ALS - Amyotrophic lateral sclerosis

WEBVTT
Kind: captions
Language: en

00:00:00.693 --> 00:00:05.122
- ALS or Amyotrophic Lateral
Sclerosis is an extremely

00:00:05.122 --> 00:00:08.292
debilitating and usually fatal disease

00:00:08.292 --> 00:00:09.521
and it's a horrible disease.

00:00:09.521 --> 00:00:12.630
You have your motor neurons,
the neurons that are

00:00:12.630 --> 00:00:14.796
essentially activating your
muscles, telling your muscles

00:00:14.796 --> 00:00:19.701
when to twitch or when to
contract and that these neurons,

00:00:19.701 --> 00:00:23.519
these cells that traverse,
that go from your brain

00:00:23.519 --> 00:00:24.566
all the way to your muscles,

00:00:24.566 --> 00:00:27.060
so there's these unusually long cells.

00:00:27.060 --> 00:00:30.313
They go through your spinal
cord, they degenerate.

00:00:30.790 --> 00:00:33.681
They degenerate and as they degenerate,

00:00:33.681 --> 00:00:36.166
there's no way for your
brain to send the signals

00:00:36.166 --> 00:00:38.397
to your muscles to do the
things that your muscles

00:00:38.397 --> 00:00:42.156
need to do and so your muscles atrophy

00:00:42.156 --> 00:00:45.129
and so that's actually
where the first part of ALS

00:00:45.129 --> 00:00:48.539
comes from, the "A" in ALS, Amyotrophic.

00:00:48.539 --> 00:00:52.238
"A" means no or not or
kind of the opposite.

00:00:52.238 --> 00:00:54.986
"Myo" means, is referring to muscle

00:00:54.986 --> 00:00:57.265
and 'trophic" is nourishment,
so this is one way to

00:00:57.265 --> 00:00:59.151
think about it is "no muscle nourishment",

00:00:59.151 --> 00:01:01.243
but what's really happening
is that the muscles

00:01:01.243 --> 00:01:03.813
start to die away because
they're no longer to be--

00:01:03.813 --> 00:01:06.972
They can no longer be activated
because, I guess you could

00:01:06.972 --> 00:01:10.966
say, the wires or motor
neurons that would normally

00:01:10.966 --> 00:01:15.042
activate them, they degenerate
and what's especially

00:01:15.042 --> 00:01:17.977
horrible about this is,
obviously, it causes paralysis,

00:01:17.977 --> 00:01:21.661
people lose control over
basic things including their

00:01:21.661 --> 00:01:25.180
speech, is the whole time that
this is happening and this is

00:01:25.180 --> 00:01:30.132
degenerating, the individual
in question has full mental

00:01:30.132 --> 00:01:33.104
awareness, so they're fully
aware of what's going on in

00:01:33.104 --> 00:01:38.104
their body so you could imagine
it's a fairly horrible thing

00:01:38.380 --> 00:01:42.478
for people to go through and
there have been many famous

00:01:42.478 --> 00:01:47.478
cases of people with ALS, most
famously probably Lou Gehrig.

00:01:47.683 --> 00:01:50.678
In fact, ALS in the U.S.,
at least, is often called

00:01:50.678 --> 00:01:53.598
Lou Gehrig's disease, one
of the most famous baseball

00:01:53.598 --> 00:01:58.213
players of all time for
the New York Yankees.

00:01:58.213 --> 00:02:02.545
He died of ALS, really,
when he was about my age

00:02:02.545 --> 00:02:04.523
in his late 30s.

00:02:04.523 --> 00:02:07.968
Stephen Hawking, he is one of
the small percentage of ALS

00:02:07.968 --> 00:02:11.684
sufferers who have been
able to live several decades

00:02:11.684 --> 00:02:13.837
past his diagnosis.

00:02:15.019 --> 00:02:20.019
It effects one to two out of
100,000 individuals each year.

00:02:21.730 --> 00:02:25.357
So if you were to imagine kind
of a large stadium of people,

00:02:25.357 --> 00:02:27.410
in a given year you would expect to--

00:02:27.410 --> 00:02:29.365
If you say there's roughly
100,000 people here,

00:02:29.365 --> 00:02:32.132
you would expect one to two of them to

00:02:32.132 --> 00:02:34.909
get afflicted with ALS.

00:02:34.909 --> 00:02:37.636
The reason why it's important
to have research behind this

00:02:37.636 --> 00:02:41.302
is it's a very, a very not
so well understood disease.

00:02:41.302 --> 00:02:42.533
People know what happens.

00:02:42.533 --> 00:02:45.334
They know that the motor
neurons degenerate,

00:02:45.334 --> 00:02:47.823
but they don't know why it's happening.

00:02:47.823 --> 00:02:48.838
There is--

00:02:48.838 --> 00:02:53.623
Scientists have identified some
genetic component to ALS for

00:02:53.623 --> 00:02:56.759
some of the sufferers, but
for the majority, they can't

00:02:56.759 --> 00:02:59.551
identify a genetic component
and it's unclear what

00:02:59.551 --> 00:03:01.498
environmental components are causing it,

00:03:01.498 --> 00:03:03.599
although people think there
must be some environmental

00:03:03.599 --> 00:03:07.508
components, but as far as
most people can tell it just

00:03:07.508 --> 00:03:11.703
randomly afflicts people once
they kind of get into their

00:03:11.703 --> 00:03:15.046
early middle age, so it's a
really, really horrible disease

00:03:15.046 --> 00:03:18.350
and it's really important that
people have awareness for it

00:03:18.350 --> 00:03:22.370
and that there is an
appropriate level of funding

00:03:22.370 --> 00:03:23.793
and in case you're wondering,

00:03:23.793 --> 00:03:26.653
yes, I did take the ice
bucket challenge as well.

00:03:26.653 --> 00:03:29.534
I have been challenged by
Shawn O'Sullivan to do the

00:03:29.534 --> 00:03:32.775
ice bucket challenge,
so I'm about to do it.

00:03:33.950 --> 00:03:34.696
Y'all ready?

00:03:34.696 --> 00:03:36.191
- Girl: Yeah
- Boy: Yeah

00:03:36.191 --> 00:03:37.901
- [Voiceover] : (giggling)

00:03:39.654 --> 00:03:40.670
- Ah.

00:03:40.670 --> 00:03:42.464
- [Voiceover] : (laughing)

00:03:45.612 --> 00:03:49.109
- I'd now like to issue the
challenge to three other people.

00:03:49.109 --> 00:03:53.368
The other Salman Khan,
famous Bollywood actor.

00:03:53.368 --> 00:03:56.744
Esther Cho, who works
with me at Khan Academy

00:03:56.744 --> 00:04:00.673
and my kids, who seemed
too happy to see myself

00:04:00.673 --> 00:04:02.021
get doused with cold water.


nth term divergence test

WEBVTT
Kind: captions
Language: en

00:00:00.285 --> 00:00:01.447
- [Voiceover] What we're going to do now

00:00:01.447 --> 00:00:04.379
is start to explore a series of tests

00:00:04.379 --> 00:00:09.379
to determine whether a series
will converge or diverge

00:00:09.993 --> 00:00:12.142
and the first one I'm going
to go through right now

00:00:12.142 --> 00:00:15.294
is perhaps the most
basic and hopefully see

00:00:15.294 --> 00:00:17.842
the most intuitive and this
is the divergence test.

00:00:17.842 --> 00:00:22.733
The divergence test won't tell
us if a series will converge

00:00:22.733 --> 00:00:26.411
but it can tell us if something
will definitely diverge.

00:00:26.411 --> 00:00:29.792
First let me write it in
kind of mathy notation

00:00:29.792 --> 00:00:32.650
and then we'll look at an
actual concrete example on it.

00:00:32.650 --> 00:00:37.650
The divergence test tells
us that if the limit

00:00:38.073 --> 00:00:43.073
as N approaches infinity of
A sub N does not equal zero,

00:00:44.930 --> 00:00:49.930
then the infinite series
going from N equals one

00:00:52.110 --> 00:00:57.110
to infinity of A sub N will diverge.

00:01:01.166 --> 00:01:03.929
We've already gone through
what it means to diverge

00:01:03.929 --> 00:01:05.090
and this sum is either going to go

00:01:05.090 --> 00:01:07.300
unbounded to positive infinity

00:01:07.300 --> 00:01:09.523
or unbounded to negative infinity

00:01:09.523 --> 00:01:12.262
or it'll just oscillate between values,

00:01:12.262 --> 00:01:17.262
it'll never really approach
a given sum or given value.

00:01:17.603 --> 00:01:19.531
That's what the divergence test tells us

00:01:19.531 --> 00:01:21.193
and you're probably
thinking, okay all right,

00:01:21.193 --> 00:01:22.965
I can kind of get what this says

00:01:22.965 --> 00:01:25.116
but where is this actually useful?

00:01:25.116 --> 00:01:27.310
To see where it can be useful

00:01:27.310 --> 00:01:28.932
let's look at a candidate series

00:01:28.932 --> 00:01:33.340
and see if we can figure out
if it's going to diverge.

00:01:33.340 --> 00:01:36.660
Let's say I had this series

00:01:36.660 --> 00:01:41.660
so the sum from N equals one to infinity

00:01:43.580 --> 00:01:46.102
and in general it doesn't
always have to be N equals one,

00:01:46.102 --> 00:01:47.607
it could be N equals five,

00:01:47.607 --> 00:01:49.720
it could be N equals zero.

00:01:49.720 --> 00:01:52.367
The key is, is that this
is an infinite series

00:01:52.367 --> 00:01:54.660
that we're talking about
so that's why we care about

00:01:54.660 --> 00:01:56.193
the limit is N approaches infinity.

00:01:56.193 --> 00:02:00.964
Let's say we have the
sum of four N squared

00:02:00.964 --> 00:02:05.964
minus N to the third over seven,

00:02:08.327 --> 00:02:12.636
minus three and to the third power.

00:02:12.636 --> 00:02:14.953
Given what we know about
the divergence test,

00:02:14.953 --> 00:02:19.697
is this series going
to converge or diverge?

00:02:21.004 --> 00:02:22.732
Well, let's just look at this,

00:02:22.732 --> 00:02:25.889
what we're taking the sums of.

00:02:25.889 --> 00:02:29.677
So this is essentially
or this is our A sub N

00:02:29.677 --> 00:02:31.899
if we're trying to match to the definition

00:02:31.899 --> 00:02:34.132
or I guess the explanation
of the divergence test.

00:02:34.132 --> 00:02:37.703
Let's think about what the limit,

00:02:37.703 --> 00:02:42.208
as N approaches infinity of four N squared

00:02:42.208 --> 00:02:46.300
minus N to the third over seven

00:02:46.300 --> 00:02:49.100
minus three N to the third is.

00:02:49.100 --> 00:02:51.549
I encourage you to pause the
video and think about that.

00:02:51.549 --> 00:02:53.844
Well there's a couple of
ways to think about it,

00:02:53.844 --> 00:02:56.090
one way is, hey look as N goes to infinity

00:02:56.090 --> 00:02:57.323
the highest degree terms

00:02:57.323 --> 00:02:58.254
of the numerator and the denominator

00:02:58.254 --> 00:02:59.371
are the ones that matter.

00:02:59.371 --> 00:03:01.318
So this is going to approach
negative N to the third

00:03:01.318 --> 00:03:03.940
over negative three N to the third

00:03:03.940 --> 00:03:06.698
which will just be which
would approach negative one

00:03:06.698 --> 00:03:10.720
over negative three which
would be positive 1/3

00:03:10.720 --> 00:03:13.140
or we could say if we want
to do a little bit more,

00:03:13.140 --> 00:03:15.451
or do a little bit more systematically.

00:03:15.866 --> 00:03:17.883
Limit as N approaches infinity,

00:03:17.883 --> 00:03:19.725
we can divide what the
numerator and the denominator

00:03:19.725 --> 00:03:20.597
by end of the third,

00:03:20.597 --> 00:03:22.191
so if we divide the
numerator by end of the third

00:03:22.191 --> 00:03:25.151
this first third is
going to be four over N

00:03:25.151 --> 00:03:30.151
minus one over seven over N
to the third, minus three.

00:03:32.911 --> 00:03:35.552
Here it becomes clear, the
limit is N approaches infinity,

00:03:35.552 --> 00:03:38.613
that's going to go to zero,
that's going to go to zero

00:03:38.613 --> 00:03:40.796
and so you're going to
be left with negative one

00:03:40.796 --> 00:03:44.739
over negative three which is equal to 1/3.

00:03:44.739 --> 00:03:49.155
Notice, the limit of A sub
N as N approaches infinity

00:03:49.155 --> 00:03:51.658
in this case is not equal to zero,

00:03:51.658 --> 00:03:56.658
therefore this sum will, this
infinite series will diverge.

00:04:00.293 --> 00:04:04.423
Now let's think for a second
why this makes a ton of sense.

00:04:04.423 --> 00:04:07.240
Well, the only way that
you're going to converge

00:04:07.240 --> 00:04:07.693
that you're going to set,

00:04:07.693 --> 00:04:09.639
remember you're taking infinite sum.

00:04:09.639 --> 00:04:11.360
You're taking an infinite sum of things

00:04:11.360 --> 00:04:12.239
so the only reasonable way

00:04:12.239 --> 00:04:15.795
that something might be able
to converge to a finite value

00:04:15.795 --> 00:04:18.883
is if every extra term you're
adding is getting smaller

00:04:18.883 --> 00:04:20.933
and smaller and smaller
as approaching zero.

00:04:20.933 --> 00:04:24.393
If as N approaches
infinity, you go unbounded

00:04:24.393 --> 00:04:26.154
or if it's even equal to 1/3,

00:04:26.154 --> 00:04:27.872
this means that for very large ends

00:04:27.872 --> 00:04:30.295
you just keep adding things
that are getting closer

00:04:30.295 --> 00:04:31.664
and closer to 1/3.

00:04:31.664 --> 00:04:34.576
Well if you add an infinite
number of one-thirds together,

00:04:34.576 --> 00:04:36.323
you're going to go to infinity,

00:04:36.323 --> 00:04:37.558
you're going to be unbounded.

00:04:37.558 --> 00:04:41.296
You are going to diverge so
that's all it's telling us.

00:04:41.296 --> 00:04:43.410
Look, in order for something to converge

00:04:43.410 --> 00:04:45.554
as you're taking infinite sum of them,

00:04:45.554 --> 00:04:46.767
at some point these things
are going to have to get

00:04:46.767 --> 00:04:48.344
really, really, really close to zero.

00:04:48.344 --> 00:04:50.081
If at some point they're
not getting close to zero,

00:04:50.081 --> 00:04:51.811
there's no way is going to converge,

00:04:51.811 --> 00:04:54.120
that thing will diverge

00:04:54.120 --> 00:04:55.599
so hopefully that makes a little sense.

00:04:55.599 --> 00:04:58.200
That also gives you I
guess another insight

00:04:58.200 --> 00:05:00.332
on what the divergence test can't do.

00:05:00.332 --> 00:05:02.391
The divergence test can be used

00:05:02.391 --> 00:05:04.550
to show that something will diverge

00:05:04.550 --> 00:05:06.890
but if something, I guess you could say

00:05:06.890 --> 00:05:08.573
passes the divergence test,

00:05:08.573 --> 00:05:11.887
it doesn't or I guess
fails the divergence test

00:05:11.887 --> 00:05:13.678
or if this isn't true,

00:05:13.678 --> 00:05:16.592
it doesn't mean that the
thing is going to converge.

00:05:16.592 --> 00:05:18.204
Let me give you an example of that.

00:05:20.465 --> 00:05:23.933
This right over here from
N equals one to infinity

00:05:23.933 --> 00:05:27.218
of one over N and this is
actually the harmonic series

00:05:27.218 --> 00:05:28.415
right over here.

00:05:28.415 --> 00:05:31.881
This, if we look at, if we try
to apply the divergence test,

00:05:31.881 --> 00:05:33.466
we would say, okay, well what's the limit

00:05:33.466 --> 00:05:37.317
as N approaches infinity of one over N.

00:05:37.317 --> 00:05:39.799
Well, hey that is zero, if
that approaches infinity

00:05:39.799 --> 00:05:42.426
this is equal to zero.

00:05:42.426 --> 00:05:46.833
We could say well it's kind
of failing the divergence test

00:05:46.833 --> 00:05:49.813
so we're not just by
using the divergence test

00:05:49.813 --> 00:05:52.650
we can't prove that this
thing is going to diverge

00:05:52.650 --> 00:05:55.128
but that doesn't mean
that it doesn't diverge.

00:05:55.128 --> 00:05:58.260
It actually turns out and we
prove this in several videos,

00:05:58.260 --> 00:06:00.566
it actually turns out that
this thing does diverge.

00:06:00.566 --> 00:06:05.566
This thing does diverge,

00:06:05.738 --> 00:06:08.918
it's just that the
divergence test isn't enough,

00:06:08.918 --> 00:06:11.816
it's not enough of a tool
to let us know for sure

00:06:11.816 --> 00:06:14.105
that this diverge, we'll
see the comparison test

00:06:14.105 --> 00:06:16.352
and the integral test can either be used

00:06:16.352 --> 00:06:19.112
to prove that this in fact does diverge.

00:06:19.112 --> 00:06:21.932
You can definitely not
say that if something,

00:06:21.932 --> 00:06:24.549
if this does not apply for something.

00:06:24.549 --> 00:06:26.448
If you think the limit
as N approaches infinity

00:06:26.448 --> 00:06:29.830
and it does go to zero,
that still might diverge.

00:06:29.830 --> 00:06:33.335
It doesn't necessarily
mean that it converges.

00:06:33.335 --> 00:06:35.076
Now there are things that do converge

00:06:35.076 --> 00:06:37.522
that where they do approach zero

00:06:37.522 --> 00:06:41.228
so for example if I take the sum from

00:06:41.228 --> 00:06:45.375
N equals one to infinity
of one over N squared.

00:06:45.375 --> 00:06:47.697
If I try to apply the divergence test,

00:06:47.697 --> 00:06:50.646
I have the limit as N approaches infinity

00:06:50.646 --> 00:06:53.334
of one over N squared,
well this does equal zero.

00:06:53.334 --> 00:06:54.535
This gets really, really large,

00:06:54.535 --> 00:06:56.281
this thing is going to approach zero

00:06:56.281 --> 00:06:59.430
and so once again this is I guess failing

00:06:59.430 --> 00:07:01.544
the test for divergence.

00:07:01.544 --> 00:07:04.523
Just with that alone
you don't know for sure

00:07:04.523 --> 00:07:06.027
that this thing is going to converge,

00:07:06.027 --> 00:07:07.205
it still might diverge,

00:07:07.205 --> 00:07:09.648
the divergence test just
might not have been enough.

00:07:09.648 --> 00:07:11.981
Now it does turn out and
once again we prove this

00:07:11.981 --> 00:07:14.838
in future videos that
this thing does converge.

00:07:14.838 --> 00:07:19.829
This thing does converge
but not because it,

00:07:19.829 --> 00:07:22.372
I guess you could say
fails the divergence test.

00:07:22.372 --> 00:07:24.704
We have to use a different test to show

00:07:24.704 --> 00:07:26.914
that this does in fact converge,

00:07:26.914 --> 00:07:28.385
just as we have to use
a different test to show

00:07:28.385 --> 00:07:30.666
that this does in fact diverge.

00:07:30.666 --> 00:07:32.601
Where the divergence test is useful

00:07:32.601 --> 00:07:36.489
is for the things that actually
pass the divergence test.

00:07:36.489 --> 00:07:38.789
When you actually find that
the limit is N approaches

00:07:38.789 --> 00:07:41.396
infinity of A sub N does not equal zero,

00:07:41.396 --> 00:07:42.991
like this case right over here.

00:07:42.991 --> 00:07:45.286
In this case the divergence test helps us

00:07:45.286 --> 00:07:48.038
because it helps us make the conclusion

00:07:48.038 --> 00:07:53.038
that this series definitely diverges.


Worked example: Integral test

WEBVTT
Kind: captions
Language: en

00:00:00.707 --> 00:00:03.127
- [Voiceover] Let's now
explain to ourselves,

00:00:03.127 --> 00:00:05.544
I guess you could say, a more formal

00:00:05.544 --> 00:00:09.094
communication of the Integral Test.

00:00:09.727 --> 00:00:11.820
Integral Test.

00:00:12.192 --> 00:00:17.192
So it tells us that if we
assume that we have sum f of x,

00:00:17.376 --> 00:00:21.310
if we have sum f of x that is positive...

00:00:22.143 --> 00:00:25.164
positive, continuous...

00:00:25.643 --> 00:00:27.058
continuous...

00:00:27.058 --> 00:00:30.475
continuous, and decreasing...

00:00:30.475 --> 00:00:32.397
and decreasing...

00:00:32.844 --> 00:00:37.844
and decreasing on some interval, on...

00:00:38.593 --> 00:00:43.593
So starting at k, and including
k, all the way to infinity.

00:00:44.344 --> 00:00:47.342
Then we can make one of two statements.

00:00:47.859 --> 00:00:52.377
We could say either that
if the improper integral

00:00:52.377 --> 00:00:57.377
from k to infinity of f
of x dx is convergent,

00:00:59.293 --> 00:01:03.677
is convergent, then...

00:01:04.627 --> 00:01:09.627
then the sum, the infinite
series from n is equal to k

00:01:11.809 --> 00:01:16.809
to infinity of f of n, is also convergent.

00:01:18.931 --> 00:01:23.685
Is also convergent...

00:01:24.433 --> 00:01:25.609
Convergent.

00:01:25.609 --> 00:01:27.352
And this is actually the
case that we saw when

00:01:27.352 --> 00:01:28.802
we looked at one over n squared,

00:01:28.802 --> 00:01:30.317
but I'll look at that in a second.

00:01:30.317 --> 00:01:32.233
But the second claim that we could make,

00:01:32.233 --> 00:01:34.884
or the second deduction that we might be

00:01:34.884 --> 00:01:36.833
able to make using the Integral Test,

00:01:36.833 --> 00:01:38.900
is if that's the other way around.

00:01:38.900 --> 00:01:42.583
That if the integral from k to infinity,

00:01:42.583 --> 00:01:47.583
the improper integral of
f of x dx, is divergent,

00:01:48.094 --> 00:01:50.695
divergent, then the same thing is true for

00:01:50.695 --> 00:01:53.078
the corresponding infinite series.

00:01:53.413 --> 00:01:57.296
Then this infinite series right over here

00:01:57.296 --> 00:01:59.496
is also going to be...

00:01:59.496 --> 00:02:04.395
is also divergent.

00:02:04.478 --> 00:02:06.463
And as I already mentioned,
in the last video

00:02:06.463 --> 00:02:08.245
we already saw this in the case of

00:02:08.245 --> 00:02:10.912
f of x is equal to one over x squared.

00:02:10.912 --> 00:02:15.912
We saw that since the
integral from one to infinity

00:02:17.094 --> 00:02:22.094
of one over x squared,
over one over x squared dx,

00:02:23.795 --> 00:02:26.972
is convergent, in fact, it equals one.

00:02:26.972 --> 00:02:28.189
It equals one.

00:02:28.189 --> 00:02:33.189
Because of that, we were
able to say that the sum

00:02:33.404 --> 00:02:38.404
from n equals the sum
from n is equal to one

00:02:39.139 --> 00:02:44.139
to infinity of one over n
squared, is also convergent.

00:02:48.555 --> 00:02:51.870
Also convergent.

00:02:52.155 --> 00:02:54.727
And now we can see an example
where we go the other way.

00:02:55.087 --> 00:02:58.321
For example, we know that this integral...

00:02:58.321 --> 00:03:00.422
Let me write the integral down.

00:03:00.422 --> 00:03:02.170
Let's start with this integral.

00:03:02.170 --> 00:03:05.272
From one to infinity,
not of f of x is equal

00:03:05.272 --> 00:03:06.903
to one over x squared, but let's say that

00:03:06.903 --> 00:03:08.780
f of x is equal to one over x.

00:03:08.780 --> 00:03:10.056
Actually let me just write that down.

00:03:10.056 --> 00:03:15.056
Let's just start with f of
x is equal to one over x.

00:03:15.138 --> 00:03:17.787
It definitely meets our
conditions that it is positive,

00:03:17.787 --> 00:03:19.555
and let's say we were going to consider it

00:03:19.555 --> 00:03:21.105
over the interval...

00:03:21.105 --> 00:03:26.105
over the interval from one to infinity.

00:03:26.939 --> 00:03:29.522
So it meets this first constraint.

00:03:29.522 --> 00:03:33.104
Over this interval,
one over x is positive,

00:03:33.104 --> 00:03:36.272
it is continuous, and it is decreasing.

00:03:36.272 --> 00:03:39.355
As x increases, f of x decreases.

00:03:39.355 --> 00:03:41.971
So the Integral Test should apply.

00:03:41.971 --> 00:03:45.105
So let's see what the improper integral

00:03:45.105 --> 00:03:47.889
from one to infinity of this would be.

00:03:47.889 --> 00:03:49.859
So if we take...

00:03:49.859 --> 00:03:54.097
If we go from one to
infinity of one over x,

00:03:54.635 --> 00:03:59.635
of one over x dx, this is equal to...

00:04:01.493 --> 00:04:04.359
We could write this as the limit

00:04:04.359 --> 00:04:08.327
as t approaches infinity
of the definite integral

00:04:08.327 --> 00:04:12.860
from one to t, of one over x dx,

00:04:12.860 --> 00:04:17.860
which is equal to the limit,
as t approaches infinity, of,

00:04:19.194 --> 00:04:22.494
take the antiderivative, is going to be of

00:04:22.494 --> 00:04:24.542
the natural log of x,

00:04:24.542 --> 00:04:29.123
the natural log of x going from one to t.

00:04:29.785 --> 00:04:31.213
One to t, we could do the...

00:04:31.213 --> 00:04:33.244
Well, it's really the absolute value of x,

00:04:33.244 --> 00:04:34.993
but we're dealing with positive x's here

00:04:34.993 --> 00:04:37.026
so it's just going to
be the natural log of x,

00:04:37.026 --> 00:04:41.193
which is going to be the
limit as t approaches

00:04:41.193 --> 00:04:45.659
infinity of the natural log of t,

00:04:45.659 --> 00:04:46.943
or I could even say the natural log of the

00:04:46.943 --> 00:04:48.543
absolute value of t, which is going to be

00:04:48.543 --> 00:04:51.495
the natural log of t,
because it's positive t's,

00:04:51.495 --> 00:04:54.109
minus the natural log of one.

00:04:54.109 --> 00:04:56.909
Minus the natural log of
the absolute value of one.

00:04:57.647 --> 00:05:00.008
The natural log of one
is zero, so it's going

00:05:00.008 --> 00:05:01.458
to be the natural log of t.

00:05:01.458 --> 00:05:03.110
The limit of that approaches infinity.

00:05:03.110 --> 00:05:04.508
But the limit as that approaches infinity

00:05:04.508 --> 00:05:05.543
is just going to be unbounded,

00:05:05.543 --> 00:05:06.875
this is going to go to infinity.

00:05:06.875 --> 00:05:10.186
This right over here is divergent.

00:05:10.494 --> 00:05:14.378
So this right over here is divergent.

00:05:15.326 --> 00:05:18.243
So this is divergent.

00:05:19.292 --> 00:05:22.841
And because this is
divergent, we can then say,

00:05:22.841 --> 00:05:25.359
we can then say, by the Integral Test,

00:05:25.359 --> 00:05:27.891
we can then say the Integral Test...

00:05:27.891 --> 00:05:30.409
Once again, our function
over this interval,

00:05:30.409 --> 00:05:32.525
positive, continuous, decreasing.

00:05:32.525 --> 00:05:34.642
We saw that this improper
integral right over here

00:05:34.642 --> 00:05:36.707
is divergent, and then by the second point

00:05:36.707 --> 00:05:39.224
of the Integral Test we can say therefore,

00:05:39.224 --> 00:05:41.126
and I haven't rigorously
proved it yet, but hopefully

00:05:41.126 --> 00:05:42.875
I gave you a good intuitive justification

00:05:42.875 --> 00:05:45.077
in the previous video,
that the integral...

00:05:45.077 --> 00:05:49.689
that the infinite series
from n equals one to infinity

00:05:50.059 --> 00:05:53.588
of one over n, which
is the harmonic series,

00:05:53.588 --> 00:05:58.413
that this is also, this is also...

00:05:59.339 --> 00:06:02.007
this is also divergent.

00:06:02.207 --> 00:06:04.021
So we've already shown
that the harmonic series

00:06:04.021 --> 00:06:06.805
is divergent using that
very beautiful, elegant

00:06:06.805 --> 00:06:10.871
proof by Oresme, I think
I'm probably mispronouncing

00:06:10.871 --> 00:06:13.739
his name, though used the
comparison test, but just

00:06:13.739 --> 00:06:16.605
like this we have used the Integral Test

00:06:16.605 --> 00:06:19.288
to show that it is also divergent.

00:06:19.288 --> 00:06:21.238
And once again, let's
remember what the whole

00:06:21.238 --> 00:06:23.419
motivation of the Integral Test is.

00:06:23.604 --> 00:06:26.672
Let me draw f of x is equal to one over x.

00:06:27.039 --> 00:06:30.089
So f of x is equal to one
over x would look like...

00:06:30.488 --> 00:06:32.671
Do my best attempt here.

00:06:32.740 --> 00:06:36.648
So let's say that's one, two, three,

00:06:37.064 --> 00:06:41.417
and that is one, two...

00:06:41.924 --> 00:06:45.588
And so let's see, when
x is one f of x is one,

00:06:45.588 --> 00:06:49.321
when x is two f of x is 1/2 or 1/3.

00:06:49.321 --> 00:06:51.872
If it's 1/2 here, it
will be two over here.

00:06:51.872 --> 00:06:53.477
So it looks like this.

00:06:53.477 --> 00:06:55.372
So this is f of x is equal to one over n,

00:06:55.372 --> 00:06:56.971
and once again we see that
then over the interval

00:06:56.971 --> 00:06:59.604
we care about, from one to
infinity, it's definitely

00:06:59.604 --> 00:07:04.604
positive, continuous, and decreasing.

00:07:05.671 --> 00:07:08.822
And if we look at this
sum right over here,

00:07:08.822 --> 00:07:11.821
we could view this sum as...

00:07:13.238 --> 00:07:15.037
Let's do that, let me write it down.

00:07:15.037 --> 00:07:16.645
So the sum...

00:07:17.355 --> 00:07:21.656
The sum from n equals one
to infinity of one over n,

00:07:21.656 --> 00:07:26.656
is equal to one plus 1/2
plus 1/3, and of course

00:07:29.920 --> 00:07:32.289
we keep going on and on and on and on.

00:07:32.289 --> 00:07:34.721
In this case, since we want
to show it's divergent,

00:07:34.721 --> 00:07:37.138
we could say, "Hey, look,
this is an overestimate of,

00:07:37.138 --> 00:07:39.338
"of this area here."

00:07:39.338 --> 00:07:40.221
Let me be clear.

00:07:40.221 --> 00:07:42.015
So we have this area.

00:07:42.354 --> 00:07:45.332
We have this area in green,

00:07:45.671 --> 00:07:49.019
which is what the improper
integral is denoting.

00:07:49.019 --> 00:07:50.740
So that right over there
is the improper integral

00:07:50.740 --> 00:07:55.740
from one to infinity of one over x dx.

00:07:57.905 --> 00:07:58.954
Now you could view this as

00:07:58.954 --> 00:08:01.322
an overestimate of that area.

00:08:01.322 --> 00:08:03.118
So this first...

00:08:03.288 --> 00:08:04.554
This one right over here, you could say

00:08:04.554 --> 00:08:08.854
that this is this one
height times one width,

00:08:09.223 --> 00:08:11.621
so that's that block right over there,

00:08:11.621 --> 00:08:13.249
that's the area of that...

00:08:13.249 --> 00:08:15.189
is going to be equal to one.

00:08:15.189 --> 00:08:18.388
Then this over here,
1/2, you could view that

00:08:18.388 --> 00:08:20.706
as the area of the next block,

00:08:20.706 --> 00:08:23.047
of the next block.

00:08:23.754 --> 00:08:26.422
So you could kind of view this
as a left-sided Riemann sum,

00:08:26.422 --> 00:08:28.255
I guess is one way to think about it.

00:08:28.255 --> 00:08:30.789
And so this is going to be 1/2...

00:08:30.789 --> 00:08:32.489
Yeah, left-sided Riemann sum.

00:08:32.489 --> 00:08:34.239
So this is going to be 1/2,

00:08:34.239 --> 00:08:37.706
and then the 1/3 is going to be this one,

00:08:37.706 --> 00:08:39.371
it's going to be this one.

00:08:39.371 --> 00:08:40.706
I noticed they're all...

00:08:40.706 --> 00:08:43.338
The actual area we care
about, the improper integral,

00:08:43.338 --> 00:08:46.122
it's all contained in these blocks.

00:08:46.122 --> 00:08:48.138
So this is going to be an over...

00:08:48.138 --> 00:08:50.721
is going to be larger than
this, but we've already

00:08:50.721 --> 00:08:53.289
seen that this is
unbounded towards infinity.

00:08:53.289 --> 00:08:54.822
This is divergent.

00:08:54.822 --> 00:08:56.353
So if this is larger than this,

00:08:56.353 --> 00:08:58.572
and this is divergent,
this goes to infinity,

00:08:58.572 --> 00:09:00.706
then this must also go to infinity.

00:09:00.706 --> 00:09:03.622
So that's exactly where the
Integral Test is coming from.


Integral test

WEBVTT
Kind: captions
Language: en

00:00:00.624 --> 00:00:04.254
- [Voiceover] Let's explore
a bit the infinite series

00:00:04.254 --> 00:00:09.254
from n equals one to infinity
of one over n squared.

00:00:09.759 --> 00:00:14.301
Which of course is equal
to one plus one fourth,

00:00:14.301 --> 00:00:16.535
that's one over two
squared, plus one over three

00:00:16.535 --> 00:00:19.737
squared, which is one
ninth, plus one sixteenth

00:00:19.890 --> 00:00:22.427
and it goes on and on and on forever.

00:00:22.808 --> 00:00:24.810
So there's a couple of
things that we know about it.

00:00:25.195 --> 00:00:29.044
The first thing is that all of
the terms here are positive.

00:00:29.475 --> 00:00:32.244
So all of the terms here are positive.

00:00:33.448 --> 00:00:37.100
So they're all positive and
that they're decreasing.

00:00:37.240 --> 00:00:40.608
It looks like they're
decreasing quite quickly

00:00:40.608 --> 00:00:44.023
here from one to one fourth
to one ninth to one sixteenth,

00:00:44.023 --> 00:00:46.544
and so they're quickly approaching zero,

00:00:46.544 --> 00:00:48.523
which makes us feel pretty good that

00:00:48.523 --> 00:00:51.983
this thing has a chance of converging.

00:00:52.372 --> 00:00:54.370
And because they're all
positive we know that

00:00:54.370 --> 00:00:57.188
this sum right over
here, if it does converge

00:00:57.188 --> 00:01:00.059
is going to be greater than zero.

00:01:00.142 --> 00:01:04.503
So the only reason why it
wouldn't converge is if

00:01:04.503 --> 00:01:07.685
somehow it goes unbounded
towards infinity,

00:01:07.685 --> 00:01:09.522
which we know if this was one over n

00:01:09.522 --> 00:01:11.481
it would be unbounded towards infinity.

00:01:11.481 --> 00:01:14.187
So this says that's a possibility here.

00:01:14.491 --> 00:01:16.493
So if we could show that this is bounded,

00:01:16.493 --> 00:01:19.051
then that will be a pretty
good argument for why

00:01:19.051 --> 00:01:21.673
this thing right over here
converges because the only

00:01:21.673 --> 00:01:24.240
reason why you could
diverge is if you went to

00:01:24.240 --> 00:01:26.024
either positive infinity
or negative infinity.

00:01:26.143 --> 00:01:27.572
We already know that this
thing isn't going to go

00:01:27.572 --> 00:01:29.602
to negative infinity because
it's all positive terms.

00:01:29.767 --> 00:01:31.534
Or you could diverge if
this thing oscillates,

00:01:31.534 --> 00:01:32.976
but it's not going to
oscillate because all of these

00:01:32.976 --> 00:01:36.114
terms are just adding
to the sum, none of them

00:01:36.114 --> 00:01:39.019
are taking away because none
of these terms are negative.

00:01:39.396 --> 00:01:42.286
So let's see if we can make
a good argument for why

00:01:42.286 --> 00:01:44.856
this sum right over here is bounded,

00:01:44.856 --> 00:01:47.181
especially if we can
come up with the bound,

00:01:47.181 --> 00:01:48.907
then that's a pretty good argument that

00:01:49.522 --> 00:01:52.656
this infinite series should converge.

00:01:53.352 --> 00:01:55.168
And the way that we're going to do that is

00:01:55.168 --> 00:01:57.342
we're going to explore a related function.

00:01:57.888 --> 00:02:00.326
So what I wanna do is I wanna explore

00:02:00.326 --> 00:02:04.135
f of x is equal to one over x squared.

00:02:04.940 --> 00:02:06.636
You could really view this right over here

00:02:06.636 --> 00:02:11.636
one over n squared as f of n
if I were to write it this way.

00:02:12.800 --> 00:02:13.967
So why is this interesting?

00:02:13.967 --> 00:02:15.192
Well let's graph it.

00:02:17.578 --> 00:02:20.739
So that's the graph of
y is equal to f of x.

00:02:24.141 --> 00:02:27.646
And notice this is a
continuous, positive, decreasing

00:02:27.646 --> 00:02:32.282
function, especially
over the interval that

00:02:32.282 --> 00:02:33.546
I care about right over here.

00:02:33.546 --> 00:02:38.034
I guess we could say for
positive values of x,

00:02:38.034 --> 00:02:43.034
it is a continuous, positive,
decreasing function.

00:02:43.863 --> 00:02:47.959
And what's interesting is
we can use this as really

00:02:47.959 --> 00:02:50.625
an underestimate for this
area right over here.

00:02:50.625 --> 00:02:51.742
What do I mean by that?

00:02:52.419 --> 00:02:55.510
Well one, this first term right over here,

00:02:56.170 --> 00:02:57.719
you could view that as the area

00:02:57.719 --> 00:03:00.688
of this block right over here.

00:03:01.383 --> 00:03:05.955
That is f of n or I guess
you can say f of one high

00:03:05.955 --> 00:03:08.346
and one wide, so it's going to be

00:03:08.346 --> 00:03:12.115
one times one over one squared or one.

00:03:15.151 --> 00:03:16.822
Let me make sure I'm
using different colors.

00:03:16.822 --> 00:03:18.960
This term right over
here, that could represent

00:03:18.960 --> 00:03:23.960
the area of this block,
which is one fourth high

00:03:24.547 --> 00:03:29.547
and one wide so it is going
to have an area of one fourth.

00:03:30.915 --> 00:03:33.143
What could this one represent?

00:03:33.509 --> 00:03:35.552
Well the area of the next
block if we're trying

00:03:35.552 --> 00:03:38.391
to estimate the area under the curve.

00:03:38.495 --> 00:03:40.852
And this might look
familiar from when we first

00:03:40.852 --> 00:03:43.392
got exposed to the integral
or even before we got

00:03:43.392 --> 00:03:45.509
exposed to the integral and
we were taking Riemann sums.

00:03:45.977 --> 00:03:48.106
So that right over here, that area,

00:03:48.106 --> 00:03:49.954
is going to be equal to one ninth.

00:03:50.827 --> 00:03:53.912
So what's intriguing about
this is we know how to

00:03:53.912 --> 00:03:58.449
find the exact area, or
the exact area from one

00:03:58.449 --> 00:04:01.414
to infinity, from x
equals one to infinity.

00:04:01.433 --> 00:04:03.255
So maybe we can use that somehow.

00:04:03.598 --> 00:04:06.668
We know what this area is right over here,

00:04:07.329 --> 00:04:11.761
which we can denote as
the improper integral

00:04:11.761 --> 00:04:16.453
from one to infinity of f of x dx.

00:04:16.626 --> 00:04:19.531
We know what that is and I'll
figure it out in a little bit.

00:04:19.638 --> 00:04:20.959
And if we know what this
is, if we can figure out

00:04:20.959 --> 00:04:22.741
the value that's going
to be an upper-bound for

00:04:22.741 --> 00:04:24.328
one fourth plus one ninth plus one

00:04:24.328 --> 00:04:26.780
sixteenth on and on and on and on.

00:04:26.862 --> 00:04:29.425
And so that will allow
us to essentially bound

00:04:29.425 --> 00:04:34.425
what this series evaluates
to and as we said earlier

00:04:34.800 --> 00:04:37.828
that would be a very good
argument for its convergence.

00:04:37.828 --> 00:04:39.692
So the whole point here,
I'm not doing a rigorous

00:04:39.692 --> 00:04:42.543
proof, but really getting
you the underlying

00:04:42.543 --> 00:04:46.279
conceptual understanding
for a very popular test

00:04:46.279 --> 00:04:47.996
for convergence or divergence

00:04:47.996 --> 00:04:50.649
which is called the integral test.

00:04:51.468 --> 00:04:53.408
Let me just write that
down just so you know

00:04:53.408 --> 00:04:57.145
what this is kind of the
mental foundations for.

00:04:58.108 --> 00:04:59.545
So what do I mean here?

00:04:59.564 --> 00:05:02.215
So let me write this sum again.

00:05:02.215 --> 00:05:03.505
Let me write it a little bit different.

00:05:03.505 --> 00:05:07.961
So our original series from n equals one

00:05:07.961 --> 00:05:11.058
to infinity of one over n squared.

00:05:11.440 --> 00:05:13.897
It's going to be equal
to this first block,

00:05:13.897 --> 00:05:17.306
the area of this first
block plus the area of

00:05:17.306 --> 00:05:19.213
all the rest of the
blocks, the one fourth plus

00:05:19.213 --> 00:05:20.851
one ninth plus one sixteenth,

00:05:24.202 --> 00:05:25.544
let me do this in a new color.

00:05:25.821 --> 00:05:30.821
Which we could write as
the sum from n equals two

00:05:32.089 --> 00:05:36.937
to infinity of one over n squared.

00:05:37.331 --> 00:05:39.832
So I just kind of expressed this as a sum

00:05:39.832 --> 00:05:41.646
of this plus all of that stuff.

00:05:42.086 --> 00:05:45.967
Now what's interesting
is that this, what I just

00:05:45.967 --> 00:05:50.131
wrote in this blue
notation that's this block

00:05:50.131 --> 00:05:53.554
plus this block plus the next block,

00:05:53.554 --> 00:05:55.734
which is going to be less than

00:05:55.734 --> 00:05:57.940
this definite integral right over here.

00:05:58.058 --> 00:06:00.048
This definite integral,
notice it's an underestimate

00:06:00.048 --> 00:06:02.635
it's always below the
curve, so it's going to

00:06:02.635 --> 00:06:04.095
be less than that definite integral.

00:06:04.095 --> 00:06:07.922
So we can write that this
thing is going to be less

00:06:07.922 --> 00:06:12.908
than one plus instead of writing this

00:06:12.908 --> 00:06:14.269
I'm gonna write the definite integral.

00:06:14.999 --> 00:06:17.472
One plus the definite integral from one

00:06:17.472 --> 00:06:22.472
to infinity of one over x squared dx.

00:06:25.444 --> 00:06:26.679
Now why is that useful?

00:06:26.907 --> 00:06:28.957
Well we know how to evaluate
this and I encourage

00:06:28.957 --> 00:06:31.602
you to review the section on Khan Academy

00:06:31.602 --> 00:06:33.877
on improper integrals if
this looks unfamiliar,

00:06:33.877 --> 00:06:37.300
but I'll evaluate this down here.

00:06:37.698 --> 00:06:41.701
We know that this is the
same thing as the limit as,

00:06:41.701 --> 00:06:43.133
I'm going to introduce a variable here,

00:06:43.133 --> 00:06:46.727
t approaches infinity
of the definite integral

00:06:46.727 --> 00:06:49.259
from one to t of and I'll just write

00:06:49.259 --> 00:06:52.191
this as x to the negative two dx.

00:06:52.449 --> 00:06:57.188
Which is equal to the limit
as t approaches infinity

00:06:57.188 --> 00:07:02.188
of negative x to the
negative one, or actually I

00:07:02.439 --> 00:07:04.145
could write that as negative one over x.

00:07:06.698 --> 00:07:09.659
And we're going to evaluate
that at t and at one,

00:07:09.659 --> 00:07:14.659
which is equal to the limit
as t approaches infinity

00:07:14.702 --> 00:07:19.702
of negative one over t
and then minus negative

00:07:20.648 --> 00:07:22.870
one over one so that
would just be plus one.

00:07:23.302 --> 00:07:25.462
And as t approaches infinity
this term right over here

00:07:25.462 --> 00:07:26.532
is going to be zero, so this is

00:07:26.532 --> 00:07:28.108
just going to simplify to one.

00:07:28.537 --> 00:07:32.468
So this whole thing evaluates to one.

00:07:33.067 --> 00:07:35.156
So just like that we were able to place

00:07:35.156 --> 00:07:37.619
an upper-bound on this series.

00:07:38.274 --> 00:07:41.894
We're able to say that
the series under question

00:07:41.894 --> 00:07:45.448
or in question, so the
infinite sum from n equals one

00:07:45.448 --> 00:07:48.221
to infinity of one over
n squared is going to be

00:07:48.221 --> 00:07:53.221
less than one plus one or it's
going to be less than two.

00:08:00.748 --> 00:08:03.084
Or another way to think about
it, it's going to be the two

00:08:03.084 --> 00:08:05.580
is this area, that's one right over there

00:08:05.580 --> 00:08:08.021
plus this area right over here.

00:08:08.358 --> 00:08:10.461
So we're saying that this
sum is going to be less

00:08:10.461 --> 00:08:12.768
than two so we have bounded it above.

00:08:12.768 --> 00:08:15.589
So we know that it cannot
go to positive infinity.

00:08:15.833 --> 00:08:18.018
Because all the terms are
positive it's definitely

00:08:18.018 --> 00:08:19.411
not going to go to negative infinity.

00:08:19.556 --> 00:08:21.364
And because all the terms
are positive we also know

00:08:21.364 --> 00:08:23.195
that this isn't going to oscillate between

00:08:23.195 --> 00:08:25.963
two different values, so
this gives us a pretty

00:08:25.963 --> 00:08:30.963
good sense that this series converges.

00:08:32.305 --> 00:08:36.946
And the logic we just
used here to argue for why

00:08:36.946 --> 00:08:39.385
this converges, once again
not a rigorous proof, but

00:08:39.385 --> 00:08:44.206
this is the underlying
logic of the integral test.


Worked example: direct comparison test

WEBVTT
Kind: captions
Language: en

00:00:00.582 --> 00:00:03.244
- [Voiceover] Let's think
about the infinite series,

00:00:03.244 --> 00:00:06.303
so we're going to go from
n equals one to infinity,

00:00:06.303 --> 00:00:11.259
of one over two to the n plus n.

00:00:11.300 --> 00:00:13.590
And what I want to do
is see if we can prove

00:00:13.590 --> 00:00:17.092
whether this thing converges or diverges.

00:00:17.092 --> 00:00:19.089
And as you can imagine based
on the context of where

00:00:19.089 --> 00:00:22.622
this video shows up on
Khan Academy that maybe

00:00:22.622 --> 00:00:25.376
we will do it using the comparison test.

00:00:25.655 --> 00:00:27.862
At any point if you feel
like you can kinda take

00:00:27.862 --> 00:00:29.482
this to the finish line, feel free

00:00:29.482 --> 00:00:31.473
to pause the video and do so.

00:00:32.451 --> 00:00:36.477
So in order to kind of
figure out or get a sense

00:00:36.477 --> 00:00:38.927
for this series right over
here, it never hurts to

00:00:38.927 --> 00:00:41.348
kind of expand it out a
little bit, so let's do that.

00:00:41.423 --> 00:00:43.538
So this would be equal
to when n equals one this

00:00:43.538 --> 00:00:46.531
is gonna be one over
two to the one plus one,

00:00:46.531 --> 00:00:48.182
so it's gonna be one over two plus

00:00:48.182 --> 00:00:51.506
one, it's gonna be one third
plus that's n equals one.

00:00:51.506 --> 00:00:54.052
When n equals two it's gonna
be one over two squared,

00:00:54.052 --> 00:00:57.763
which is four plus two plus one over six.

00:00:58.224 --> 00:01:03.224
Plus let's see we go
to three, n equals one,

00:01:04.641 --> 00:01:06.526
n equals two, n equals
three is gonna be one over

00:01:06.526 --> 00:01:09.457
two to the third, which
is eight plus three is 11.

00:01:09.574 --> 00:01:13.173
So one over 11, maybe
I'll do one more term.

00:01:13.388 --> 00:01:18.141
Two the fourth power is going
to be 16 plus four is 20.

00:01:18.365 --> 00:01:22.040
Plus one over 20 and obviously we

00:01:22.040 --> 00:01:24.351
just keep going on and on and on.

00:01:25.394 --> 00:01:30.394
So it looks, it feels like
this thing could converge.

00:01:32.225 --> 00:01:35.991
All of our terms are
positive, but they are getting

00:01:35.991 --> 00:01:38.604
smaller and smaller quite fast.

00:01:38.960 --> 00:01:43.960
And if we really look at
the behavior of the terms

00:01:45.235 --> 00:01:47.079
as n gets larger and
larger, we see that the two

00:01:47.079 --> 00:01:48.536
to the n in the denominator will grow

00:01:48.536 --> 00:01:51.238
much, much faster than the n will.

00:01:51.504 --> 00:01:55.316
So this kind of behaves
like one over two to the n,

00:01:55.790 --> 00:01:57.502
which is a clue of something that we might

00:01:57.502 --> 00:01:59.818
be able to use for the comparison test.

00:02:00.365 --> 00:02:01.655
So let me just write that down.

00:02:01.674 --> 00:02:05.573
So we have one over, so
the infinite series from

00:02:05.573 --> 00:02:08.962
n equals one to infinity
of one over two to the n,

00:02:09.601 --> 00:02:12.001
and so when n equals one this is

00:02:12.001 --> 00:02:16.396
going to be equal to one half.

00:02:16.803 --> 00:02:19.176
When n is equal to two this is

00:02:19.176 --> 00:02:21.271
going to be equal to one fourth.

00:02:21.584 --> 00:02:24.305
When n is equal to three
this is equal to one eighth.

00:02:24.540 --> 00:02:27.422
When n is equal to four this
is equal to one sixteenth.

00:02:27.745 --> 00:02:29.860
And we go on and on and on and on.

00:02:30.689 --> 00:02:32.958
And what's interesting about
this is we recognize it.

00:02:32.958 --> 00:02:35.907
This is a geometric
series, so let me be clear.

00:02:35.907 --> 00:02:38.849
This thing right over here,
that is the same thing

00:02:38.849 --> 00:02:43.849
as the sum of from n
equals one to infinity of

00:02:44.415 --> 00:02:48.691
one half to the n power, just
writing it in a different way.

00:02:49.012 --> 00:02:51.495
And since the absolute
value of one half, which

00:02:51.495 --> 00:02:56.495
is just one half, so
because the absolute value

00:02:57.728 --> 00:03:01.786
of one half is less than one we know that

00:03:01.786 --> 00:03:05.947
this geometric series converges,
we know that it converges

00:03:05.947 --> 00:03:08.774
And actually we even
have formulas for finding

00:03:08.774 --> 00:03:13.774
the exact sum of or to figure
out what it converges to.

00:03:15.138 --> 00:03:17.980
And so we know this thing
converges and we see that

00:03:17.980 --> 00:03:20.516
actually these two series
combined meet all of the

00:03:20.516 --> 00:03:23.361
constraints we need for
the comparison test.

00:03:23.920 --> 00:03:26.813
So let's go back to what we
wrote about the comparison test.

00:03:27.276 --> 00:03:31.287
So the comparison test,
we have two series, all

00:03:31.287 --> 00:03:33.420
of their terms are greater
than or equal to zero.

00:03:33.572 --> 00:03:35.714
All of these terms are
greater than or equal to zero.

00:03:36.199 --> 00:03:39.706
And then for the corresponding
terms in one series,

00:03:39.706 --> 00:03:41.337
all of them are going
to be less than or equal

00:03:41.337 --> 00:03:42.973
to the corresponding
terms in the next one.

00:03:43.421 --> 00:03:46.847
And so if we look over here,
we can consider this one,

00:03:46.926 --> 00:03:49.377
the magenta series, this
is kind of our infinite

00:03:49.377 --> 00:03:53.008
series of dealing with A
sub n and that this right

00:03:53.008 --> 00:03:56.490
over here is, well I
already did it in blue

00:03:56.490 --> 00:03:58.087
this is kind of the blue series.

00:03:58.264 --> 00:04:02.313
And notice all their
terms are nonnegative and

00:04:02.313 --> 00:04:07.005
the corresponding terms
one half is greater than

00:04:07.005 --> 00:04:09.321
one third, one fourth is
greater than one sixth,

00:04:09.321 --> 00:04:11.551
one eighth is greater than one eleventh.

00:04:11.878 --> 00:04:14.188
One over two to the n is
always going to be greater

00:04:14.188 --> 00:04:16.961
than one over two to the n plus n for

00:04:16.961 --> 00:04:19.579
the n that we care about here.

00:04:19.975 --> 00:04:22.109
And since we know that
this converges, since we

00:04:22.109 --> 00:04:23.921
know that the larger one converges.

00:04:24.511 --> 00:04:26.913
It's a geometric series
where the common ratio

00:04:26.913 --> 00:04:29.783
the absolute value of
the common ratio is less

00:04:29.783 --> 00:04:32.339
than one, since we know
that the larger series

00:04:32.339 --> 00:04:35.637
converges therefore, the
smaller series or the

00:04:35.637 --> 00:04:38.207
one where every corresponding
term is less than

00:04:38.207 --> 00:04:42.167
the one in the blue one,
that one must also converge.

00:04:42.473 --> 00:04:47.473
So by the comparison test the series

00:04:47.633 --> 00:04:52.633
in question must also converge.


Proof: harmonic series diverges

WEBVTT
Kind: captions
Language: en

00:00:01.144 --> 00:00:04.360
- [Voiceover] This is a
painting of Nicole Oresme.

00:00:04.360 --> 00:00:06.610
I looked up the phonetic spelling
before making this video.

00:00:06.610 --> 00:00:09.277
I'm assuming I'm still mispronouncing it.

00:00:09.277 --> 00:00:11.654
But I apologize to all the French speakers

00:00:11.654 --> 00:00:12.978
out there ahead of time.

00:00:12.978 --> 00:00:16.277
He was a famous French
philosopher mathematician,

00:00:16.277 --> 00:00:20.094
who lived in medieval France,
he lived in the 1300s.

00:00:20.094 --> 00:00:22.260
And he is famous for his proof

00:00:22.260 --> 00:00:26.719
that the harmonic series
actually diverges.

00:00:26.843 --> 00:00:29.811
And just as a little bit of
review, this is a harmonic series.

00:00:29.811 --> 00:00:33.552
One plus 1/2, plus 1/3,
plus 1/4, plus 1/5.

00:00:33.552 --> 00:00:35.443
And it's always been in my brain,

00:00:35.443 --> 00:00:37.212
the first time that I
saw the harmonic series,

00:00:37.212 --> 00:00:39.793
it wasn't obvious to me whether
it converged or diverged.

00:00:39.793 --> 00:00:42.500
It looks like, well, gee,
all these terms are positive,

00:00:42.500 --> 00:00:45.920
but they're going towards
zero, so I could imagine

00:00:45.920 --> 00:00:48.943
that this thing could converge.

00:00:48.943 --> 00:00:50.277
But he proved it otherwise.

00:00:50.277 --> 00:00:52.811
He proved one of the most
famous and most elegant proofs

00:00:52.811 --> 00:00:56.127
in mathematics that it
does indeed diverge.

00:00:56.127 --> 00:00:58.032
And the way that he did this,

00:00:58.032 --> 00:01:02.107
is he replaced every term
in the harmonic series

00:01:02.107 --> 00:01:06.212
with a term that is less
than or equal to that term.

00:01:06.212 --> 00:01:11.212
And then by proving that
his new series diverges

00:01:12.806 --> 00:01:15.566
and it's less than or
equal to this series,

00:01:15.566 --> 00:01:17.302
or each of the terms
are less than or equal

00:01:17.302 --> 00:01:19.819
to each of the
corresponding terms up here,

00:01:19.819 --> 00:01:21.968
then he says, therefore,
by the comparison test,

00:01:21.968 --> 00:01:23.707
this must diverge.

00:01:23.707 --> 00:01:25.672
So, how did he construct that?

00:01:25.672 --> 00:01:29.256
One way to think about it is
he replaced each of the terms

00:01:29.256 --> 00:01:34.236
in the harmonic series with
the largest power of 1/2

00:01:34.236 --> 00:01:38.861
that is less than or equal to that term.

00:01:38.861 --> 00:01:41.335
So, what's the largest power of 1/2

00:01:41.335 --> 00:01:44.676
that is less than or equal to one?

00:01:44.676 --> 00:01:47.586
Well, one is a power of 1/2, so that is,

00:01:47.586 --> 00:01:49.619
1/2 to the zero power is one.

00:01:49.619 --> 00:01:51.848
So, one is the largest power of 1/2

00:01:51.848 --> 00:01:54.043
that is less than or equal to one.

00:01:54.043 --> 00:01:56.389
I'll just write the one there.

00:01:56.389 --> 00:01:58.472
And now, what's the largest power of 1/2

00:01:58.472 --> 00:02:00.769
that is less than or equal to 1/2?

00:02:00.769 --> 00:02:02.264
That's just going to be 1/2,

00:02:02.264 --> 00:02:04.742
that's just 1/2 to the first power.

00:02:04.742 --> 00:02:09.223
Now, what's the largest power of 1/2

00:02:09.223 --> 00:02:12.534
that is less than or equal to 1/3?

00:02:12.534 --> 00:02:16.448
1/2 is larger than 1/3,
it's not less than 1/3.

00:02:16.448 --> 00:02:21.448
We want it to be less than 1/3,
so the largest power of 1/2

00:02:24.295 --> 00:02:27.462
that is less than or equal to 1/3 is 1/4.

00:02:27.462 --> 00:02:32.462
I replace 1/3 with 1/4 and, of
course, replace 1/4 with 1/4.

00:02:33.842 --> 00:02:38.842
And then we get to 1/5.

00:02:39.416 --> 00:02:41.227
What's the largest power of 1/2

00:02:41.227 --> 00:02:43.385
that is less than or equal to 1/5?

00:02:43.385 --> 00:02:46.594
Once again, 1/4 is greater than 1/5.

00:02:46.594 --> 00:02:48.720
The largest power of 1/2 that is less than

00:02:48.720 --> 00:02:52.018
or equal to 1/5 is 1/8.

00:02:52.018 --> 00:02:54.831
So, you replace that with 1/8.

00:02:54.831 --> 00:02:58.666
Of course, we replace for the
same reason that with 1/8.

00:02:58.666 --> 00:03:01.902
You would replace that one with 1/8.

00:03:01.902 --> 00:03:03.723
And, of course, 1/8.

00:03:03.723 --> 00:03:07.607
The largest power of 1/2 that is less than

00:03:07.607 --> 00:03:10.236
or equal to 1/8 is 1/8.

00:03:10.236 --> 00:03:12.508
And then what would you replace 1/9 with?

00:03:12.508 --> 00:03:16.502
You would replace 1/9 with 1/16
by the exact same argument.

00:03:16.502 --> 00:03:18.804
And you would keep going all
the way until you get to 1/16,

00:03:18.804 --> 00:03:23.220
so you would extensionally
have eight 1/16s in a row.

00:03:23.220 --> 00:03:24.810
What's interesting here?

00:03:24.810 --> 00:03:27.442
Let's first verify that we can use

00:03:27.442 --> 00:03:29.349
the comparison test here.

00:03:29.349 --> 00:03:33.018
In this first series each of
the terms are non-negative.

00:03:33.018 --> 00:03:36.407
In the second series each of
the terms are non-negative.

00:03:36.407 --> 00:03:40.095
And we also see that each
of the corresponding terms

00:03:40.095 --> 00:03:43.419
in a harmonic series is greater than

00:03:43.419 --> 00:03:45.918
or equal to the corresponding
term in this series.

00:03:45.918 --> 00:03:47.356
We constructed it this way.

00:03:47.356 --> 00:03:49.644
These are equal, this one is equal,

00:03:49.644 --> 00:03:51.957
this is greater than this,
1/3 is greater than 1/4.

00:03:51.957 --> 00:03:54.677
1/4 is equal to 1/4,
1/5 is greater than 1/8,

00:03:54.677 --> 00:03:58.046
1/6 is greater than 1/8,
1/7 is greater than 1/8.

00:03:58.046 --> 00:04:00.593
1/8 is equal to 1/8.

00:04:00.593 --> 00:04:03.310
One way to think about it is,

00:04:03.310 --> 00:04:05.259
each of the corresponding terms

00:04:05.259 --> 00:04:08.651
in this new constructed
series are smaller.

00:04:08.651 --> 00:04:10.805
Let me just call it S.

00:04:10.805 --> 00:04:13.163
In this infinite sum, and, of course,

00:04:13.163 --> 00:04:14.632
we keep going on and on and on.

00:04:14.632 --> 00:04:17.143
Maybe i should do that in magenta.

00:04:19.171 --> 00:04:22.320
We see that each of the
corresponding terms here

00:04:22.320 --> 00:04:24.640
are smaller than the
corresponding terms up here.

00:04:24.640 --> 00:04:26.670
And they are all positive.

00:04:26.670 --> 00:04:31.670
If we can prove that S, this
sum right over here, diverges,

00:04:34.650 --> 00:04:37.910
then, by the comparison
test, the larger series,

00:04:37.910 --> 00:04:39.018
the harmonic series here,

00:04:39.018 --> 00:04:40.932
the one where the
corresponding terms are larger,

00:04:40.947 --> 00:04:43.436
that must also diverge.

00:04:43.436 --> 00:04:44.994
And how do we do that?

00:04:44.994 --> 00:04:46.768
Let's just actually take these sums.

00:04:46.768 --> 00:04:49.094
This is going to be, let me write it.

00:04:49.094 --> 00:04:54.094
S is going to be equal to one plus 1/2.

00:04:56.254 --> 00:04:58.418
1/4 plus 1/4, what's that?

00:04:58.418 --> 00:05:01.306
That's 2/4, or 1/2.

00:05:01.306 --> 00:05:02.858
Did you see what's going on here?

00:05:02.858 --> 00:05:04.029
This is exciting.

00:05:04.029 --> 00:05:06.030
What's 1/8 plus 1/8 plus 1/8 plus 1/8?

00:05:06.030 --> 00:05:10.055
That's 4/8, or 1/2.

00:05:10.055 --> 00:05:13.352
What's 1/16 plus 1/16, and we
are going to go all the way

00:05:13.352 --> 00:05:15.736
until we are going to have eight of these.

00:05:15.736 --> 00:05:20.736
That's going to be 8/16, or 1/2.

00:05:21.536 --> 00:05:26.193
And then you are going to
have sixteen 1/32nds, or 1/2.

00:05:26.193 --> 00:05:29.965
And so, we are essentially
just going to be adding 1/2.

00:05:29.965 --> 00:05:32.067
We start with one, we
just keep adding plus 1/2,

00:05:32.067 --> 00:05:34.581
plus 1/2, plus 1/2, plus 1/2.

00:05:34.581 --> 00:05:38.939
This is clearly going to be
equal to, or this is unbounded.

00:05:38.939 --> 00:05:43.924
You could say, this is equal to infinity.

00:05:43.924 --> 00:05:48.924
Or, another way to think about
this, is S clearly diverges.

00:05:52.474 --> 00:05:57.474
And since each of its terms are smaller

00:05:58.693 --> 00:06:01.261
than the corresponding terms
in the harmonic series,

00:06:01.261 --> 00:06:06.261
we can then say the
harmonic series diverges.

00:06:13.493 --> 00:06:16.449
There is no way that this
thing over here can converge.

00:06:16.449 --> 00:06:19.696
If each of its corresponding
terms are smaller,

00:06:19.696 --> 00:06:22.284
you could even think of
the sum as being smaller.

00:06:22.284 --> 00:06:23.845
But this sum goes to infinity,

00:06:23.845 --> 00:06:27.701
so this one must also go to infinity.

00:06:27.701 --> 00:06:31.177
Hopefully, you found that
as interesting as I did.


Direct comparison test

WEBVTT
Kind: captions
Language: en

00:00:00.937 --> 00:00:02.954
- [Voiceover] So let's
get a basic understanding

00:00:02.954 --> 00:00:07.263
of the comparison test when
we are trying to decide

00:00:07.263 --> 00:00:10.823
whether a series is
converging or diverging.

00:00:10.823 --> 00:00:12.661
So let's think of two series.

00:00:12.661 --> 00:00:16.630
So let's say that I have
this magenta series here.

00:00:16.630 --> 00:00:17.690
It's an infinite series from n equals one

00:00:17.690 --> 00:00:19.474
to infinity of a sub n.

00:00:19.474 --> 00:00:21.773
We're speaking in generalities here,

00:00:21.773 --> 00:00:23.743
and let's have another one.

00:00:23.743 --> 00:00:27.818
That's the series b sub n
from n equals one to infinity,

00:00:27.818 --> 00:00:30.872
and we know some things
about these series.

00:00:30.872 --> 00:00:33.768
The first thing we know
is that all the terms

00:00:33.768 --> 00:00:36.350
in these series are non-negative.

00:00:36.350 --> 00:00:41.350
So a sub n and b sub n
are greater than or equal

00:00:41.857 --> 00:00:45.948
to zero, which tells us
that these are either

00:00:45.948 --> 00:00:49.527
going to diverge to positive infinity

00:00:49.527 --> 00:00:54.527
or they're going to converge
to some finite value.

00:00:55.894 --> 00:00:57.286
They're not going to
oscillate, because you're not

00:00:57.286 --> 00:00:58.551
going to have negative values here.

00:00:58.551 --> 00:00:59.964
You can't go to negative
infinity, because you

00:00:59.964 --> 00:01:02.318
don't have negative values here.

00:01:02.318 --> 00:01:05.640
Now, let's say we also know that each

00:01:05.640 --> 00:01:08.130
of the corresponding
terms in the first series

00:01:08.130 --> 00:01:11.221
are less than or equal
to the corresponding term

00:01:11.221 --> 00:01:13.590
in the second series.

00:01:13.590 --> 00:01:16.401
Less than or equal to b sub n.

00:01:16.401 --> 00:01:18.528
Once again, this is true for all the ns

00:01:18.528 --> 00:01:19.689
that we care about.

00:01:19.689 --> 00:01:22.329
So n equals one, two, three,

00:01:22.329 --> 00:01:25.350
all the way on, and on, and on.

00:01:25.350 --> 00:01:29.328
So the comparison test
tells us that because

00:01:29.328 --> 00:01:31.875
all the corresponding terms

00:01:31.875 --> 00:01:34.094
of this series are less
than the corresponding

00:01:34.094 --> 00:01:36.971
terms here, but they're greater than zero,

00:01:36.971 --> 00:01:40.568
that if this series converges,
the one that's larger,

00:01:40.568 --> 00:01:41.849
if this one converges, well then the one

00:01:41.849 --> 00:01:44.970
that is smaller than
it, or I guess when we

00:01:44.970 --> 00:01:46.566
think about it is kind
of bounded by this one,

00:01:46.566 --> 00:01:47.915
must also converge.

00:01:47.915 --> 00:01:49.255
I'm not doing a formal
proof here, but hopefully

00:01:49.255 --> 00:01:51.409
that gives you a little bit of intuition.

00:01:51.409 --> 00:01:56.409
So the comparison test tells
us if, I guess in my brain

00:01:58.107 --> 00:02:01.732
the larger series, the one
whose corresponding terms

00:02:01.732 --> 00:02:04.370
are at least as large as
the ones here, if this one

00:02:04.370 --> 00:02:06.830
converges, if this one
doesn't go unbounded

00:02:06.830 --> 00:02:11.413
towards infinity, it sums
to some finite value,

00:02:11.413 --> 00:02:15.626
then that tells us that the
one that is in some ways

00:02:15.626 --> 00:02:19.610
I guess you could say
smaller must also converge.

00:02:20.960 --> 00:02:25.215
So this one right over
here must also converge.

00:02:26.384 --> 00:02:28.114
So that must also converge.

00:02:28.114 --> 00:02:29.580
So why is that useful?

00:02:29.580 --> 00:02:31.152
We'll see this in future videos.

00:02:31.152 --> 00:02:33.628
Well if you find if you're
looking you have your

00:02:33.628 --> 00:02:35.684
a sub n and you're like
gee, I wish I could prove

00:02:35.684 --> 00:02:37.615
that it converges, I kind
of have a gut feeling

00:02:37.615 --> 00:02:40.372
it converges, the comparison
test tells us, well,

00:02:40.372 --> 00:02:45.372
just find another series
that whose corresponding

00:02:45.870 --> 00:02:47.182
terms are at least as
large as the corresponding

00:02:47.182 --> 00:02:50.309
terms here, and if you can
prove that one converges,

00:02:50.309 --> 00:02:52.320
then you're good with this one.

00:02:52.320 --> 00:02:53.773
Of course it would only apply to the case

00:02:53.773 --> 00:02:57.755
where your original
series, each of the terms

00:02:57.755 --> 00:02:59.916
are non-negative.

00:02:59.916 --> 00:03:01.847
Now what if you went the other way around?

00:03:01.847 --> 00:03:04.617
What if you could prove
that the magenta series,

00:03:04.617 --> 00:03:06.726
the smaller one, and I
guess I could kind of

00:03:06.726 --> 00:03:08.857
put them in quotes,
this one right over here

00:03:08.857 --> 00:03:13.857
is the smaller, I guess
each of its corresponding

00:03:14.330 --> 00:03:15.868
terms are smaller, what if you could prove

00:03:15.868 --> 00:03:18.336
this one diverges?

00:03:18.336 --> 00:03:20.103
Well if this one
diverges, it's going to go

00:03:20.103 --> 00:03:21.684
unbounded to infinity.

00:03:21.684 --> 00:03:23.370
It's not going to go to negative infinity.

00:03:23.370 --> 00:03:24.160
All the terms are positive.

00:03:24.160 --> 00:03:26.032
It's not going to diverge because

00:03:26.032 --> 00:03:27.451
it oscillates between two values.

00:03:27.451 --> 00:03:29.526
Once again, if it's
oscillating between values

00:03:29.526 --> 00:03:31.394
the only way you could do that is if you

00:03:31.394 --> 00:03:33.640
had negative terms here,
so this would kind of

00:03:33.640 --> 00:03:36.597
be unbounded towards infinity.

00:03:36.597 --> 00:03:38.632
Well if this one is
unbounded, each of these

00:03:38.632 --> 00:03:40.424
corresponding terms
are larger, so this one

00:03:40.424 --> 00:03:41.758
must also be unbounded.

00:03:41.758 --> 00:03:43.743
So let's write that down.

00:03:43.743 --> 00:03:48.743
So the comparison test
tells us if our smaller

00:03:50.481 --> 00:03:54.977
series diverges, if this one diverges,

00:03:54.977 --> 00:03:58.495
then the larger one must also diverge.

00:04:04.714 --> 00:04:06.669
So once again, if you wanted to prove that

00:04:06.669 --> 00:04:09.188
this thing right over
here is going to diverge,

00:04:09.188 --> 00:04:11.262
and if you have a, once
again you know that

00:04:11.262 --> 00:04:12.930
all the b sub ns are greater
than or equal to zero

00:04:12.930 --> 00:04:14.772
and you want to prove it diverges,

00:04:14.772 --> 00:04:18.241
well, maybe you could try
to find another series

00:04:18.241 --> 00:04:19.893
where each of the
corresponding terms are less

00:04:19.893 --> 00:04:21.912
than the corresponding
terms here and you could

00:04:21.912 --> 00:04:25.531
prove this one diverges,
then you would be all set.

00:04:25.531 --> 00:04:29.209
We're going to start doing
that in the next few videos.


Justification for polar arc length formula

WEBVTT
Kind: captions
Language: en

00:00:00.566 --> 00:00:01.033
- [Voiceover] What I want to do with

00:00:01.033 --> 00:00:04.433
this video is come up with
the formula for the arc

00:00:04.433 --> 00:00:08.433
length of a curve that's
defined in polar coordinates.

00:00:08.433 --> 00:00:10.433
So, if this curve right over here is r

00:00:10.433 --> 00:00:12.900
is equal to F of theta, how do we figure

00:00:12.900 --> 00:00:16.833
out the length of this curve between two

00:00:16.833 --> 00:00:19.300
thetas, say between theta is equal to,

00:00:19.300 --> 00:00:20.433
well let's say, in this case, it looks

00:00:20.433 --> 00:00:22.567
like between theta zero radius and say, pi

00:00:22.567 --> 00:00:27.466
over two radius, but between
any two bounds for r theta.

00:00:27.966 --> 00:00:29.233
So the way we're going to do this, an if

00:00:29.233 --> 00:00:30.833
at any point you get excited or

00:00:30.833 --> 00:00:32.567
inspired, you definitely should pause

00:00:32.567 --> 00:00:34.633
the video and see if you can run with the

00:00:34.633 --> 00:00:36.100
formula for arc length when you're

00:00:36.100 --> 00:00:37.933
dealing with something in polar form.

00:00:38.301 --> 00:00:39.433
But the way that we're going to tackle

00:00:39.433 --> 00:00:41.166
it is the exact same way that we tackled

00:00:41.166 --> 00:00:42.900
arc length when we were dealing with

00:00:42.900 --> 00:00:44.800
standard rectangular coordinates.

00:00:45.166 --> 00:00:48.100
So, let's take a little, small section of

00:00:48.100 --> 00:00:50.966
the arc length, let's take a little small

00:00:50.966 --> 00:00:52.600
section, I'm going to blow it up.

00:00:52.833 --> 00:00:55.367
So, let's call this right over here,

00:00:55.367 --> 00:00:58.366
this is our infinitesitely, this is our

00:00:58.366 --> 00:01:02.500
infinitely smaller, infinitesimal sized,

00:01:02.500 --> 00:01:06.000
our piece of our arc length,
I'm going to call it DS.

00:01:06.300 --> 00:01:08.135
And obviously, this is
a lot bigger than maybe

00:01:08.135 --> 00:01:09.766
you would imagine when you think of

00:01:09.766 --> 00:01:11.700
infinitesimal, but then if you integrate

00:01:11.700 --> 00:01:13.766
together all of the DSes, if you

00:01:13.766 --> 00:01:16.367
integrate together all of the DSes, then

00:01:16.367 --> 00:01:18.500
you're going to have, you're going to have

00:01:18.500 --> 00:01:21.866
the length of the actual
curve that you care about.

00:01:22.100 --> 00:01:24.033
So we can say that the length is going

00:01:24.033 --> 00:01:26.433
to be all of the DSes integrated, all of

00:01:26.433 --> 00:01:28.767
the infinitely, the infinite sum of all

00:01:28.767 --> 00:01:33.733
these infinitely small, all of
these infinitely small DSes.

00:01:33.900 --> 00:01:36.100
Now, to actually put this in terms we

00:01:36.100 --> 00:01:39.033
can relate to in terms of Rs and thetas,

00:01:39.033 --> 00:01:41.300
I'm first going to relate this to Xs and

00:01:41.300 --> 00:01:43.586
Ys and then relate the Rs and thetas to

00:01:43.586 --> 00:01:45.500
Xs and Ys, which we have seen before when

00:01:45.500 --> 00:01:48.800
we have converted between polar
form and rectangular form.

00:01:49.166 --> 00:01:52.966
So we know that this DS is going to be

00:01:52.966 --> 00:01:57.868
equal to our infinitely
small change in X squared.

00:01:58.100 --> 00:02:00.566
So if this is our,
going from this point to

00:02:00.566 --> 00:02:02.734
that point, that's our
change in our arc length.

00:02:02.900 --> 00:02:04.766
But this distance, right over here would

00:02:04.766 --> 00:02:08.500
be our change in our X, I'll write that

00:02:08.500 --> 00:02:10.366
as DX and I'm writing everything as

00:02:10.366 --> 00:02:11.566
differentials, which is a little bit

00:02:11.566 --> 00:02:13.900
mathematically, I guess we could say,

00:02:13.900 --> 00:02:15.633
hand-wavy or loosey
goosey, but it gives you

00:02:15.633 --> 00:02:18.266
a good conceptual understanding
of where this comes from.

00:02:18.366 --> 00:02:19.766
You could, if we were a little bit more

00:02:19.766 --> 00:02:21.501
precise, we could take DX, we could do

00:02:21.501 --> 00:02:23.438
delta Xs and then eventually take limits

00:02:23.438 --> 00:02:24.900
and all the rest, but I'll just go with

00:02:24.900 --> 00:02:26.366
this because it makes a little bit, at

00:02:26.366 --> 00:02:28.800
least for my brain, more conceptual sense.

00:02:29.233 --> 00:02:30.900
So that's, that's our change in X when we

00:02:30.900 --> 00:02:32.800
go from that point to that point.

00:02:32.985 --> 00:02:36.833
Then this is our change in Y when we go

00:02:36.833 --> 00:02:39.400
from this point to this point, DY.

00:02:39.500 --> 00:02:41.433
And we've seen this before when we got

00:02:41.433 --> 00:02:43.166
our justification for the arc length

00:02:43.166 --> 00:02:44.967
formula rectangular coordinates, we

00:02:44.967 --> 00:02:47.633
could say that DS is going to be equal

00:02:47.633 --> 00:02:52.633
to the square root of DX squared, DX

00:02:53.766 --> 00:02:55.833
squared, plus DY squared and this just

00:02:55.833 --> 00:02:57.666
comes straight out of
the Pythagorean Theorum.

00:02:58.100 --> 00:03:01.233
Plus DY squared, and then if we can

00:03:01.233 --> 00:03:04.266
integrate these, then we're
kind of in the same place.

00:03:04.566 --> 00:03:09.000
But how do we get these
in terms of Rs and thetas?

00:03:09.166 --> 00:03:11.166
Well, to do that, we just have to remind

00:03:11.166 --> 00:03:13.833
ourselves what X is in terms of r and

00:03:13.833 --> 00:03:16.466
theta, what Y is in terms of r and theta.

00:03:16.566 --> 00:03:20.766
So X, we know, is going to be equal to r

00:03:20.766 --> 00:03:24.233
cosine of theta and we first saw this when

00:03:24.233 --> 00:03:26.700
we just first were going back and forth

00:03:26.700 --> 00:03:28.666
between polar and rectangular coordinates.

00:03:28.767 --> 00:03:33.767
And Y is going to be r sine of theta.

00:03:35.500 --> 00:03:37.766
And now we can use this to say what

00:03:37.766 --> 00:03:40.066
DX and what DY are going to be.

00:03:40.367 --> 00:03:43.566
DX is then going to be equal to, and we

00:03:43.566 --> 00:03:45.900
have to remember that r is going to be a

00:03:45.900 --> 00:03:48.233
function of theta, so actually, let me

00:03:48.233 --> 00:03:49.566
write it this way, let me just rewrite it.

00:03:49.566 --> 00:03:53.766
So X, we could also
write it as, F of theta

00:03:53.766 --> 00:03:58.766
times cosine theta and Y is equal to

00:03:59.300 --> 00:04:03.200
F of theta times sine of theta.

00:04:03.700 --> 00:04:07.700
So now, what's a DX? DX is going to be,

00:04:07.700 --> 00:04:08.834
this is just, we're just going

00:04:08.834 --> 00:04:10.633
to apply the product rule here.

00:04:10.633 --> 00:04:12.434
It's going to be F prime of theta,

00:04:12.434 --> 00:04:14.033
derivative of the first expression times

00:04:14.033 --> 00:04:17.234
the second one, times cosine to theta plus

00:04:17.234 --> 00:04:18.566
the derivative of the second one.

00:04:18.566 --> 00:04:19.833
Well, the derivative of cosine of theta is

00:04:19.833 --> 00:04:24.200
negative sine of theta, so
we'll say minus sine theta.

00:04:25.166 --> 00:04:26.833
Minus sine theta times the first

00:04:26.833 --> 00:04:29.366
expressions, so F of theta, that was just

00:04:29.366 --> 00:04:31.333
the product rule, that's our DX.

00:04:31.816 --> 00:04:35.100
And then, of course, D theta, D theta.

00:04:35.100 --> 00:04:36.633
Another way you could have said if you

00:04:36.633 --> 00:04:37.833
treated these differentials like

00:04:37.833 --> 00:04:39.500
numbers, you could divide both sides by

00:04:39.500 --> 00:04:40.766
D theta, you would have the derivative of

00:04:40.766 --> 00:04:42.633
X with respect to theta is this business

00:04:42.633 --> 00:04:45.798
right over here, so those
are equivalent statements.

00:04:45.798 --> 00:04:47.800
And also do the same thing for DY.

00:04:48.100 --> 00:04:51.433
So DY, same again, by the product rule

00:04:51.433 --> 00:04:56.433
is going to be F prime, F prime of theta

00:04:58.033 --> 00:05:03.033
times sine of theta plus F of theta times

00:05:04.566 --> 00:05:07.200
the derivative of sine of
theta, which is cosine of theta.

00:05:08.367 --> 00:05:11.566
Cosine of theta, and now if we want to

00:05:11.566 --> 00:05:12.766
figure out what DS is,
we're going to have to

00:05:12.766 --> 00:05:15.867
take the sum of DX squared and
DY squared, so let's do that.

00:05:16.233 --> 00:05:21.233
So, DX, DX squared is going to be equal

00:05:23.433 --> 00:05:24.766
to, we just need to square all of this

00:05:24.766 --> 00:05:26.433
business, so I'm just going to square

00:05:26.433 --> 00:05:28.233
this and then multiply that times D theta

00:05:28.233 --> 00:05:30.800
squared, so that's going to be equal to,

00:05:30.900 --> 00:05:35.466
this is going to be F prime of theta

00:05:36.033 --> 00:05:40.100
squared, cosine squared theta, minus 2

00:05:40.100 --> 00:05:44.900
times the product of
these, minus 2 times F

00:05:44.900 --> 00:05:49.900
prime of theta, F of theta, cosine theta,

00:05:51.766 --> 00:05:55.900
sine theta, and then this one squared, so

00:05:55.900 --> 00:05:57.367
negative times negative is a positive, so

00:05:57.367 --> 00:06:02.367
plus F of theta squared, sine
squared, sine squared theta.

00:06:05.166 --> 00:06:07.433
So that's DX squared, and then, of

00:06:07.433 --> 00:06:09.233
course, we have the D theta, well not done

00:06:09.233 --> 00:06:12.500
yet, then we have the D theta squared and

00:06:12.500 --> 00:06:14.600
now let's figure out what DY squared is.

00:06:15.233 --> 00:06:20.233
So, DY squared is going to be equal to...

00:06:20.767 --> 00:06:23.033
Well, this term, squared, oh I have to

00:06:23.033 --> 00:06:26.033
forget, this DY is going to have a D theta

00:06:26.033 --> 00:06:27.866
at the end, don't want to forget that.

00:06:28.100 --> 00:06:31.300
And so over here, this is going to be F

00:06:31.300 --> 00:06:36.300
prime of theta squared, sine squared,

00:06:38.566 --> 00:06:41.033
sine squared theta, and then two times

00:06:41.033 --> 00:06:45.067
the products of these, so plus two times

00:06:45.166 --> 00:06:50.166
F prime of, let me, F prime of theta, F of

00:06:50.367 --> 00:06:51.633
theta, it's a little bit hairy, but we'll

00:06:51.633 --> 00:06:54.533
see in a few it's going
to clean up nicely.

00:06:54.556 --> 00:06:59.066
F of theta, cosine sine, cosine theta,

00:06:59.300 --> 00:07:01.566
sine theta and then we just want to square

00:07:01.566 --> 00:07:06.566
this plus F of theta
squared cosine, cosine

00:07:09.033 --> 00:07:13.666
squared theta, and then D theta squared.

00:07:14.166 --> 00:07:17.666
D theta squared, now let's
add these two together.

00:07:18.366 --> 00:07:21.200
So let's add them together,
and what are we going to get?

00:07:21.300 --> 00:07:25.833
So, if we add DX squared and DY squared,

00:07:25.833 --> 00:07:30.100
we're going to get, so DX squared plus DY

00:07:30.100 --> 00:07:35.000
squared plus DY squared is equal to.

00:07:35.433 --> 00:07:38.033
So over here we have cosine squared

00:07:38.033 --> 00:07:41.833
theta times F prime of theta squared and

00:07:41.833 --> 00:07:45.533
then sine squared there times
F prime of theta squared.

00:07:45.966 --> 00:07:48.300
So we can factor out an F prime of theta

00:07:48.300 --> 00:07:49.972
squared, so it's going to be equal to,

00:07:49.972 --> 00:07:52.900
so if we factor these characters out,

00:07:52.900 --> 00:07:57.900
it's going to be F prime of theta squared

00:07:58.433 --> 00:08:03.433
times, times cosine squared theta, cosine

00:08:04.100 --> 00:08:06.400
squared theta plus sine squared theta.

00:08:07.233 --> 00:08:10.969
Plus sine squared theta and we see that

00:08:10.969 --> 00:08:12.566
that's going to simplify nicely, that this

00:08:12.566 --> 00:08:15.766
is just going to be equal to one,

00:08:15.766 --> 00:08:18.333
that's just one basic trig identity.

00:08:18.433 --> 00:08:19.833
And then, these middle two terms

00:08:19.833 --> 00:08:21.900
actually cancel out, this is negative of

00:08:21.900 --> 00:08:26.033
this, so these two
cancel out and then over

00:08:26.033 --> 00:08:29.733
here, we can factor out
an F of theta squared.

00:08:30.366 --> 00:08:33.500
So, we could factor out an F of theta

00:08:33.500 --> 00:08:38.500
squared, so it becomes plus F of theta

00:08:38.900 --> 00:08:43.900
squared times sine squared theta, sine

00:08:44.966 --> 00:08:47.233
squared theta plus cosine squared

00:08:47.233 --> 00:08:51.334
theta, plus cosine squared theta.

00:08:51.442 --> 00:08:53.300
Well, that simplifies nicely.

00:08:53.300 --> 00:08:55.866
This is just going to be equal to one.

00:08:56.100 --> 00:08:59.900
And then, we have these and then this D, D

00:08:59.900 --> 00:09:03.200
theta squared is multiplied
times everything.

00:09:03.567 --> 00:09:07.033
So, so everything right over here is going

00:09:07.033 --> 00:09:11.100
to be, so times D theta, D theta squared.

00:09:11.100 --> 00:09:12.566
You can almost view these as the

00:09:12.566 --> 00:09:14.100
coefficients on D theta squared and

00:09:14.100 --> 00:09:15.666
we added those two coefficients.

00:09:15.966 --> 00:09:17.634
So, this is going to clean up nicely now,

00:09:17.634 --> 00:09:22.634
so this simplifies as DX squared plus DY

00:09:23.300 --> 00:09:28.300
squared plus DY squared is equal to, is

00:09:29.633 --> 00:09:34.633
equal to F prime of
theta squared plus F of

00:09:37.567 --> 00:09:42.567
theta plus F of theta squared.

00:09:44.633 --> 00:09:46.858
And then all of that
times, I'm going to do

00:09:46.858 --> 00:09:48.634
this in a new color, actually.

00:09:48.634 --> 00:09:50.730
All of that times D theta squared.

00:09:50.730 --> 00:09:52.500
Actually, that, I've
already used that color.

00:09:52.500 --> 00:09:54.325
I'll use the magenta.

00:09:54.325 --> 00:09:59.236
All of that times D theta
squared, D theta squared.

00:10:00.151 --> 00:10:01.900
Now we know that DS is going to be the

00:10:01.900 --> 00:10:03.633
square root of this, so let's write that,

00:10:03.633 --> 00:10:08.633
so DS is equal to the square root of this,

00:10:08.766 --> 00:10:11.666
which is equal to the square root of this.

00:10:12.233 --> 00:10:14.633
Which is going to be equal to, well we

00:10:14.633 --> 00:10:17.946
can factor out a, the square root of D

00:10:17.946 --> 00:10:19.248
theta squared is just going to be D

00:10:19.248 --> 00:10:20.866
theta, so we can just take that out.

00:10:21.100 --> 00:10:25.566
And we are left with, we are left with F

00:10:25.566 --> 00:10:30.566
prime of theta squared. F prime of theta

00:10:32.500 --> 00:10:37.500
squared plus F of theta squared,
plus F of theta squared.

00:10:39.500 --> 00:10:41.566
And now we take a D theta out of the

00:10:41.566 --> 00:10:43.207
radical, if you put it
in as D theta squared,

00:10:43.207 --> 00:10:47.600
you take it out, it's going
to be D theta, D theta.

00:10:48.033 --> 00:10:49.233
So this is interesting, so if we wanted

00:10:49.233 --> 00:10:51.033
to integrate them, so if you wanted to

00:10:51.033 --> 00:10:52.833
integrate this, if you want to integrate

00:10:52.833 --> 00:10:55.300
this, if you want to integrate this, you

00:10:55.300 --> 00:10:58.467
just want to integrate
this right over here.

00:10:58.766 --> 00:11:00.100
And you would integrate it from your

00:11:00.100 --> 00:11:01.700
starting theta, maybe we could call that

00:11:01.700 --> 00:11:04.333
alpha, to your ending theta, beta.

00:11:04.433 --> 00:11:05.766
And just like that, we have given

00:11:05.766 --> 00:11:08.366
ourselves a reasonable justification, or

00:11:08.366 --> 00:11:11.033
hopefully a conceptual understanding, for

00:11:11.033 --> 00:11:12.900
the formula for arc length when we're

00:11:12.900 --> 00:11:15.200
dealing with something in polar form.

00:11:15.366 --> 00:11:19.033
If you have r is equal to F of theta,

00:11:19.033 --> 00:11:21.100
you find what F prime of theta is, or

00:11:21.100 --> 00:11:22.633
you could think of it as the derivative

00:11:22.633 --> 00:11:25.033
of r with respect to theta, square that,

00:11:25.033 --> 00:11:26.966
add that to F of theta squared, take the

00:11:26.966 --> 00:11:28.766
square root and then integrate with

00:11:28.766 --> 00:11:32.600
respect to theta from alpha to beta.

00:11:32.833 --> 00:11:36.566
And so this right over
here, our arc length,

00:11:36.566 --> 00:11:41.033
is going to be equal to
this right over here.

00:11:41.033 --> 00:11:44.600
And the next few videos, we
will actually apply this.


Arc length of petal of polar graph

WEBVTT
Kind: captions
Language: en

00:00:00.476 --> 00:00:02.167
- What I want to do in this video

00:00:02.167 --> 00:00:06.016
is find the arc length of one petal,

00:00:06.016 --> 00:00:07.712
I guess we could call it,

00:00:07.712 --> 00:00:12.712
of the graph of r is equal
to four sine of two theta.

00:00:13.092 --> 00:00:16.529
So I want to find the length
of this portion of the curve

00:00:16.529 --> 00:00:18.520
that is in red right over here.

00:00:18.931 --> 00:00:21.280
We'll do this in two phases.

00:00:21.280 --> 00:00:23.017
First of all I want to set
up the definite integral

00:00:23.017 --> 00:00:26.437
for finding that arc length,
and then we'll evaluate it.

00:00:26.437 --> 00:00:28.092
I'll actually use a
calculator to evaluate it

00:00:28.092 --> 00:00:30.124
because it'll be a little
bit more straightforward

00:00:30.124 --> 00:00:32.179
at least for this integral.

00:00:32.179 --> 00:00:34.060
Let's just start with that.

00:00:34.060 --> 00:00:35.274
I encourage you to pause the video.

00:00:35.274 --> 00:00:37.831
Knowing what we know about
the formula for arc length,

00:00:37.831 --> 00:00:39.203
when we have it in polar form,

00:00:39.203 --> 00:00:41.895
see if you can apply it to
figure out this arc length

00:00:41.895 --> 00:00:43.303
right over here.

00:00:43.833 --> 00:00:45.170
I'm assuming you've had a go at it,

00:00:45.170 --> 00:00:48.653
so let's remind ourselves
that the arc length

00:00:48.653 --> 00:00:53.223
is going to be the integral
from our starting angle

00:00:53.223 --> 00:00:56.454
to our ending angle, we'll
call it from alpha to beta,

00:00:56.454 --> 00:01:01.454
of the square root of the
derivative of our function.

00:01:06.668 --> 00:01:11.668
Which respect to theta squared,
plus our function squared.

00:01:14.029 --> 00:01:17.027
Plus our function squared d theta.

00:01:17.904 --> 00:01:19.563
Let's figure out what r prime of theta

00:01:19.563 --> 00:01:21.806
and what r of theta actually are.

00:01:22.781 --> 00:01:24.127
I'll color code it.

00:01:24.127 --> 00:01:28.716
R prime of theta, we
know what r of theta is

00:01:28.716 --> 00:01:30.688
let me just rewrite it.

00:01:31.000 --> 00:01:36.000
We know r of theta is equal
to four sine of two theta.

00:01:42.448 --> 00:01:47.448
Our prime of theta is just
going to be derivative

00:01:47.909 --> 00:01:51.290
of two theta with respect
to theta is two times four

00:01:51.290 --> 00:01:56.290
is eight, derivative of sine
is cosine, cosine of two theta.

00:01:57.682 --> 00:02:01.515
If we wanted to write the
definite integral for arc length

00:02:01.515 --> 00:02:05.966
it's going to be equal
to the definite integral.

00:02:05.966 --> 00:02:08.021
What's our starting bound?

00:02:08.021 --> 00:02:09.340
For the petal that we care about,

00:02:09.340 --> 00:02:13.023
we're going to start at
an angle of zero radians.

00:02:13.399 --> 00:02:16.597
When we have an angle of
zero radians, r is zero,

00:02:16.597 --> 00:02:18.662
that's this point right over here.

00:02:19.016 --> 00:02:20.599
We're going to start at zero radians.

00:02:20.599 --> 00:02:23.703
And we're going to go all
the way up to pi over two.

00:02:23.703 --> 00:02:27.680
When sine of two times pi
over two is sine of pi,

00:02:27.680 --> 00:02:30.094
which is zero, so we'll
get back to this point

00:02:30.124 --> 00:02:31.558
right over here.

00:02:31.558 --> 00:02:34.337
We're going to go from
zero to pi over two,

00:02:34.907 --> 00:02:36.921
to pi over two radians.

00:02:36.921 --> 00:02:39.550
It's going to be the square root of,

00:02:39.550 --> 00:02:41.665
give myself some real estate here,

00:02:41.665 --> 00:02:46.665
r prime of theta squared is
going to be this thing squared.

00:02:47.522 --> 00:02:50.242
It's going to be 60--

00:02:50.356 --> 00:02:52.236
Let me do it in that same blue color,

00:02:52.948 --> 00:02:57.948
it's going to be 64 cosine
squared of two theta,

00:03:01.426 --> 00:03:06.426
then plus this squared
which is going to be

00:03:07.076 --> 00:03:12.076
16 sine squared of two theta,

00:03:13.509 --> 00:03:15.690
then of course we have our d theta.

00:03:17.942 --> 00:03:21.077
You can attempt to solve this
or evaluate this analytically,

00:03:21.077 --> 00:03:22.907
it's not too straightforward

00:03:22.907 --> 00:03:24.518
so I'm going to use a calculator.

00:03:24.518 --> 00:03:25.812
It just gives us a
practice with knowing that

00:03:25.812 --> 00:03:27.562
hey, there are tools out
there that can help us

00:03:27.562 --> 00:03:29.023
do these types of things.

00:03:29.472 --> 00:03:31.623
I'll go to the definite integral,

00:03:31.623 --> 00:03:34.143
I'll go to second calculus on my t 85,

00:03:34.143 --> 00:03:37.192
I picked this choice, which
is for the definite integral.

00:03:38.868 --> 00:03:41.350
Let me express when I'm taking
the definite integral of,

00:03:41.350 --> 00:03:46.350
it's going to be the square root of 64

00:03:48.241 --> 00:03:52.330
times cosine of two theta squared.

00:03:54.273 --> 00:03:56.443
I'm going to use x instead of theta

00:03:56.443 --> 00:03:58.245
just because it's an easier variable

00:03:58.245 --> 00:03:59.990
to use on my calculator.

00:04:00.131 --> 00:04:02.378
Cosine of two--

00:04:03.256 --> 00:04:05.483
Actually let me make sure
my parenthesis are good.

00:04:05.483 --> 00:04:10.483
Cosine of two x, alright I'll close that.

00:04:13.366 --> 00:04:16.286
Now I want to square that.

00:04:17.523 --> 00:04:18.930
I did that first part.

00:04:19.187 --> 00:04:22.539
Plus 16 times

00:04:22.847 --> 00:04:25.072
plus 16

00:04:25.072 --> 00:04:26.549
times

00:04:26.918 --> 00:04:29.097
16 times

00:04:29.574 --> 00:04:34.574
sine of two x.

00:04:34.877 --> 00:04:37.024
Close that, close that.

00:04:37.024 --> 00:04:38.738
Squared.

00:04:40.231 --> 00:04:42.772
I'm not sure if the calculator
knows to interpret that

00:04:42.772 --> 00:04:46.193
as multiplication, so let me
insert a times right over here.

00:04:46.193 --> 00:04:49.998
64 times cosine of two
x, that thing squared.

00:04:49.998 --> 00:04:53.608
Plus 16 times sine of two
x, that thing squared.

00:04:54.247 --> 00:04:56.111
Let me go to the end.

00:04:58.901 --> 00:05:01.680
Let me close my radical,

00:05:01.680 --> 00:05:05.711
so that's going to close
that right over there,

00:05:05.711 --> 00:05:08.019
that closes what I'm
taking the square root of.

00:05:08.019 --> 00:05:11.746
Then comma, the variable
that I'm integrating

00:05:11.746 --> 00:05:14.156
with respect to is x, in this case.

00:05:14.156 --> 00:05:15.371
Everywhere I saw theta here

00:05:15.371 --> 00:05:17.329
I'm just replacing with an x there.

00:05:17.329 --> 00:05:21.298
I want to do it from,
instead of theta equals zero

00:05:21.298 --> 00:05:22.705
to theta equals pi over two,

00:05:22.705 --> 00:05:27.705
we'll say x is equal to zero
to x is equal to pi over two.

00:05:31.001 --> 00:05:33.367
Hopefully I have not made a
mistake when I've typed this in,

00:05:33.367 --> 00:05:35.969
And I get...

00:05:37.937 --> 00:05:39.712
It's chugging on a little bit.

00:05:40.408 --> 00:05:42.494
And we get...

00:05:43.631 --> 00:05:44.711
Will it actually come up?

00:05:44.711 --> 00:05:45.941
Oh finally!

00:05:46.024 --> 00:05:49.139
If we were to round it
to the nearest thousandth

00:05:49.139 --> 00:05:52.754
it'll be nine point six eight eight.

00:05:54.342 --> 00:05:58.582
Approximately nine point six eight eight.

00:05:59.319 --> 00:06:01.950
Let's just see, does that
make intuitive sense?

00:06:01.950 --> 00:06:05.950
It goes as far as four away,

00:06:05.950 --> 00:06:09.755
so if you just went out
to four and then back down

00:06:09.755 --> 00:06:11.442
that would be about eight.

00:06:11.442 --> 00:06:13.607
Of course we have actually gone out some,

00:06:13.607 --> 00:06:15.985
so it actually makes intuitive
sense that this arc length

00:06:15.985 --> 00:06:19.495
would be nine point six eight eight.

00:06:20.167 --> 00:06:21.653
Hopefully you enjoyed that.


Growing your mind

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.700
What determines our intelligence?

00:00:02.700 --> 00:00:04.700
One school of thought is that,

00:00:04.700 --> 00:00:06.700
ok maybe, you are either just

00:00:06.700 --> 00:00:08.340
born with it, either have it or you don't.

00:00:08.340 --> 00:00:09.640
Some people are smart,

00:00:09.640 --> 00:00:11.320
some people are less smart,

00:00:11.320 --> 00:00:12.920
or maybe you can change it.

00:00:12.920 --> 00:00:14.920
And it turns out there's actually been a lot

00:00:14.920 --> 00:00:16.920
of research on this in the last few decades.

00:00:16.920 --> 00:00:18.380
And

00:00:18.380 --> 00:00:20.100
the answer is pretty clear that

00:00:20.100 --> 00:00:21.220
your intelligence

00:00:21.220 --> 00:00:23.680
can actually be changed.

00:00:23.680 --> 00:00:26.340
What we've learned, what researchers have taught us,

00:00:26.340 --> 00:00:28.640
is that our brains are actually a lot like a muscle.

00:00:28.640 --> 00:00:30.660
We know that you can grow your muscles

00:00:30.660 --> 00:00:32.320
by going into the gym,

00:00:32.340 --> 00:00:33.660
and doing exercise,

00:00:33.660 --> 00:00:35.340
and straining your muscles.

00:00:35.340 --> 00:00:38.140
You don't just work on things that are easy for your muscles to do;

00:00:38.140 --> 00:00:40.380
you do things that your muscles have to struggle with,

00:00:40.380 --> 00:00:42.520
that your muscles have to strain with

00:00:42.520 --> 00:00:44.340
and then they rebuild themselves

00:00:44.340 --> 00:00:45.920
and they come back stronger.

00:00:45.920 --> 00:00:48.340
By struggling, its a signal to your body

00:00:48.340 --> 00:00:50.020
to devote more resources

00:00:50.020 --> 00:00:51.720
to that part of the body.

00:00:51.720 --> 00:00:54.340
And we see that exact same thing with the brain .

00:00:54.340 --> 00:00:56.940
And here, just a couple of examples

00:00:56.940 --> 00:00:57.960
of it.

00:00:57.960 --> 00:01:00.340
This is first one,

00:01:00.340 --> 00:01:01.680
This shows how

00:01:01.680 --> 00:01:03.780
the human brain develops

00:01:03.780 --> 00:01:05.240
in early childhood.

00:01:05.240 --> 00:01:08.340
This is a depiction of the

00:01:08.340 --> 00:01:10.340
neurons in a brain

00:01:10.340 --> 00:01:11.740
at birth.

00:01:11.740 --> 00:01:14.340
And then over time, as a child develops

00:01:14.340 --> 00:01:16.340
It interacts with its environment,

00:01:16.340 --> 00:01:17.960
it tries things out,

00:01:17.960 --> 00:01:20.340
it struggles to talk,

00:01:20.340 --> 00:01:22.340
to converse,to interact with folks,

00:01:22.340 --> 00:01:24.340
to understand the world around it.

00:01:24.340 --> 00:01:26.340
And as it struggles,

00:01:26.340 --> 00:01:28.340
by age 6 you have a much

00:01:28.340 --> 00:01:30.500
deeper and much stronger connectedness

00:01:30.500 --> 00:01:33.840
between the different neurons.

00:01:33.840 --> 00:01:35.280
Similarly, similarly,

00:01:35.280 --> 00:01:37.180
I can never say that word,

00:01:37.180 --> 00:01:40.660
These are depictions of, these are the neurons

00:01:40.660 --> 00:01:42.660
of two different scenarios.

00:01:42.660 --> 00:01:44.660
This is the nerves

00:01:44.660 --> 00:01:46.660
of animals ,

00:01:46.660 --> 00:01:48.660
that were in un stimulating environments:

00:01:48.660 --> 00:01:50.660
where they are not around other animals.

00:01:50.660 --> 00:01:52.660
They are in a bear cage.

00:01:52.660 --> 00:01:54.660
This is the brain

00:01:54.660 --> 00:01:56.080
of an animal that is

00:01:56.080 --> 00:01:57.800
in a stimulating environment,

00:01:57.800 --> 00:01:59.800
that is constantly being challenged,

00:01:59.800 --> 00:02:01.160
that is constantly looking at

00:02:01.160 --> 00:02:02.720
stimulating new things.

00:02:02.720 --> 00:02:04.320
And so the big take away

00:02:04.320 --> 00:02:06.320
from this whole area of research

00:02:06.320 --> 00:02:07.800
is you absolutely can change

00:02:07.800 --> 00:02:09.380
your intelligence.

00:02:09.380 --> 00:02:11.120
That your brain is like a muscle

00:02:11.120 --> 00:02:13.660
the more you use it,the stronger it gets

00:02:13.660 --> 00:02:15.680
and that the best way to grow it

00:02:15.680 --> 00:02:18.000
isn't to do things that are easy for you.

00:02:18.000 --> 00:02:19.520
That might help a little bit

00:02:19.520 --> 00:02:21.300
but what really helps your brain

00:02:21.300 --> 00:02:22.920
is when you struggle with things.

00:02:22.920 --> 00:02:24.900
And actually research shows that your brain

00:02:24.900 --> 00:02:27.040
grows the most not when you get a question right,

00:02:27.040 --> 00:02:29.280
but when you get a question wrong.

00:02:29.280 --> 00:02:31.540
So at least for me this is incridibly exciting

00:02:31.540 --> 00:02:33.220
because, unless we know

00:02:33.220 --> 00:02:34.960
that when I'm going through something.

00:02:34.960 --> 00:02:37.540
And I'm facing those times of maybe a little bit of adversity,

00:02:37.540 --> 00:02:39.540
or a little bit of frustration.

00:02:39.540 --> 00:02:40.400
I can -

00:02:40.400 --> 00:02:42.140
feel good about the fact that those

00:02:42.140 --> 00:02:44.280
are actually the times that I'm growing the most.

00:02:44.280 --> 00:02:46.200
And this isn't just something that I'm saying

00:02:46.200 --> 00:02:47.880
nice words. Research tells us that when

00:02:47.880 --> 00:02:49.260
you get something wrong,

00:02:49.260 --> 00:02:50.620
when you challenge your brain,

00:02:50.620 --> 00:02:52.560
when you review why you got it wrong,

00:02:52.560 --> 00:02:54.360
when you really process that feedback ,

00:02:54.360 --> 00:02:56.300
that's when your brain grows the most.

00:02:56.300 --> 00:02:57.700
And then if you keep doing that

00:02:57.700 --> 00:03:00.120
you are well on your way to having a stronger, more able,

00:03:00.120 --> 00:03:05.540
and I guess ,smarter brain.