Reverse chain rule example

WEBVTT
Kind: captions
Language: en

00:00:00.310 --> 00:00:02.390
- [Voiceover] Let's see if we
can evaluate the indefinite

00:00:02.390 --> 00:00:05.200
integral x over two times sine of

00:00:05.200 --> 00:00:07.638
two x squared plus two, dx.

00:00:07.638 --> 00:00:08.976
And try to pause the video and

00:00:08.976 --> 00:00:11.669
see if you can work
through it on your own.

00:00:11.669 --> 00:00:13.968
So, let's see what is going on here.

00:00:13.968 --> 00:00:16.940
So, I have this x over
two, and then I have

00:00:16.940 --> 00:00:19.215
sine of two x squared plus two.

00:00:19.215 --> 00:00:22.814
Now, if I were just taking
the indefinite integral of

00:00:22.814 --> 00:00:26.576
sine of x, that is pretty straightforward.

00:00:26.576 --> 00:00:28.433
The indefinite integral of sine of x.

00:00:28.433 --> 00:00:32.682
Well, we know that the
derivative of cosine of x

00:00:32.682 --> 00:00:35.297
is equal to negative sine of x.

00:00:35.297 --> 00:00:39.105
So if I were to take the
derivative of negative cosine of x,

00:00:39.105 --> 00:00:41.627
that's going to be positive sine of x.

00:00:41.627 --> 00:00:44.019
So this is just going to
be negative cosine of x.

00:00:44.019 --> 00:00:46.782
And I could have made that even clearer.

00:00:46.782 --> 00:00:49.661
I could have put a negative
here and then a negative here.

00:00:49.661 --> 00:00:51.774
And you see, well look,
the anti-derivative of

00:00:51.774 --> 00:00:54.119
negative sine of x is just
cosine of x, and then I have

00:00:54.119 --> 00:00:56.975
this negative out here,
negative cosine of x.

00:00:56.975 --> 00:00:58.113
But that's not what I have here.

00:00:58.113 --> 00:00:59.877
I don't have sine of x.

00:00:59.877 --> 00:01:03.221
I have sine of two x squared plus two.

00:01:03.221 --> 00:01:05.984
But then I have this other
thing with an x here,

00:01:05.984 --> 00:01:08.445
and so what your brain
might be doing, or it's good

00:01:08.445 --> 00:01:09.760
once you get enough
practice when your brain

00:01:09.760 --> 00:01:13.684
will start doing this, say
okay, this is interesting.

00:01:13.684 --> 00:01:17.552
This kind of looks like
the derivative of this.

00:01:17.552 --> 00:01:20.246
And this is, if we were to call this...

00:01:20.246 --> 00:01:21.894
I'm tired of that orange.

00:01:21.894 --> 00:01:24.263
If we were to call a...

00:01:24.263 --> 00:01:27.583
Why am I doing this,
color changing this...

00:01:27.583 --> 00:01:30.114
If we were to call this f of x.

00:01:30.114 --> 00:01:33.457
If two x squared plus two is f of x,

00:01:33.457 --> 00:01:36.917
Two x squared plus two is f of x.

00:01:36.917 --> 00:01:38.891
What is f prime of x?

00:01:38.891 --> 00:01:42.211
Well, then f prime of x, f prime of x

00:01:42.211 --> 00:01:45.020
is going to be four x.

00:01:45.020 --> 00:01:48.364
And this thing right over
here isn't exactly four x,

00:01:48.364 --> 00:01:50.941
but we can make it, we can
do a little rearranging,

00:01:50.941 --> 00:01:52.683
multiplying and dividing by a constant,

00:01:52.683 --> 00:01:54.192
so this becomes four x.

00:01:54.192 --> 00:01:57.907
What if, what if we were to...

00:01:57.907 --> 00:02:00.127
What if we were to multiply
and divide by four,

00:02:00.127 --> 00:02:03.981
so we multiply by four there
and then we divide by four,

00:02:03.981 --> 00:02:06.086
and then we take it out
of the integral sign.

00:02:06.086 --> 00:02:08.918
And even better let's take this
two out so let's just take.

00:02:08.918 --> 00:02:10.953
So, let's take the one half out of here,

00:02:10.953 --> 00:02:12.415
so this is going to be one half.

00:02:12.415 --> 00:02:15.713
And so I could have rewritten
the original integral as

00:02:15.713 --> 00:02:18.336
one half times one
fourth, so it's one eighth

00:02:18.336 --> 00:02:22.121
times the integral, times the integral

00:02:22.121 --> 00:02:25.243
of four x

00:02:25.243 --> 00:02:27.403
times sine

00:02:27.403 --> 00:02:32.046
of two x squared plus two, dx.

00:02:32.046 --> 00:02:33.779
Well, now this is interesting,

00:02:33.779 --> 00:02:37.448
because if this is f of x, if...

00:02:37.448 --> 00:02:40.620
If this business right
over here if f of x,

00:02:40.620 --> 00:02:43.127
so we're essentially
taking sine of f of x,

00:02:43.127 --> 00:02:46.678
then this business right over here is

00:02:46.678 --> 00:02:50.090
f prime of x, which is a
good signal to us that, hey,

00:02:50.090 --> 00:02:52.574
the reverse chain rule
is applicable over here.

00:02:52.574 --> 00:02:55.593
We can rewrite this, we
can also rewrite this as,

00:02:55.593 --> 00:02:58.356
this is going to be equal to one.

00:02:58.356 --> 00:03:02.628
We can rewrite this as, the
same things as one eighth...

00:03:02.628 --> 00:03:03.284
They're the same colors.

00:03:03.284 --> 00:03:06.372
I'm using a new art program,
and sometimes the color

00:03:06.372 --> 00:03:08.763
changing isn't as obvious as it should be.

00:03:08.763 --> 00:03:13.443
So one eighth times the
integral of f prime of x,

00:03:14.754 --> 00:03:19.005
f prime of x

00:03:19.005 --> 00:03:22.377
times sine,

00:03:22.422 --> 00:03:25.162
sine of f of x,

00:03:25.162 --> 00:03:28.180
sine of f of x, dx,

00:03:28.180 --> 00:03:30.502
throw that f of x in there.

00:03:30.502 --> 00:03:33.033
So, sine of f of x.

00:03:33.033 --> 00:03:35.007
And so when you view it
this way, you say, hey,

00:03:35.007 --> 00:03:37.352
by the reverse chain rule, I have...

00:03:37.352 --> 00:03:41.253
I have a function, and I have
its derivative here, so I can

00:03:41.253 --> 00:03:43.775
really just take the antiderivative
with respect to this.

00:03:43.775 --> 00:03:45.470
This is essentially what
we're doing in u-substitution.

00:03:45.470 --> 00:03:47.745
You could do u-substitution
here, you could set u

00:03:47.745 --> 00:03:51.632
equalling this, and then du
is going to be four x dx.

00:03:51.632 --> 00:03:53.207
But now we're getting a little
practice, starting to do a

00:03:53.207 --> 00:03:55.366
little bit more in our heads.

00:03:55.366 --> 00:03:58.756
So, what would this interval
integrate out to be?

00:03:58.756 --> 00:04:02.925
Well, this would be one eighth times...

00:04:04.166 --> 00:04:07.571
Well, if you take the
antiderivative of sine of f of x

00:04:07.571 --> 00:04:10.148
with respect to f of x,
well, we already saw that

00:04:10.148 --> 00:04:13.004
that's negative cosine of
x, so this is going to be

00:04:13.004 --> 00:04:17.160
times negative cosine,

00:04:17.160 --> 00:04:20.178
negative cosine of f of x.

00:04:20.178 --> 00:04:23.644
Negative cosine of f of x,

00:04:23.644 --> 00:04:25.478
negative cosine of f of x.

00:04:25.478 --> 00:04:28.068
Woops, I was going for the blue there.

00:04:28.752 --> 00:04:32.087
And then of course you have your plus c.

00:04:32.087 --> 00:04:34.408
So what is this going to be?

00:04:34.408 --> 00:04:35.686
Well, instead of just saying f pri..

00:04:35.686 --> 00:04:38.170
Instead of saying in terms
of f of x, we just say it

00:04:38.170 --> 00:04:39.517
in terms of two x squared.

00:04:39.517 --> 00:04:40.654
This is going to be...

00:04:40.654 --> 00:04:44.253
Or two x squared plus two
is going to be one eighth.

00:04:44.253 --> 00:04:47.040
Times, I have a function
and I have its derivative,

00:04:47.040 --> 00:04:48.502
so I can really just
integrate with respect

00:04:48.502 --> 00:04:51.080
to that function, so
it's going to be times

00:04:51.080 --> 00:04:55.073
negative negative cosine

00:04:55.073 --> 00:04:57.851
of two x squared plus two,

00:04:57.851 --> 00:05:01.357
two x squared plus two,

00:05:01.357 --> 00:05:04.561
and then, of course, I have my...

00:05:04.561 --> 00:05:06.163
I keep switching to that color.

00:05:06.163 --> 00:05:09.948
I have my plus c, and of
course, I could just take the

00:05:09.948 --> 00:05:12.409
negative out, it would be
negative one eighth cosine

00:05:12.409 --> 00:05:15.195
of this business and then plus c.

00:05:15.195 --> 00:05:16.403
And we're done.

00:05:16.403 --> 00:05:18.237
We have just employed
the reverse chain rule.

00:05:18.237 --> 00:05:21.302
We could have used
substitution, but hopefully

00:05:21.302 --> 00:05:22.765
we're getting a little
bit of practice here.

00:05:22.765 --> 00:05:24.576
Hey, I'm seeing something
here, and I'm seeing

00:05:24.576 --> 00:05:27.292
it's derivative, so let me
just integrate with respect

00:05:27.292 --> 00:05:29.104
to this thing, which is
really what you would set

00:05:29.104 --> 00:05:31.843
u to be equal to here,
integrating with respect to the u,

00:05:31.843 --> 00:05:33.329
and you have your du here.

00:05:33.329 --> 00:05:37.277
This times this is du, so you're, like,

00:05:37.277 --> 00:05:39.738
integrating sine of u, du.

00:05:39.738 --> 00:05:42.199
I encourage you to try to
use u-substitution here,

00:05:42.199 --> 00:05:45.727
and you'll see it's the exact
same thing that we just did.


Reverse chain rule introduction

WEBVTT
Kind: captions
Language: en

00:00:00.173 --> 00:00:02.695
- [Voiceover] Hopefully we
all remember our good friend

00:00:02.695 --> 00:00:06.571
the chain rule from differential calculus

00:00:06.571 --> 00:00:07.928
that tells us that if I were to take

00:00:07.928 --> 00:00:10.146
the derivative with respect to x

00:00:10.146 --> 00:00:15.146
of g of f of x, g of, let me write those

00:00:15.165 --> 00:00:20.165
parentheses a little
bit closer, g of f of x,

00:00:21.117 --> 00:00:24.218
g of f of x, that this is just going

00:00:24.218 --> 00:00:27.610
to be equal to the derivative of g

00:00:27.610 --> 00:00:30.046
with respect to f of
x, so we can write that

00:00:30.046 --> 00:00:33.338
as g prime of f of x.

00:00:33.338 --> 00:00:37.592
G prime of f of x, times the derivative

00:00:37.592 --> 00:00:42.592
of f with respect to
x, times f prime of x.

00:00:43.150 --> 00:00:45.443
And if you want to see it in the other

00:00:45.443 --> 00:00:48.653
notation, I guess you
could say, it would be,

00:00:48.653 --> 00:00:49.851
you could write this part right

00:00:49.851 --> 00:00:52.736
over here as the derivative of g

00:00:52.736 --> 00:00:57.643
with respect to f times
the derivative of f.

00:00:57.643 --> 00:00:59.995
The derivative of f with respect to x,

00:00:59.995 --> 00:01:01.138
and that's going to give you the

00:01:01.138 --> 00:01:03.899
derivative of g with respect to x.

00:01:03.899 --> 00:01:05.974
This is just a review,
this is the chain rule

00:01:05.974 --> 00:01:07.854
that you remember from, or hopefully

00:01:07.854 --> 00:01:10.663
remember, from differential calculus.

00:01:10.663 --> 00:01:12.390
It's hard to get, it's hard to get too far

00:01:12.390 --> 00:01:15.378
in calculus without really grokking,

00:01:15.378 --> 00:01:18.243
really understanding the chain rule.

00:01:18.243 --> 00:01:20.481
So what I want to do here
is, well if this is true,

00:01:20.481 --> 00:01:22.541
then can't we go the other way around?

00:01:22.541 --> 00:01:26.109
If I wanted to take the integral of this,

00:01:26.109 --> 00:01:30.377
if I wanted to take
the integral of g prime

00:01:30.377 --> 00:01:35.377
of f of x, g prime of f of x, times

00:01:37.006 --> 00:01:42.006
f prime of x, dx, well, this
should just be equal to,

00:01:42.753 --> 00:01:46.979
this should just be equal to g of f of x,

00:01:46.979 --> 00:01:49.841
g of f of x, and then
of course whenever I'm

00:01:49.841 --> 00:01:52.603
taking an indefinite integral g of,

00:01:52.603 --> 00:01:53.782
let me make sure they're the same color,

00:01:53.782 --> 00:01:56.777
g of f of x, so I just swapped sides,

00:01:56.777 --> 00:01:58.189
I'm going the other way.

00:01:58.189 --> 00:01:59.989
So if I'm taking the indefinite integral,

00:01:59.989 --> 00:02:01.143
wouldn't it just be equal to this?

00:02:01.143 --> 00:02:02.833
And of course I can't forget that

00:02:02.833 --> 00:02:04.364
I could have a constant
here now that might

00:02:04.364 --> 00:02:06.085
have been introduced, because if I take

00:02:06.085 --> 00:02:08.603
the derivative, the constant disappears.

00:02:08.603 --> 00:02:11.238
And so this idea, you
could really just call

00:02:11.253 --> 00:02:13.357
the reverse chain rule.

00:02:13.357 --> 00:02:18.357
Reverse, reverse chain,
the reverse chain rule.

00:02:19.555 --> 00:02:21.243
Which is essentially, or it's exactly

00:02:21.243 --> 00:02:23.339
what we did with
u-substitution, we just did it

00:02:23.339 --> 00:02:25.174
a little bit more methodically
with u-substitution.

00:02:25.174 --> 00:02:26.924
And we'll see that in a second, but before

00:02:26.924 --> 00:02:28.834
we see how u-substitution relates to

00:02:28.834 --> 00:02:30.700
what I just wrote down
here, let's actually

00:02:30.700 --> 00:02:32.763
apply it and see where it's useful.

00:02:32.763 --> 00:02:34.902
And this is really a way
of doing u-substitution

00:02:34.902 --> 00:02:37.153
without having to do
u-substitution, or doing

00:02:37.153 --> 00:02:39.071
u-substitution in your head, or doing

00:02:39.071 --> 00:02:42.201
u-substitution-like problems
a little bit faster.

00:02:42.201 --> 00:02:44.094
So let me give you an example.

00:02:44.094 --> 00:02:46.498
So let's say that we had, and I'm going to

00:02:46.498 --> 00:02:47.989
color code it so that it jumps out at you

00:02:47.989 --> 00:02:51.000
a little bit more, let's say that we had

00:02:51.000 --> 00:02:53.232
sine of x, and I'm going
to write it this way,

00:02:53.232 --> 00:02:56.820
I could write it, so let's say sine of x,

00:02:58.789 --> 00:03:02.836
sine of x squared, and
obviously the typical

00:03:02.836 --> 00:03:07.254
convention, the typical,
the sine of x squared,

00:03:07.254 --> 00:03:09.024
the typical convention
would be to put the squared

00:03:09.024 --> 00:03:10.824
right over here, but I'm
going to write it like this,

00:03:10.824 --> 00:03:12.669
and I think you might
be able to guess why.

00:03:12.669 --> 00:03:16.231
Sine of x squared times cosine of x.

00:03:16.231 --> 00:03:17.998
Times, actually, I'll do this in a,

00:03:17.998 --> 00:03:19.600
let me do this in a different color.

00:03:19.600 --> 00:03:24.600
Times cosine of x, times cosine of x.

00:03:24.615 --> 00:03:27.413
So I encourage you to pause this video

00:03:27.413 --> 00:03:29.827
and think about, does it
meet this pattern here,

00:03:29.827 --> 00:03:33.090
and if so, what is this
indefinite integral going to be?

00:03:34.351 --> 00:03:35.760
Well let's think about it.

00:03:35.760 --> 00:03:39.383
If f of x is sine of x,
what's the derivative of that?

00:03:39.383 --> 00:03:41.666
What's f prime of x?

00:03:41.666 --> 00:03:44.130
Well f prime of x in that circumstance

00:03:44.130 --> 00:03:48.417
is going to be cosine of x, and what is g?

00:03:48.417 --> 00:03:52.222
Well g is whatever you
input into g squared.

00:03:52.222 --> 00:03:53.812
So what's this going to be if we

00:03:53.812 --> 00:03:56.066
just do the reverse chain rule?

00:03:56.066 --> 00:03:58.594
Well this is going to be, well we take

00:03:58.594 --> 00:04:01.834
sorry, g prime is taking
whatever this thing is,

00:04:01.834 --> 00:04:05.577
squared, so g is going
to be the anti-derivative

00:04:05.577 --> 00:04:06.640
of that, so it's going to be taking

00:04:06.640 --> 00:04:09.388
something to the third power and then

00:04:09.388 --> 00:04:11.783
dividing it by three, so let's do that.

00:04:11.783 --> 00:04:15.680
So if we essentially
take the anti-derivative

00:04:15.680 --> 00:04:16.989
here with respect to sine of x,

00:04:16.989 --> 00:04:20.116
instead of with respect
to x, you're going to get

00:04:20.116 --> 00:04:22.977
you're going to get sine of x,

00:04:23.793 --> 00:04:28.793
sine of x to the, to the third power

00:04:29.949 --> 00:04:33.388
over three, and then of course

00:04:33.388 --> 00:04:37.616
you have the, you have the plus c.

00:04:37.616 --> 00:04:38.902
And if you don't believe this, just take

00:04:38.902 --> 00:04:40.610
the derivative of this,
you'll have to employ

00:04:40.610 --> 00:04:42.994
the chain rule and
you'll get exactly this.

00:04:42.994 --> 00:04:44.726
And you say well wait,
how does this relate

00:04:44.726 --> 00:04:46.156
to u-substitution?

00:04:46.156 --> 00:04:47.953
Well in u-substitution you would have said

00:04:47.953 --> 00:04:51.076
u equals sine of x,
then du would have been

00:04:51.076 --> 00:04:54.305
cosine of x, dx, and
actually let me just do that.

00:04:54.305 --> 00:04:56.277
That actually might clear
things up a little bit.

00:04:56.277 --> 00:05:00.578
You would set this to be u, and then this,

00:05:00.578 --> 00:05:02.634
all of this business right over here,

00:05:02.634 --> 00:05:05.723
would then be du, and then you would have

00:05:05.723 --> 00:05:08.858
the integral, you would have the integral

00:05:08.858 --> 00:05:13.858
u squared, u squared, I don't have to

00:05:14.118 --> 00:05:19.118
put parentheses around it, u squared, du.

00:05:19.562 --> 00:05:22.428
U squared, du, well, let me do that

00:05:22.428 --> 00:05:26.089
in that orange color, u squared, du.

00:05:26.089 --> 00:05:27.471
Well that's pretty straightforward,

00:05:27.471 --> 00:05:31.464
this is going to be equal to u, this is

00:05:31.464 --> 00:05:34.236
going to be equal to u to the third power

00:05:35.359 --> 00:05:39.463
over three, plus c,
which is equal to what?

00:05:39.463 --> 00:05:41.146
Well we just said u is equal to sine of x,

00:05:41.146 --> 00:05:43.073
you reverse substitute, and you're going

00:05:43.073 --> 00:05:45.244
to get exactly that right over here.

00:05:45.244 --> 00:05:47.253
So when we talk about
the reverse chain rule,

00:05:47.269 --> 00:05:51.120
it's essentially just doing
u-substitution in our head.

00:05:51.120 --> 00:05:55.056
So in the next few examples,
I will do exactly that.


Integral of tan x

WEBVTT
Kind: captions
Language: en

00:00:00.279 --> 00:00:01.547
- [Voiceover] Let's see if we can evaluate

00:00:01.547 --> 00:00:06.547
the indefinite integral of tangent x dx.

00:00:08.102 --> 00:00:09.748
Like always, pause the video and see

00:00:09.748 --> 00:00:11.512
if you can figure it out on your own

00:00:11.512 --> 00:00:15.009
and I will give you a hint,
think reverse chain rule.

00:00:15.441 --> 00:00:17.258
Alright, so you have attempted it

00:00:17.258 --> 00:00:18.777
and you would say well reverse chain rule,

00:00:18.777 --> 00:00:20.370
that's kind of your seeing a function

00:00:20.370 --> 00:00:21.392
and your seeing it's derivative

00:00:21.392 --> 00:00:22.692
and you can integrate with
respect to that function.

00:00:22.707 --> 00:00:26.345
All you see here is a tangent
of x, what am I talking about?

00:00:26.622 --> 00:00:28.464
Well, whenever you see a tangent of x

00:00:28.464 --> 00:00:31.654
or a cosecant or a secant,
at least in my brain,

00:00:31.654 --> 00:00:34.259
I always like to break it
down into how it's defined

00:00:34.259 --> 00:00:36.973
in terms of sine and cosine because we do

00:00:36.973 --> 00:00:39.363
at least have some tools at our disposal

00:00:39.363 --> 00:00:41.734
for dealing with sines and cosines.

00:00:41.734 --> 00:00:43.700
Or at least our brains, at least my brain,

00:00:43.700 --> 00:00:45.300
has an easier time processing them.

00:00:45.300 --> 00:00:47.632
We know tangent of x is
the same thing as sine of x

00:00:47.632 --> 00:00:50.963
over cosine of x so let
me rewrite it that way.

00:00:51.200 --> 00:00:56.200
This is equal to the indefinite
integral of sine of x

00:00:56.447 --> 00:01:01.119
over cosine of x dx and you
could even write it this way

00:01:01.119 --> 00:01:02.672
and this is a little bit of a hint.

00:01:04.654 --> 00:01:06.939
You could even write it as sine of x

00:01:07.324 --> 00:01:10.959
times one over cosine of x.

00:01:10.959 --> 00:01:12.800
If you couldn't figure
it out the first time,

00:01:12.800 --> 00:01:14.811
I encourage you to pause the video again

00:01:14.811 --> 00:01:19.100
and, once again, think reverse chain rule.

00:01:20.225 --> 00:01:21.533
So, what am I talking about when

00:01:21.533 --> 00:01:23.553
I keep saying reverse chain rule?

00:01:23.553 --> 00:01:27.919
Let's just review that before
I proceed with this example.

00:01:27.919 --> 00:01:30.515
If I were to just tell you, well what's

00:01:30.515 --> 00:01:34.480
the indefinite integral of one over x dx?

00:01:34.480 --> 00:01:36.017
We know that, that's going to be

00:01:36.017 --> 00:01:39.687
the natural log of the
absolute value of x plus c.

00:01:40.502 --> 00:01:43.306
Now, if I were to ask you what is

00:01:43.306 --> 00:01:47.379
the indefinite integral of f prime

00:01:47.379 --> 00:01:52.379
of x times one over f of x dx.

00:01:53.493 --> 00:01:55.164
What is that going to be?

00:01:55.164 --> 00:01:57.758
Here's where the reverse
chain rule applies.

00:01:57.758 --> 00:02:02.505
Where I have one over f of x,
if only I had it's derivative

00:02:02.505 --> 00:02:04.389
being multiplied by this
thing then I could just

00:02:04.389 --> 00:02:06.067
integrate with respect to f of x.

00:02:06.067 --> 00:02:08.151
Well, I do, I have it's
derivative right over here.

00:02:08.151 --> 00:02:10.828
It's multiplying times this thing.

00:02:10.828 --> 00:02:12.855
I can use the reverse chain rule to say

00:02:12.855 --> 00:02:14.083
that this is going to be equal

00:02:14.083 --> 00:02:17.722
to the natural log of the absolute value

00:02:17.722 --> 00:02:19.895
of the thing that I
have in the denominator,

00:02:19.895 --> 00:02:24.766
which is f of x plus c
and that is exactly...

00:02:25.974 --> 00:02:27.940
I've made it too wide so
you can't see everything...

00:02:27.940 --> 00:02:31.228
but that's exactly what's
going on right over here.

00:02:31.228 --> 00:02:35.186
I have, if I say this
thing right over here,

00:02:35.186 --> 00:02:38.401
cosine of x is f of x,

00:02:38.862 --> 00:02:42.000
then sine of x is not quite

00:02:42.000 --> 00:02:44.831
the derivative, it's the
negative of the derivative.

00:02:45.724 --> 00:02:47.915
F prime of x would be negative sine of x.

00:02:47.915 --> 00:02:50.388
How do I get that, how do I engineer it?

00:02:50.388 --> 00:02:52.483
What if I just threw a negative there

00:02:52.483 --> 00:02:54.590
and a negative there so it is essentially

00:02:54.590 --> 00:02:56.552
multiplying by negative one twice

00:02:56.552 --> 00:02:58.851
which is still going to stay positive.

00:02:58.851 --> 00:03:01.592
Negative sine of x, right over here,

00:03:02.484 --> 00:03:04.450
I'm trying to squeeze it in
between the integral sine

00:03:04.450 --> 00:03:06.784
and the sine of x, this right over here,

00:03:06.784 --> 00:03:08.307
now that I put a negative sine of x,

00:03:08.307 --> 00:03:10.420
that is the derivative of cosine of x.

00:03:10.420 --> 00:03:14.927
This is f prime of x so I can just

00:03:14.927 --> 00:03:17.015
apply the reverse chain rule.

00:03:17.015 --> 00:03:18.970
This is going to be, we deserve

00:03:18.970 --> 00:03:20.820
a little mini drum roll
here, this is going

00:03:20.820 --> 00:03:24.004
to be equal to the natural
log of the absolute value

00:03:24.004 --> 00:03:28.500
of our f of x, which is
going to be cosine of x.

00:03:28.757 --> 00:03:31.275
Then, of course, we have our plus c

00:03:31.275 --> 00:03:34.967
and we can't forget we
had this little negative

00:03:34.967 --> 00:03:36.542
sitting out here so we're going

00:03:36.542 --> 00:03:39.156
to have to put the
negative right over there.

00:03:39.156 --> 00:03:42.020
And we are done, we just figured out

00:03:42.020 --> 00:03:43.931
that's kind of a neat result because

00:03:43.931 --> 00:03:45.312
it feels like that's something we

00:03:45.312 --> 00:03:47.296
should know how to take
the indefinite integral of.

00:03:47.296 --> 00:03:49.939
The indefinite integral
of tangent of x is,

00:03:49.939 --> 00:03:52.514
and it's neat they're connected
in this way, is the negative

00:03:52.514 --> 00:03:57.166
natural log of the absolute
value of cosine of x plus c.


Trig identities for trig integral

WEBVTT
Kind: captions
Language: en

00:00:00.266 --> 00:00:01.023
- [Voiceover] Let's see if we can take

00:00:01.023 --> 00:00:06.023
the indefinite integral of
sine of x to the fourth dx.

00:00:09.800 --> 00:00:11.000
And like always, pause the video

00:00:11.000 --> 00:00:12.848
and see if you can work
through it on your own.

00:00:13.433 --> 00:00:17.033
So if we had an odd exponent up here,

00:00:17.033 --> 00:00:19.233
whether it was a sine or a cosine,

00:00:19.233 --> 00:00:20.833
then the technique I would use,

00:00:20.833 --> 00:00:24.233
so if this was sine to the third of x,

00:00:24.233 --> 00:00:26.400
I would separate one
of the sine of xs out,

00:00:26.400 --> 00:00:30.966
so I would rewrite it as sine
squared x times sine of x,

00:00:30.966 --> 00:00:32.133
and then I would convert this

00:00:32.133 --> 00:00:34.300
using the Pythagorean Identity,

00:00:34.300 --> 00:00:35.633
and then when I distribute the sine of x,

00:00:35.633 --> 00:00:37.966
I'd be able to use u-substitution.

00:00:37.966 --> 00:00:39.633
We've done that in
previous example videos.

00:00:39.633 --> 00:00:40.600
You could have done this if it was

00:00:40.600 --> 00:00:44.127
cosine to the third of x
as well, or to the fifth,

00:00:44.127 --> 00:00:46.566
or to the seventh, if
you had an odd exponent.

00:00:46.566 --> 00:00:48.200
But here we have an even exponent.

00:00:48.200 --> 00:00:49.586
So what do we do?

00:00:50.233 --> 00:00:51.533
So the technique we will use,

00:00:51.533 --> 00:00:53.433
and I guess you could call it a trick,

00:00:53.433 --> 00:00:54.994
the technique or trick to use

00:00:54.994 --> 00:00:58.133
is once again you want to
algebraically manipulate this

00:00:58.133 --> 00:01:00.000
so you can use integration techniques

00:01:00.000 --> 00:01:01.261
that we are familiar with.

00:01:01.261 --> 00:01:02.700
And in this case we would want to use

00:01:02.700 --> 00:01:05.166
the Double Angle Identity.

00:01:05.166 --> 00:01:09.935
The Double Angle Identity
tells us that sine squared of x

00:01:09.935 --> 00:01:14.935
is equal to 1/2 times one
minus cosine of two x.

00:01:19.101 --> 00:01:21.509
So how can I apply this over here?

00:01:22.002 --> 00:01:25.011
Our original integral is
just the same thing as,

00:01:25.011 --> 00:01:26.728
this is going to be the same thing as

00:01:26.728 --> 00:01:31.728
the integral of sine of x
squared, all of that squared, dx.

00:01:33.432 --> 00:01:36.009
Now I can make this substitution.

00:01:36.009 --> 00:01:41.009
So this is the same thing as this,

00:01:41.246 --> 00:01:42.715
which is of course the same thing as this

00:01:42.715 --> 00:01:44.566
by the Double Angle Identity.

00:01:44.566 --> 00:01:49.566
So I could rewrite it
as the integral of 1/2

00:01:52.433 --> 00:01:55.633
times one minus cosine of two x,

00:01:55.633 --> 00:02:00.633
and then all of that squared dx.

00:02:01.346 --> 00:02:02.966
And what is that going to be equal to?

00:02:02.966 --> 00:02:04.700
That's going to be equal to, let's see.

00:02:04.700 --> 00:02:07.396
1/2 squared is 1/4, so
I can take that out.

00:02:07.766 --> 00:02:12.766
So we get 1/4 times the integral of...

00:02:13.861 --> 00:02:16.089
I'm just going to square
all this business.

00:02:16.474 --> 00:02:21.474
One squared, which is one,
minus two cosine of two x

00:02:22.233 --> 00:02:27.233
plus cosine squared of two x dx.

00:02:30.283 --> 00:02:35.283
Fairly straightforward to
take the indefinite integral,

00:02:36.466 --> 00:02:38.066
or to take the antiderivative
of these two pieces.

00:02:38.066 --> 00:02:39.666
But what do I do here?

00:02:39.666 --> 00:02:42.566
Once again I've got an even exponent.

00:02:42.566 --> 00:02:45.655
Let's apply the Double
Angle Identity for cosine.

00:02:45.979 --> 00:02:50.692
We know that cosine squared of two x

00:02:51.200 --> 00:02:55.133
is going to be equal to 1/2 times one plus

00:02:55.133 --> 00:03:00.100
cosine of double this
angle, so cosine of four x.

00:03:00.100 --> 00:03:02.667
Once again just make the substitution.

00:03:02.667 --> 00:03:06.751
This is going to be equal to...

00:03:06.751 --> 00:03:08.433
Actually let me just write it this way.

00:03:08.433 --> 00:03:12.200
1/4, and maybe I'll switch...

00:03:12.200 --> 00:03:13.433
I'll just keep it in this color.

00:03:13.433 --> 00:03:18.433
1/4 times one minus two
cosine two x plus 1/2,

00:03:25.792 --> 00:03:26.967
I'll just distribute the 1/2.

00:03:26.967 --> 00:03:31.967
Plus 1/2 plus 1/2 cosine of four x dx.

00:03:37.119 --> 00:03:42.119
Let's see, I could take this
1/2 and add it to this one,

00:03:43.081 --> 00:03:45.566
and that's going to get me 3/2,

00:03:45.566 --> 00:03:48.733
so add those together,
I'm going to get 3/2.

00:03:48.733 --> 00:03:52.000
So let me rewrite this as, I'm
in the home stretch really,

00:03:52.000 --> 00:03:57.000
this is going to be equal to
1/4 times the integral of 3/2

00:03:58.027 --> 00:04:01.500
minus two cosine of two x.

00:04:01.500 --> 00:04:04.507
The derivative of four x is four,

00:04:04.507 --> 00:04:06.067
so it'd be great if I had a four out here.

00:04:06.067 --> 00:04:09.802
Let me rewrite 1/2 as four over eight.

00:04:09.802 --> 00:04:11.800
Really I'm just multiplying
and dividing by four.

00:04:11.800 --> 00:04:12.866
That's one way to think about it.

00:04:12.866 --> 00:04:17.866
So I could say, plus 1/8
times four cosine of four x.

00:04:22.766 --> 00:04:24.712
I'm just doing this so that it's...

00:04:24.712 --> 00:04:26.066
In theory you could do u-substitution,

00:04:26.066 --> 00:04:27.742
but we've have a lot
of practice with this.

00:04:27.742 --> 00:04:29.619
I have a function, I have its derivative,

00:04:29.619 --> 00:04:30.833
I could just integrate with respect

00:04:30.833 --> 00:04:33.366
to the four x right over there.

00:04:33.366 --> 00:04:34.833
I guess you could say it's
the reverse chain rule,

00:04:34.833 --> 00:04:37.766
which is really what
u-substitution is all about.

00:04:37.766 --> 00:04:40.066
Now I'm ready to integrate.

00:04:40.066 --> 00:04:41.466
So this is going to be equal to,

00:04:41.466 --> 00:04:43.166
I think we deserve a
little bit of a drumroll,

00:04:43.166 --> 00:04:48.166
1/4 times 3/2 x,

00:04:48.966 --> 00:04:51.233
c-derivative of two x is
sitting right over here too,

00:04:51.233 --> 00:04:54.733
so this is going to be
minus sine of two x.

00:04:54.733 --> 00:04:56.766
We can verify that this
is a c-derivative of this.

00:04:56.766 --> 00:04:59.215
It's going to be two cosine of two x,

00:04:59.215 --> 00:05:00.716
we have it right over there,

00:05:01.209 --> 00:05:06.209
plus 1/8 times sine of four x.

00:05:08.600 --> 00:05:09.700
Derivative of sine of four x

00:05:09.700 --> 00:05:12.500
is going to be four cosine of four x,

00:05:12.500 --> 00:05:15.100
which is exactly what we have there.

00:05:15.100 --> 00:05:17.863
And then home stretch, we just write

00:05:17.863 --> 00:05:19.866
the plus C, plus sub constant.

00:05:19.866 --> 00:05:21.500
This is an indefinite integral.

00:05:21.500 --> 00:05:23.567
And we're all done.

00:05:23.567 --> 00:05:25.630
This wasn't the simplest of problems,

00:05:25.630 --> 00:05:27.496
but it also wasn't too bad.

00:05:27.496 --> 00:05:30.333
And it's strangely
satisfying to get it done.


Integral of odd powered trig function with u substitution

WEBVTT
Kind: captions
Language: en

00:00:00.535 --> 00:00:01.949
- [Voiceover] Let's see if
we can take the indefinite

00:00:01.949 --> 00:00:05.211
integral of sine squared x
cosine to the third x dx.

00:00:05.211 --> 00:00:06.838
Like always, pause the
video and see if you

00:00:06.838 --> 00:00:09.602
can work it through on your own.

00:00:09.602 --> 00:00:13.134
All right, so right when
you look at it you're like,

00:00:13.134 --> 00:00:15.267
"Oh wow, if this was just a sine of x,

00:00:15.267 --> 00:00:16.877
"not a sine squared of x.

00:00:16.877 --> 00:00:19.094
"Well that's going to be the negative of

00:00:19.094 --> 00:00:20.825
"the derivative of cosine of x.

00:00:20.825 --> 00:00:22.948
"Maybe I could have used u-substitution.

00:00:22.948 --> 00:00:25.322
"Likewise, if this was just
a cosine of x not a cosine

00:00:25.322 --> 00:00:28.101
"to the third of x, I could
have used u-substitution.

00:00:28.101 --> 00:00:29.899
"I could have said u
is equal to sine of x,

00:00:29.899 --> 00:00:32.877
"but I can't do that over here."

00:00:32.877 --> 00:00:36.549
So the general ideas here,
if one of these has an

00:00:36.549 --> 00:00:39.505
odd exponent and you see
that this one does have

00:00:39.505 --> 00:00:42.153
an odd exponent, the
cosine has an odd exponent.

00:00:42.153 --> 00:00:44.458
What you do is you try to

00:00:44.458 --> 00:00:47.016
algebraically engineer
this expression inside,

00:00:47.016 --> 00:00:49.208
so that you can use u-substitution.

00:00:49.208 --> 00:00:51.597
The way that you can do
that, if you have an odd

00:00:51.597 --> 00:00:54.685
exponent like this, is to
separate out one of the

00:00:54.685 --> 00:00:59.313
cosine's x and then use
the Pythagorean Identity

00:00:59.313 --> 00:01:02.888
with the remaining cosine squared
x, what do I mean by that?

00:01:02.888 --> 00:01:07.141
Let me just rewrite this,
so this can be rewritten as

00:01:07.141 --> 00:01:11.160
sine squared x times...

00:01:11.175 --> 00:01:12.890
Let me write it this way.

00:01:12.890 --> 00:01:16.827
Times cosine squared x

00:01:16.827 --> 00:01:18.996
cosine x.

00:01:18.996 --> 00:01:21.316
All I did is I rewrote
sometime to the third power,

00:01:21.316 --> 00:01:23.245
something to the second
power times that thing

00:01:23.245 --> 00:01:26.232
to the first power, dx.

00:01:27.325 --> 00:01:30.513
And this could be rewritten as,

00:01:30.513 --> 00:01:33.779
sine squared x,

00:01:33.779 --> 00:01:36.425
then I'm going to use
the Pythagorean Identity

00:01:36.425 --> 00:01:40.759
to convert this into
one minus sine squared.

00:01:40.759 --> 00:01:43.151
So that, by the Pythagorean
Identity, is the same thing

00:01:43.151 --> 00:01:48.151
as one minus sine squared x

00:01:48.157 --> 00:01:50.017
and then we have cosine x,

00:01:50.017 --> 00:01:53.934
times cosine x dx.

00:01:56.842 --> 00:02:00.150
Now I can distribute this
sine squared times one

00:02:00.150 --> 00:02:02.852
minus sine squared and I am left with...

00:02:03.576 --> 00:02:07.451
Let me do this in a new color,
this business right here.

00:02:07.451 --> 00:02:11.569
Now I'm left with the
indefinite integral of,

00:02:11.569 --> 00:02:16.144
Sine squared x times one is going to be

00:02:16.452 --> 00:02:19.635
sine squared x and then sine squared x

00:02:19.635 --> 00:02:22.264
times negative sine squared x is

00:02:22.264 --> 00:02:26.340
negative sine to the fourth.

00:02:26.340 --> 00:02:29.833
Then all of that times cosine x.

00:02:29.833 --> 00:02:34.639
All of that times cosine x dx.

00:02:34.639 --> 00:02:37.695
Now this is starting to look interesting,

00:02:37.695 --> 00:02:41.355
cause I have sine squared x
minus sine to the fourth x,

00:02:41.355 --> 00:02:44.052
but I have the derivative
of sine sitting out here.

00:02:44.052 --> 00:02:45.963
I have cosine x, that's
the whole reason why we

00:02:45.963 --> 00:02:48.450
did this little algebraic manipulation.

00:02:48.450 --> 00:02:51.032
So u-substitution works
out quite well now.

00:02:51.032 --> 00:02:54.459
Because if we said that..

00:02:54.459 --> 00:02:57.250
Let me do another color, I'll do purple.

00:02:57.250 --> 00:02:59.432
If we say that u

00:03:00.630 --> 00:03:02.752
is equal to sine of x,

00:03:02.752 --> 00:03:07.296
then du is going to be equal to cosine

00:03:07.296 --> 00:03:10.773
of x dx, and that works out quite well

00:03:10.773 --> 00:03:14.072
because we have the du right over here.

00:03:14.072 --> 00:03:17.618
Then this would be u squared
minus u to the fourth.

00:03:17.618 --> 00:03:20.238
Which we know how to take
the anti-derivative of,

00:03:20.238 --> 00:03:21.831
it's over the whole stretch.

00:03:21.831 --> 00:03:24.814
We can rewrite this as
the indefinite integral.

00:03:24.814 --> 00:03:27.341
Instead of sine squared
x we're saying sine is

00:03:27.341 --> 00:03:30.745
the same thing as u so
we can rewrite that as,

00:03:32.191 --> 00:03:37.033
u squared minus

00:03:37.985 --> 00:03:41.408
u to the fourth

00:03:41.408 --> 00:03:44.185
times du.

00:03:47.509 --> 00:03:49.901
This is pretty straight
forward, now this is going

00:03:49.901 --> 00:03:54.293
to be u to the third over three

00:03:54.293 --> 00:03:59.293
minus u to the fifth over five plus c.

00:04:02.449 --> 00:04:04.438
Then we just do the reverse substitution

00:04:04.438 --> 00:04:06.724
and then that gets us to be, instead of u

00:04:06.724 --> 00:04:08.414
we want to put a sine of x there.

00:04:08.414 --> 00:04:12.146
So we're gonna get sine to the third,

00:04:12.146 --> 00:04:14.466
sine of x to the third over three

00:04:14.466 --> 00:04:18.325
minus sine of x to the fifth power.

00:04:18.325 --> 00:04:20.069
Let me rewrite that five a little bit.

00:04:20.069 --> 00:04:24.080
Sine of x to the fifth power over five

00:04:24.080 --> 00:04:27.385
plus c and we

00:04:27.385 --> 00:04:30.747
are done.


Using trig identity to use u substitution

WEBVTT
Kind: captions
Language: en

00:00:00.333 --> 00:00:01.600
- [Voiceover] Let's see if we can take

00:00:01.600 --> 00:00:06.466
the indefinite integral of
cosine of X to the third power.

00:00:06.466 --> 00:00:08.433
I encourage you to pause the video

00:00:08.433 --> 00:00:12.267
and see if you can figure
this out on your own.

00:00:13.633 --> 00:00:18.066
You have given it a go and
you might have gotten stuck.

00:00:18.066 --> 00:00:19.533
Some of you all might have
been able to figure it out,

00:00:19.533 --> 00:00:21.167
but some of you all
might have gotten stuck.

00:00:21.167 --> 00:00:23.433
You're like, "Okay,
cosine to the third power.

00:00:23.433 --> 00:00:25.667
"Well, gee, if I only had a
derivative of cosine here,

00:00:25.667 --> 00:00:28.000
"if I had a negative sign
of X or a sin of X here,

00:00:28.000 --> 00:00:30.200
"maybe I could've used U substitution,

00:00:30.200 --> 00:00:31.933
"but how do I take the anti-derivative

00:00:31.933 --> 00:00:34.334
"of cosine of X to the third power?"

00:00:34.701 --> 00:00:36.900
The key here is, is to use

00:00:36.900 --> 00:00:39.466
some basic trigonometric identities.

00:00:39.466 --> 00:00:40.967
What do I mean by that?

00:00:40.967 --> 00:00:45.967
We know that sin squared
X plus cosine squared X

00:00:49.300 --> 00:00:52.067
is equal to one, or if
we subtract sin squared

00:00:52.067 --> 00:00:57.067
from both sides, we know
that cosine squared X

00:00:58.233 --> 00:01:00.600
is equal to one, write
it this way, is equal to

00:01:00.600 --> 00:01:03.600
one minus sin squared X.

00:01:04.266 --> 00:01:06.966
What would happen if
cosine to the third power,

00:01:06.966 --> 00:01:08.700
that's cosine squared times cosine.

00:01:08.700 --> 00:01:11.233
What happens if we were to take

00:01:14.833 --> 00:01:16.333
that cosine squared?

00:01:16.333 --> 00:01:17.433
Let me just rewrite it.

00:01:17.433 --> 00:01:19.500
This is the same thing as cosine of X

00:01:19.500 --> 00:01:24.500
times cosine squared of X, DX.

00:01:24.901 --> 00:01:27.266
What if we were to take this thing

00:01:27.266 --> 00:01:30.133
right over here, let me
do that magenta color.

00:01:30.500 --> 00:01:32.467
What if we were to take
this right over here

00:01:32.467 --> 00:01:34.367
and replace it with this.

00:01:34.367 --> 00:01:35.501
I now what you're thinking.

00:01:35.501 --> 00:01:37.167
"Sal, what's that going to do for me?

00:01:37.167 --> 00:01:41.633
"This feels like I'm making this integral

00:01:41.633 --> 00:01:43.733
"even more convoluted."

00:01:43.733 --> 00:01:45.733
What I would tell you, I would say,

00:01:45.733 --> 00:01:50.166
"This might seem like it's
getting more complicated,

00:01:50.166 --> 00:01:51.866
"but as you explore and you play with it,

00:01:51.866 --> 00:01:53.467
"you'll see that this actually makes

00:01:53.467 --> 00:01:55.466
"the integral more solvable."

00:01:55.466 --> 00:01:56.966
Let's try it out.

00:01:56.966 --> 00:01:59.900
If we do that, this is
going to be equal to

00:02:01.838 --> 00:02:06.838
the indefinite integral,
cosine of X times one

00:02:08.906 --> 00:02:13.906
minus sin squared X, DX.

00:02:14.437 --> 00:02:16.333
What is this going to be equal to?

00:02:16.333 --> 00:02:18.567
This is going to be equal to,

00:02:18.667 --> 00:02:21.433
let me do this in that green color.

00:02:21.933 --> 00:02:23.566
This is going to be equal to

00:02:23.566 --> 00:02:26.366
the indefinite integral of cosine X.

00:02:26.366 --> 00:02:29.000
I'm just going to
distribute the cosine of X.

00:02:29.000 --> 00:02:34.000
Cosine of X minus, minus cosine of X,

00:02:35.834 --> 00:02:40.834
cosign of X sin squared of X, sin squared,

00:02:43.100 --> 00:02:48.100
sin squared X and then I can
close the parentheses, DX.

00:02:49.766 --> 00:02:53.800
This, of course, is going
to be equal to the integral

00:02:53.800 --> 00:02:55.500
of cosine of X, DX, and we know

00:02:55.500 --> 00:02:59.433
what that's going to
be, minus the integral.

00:02:59.433 --> 00:03:02.866
I'll switch to one color
now, of cosine of X,

00:03:02.866 --> 00:03:07.866
sin squared X, sin squared X, DX.

00:03:09.400 --> 00:03:11.567
Now, this is where it gets interesting.

00:03:11.567 --> 00:03:15.200
This part right over here
is pretty straight forward.

00:03:15.200 --> 00:03:18.133
The anti-derivative of
cosine of X is just sin of X.

00:03:18.133 --> 00:03:20.100
This right over here is
going to be sin of X.

00:03:20.100 --> 00:03:23.800
I'll worry about the plus C at
the end because both of these

00:03:23.800 --> 00:03:25.433
are going to have a
plus C, so might as well

00:03:25.433 --> 00:03:27.233
just put one big plus c at the end.

00:03:27.233 --> 00:03:29.933
That's sin of X and then what do we have

00:03:29.933 --> 00:03:32.000
going on over here?

00:03:32.000 --> 00:03:35.466
Well, you might recognize, I
have a function of sin of X.

00:03:35.466 --> 00:03:37.433
I'm taking sin of X and I'm squaring it

00:03:37.433 --> 00:03:39.933
and then I have sin of X's
derivative right over here.

00:03:39.933 --> 00:03:44.933
This fits the, I have some
derivative of a function

00:03:45.733 --> 00:03:49.000
and then I have another and then I have a,

00:03:49.000 --> 00:03:51.866
I guess you could say, a
function of that function.

00:03:51.866 --> 00:03:53.466
G of F of X.

00:03:53.466 --> 00:03:56.800
That's a sin that maybe U
substitution is in order,

00:03:56.800 --> 00:03:59.166
or we've seen the pattern,
we've seen this show

00:03:59.166 --> 00:04:02.133
multiple times already,
that you could just say,

00:04:02.133 --> 00:04:05.966
"Okay, if I have a function of a function

00:04:05.966 --> 00:04:07.767
"and I have that functions derivative,

00:04:07.767 --> 00:04:09.433
"then essentially I can just
take the anti-derivative

00:04:09.433 --> 00:04:11.166
"with respect to this function."

00:04:11.166 --> 00:04:15.300
This would be equal to, say, capital G

00:04:15.300 --> 00:04:17.600
is the anti-derivative of lower case G.

00:04:17.600 --> 00:04:22.300
Capital G of F of X plus C.

00:04:22.300 --> 00:04:24.633
Now, if what I said didn't
make sense, then we could do

00:04:24.633 --> 00:04:26.000
U substitution and go through it

00:04:26.000 --> 00:04:28.233
a little bit more step by step.

00:04:28.233 --> 00:04:30.666
Let's just do that because
we want things to make sense.

00:04:30.666 --> 00:04:33.100
That's the whole point of these videos.

00:04:33.100 --> 00:04:38.100
We could say U is equal
to sin of X and then DU

00:04:39.066 --> 00:04:44.066
is going to be equal to cosine of X, DX.

00:04:44.133 --> 00:04:49.067
This part and that part is going to be DU

00:04:49.067 --> 00:04:52.433
and then this is going to be U squared.

00:04:52.433 --> 00:04:54.200
This is going to be minus.

00:04:54.200 --> 00:04:59.200
We have the integral of U squared, DU.

00:04:59.700 --> 00:05:01.501
What is this going to be?

00:05:01.501 --> 00:05:04.800
This is going to be, we're
going to have negative U

00:05:04.800 --> 00:05:07.600
to the third power over three.

00:05:07.600 --> 00:05:09.933
Then, we know what U is.

00:05:09.933 --> 00:05:13.166
The U is equal to sin of X.

00:05:13.166 --> 00:05:16.933
We have our sin of X
here for the first part

00:05:16.933 --> 00:05:21.534
of the integral, for the first integral.

00:05:21.534 --> 00:05:26.256
We have the sin of X and then
this is going to be minus.

00:05:26.256 --> 00:05:27.600
Let me just write it this way.

00:05:27.600 --> 00:05:31.334
Minus 1/3 minus 1/3.

00:05:31.334 --> 00:05:34.133
Instead of U to the third,
we know U is sin of X.

00:05:34.133 --> 00:05:38.367
Sin of X to the third power.

00:05:38.367 --> 00:05:43.367
Then now, we can throw that plus C there.

00:05:43.800 --> 00:05:45.166
We're done.

00:05:45.166 --> 00:05:47.600
We've just evaluated
that indefinite integral.

00:05:47.600 --> 00:05:51.068
The key to it is to just
play around a little bit

00:05:51.068 --> 00:05:54.433
with trigonometric identities
so that you can get

00:05:54.433 --> 00:05:56.800
the integral to a point that you can use

00:05:56.800 --> 00:05:59.567
the reverse chain rule or
you can use U substitution,

00:05:59.567 --> 00:06:00.966
which is just really another way

00:06:00.966 --> 00:06:04.633
of expressing the reverse chain rule.


Worked example: Definite integral by thinking about the function's graph

WEBVTT
Kind: captions
Language: en

00:00:00.361 --> 00:00:02.379
- [Voiceover] What I want to
do in this video is see if

00:00:02.379 --> 00:00:04.723
we can evaluate the definite
integral from negative three to

00:00:04.723 --> 00:00:09.287
three of the square root
of nine minus x squared dx.

00:00:09.287 --> 00:00:11.383
I encourage you to pause this
video and try it on your own.

00:00:11.383 --> 00:00:15.623
I'll give you a hint,
you can do this purely

00:00:15.623 --> 00:00:19.499
by looking at the graph of this function.

00:00:20.084 --> 00:00:22.718
All right, I'm assuming
you've had a go at it.

00:00:22.718 --> 00:00:24.414
So let's just think about,
I just told you that you

00:00:24.414 --> 00:00:26.184
could do this by using
the graph of the function.

00:00:26.184 --> 00:00:28.890
Let's graph this function.

00:00:31.889 --> 00:00:34.564
Let's get a y-axis here.

00:00:34.564 --> 00:00:38.067
This is my y-axis.

00:00:38.067 --> 00:00:40.571
This is my x-axis.

00:00:42.756 --> 00:00:44.985
You might be saying, "Oh
well, what is the graph

00:00:44.985 --> 00:00:46.738
"of this thing?", it
might not jump out at you.

00:00:46.738 --> 00:00:49.518
It's been a little while
(mumble) of a hint since you've

00:00:49.518 --> 00:00:52.991
done conic sections, maybe
in your Algebra class.

00:00:52.991 --> 00:00:55.427
Let's just remind ourselves.

00:00:55.427 --> 00:00:58.852
If this function, if we said y is equal to

00:00:58.852 --> 00:01:03.001
some function of x, which
we see is the square root

00:01:03.001 --> 00:01:06.365
of nine minus x squared.

00:01:06.365 --> 00:01:09.170
Then we could say, "Well,
that means that y squared must

00:01:09.170 --> 00:01:10.865
"must be equal to this thing squared."

00:01:10.865 --> 00:01:13.665
Which is nine minus x squared.

00:01:13.665 --> 00:01:18.452
Then we could say, "Y squared
plus x squared is equal

00:01:18.452 --> 00:01:22.836
"to nine." and you might
recognize this as a circle

00:01:24.420 --> 00:01:29.219
centered at origin with radius

00:01:30.358 --> 00:01:32.916
equal to three, the square root of nine.

00:01:32.916 --> 00:01:36.797
So radius is equal to three,
centered at the origin.

00:01:36.797 --> 00:01:39.112
Now, they graph of this is
not going to be a circle.

00:01:39.112 --> 00:01:41.437
This is a function.

00:01:42.098 --> 00:01:44.007
It would be a circle and it would not be

00:01:44.007 --> 00:01:45.961
a function anymore if
you said the positive and

00:01:45.961 --> 00:01:48.035
negative square roots of here.

00:01:51.203 --> 00:01:53.424
When we took the squares
of both sides we got

00:01:53.424 --> 00:01:55.925
the bottom back I guess you could say.

00:01:55.925 --> 00:01:59.147
But up here, we're only talking
about the principal root.

00:01:59.147 --> 00:02:00.106
When you're talking
about the principal root

00:02:00.106 --> 00:02:01.870
you're really talking about the top.

00:02:01.870 --> 00:02:05.091
This is the top of a circle centered at

00:02:05.091 --> 00:02:07.783
the origin with radius three.

00:02:07.783 --> 00:02:11.480
So this is top of circle
cause it's the positive

00:02:11.480 --> 00:02:13.326
square root, so let's draw that.

00:02:13.326 --> 00:02:16.667
It's gonna have a radius of
three, centered at the origin.

00:02:16.667 --> 00:02:21.010
This is gonna be negative three,
this is going to be three.

00:02:21.010 --> 00:02:23.281
This is going to be three right over here.

00:02:23.281 --> 00:02:27.185
So this function is
going to look like this.

00:02:30.416 --> 00:02:33.280
It's actually only defined
between negative three and three.

00:02:33.280 --> 00:02:36.238
The absolute value of x
is greater than three,

00:02:36.238 --> 00:02:38.135
then you're going to get
a negative value in here.

00:02:38.135 --> 00:02:40.084
Then you can't take the principal root,

00:02:40.084 --> 00:02:42.170
as if we're defining it over

00:02:44.446 --> 00:02:47.513
a positive or non-negative
values I should say.

00:02:47.513 --> 00:02:49.219
So this is the graph.

00:02:49.219 --> 00:02:52.508
What is the definite integral
from negative three to three.

00:02:52.508 --> 00:02:55.361
Well it's just the area under the curve

00:02:55.361 --> 00:02:57.884
and above the x-axis, it's the stuff that

00:02:57.884 --> 00:03:00.063
I am shading in in green.

00:03:00.063 --> 00:03:01.083
Well, what's that?

00:03:01.083 --> 00:03:02.729
We don't need Calculus to figure that out.

00:03:02.729 --> 00:03:05.447
You can do this with just
traditional geometry.

00:03:05.447 --> 00:03:09.037
The area of the entire circle,

00:03:09.037 --> 00:03:11.659
if there were an entire circle,
would just be pi r squared.

00:03:11.659 --> 00:03:14.393
So it would be pi times three squared,

00:03:14.393 --> 00:03:16.868
which is equal to nine pi.

00:03:16.868 --> 00:03:19.082
Now this is only half
of the entire circle.

00:03:19.082 --> 00:03:21.591
So we're gonna divide that by two.

00:03:21.591 --> 00:03:24.543
The area is nine pi over two.

00:03:24.543 --> 00:03:28.846
So this thing is nine pi over two.


Statistical significance on bus speeds

WEBVTT
Kind: captions
Language: en

00:00:00.536 --> 00:00:02.816
- [Voiceover] "Giovanna
usually takes bus B to work,

00:00:02.816 --> 00:00:06.564
"but now she thinks that bus
A gets her to work faster.

00:00:06.564 --> 00:00:09.424
"She randomized 50 workdays
between a treatment group

00:00:09.424 --> 00:00:10.780
"and a control group.

00:00:10.780 --> 00:00:13.140
"For each day from the
treatment group, she took bus A;

00:00:13.140 --> 00:00:15.773
"and for each day from the
control group, she took bus B.

00:00:15.773 --> 00:00:19.513
"Each day she timed the
length of her drive."

00:00:19.513 --> 00:00:20.760
This is really interesting
what she did, it's very

00:00:20.760 --> 00:00:23.910
important, she randomized
the 50 work days.

00:00:23.910 --> 00:00:25.693
Before she did this, instead
of just kinda waking up in

00:00:25.693 --> 00:00:29.300
the morning and just deciding
on her own which bus to take.

00:00:29.300 --> 00:00:32.238
Because humans are infamously
bad at being random.

00:00:32.238 --> 00:00:33.565
Even when we think we're being random,

00:00:33.565 --> 00:00:35.017
we're actually not that random.

00:00:35.017 --> 00:00:38.686
She might inadvertently be
taking bus A earlier in the week.

00:00:38.686 --> 00:00:40.777
Or maybe the commute times are shorter.

00:00:40.777 --> 00:00:43.542
Or maybe she inadvertently
takes bus A when

00:00:43.542 --> 00:00:45.442
the weather is better,
when there's less traffic.

00:00:45.442 --> 00:00:49.089
Remember, there's a natural
tendency for human beings

00:00:49.089 --> 00:00:51.569
to want to confirm their hypothesis.

00:00:51.569 --> 00:00:54.089
So, if she thinks that
bus A is faster maybe

00:00:54.089 --> 00:00:56.865
she'll want to pick the days where she'll

00:00:56.865 --> 00:00:58.705
get data to confirm her hypothesis.

00:00:58.705 --> 00:01:01.940
It's really important that
she randomize the 50 workdays.

00:01:01.940 --> 00:01:04.030
What I could imagine she did
is maybe she wrote each of

00:01:04.030 --> 00:01:06.483
the work days, the dates,
on a piece of paper.

00:01:06.483 --> 00:01:09.293
She would have 50 pieces of
paper and then she turned

00:01:09.293 --> 00:01:11.634
them all upside down or
maybe she closed her eyes

00:01:11.634 --> 00:01:13.509
and then she moved them
all over her table.

00:01:13.509 --> 00:01:15.924
Then with her eyes closed
she randomly moved them to

00:01:15.924 --> 00:01:17.973
either the left or the right of the table.

00:01:17.973 --> 00:01:19.683
If they moved to the left
of the table then those are

00:01:19.683 --> 00:01:21.591
the days she'll take bus A,
if she moves them to the right

00:01:21.591 --> 00:01:23.779
of the table those are
the days she takes bus B.

00:01:23.779 --> 00:01:28.043
That's how she can make sure
that this is truly random.

00:01:28.043 --> 00:01:31.125
So then they tell us, this is important,

00:01:31.125 --> 00:01:32.997
"The results of the experiment
showed that the median

00:01:32.997 --> 00:01:37.252
"travel duration for bus A
is eight minutes less than

00:01:37.252 --> 00:01:41.636
"the median travel duration for bus B."

00:01:41.636 --> 00:01:44.243
Or one way to think about, if we said,

00:01:44.256 --> 00:01:49.012
"The treatment group

00:01:49.672 --> 00:01:54.672
"median minus the control group median.

00:01:58.677 --> 00:02:00.955
"What would we get?"

00:02:00.955 --> 00:02:04.494
Well, the treatment group
is eight minutes less

00:02:04.494 --> 00:02:06.253
than the control group? Right?

00:02:06.253 --> 00:02:09.508
This is A, this is B, so if
this is eight less than this,

00:02:09.508 --> 00:02:12.555
then this is going to be
equal to negative eight.

00:02:12.555 --> 00:02:14.958
This is just another
way of restating what I

00:02:14.958 --> 00:02:17.060
have underlined right over here.

00:02:17.060 --> 00:02:19.394
Someone's car alarm went off,
hope you're not hearing that.

00:02:19.394 --> 00:02:20.371
Anyway, I'll try to pay
attention while it's

00:02:20.371 --> 00:02:22.031
going off (chuckles).

00:02:22.031 --> 00:02:23.954
"To test whether the
results could be explained

00:02:23.954 --> 00:02:27.229
"by random chance, she
created the table below,

00:02:27.229 --> 00:02:31.505
"Which summarizes the results
of 1000 re-randomizations

00:02:31.505 --> 00:02:34.354
"of the data, with
differences between medians

00:02:34.354 --> 00:02:36.257
"rounded to the nearest five minutes."

00:02:36.257 --> 00:02:37.849
What is going on over here?

00:02:37.849 --> 00:02:39.646
You might say well look,
"She got her result that she

00:02:39.646 --> 00:02:44.027
"wanted to get, this data
seems to confirm that

00:02:44.027 --> 00:02:46.429
"bus A gets her to work faster.

00:02:46.429 --> 00:02:48.578
"What's all this other
business with re-randomization

00:02:48.578 --> 00:02:50.016
"she's doing?"

00:02:50.016 --> 00:02:53.251
The important thing to realize
is, and she realizes this,

00:02:53.251 --> 00:02:55.052
is that she might have
just gotten this data that

00:02:55.052 --> 00:02:56.799
I underlined, by random chance.

00:02:56.799 --> 00:02:59.615
There's some chance maybe A
and B are completely similar,

00:02:59.615 --> 00:03:01.857
in terms of how long they take in reality.

00:03:01.857 --> 00:03:05.120
She just happened to pick bus A on days

00:03:05.120 --> 00:03:06.836
where bus A got to work faster.

00:03:06.836 --> 00:03:09.871
Maybe bus B is faster but
she just happened to take

00:03:09.871 --> 00:03:12.355
bus A on the days that it was faster.

00:03:12.355 --> 00:03:15.455
The days it just happened
to have less traffic.

00:03:15.455 --> 00:03:18.799
What she's doing here is
she re-randomized the data

00:03:18.799 --> 00:03:21.178
and she wants to see that
with all this re-randomized

00:03:21.178 --> 00:03:24.615
data, out of these 1000 re-randomizations,

00:03:24.615 --> 00:03:27.565
what fraction of them do
I get a result like this?

00:03:27.565 --> 00:03:32.209
Do I get a result where A is
eight minutes or more faster?

00:03:32.209 --> 00:03:35.142
Or you could say that the
median travel duration for bus A

00:03:35.142 --> 00:03:38.151
is eight minutes less,
or even less than that,

00:03:38.151 --> 00:03:39.760
than the median travel for bus B.

00:03:39.760 --> 00:03:42.006
So if it was nine minutes
less, or 10 minutes less,

00:03:42.006 --> 00:03:44.719
or 15 minutes less, those
are all the interesting ones.

00:03:44.719 --> 00:03:46.939
Those are the ones that
confirm our hypothesis,

00:03:46.939 --> 00:03:49.873
that bus A gets to work faster.

00:03:49.873 --> 00:03:51.705
Let's look at this table, it's not below,

00:03:51.705 --> 00:03:53.226
it's actually to the right.

00:03:53.226 --> 00:03:56.468
Let's just remind ourselves
what she did here,

00:03:56.468 --> 00:03:58.796
cause the first time
you try to process this

00:03:58.796 --> 00:04:00.949
it can seem a little bit daunting.

00:04:00.949 --> 00:04:03.160
So, in her experiment,

00:04:03.160 --> 00:04:06.968
let me write this down, experiment...

00:04:06.968 --> 00:04:09.010
The car alarm outside which
you probably, hopefully

00:04:09.010 --> 00:04:11.146
are not hearing, it's actually
a surprisingly pleasant

00:04:11.146 --> 00:04:13.351
sounding car alarm, sounds like a

00:04:13.351 --> 00:04:16.750
slightly obnoxious bird,
but anyway (laughs).

00:04:16.750 --> 00:04:20.450
Her experiment is, the way I described it,

00:04:20.450 --> 00:04:24.334
25 days she would take bus A,
25 days she would take bus B.

00:04:24.334 --> 00:04:26.238
She would record all the
travel times and let's say

00:04:26.238 --> 00:04:28.587
that I have 25 data points in each column.

00:04:28.587 --> 00:04:33.015
Let's say they get 12 minutes,
20 minutes, 25 minutes,

00:04:33.015 --> 00:04:35.982
and you just keep going,
there's 25 data points.

00:04:35.982 --> 00:04:39.500
Let's just say that there are
12 data points less than 20

00:04:39.500 --> 00:04:42.674
minutes and 12 data points
more than 20 minutes.

00:04:42.674 --> 00:04:45.446
In this circumstance,
her median time for bus A

00:04:45.446 --> 00:04:48.084
would be 20 minutes and I
just made this number up.

00:04:48.084 --> 00:04:50.645
So in order for this to
be eight minutes less

00:04:50.645 --> 00:04:53.075
than the median time for
bus B, the median for bus B

00:04:53.075 --> 00:04:56.719
would have to be 28 and maybe
you have data points here.

00:04:56.719 --> 00:05:01.284
Maybe this is 18 and you have
12 more that are less than 28.

00:05:01.284 --> 00:05:05.744
Then you have 12 more
that are greater than 28.

00:05:05.744 --> 00:05:09.684
So the median time for bus B

00:05:12.424 --> 00:05:16.133
would be 28, once again
I just made this data up.

00:05:16.133 --> 00:05:18.998
If you took treatment group median.

00:05:18.998 --> 00:05:22.547
I 'll just write TGM for short.

00:05:22.547 --> 00:05:27.153
TGM minus control group median.

00:05:28.258 --> 00:05:31.568
What do you get? 20 minus
28 is negative eight.

00:05:31.568 --> 00:05:34.280
This is the actual results of....

00:05:34.280 --> 00:05:37.128
These are theoretical,
potential results, hypothetical

00:05:37.128 --> 00:05:39.708
results for her actual experiment.

00:05:39.708 --> 00:05:42.006
Now what's all of this business over here?

00:05:42.006 --> 00:05:44.650
What she did is she took
these times and she said,

00:05:44.650 --> 00:05:47.108
"You know what, let's just
imagine a world where I could

00:05:47.108 --> 00:05:49.905
"have gotten any of these
times randomly on either bus."

00:05:49.905 --> 00:05:54.538
So she just randomly
re-sorted them between

00:05:55.691 --> 00:05:58.885
A and B, she did that a thousand times.

00:05:58.885 --> 00:06:02.753
The first time, the second
time, the third time.

00:06:02.753 --> 00:06:04.722
She does this

00:06:04.722 --> 00:06:07.785
1000 times.

00:06:07.785 --> 00:06:10.008
I'm assuming she used some
type of computer program

00:06:10.008 --> 00:06:12.583
to do it and each time, once again,

00:06:12.583 --> 00:06:15.421
she just took the data
that she had and she just

00:06:15.421 --> 00:06:18.967
rearranged it, she just reshuffled it.

00:06:18.967 --> 00:06:22.139
Maybe A on one day.

00:06:24.133 --> 00:06:27.063
Maybe it got this 18.

00:06:27.063 --> 00:06:29.781
Maybe it gets the 25.

00:06:29.781 --> 00:06:32.514
Maybe it gets a 30.

00:06:32.514 --> 00:06:34.971
Once again, I got the 18, the 25, the 30

00:06:34.971 --> 00:06:37.041
and maybe B gets the...

00:06:37.041 --> 00:06:38.957
You know she's reshuffling
all these other data points

00:06:38.957 --> 00:06:41.280
that I just have with dots and maybe B...

00:06:41.280 --> 00:06:45.546
Let's see she had the 18,
25, 30, maybe 12, 20, and 28.

00:06:49.253 --> 00:06:52.037
So in this circumstance,
this random reshuffling

00:06:52.037 --> 00:06:54.007
and she keeps doing it
over and over again.

00:06:54.007 --> 00:06:58.658
In this random reshuffling,
the treatment group median

00:06:58.658 --> 00:07:01.997
minus the control group
median is going to be what?

00:07:01.997 --> 00:07:04.274
It's going to be equal to positive five.

00:07:04.274 --> 00:07:08.012
In this random shuffling,
this hypothetical scenario,

00:07:08.012 --> 00:07:11.621
Bus A's median would
have been five minutes

00:07:11.621 --> 00:07:14.648
longer than bus B's.

00:07:14.648 --> 00:07:17.728
If she gets this result
with this random re-sorting,

00:07:17.728 --> 00:07:19.875
this would have been...

00:07:20.506 --> 00:07:23.278
She would have had a column here for five.

00:07:23.278 --> 00:07:25.585
Then she would have

00:07:26.446 --> 00:07:28.081
put one notch right over here.

00:07:28.081 --> 00:07:31.373
It looks like she classified
things or maybe she didn't even

00:07:31.373 --> 00:07:35.284
get the data but she classified
them by multiples of two.

00:07:35.284 --> 00:07:37.640
If she got this again then
she would have put a two here.

00:07:37.640 --> 00:07:39.901
Then she would have said,
"Okay, in how many of these

00:07:39.901 --> 00:07:43.499
"random reshufflings am I
getting a scenario where

00:07:43.499 --> 00:07:46.904
"there's a five minute difference?

00:07:46.904 --> 00:07:48.999
"Or where the treatment group
was five minutes longer?"

00:07:48.999 --> 00:07:50.854
What is this saying?

00:07:50.854 --> 00:07:55.024
For example, this is saying that 18

00:07:55.024 --> 00:07:57.657
out of the 1000 reshufflings,
which she just randomly

00:07:57.657 --> 00:08:01.383
re-shuffled the data, 18
out of those 1000 times,

00:08:01.383 --> 00:08:05.240
she found a scenario where
her treatment group median

00:08:05.240 --> 00:08:08.155
was 10 minutes longer
than her control group.

00:08:08.155 --> 00:08:11.927
Where bus A's median
was in this hypothetical

00:08:11.927 --> 00:08:14.401
re-randomization where
the treatment group is

00:08:14.401 --> 00:08:17.714
10 minutes slower than the control group.

00:08:17.714 --> 00:08:21.828
There were 159 times where
the treatment group...

00:08:21.828 --> 00:08:24.020
Once again, in her random
reshuffling, these aren't based

00:08:24.020 --> 00:08:27.001
on observations, these
are random reshufflings.

00:08:27.001 --> 00:08:30.877
There's 159 times where
her treatment group is

00:08:30.877 --> 00:08:33.630
four minutes slower
than her control group.

00:08:33.630 --> 00:08:35.648
The whole reason for
doing this is she says,

00:08:35.648 --> 00:08:38.799
"Okay, what's the probability
of getting a result

00:08:38.799 --> 00:08:41.791
"like this or better?"

00:08:41.791 --> 00:08:44.478
I say, "better", as one
that even more confirms her

00:08:44.478 --> 00:08:47.067
hypothesis, that the
treatment group is faster

00:08:47.067 --> 00:08:48.626
than the control group.

00:08:48.626 --> 00:08:50.974
Well, the scenario, this
scenario is this one right

00:08:50.974 --> 00:08:54.854
over here and then another
one that the treatment group

00:08:54.854 --> 00:08:57.349
is even faster, is this right over here.

00:08:57.349 --> 00:09:00.743
Here, the treatment group
median is 10 less than

00:09:00.743 --> 00:09:02.704
the control group median.

00:09:02.704 --> 00:09:06.601
In how many of these scenarios,
out of the thousands,

00:09:06.601 --> 00:09:07.972
is this occurring?

00:09:07.972 --> 00:09:11.090
Well, this one occurs 85
times, this one occurs eight.

00:09:11.090 --> 00:09:15.329
If you add these two together,
93 out of the thousand times,

00:09:15.329 --> 00:09:17.883
out of her re-randomization
or I guess you could

00:09:17.883 --> 00:09:21.896
say 9.3 percent of the time, the data...

00:09:21.896 --> 00:09:26.896
9.3 percent of the randomized,
the 1000 re-randomizations,

00:09:28.348 --> 00:09:31.933
9.3 percent of the time she got data

00:09:31.933 --> 00:09:35.974
that was as validating
of a hypothesis or more

00:09:35.974 --> 00:09:38.335
than the actual experiment.

00:09:38.335 --> 00:09:42.010
One way to think about
this is, the probability of

00:09:42.010 --> 00:09:45.730
randomly getting the
results from her experiment

00:09:45.730 --> 00:09:48.511
or better results from her experiment

00:09:48.511 --> 00:09:51.376
are 9.3 percent.

00:09:51.376 --> 00:09:55.288
They're low, it's a reasonably
low probability that

00:09:55.288 --> 00:09:57.430
this happened purely by chance.

00:09:57.430 --> 00:09:59.553
Now, a question is,
"What's the threshold?"

00:09:59.553 --> 00:10:01.559
If it was a 50 percent you
say, "Okay, this was very

00:10:01.559 --> 00:10:03.287
"likely to happen by chance."

00:10:03.287 --> 00:10:06.333
If this was a 25 percent
you're like, "Okay, it's less

00:10:06.333 --> 00:10:08.692
"likely to happen by chance
but it could happen."

00:10:08.692 --> 00:10:12.534
9.3 percent, it's roughly 10 percent.

00:10:12.534 --> 00:10:17.109
For every 10 people who do
an experiment like she did,

00:10:17.109 --> 00:10:20.671
even if it was random, one
person would get data like this?

00:10:20.671 --> 00:10:23.133
What typically happens amongst
statisticians is they draw

00:10:23.133 --> 00:10:24.881
a threshold and the threshold for

00:10:24.881 --> 00:10:28.894
statistical significance
is usually five percent.

00:10:28.894 --> 00:10:30.901
One way to think about it,
the probability of her getting

00:10:30.901 --> 00:10:32.745
this result by chance,

00:10:32.745 --> 00:10:35.484
this result or a more extreme result?

00:10:35.484 --> 00:10:38.224
One that more confirms
her hypothesis by chance

00:10:38.224 --> 00:10:40.377
is 9.3 percent.

00:10:40.377 --> 00:10:44.211
If you're cut-off for
significance is five percent.

00:10:44.211 --> 00:10:46.366
If you said, "Okay, this has
to be five percent or less."

00:10:46.366 --> 00:10:49.167
Then you say, "Okay, this is
not statistically significant."

00:10:49.167 --> 00:10:52.001
There's more than a five
percent chance that I could

00:10:52.001 --> 00:10:54.909
have gotten this result
purely through random chance.

00:10:54.909 --> 00:10:57.080
Once again, that just depends on where you

00:10:57.080 --> 00:10:58.771
have that threshold.

00:10:58.771 --> 00:11:00.685
When we go back, I think
we've already answered

00:11:00.685 --> 00:11:03.183
the final question,
"According to the simulations,

00:11:03.183 --> 00:11:05.495
"what is the probability of
the treatment group's median

00:11:05.495 --> 00:11:08.032
"being lower than the
control group's median

00:11:08.032 --> 00:11:09.988
"by eight minutes or more?"

00:11:09.988 --> 00:11:12.263
Which once again, eight
minutes or more, that would be

00:11:12.263 --> 00:11:15.132
negative eight and negative 10.

00:11:15.132 --> 00:11:18.313
We just figured that out,
that was 93 out of the 1000

00:11:18.313 --> 00:11:22.704
re-randomizations, so
it's a 9.3 percent chance.

00:11:22.704 --> 00:11:25.153
If you set five percent as
your cut-off for statistical

00:11:25.153 --> 00:11:28.929
significance, you say, "Okay,
this doesn't quite meet my

00:11:28.929 --> 00:11:32.078
"cut-off so maybe this
is not a statistically

00:11:32.078 --> 00:11:35.089
"significant result."


Rotation by radians and quadrants

WEBVTT
Kind: captions
Language: en

00:00:00.418 --> 00:00:01.900
- [Voiceover] What I
want to do in this video

00:00:01.900 --> 00:00:05.190
is get some practice, or
become familiar with what

00:00:05.190 --> 00:00:08.182
different angle measures in
radians actually represent.

00:00:08.182 --> 00:00:10.095
And to get our familiarity,

00:00:10.095 --> 00:00:13.788
we're gonna start with a ray
that starts at the origin,

00:00:13.788 --> 00:00:16.067
and moves along, and...

00:00:16.067 --> 00:00:19.213
Not moves, and points
along the positive X axis.

00:00:19.213 --> 00:00:21.076
We're gonna start with this magenta ray,

00:00:21.076 --> 00:00:25.753
and we're gonna rotate
it around the origin

00:00:25.753 --> 00:00:29.098
counterclockwise by
different angle measures.

00:00:29.098 --> 00:00:31.593
And think about what
quadrant do we fall into

00:00:31.593 --> 00:00:35.149
if we start with this and we
were to rotate counterclockwise

00:00:35.149 --> 00:00:36.636
by three pi over five radians?

00:00:36.636 --> 00:00:37.840
And then, if we start with this,

00:00:37.840 --> 00:00:39.281
and we were to rotate counterclockwise

00:00:39.281 --> 00:00:41.381
by two pi over seven radians?

00:00:41.381 --> 00:00:42.550
Or, if we were to start with this,

00:00:42.550 --> 00:00:45.329
and then rotate counterclockwise
by three radians?

00:00:45.329 --> 00:00:47.571
We encourage you to pause
the video and think about,

00:00:47.571 --> 00:00:49.032
starting with this,

00:00:49.032 --> 00:00:52.891
if we were to rotate
counterclockwise by each of these,

00:00:52.891 --> 00:00:56.433
what quadrant are we going to end up in?

00:00:57.128 --> 00:00:59.422
Assume you've paused the video,

00:00:59.422 --> 00:01:00.538
and you've tried it out on your own,

00:01:00.538 --> 00:01:03.651
so let's try this first
one, three pi over five.

00:01:03.651 --> 00:01:06.700
Three pi over five, so
we're gonna start rotating.

00:01:07.363 --> 00:01:09.721
If we go straight up,

00:01:09.721 --> 00:01:11.763
if we rotate it, essentially,

00:01:11.763 --> 00:01:12.967
if you want to think in degrees,

00:01:12.967 --> 00:01:15.483
if you rotate it
counterclockwise 90 degrees,

00:01:15.483 --> 00:01:18.100
that is going to get us to pi over two.

00:01:18.827 --> 00:01:20.531
That would have been

00:01:20.562 --> 00:01:24.833
a counterclockwise rotation
of pi over two radians.

00:01:25.281 --> 00:01:27.837
Now is three pi over five
greater or less than that?

00:01:28.207 --> 00:01:29.866
Well, three pi over five,

00:01:30.385 --> 00:01:32.176
three pi over five

00:01:32.176 --> 00:01:33.620
is greater than,

00:01:33.620 --> 00:01:35.404
or I guess another way I can say it is,

00:01:35.404 --> 00:01:39.192
three pi over six is less
than three pi over five.

00:01:39.192 --> 00:01:40.419
You make the denominator smaller,

00:01:40.419 --> 00:01:42.671
making the fraction larger.

00:01:43.332 --> 00:01:46.069
Three pi over six is the
same thing as pi over two.

00:01:46.069 --> 00:01:47.296
So, let me write it this way.

00:01:47.788 --> 00:01:51.322
Pi over two is less
than three pi over five.

00:01:51.860 --> 00:01:53.374
It's definitely past this.

00:01:53.374 --> 00:01:55.017
We're gonna go past this.

00:01:55.017 --> 00:01:56.921
Does that get us all the way over here?

00:01:56.921 --> 00:01:58.407
If we were to go, essentially,

00:01:58.407 --> 00:02:00.624
be pointed in the opposite direction.

00:02:00.624 --> 00:02:02.827
Instead of being pointed to
the right, making a full,

00:02:02.830 --> 00:02:05.838
I guess you could say 180 degree
counterclockwise rotation,

00:02:05.838 --> 00:02:07.892
that would be pi radians.

00:02:07.892 --> 00:02:08.657
That would be pi radians.

00:02:08.657 --> 00:02:10.353
But this thing is less than pi.

00:02:10.353 --> 00:02:12.410
Pi would be five pi over five.

00:02:12.595 --> 00:02:15.508
This is less than pi radians.

00:02:15.508 --> 00:02:17.459
We are going to sit,

00:02:17.459 --> 00:02:19.897
we are going to sit someplace,

00:02:19.897 --> 00:02:22.637
someplace, and I'm just estimating it.

00:02:22.637 --> 00:02:25.993
We are gonna sit someplace like that.

00:02:26.270 --> 00:02:29.680
And so we are going to sit
in the second quadrant.

00:02:30.280 --> 00:02:32.093
Let's think about two pi seven.

00:02:32.550 --> 00:02:36.045
Two pi over seven, do we
even get past pi over two?

00:02:36.276 --> 00:02:41.276
Pi over two here would
be 3.5 pi over seven.

00:02:42.174 --> 00:02:43.846
We don't even get to pi over two.

00:02:43.846 --> 00:02:44.995
We're gonna end up,

00:02:44.995 --> 00:02:46.714
we're gonna end up someplace,

00:02:46.714 --> 00:02:48.315
someplace over here.

00:02:48.315 --> 00:02:50.428
This thing is, it's greater than zero,

00:02:50.428 --> 00:02:52.948
so we're gonna definitely
start moving counterclockwise,

00:02:52.948 --> 00:02:54.054
but we're not even gonna get to...

00:02:54.054 --> 00:02:56.034
This thing is less than pi over two.

00:02:56.034 --> 00:02:58.993
This is gonna throw us
in the first quadrant.

00:02:59.517 --> 00:03:01.769
What about three radians?

00:03:01.769 --> 00:03:03.278
One way to think about it is,

00:03:03.278 --> 00:03:05.865
three is a little bit less than pi.

00:03:06.403 --> 00:03:10.264
Right? Three is less than pi

00:03:10.264 --> 00:03:12.438
but it's greater than pi over two.

00:03:12.500 --> 00:03:13.294
How do we know that?

00:03:13.294 --> 00:03:18.294
Well, pi is approximately 3.14159

00:03:18.517 --> 00:03:20.728
and it just keeps going on and on forever.

00:03:20.728 --> 00:03:22.388
So, three is definitely closer to that

00:03:22.388 --> 00:03:24.258
than it is to half of that.

00:03:24.258 --> 00:03:27.776
It's going to be between
pi over two, and pi.

00:03:27.776 --> 00:03:31.947
It's gonna be, if we start
with this magenta ray,

00:03:31.947 --> 00:03:35.752
we rotate counterclockwise
by three radians,

00:03:35.752 --> 00:03:37.253
we are gonna get...

00:03:37.253 --> 00:03:38.594
Actually, it's probably gonna be,

00:03:38.594 --> 00:03:39.756
it's gonna look something,

00:03:39.756 --> 00:03:41.254
it's gonna be something like this.

00:03:41.254 --> 00:03:43.308
But for the sake of this exercise,

00:03:43.308 --> 00:03:45.793
we have gotten ourselves, once again,

00:03:45.793 --> 00:03:48.800
into the second quadrant.


Comparing models to fit data

WEBVTT
Kind: captions
Language: en

00:00:00.236 --> 00:00:03.575
- Christine works in a
movie store in her hometown.

00:00:03.575 --> 00:00:05.956
Using the store's total selection,

00:00:05.956 --> 00:00:09.009
she documented the price
of each movie title

00:00:09.009 --> 00:00:11.537
and how many years it has been since

00:00:11.537 --> 00:00:13.893
it was featured in movie theatres.

00:00:13.893 --> 00:00:16.424
She plotted the points below.

00:00:16.424 --> 00:00:18.970
So let's see what's going on below here.

00:00:19.554 --> 00:00:21.279
Looks like there's two
curves that she tries to fit.

00:00:21.279 --> 00:00:23.718
I'm assuming we're going to
read about it in a second.

00:00:23.718 --> 00:00:25.877
But these blue points are the data points.

00:00:25.877 --> 00:00:29.457
So, for example, this
data point right over here

00:00:29.457 --> 00:00:33.852
shows a movie that the
title costs six dollars,

00:00:33.852 --> 00:00:37.076
and it has been released
for almost two years,

00:00:37.076 --> 00:00:38.526
a little under two years.

00:00:38.526 --> 00:00:40.551
This data point right
over here, this is a movie

00:00:40.551 --> 00:00:43.543
that has been released
for almost four years,

00:00:43.543 --> 00:00:46.076
looks like maybe three
and three quarters years.

00:00:46.076 --> 00:00:49.355
And they're selling that,
looks like for a dollar

00:00:49.355 --> 00:00:50.748
or even a little bit less than a dollar.

00:00:50.748 --> 00:00:52.648
So those are her data points.

00:00:52.648 --> 00:00:53.958
So once, again, she documented the price

00:00:53.958 --> 00:00:58.958
of each movie title as
a function of how many

00:00:59.539 --> 00:01:03.043
years it's been since it was
featured in movie theatres.

00:01:03.043 --> 00:01:05.856
She is looking for a function
that models her data.

00:01:05.856 --> 00:01:08.586
Since the trend of the data
is decreasing and convex,

00:01:08.586 --> 00:01:11.430
and you see it here, it's
definitely decreasing,

00:01:11.430 --> 00:01:15.361
and convex, it's opening
upwards, if you imagine a curve,

00:01:15.361 --> 00:01:18.378
it looks like it's opening
upwards a little bit like that,

00:01:18.378 --> 00:01:22.883
so decreasing and convex,
she found a decreasing convex

00:01:22.883 --> 00:01:27.883
exponential model and a
decreasing convex quadratic model.

00:01:27.899 --> 00:01:32.243
Which of the following
functions better fits the data?

00:01:32.703 --> 00:01:34.851
Function A, this is an exponential.

00:01:34.851 --> 00:01:37.999
This is the one in green right over here.

00:01:37.999 --> 00:01:42.999
And Function B, this one right
over here is a quadratic.

00:01:43.747 --> 00:01:48.180
And you can see this one in purple.

00:01:48.180 --> 00:01:51.612
And so, which one of those
better fits the data?

00:01:52.642 --> 00:01:55.098
If we look at what's going on here,

00:01:55.098 --> 00:01:58.083
the green function, the exponential one,

00:01:58.083 --> 00:02:00.764
most of the data points
for any given duration,

00:02:00.764 --> 00:02:02.899
for how long the title's been out,

00:02:02.899 --> 00:02:05.601
it looks like it's
consistently underestimating.

00:02:05.601 --> 00:02:09.123
That it's always, the model's guess,

00:02:09.123 --> 00:02:11.816
or what the model would say the price is,

00:02:11.816 --> 00:02:16.530
is always, essentially except
for only one data point

00:02:16.530 --> 00:02:18.822
right over here, for all
of these other data points

00:02:18.822 --> 00:02:21.313
it's underestimating
what the price would be.

00:02:21.313 --> 00:02:25.173
The purple model or the purple
function right over here,

00:02:26.033 --> 00:02:29.090
it has more of a balance
between overestimating,

00:02:29.090 --> 00:02:33.060
right over here, it's
overestimating by a little bit,

00:02:33.060 --> 00:02:34.639
and underestimating.

00:02:34.639 --> 00:02:36.333
And its underestimates are closer,

00:02:36.333 --> 00:02:39.844
and its overestimates are
closer than this green model.

00:02:39.844 --> 00:02:44.844
So I would say that Function B
is definitely a better model.

00:02:45.657 --> 00:02:47.977
Use the function of best
fit, so we're going to say

00:02:47.977 --> 00:02:50.907
Function B, to predict
the price of a movie

00:02:50.907 --> 00:02:54.281
that was featured in
theatres 5.5 years ago.

00:02:54.281 --> 00:02:56.604
Round your answer to the nearest cent.

00:02:56.604 --> 00:02:59.677
So 5.5 years ago, that's
going to be right over here.

00:02:59.677 --> 00:03:02.477
We're going to go to Function
B, which is this purple one.

00:03:02.477 --> 00:03:05.307
So it's going to be under a dollar.

00:03:05.307 --> 00:03:07.083
But we want to get something
to the nearest cent,

00:03:07.083 --> 00:03:11.417
so let's actually use the actual
definition of the function.

00:03:11.417 --> 00:03:14.486
So this is price as a function of how long

00:03:14.486 --> 00:03:16.229
the movie has been released.

00:03:16.229 --> 00:03:18.320
Where x is how long it's been released,

00:03:18.320 --> 00:03:20.283
and y is its price.

00:03:21.652 --> 00:03:25.475
If x is 5.5, let's figure
out what y is going to be.

00:03:26.628 --> 00:03:31.628
So y is going to be equal
to 0.5 times x squared.

00:03:32.688 --> 00:03:37.688
So x is 5.5 squared.

00:03:40.388 --> 00:03:43.957
So then we have minus five times x again.

00:03:43.957 --> 00:03:46.427
So minus five times 5.5.

00:03:46.427 --> 00:03:48.890
And then we have plus 13.

00:03:48.890 --> 00:03:51.189
And what does that get us?

00:03:51.189 --> 00:03:53.790
That gets us 62 1/2 cents.

00:03:53.790 --> 00:03:56.448
If we were to round our
answer to the nearest cent,

00:03:56.448 --> 00:04:00.053
that's going to be 63 cents.

00:04:00.391 --> 00:04:02.298
And we got it right.


Rotating 2D shapes in 3D

WEBVTT
Kind: captions
Language: en

00:00:00.325 --> 00:00:01.123
- What I want to do in this video

00:00:01.123 --> 00:00:04.284
is get some practice
visualizing what happens

00:00:04.284 --> 00:00:07.213
if we were to try to rotate
two dimensional shapes

00:00:07.213 --> 00:00:08.505
in three dimensions.

00:00:08.505 --> 00:00:09.757
Well what do I mean by that?

00:00:09.922 --> 00:00:11.963
Let's say I started with a right triangle.

00:00:12.632 --> 00:00:16.112
So let's say my right
triangle looks like this.

00:00:17.461 --> 00:00:20.036
So let's say it looks like that.

00:00:20.571 --> 00:00:21.684
Right over there.

00:00:22.747 --> 00:00:24.749
And so this is a right angle.

00:00:24.983 --> 00:00:28.721
And let's say that this
width right over here

00:00:28.721 --> 00:00:33.721
is three units and let's say
that this length is five units

00:00:34.340 --> 00:00:35.987
and now I'm gonna do
something interesting.

00:00:36.422 --> 00:00:39.696
I'm gonna take this two
dimensional right triangle

00:00:39.696 --> 00:00:42.064
and I'm gonna try to rotate
it in three dimensions

00:00:42.064 --> 00:00:47.064
around this line, around
the line that I'm doing

00:00:47.224 --> 00:00:48.759
as a dotted magenta line.

00:00:48.759 --> 00:00:52.519
So I'm gonna rotate it around
this line right over there.

00:00:52.822 --> 00:00:56.073
So if I were to rotate
it around this line,

00:00:56.073 --> 00:00:58.370
what type of a shape am I going to get?

00:00:58.604 --> 00:00:59.455
And I encourage you --

00:00:59.455 --> 00:01:00.887
It's going to be a
three dimensional shape.

00:01:00.887 --> 00:01:02.041
I encourage you to think about it,

00:01:02.041 --> 00:01:03.595
maybe take out a piece of paper, draw it,

00:01:03.595 --> 00:01:05.474
or just try to imagine it in your head.

00:01:06.946 --> 00:01:08.456
Well to think about it
in three dimensions,

00:01:08.456 --> 00:01:09.206
what I'm going to do is try to

00:01:09.206 --> 00:01:10.638
look at this thing in three dimensions.

00:01:10.638 --> 00:01:12.379
So let me draw this same line

00:01:12.379 --> 00:01:13.610
but I'm gonna draw it at an angle

00:01:13.610 --> 00:01:15.884
so we can visualize the whole
thing in three dimensions.

00:01:16.959 --> 00:01:18.894
So imagine if this was
sitting on the ground.

00:01:19.469 --> 00:01:22.021
So that's our magenta line,

00:01:22.301 --> 00:01:24.552
and then I can draw my triangle.

00:01:24.662 --> 00:01:27.005
So my triangle would
look something like this.

00:01:27.541 --> 00:01:30.528
So it would look like this.

00:01:32.038 --> 00:01:34.545
So once again this is five units,

00:01:34.545 --> 00:01:37.469
this is three units,
this a right triangle.

00:01:37.994 --> 00:01:39.429
I'm gonna rotate it around the line,

00:01:39.429 --> 00:01:40.541
so what's it gonna look like?

00:01:41.175 --> 00:01:43.252
Well this and this right over here

00:01:43.252 --> 00:01:46.015
is gonna rotate around and it's gonna form

00:01:46.015 --> 00:01:49.258
a circle with a radius of three, right?

00:01:49.258 --> 00:01:51.649
So it's gonna form, so it intersects,

00:01:51.649 --> 00:01:53.923
if that was on the ground
it's gonna be three again.

00:01:54.273 --> 00:01:56.918
And let me draw it down so
it's gonna keep going down.

00:01:57.330 --> 00:01:58.087
Whoops.

00:01:58.243 --> 00:02:00.743
We don't want to press the wrong button.

00:02:00.743 --> 00:02:02.500
So it's gonna look something like this.

00:02:02.500 --> 00:02:04.293
That's what the base is gonna look like.

00:02:05.151 --> 00:02:06.168
But then this end right over here

00:02:06.168 --> 00:02:07.306
is just gonna stay at a point

00:02:07.306 --> 00:02:09.271
because this is right
on that magenta line.

00:02:09.271 --> 00:02:10.756
So it's gonna stay at a point.

00:02:10.990 --> 00:02:12.522
And so if you were to
look at the intersect

00:02:12.522 --> 00:02:14.447
so it would look something like this.

00:02:14.597 --> 00:02:16.786
So it would look like
this and then you'd have

00:02:16.786 --> 00:02:19.136
another thing that goes like this

00:02:19.136 --> 00:02:21.422
and so if you were to
take a section like this

00:02:21.422 --> 00:02:23.535
it would have a little smaller circle here

00:02:23.535 --> 00:02:25.902
based on what this distance is.

00:02:26.647 --> 00:02:30.638
So what is the shape, what is
the shape that I am drawing?

00:02:31.310 --> 00:02:33.742
Well what you see, what
it is, it's a cone.

00:02:34.301 --> 00:02:35.462
It's a cone and if I shade it in

00:02:35.462 --> 00:02:37.335
you might see the cone
a little bit better.

00:02:37.335 --> 00:02:41.884
So let me shade it in so you see the cone.

00:02:42.652 --> 00:02:45.938
So what you end up getting is a cone

00:02:45.938 --> 00:02:48.650
where it's base, so I'm shading it in

00:02:48.650 --> 00:02:51.529
so that hopefully helps a little bit,

00:02:51.529 --> 00:02:52.899
so what you end up getting is a cone

00:02:52.899 --> 00:02:56.427
where the base has a
radius of three units.

00:02:56.867 --> 00:02:58.052
So let me draw this.

00:02:58.486 --> 00:03:01.559
This right over here is
the radius of the base

00:03:01.559 --> 00:03:03.200
and it is three units.

00:03:03.200 --> 00:03:04.707
I could also draw it like this.

00:03:05.127 --> 00:03:07.053
So the cone is gonna look like this.

00:03:07.681 --> 00:03:10.403
And this is the tip of the cone

00:03:10.403 --> 00:03:15.258
and it's gonna look just like this.

00:03:15.258 --> 00:03:16.860
And once again let me
shade it a little bit

00:03:16.860 --> 00:03:18.849
so that you can appreciate that

00:03:18.849 --> 00:03:20.798
this is a three dimensional shape.

00:03:21.287 --> 00:03:24.190
So draw the cone so you can shade it

00:03:24.943 --> 00:03:26.914
and we can even construct the original

00:03:26.914 --> 00:03:29.755
so that, well or we can
construct the original shape

00:03:29.755 --> 00:03:32.195
so you see how it
constructs so it makes this,

00:03:32.195 --> 00:03:34.026
the line, that magenta line,

00:03:34.026 --> 00:03:35.719
is gonna do this type of thing.

00:03:36.116 --> 00:03:38.358
It's gonna go through
the center of the base,

00:03:38.561 --> 00:03:41.021
it's gonna go through
the center of the base

00:03:41.255 --> 00:03:42.971
just like that.

00:03:43.825 --> 00:03:47.779
And our original shape, our
original right triangle,

00:03:47.779 --> 00:03:50.485
if you just took a cross section of it

00:03:50.485 --> 00:03:55.287
that included that line you
would have your original shape.

00:03:55.287 --> 00:03:56.562
Let me do this in orange.

00:03:57.167 --> 00:04:01.734
So the original shape is right over there.

00:04:01.734 --> 00:04:02.755
So what do you get?

00:04:02.755 --> 00:04:07.755
You get a cone where the radius
of the base is three units.

00:04:08.699 --> 00:04:09.719
Interesting.


Arc length as fraction of circumference

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.202
- Let's say that I have a circle.

00:00:02.202 --> 00:00:05.667
My best attempt to draw a
reasonably perfect circle.

00:00:05.667 --> 00:00:09.700
So, there you go, not too
bad, it's a little bit

00:00:09.700 --> 00:00:11.366
of a hairy circle but you get the idea.

00:00:11.366 --> 00:00:16.067
So, this is a circle, this
is the center of the circle,

00:00:16.067 --> 00:00:19.267
and let's say that I have
an arc along this circle.

00:00:19.267 --> 00:00:21.900
So, I'll do the arc in green.

00:00:21.900 --> 00:00:26.700
So, I have an arc that
is part of the circle,

00:00:26.700 --> 00:00:30.800
and it subtends an
angle, so that's my arc.

00:00:30.800 --> 00:00:35.400
Right over there, and
it subtends an angle,

00:00:35.400 --> 00:00:38.068
and the angle that it subtends,
so what I mean subtends,

00:00:38.068 --> 00:00:40.033
you take each of the endpoints of the arc,

00:00:40.033 --> 00:00:42.733
go to the center of the circle,

00:00:42.733 --> 00:00:47.733
go to the center of the
circle just like this,

00:00:48.966 --> 00:00:53.966
and so it subtends angle
theta, right over here,

00:00:54.100 --> 00:00:57.033
so it subtends angle theta,
and let's say that we know

00:00:57.033 --> 00:01:02.033
that angle theta is equal to two radians.

00:01:02.367 --> 00:01:06.300
So my question to you is what fraction

00:01:06.300 --> 00:01:11.300
of the entire circumference
is this green arc?

00:01:11.333 --> 00:01:16.333
What fraction of the entire
circumference is this green arc?

00:01:17.267 --> 00:01:21.000
And like always, pause the
video, and give it a go.

00:01:21.000 --> 00:01:24.600
(laughs) All right, so let's
think through it a little bit.

00:01:24.600 --> 00:01:26.067
So, you might say well how do I know that,

00:01:26.067 --> 00:01:28.300
I don't know what the
radius of this thing is,

00:01:28.300 --> 00:01:32.166
I don't, how do I think through this?

00:01:32.166 --> 00:01:36.100
And we just have to remind
ourselves what radians mean,

00:01:36.100 --> 00:01:37.633
what radians mean.

00:01:37.633 --> 00:01:42.133
If an arc subtends the
angle of two radians,

00:01:42.133 --> 00:01:47.133
that means that the arc itself
is two "radiuseseses" long.

00:01:47.600 --> 00:01:49.066
(laughs) So, this right over here,

00:01:49.066 --> 00:01:51.766
let me make this a
little clearer, so this,

00:01:51.766 --> 00:01:56.533
if the radius is r, if this radius is,

00:01:56.533 --> 00:01:58.866
I already used that
color, if this radius...

00:01:58.866 --> 00:02:01.293
I have trouble switching
colors (laughs) all right.

00:02:01.293 --> 00:02:05.133
If this radius is length r,

00:02:05.133 --> 00:02:09.600
then the length, if this
angle is two radians,

00:02:09.600 --> 00:02:12.833
then the arc that
subtends it is going to be

00:02:12.833 --> 00:02:17.600
two radiuses long, so this
length right over here,

00:02:17.600 --> 00:02:20.333
is two radiuses.

00:02:20.333 --> 00:02:23.700
Now, what fraction of the
entire circumference is that?

00:02:23.700 --> 00:02:26.134
Well, the entire circumference, we know,

00:02:26.134 --> 00:02:27.400
we know this from basic geometry,

00:02:27.400 --> 00:02:31.233
the entire circumference
is two pi times the radius,

00:02:31.233 --> 00:02:34.800
or you can say it's two pi
radii, two pi "radiuseses",

00:02:34.800 --> 00:02:36.700
(laughs) two pi radii is
the correct way to say it.

00:02:36.700 --> 00:02:38.233
So, what fraction is it?

00:02:38.233 --> 00:02:41.800
It's two radii, it's two radii,

00:02:41.800 --> 00:02:46.800
over two pi radii, over two pi radii,

00:02:47.068 --> 00:02:51.666
twos cancel out, rs cancel
out, and so it is one "pith",

00:02:51.666 --> 00:02:53.172
(laughs) I guess you could say,

00:02:53.172 --> 00:02:57.172
it is one over pi of
the total circumference.


Modeling with combined functions

WEBVTT
Kind: captions
Language: en

00:00:00.493 --> 00:00:03.032
- [Voiceover] Ify is
building a tree tower,

00:00:03.032 --> 00:00:06.933
which is a tower built on top of a tree.

00:00:06.933 --> 00:00:09.831
The tree is currently five meters tall

00:00:09.831 --> 00:00:13.181
and Ify has found, I don't
know if it's Ify or Ify,

00:00:13.181 --> 00:00:15.015
and Ify has found that it is growing

00:00:15.015 --> 00:00:16.945
by zero point one meters a month

00:00:16.945 --> 00:00:19.225
or a tenth of a meter a month.

00:00:19.225 --> 00:00:21.862
The tower is currently two meters tall.

00:00:21.862 --> 00:00:24.226
So this tower that sits
on top of the tree,

00:00:24.226 --> 00:00:27.948
is two meters tall
currently and Ify, or Ify,

00:00:27.948 --> 00:00:31.509
adds to it at about zero
point two meters a month.

00:00:31.509 --> 00:00:34.695
So, I guess, he or she, I don't know,

00:00:34.710 --> 00:00:36.998
Ify, let's just say, "he."

00:00:36.998 --> 00:00:40.107
Is continuously building this tower

00:00:40.107 --> 00:00:43.632
on this continuously growing
tree, which is fascinating.

00:00:43.632 --> 00:00:47.204
Alright, the function A
of m, returns the tree's

00:00:47.204 --> 00:00:50.590
height in meters and months from now.

00:00:50.590 --> 00:00:51.827
Fascinating.

00:00:51.827 --> 00:00:54.390
The function B of m,
returns the tower's height

00:00:54.390 --> 00:00:56.051
in meters and months from now.

00:00:56.051 --> 00:00:58.453
So, this is the tree's height,
A of m, is the tree's height.

00:00:58.453 --> 00:01:01.215
B of m is the tower's height.

00:01:01.215 --> 00:01:03.981
Find the formula of the two functions.

00:01:03.981 --> 00:01:06.337
So, A of m, so they tell us the tree

00:01:06.337 --> 00:01:08.185
is currently five meters tall.

00:01:08.185 --> 00:01:11.293
So, it's going to be five
meters tall right at the start.

00:01:11.293 --> 00:01:16.293
And then every month it is
growing by zero point one meters,

00:01:16.874 --> 00:01:21.874
so it's going to be five
plus zero point one times m.

00:01:22.183 --> 00:01:24.642
And this m here, this is not meters,

00:01:24.642 --> 00:01:25.718
this is actually the months.

00:01:25.718 --> 00:01:29.461
Remember, m returns, m
is the number of months.

00:01:29.461 --> 00:01:31.699
So after zero months,which is right now,

00:01:31.699 --> 00:01:33.527
well this is just going to be five.

00:01:33.527 --> 00:01:35.417
After one month, it's
going to be five point one.

00:01:35.417 --> 00:01:37.368
After two months it's
going to be five point two.

00:01:37.368 --> 00:01:39.593
Which is exactly what we want.

00:01:39.593 --> 00:01:41.946
Alright, now let's think about the tower.

00:01:41.946 --> 00:01:43.817
So, the formula for B of m,

00:01:43.817 --> 00:01:46.365
so the tower is currently two meters tall.

00:01:46.365 --> 00:01:47.941
So, it's currently two meters tall

00:01:47.941 --> 00:01:52.941
and it grows at two tenths
of a meter per month.

00:01:54.403 --> 00:01:56.719
So, two tenths times the number of months.

00:01:56.719 --> 00:01:59.489
And once again, this m right
over here is not meters,

00:01:59.489 --> 00:02:00.477
I'm not writing the units here,

00:02:00.477 --> 00:02:04.060
we're just assuming whatever
this returns is in meters.

00:02:04.060 --> 00:02:07.667
This m right over here is the
number of months that pass by,

00:02:07.667 --> 00:02:10.550
the number of months from now.

00:02:10.550 --> 00:02:14.085
Alright, the function C of m
returns the vertical distance

00:02:14.085 --> 00:02:17.188
between the ground and
the top end of the tower.

00:02:17.188 --> 00:02:19.473
Makes sense, that would be
from the bottom of the tree

00:02:19.473 --> 00:02:21.511
to the top of the tower.

00:02:21.511 --> 00:02:25.927
Wright the formula, C of m,
in terms of A of m and B of m.

00:02:25.927 --> 00:02:28.786
Well the total height is going
to be the height of the tree,

00:02:28.786 --> 00:02:33.462
which is A of m plus
the height of the tower.

00:02:33.462 --> 00:02:37.346
Plus B of m, plus B of m.

00:02:37.346 --> 00:02:39.325
That's what C of m is going to be.

00:02:39.325 --> 00:02:39.899
And then they say,

00:02:39.899 --> 00:02:42.615
"Wright the formula of
C of m in terms of m."

00:02:42.615 --> 00:02:44.301
Well, we just need to
add these two functions.

00:02:44.301 --> 00:02:46.543
So, if we add five plus zero point one m

00:02:46.543 --> 00:02:49.040
to two plus zero point two m,

00:02:49.040 --> 00:02:51.671
that's going to be, we
could add five plus two,

00:02:51.671 --> 00:02:54.859
and we'er going to get seven, plus,

00:02:54.859 --> 00:02:57.577
and if I have point one m
and I add another point 2 m,

00:02:57.577 --> 00:02:59.000
that's going to be point 3 m.

00:02:59.000 --> 00:03:04.000
Zero point 3 m, and we are done.

00:03:04.544 --> 00:03:06.609
And we got it right.