Area models to visualize division using place value

WEBVTT
Kind: captions
Language: en

00:00:00.413 --> 00:00:02.061
- [Voiceover] Let's say
that this rectangle,

00:00:02.061 --> 00:00:04.530
this green rectangle right over here,

00:00:04.530 --> 00:00:06.857
let's say it had an area of

00:00:06.857 --> 00:00:10.647
268 square units,

00:00:10.647 --> 00:00:12.253
whatever those units are.

00:00:12.253 --> 00:00:14.366
You could imagine them
being square centimeters,

00:00:14.366 --> 00:00:16.862
or if you imagine this being a big field

00:00:16.862 --> 00:00:17.942
that you're looking at from space,

00:00:17.942 --> 00:00:19.928
it could be square miles or something.

00:00:19.928 --> 00:00:23.410
So I'm just gonna write square units.

00:00:23.410 --> 00:00:24.731
And lets say you knew the dimensions

00:00:24.731 --> 00:00:26.766
of one side of the field.

00:00:26.766 --> 00:00:30.439
So let's say that you
knew, you knew that this...

00:00:30.439 --> 00:00:31.793
let me just hit a color,

00:00:31.793 --> 00:00:33.290
let's say you knew

00:00:33.290 --> 00:00:36.551
that this side of the
field right over here,

00:00:36.551 --> 00:00:40.406
the length of this side
of the field is two units.

00:00:40.406 --> 00:00:41.665
Two units.

00:00:41.665 --> 00:00:43.366
And I haven't really drawn this to scale.

00:00:43.366 --> 00:00:45.491
If I wanted to draw it to
scale, it would be like,

00:00:45.491 --> 00:00:47.934
it would be much shorter,
it would be like that.

00:00:47.934 --> 00:00:49.589
But you would have trouble
seeing the rectangle then.

00:00:49.589 --> 00:00:51.250
But let's just assume it's two units.

00:00:51.250 --> 00:00:54.303
So if you know the whole
area is 268 square units,

00:00:54.303 --> 00:00:56.020
and one side is two units,

00:00:56.020 --> 00:00:58.330
what's the other side going to be?

00:00:58.330 --> 00:01:00.710
What is the other side going to be?

00:01:00.710 --> 00:01:03.567
What is, what is...

00:01:03.567 --> 00:01:05.006
let me pick another color.

00:01:05.006 --> 00:01:06.979
What is this side,

00:01:06.979 --> 00:01:10.706
what is this side of this rectangular,

00:01:10.706 --> 00:01:13.504
of this rectangle, this
field, whatever this might be,

00:01:13.504 --> 00:01:16.359
what is the length of that side?

00:01:16.359 --> 00:01:19.459
Well if you multiply these
two sides, you get the area,

00:01:19.459 --> 00:01:20.980
so if you start with the area,

00:01:20.980 --> 00:01:24.103
if you start with 268,

00:01:24.103 --> 00:01:27.864
268, and you divide by the other side,

00:01:27.864 --> 00:01:30.661
divide by the other side,

00:01:30.661 --> 00:01:33.028
you're going to get the length
of this side right over here.

00:01:33.028 --> 00:01:35.028
So if we wanted to figure
out the length of that side,

00:01:35.028 --> 00:01:38.858
it would be 268 divided by two.

00:01:38.858 --> 00:01:40.867
And you've already seen,
or we've already seen,

00:01:40.867 --> 00:01:45.139
multiple ways to figure out
what 268 divided by two is,

00:01:45.139 --> 00:01:47.844
but the whole reason of
me drawing this rectangle,

00:01:47.844 --> 00:01:49.585
or this aerial view of this field,

00:01:49.585 --> 00:01:50.978
or whatever you want to call it,

00:01:50.978 --> 00:01:54.206
is so that we visualize it using area.

00:01:54.206 --> 00:01:55.518
And so one way to do it is,

00:01:55.518 --> 00:01:59.359
is to break up this 268 square unit area

00:01:59.359 --> 00:02:04.359
into areas that are easier
to imagine dividing by two.

00:02:04.840 --> 00:02:09.576
So here I have the same field,
but I've just broken it up.

00:02:09.576 --> 00:02:11.097
So it's the same field,

00:02:11.097 --> 00:02:15.069
this dimension right over
here is still two units,

00:02:15.069 --> 00:02:16.459
two units,

00:02:16.459 --> 00:02:18.062
but I've broken it up.

00:02:18.062 --> 00:02:21.660
This blue area is 200...

00:02:21.660 --> 00:02:24.095
let me do that in another color.

00:02:24.095 --> 00:02:28.859
So this blue area is 200 units,

00:02:28.859 --> 00:02:31.877
this yellow area is 60 units,

00:02:31.877 --> 00:02:36.660
and this magenta area is eight units.

00:02:36.660 --> 00:02:38.633
Now why is that useful?

00:02:38.633 --> 00:02:42.615
Well, now each of these, it's
much easier to divide by two.

00:02:42.615 --> 00:02:47.352
All I did is I took the 268, and I said,

00:02:47.352 --> 00:02:50.253
"Well look, this is the same thing as

00:02:50.253 --> 00:02:53.992
"200 plus 60...

00:02:53.992 --> 00:02:57.173
"plus 60, plus eight."

00:02:57.173 --> 00:02:59.518
I just broke up the 268
into things that are

00:02:59.518 --> 00:03:03.741
easier to divide by two, and
now I can take those things

00:03:03.741 --> 00:03:05.678
and divide by two.

00:03:05.678 --> 00:03:08.340
I can take each of them and divide by two.

00:03:08.340 --> 00:03:10.234
So what is, what is this,

00:03:10.234 --> 00:03:14.299
what is this dimension going
to be, right over here?

00:03:14.299 --> 00:03:16.200
Actually let me do that
in a different color.

00:03:16.200 --> 00:03:19.730
What is this dimension
right over here going to be?

00:03:20.438 --> 00:03:23.070
Well two times that is
going to be a hundred,

00:03:23.070 --> 00:03:24.801
so this is going to be 100.

00:03:24.801 --> 00:03:25.989
And how did we get that?

00:03:25.989 --> 00:03:28.542
Well we got that by 200 divided by two.

00:03:28.542 --> 00:03:31.526
200 divided by two is 100.

00:03:31.526 --> 00:03:33.771
What's 60 divided by two?

00:03:33.771 --> 00:03:37.968
Well, 60 divided by two is going to be 30.

00:03:37.968 --> 00:03:40.129
So this part of the field

00:03:40.129 --> 00:03:42.102
is going to be 30 in that direction

00:03:42.102 --> 00:03:43.208
and two in this direction.

00:03:43.208 --> 00:03:46.640
And once again, not drawn to scale.

00:03:46.640 --> 00:03:49.184
And then finally, what's
this section going to be?

00:03:49.184 --> 00:03:52.457
It's going to be, it's going to be,

00:03:52.457 --> 00:03:55.720
eight divided by two, which is four.

00:03:55.720 --> 00:04:00.038
Notice, hundred times two
is 200, 30 times two is 60,

00:04:00.038 --> 00:04:03.405
four times two is eight.

00:04:03.405 --> 00:04:05.382
And so this whole length up here

00:04:05.382 --> 00:04:09.140
is going to be 100, plus 30, plus four,

00:04:09.140 --> 00:04:12.146
or 134.

00:04:12.146 --> 00:04:14.143
Now we've already seen other
ways of coming up with this.

00:04:14.143 --> 00:04:17.382
You say, "Look, two hundreds
divided by two is 100,

00:04:17.382 --> 00:04:20.923
"six tens divided by two is three tens,

00:04:20.923 --> 00:04:24.081
"eight ones divided by two is four ones."

00:04:24.081 --> 00:04:26.484
And that's exactly what
we just did over here,

00:04:26.484 --> 00:04:30.506
but we visualized it using
this kind of a rectangle,

00:04:30.506 --> 00:04:32.511
breaking it up into chunks that are,

00:04:32.511 --> 00:04:34.953
maybe easier to imagine dividing by two.

00:04:34.953 --> 00:04:37.895
We broke it up into two hundreds,

00:04:37.895 --> 00:04:40.055
two hundreds right over here.

00:04:40.055 --> 00:04:43.863
We broke it up into six tens, or 60,

00:04:43.863 --> 00:04:46.799
or 60 right over here.

00:04:46.799 --> 00:04:49.826
And we broke it up into
eight ones, eight ones.

00:04:49.826 --> 00:04:52.754
We broke up the area, and then
we took each of those areas

00:04:52.754 --> 00:04:55.738
and we divided it by two to
find that part of the length.

00:04:55.738 --> 00:04:59.081
And when you add them all
together, you get the entire,

00:04:59.081 --> 00:05:01.717
you get the entire length.

00:05:01.717 --> 00:05:03.323
Now, that's just one way to do it,

00:05:03.323 --> 00:05:04.955
where you take each of the place values

00:05:04.955 --> 00:05:07.598
and you break up your field
or your rectangle like that,

00:05:07.598 --> 00:05:09.077
but you could do it other ways.

00:05:09.077 --> 00:05:11.074
You don't always have to
break it up that much.

00:05:11.074 --> 00:05:16.074
For example, let's say,
let's say that this...

00:05:16.213 --> 00:05:19.188
let's say that this area

00:05:19.188 --> 00:05:23.767
is 856 square units.

00:05:23.767 --> 00:05:27.544
Square units...

00:05:27.544 --> 00:05:31.003
Now let's say that this
dimension right over here

00:05:31.003 --> 00:05:33.801
is eight, is eight units.

00:05:33.801 --> 00:05:36.239
Is eight units.

00:05:36.239 --> 00:05:37.864
So how could we break this up

00:05:37.864 --> 00:05:39.025
so it's easier to think about

00:05:39.025 --> 00:05:41.278
what the other dimension would be,

00:05:41.278 --> 00:05:43.053
what the length is going to be?

00:05:43.053 --> 00:05:43.952
And this length, once again,

00:05:43.952 --> 00:05:47.859
is going to be 856 divided by eight.

00:05:47.859 --> 00:05:52.648
856 divided by eight

00:05:52.648 --> 00:05:55.394
is this length right over here.

00:05:55.394 --> 00:05:56.740
Well you could break it up in to

00:05:56.740 --> 00:05:58.780
eight hundreds, five tens, and six ones,

00:05:58.780 --> 00:05:59.876
but you might notice,

00:05:59.876 --> 00:06:02.940
"Well five tens, it's not so
easy to divide that by eight."

00:06:02.940 --> 00:06:05.169
But we can divide 56 by eight.

00:06:05.169 --> 00:06:08.001
We know that eight times seven is 56.

00:06:08.001 --> 00:06:10.474
So what you could do is, you
could break this rectangle

00:06:10.474 --> 00:06:13.296
into the 856 square units,

00:06:13.296 --> 00:06:17.672
you could break it up
in to 800 square units,

00:06:17.672 --> 00:06:22.653
and then another 56, and
another 56 square units.

00:06:22.653 --> 00:06:25.473
So once again, we broke it
up in to eight hundreds,

00:06:25.473 --> 00:06:28.120
and then 56 ones.

00:06:28.120 --> 00:06:30.140
And then 50...

00:06:30.140 --> 00:06:32.113
56...

00:06:32.113 --> 00:06:33.611
856 ones.

00:06:33.611 --> 00:06:35.759
Same area, I just broke it up.

00:06:35.759 --> 00:06:37.995
And now if you say, "Look, this is eight."

00:06:37.995 --> 00:06:42.074
if you say, if you say, "This
right over here is eight,

00:06:42.074 --> 00:06:47.074
"well what is going to be, what
is going to be this length?

00:06:47.112 --> 00:06:48.994
"What is this length up here going to be?"

00:06:48.994 --> 00:06:52.035
Well it's going to be 800 divided by,

00:06:52.035 --> 00:06:54.310
it's going to be 800 divided by eight.

00:06:54.310 --> 00:06:55.573
So actually, let me write it below it.

00:06:55.573 --> 00:06:57.155
So 800 divided by eight,

00:06:57.155 --> 00:06:59.232
this length right over
here is going to be,

00:06:59.232 --> 00:07:01.694
is going to be 100.

00:07:01.694 --> 00:07:02.502
How did I get that?

00:07:02.502 --> 00:07:05.618
I got eight hundreds
divided by eight is 100.

00:07:05.618 --> 00:07:08.544
800 divided by eight is 100.

00:07:08.544 --> 00:07:10.517
And then you have this other magenta part.

00:07:10.517 --> 00:07:12.711
What is 56 divided by eight?

00:07:12.711 --> 00:07:13.894
Well that's seven.

00:07:13.894 --> 00:07:16.797
56 divided by eight, this
length right over here,

00:07:16.797 --> 00:07:18.167
is going to be seven.

00:07:18.167 --> 00:07:20.036
So what's this entire length?

00:07:20.036 --> 00:07:21.649
What's this entire length?

00:07:21.649 --> 00:07:23.611
It's going to be a hundred plus seven,

00:07:23.611 --> 00:07:26.783
which is equal to 107.

00:07:26.783 --> 00:07:27.659
So once again, you could have said,

00:07:27.659 --> 00:07:30.243
"Look, eight goes into
800 a hundred times,

00:07:30.243 --> 00:07:32.587
"eight goes into 56 seven times, 107,

00:07:32.587 --> 00:07:34.945
"why did Sal draw these rectangles?"

00:07:34.945 --> 00:07:36.488
Well just so that you can help...

00:07:36.488 --> 00:07:38.578
it's sometimes helpful
to visualize, and say,

00:07:38.578 --> 00:07:41.132
"Look, you're trying,
this is an area problem.

00:07:41.132 --> 00:07:44.117
"If this is the total area,
if 856 is the total area,

00:07:44.117 --> 00:07:46.426
"and eight is one of the dimensions,

00:07:46.426 --> 00:07:48.745
"well then 856 divided by eight

00:07:48.745 --> 00:07:50.670
"is going to be the other dimension."

00:07:50.670 --> 00:07:53.559
And one way, when you're
breaking this up by place value,

00:07:53.559 --> 00:07:54.743
you're breaking up the numbers into

00:07:54.743 --> 00:07:56.914
pieces that are easier to divide by eight,

00:07:56.914 --> 00:07:59.402
you can think of it as you're
just breaking up that area

00:07:59.402 --> 00:08:02.305
and you're just trying to
figure out parts of this length.

00:08:02.305 --> 00:08:04.082
So that's a hundred right over there,

00:08:04.082 --> 00:08:06.543
and then this right over here is seven.

00:08:06.543 --> 00:08:09.468
So anyway, hopefully this broadens your

00:08:09.468 --> 00:08:11.685
visualization capabilities

00:08:11.685 --> 00:08:14.146
when you are dividing multiple,

00:08:14.146 --> 00:08:17.024
when you're doing division.


Division using understanding of place value

WEBVTT
Kind: captions
Language: en

00:00:00.074 --> 00:00:01.242
- [Voiceover] Let's say
you want to figure out

00:00:01.242 --> 00:00:04.039
what 5,600 divided by 8 is.

00:00:04.039 --> 00:00:06.721
Now, you might be tempted to
do some type of long division

00:00:06.721 --> 00:00:09.681
8 into 5,600,

00:00:09.681 --> 00:00:11.978
and you could do it this
way, and you using paper.

00:00:11.978 --> 00:00:14.159
What I want to show you
is that you could maybe

00:00:14.159 --> 00:00:15.367
do this in your head,

00:00:15.367 --> 00:00:17.561
or maybe even just use
less paper to do it.

00:00:17.837 --> 00:00:19.591
The key realization here is like look

00:00:19.591 --> 00:00:21.751
if you got these two zeros here,

00:00:21.751 --> 00:00:23.852
this is really the same thing as

00:00:23.852 --> 00:00:28.852
56 times 100 divided 8.

00:00:30.899 --> 00:00:34.550
56 times 100 is 5,600.

00:00:34.550 --> 00:00:36.350
You got the two zeros here,

00:00:36.350 --> 00:00:37.789
you got the two zeros here.

00:00:37.789 --> 00:00:38.986
Let me make it very clear.

00:00:38.986 --> 00:00:40.089
Two zeros here,

00:00:40.089 --> 00:00:41.969
you got the two zeros here.

00:00:41.969 --> 00:00:43.827
In fact, sometimes people
will call this number

00:00:43.827 --> 00:00:46.567
56 hundred, 56 hundreds.

00:00:46.567 --> 00:00:51.567
So, 5,600 is the same
thing as 56 times 100.

00:00:52.151 --> 00:00:53.919
When you write it this
way, then you might say hey

00:00:53.919 --> 00:00:57.019
instead of doing 56
times 100 divided by 8,

00:00:57.019 --> 00:00:58.598
I can switch the order.

00:00:58.598 --> 00:01:00.235
This is going to be the same thing.

00:01:00.235 --> 00:01:03.356
This is going to be the same thing as

00:01:07.586 --> 00:01:09.668
56 divided by 8

00:01:15.034 --> 00:01:16.154
times 100.

00:01:20.845 --> 00:01:22.083
Why'd I do this?

00:01:22.083 --> 00:01:23.848
Well, it's pretty straight forward

00:01:23.848 --> 00:01:25.926
if you remember your multiplication tables

00:01:25.926 --> 00:01:28.004
what 56 divided by 8 is.

00:01:28.004 --> 00:01:30.268
8 times 7 is 56.

00:01:30.944 --> 00:01:33.430
This right over here is going to be 7.

00:01:33.430 --> 00:01:35.601
Then, you're just left with figuring out

00:01:35.601 --> 00:01:37.912
what 7 times 100 is.

00:01:37.912 --> 00:01:41.114
7 times 100.

00:01:41.546 --> 00:01:43.485
Well, that's going to be 700s.

00:01:44.438 --> 00:01:47.874
700s is just 700.

00:01:48.294 --> 00:01:52.309
700 and you're done.

00:01:52.682 --> 00:01:54.087
Let's do more of these,

00:01:54.087 --> 00:01:57.187
where we might be tempted to
do some type of long division,

00:01:57.187 --> 00:01:59.931
but if we really think about
ways to break up the numbers,

00:01:59.931 --> 00:02:01.788
we might be able to do it with less paper

00:02:01.788 --> 00:02:03.876
or possibly even in our heads.

00:02:04.249 --> 00:02:06.952
Let's say that you had the number

00:02:08.610 --> 00:02:09.787
800.

00:02:09.787 --> 00:02:11.366
Let me do this in a different color.

00:02:11.366 --> 00:02:16.366
Let's say you had the number 800 and 46.

00:02:17.844 --> 00:02:21.661
846 and you want to divide that

00:02:22.058 --> 00:02:23.519
by 2.

00:02:23.996 --> 00:02:26.736
Well, the realization here,
or what you might think about

00:02:26.736 --> 00:02:28.361
is let's break this number up.

00:02:28.361 --> 00:02:30.292
Let's break up this 846.

00:02:30.292 --> 00:02:34.051
846 is the same thing ...

00:02:34.274 --> 00:02:35.191
this is going to be ...

00:02:35.191 --> 00:02:36.433
actually, let me do it this way.

00:02:36.433 --> 00:02:39.335
846 is the same thing as

00:02:39.996 --> 00:02:41.513
this is the same thing as

00:02:43.159 --> 00:02:46.170
800 plus 40

00:02:48.820 --> 00:02:50.175
plus 6

00:02:54.320 --> 00:02:57.955
and so 846 is the same
things as 800 plus 40 plus 6.

00:02:57.955 --> 00:02:59.975
We can just divide that by 2.

00:02:59.975 --> 00:03:02.540
We can just divide that by 2.

00:03:02.540 --> 00:03:04.703
Divided by 2.

00:03:04.703 --> 00:03:07.060
Notice, I didn't do
anything too fancy here.

00:03:07.060 --> 00:03:10.844
This is just the same
thing as this over here.

00:03:10.844 --> 00:03:11.796
I just broke it up.

00:03:11.796 --> 00:03:15.418
You have 800s, four
10s which is 40, and 6,

00:03:15.418 --> 00:03:16.997
and you can divide that by 2.

00:03:16.997 --> 00:03:18.530
Well, how does that help us?

00:03:18.883 --> 00:03:21.694
Well, now we can divide each
of these separately by 2.

00:03:21.694 --> 00:03:22.750
one way to think about it is

00:03:22.750 --> 00:03:24.759
we can distribute

00:03:25.297 --> 00:03:26.619
the division by 2.

00:03:26.619 --> 00:03:28.721
Where, you might be familiar using the

00:03:28.721 --> 00:03:30.299
distributive property in multiplication,

00:03:30.299 --> 00:03:32.122
but you could also do it with division.

00:03:32.122 --> 00:03:34.781
This thing right over here is
going to be the same thing.

00:03:34.781 --> 00:03:36.696
This is going to be equal to

00:03:36.696 --> 00:03:39.016
800 divided by 2.

00:03:39.285 --> 00:03:41.918
800 divided by 2

00:03:44.322 --> 00:03:46.600
plus 40 divided by 2.

00:03:47.031 --> 00:03:49.647
plus 40 divided by 2.

00:03:49.647 --> 00:03:51.829
I'm going to add parenthesis in a second

00:03:51.829 --> 00:03:53.360
to make it a little bit clearer.

00:03:53.385 --> 00:03:55.184
Plus 6 divided by 2.

00:03:55.184 --> 00:03:59.026
Plus 6 divided by 2.

00:04:07.770 --> 00:04:10.195
Once again, you have

00:04:14.087 --> 00:04:15.414
800 divided by 2.

00:04:15.414 --> 00:04:16.877
That's that right over there.

00:04:16.877 --> 00:04:20.452
Plus 40 divided by 2.

00:04:20.452 --> 00:04:21.996
40 divided by 2.

00:04:21.996 --> 00:04:23.738
Plus 6 divided by 2.

00:04:23.738 --> 00:04:26.512
Plus 6 divided by 2.

00:04:26.512 --> 00:04:29.194
We distributed the division by 2.

00:04:29.194 --> 00:04:30.738
Well, 800 divided by 2,

00:04:30.738 --> 00:04:32.619
you should do in your head.

00:04:32.619 --> 00:04:34.894
This is 800, divided that by 2,

00:04:34.894 --> 00:04:36.223
and you're going to get 4 hundreds.

00:04:37.657 --> 00:04:40.404
4 tens divided by 2 is going to be 2 tens

00:04:42.065 --> 00:04:43.421
or 20.

00:04:44.096 --> 00:04:46.754
800s divided by 2 is 4 hundreds.

00:04:46.754 --> 00:04:48.995
4 tens divided by 2 is 2 tens.

00:04:48.995 --> 00:04:51.723
Then, 6 divided by 2 is 3.

00:04:51.723 --> 00:04:56.668
You are left with 400 plus 20 plus 3,

00:04:56.668 --> 00:05:01.668
which of course is equal to 400 and 23.

00:05:03.982 --> 00:05:08.068
I used a lot of my screen
here to work this out for you.

00:05:08.068 --> 00:05:08.962
You really

00:05:09.915 --> 00:05:11.482
digest what's going on here,

00:05:11.482 --> 00:05:12.944
but once you get good at it,

00:05:12.944 --> 00:05:14.070
you could say okay look

00:05:14.070 --> 00:05:15.294
8 divided by 2,

00:05:15.551 --> 00:05:18.210
800s divided by 2 is 400s.

00:05:18.210 --> 00:05:19.232
Actually I could do it right over here.

00:05:19.232 --> 00:05:21.426
8 hundreds divided by 2 is 400s.

00:05:21.426 --> 00:05:25.245
4 tens divided by 2 is 2 tens.

00:05:26.260 --> 00:05:29.389
6 ones divided by 2 is 3 ones.

00:05:29.389 --> 00:05:30.078
You could say hey look

00:05:30.078 --> 00:05:31.982
if I just divided each
of these numbers by 2

00:05:31.982 --> 00:05:34.603
I get 423.

00:05:34.655 --> 00:05:36.118
This is what you are really doing,

00:05:36.118 --> 00:05:38.219
you are separating it out by place value.

00:05:38.219 --> 00:05:41.017
You are dividing each of those places

00:05:41.017 --> 00:05:44.558
the hundreds, the tens, and the ones by 2.

00:05:44.558 --> 00:05:46.357
Then, you got 423.

00:05:46.357 --> 00:05:48.214
By the way, that's
exactly what you would do

00:05:48.214 --> 00:05:50.316
if you actually did it
through long division.

00:05:50.316 --> 00:05:54.042
You are actually dividing
2 into each of the places.

00:05:54.380 --> 00:05:56.367
Let's do one more just to really make sure

00:05:56.367 --> 00:05:59.270
that we are fully enjoying this.

00:05:59.608 --> 00:06:01.796
Let's say that I have

00:06:04.672 --> 00:06:07.144
963

00:06:09.421 --> 00:06:14.055
and I want to divide that by 9.

00:06:14.525 --> 00:06:15.833
Well, over here,

00:06:15.833 --> 00:06:17.217
you might make the realization

00:06:17.217 --> 00:06:20.361
well I can see parts of this
that I know how to divide by 9.

00:06:20.361 --> 00:06:23.695
I could say this is the same thing as 900,

00:06:23.695 --> 00:06:25.515
and I'm breaking out the 900,

00:06:25.515 --> 00:06:28.290
because I know how to
divide 900 divided by 9.

00:06:28.290 --> 00:06:29.590
That's 900.

00:06:29.590 --> 00:06:32.957
I know that 63 is a multiple of 9.

00:06:32.957 --> 00:06:35.209
Why don't I break out 63 separately.

00:06:35.209 --> 00:06:36.052
I don't even have to break out the

00:06:36.052 --> 00:06:37.387
tens and the ones separately.

00:06:37.387 --> 00:06:38.665
I could say this is the same thing as

00:06:38.665 --> 00:06:40.952
900 plus 63

00:06:40.952 --> 00:06:45.952
and all of that divided by 9.

00:06:45.967 --> 00:06:48.161
Well, this is going to
be the same thing as

00:06:48.161 --> 00:06:49.809
900 divided by 9.

00:06:50.547 --> 00:06:52.332
Let me do that in the brown color.

00:06:52.932 --> 00:06:57.932
900 divided by 9 plus 63 divided by 9.

00:07:02.179 --> 00:07:05.059
I just distributed the division by 9.

00:07:05.059 --> 00:07:07.565
Let me put some parenthesis around this.

00:07:07.764 --> 00:07:09.556
What is this going to be equal to?

00:07:09.740 --> 00:07:12.450
Well, 900 divided by 9 is 100.

00:07:12.450 --> 00:07:15.815
This is going to be 100,

00:07:15.979 --> 00:07:19.054
and 63 divided by 9 is 7.

00:07:21.045 --> 00:07:23.130
This is going to be 100 plus 7,

00:07:23.130 --> 00:07:23.973
or

00:07:27.840 --> 00:07:30.028
100 and 7.

00:07:30.028 --> 00:07:32.188
Once again, I wrote it all out like this.

00:07:32.188 --> 00:07:33.593
Once you get some practice,

00:07:33.593 --> 00:07:37.993
you'll say hey look 9 goes
into 900 a hundred times.

00:07:37.993 --> 00:07:39.872
9 goes into 63 seven times.

00:07:39.872 --> 00:07:43.704
So, 9 goes into 963 one
hundred and 7 times.

00:07:43.704 --> 00:07:45.272
Hopefully you found that fun.

00:07:45.272 --> 00:07:47.186
This is useful stuff.

00:07:47.186 --> 00:07:48.844
One of the most important things you're

00:07:48.844 --> 00:07:49.821
going to have to find is that

00:07:49.821 --> 00:07:51.143
all the time you're going
to find these numbers

00:07:51.143 --> 00:07:52.964
while you're doing, you
know, your finances,

00:07:52.964 --> 00:07:55.624
or you're trying to calculate
the check at a restaurant.

00:07:55.624 --> 00:07:57.133
You're going to find it really valuable

00:07:57.133 --> 00:07:59.326
to be able to do this type of division.

00:07:59.326 --> 00:08:01.902
With some practice, even without paper.


Introduction to trigonometric substitution

WEBVTT
Kind: captions
Language: en

00:00:00.441 --> 00:00:02.090
- [Voiceover] Let's say
that we want to evaluate

00:00:02.090 --> 00:00:04.481
this indefinite integral right over here.

00:00:04.481 --> 00:00:06.385
And you immediately say hey,
you've got the square root

00:00:06.385 --> 00:00:09.217
of four mins X squared in the denominator,

00:00:09.217 --> 00:00:10.982
you could try to use substitution,

00:00:10.982 --> 00:00:13.094
but it really doesn't simplify this in any

00:00:13.094 --> 00:00:14.256
reasonable way.

00:00:14.256 --> 00:00:15.974
How do you tackle this?

00:00:15.974 --> 00:00:17.763
And an insight that you might have is well

00:00:17.763 --> 00:00:20.780
this thing, this square root
of four minus X squared,

00:00:20.780 --> 00:00:22.801
this looks like something that I might get

00:00:22.801 --> 00:00:24.937
if I'm dealing with the
Pythagorean theorem,

00:00:24.937 --> 00:00:27.398
if I'm solving for non-hypotenuse sides

00:00:27.398 --> 00:00:28.535
of the Pythagorean theorem,

00:00:28.535 --> 00:00:31.044
especially if the length of the hypotenuse

00:00:31.044 --> 00:00:33.041
is two, this would be two squared

00:00:33.041 --> 00:00:34.968
and then the other side is X,

00:00:34.968 --> 00:00:36.454
then two squared minus X squared,

00:00:36.454 --> 00:00:38.776
this would be the length
of the other side.

00:00:38.776 --> 00:00:40.657
Well let's just run with that then.

00:00:40.657 --> 00:00:42.995
So, if we have,

00:00:43.740 --> 00:00:45.830
if we have a,

00:00:45.830 --> 00:00:48.546
and if any of you didn't
feel that inside immediately,

00:00:48.546 --> 00:00:49.746
it's okay.

00:00:50.723 --> 00:00:52.673
Because to be honest, I
actually did not feel that

00:00:52.673 --> 00:00:55.754
inside probably the first
time I saw things like this.

00:00:56.593 --> 00:00:58.078
So let's say you have a right triangle.

00:00:58.078 --> 00:01:01.307
So let's just visualize that
insight that we talked about.

00:01:01.307 --> 00:01:04.991
So if the hypotenuse of the
right triangle is of length two,

00:01:05.783 --> 00:01:08.028
the hypotenuse is of length two,

00:01:09.074 --> 00:01:11.094
hypotenuse is of length two

00:01:11.094 --> 00:01:14.183
and let's say this side, right over here.

00:01:14.183 --> 00:01:15.785
Actually, let me do this side.

00:01:15.785 --> 00:01:17.317
Let's say this side...

00:01:17.317 --> 00:01:18.432
Actually, I'm gonna go with this side.

00:01:18.432 --> 00:01:21.885
This side right over here is of length X.

00:01:22.723 --> 00:01:25.276
This side right over here is of length X.

00:01:25.276 --> 00:01:26.670
It's a little counter
intuitive, because normally

00:01:26.670 --> 00:01:28.597
we associate this side over here with Y,

00:01:28.597 --> 00:01:29.874
but let's just go with it.

00:01:29.874 --> 00:01:32.057
This side right over here is of length X.

00:01:32.057 --> 00:01:33.983
Then what would this
side right over here be?

00:01:33.983 --> 00:01:37.668
What would this other
non-hypotenuse length be?

00:01:38.436 --> 00:01:41.362
Well, just if you solve for it
using the Pythagorean Theorem

00:01:41.362 --> 00:01:44.008
you get that it is going to be

00:01:44.008 --> 00:01:48.073
the square root of the hypotenuse squared,

00:01:48.073 --> 00:01:50.202
two squared, which is just four,

00:01:51.156 --> 00:01:52.781
minus the other side squared.

00:01:52.781 --> 00:01:54.639
so minus X squared.

00:01:54.639 --> 00:01:56.473
Well that's interesting,
that is this expression

00:01:56.473 --> 00:01:58.701
that insight, that intuition that we

00:01:58.701 --> 00:02:00.815
may have had when we saw
this right over here,

00:02:00.815 --> 00:02:02.882
but still, how does that help us?

00:02:02.882 --> 00:02:05.436
Well this is where the
trigonometry comes in.

00:02:05.436 --> 00:02:07.843
Because if we define,

00:02:08.751 --> 00:02:10.306
if this angle right over here,

00:02:10.306 --> 00:02:12.303
we say this is theta,

00:02:12.303 --> 00:02:14.973
then what is sine and
cosine of theta going to be

00:02:14.973 --> 00:02:16.971
in terms of these sides?

00:02:16.971 --> 00:02:17.876
Well let's see,

00:02:17.876 --> 00:02:19.432
the sine of theta,

00:02:19.432 --> 00:02:21.801
sine of theta is equal to the opposite

00:02:21.801 --> 00:02:22.938
over the hypotenuse.

00:02:22.938 --> 00:02:24.866
Is equal to X over two.

00:02:24.866 --> 00:02:26.561
Or if you want to solve for X,

00:02:26.561 --> 00:02:30.810
we get X is equal to two sine theta.

00:02:30.810 --> 00:02:32.229
Well that's interesting.

00:02:32.229 --> 00:02:34.017
What about the cosine of theta?

00:02:34.017 --> 00:02:37.522
Cosine of theta is equal
to the adjacent side,

00:02:37.522 --> 00:02:40.727
square root of four minus X squared

00:02:40.727 --> 00:02:42.528
over the hypotenuse.

00:02:42.528 --> 00:02:44.386
Or if you want to solve for this side,

00:02:44.386 --> 00:02:46.011
it's going to be,

00:02:46.011 --> 00:02:48.401
we could say that the square root of four

00:02:48.401 --> 00:02:50.330
minus X squared is going to be equal to

00:02:50.330 --> 00:02:54.044
the hypotenuse times the cosine of theta.

00:02:54.044 --> 00:02:55.155
So that's interesting.

00:02:55.155 --> 00:02:57.406
If X is equal to two sine theta,

00:02:57.406 --> 00:02:59.612
then this other side, then
this entire expression

00:02:59.612 --> 00:03:02.491
simplifies to two cosine theta.

00:03:02.491 --> 00:03:04.901
And that seems pretty interesting now.

00:03:04.901 --> 00:03:06.736
So let's make the substitution.

00:03:06.736 --> 00:03:11.736
Let's say that X is
equal to two sine theta.

00:03:12.419 --> 00:03:14.067
And if X is equal to two sine theta,

00:03:14.067 --> 00:03:17.435
then DX is going to be equal to

00:03:17.435 --> 00:03:21.956
two cosine theta D theta.

00:03:22.700 --> 00:03:24.924
And then if we have X as
equal to two sine theta,

00:03:24.924 --> 00:03:26.550
then what is this thing right over here?

00:03:26.550 --> 00:03:27.526
Well we just figured out,

00:03:27.526 --> 00:03:29.917
this thing is two cosine theta.

00:03:29.917 --> 00:03:32.007
This thing is equal to two,

00:03:32.007 --> 00:03:33.463
actually, let me do it
in that orange color.

00:03:34.165 --> 00:03:36.066
Two cosine theta.

00:03:36.066 --> 00:03:39.432
This is equal to two cosine theta.

00:03:39.432 --> 00:03:40.988
And we were able to do this,

00:03:40.988 --> 00:03:43.671
drawing this right triangle and using

00:03:43.671 --> 00:03:47.687
the sohcahtoa definition
of these trig functions,

00:03:47.687 --> 00:03:49.080
and obviously we could use units circle,

00:03:49.080 --> 00:03:50.543
that's kind of an extension of these.

00:03:50.543 --> 00:03:51.867
But you could also do it if you say

00:03:51.867 --> 00:03:53.841
hey, look you know, if
this is two sine theta

00:03:53.841 --> 00:03:55.767
you could use the Pythagorean identity,

00:03:55.767 --> 00:03:57.719
our trigonometric identities and you would

00:03:57.719 --> 00:03:59.273
see that this entire expression,

00:03:59.273 --> 00:04:02.827
if you did it that way, would
simplify to two cosine theta.

00:04:02.827 --> 00:04:04.081
But now let's just run forward

00:04:04.081 --> 00:04:05.172
and let's see if we can evaluate it

00:04:05.172 --> 00:04:06.821
using this substitution.

00:04:06.821 --> 00:04:10.140
So this is going to be
the indefinite integral,

00:04:10.140 --> 00:04:13.067
so DX, so one times DX, that's going to be

00:04:13.067 --> 00:04:15.017
DX is two cosine theta D theta.

00:04:15.017 --> 00:04:16.619
So let me right that, that is

00:04:16.619 --> 00:04:20.218
two cosine theta

00:04:20.218 --> 00:04:22.285
D, I'll right the D theta out here,

00:04:22.285 --> 00:04:23.864
and then what's the denominator?

00:04:23.864 --> 00:04:25.325
Square root of four minus X squared?

00:04:25.325 --> 00:04:27.064
Well that's two cosine theta again.

00:04:28.015 --> 00:04:30.336
So that is two

00:04:30.336 --> 00:04:32.334
cosine theta.

00:04:32.334 --> 00:04:34.725
Well this seems to work out quite nicely.

00:04:34.725 --> 00:04:38.022
If you two cosine theta
over two cosine theta,

00:04:38.022 --> 00:04:39.601
that's just going to be one.

00:04:39.601 --> 00:04:41.296
This simplifies.

00:04:41.296 --> 00:04:43.286
This simplifies to

00:04:44.060 --> 00:04:46.841
D theta, which is, if
you just evaluate this,

00:04:46.841 --> 00:04:48.953
this is just going to be equal to

00:04:48.953 --> 00:04:51.648
theta plus C.

00:04:51.648 --> 00:04:55.013
Well, this is kind of nice,
but we're still not done.

00:04:55.013 --> 00:04:58.056
We want our indefinite integral

00:04:58.056 --> 00:05:00.657
in terms of X.

00:05:00.657 --> 00:05:02.723
So now let's just solve for X here.

00:05:02.723 --> 00:05:04.951
So if X is two cosine theta,

00:05:04.951 --> 00:05:07.274
sorry, if X is two sine theta,

00:05:07.274 --> 00:05:10.502
so X is equal to two sine,

00:05:10.502 --> 00:05:15.168
X is equal to two sine theta,

00:05:15.168 --> 00:05:17.072
then let's see, divide both sides by two.

00:05:17.072 --> 00:05:21.043
X over two is equal to sine theta.

00:05:21.043 --> 00:05:22.691
And then if you want to solve for theta,

00:05:22.691 --> 00:05:24.502
theta's the angle that if
you take the sine of it

00:05:24.502 --> 00:05:25.942
you get X over two.

00:05:25.942 --> 00:05:27.173
So we could say,

00:05:27.173 --> 00:05:29.216
let's give ourselves a
little bit more real estate,

00:05:29.216 --> 00:05:30.369
that theta

00:05:31.045 --> 00:05:33.320
is equal to the inverse sine,

00:05:33.320 --> 00:05:36.175
the inverse sine of this thing,

00:05:36.175 --> 00:05:37.291
X over two.

00:05:37.291 --> 00:05:38.405
We could write it that way,

00:05:38.405 --> 00:05:39.938
or we could write that theta

00:05:39.938 --> 00:05:42.282
is equal to the arcsine.

00:05:42.282 --> 00:05:45.603
The arcsine of,

00:05:45.603 --> 00:05:48.506
the arcsine of X over two.

00:05:48.506 --> 00:05:52.034
So this is going to be equal to the,

00:05:52.034 --> 00:05:53.938
theta is arcsine of X over two,

00:05:53.938 --> 00:05:55.215
so let's just do that.

00:05:55.215 --> 00:05:59.017
Arcsine of X over two

00:05:59.854 --> 00:06:01.085
plus C.

00:06:01.085 --> 00:06:01.898
And we're done.

00:06:01.898 --> 00:06:04.383
We've just evaluated
that indefinite integral.

00:06:04.383 --> 00:06:06.264
Now some of you might
have noticed something.

00:06:06.264 --> 00:06:08.841
I kind of brushed past it really fast

00:06:08.841 --> 00:06:10.744
just to give you the big,

00:06:10.744 --> 00:06:12.440
to see the forest for the trees,

00:06:12.440 --> 00:06:13.741
but there's some interesting details

00:06:13.741 --> 00:06:15.338
and it's, I think, worth digging

00:06:15.338 --> 00:06:17.265
into a little bit over here.

00:06:17.265 --> 00:06:19.262
So the first one is,
you might see that look

00:06:19.262 --> 00:06:21.467
the restriction on X here is,

00:06:21.467 --> 00:06:23.464
or the domain here on this expression

00:06:23.464 --> 00:06:24.626
is restricted.

00:06:24.626 --> 00:06:25.926
So let's just keep track of that

00:06:25.926 --> 00:06:27.551
to make sure that we
didn't do anything strange

00:06:27.551 --> 00:06:29.268
when we did that substitution.

00:06:29.268 --> 00:06:31.962
So the domain here, X has to be

00:06:32.678 --> 00:06:35.046
X has to be greater than negative two

00:06:35.046 --> 00:06:36.718
and less than two.

00:06:36.718 --> 00:06:38.878
If the absolute value
of X were equal to two,

00:06:38.878 --> 00:06:40.757
then you would have a
zero in the denominator.

00:06:40.757 --> 00:06:42.871
If the absolute value of
X is greater than two,

00:06:42.871 --> 00:06:45.262
then you're gonna have a
negative in the denominator

00:06:45.262 --> 00:06:46.726
and that's not defined.

00:06:46.726 --> 00:06:49.372
So this right over here is the domain.

00:06:49.372 --> 00:06:50.974
So let's make sure that our substitution

00:06:50.974 --> 00:06:52.948
didn't do anything weird with that.

00:06:52.948 --> 00:06:55.455
So if X has to be between
negative two and two,

00:06:55.455 --> 00:06:57.058
and we're saying X is two sine theta,

00:06:57.058 --> 00:06:58.475
that means two sine theta would have to be

00:06:58.475 --> 00:07:00.215
between negative two and two.

00:07:00.215 --> 00:07:03.583
So negative two would have to be less than

00:07:03.583 --> 00:07:05.092
two sine theta.

00:07:05.092 --> 00:07:07.530
Two sine theta.

00:07:07.530 --> 00:07:10.710
Which would have to be less than two.

00:07:10.710 --> 00:07:13.191
We could divide all the
different parts of this

00:07:13.191 --> 00:07:15.652
compound inequality by two

00:07:15.652 --> 00:07:19.663
and you're gonna get negative
one is less than sine theta.

00:07:20.500 --> 00:07:22.635
Is less than sine theta,

00:07:22.635 --> 00:07:25.074
which is less than one.

00:07:25.074 --> 00:07:29.834
And the way we can do that is if theta

00:07:29.834 --> 00:07:33.085
is less than pi over two.

00:07:33.085 --> 00:07:35.987
At pi over two, sine of
theta would be equal to one.

00:07:35.987 --> 00:07:39.098
And if theta is greater
than negative pi over two.

00:07:39.098 --> 00:07:40.562
So if we restrict it in this way,

00:07:40.562 --> 00:07:42.141
if we say theta is going to be

00:07:42.141 --> 00:07:44.438
in this range right over here,

00:07:44.438 --> 00:07:48.932
then we are restricting our
domain in a reasonable way.

00:07:49.762 --> 00:07:52.626
And this works out well
because this typically is

00:07:52.626 --> 00:07:56.736
the range for the arcsine function.

00:07:56.736 --> 00:07:58.617
So we could feel good about that.

00:07:58.617 --> 00:08:00.033
Now another question that you might have

00:08:00.033 --> 00:08:01.518
is okay well look, you know we divided

00:08:01.518 --> 00:08:03.261
by cosine of theta here, but that's okay

00:08:03.261 --> 00:08:05.744
as long as cosine of
theta does not equal zero,

00:08:05.744 --> 00:08:07.626
'cause you don't want a
zero in the denominator.

00:08:07.626 --> 00:08:10.273
And the good thing about
this restriction on theta

00:08:10.273 --> 00:08:13.708
is as long as theta is greater than

00:08:13.708 --> 00:08:16.310
negative pi over two and
less than pi over two,

00:08:16.310 --> 00:08:18.330
cosine of theta is going to be,

00:08:18.330 --> 00:08:19.584
it's not going to be zero, and actually

00:08:19.584 --> 00:08:21.696
it's going to be positive.

00:08:21.696 --> 00:08:25.133
If negative pi over two or
pi over two were allowed,

00:08:25.133 --> 00:08:26.804
then you would get a zero down here

00:08:26.804 --> 00:08:27.780
and we would have to think about

00:08:27.780 --> 00:08:29.731
restricting things in some other ways.

00:08:29.731 --> 00:08:31.936
So it looks like everything is cool.

00:08:31.936 --> 00:08:33.514
We dug a little deeper.

00:08:33.514 --> 00:08:34.839
We said, okay we haven't done anything

00:08:34.839 --> 00:08:36.666
strange to the domain or to,

00:08:38.061 --> 00:08:40.336
unrestricted it some strange way,

00:08:40.336 --> 00:08:41.729
and so we can feel,

00:08:41.729 --> 00:08:44.120
we can feel good about,

00:08:44.120 --> 00:08:46.559
we can feel good about this,

00:08:46.559 --> 00:08:47.823
about this,

00:08:48.805 --> 00:08:51.404
this answer that we've gotten.


Conditional & absolute convergence

WEBVTT
Kind: captions
Language: en

00:00:00.511 --> 00:00:02.380
- [Voiceover] In the
video where we introduced

00:00:02.380 --> 00:00:06.153
the alternating series test,
we in fact used the series,

00:00:06.153 --> 00:00:09.357
we used the infinite
series from n equals one

00:00:09.357 --> 00:00:12.039
to infinity of negative one,

00:00:12.039 --> 00:00:15.231
to the n plus one over n.

00:00:15.231 --> 00:00:17.611
We used this as our example

00:00:17.611 --> 00:00:19.852
to apply the alternating series test,

00:00:19.852 --> 00:00:22.719
and we proved that this thing
right over here converges.

00:00:22.719 --> 00:00:25.215
So this series, which is one,

00:00:27.486 --> 00:00:31.206
minus 1/2, plus 1/3,

00:00:31.206 --> 00:00:33.634
minus 1/4, and it just keeps

00:00:33.634 --> 00:00:36.065
going on and on and on forever.

00:00:36.065 --> 00:00:39.070
We used the alternating
series test in that video

00:00:39.070 --> 00:00:41.044
to prove that it converges.

00:00:41.044 --> 00:00:43.308
So this thing converges.

00:00:43.308 --> 00:00:47.324
So this converges by

00:00:47.324 --> 00:00:49.669
alternating series test.

00:00:58.067 --> 00:01:01.699
Alternating series test, and
if you wanna review that,

00:01:01.699 --> 00:01:04.242
go watch the video on the
alternating series test.

00:01:04.242 --> 00:01:06.161
Now let's think a little bit
about what happens if we were

00:01:06.161 --> 00:01:08.947
to take the absolute value
of each of these terms.

00:01:08.947 --> 00:01:10.533
So if we were to take the absolute value

00:01:10.533 --> 00:01:13.133
of each of these terms, so
if you were to take the sum

00:01:13.133 --> 00:01:17.498
from n equals one to infinity
of the absolute value

00:01:17.498 --> 00:01:20.412
of negative one to the

00:01:20.412 --> 00:01:23.210
n plus one over n,

00:01:23.210 --> 00:01:25.578
well what is this going to be equal to?

00:01:25.578 --> 00:01:28.097
Well, this numerator is either gonna be

00:01:28.097 --> 00:01:29.897
one or negative one, the
absolute value of that

00:01:29.897 --> 00:01:32.300
is always gonna be one, so
it's going to be that over.

00:01:32.300 --> 00:01:35.098
And n is always positive, we're
going from one to infinity,

00:01:35.098 --> 00:01:38.174
so it's just going to be equal to the sum,

00:01:38.174 --> 00:01:41.945
it's going to be equal to
the sum from n equals one

00:01:41.945 --> 00:01:45.265
to infinity of one over n.

00:01:45.265 --> 00:01:47.773
And this is just the
famous harmonic series.

00:01:47.773 --> 00:01:49.769
And there's this video that we have,

00:01:49.769 --> 00:01:51.430
and you should look it up on Khan Academy

00:01:51.430 --> 00:01:53.972
if you don't believe
me, on the famous proof

00:01:53.972 --> 00:01:56.131
that the harmonic series diverges.

00:01:56.131 --> 00:01:58.766
So the harmonic series is one plus 1/2,

00:01:58.766 --> 00:02:01.843
plus 1/3, this thing right over here,

00:02:01.843 --> 00:02:06.231
this thing right over here diverges.

00:02:06.231 --> 00:02:10.224
And so when you see a
series that converges,

00:02:10.224 --> 00:02:12.593
but if you were to take the absolute value

00:02:12.593 --> 00:02:15.611
of each of its terms,
and then that diverges,

00:02:15.611 --> 00:02:19.012
we say that this series
converges conditionally.

00:02:20.622 --> 00:02:22.758
You can say it converges,
but you could also

00:02:22.758 --> 00:02:25.126
say it converges conditionally.

00:02:26.933 --> 00:02:28.616
And the condition is,
I guess you could say,

00:02:28.616 --> 00:02:30.288
that we're not taking the absolute value

00:02:30.288 --> 00:02:31.751
of each of the terms.

00:02:31.751 --> 00:02:35.431
And if something converges when

00:02:35.431 --> 00:02:37.195
you take the absolute value as well,

00:02:37.195 --> 00:02:39.726
then you say it converges absolutely.

00:02:39.726 --> 00:02:41.491
And so let's look at an example of that.

00:02:41.491 --> 00:02:43.126
If I were to take...

00:02:46.970 --> 00:02:49.989
This series, let's do a geometric series,

00:02:49.989 --> 00:02:51.567
that might be fun.

00:02:51.567 --> 00:02:53.750
Actually I'm using these colors too much,

00:02:53.750 --> 00:02:55.375
let me use another color.

00:02:55.375 --> 00:02:59.113
Let's say, let's take the sum

00:02:59.113 --> 00:03:02.584
from n equals one to infinity of

00:03:02.584 --> 00:03:06.926
negative 1/2 to

00:03:06.926 --> 00:03:10.409
the n plus one power.

00:03:10.409 --> 00:03:12.870
We know this is a geometric series

00:03:12.870 --> 00:03:15.203
where the absolute value
of our common ratio

00:03:15.203 --> 00:03:18.268
is less than one, we
know that this converges.

00:03:21.774 --> 00:03:23.434
And if we were to take the absolute value

00:03:23.434 --> 00:03:26.197
of each of these terms, so
if you were to take the sum,

00:03:26.197 --> 00:03:27.811
Let me do that in a different color,

00:03:27.811 --> 00:03:30.597
just to mix things up a little bit.

00:03:30.597 --> 00:03:33.917
If you were to take the absolute value

00:03:33.917 --> 00:03:36.610
of each of these terms,
so the absolute value

00:03:36.610 --> 00:03:39.234
of negative 1/2,

00:03:42.484 --> 00:03:45.596
to the n plus one power,

00:03:45.596 --> 00:03:48.962
this is going to be the
same thing as the sum,

00:03:48.962 --> 00:03:52.515
from n equals one

00:03:52.515 --> 00:03:55.823
to infinity of 1/2

00:03:55.823 --> 00:03:58.157
to the n plus one.

00:03:58.157 --> 00:04:00.861
And here once again, the common ratio,

00:04:00.861 --> 00:04:03.334
the absolute value of the
common ratio is less than one,

00:04:03.334 --> 00:04:04.762
and we've studied this when we looked

00:04:04.762 --> 00:04:06.097
at geometric series.

00:04:06.097 --> 00:04:08.256
This also converges.

00:04:09.924 --> 00:04:11.805
So when we took the
absolute value of the terms,

00:04:11.805 --> 00:04:13.349
it still converged.

00:04:13.349 --> 00:04:15.926
So for this one, we can
say that this converges

00:04:15.926 --> 00:04:18.524
absolutely.

00:04:20.360 --> 00:04:22.322
So we've talked a lot already

00:04:22.322 --> 00:04:24.807
about convergence or divergence,

00:04:24.807 --> 00:04:26.304
and that's all been good.

00:04:26.304 --> 00:04:27.546
And what we're doing in this video is

00:04:27.546 --> 00:04:30.901
we're introducing a nuance
or flavors of convergence.

00:04:30.901 --> 00:04:33.699
So you can converge, but it
might be interesting to say

00:04:33.699 --> 00:04:36.160
well, would it still converge if we

00:04:36.160 --> 00:04:38.087
took the absolute value of the terms?

00:04:38.087 --> 00:04:40.409
If it won't, if you converge,
but it doesn't converge

00:04:40.409 --> 00:04:42.209
when you take the absolute
value of the terms,

00:04:42.209 --> 00:04:44.623
then you say it converges conditionally.

00:04:44.623 --> 00:04:47.584
If it converges, and
it still converges when

00:04:47.584 --> 00:04:49.604
you take the absolute value of the terms,

00:04:49.604 --> 00:04:52.517
then we say it converges absolutely.

00:04:52.517 --> 00:04:53.980
Because even if you
take the absolute value

00:04:53.980 --> 00:04:56.197
of the terms, it converges.

00:04:56.197 --> 00:04:58.645
Hopefully you find that interesting.


Infinite series as limit of partial sums

WEBVTT
Kind: captions
Language: en

00:00:00.173 --> 00:00:03.245
- [Voiceover] Let's say that
we have an infinite series S

00:00:03.245 --> 00:00:06.312
so that's the sum from n = 1

00:00:06.312 --> 00:00:09.356
to infinity of a sub n.

00:00:09.356 --> 00:00:13.011
We could write it out a sub 1 plus a sub 2

00:00:13.011 --> 00:00:16.303
and we're just going to go on
and on and on for infinity.

00:00:16.303 --> 00:00:19.179
We're going to go on
and on and on forever.

00:00:19.179 --> 00:00:21.629
So, let's say, and I've written
it in very general terms

00:00:21.629 --> 00:00:25.740
let's say we have a formula
for the partial sums of S.

00:00:25.740 --> 00:00:29.604
We know that S sub n is equal

00:00:29.604 --> 00:00:32.411
to 2n to the third

00:00:32.411 --> 00:00:35.609
over n plus 1

00:00:35.609 --> 00:00:38.290
times n plus 2.

00:00:38.290 --> 00:00:41.617
Now, my question to you is,
based on what I've just told you

00:00:41.974 --> 00:00:43.711
S is the sum in a very general way written

00:00:43.711 --> 00:00:46.364
this infinite series but
I have the partial sum.

00:00:46.364 --> 00:00:49.807
The sum of the first n terms of S is given

00:00:49.807 --> 00:00:52.667
by this formula right over here does

00:00:52.667 --> 00:00:56.002
this series converge or diverge?

00:00:56.002 --> 00:00:58.942
Does this thing converge
to some finite value

00:00:58.942 --> 00:01:02.780
or is it unbounded and does it diverge?

00:01:02.780 --> 00:01:05.546
Well, one way to think about this is

00:01:07.013 --> 00:01:11.374
the idea that our infinite series S

00:01:11.374 --> 00:01:15.015
is just the limit as n approaches

00:01:15.015 --> 00:01:18.413
infinity of our partial sums.

00:01:18.413 --> 00:01:19.742
So, what do we mean by that?

00:01:19.742 --> 00:01:22.977
Well, you could a sequence
of partial sums here.

00:01:22.977 --> 00:01:27.644
You have S sub 1, S sub 2, S sub 3

00:01:27.644 --> 00:01:28.667
and you keep going so this

00:01:28.667 --> 00:01:30.408
would be the sum of the first term.

00:01:30.408 --> 00:01:31.981
This would be the sum
of the first 2 terms.

00:01:31.981 --> 00:01:34.040
This would be the sum of the first 3 terms

00:01:34.040 --> 00:01:36.396
and just think about what
happens to this sequence

00:01:36.396 --> 00:01:40.635
as n right over here

00:01:40.635 --> 00:01:43.340
approaches infinity

00:01:43.340 --> 00:01:44.812
because that's what this series is.

00:01:44.812 --> 00:01:47.107
It's the sum of the first, I guess

00:01:47.107 --> 00:01:49.575
you could say the first, infinite terms.

00:01:49.575 --> 00:01:51.574
It's the sum of all, you have

00:01:51.574 --> 00:01:54.461
an infinite number of terms here.

00:01:54.477 --> 00:01:56.067
Well, let's think about what this.

00:01:56.067 --> 00:01:59.910
The limit is n approaches
infinity of S sub n.

00:01:59.910 --> 00:02:02.109
That's just going to be

00:02:03.369 --> 00:02:06.244
the limit as n approaches

00:02:06.244 --> 00:02:10.206
infinity of this business right over here.

00:02:10.206 --> 00:02:15.206
2n to the third power over n plus 1

00:02:17.303 --> 00:02:20.735
times n plus 2 and there's

00:02:20.735 --> 00:02:22.777
several ways you could evaluate this.

00:02:22.777 --> 00:02:23.942
One way is you could just realize,

00:02:23.942 --> 00:02:25.470
"Hey, look in the bottom this is going

00:02:25.470 --> 00:02:29.108
"to be a second degree polynomial."

00:02:29.108 --> 00:02:30.637
On up here, you have a third degree

00:02:30.637 --> 00:02:32.338
so the numerator is gonna grow faster

00:02:32.338 --> 00:02:34.933
than the denominator so this
is going to be unbounded.

00:02:34.933 --> 00:02:36.311
So that will immediately tell you

00:02:36.311 --> 00:02:38.135
well this is gonna approach infinity

00:02:38.135 --> 00:02:41.513
so S is going to

00:02:42.806 --> 00:02:44.029
diverge

00:02:44.029 --> 00:02:45.864
but if you wanna do it a little bit

00:02:45.864 --> 00:02:47.539
less hand wavy than that we can

00:02:47.539 --> 00:02:50.004
actually do a little bit more algebra.

00:02:50.004 --> 00:02:54.737
Limit as n approaches infinity, 2n

00:02:54.737 --> 00:02:56.484
to the third power

00:02:56.484 --> 00:02:58.797
over, let's multiply this out,

00:02:58.797 --> 00:03:02.493
n squared plus 3n

00:03:02.494 --> 00:03:04.049
plus 2

00:03:04.049 --> 00:03:05.746
and, let's see, we can
divide the numerator

00:03:05.746 --> 00:03:08.981
and the denominator by n squared.

00:03:08.981 --> 00:03:11.643
So, this is going to be the limit

00:03:11.643 --> 00:03:14.802
as n approaches infinity of, if we divide

00:03:14.802 --> 00:03:17.040
the numerator by n squared, you're going

00:03:17.040 --> 00:03:19.611
to have, actually, let's
divide the numerator

00:03:19.611 --> 00:03:23.478
and, well yeah let's
divide it by n squared

00:03:23.478 --> 00:03:25.705
so, if we divide the
numerator by n squared,

00:03:25.705 --> 00:03:28.080
we're gonna have 2n and
then the denominator

00:03:28.080 --> 00:03:29.437
divided by n squared you're gonna have

00:03:29.437 --> 00:03:33.208
1 plus 3 over n

00:03:33.208 --> 00:03:36.545
plus 2 over n squared.

00:03:37.537 --> 00:03:39.341
Now, when you look at it like this,

00:03:39.341 --> 00:03:41.212
it becomes pretty clear this thing

00:03:41.212 --> 00:03:43.239
as n approaches infinity,
this thing is gonna

00:03:43.239 --> 00:03:45.847
towards infinity but this thing down here

00:03:45.847 --> 00:03:49.106
the denominator this
is gonna go towards 0.

00:03:49.106 --> 00:03:50.766
This is gonna go towards 0

00:03:50.766 --> 00:03:52.876
so the denominator's gonna go towards 1.

00:03:52.876 --> 00:03:56.275
So, this whole thing, is the limit

00:03:56.275 --> 00:03:59.705
is gonna go to infinity
and since the limit

00:03:59.705 --> 00:04:02.914
of the partial sums goes to infinity

00:04:02.914 --> 00:04:05.252
that mean that this infinite series

00:04:05.252 --> 00:04:07.044
is not going to be a finite value.

00:04:07.044 --> 00:04:09.276
It's just going to diverge.

00:04:09.276 --> 00:04:12.700
So, this character right over here

00:04:12.700 --> 00:04:14.177
is going

00:04:14.177 --> 00:04:18.003
to diverge.

00:04:18.140 --> 00:04:19.866
In order for it to have converged,

00:04:19.866 --> 00:04:22.410
this thing should have come, this limit

00:04:22.410 --> 00:04:25.107
should have been some finite value.

00:04:25.107 --> 00:04:26.901
So, hopefully, that makes sense.

00:04:26.901 --> 00:04:29.211
All we say is, "Look, infinite series,

00:04:29.211 --> 00:04:31.738
"we had a formula for the partial sum

00:04:31.738 --> 00:04:34.033
"of the first n terms
and then we said oh look

00:04:34.033 --> 00:04:36.511
"the series itself, the infinite series,

00:04:36.511 --> 00:04:39.335
"you could view it as a limit of,

00:04:39.335 --> 00:04:41.412
"as n approaches infinity,
of the partial sum

00:04:41.412 --> 00:04:43.905
"S sub n and we said hey,
that approach infinity

00:04:43.905 --> 00:04:45.846
"this thing is diverging."


Partial sums intro

WEBVTT
Kind: captions
Language: en

00:00:00.150 --> 00:00:01.033
- [Voiceover] Let's say that you have

00:00:01.033 --> 00:00:03.738
an infinite series, S, which

00:00:03.738 --> 00:00:08.738
is equal to the sum from n equals one,

00:00:09.554 --> 00:00:11.179
let me write that a little bit neater.

00:00:11.179 --> 00:00:13.837
n equals one to infinity

00:00:13.837 --> 00:00:15.590
of a sub n.

00:00:15.590 --> 00:00:17.529
This is all a little bit of review.

00:00:17.529 --> 00:00:19.190
We would say, well this is the same thing

00:00:19.190 --> 00:00:23.473
as a sub one, plus a sub two,

00:00:23.473 --> 00:00:25.690
plus a sub three, and we would just

00:00:25.690 --> 00:00:29.394
keep going on and on and on forever.

00:00:29.394 --> 00:00:30.758
Now what I want to introduce to you

00:00:30.758 --> 00:00:32.962
is the idea of a partial sum.

00:00:32.962 --> 00:00:35.725
This right over here
is an infinite series.

00:00:35.725 --> 00:00:38.071
But we can define a partial sum,

00:00:38.071 --> 00:00:42.435
so if we say S sub six,

00:00:42.435 --> 00:00:43.572
this notation says, okay,

00:00:43.572 --> 00:00:45.291
if S is an infinite series,

00:00:45.291 --> 00:00:50.291
S sub six is the partial
sum of the first six terms.

00:00:50.573 --> 00:00:52.871
So in this case, this is going to be

00:00:52.871 --> 00:00:54.450
we're not going to just
keep going on forever,

00:00:54.450 --> 00:00:56.493
this is going to be a sub one,

00:00:56.493 --> 00:01:00.012
plus a sub two, plus a sub three,

00:01:00.012 --> 00:01:03.797
plus a sub four, plus a sub five,

00:01:03.797 --> 00:01:06.477
plus a sub six.

00:01:06.477 --> 00:01:09.298
And I can make this a little
bit more tangible if you like.

00:01:09.310 --> 00:01:13.686
So let's say that S,
the infinite series S,

00:01:13.686 --> 00:01:18.260
is equal to the sum from n equals one,

00:01:18.260 --> 00:01:23.260
to infinity of one over n squared.

00:01:23.299 --> 00:01:26.282
In this case it would
be one over one squared,

00:01:26.282 --> 00:01:28.801
plus one over two squared,

00:01:28.801 --> 00:01:30.902
plus one over three squared,

00:01:30.902 --> 00:01:34.304
and we would just keep going
on and on and on forever.

00:01:34.304 --> 00:01:37.462
But what would S sub --

00:01:37.462 --> 00:01:39.899
I should do that in that same color.

00:01:39.899 --> 00:01:42.243
What would S --

00:01:42.243 --> 00:01:44.323
I said I would change color, and I didn't.

00:01:44.323 --> 00:01:48.780
What would S sub three be equal to?

00:01:48.780 --> 00:01:50.580
The partial sum of the first three terms,

00:01:50.580 --> 00:01:52.426
and I encourage you to pause the video

00:01:52.426 --> 00:01:55.885
and try to work through it on your own.

00:01:55.885 --> 00:01:58.359
Well, it's just going to be

00:01:58.359 --> 00:02:01.225
the first term one, plus the second term,

00:02:01.225 --> 00:02:04.684
1/4, plus the third term, 1/9,

00:02:04.684 --> 00:02:07.830
is going to be the sum
of the first three terms,

00:02:07.830 --> 00:02:09.130
and we can figure that out,

00:02:09.130 --> 00:02:11.266
that's to see if you have
a common denominator here,

00:02:11.266 --> 00:02:13.089
it's going to be 36.

00:02:13.089 --> 00:02:15.550
It's going to be 36/36,

00:02:15.550 --> 00:02:19.660
plus 9/36, plus 4/36,

00:02:19.660 --> 00:02:24.453
so this is going to be 49/36.

00:02:25.683 --> 00:02:28.221
49/36.

00:02:28.221 --> 00:02:29.267
So the whole point of this video,

00:02:29.267 --> 00:02:33.621
is just to appreciate this
idea of a partial sum.

00:02:33.621 --> 00:02:34.585
And what we'll see is,

00:02:34.585 --> 00:02:37.186
that you can actually
express what a partial sum

00:02:37.186 --> 00:02:38.706
might be algebraically.

00:02:38.706 --> 00:02:40.367
So for example,

00:02:40.367 --> 00:02:42.178
for example, let's give
ourselves a little bit

00:02:42.178 --> 00:02:44.523
more real estate here.

00:02:44.523 --> 00:02:47.623
Let's say, let's go back to just saying

00:02:47.623 --> 00:02:49.817
we have an infinite series, S,

00:02:49.817 --> 00:02:54.042
that is equal to the sum from n equals one

00:02:54.042 --> 00:02:57.362
to infinity of a sub n.

00:02:57.362 --> 00:03:00.798
And let's say we know the partial sum,

00:03:00.798 --> 00:03:04.096
S sub n, so the sum of the first n terms

00:03:04.096 --> 00:03:08.078
of this is equal to n squared

00:03:08.078 --> 00:03:12.559
minus three over

00:03:12.559 --> 00:03:16.164
n to the third plus four.

00:03:16.164 --> 00:03:17.882
So just as a bit of a reminder

00:03:17.882 --> 00:03:18.822
of what this is saying.

00:03:18.822 --> 00:03:20.166
S sub n...

00:03:20.166 --> 00:03:22.964
S sub n is the same thing as a sub one,

00:03:22.964 --> 00:03:26.145
plus a sub two, plus you keep going

00:03:26.145 --> 00:03:29.163
all the way to a sub n,

00:03:29.163 --> 00:03:31.160
and that's going to be
equal to this business,

00:03:31.160 --> 00:03:33.307
n squared minus three

00:03:33.307 --> 00:03:37.324
over n to the third plus four.

00:03:37.324 --> 00:03:39.182
Now, given that, if someone were

00:03:39.182 --> 00:03:40.945
to walk up to you on the street and say,

00:03:40.945 --> 00:03:42.571
okay now that you know the notation

00:03:42.571 --> 00:03:45.868
for a partial sum, I
have a little question

00:03:45.868 --> 00:03:47.668
to ask of you.

00:03:47.668 --> 00:03:51.510
If S is the infinite series,

00:03:51.510 --> 00:03:53.959
and I'm writing it in very
general terms right over here,

00:03:53.959 --> 00:03:55.445
so S is the infinite series

00:03:55.445 --> 00:03:58.465
from n equals one to infinity of a sub n,

00:03:58.465 --> 00:04:00.426
and the partial sum, S sub n,

00:04:00.426 --> 00:04:02.155
is defined this way,

00:04:02.155 --> 00:04:05.104
so someone, they tell
you these two things,

00:04:05.104 --> 00:04:09.365
and then they say find what

00:04:09.365 --> 00:04:13.533
the sum from n equals one to six

00:04:13.533 --> 00:04:15.703
of a sub n is, and I encourage you

00:04:15.703 --> 00:04:19.254
to pause the video and
try to figure it out.

00:04:19.430 --> 00:04:22.750
Well, this is just going to be a sub one,

00:04:22.750 --> 00:04:25.804
plus a sub two, plus a sub three,

00:04:25.804 --> 00:04:28.648
plus a sub four, and when I say sub

00:04:28.648 --> 00:04:30.227
that just means subscript,

00:04:30.227 --> 00:04:35.227
plus a sub five, plus a sub six,

00:04:35.312 --> 00:04:36.472
well that's just the same thing

00:04:36.472 --> 00:04:39.722
as the partial sum, this
is just the same thing

00:04:39.722 --> 00:04:43.903
as the partial sum of the first six terms

00:04:43.903 --> 00:04:45.389
for our infinite series.

00:04:45.389 --> 00:04:48.233
It's just going to be the
partial sum S sub six.

00:04:48.233 --> 00:04:50.926
And we know how to algebraically evaluate

00:04:50.926 --> 00:04:52.087
what S sub six is.

00:04:52.087 --> 00:04:55.547
We can apply this formula
that we were given.

00:04:55.547 --> 00:04:58.948
S sub six is equal to, well,

00:04:58.948 --> 00:04:59.980
everywhere we see an n,

00:04:59.980 --> 00:05:01.606
we replaced with a six,

00:05:01.606 --> 00:05:04.787
it's going to be six squared minus three

00:05:04.787 --> 00:05:07.967
over six to the third

00:05:07.967 --> 00:05:10.081
plus four, so what is this going to be?

00:05:10.081 --> 00:05:13.610
Six squared is 36 minus three,

00:05:13.610 --> 00:05:15.780
so that's 33,

00:05:15.780 --> 00:05:17.104
and six to the third, let's see,

00:05:17.104 --> 00:05:19.183
36 times six, I always forget,

00:05:19.183 --> 00:05:20.900
my brain wants to say 216,

00:05:20.900 --> 00:05:23.130
but let me make sure that
that's actually the case.

00:05:23.130 --> 00:05:26.019
Six times 30 is 180,

00:05:26.019 --> 00:05:28.946
plus 36, yes, it is 216,

00:05:28.946 --> 00:05:30.422
so I guess I have, inadvertently,

00:05:30.422 --> 00:05:32.314
by seeing six to the third so many times

00:05:32.314 --> 00:05:34.914
in my life, I have inadvertently memorized

00:05:34.914 --> 00:05:37.804
six to the third power,
never a horrible thing

00:05:37.804 --> 00:05:39.221
to have that in your brain.

00:05:39.221 --> 00:05:42.297
So this is going to be 216 plus four,

00:05:42.297 --> 00:05:45.502
so 220.

00:05:45.502 --> 00:05:49.843
So, S sub six, or the sum
of the first six terms

00:05:49.843 --> 00:05:52.014
of the series right over here,

00:05:52.014 --> 00:05:55.996
is 33/220, and we're done.

00:05:55.996 --> 00:05:57.761
And the whole point of this is just so you

00:05:57.761 --> 00:06:01.258
kind of appreciate, or
really do appreciate

00:06:01.258 --> 00:06:03.917
this partial sum notation,

00:06:03.917 --> 00:06:06.950
and understand what it really means.


Proof: Differentiability implies continuity

WEBVTT
Kind: captions
Language: en

00:00:00.235 --> 00:00:01.469
- [Voiceover] What I
hope to do in this video

00:00:01.469 --> 00:00:04.274
is prove that if a
function is differentiable

00:00:04.274 --> 00:00:06.442
at some point, C, that it's also

00:00:06.442 --> 00:00:09.002
going to be continuous at that point C.

00:00:09.002 --> 00:00:11.435
But, before we do the proof,
let's just remind ourselves

00:00:11.435 --> 00:00:15.008
what differentiability means
and what continuity means.

00:00:15.008 --> 00:00:17.246
So, first, differentiability.

00:00:17.442 --> 00:00:21.469
Differentiability

00:00:21.947 --> 00:00:24.210
So, let's think about that, first.

00:00:24.210 --> 00:00:27.440
And it's always helpful to draw ourselves

00:00:27.440 --> 00:00:29.242
a function.

00:00:29.374 --> 00:00:32.744
So, that's our Y-axis.

00:00:32.744 --> 00:00:36.269
This is our X-axis.

00:00:36.269 --> 00:00:38.507
And let's just draw some function, here.

00:00:38.507 --> 00:00:40.746
So, let's say my function looks like this

00:00:40.746 --> 00:00:44.133
and we care about the point X equals C,

00:00:44.133 --> 00:00:45.773
which is right over here.

00:00:45.773 --> 00:00:48.470
So, that's the point X equals C,

00:00:48.470 --> 00:00:50.434
and then, this value, of course,

00:00:50.434 --> 00:00:53.337
is going to be F of C.

00:00:53.337 --> 00:00:54.709
F of C.

00:00:54.709 --> 00:00:57.813
And one way that we
can find the derivative

00:00:57.813 --> 00:01:00.901
at X equals C, or the
slope of the tangent line

00:01:00.901 --> 00:01:05.270
at X equals C is, we could
start with some other point.

00:01:05.270 --> 00:01:08.547
Say, some arbitrary X out here.

00:01:08.547 --> 00:01:11.479
So, let's say this is
some arbitrary X out here.

00:01:11.479 --> 00:01:13.944
So, then, this point right
over there, this value,

00:01:13.944 --> 00:01:17.629
this Y value, would be F of X.

00:01:17.629 --> 00:01:19.108
Would be F of X.

00:01:19.108 --> 00:01:23.267
This graph, of course, is
a graph of Y equals F of X.

00:01:23.267 --> 00:01:27.513
And we can think about finding
the slope of this line,

00:01:27.913 --> 00:01:30.802
this secant line between these two points,

00:01:30.802 --> 00:01:33.609
but then, we can find the
limit as X approaches C.

00:01:33.609 --> 00:01:35.971
And as X approaches C, this secant,

00:01:35.971 --> 00:01:37.947
the slope of the secant
line is going to approach

00:01:37.947 --> 00:01:39.303
the slope of the tangent line, or,

00:01:39.303 --> 00:01:40.972
it's going to be the derivative.

00:01:40.972 --> 00:01:43.613
And so, we could take the limit...

00:01:44.811 --> 00:01:48.134
The limit as X approaches C,

00:01:48.134 --> 00:01:50.846
as X approaches C,

00:01:50.846 --> 00:01:54.301
of the slope of this secant line.

00:01:54.301 --> 00:01:55.206
So, what's the slope?

00:01:55.206 --> 00:01:57.641
Well, it's gonna be change
in Y over change in X.

00:01:57.641 --> 00:02:01.711
The change in Y is F of X minus F of C,

00:02:01.711 --> 00:02:03.678
that's our change in Y right over here.

00:02:03.678 --> 00:02:06.266
This is all review, this
is just one definition

00:02:06.266 --> 00:02:08.700
of the derivative, or one way
to think about the derivative.

00:02:08.700 --> 00:02:12.506
So, it's going to be F of X minus F of C,

00:02:12.506 --> 00:02:16.577
that's our change in Y,
over our change in X.

00:02:16.577 --> 00:02:20.371
Over our change in X, which is X minus C.

00:02:20.371 --> 00:02:23.974
It is X minus,

00:02:23.974 --> 00:02:26.480
X minus C.

00:02:26.480 --> 00:02:29.445
So, if this limit exists,
then, we're able to find

00:02:29.445 --> 00:02:32.066
the slope of the tangent
line at this point,

00:02:32.066 --> 00:02:34.172
and we call that slope
of the tangent line,

00:02:34.172 --> 00:02:37.510
we call that the derivative at X equals C.

00:02:37.510 --> 00:02:41.908
We say that this is going
to be equal to F prime,

00:02:41.908 --> 00:02:42.877
F prime of C.

00:02:42.877 --> 00:02:44.779
All of this is review.

00:02:44.779 --> 00:02:47.370
So, if we're saying, one
way to think about it,

00:02:47.370 --> 00:02:49.240
if we're saying that the function, F,

00:02:49.240 --> 00:02:51.362
is differentiable at X equals C,

00:02:51.362 --> 00:02:53.000
we're really just saying that this limit

00:02:53.000 --> 00:02:55.642
right over here actually exists.

00:02:55.642 --> 00:02:56.978
And if this limit actually exists,

00:02:56.978 --> 00:02:59.573
we just call that value F prime of C.

00:02:59.573 --> 00:03:02.079
So, that's just a review
of differentiability.

00:03:02.079 --> 00:03:06.338
Now, let's give ourselves
a review of continuity.

00:03:06.338 --> 00:03:10.581
Continuity.

00:03:10.581 --> 00:03:13.671
So, the definition for continuity is

00:03:13.671 --> 00:03:18.671
if the limit as X approaches C of F of X

00:03:19.412 --> 00:03:22.813
is equal to F of C.

00:03:22.813 --> 00:03:25.218
Now, this might seem a
little bit, you know,

00:03:25.218 --> 00:03:27.908
well, it might pop out
to you as being intuitive

00:03:27.908 --> 00:03:29.900
or it might seem a little,
well, where did this come from,

00:03:29.900 --> 00:03:31.277
well, let's visualize it and hopefully

00:03:31.277 --> 00:03:33.471
it'll make some intuitive sense.

00:03:33.471 --> 00:03:36.258
So, if you have a function,

00:03:36.258 --> 00:03:37.300
so, let's actually look at some cases

00:03:37.300 --> 00:03:39.080
where you're not continuous.

00:03:39.080 --> 00:03:41.966
And that actually might make
it a little bit more clear.

00:03:41.966 --> 00:03:45.443
So, if you had a point
discontinuity at X equals C,

00:03:45.443 --> 00:03:48.459
so, this is X equals C, so, if you had

00:03:48.459 --> 00:03:50.608
a point discontinuity, so,

00:03:50.608 --> 00:03:53.026
lemme draw it like this, actually.

00:03:53.026 --> 00:03:55.645
So, you have a gap, here, and X equals,

00:03:55.645 --> 00:03:59.779
when X equals C, F of C
is actually way up here.

00:03:59.779 --> 00:04:03.475
So, this is F of C, and then,

00:04:03.475 --> 00:04:05.744
the function continues like this.

00:04:05.744 --> 00:04:09.170
The limit, as X approaches C of F of X

00:04:09.170 --> 00:04:13.512
is going to be this value,
which is clearly different

00:04:13.512 --> 00:04:14.646
than F of C.

00:04:14.646 --> 00:04:17.433
This value right over here,
if you take the limit,

00:04:17.433 --> 00:04:22.414
if you take the limit as
X approaches C of F of X,

00:04:23.043 --> 00:04:24.272
you're approaching this value.

00:04:24.272 --> 00:04:26.144
This, right over here, is the limit,

00:04:26.144 --> 00:04:30.945
as X approaches C of F of X,
which is different than F of C.

00:04:30.945 --> 00:04:33.410
So, it makes it, so, this
definition of continuity

00:04:33.410 --> 00:04:35.204
seems to be good, at least for this case,

00:04:35.204 --> 00:04:38.980
because this is not a continuous function,

00:04:38.980 --> 00:04:40.900
you have a point discontinuity.

00:04:40.900 --> 00:04:43.399
So, for at least in this case, our,

00:04:43.399 --> 00:04:45.671
this definition of
continuity would properly

00:04:45.671 --> 00:04:49.371
identify this as not
a continuous function.

00:04:49.371 --> 00:04:52.533
Now, you could also think
about a jump discontinuity.

00:04:52.533 --> 00:04:55.679
You can also think about
a jump discontinuity.

00:04:55.679 --> 00:04:57.068
So, let's look at this.

00:04:57.068 --> 00:05:00.110
And all this is, hopefully,
a little bit of review.

00:05:00.110 --> 00:05:03.045
So, a jump discontinuity
at C, at X equals C,

00:05:03.045 --> 00:05:04.900
might look like this.

00:05:04.900 --> 00:05:07.598
Might look like this.

00:05:07.598 --> 00:05:11.343
So, this is at X equals C.

00:05:11.343 --> 00:05:13.870
So, this is X equals C right over here.

00:05:13.870 --> 00:05:17.145
This would be F of C.

00:05:17.145 --> 00:05:19.501
But, if you tried to
find a value at the limit

00:05:19.501 --> 00:05:21.935
as X approaches C of F of X,

00:05:21.935 --> 00:05:24.041
you'd get a different
value as you approach

00:05:24.041 --> 00:05:27.333
C from the negative side, you
would approach this value,

00:05:27.333 --> 00:05:29.143
and as you approach C
from the positive side,

00:05:29.143 --> 00:05:32.246
you would approach F of C, and
so, the limit wouldn't exist.

00:05:32.246 --> 00:05:34.103
So, this limit right
over here wouldn't exist

00:05:34.103 --> 00:05:37.503
in the case of jump, of this
type of a jump discontinuity.

00:05:37.503 --> 00:05:40.547
So, once again, this
definition would properly

00:05:40.547 --> 00:05:43.808
say that this is not,
this one right over here,

00:05:43.808 --> 00:05:47.674
is not continuous, this limit
actually would not even exist.

00:05:47.674 --> 00:05:49.671
And then, you could even look at a,

00:05:49.671 --> 00:05:54.245
you could look at a function
that is truly continuous.

00:05:54.245 --> 00:05:57.346
If you look at a function
that is truly continuous.

00:05:57.346 --> 00:06:00.077
So, something like this.

00:06:00.077 --> 00:06:01.403
Something like this.

00:06:01.403 --> 00:06:03.337
That is X equals C.

00:06:03.337 --> 00:06:05.100
Well, this is F of C.

00:06:05.100 --> 00:06:06.940
This is F of C.

00:06:06.940 --> 00:06:09.843
And if you were to take the
limit as X approaches C,

00:06:09.843 --> 00:06:13.404
as X approaches C from
either side of F of X,

00:06:13.404 --> 00:06:15.371
you're going to approach F of C.

00:06:15.371 --> 00:06:19.234
So, here, you have the
limit as X approaches C

00:06:19.234 --> 00:06:22.873
of F of X, indeed, is equal to F of C.

00:06:22.873 --> 00:06:25.537
So, it's what you would expect
for a continuous function.

00:06:25.537 --> 00:06:27.346
So, now that we've done that review

00:06:27.346 --> 00:06:29.669
of differentiability and continuity,

00:06:29.669 --> 00:06:32.370
let's prove that differentiability

00:06:32.370 --> 00:06:35.180
actually implies continuity,
and I think it's important

00:06:35.180 --> 00:06:36.767
to kinda do this review,
just so that you can

00:06:36.767 --> 00:06:38.703
really visualize things.

00:06:38.703 --> 00:06:42.368
So, differentiability implies this limit

00:06:42.368 --> 00:06:44.101
right over here exists.

00:06:44.101 --> 00:06:46.378
So, let's start with a
slightly different limit.

00:06:46.378 --> 00:06:48.941
Lemme draw a line, here, actually.

00:06:48.941 --> 00:06:51.744
Lemme draw a line just so we're
doing something different.

00:06:51.744 --> 00:06:56.101
So, let's take, let us take the limit

00:06:56.101 --> 00:07:00.844
as X approaches C of F of X,

00:07:01.573 --> 00:07:04.511
of F of X minus F of C.

00:07:04.511 --> 00:07:07.175
Of F of X minus F of C.

00:07:07.175 --> 00:07:09.046
Well, can we rewrite this?

00:07:09.046 --> 00:07:11.744
Well, we could rewrite this as the limit,

00:07:11.744 --> 00:07:14.739
as X approaches C, and
we could essentially

00:07:14.739 --> 00:07:16.269
take this expression and multiply

00:07:16.269 --> 00:07:18.236
and divide it by X minus C.

00:07:18.236 --> 00:07:20.842
So, let's multiply it times X minus C.

00:07:20.842 --> 00:07:24.536
X minus C, and divide it by X minus C.

00:07:24.536 --> 00:07:28.608
So, we have F of X minus F of C,

00:07:28.608 --> 00:07:31.635
all of that over X minus C.

00:07:31.635 --> 00:07:33.980
So, all I did is I
multiplied and I divided

00:07:33.980 --> 00:07:35.900
by X minus C.

00:07:35.900 --> 00:07:38.644
Well, what's this limit
going to be equal to?

00:07:38.644 --> 00:07:40.574
This is going to be equal to,

00:07:40.574 --> 00:07:42.700
it's going to be the limit,
and I'm just applying

00:07:42.700 --> 00:07:47.507
the property of limit, applying
a property of limits, here.

00:07:47.507 --> 00:07:49.468
So, the limit of the product is equal to

00:07:49.468 --> 00:07:51.581
the same thing as a product of the limits.

00:07:51.581 --> 00:07:55.365
So, it's the limit as X
approaches C of X minus C,

00:07:55.365 --> 00:07:58.631
times the limit, lemme write this way,

00:07:58.631 --> 00:08:03.631
times the limit as X approaches C

00:08:03.802 --> 00:08:08.274
of F of X minus F of C,

00:08:08.274 --> 00:08:11.376
all of that over X minus C.

00:08:11.376 --> 00:08:15.344
Now, what is this thing right over here?

00:08:15.511 --> 00:08:19.707
Well, if we assume that
F is differentiable at C,

00:08:19.707 --> 00:08:20.944
and we're going to do that, actually,

00:08:20.944 --> 00:08:22.443
I should have started off there.

00:08:22.443 --> 00:08:26.867
Let's assume 'cause we wanted
to show the differentiability,

00:08:26.867 --> 00:08:28.475
it proves continuity.

00:08:28.475 --> 00:08:32.946
If we assume F differentiable,

00:08:34.146 --> 00:08:37.667
differentiable at C, well then,

00:08:37.667 --> 00:08:41.431
this right over here is just
going to be F prime of C.

00:08:41.807 --> 00:08:43.441
This right over here, we
just saw it right over here,

00:08:43.441 --> 00:08:45.407
that's this exact same thing.

00:08:45.407 --> 00:08:49.702
This is F prime, F prime of C.

00:08:50.104 --> 00:08:51.334
And what is this thing right over here?

00:08:51.334 --> 00:08:54.413
The limit as X approaches C of X minus C?

00:08:54.413 --> 00:08:55.679
Well, that's just gonna be zero.

00:08:55.679 --> 00:08:57.180
As X approaches C, there's gonna become,

00:08:57.180 --> 00:09:00.900
approach C minus C, that's
just going to be zero.

00:09:01.474 --> 00:09:04.645
So, what's zero times F prime of C?

00:09:04.645 --> 00:09:06.342
Well, F prime of C is just
going to be some value,

00:09:06.342 --> 00:09:09.276
so, zero times anything
is just going to be zero.

00:09:09.276 --> 00:09:10.942
So, I did all that work to get a zero.

00:09:10.942 --> 00:09:12.909
Now, why is this interesting?

00:09:12.909 --> 00:09:14.874
Well, we just said, we just assumed

00:09:14.874 --> 00:09:16.714
that if F is differentiable at C,

00:09:16.714 --> 00:09:20.803
and we evaluate this limit, we get zero.

00:09:20.803 --> 00:09:23.766
So, if we assume F is differentiable at C,

00:09:23.766 --> 00:09:27.946
we can write, we can write the limit,

00:09:27.946 --> 00:09:32.033
I'm just rewriting it, the
limit as X approaches C

00:09:32.033 --> 00:09:36.811
of F of X minus F of
C, and I could even put

00:09:36.811 --> 00:09:37.812
parenthesis around it like that,

00:09:37.812 --> 00:09:39.545
which I already did up here,

00:09:39.545 --> 00:09:42.008
is equal to zero.

00:09:42.008 --> 00:09:43.100
Well, this is the same thing,

00:09:43.100 --> 00:09:44.379
I could use limit properties again,

00:09:44.379 --> 00:09:46.066
this is the same thing as saying,

00:09:46.066 --> 00:09:47.406
and I'll do it over, well, actually,

00:09:47.406 --> 00:09:49.514
lemme do it down here.

00:09:49.514 --> 00:09:54.514
The limit as X approaches C of F of X

00:09:55.235 --> 00:09:59.373
minus the limit as X approaches C

00:09:59.634 --> 00:10:04.634
of F of C, of F of C, is equal to zero.

00:10:05.375 --> 00:10:06.967
The different, the limit of the difference

00:10:06.967 --> 00:10:09.233
is the same thing as the
difference of the limits.

00:10:09.233 --> 00:10:11.834
Well, what's this thing
over here going to be?

00:10:11.834 --> 00:10:13.301
Well, F of C is just a number,

00:10:13.301 --> 00:10:14.844
it's not a function of X anymore,

00:10:14.844 --> 00:10:17.637
it's just, F of C is going
to valuate it to something.

00:10:17.637 --> 00:10:20.668
So, this is just going to be F of C.

00:10:20.900 --> 00:10:22.646
This is just going to be F of C.

00:10:22.646 --> 00:10:25.303
So, the limit of F of X as X approaches C,

00:10:25.303 --> 00:10:27.878
minus F of C is equal to zero.

00:10:27.878 --> 00:10:30.733
Well, just add F of C to both
sides and what do you get?

00:10:30.733 --> 00:10:34.933
Well, you get the limit as X approaches C

00:10:34.933 --> 00:10:38.804
of F of X is equal to F of C.

00:10:38.911 --> 00:10:41.033
And this is the definition of continuity.

00:10:41.033 --> 00:10:44.402
The limit of my function as X approaches C

00:10:44.402 --> 00:10:47.167
is equal to the function,
is equal to the value

00:10:47.167 --> 00:10:49.068
of the function at C.

00:10:49.068 --> 00:10:53.340
This is, this means that
our function is continuous.

00:10:54.101 --> 00:10:57.347
Continuous at C.

00:10:57.347 --> 00:11:01.012
So, just a reminder, we started assuming

00:11:01.012 --> 00:11:03.609
F differentiable at C, we use that fact

00:11:03.609 --> 00:11:06.847
to evaluate this limit right over here,

00:11:06.847 --> 00:11:08.875
which, we got to be equal to zero,

00:11:08.875 --> 00:11:11.075
and if that limit is equal to zero,

00:11:11.075 --> 00:11:13.681
then, it just follows, just
doing a little bit of algebra

00:11:13.681 --> 00:11:15.380
and using properties of limits,

00:11:15.380 --> 00:11:17.736
that the limit as X approaches C of F of X

00:11:17.736 --> 00:11:20.771
is equal to F of C, and
that's our definition

00:11:20.771 --> 00:11:23.379
of being continuous.

00:11:23.379 --> 00:11:26.345
Continuous at the point C.

00:11:26.345 --> 00:11:27.714
So, hopefully, that satisfies you.

00:11:27.714 --> 00:11:31.007
If we know that the
derivative exists at a point,

00:11:31.007 --> 00:11:32.707
if it's differentiable at a point C,

00:11:32.707 --> 00:11:35.780
that means it's also
continuous at that point C.

00:11:35.780 --> 00:11:38.513
The function is also
continuous at that point.


If function u is continuous at x, then _u_0 as _x_0

WEBVTT
Kind: captions
Language: en

00:00:00.452 --> 00:00:02.170
- [Voiceover] The result
that I hope to show you,

00:00:02.170 --> 00:00:04.104
or you give you an
intuition for, in this video

00:00:04.104 --> 00:00:06.001
is something that we will use

00:00:06.001 --> 00:00:08.102
in the proof of the chain rule,

00:00:08.102 --> 00:00:09.569
or in a proof of the chain rule actually.

00:00:09.569 --> 00:00:13.307
We may do more than
proof of the chain rule.

00:00:13.307 --> 00:00:14.886
But the result we're gonna look at

00:00:14.886 --> 00:00:16.929
is if we have some function u

00:00:16.929 --> 00:00:18.474
which is a function of x,

00:00:18.474 --> 00:00:22.188
and we know that it is continuous

00:00:23.871 --> 00:00:26.729
at x equals c,

00:00:26.729 --> 00:00:30.084
so if we know this, then
that's going to imply

00:00:30.084 --> 00:00:34.159
that the change in u goes

00:00:34.159 --> 00:00:39.114
to zero as our change in x in this region

00:00:39.114 --> 00:00:41.424
around c goes to zero.

00:00:41.424 --> 00:00:43.351
This is what I want to
get an intuition for.

00:00:43.351 --> 00:00:45.686
That if u is continuous at c,

00:00:45.686 --> 00:00:48.667
then as our change in x around c

00:00:48.667 --> 00:00:51.745
gets smaller, and smaller, and smaller

00:00:51.745 --> 00:00:54.566
as it approaches zero,
then our change in u

00:00:54.566 --> 00:00:57.110
approaches zero as well.

00:00:58.296 --> 00:01:00.282
To think about this, or
to even kind of prove it

00:01:00.282 --> 00:01:03.450
to ourselves a little bit more rigorously,

00:01:03.450 --> 00:01:04.785
let's think about what it means

00:01:04.785 --> 00:01:07.409
to be continuous at x equals c.

00:01:07.409 --> 00:01:10.984
Well the definition of continuity is,

00:01:10.984 --> 00:01:14.003
so this literally is
the same thing as saying

00:01:14.003 --> 00:01:18.297
that the limit as x approaches c

00:01:18.297 --> 00:01:23.104
of u of x is equal to u of c.

00:01:23.104 --> 00:01:26.215
The limit that our function approaches

00:01:26.215 --> 00:01:27.556
as x approaches c is equal

00:01:27.556 --> 00:01:29.408
to the value of the function at c.

00:01:29.408 --> 00:01:31.811
We don't have a point discontinuity

00:01:31.811 --> 00:01:33.471
or a jump discontinuity.

00:01:33.471 --> 00:01:34.574
If we had a jump discontinuity,

00:01:34.574 --> 00:01:35.932
then the limit wouldn't exist,

00:01:35.932 --> 00:01:38.231
and we've seen that in previous videos.

00:01:38.231 --> 00:01:40.750
Now I'm just gonna
manipulate this algebraically

00:01:40.750 --> 00:01:43.108
so it essentially gives us this conclusion

00:01:43.108 --> 00:01:44.732
right over here.

00:01:44.732 --> 00:01:46.752
So this we can rewrite.

00:01:46.752 --> 00:01:48.389
It's important to realize that u of c,

00:01:48.389 --> 00:01:50.003
this is just going to be some value.

00:01:50.003 --> 00:01:53.207
It looks like maybe this is a function

00:01:53.207 --> 00:01:54.251
of x or something, but no,

00:01:54.251 --> 00:01:55.970
this is just going to be some value.

00:01:55.970 --> 00:01:57.769
I've inputted c here and I've evaluated

00:01:57.769 --> 00:01:59.267
the function of that, and so this is going

00:01:59.267 --> 00:01:59.864
to be some number.

00:01:59.864 --> 00:02:01.719
It could be five, or seven, or pi,

00:02:01.719 --> 00:02:03.947
or negative one, but it's just going

00:02:03.947 --> 00:02:06.878
to be some value, some constant.

00:02:06.878 --> 00:02:08.796
So I can treat it like a constant.

00:02:08.796 --> 00:02:10.408
So this is going to be the same thing

00:02:10.408 --> 00:02:14.386
as saying the limit as x

00:02:14.386 --> 00:02:19.188
approaches c of u of x

00:02:19.188 --> 00:02:21.547
minus u of c

00:02:21.547 --> 00:02:24.101
is equal to zero.

00:02:24.101 --> 00:02:26.028
And actually, in the video where we prove

00:02:26.028 --> 00:02:28.501
that differentiability implies continuity,

00:02:28.501 --> 00:02:31.764
we started with this and we proved

00:02:31.764 --> 00:02:32.622
that right over there.

00:02:32.622 --> 00:02:35.048
We show that these two
are equivalent things.

00:02:35.048 --> 00:02:36.036
But hopefully you could even think

00:02:36.036 --> 00:02:36.656
about the intuition.

00:02:36.656 --> 00:02:41.462
This is just if the limit of u of x

00:02:41.462 --> 00:02:43.564
is x approaches c is equal to this,

00:02:43.564 --> 00:02:45.468
then when you evaluate this limit,

00:02:45.468 --> 00:02:46.966
the limit is x approaches c,

00:02:46.966 --> 00:02:50.611
well this thing is going
to approach u of c,

00:02:50.611 --> 00:02:51.892
because we saw it right up here,

00:02:51.892 --> 00:02:53.831
u of c minus u of c is indeed

00:02:53.831 --> 00:02:55.758
going to be equal to zero.

00:02:55.758 --> 00:02:57.871
So hopefully you don't
feel like it's too much

00:02:57.871 --> 00:02:59.810
of a stretch, and you can just subtract

00:02:59.810 --> 00:03:02.016
u of c from both sides
and apply properties

00:03:02.016 --> 00:03:05.429
of limits, and you can
get this result as well.

00:03:05.429 --> 00:03:07.750
But this is interesting
because this essentially

00:03:07.750 --> 00:03:09.955
can take us to this,

00:03:09.955 --> 00:03:11.483
the idea that as our change in x

00:03:11.483 --> 00:03:13.285
gets smaller, and smaller, and smaller

00:03:13.285 --> 00:03:15.723
as it approaches zero, then our change

00:03:15.723 --> 00:03:18.627
in our function is also
going to approach zero.

00:03:18.627 --> 00:03:22.085
Now let's just graph
this or visualize this

00:03:22.085 --> 00:03:24.152
to get a sense of that.

00:03:24.152 --> 00:03:29.152
So this is our x axis.

00:03:29.512 --> 00:03:30.676
Woops.

00:03:30.676 --> 00:03:32.475
That's our x axis.

00:03:32.475 --> 00:03:35.578
Let's call that our u axis maybe.

00:03:35.578 --> 00:03:36.925
And I did u intentionally because

00:03:36.925 --> 00:03:38.121
that's the variable we'll use

00:03:38.121 --> 00:03:41.766
in our proof of the chain rule video.

00:03:41.766 --> 00:03:43.067
Let's say this right over here

00:03:43.067 --> 00:03:44.892
is our function.

00:03:45.318 --> 00:03:47.626
Let's say this right over here,

00:03:47.626 --> 00:03:50.089
that is c.

00:03:50.089 --> 00:03:52.889
This right over here is u of c.

00:03:54.011 --> 00:03:56.521
U of c.

00:03:56.521 --> 00:04:00.734
Then let's just take some
arbitrary x over here.

00:04:01.115 --> 00:04:03.429
So some arbitrary x,

00:04:03.429 --> 00:04:07.949
and then this right over here is u of x.

00:04:11.787 --> 00:04:15.577
So if we define our change,

00:04:16.499 --> 00:04:17.225
let me do this.

00:04:17.225 --> 00:04:21.295
If we define our change in u is equal

00:04:21.295 --> 00:04:23.396
to u of x minus u of c,

00:04:23.396 --> 00:04:24.569
which makes sense because this

00:04:24.569 --> 00:04:27.790
is our change in u.

00:04:27.790 --> 00:04:31.206
So let's say this is going to be u of x

00:04:31.206 --> 00:04:34.246
minus u of c.

00:04:34.246 --> 00:04:38.392
If we define our change in x

00:04:38.392 --> 00:04:40.043
is equal to x minus c,

00:04:40.043 --> 00:04:41.098
which it is in this case,

00:04:41.098 --> 00:04:43.492
it is x minus c,

00:04:46.504 --> 00:04:48.580
then we can rewrite this limit

00:04:48.580 --> 00:04:49.527
right over here.

00:04:49.527 --> 00:04:52.568
Instead of saying the
limit as x approaches c,

00:04:52.568 --> 00:04:55.192
we could write the limit as delta x

00:04:55.192 --> 00:04:58.372
approaches zero, because
if x approaches c,

00:04:58.372 --> 00:05:01.090
then delta x is going to approach zero.

00:05:01.090 --> 00:05:02.668
So we could write this the limit

00:05:02.668 --> 00:05:07.640
as delta x approaches zero of delta u

00:05:10.580 --> 00:05:12.202
is going to be equal to zero.

00:05:12.202 --> 00:05:13.549
We define this as our change in u,

00:05:13.549 --> 00:05:15.104
and it is our change in u.

00:05:15.104 --> 00:05:18.005
So this is equal to zero.

00:05:18.005 --> 00:05:19.979
So another way of thinking about this

00:05:19.979 --> 00:05:23.648
is as delta x approaches zero,

00:05:23.648 --> 00:05:25.679
our change in u, our
change in the function,

00:05:25.679 --> 00:05:27.340
is going to approach zero.

00:05:27.340 --> 00:05:32.340
So as delta x approaches zero,

00:05:35.423 --> 00:05:38.702
delta u approaches zero.

00:05:38.702 --> 00:05:39.742
That's what we wrote over here.

00:05:39.742 --> 00:05:41.425
Delta u approaches zero

00:05:41.425 --> 00:05:43.200
as delta x approaches zero.

00:05:43.200 --> 00:05:47.124
In a lot of ways, this is
hopefully common sense.

00:05:47.124 --> 00:05:49.040
We're dealing with a continuous function.

00:05:49.040 --> 00:05:50.598
As you get smaller and smaller,

00:05:50.598 --> 00:05:51.639
and you can just think of it this way,

00:05:51.639 --> 00:05:54.357
as you get smaller and
smaller changes in x's,

00:05:54.357 --> 00:05:56.759
as our change in x gets smaller,

00:05:56.759 --> 00:05:58.937
and smaller, and smaller, and smaller,

00:05:58.937 --> 00:06:00.375
well it's because it's continuous.

00:06:00.375 --> 00:06:00.982
You wouldn't be able to say this

00:06:00.982 --> 00:06:02.677
for a discontinuous function,

00:06:02.677 --> 00:06:04.177
but because it's continuous,

00:06:04.177 --> 00:06:05.094
or you wouldn't be able to say this

00:06:05.094 --> 00:06:07.962
for some discontinuous functions,

00:06:07.962 --> 00:06:09.621
as our change in x gets smaller,

00:06:09.621 --> 00:06:11.014
and smaller, and smaller,

00:06:11.014 --> 00:06:13.104
then our change in u is
going to get smaller,

00:06:13.104 --> 00:06:14.057
and smaller, and smaller.

00:06:14.057 --> 00:06:15.317
So it makes intuitive sense,

00:06:15.317 --> 00:06:16.338
but hopefully this makes you feel

00:06:16.338 --> 00:06:18.013
even better about it
because we're going to use

00:06:18.013 --> 00:06:21.112
this idea to prove the chain rule

00:06:21.112 --> 00:06:23.027
in the next video.


Chain rule proof

WEBVTT
Kind: captions
Language: en

00:00:00.301 --> 00:00:02.101
- What I hope to do in this video

00:00:02.101 --> 00:00:05.340
is a proof of the famous and useful

00:00:05.340 --> 00:00:10.181
and somewhat elegant and
sometimes infamous chain rule.

00:00:10.501 --> 00:00:13.342
And, if you've been
following some of the videos

00:00:13.342 --> 00:00:16.122
on "differentiability implies continuity",

00:00:16.122 --> 00:00:18.282
and what happens to a continuous function

00:00:18.282 --> 00:00:21.962
as our change in x, if x is
our independent variable,

00:00:21.962 --> 00:00:24.722
as that approaches zero, how the change

00:00:24.722 --> 00:00:26.482
in our function approaches zero,

00:00:26.482 --> 00:00:31.182
then this proof is actually
surprisingly straightforward,

00:00:31.182 --> 00:00:32.522
so let's just get to it, and this is just

00:00:32.522 --> 00:00:35.702
one of many proofs of the chain rule.

00:00:35.702 --> 00:00:38.781
So the chain rule tells us that if

00:00:40.410 --> 00:00:43.343
y is a function of u,

00:00:43.343 --> 00:00:45.356
which is a function of x,

00:00:45.541 --> 00:00:49.702
and we want to figure out
the derivative of this,

00:00:49.702 --> 00:00:53.162
so we want to differentiate
this with respect to x,

00:00:53.162 --> 00:00:56.523
so we're gonna differentiate
this with respect to x,

00:00:56.802 --> 00:00:58.143
we could write this as the derivative

00:00:58.143 --> 00:01:00.343
of y with respect to x,

00:01:00.343 --> 00:01:02.262
which is going to be
equal to the derivative

00:01:02.262 --> 00:01:05.003
of y with respect to u,

00:01:05.003 --> 00:01:09.582
times the derivative
of u with respect to x.

00:01:09.603 --> 00:01:12.263
This is what the chain rule tells us.

00:01:12.263 --> 00:01:15.718
But how do we actually
go about proving it?

00:01:15.903 --> 00:01:18.203
Well we just have to remind ourselves

00:01:18.203 --> 00:01:21.315
that the derivative of
y with respect to x...

00:01:21.423 --> 00:01:24.866
the derivative of y with respect to x,

00:01:24.866 --> 00:01:29.866
is equal to the limit as
delta x approaches zero

00:01:30.703 --> 00:01:35.026
of change in y over change in x.

00:01:35.303 --> 00:01:38.355
Now we can do a little bit of
algebraic manipulation here

00:01:38.355 --> 00:01:41.058
to introduce a change
in u, so let's do that.

00:01:41.058 --> 00:01:43.743
So this is going to be the same thing as

00:01:43.743 --> 00:01:48.056
the limit as delta x approaches zero,

00:01:48.303 --> 00:01:51.059
and I'm gonna rewrite
this part right over here.

00:01:51.059 --> 00:01:53.387
I'm gonna essentially divide and multiply

00:01:53.387 --> 00:01:55.183
by a change in u.

00:01:55.183 --> 00:01:58.442
So I could rewrite this as delta y

00:01:58.442 --> 00:02:00.638
over delta u

00:02:00.918 --> 00:02:04.802
times delta u, whoops...

00:02:05.062 --> 00:02:09.741
times delta u over delta x.

00:02:09.741 --> 00:02:11.301
Change in y over change in u,

00:02:11.301 --> 00:02:13.681
times change in u over change in x.

00:02:13.681 --> 00:02:17.322
And you can see, these are
just going to be numbers here,

00:02:17.322 --> 00:02:19.349
so our change in u, this
would cancel with that,

00:02:19.349 --> 00:02:22.202
and you'd be left with
change in y over change x,

00:02:22.202 --> 00:02:24.342
which is exactly what we had here.

00:02:24.342 --> 00:02:28.202
So nothing earth-shattering just yet.

00:02:28.602 --> 00:02:31.105
But what's this going to be equal to?

00:02:31.382 --> 00:02:33.114
What's this going to be equal to?

00:02:33.114 --> 00:02:34.442
Well the limit of the product

00:02:34.442 --> 00:02:36.328
is the same thing as the
product of the limit,

00:02:36.328 --> 00:02:39.021
so this is going to be the same thing as

00:02:39.021 --> 00:02:42.322
the limit as delta x

00:02:42.322 --> 00:02:45.861
approaches zero of,
and I'll color-coat it,

00:02:45.861 --> 00:02:49.141
of this stuff, of delta y

00:02:49.141 --> 00:02:52.434
over delta u, times--

00:02:52.773 --> 00:02:54.742
maybe I'll put parentheses around it,

00:02:56.971 --> 00:02:59.728
times the limit...

00:03:00.267 --> 00:03:03.247
the limit as delta x approaches zero,

00:03:03.455 --> 00:03:05.796
delta x approaches zero,

00:03:05.796 --> 00:03:08.492
of this business.

00:03:08.492 --> 00:03:10.626
So let me put some parentheses around it.

00:03:10.933 --> 00:03:14.154
Delta u over delta x.

00:03:14.695 --> 00:03:16.855
So what does this simplify to?

00:03:16.855 --> 00:03:18.776
Well this right over here,
this is the definition,

00:03:18.776 --> 00:03:21.696
and if we're assuming, in
order for this to even be true,

00:03:21.696 --> 00:03:24.955
we have to assume that u and y

00:03:24.955 --> 00:03:27.235
are differentiable at x.

00:03:27.235 --> 00:03:29.955
So we assume, in order
for this to be true,

00:03:29.955 --> 00:03:31.497
we're assuming...

00:03:32.976 --> 00:03:37.717
we're assuming y comma
u are differentiable...

00:03:42.535 --> 00:03:44.322
are differentiable at x.

00:03:44.322 --> 00:03:46.001
And remember also, if
they're differentiable at x,

00:03:46.001 --> 00:03:48.062
that means they're continuous at x.

00:03:48.062 --> 00:03:49.662
But if u is differentiable at x,

00:03:49.662 --> 00:03:53.371
then this limit exists, and
this is the derivative of...

00:03:53.371 --> 00:03:56.442
this is u prime of x, or du/dx,

00:03:56.442 --> 00:03:58.542
so this right over here...

00:03:58.542 --> 00:04:02.775
we can rewrite as du/dx,

00:04:02.775 --> 00:04:04.903
I think you see where this is going.

00:04:04.903 --> 00:04:06.703
Now this right over here,

00:04:06.703 --> 00:04:08.743
just looking at it the way
it's written out right here,

00:04:08.743 --> 00:04:12.043
we can't quite yet call this dy/du,

00:04:12.043 --> 00:04:15.123
because this is the limit
as delta x approaches zero,

00:04:15.123 --> 00:04:17.883
not the limit as delta u approaches zero.

00:04:17.883 --> 00:04:19.323
But we just have to remind ourselves

00:04:19.323 --> 00:04:22.091
the results from, probably,
the previous video

00:04:22.091 --> 00:04:23.902
depending on how you're watching it,

00:04:23.902 --> 00:04:26.263
which is, if we have a function u

00:04:26.263 --> 00:04:29.362
that is continuous at a point,

00:04:29.362 --> 00:04:31.703
that, as delta x approaches zero,

00:04:31.703 --> 00:04:33.816
delta u approaches zero.

00:04:34.123 --> 00:04:36.544
So we can actually rewrite this...

00:04:36.544 --> 00:04:40.144
we can rewrite this right over here,

00:04:40.144 --> 00:04:41.843
instead of saying delta x approaches zero,

00:04:41.843 --> 00:04:43.384
that's just going to have the effect,

00:04:43.384 --> 00:04:45.284
because u is differentiable at x,

00:04:45.284 --> 00:04:47.124
which means it's continuous at x,

00:04:47.124 --> 00:04:49.943
that means that delta u
is going to approach zero.

00:04:49.943 --> 00:04:52.637
As our change in x gets smaller
and smaller and smaller,

00:04:52.637 --> 00:04:53.845
our change in u is going to get

00:04:53.845 --> 00:04:56.166
smaller and smaller and smaller.

00:04:56.166 --> 00:04:57.885
So we can rewrite this,

00:04:57.885 --> 00:05:00.425
as our change in u approaches zero,

00:05:00.425 --> 00:05:02.024
and when we rewrite it like that,

00:05:02.024 --> 00:05:04.225
well then this is just dy/du.

00:05:04.225 --> 00:05:08.365
This is just dy, the derivative
of y, with respect to u.

00:05:08.565 --> 00:05:10.845
So just like that, if we assume

00:05:10.845 --> 00:05:14.185
y and u are differentiable at x,

00:05:14.185 --> 00:05:16.165
or you could say that
y is a function of u,

00:05:16.165 --> 00:05:18.485
which is a function of x,

00:05:18.485 --> 00:05:21.381
we've just shown, in
fairly simple algebra here,

00:05:21.381 --> 00:05:23.044
and using some assumptions about

00:05:23.044 --> 00:05:25.225
differentiability and continuity,

00:05:25.225 --> 00:05:27.004
that it is indeed the case

00:05:27.004 --> 00:05:29.346
that the derivative of y with respect to x

00:05:29.346 --> 00:05:32.205
is equal to the derivative
of y with respect to u

00:05:32.205 --> 00:05:35.165
times the derivative
of u with respect to x.

00:05:35.165 --> 00:05:37.844
Hopefully you find that convincing.