

00:02:54.995 --> 00:02:58.965
variables, on a given day I
wouldn't see 2.2 dogs or pi

00:02:58.965 --> 00:03:02.320
dogs, the expected value itself
does not have to be a whole

00:03:02.320 --> 00:03:05.348
number 'cause you could of
course average it over many days.

00:03:05.348 --> 00:03:09.515
But this same idea that the
mean of a sum is the same thing

00:03:10.500 --> 00:03:14.256
as a sum of means and the
mean of a difference of random

00:03:14.256 --> 00:03:18.117
variables is the same as
the difference of the means.

00:03:18.117 --> 00:03:20.872
In a future video I'll do a proof of this.


Worked example: Parametric arc length

WEBVTT
Kind: captions
Language: en

00:00:00.615 --> 00:00:02.679
- [Instructor] Let's
say that X is a function

00:00:02.679 --> 00:00:06.940
of the parameter T and
it's equal to cosine of T

00:00:06.940 --> 00:00:10.273
and Y is also defined as a function of T

00:00:12.321 --> 00:00:15.217
and it's equal to sin of T

00:00:15.217 --> 00:00:17.136
and we wanna find the arc length

00:00:17.136 --> 00:00:18.943
of the curve traced out,

00:00:18.943 --> 00:00:22.110
so length of curve

00:00:24.308 --> 00:00:28.319
from T is equal to zero

00:00:28.319 --> 00:00:31.532
to T is equal to pi over two.

00:00:31.532 --> 00:00:34.001
So, pause this video and
see if you can work that out

00:00:34.001 --> 00:00:37.418
based on formulas that we
have seen in other videos.

00:00:37.418 --> 00:00:39.617
Alright, so first, I'm
gonna look at the formula

00:00:39.617 --> 00:00:41.018
and then we're gonna visualize it

00:00:41.018 --> 00:00:43.698
and appreciate why what
we got from the formula

00:00:43.698 --> 00:00:45.111
actually makes sense.

00:00:45.111 --> 00:00:47.521
So, the formula tells us that arc length

00:00:47.521 --> 00:00:49.529
of a parametric curve,

00:00:49.529 --> 00:00:52.446
arc length is equal to the integral

00:00:54.354 --> 00:00:56.403
from our starting point of our parameter,

00:00:56.403 --> 00:00:59.201
T equals A to our ending
point of our parameter,

00:00:59.201 --> 00:01:03.118
T equals B of the square
root of the derivative

00:01:04.610 --> 00:01:07.110
of X with respect to T squared

00:01:08.039 --> 00:01:12.206
plus the derivative of Y with
respect to T squared DT, DT.

00:01:15.882 --> 00:01:18.331
This could also be
rewritten as this is equal

00:01:18.331 --> 00:01:21.147
to the integral from A to B

00:01:21.147 --> 00:01:24.980
of the square root of DX/DT squared

00:01:27.429 --> 00:01:30.096
plus DY/DT squared DT

00:01:32.854 --> 00:01:35.641
but either way we can now
apply it in this context.

00:01:35.641 --> 00:01:37.385
What is DX/DT?

00:01:37.385 --> 00:01:41.552
So, DX/DT is equal to
the derivative of cosine

00:01:43.772 --> 00:01:46.689
of T is equal to negative sin of T,

00:01:47.969 --> 00:01:49.542
negative sin of T

00:01:49.542 --> 00:01:51.740
and what is DY/DT?

00:01:51.740 --> 00:01:54.337
The derivative of Y with respect to T.

00:01:54.337 --> 00:01:57.971
The derivative of sin of T
is cosine of T, cosine of T.

00:01:57.971 --> 00:02:01.040
So, our arc length up
here is going to be equal

00:02:01.040 --> 00:02:04.496
to the integral from T is equal to zero

00:02:04.496 --> 00:02:07.005
to pi over two, that's what we care about,

00:02:07.005 --> 00:02:08.033
our parameter's going from zero

00:02:08.033 --> 00:02:10.704
to pi over two of the square root

00:02:10.704 --> 00:02:14.503
of the derivative of X
with respect to T squared.

00:02:14.503 --> 00:02:17.322
That's a negative sin of T squared,

00:02:17.322 --> 00:02:19.434
well, if you square it the
negative's gonna go away,

00:02:19.434 --> 00:02:21.093
we're gonna have negative
sin times negative sin

00:02:21.093 --> 00:02:23.068
is positive sin squared,

00:02:23.068 --> 00:02:26.485
so I could write this as sin squared of T

00:02:27.874 --> 00:02:30.412
and then DY/DT squared,

00:02:30.412 --> 00:02:32.277
that's just cosine squared T,

00:02:32.277 --> 00:02:34.598
plus cosine squared T

00:02:34.598 --> 00:02:37.434
and then we have our DT out here.

00:02:37.434 --> 00:02:41.049
Now lucky for us, sin
squared plus cosine squared

00:02:41.049 --> 00:02:45.018
of some variable is always
going to be equal to one.

00:02:45.018 --> 00:02:49.062
So, that's one of our most
basic trig identities,

00:02:49.062 --> 00:02:51.236
comes straight out of the
unit circle definition

00:02:51.236 --> 00:02:52.893
of sin and cosine

00:02:52.893 --> 00:02:54.843
and so, we have the square root of one,

00:02:54.843 --> 00:02:55.787
the principle root of one

00:02:55.787 --> 00:02:56.976
which is just going to be one,

00:02:56.976 --> 00:02:59.692
so everything here has just simplified

00:02:59.692 --> 00:03:03.359
to the integral from
zero to pi over two DT.

00:03:04.376 --> 00:03:06.755
Well, this is going to be equal to,

00:03:06.755 --> 00:03:08.320
so you could view it, this is a one here,

00:03:08.320 --> 00:03:10.646
the antiderivative of
one with respect to T

00:03:10.646 --> 00:03:12.139
is just gonna be T

00:03:12.139 --> 00:03:16.013
and we're gonna evaluate
that from pi over two.

00:03:16.013 --> 00:03:17.463
We're gonna evaluate that at pi over two

00:03:17.463 --> 00:03:18.616
and then subtract.

00:03:18.616 --> 00:03:19.730
It evaluated at zero,

00:03:19.730 --> 00:03:23.754
so this is gonna be equal to pi over two

00:03:23.754 --> 00:03:25.437
minus zero,

00:03:25.437 --> 00:03:28.098
that's going to be equal to pi over two.

00:03:28.098 --> 00:03:32.265
Now let's think about why
that actually does make sense.

00:03:33.447 --> 00:03:35.280
Let's plot this curve.

00:03:36.609 --> 00:03:40.192
So, that is my Y axis
and this is my X axis

00:03:42.721 --> 00:03:43.995
right over here.

00:03:43.995 --> 00:03:45.995
When T is equal to zero,

00:03:47.536 --> 00:03:50.342
you have X of zero is cosine of zero,

00:03:50.342 --> 00:03:52.148
X is equal to one,

00:03:52.148 --> 00:03:53.579
X is equal to one,

00:03:53.579 --> 00:03:56.018
and Y, sin of zero is just zero,

00:03:56.018 --> 00:03:57.048
so Y is equal to zero,

00:03:57.048 --> 00:03:58.968
so we're at this point right here

00:03:58.968 --> 00:04:00.718
at T is equal to zero

00:04:02.160 --> 00:04:04.889
and then as T increases up to pi over two,

00:04:04.889 --> 00:04:07.518
we trace out the top right corner

00:04:07.518 --> 00:04:09.462
of the unit circle

00:04:09.462 --> 00:04:10.675
and we end up right over here

00:04:10.675 --> 00:04:13.258
when T is equal to pi over two.

00:04:14.273 --> 00:04:15.966
You could view T in this case

00:04:15.966 --> 00:04:19.208
as some type of an angle in radians.

00:04:19.208 --> 00:04:22.397
And so, the arc length
is really just the length

00:04:22.397 --> 00:04:25.034
of quarter of a quarter of a unit circle.

00:04:25.034 --> 00:04:26.261
Well, we know what the circumference

00:04:26.261 --> 00:04:28.839
of a circle is, it is two pi R.

00:04:28.839 --> 00:04:31.860
In the unit circle
case, the radius is one,

00:04:31.860 --> 00:04:34.678
so the circumference of the
entire circle is two pie.

00:04:34.678 --> 00:04:38.700
One fourth of that is
going to be pi over two.

00:04:38.700 --> 00:04:40.798
So, it's nice when we this fancy thing

00:04:40.798 --> 00:04:42.856
that we feel good about it in calculus

00:04:42.856 --> 00:04:47.023
is consistent with what we
first learned in basic geometry.


Parametric curve arc length

WEBVTT
Kind: captions
Language: en

00:00:00.571 --> 00:00:02.785
- [Instructor] Let's say we're
going to trace out a curve

00:00:02.785 --> 00:00:05.159
where our X coordinate
and our Y coordinate

00:00:05.159 --> 00:00:07.328
that they are each defined by

00:00:07.328 --> 00:00:10.467
or they're functions
of a third parameter T.

00:00:10.467 --> 00:00:13.476
So, we could say that X is a function of T

00:00:13.476 --> 00:00:17.004
and we could also say
that Y is a function T.

00:00:17.004 --> 00:00:19.561
If this notion is completely
unfamiliar to you,

00:00:19.561 --> 00:00:22.523
I encourage you to review the
videos on parametric equations

00:00:22.523 --> 00:00:24.202
on Khan Academy.

00:00:24.202 --> 00:00:25.678
But what we're going to think about

00:00:25.678 --> 00:00:27.758
and I'm gonna talk about in
generalities in this video.

00:00:27.758 --> 00:00:29.786
In future videos we're going to be dealing

00:00:29.786 --> 00:00:32.769
with more concrete examples

00:00:32.769 --> 00:00:34.726
but we're gonna think
about what is the path

00:00:34.726 --> 00:00:35.858
that is traced out

00:00:35.858 --> 00:00:38.025
from when T is equal to A,

00:00:39.148 --> 00:00:42.188
so this is where we are
when T is equal to A,

00:00:42.188 --> 00:00:45.917
so in this case this
point would be X of A,

00:00:45.917 --> 00:00:47.000
comma Y of A,

00:00:49.083 --> 00:00:50.055
that's this point

00:00:50.055 --> 00:00:52.603
and then as we increase from T equals A

00:00:52.603 --> 00:00:54.186
to T is equal to B,

00:00:55.818 --> 00:00:59.668
so our curve might do something like this,

00:00:59.668 --> 00:01:02.355
so this is when T is equal to B,

00:01:02.355 --> 00:01:03.224
T is equal to B,

00:01:03.224 --> 00:01:06.087
so this point right over here is X of B,

00:01:06.087 --> 00:01:07.170
comma Y of B.

00:01:08.731 --> 00:01:10.611
Let's think about how do we figure out

00:01:10.611 --> 00:01:12.806
the length of this actual curve,

00:01:12.806 --> 00:01:17.311
this actual arc length from
T equals A to T equals B?

00:01:17.311 --> 00:01:18.400
Well, to think about that

00:01:18.400 --> 00:01:20.686
we're gonna zoom in and
think about what happens

00:01:20.686 --> 00:01:24.546
when we have a very small change in T?

00:01:24.546 --> 00:01:26.045
So, a very small change in T.

00:01:26.045 --> 00:01:28.616
Let's say we're starting at
this point right over here

00:01:28.616 --> 00:01:30.636
and we have a very small change in T,

00:01:30.636 --> 00:01:34.957
so we go from this point
to let's say this point

00:01:34.957 --> 00:01:36.803
over that very small change in T.

00:01:36.803 --> 00:01:38.091
It actually would be
much smaller than this

00:01:38.091 --> 00:01:39.102
but if I drew it any smaller,

00:01:39.102 --> 00:01:40.487
you would have trouble seeing it.

00:01:40.487 --> 00:01:43.085
So, let's say that that
is our very small change

00:01:43.085 --> 00:01:46.752
in our path in our arc
that we are traveling

00:01:47.866 --> 00:01:50.179
and so, we wanna find this length.

00:01:50.179 --> 00:01:52.246
Well, we could break it down

00:01:52.246 --> 00:01:54.244
into how far we've
moved in the X direction

00:01:54.244 --> 00:01:56.603
and how far we've moved
in the Y direction.

00:01:56.603 --> 00:01:58.792
So, in the X direction,

00:01:58.792 --> 00:02:00.490
the X direction right over here,

00:02:00.490 --> 00:02:03.002
we would have moved a
very small change in X

00:02:03.002 --> 00:02:04.640
and what would that be equal to?

00:02:04.640 --> 00:02:05.823
Well, that would be the rate of change

00:02:05.823 --> 00:02:08.794
with which we are
changing with respect to T

00:02:08.794 --> 00:02:11.963
with which X is changing with respect to T

00:02:11.963 --> 00:02:14.368
times our very small change in T

00:02:14.368 --> 00:02:15.860
and this is a little hand wavy,

00:02:15.860 --> 00:02:17.673
I'm using differential notion

00:02:17.673 --> 00:02:20.002
and I'm conceptually using the idea

00:02:20.002 --> 00:02:23.813
of a differential as an
infinitesimally small change

00:02:23.813 --> 00:02:25.731
in that variable.

00:02:25.731 --> 00:02:27.449
And so, this isn't a formal proof

00:02:27.449 --> 00:02:29.011
but it's to give us the intuition

00:02:29.011 --> 00:02:30.898
for how we derive arc length

00:02:30.898 --> 00:02:33.163
when we're dealing with
parametric equations.

00:02:33.163 --> 00:02:35.736
So, this will hopefully
make conceptual sense

00:02:35.736 --> 00:02:36.868
that this is our DX.

00:02:36.868 --> 00:02:38.184
In fact, we could even write it this way,

00:02:38.184 --> 00:02:42.351
DX/DT, that's the same thing
as X prime of T times DT

00:02:43.431 --> 00:02:44.752
and then our change in Y

00:02:44.752 --> 00:02:47.162
is going to be the same idea.

00:02:47.162 --> 00:02:49.686
Our change in Y, our
infinitesimally small change in Y

00:02:49.686 --> 00:02:52.117
when we have an infinitesimally
small change in T,

00:02:52.117 --> 00:02:53.482
well, you could view that
as your rate of change

00:02:53.482 --> 00:02:55.392
of Y with respect to T

00:02:55.392 --> 00:02:57.189
times your change in T,

00:02:57.189 --> 00:02:59.275
your very small change in T

00:02:59.275 --> 00:03:00.588
which is going to be equal to,

00:03:00.588 --> 00:03:03.838
we could write that as Y prime of T DT.

00:03:05.351 --> 00:03:07.816
Now, based on this,
what would be the length

00:03:07.816 --> 00:03:12.498
of our infinitesimally small
arc length right over here?

00:03:12.498 --> 00:03:15.219
Well, that we could just
use the Pythagorean theorem.

00:03:15.219 --> 00:03:18.392
That is going to be the square root of,

00:03:18.392 --> 00:03:20.094
that's the hypotenuse
of this right triangle

00:03:20.094 --> 00:03:21.002
right over here.

00:03:21.002 --> 00:03:22.006
So, it's gonna be the square root

00:03:22.006 --> 00:03:24.357
of this squared plus this squared.

00:03:24.357 --> 00:03:26.497
So, it is the square root of,

00:03:26.497 --> 00:03:28.390
I'm gonna give myself a
little bit more space here

00:03:28.390 --> 00:03:30.110
because I think I'm gonna use a lot of it,

00:03:30.110 --> 00:03:32.190
so the stuff in blue squared,

00:03:32.190 --> 00:03:36.216
DX squared we could
rewrite that as X prime

00:03:36.216 --> 00:03:37.466
of T DT squared

00:03:39.369 --> 00:03:43.536
plus this squared which
is Y prime of T DT squared

00:03:47.979 --> 00:03:50.524
and now let's just try to
simplify this a little bit.

00:03:50.524 --> 00:03:52.963
Remember, this is this
infinitesimally small arc length

00:03:52.963 --> 00:03:54.444
right over here.

00:03:54.444 --> 00:03:58.798
So, we can actually
factor out a DT squared,

00:03:58.798 --> 00:04:00.654
it's a term in both of these

00:04:00.654 --> 00:04:02.985
and so, we can rewrite this as,

00:04:02.985 --> 00:04:05.540
let me, so I can rewrite this

00:04:05.540 --> 00:04:07.642
and then write my big radical sign,

00:04:07.642 --> 00:04:10.828
so I'm gonna factor out a DT squared here,

00:04:10.828 --> 00:04:14.224
so we could write this as DT squared

00:04:14.224 --> 00:04:16.391
times X prime of T squared

00:04:19.830 --> 00:04:21.913
plus Y prime of T squared

00:04:26.449 --> 00:04:27.282
and then to be clear

00:04:27.282 --> 00:04:29.850
this is being multiplied
by all of this stuff

00:04:29.850 --> 00:04:31.069
right over there.

00:04:31.069 --> 00:04:34.624
Well, now if we have this DT
squared under the radical,

00:04:34.624 --> 00:04:35.595
we can take it out

00:04:35.595 --> 00:04:36.910
and so, we will have a DT

00:04:36.910 --> 00:04:40.750
and so, this is all going to
be equal to the square root

00:04:40.750 --> 00:04:44.127
of, so the stuff that's
still under the radical

00:04:44.127 --> 00:04:47.044
is going to be X prime of T squared

00:04:50.381 --> 00:04:52.464
plus Y prime of T squared

00:04:55.414 --> 00:04:57.522
and now we took out a DT

00:04:57.522 --> 00:04:59.863
and now we took out a DT.

00:04:59.863 --> 00:05:03.029
I could have written it right over here

00:05:03.029 --> 00:05:04.260
but I'm just writing it on the other side,

00:05:04.260 --> 00:05:06.268
we're just multiplying the two.

00:05:06.268 --> 00:05:08.655
So, this is once again just
rewriting the expression

00:05:08.655 --> 00:05:12.830
for this infinitesimally
small change in arc length.

00:05:12.830 --> 00:05:15.512
Well, what's lucky for us is in calculus

00:05:15.512 --> 00:05:17.856
we have the tools for adding up

00:05:17.856 --> 00:05:21.559
all of these infinitesimally small changes

00:05:21.559 --> 00:05:24.141
and that's what the definite
integral does for us.

00:05:24.141 --> 00:05:26.177
So, what we can do if we wanna add up that

00:05:26.177 --> 00:05:27.933
plus that plus that plus that

00:05:27.933 --> 00:05:30.016
and remember, these are
infinitesimally small changes.

00:05:30.016 --> 00:05:32.321
I'm just showing them
as not infinitesimally

00:05:32.321 --> 00:05:34.284
just so that you can
kind of think about them

00:05:34.284 --> 00:05:35.857
but if you were to add them all up,

00:05:35.857 --> 00:05:38.614
then we are essentially
taking the integral

00:05:38.614 --> 00:05:40.647
and we're integrating with respect to T

00:05:40.647 --> 00:05:43.862
and so, we're starting at T is equal to A,

00:05:43.862 --> 00:05:46.600
all the way to T is equal to B

00:05:46.600 --> 00:05:50.600
and just like that we
have been able to at least

00:05:51.466 --> 00:05:52.932
feel good conceptually

00:05:52.932 --> 00:05:55.522
for the formula of arc length

00:05:55.522 --> 00:05:59.456
when we're dealing with
parametric equations.

00:05:59.456 --> 00:06:00.840
In the next few videos

00:06:00.840 --> 00:06:04.224
we'll actually apply it
to figure out arc lengths.