Subtracting complex numbers

WEBVTT
Kind: captions
Language: en

00:00:00.716 --> 00:00:02.305
Voiceover:So we've plotted two

00:00:02.305 --> 00:00:03.881
complex numbers right over here

00:00:03.881 --> 00:00:05.609
on this argand diagram.

00:00:05.609 --> 00:00:07.234
And what I want to think about is

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what if we defined a
third complex number, C,

00:00:09.810 --> 00:00:13.468
as being equal to the complex number, A,

00:00:13.468 --> 00:00:16.262
minus the complex number, B.

00:00:16.262 --> 00:00:18.103
What is C going to look like

00:00:18.103 --> 00:00:19.982
if we were to write it as just a

00:00:19.982 --> 00:00:22.366
sum of its real and imaginary parts

00:00:22.366 --> 00:00:23.734
and what would it look like

00:00:23.734 --> 00:00:25.764
on this argand diagram?

00:00:25.764 --> 00:00:27.860
So let's just first think about this.

00:00:27.860 --> 00:00:29.525
So C is...

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Let me do that yellow color.

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C is equal to A,

00:00:34.157 --> 00:00:41.949
which is the same thing as
six plus two i, minus B.

00:00:41.949 --> 00:00:43.094
And what is B?

00:00:43.094 --> 00:00:46.885
B is two minus i.

00:00:46.885 --> 00:00:48.622
So when we're subtracting anything,

00:00:48.622 --> 00:00:51.574
we can distribute this negative sign.

00:00:51.574 --> 00:00:53.469
And so this would be the same thing...

00:00:53.469 --> 00:00:59.325
This would be equivalant
to six plus two i,

00:00:59.325 --> 00:01:03.702
minus two, and then
distributing the negative

00:01:03.702 --> 00:01:06.109
on a negative i, that's
going to be a positive i.

00:01:06.109 --> 00:01:08.317
Plus i, just like that.

00:01:08.317 --> 00:01:13.293
And now this is going to be equal to...

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I'll do it right over here.

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This is going to be equal to...

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We can add the real parts right over here.

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We have six minus two.

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And so that's going to be four.

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And then we can add the two i to the i.

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Which is going to be three i's.

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Plus four, plus three i.

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And that is the complex number C.

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Let's see why that actually makes sense

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when we visualize A and B as vectors.

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So this is B.

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When we do A minus B,

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that's essentially the same thing as

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A plus negative B.

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So what's negative B going to be equal to?

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Negative B is going to be equal to

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negative two plus i.

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I just multiplied all the
terms by negative one.

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So what does that look like?

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Let's see, negative two plus i

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gets us right over there.

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And we see we've essentially just flipped

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this vector over the origin.

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So this right over here is negative B.

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That is negative B.

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And we've already seen,

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if you're adding two things,

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we're going to add A to negative B.

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So we've plotted A so that
its tail's at the origin

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and its head is right over here.

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And then we can take
the tail of negative B,

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put it right over here

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and then if we just shift it over,

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it goes two to the left and one up,

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it gets to that point right over here.

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It gets to this point right over there.

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This vector is another way to visualize

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negative B as a vector.

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And so where our new head is,

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once we've put the tail
of negative B here,

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where our new head is going to be

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the head of our vector C.

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So C is going to be this
vector right over here.

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C can be visualized as this
vector right over here.

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And we see that is one, two, three, four,

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that is four plus one, two, three i.

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We see that right over there.

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So on one level,

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it's kind of straight forward,

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manipulating expressions
and just recognizing

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that you can only add the

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imaginary parts to the imaginary parts

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and the real parts to the real parts.

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And if you want to visualize
it on an argand diagram,

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it really is analogous
to subtracting B from A

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is equivalent to adding
the negative of B to A.


Introduction to the FAFSA

WEBVTT
Kind: captions
Language: en

00:00:00.749 --> 00:00:02.799
Sal: So I'm here with Sean Logan,

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who's the Director of College Counseling

00:00:04.665 --> 00:00:07.531
at Phillips Andover
Academy, and Jim Ventre,

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Ven-tree, am I pronouncing it right, Jim?

00:00:09.097 --> 00:00:09.931
Jim: Ven-tree, yes.

00:00:09.931 --> 00:00:11.215
Sal: Ven-tree, who's the Director of

00:00:11.215 --> 00:00:13.665
Admissions and Financial
Aid at Phillips Andover,

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and what we're hoping to talk, or start

00:00:15.916 --> 00:00:19.298
talking about in this
video, is really just the

00:00:19.298 --> 00:00:22.883
financial aid process, and
in particular, the FAFSA,

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so, I guess, wait a
minute, I should probably

00:00:25.014 --> 00:00:28.082
do that in all caps, my first question for

00:00:28.082 --> 00:00:31.048
either of you is, I
guess, "what is the FAFSA,

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"and what does it actually stand for?"

00:00:33.514 --> 00:00:36.599
Sean: Right, so the FAFSA
is the free application

00:00:36.599 --> 00:00:39.532
for federal student aid,
and it's really important

00:00:39.532 --> 00:00:43.848
for students to realize
that free part of this name,

00:00:43.848 --> 00:00:46.648
because there is an
opportunity to get on the web

00:00:46.648 --> 00:00:49.698
and go to FAFSA.com and
somebody will happily

00:00:49.698 --> 00:00:51.764
help you with it, but
you need to pay for it.

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So the first thing we talk
with students a lot about

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is the FAFSA is a free form.

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You should not be paying for it.

00:00:57.097 --> 00:00:59.799
And the title again, it's
by the federal government,

00:00:59.799 --> 00:01:02.882
they use a rubric to sort of figure out,

00:01:02.882 --> 00:01:05.498
based on this form, how
much your family should

00:01:05.498 --> 00:01:07.531
contribute to a college education.

00:01:07.531 --> 00:01:10.931
Sal: And it's Free Application
for Federal Student Aid?

00:01:10.931 --> 00:01:12.583
Sean: Student, yes, student aid.

00:01:12.583 --> 00:01:14.109
Sal: And just to hit
the point home of what

00:01:14.109 --> 00:01:16.716
you just said, it should
absolutely be free,

00:01:16.716 --> 00:01:18.666
and there are places where
people will charge you

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to kind of optimize it, or
fill it out for you, or...

00:01:21.243 --> 00:01:23.666
Sean: That's correct, so the first thing,

00:01:23.666 --> 00:01:27.197
normally when you start to type in on

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a search engine, FAFSA, a lot of times

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FAFSA.com pops up.

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It's a legitimate site,
but they charge you

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something like 80, 85 dollars

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to help you fill out the form.

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So you want to go to FAFSA.ed.gov

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to start this process.

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Sal: .ed.gov, and there it will be free,

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and what is, when you
fill out this form, in a

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future video we'll actually

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take a look at the form itself,

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but what do you input
into it, and what comes

00:01:54.848 --> 00:01:57.415
out of it, and then who
uses that information?

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Jim: I think for many
students who are applying

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to college right now, they are accustomed

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to telling their story,
in essays and in various

00:02:04.981 --> 00:02:07.715
pieces of their application,
and they have to

00:02:07.715 --> 00:02:11.833
think of the FAFSA as just their family's

00:02:11.833 --> 00:02:15.384
financial story, and
their financial story has

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many variables, but most
importantly, the things

00:02:18.822 --> 00:02:23.832
that the FAFSA asks for are
simplified as things that

00:02:23.832 --> 00:02:26.465
would be questions that
students could probably

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answer right off the top of their head,

00:02:28.748 --> 00:02:31.565
in terms of their family
income, and whether they

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own a home, or whether
they rent where they live,

00:02:34.582 --> 00:02:36.382
it's things like that.

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Sal: And, I mean, I
remember filling this, we

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were talking about this
before, I guess clicking

00:02:39.882 --> 00:02:42.803
record, all of us were
significant recipients

00:02:42.803 --> 00:02:45.248
of financial aid, of
need-based financial aid,

00:02:45.248 --> 00:02:46.382
and I remember filling it out, I remember

00:02:46.382 --> 00:02:49.048
I kind of had my mom's
tax form, which I had to

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fill out for her, right
before filling out the FAFSA,

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Sean: Exactly (laughing)

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Sal: But that's essentially, I remember,

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at least this was in the early nineties,

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kind of the inputs that I
needed to be able to fill out

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most of the form.

00:03:01.748 --> 00:03:03.632
Sean: And that's correct,
yeah, I mean, one of the

00:03:03.632 --> 00:03:06.739
things that specifically
looking at, in working with

00:03:06.739 --> 00:03:09.498
a lot of low-income
families, one of the things

00:03:09.498 --> 00:03:10.932
to realize, is that it generally is

00:03:10.932 --> 00:03:12.865
actually a pretty simple
form, because there

00:03:12.865 --> 00:03:14.882
aren't a lot of things
that make it complicated.

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There aren't trust funds,
there aren't investments,

00:03:17.820 --> 00:03:19.582
there aren't probably, in a lot of times,

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sometimes there are homes that are owned,

00:03:21.481 --> 00:03:23.148
but a lot of times they're just renting.

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Usually it's a paycheck that comes in,

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and it's sort of a standard
paycheck that comes

00:03:28.715 --> 00:03:30.671
in for the year, so it's actually a pretty

00:03:30.671 --> 00:03:33.397
simple form, it's just
getting students to step up

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and do it, is the real challenge.

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Sal: And the reason obviously, it's

00:03:37.432 --> 00:03:38.800
pretty obvious why they care about income,

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that's kind of the total amount of money

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your family's making, but then the reason

00:03:42.999 --> 00:03:44.332
why they care about things
like home ownership,

00:03:44.332 --> 00:03:46.465
and if I remember
properly, they ask things

00:03:46.465 --> 00:03:48.064
about the number of siblings you have,

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and they try to understand your family's

00:03:49.431 --> 00:03:52.498
expenses too, and the
whole reason is to kind of

00:03:52.498 --> 00:03:55.847
figure out their ability
to pay for college?

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Jim: That's right, so at
the outset, students are

00:03:59.798 --> 00:04:02.134
asked questions to fill
out on their family's

00:04:02.134 --> 00:04:04.965
financial circumstances,
basically what the colleges

00:04:04.965 --> 00:04:07.998
are trying to do, is come to a conclusion,

00:04:07.998 --> 00:04:12.048
which allows them to have a
picture of what the family

00:04:12.048 --> 00:04:14.764
is up against every month, every week,

00:04:14.764 --> 00:04:17.549
and what's left in the bank, so to speak,

00:04:17.549 --> 00:04:20.981
at the end of all the
family operating expenses.

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Sal: And so what, after
you fill out the form,

00:04:23.415 --> 00:04:26.082
where does that go, this is at the

00:04:26.082 --> 00:04:28.299
Department of Education site, but do they

00:04:28.299 --> 00:04:30.231
process it, or do they just give the form

00:04:30.231 --> 00:04:32.749
to the universities,
what happens after that?

00:04:32.749 --> 00:04:34.332
Sean: That's a great point Sal, I think

00:04:34.332 --> 00:04:37.431
one of the misperceptions
about the FAFSA is,

00:04:37.431 --> 00:04:40.032
I'm gonna submit this form to the

00:04:40.032 --> 00:04:42.466
government, basically, and
then they're gonna crunch

00:04:42.466 --> 00:04:44.466
all these numbers, and then
they're gonna determine

00:04:44.466 --> 00:04:46.332
a financial aid package for my family.

00:04:46.332 --> 00:04:48.499
I think that's a really
common misperception.

00:04:48.499 --> 00:04:52.165
All the FAFSA does, is it
looks again at your family's

00:04:52.165 --> 00:04:54.115
income, number of kids in the family,

00:04:54.115 --> 00:04:55.915
other assets, those
sorts of things, and it

00:04:55.915 --> 00:04:58.998
determines an Estimated
Family Contribution.

00:04:58.998 --> 00:05:02.165
What a college does with
that varies greatly.

00:05:02.165 --> 00:05:04.165
And I think that's something that families

00:05:04.165 --> 00:05:05.248
need to understand.

00:05:05.248 --> 00:05:07.681
So all it is going to do
is determine a number.

00:05:07.681 --> 00:05:10.620
That number might mean
your family should be able

00:05:10.620 --> 00:05:13.156
to contribute 1000 dollars towards your

00:05:13.156 --> 00:05:15.415
son or daughter's education, 5000 dollars,

00:05:15.415 --> 00:05:17.582
or zero dollars for that matter, but

00:05:17.582 --> 00:05:19.000
that's all it does.

00:05:19.000 --> 00:05:21.514
Sal: I see, and the reason
why this exists is so

00:05:21.514 --> 00:05:24.625
that colleges would
have a somewhat uniform

00:05:24.625 --> 00:05:27.193
way of looking at this,
and they can feel good

00:05:27.193 --> 00:05:30.599
that the data is real?

00:05:30.599 --> 00:05:32.182
Jim: That's right, and so
one of the things in this

00:05:32.182 --> 00:05:35.299
Estimated Family Contribution
is sort of this sense

00:05:35.299 --> 00:05:38.614
that the family is
self-reporting where they stand

00:05:38.614 --> 00:05:41.018
in their family circumstances financially.

00:05:41.018 --> 00:05:42.715
And one of the things
we want to caution about

00:05:42.715 --> 00:05:44.564
is that when you receive a sort of

00:05:44.564 --> 00:05:46.697
Estimated Family Contribution,
or what they call

00:05:46.697 --> 00:05:50.614
the EFC, you want to make
sure that's in line with

00:05:50.614 --> 00:05:52.748
your expectations, because
sometimes that number

00:05:52.748 --> 00:05:57.249
can seem odd to the family
when they understand

00:05:57.249 --> 00:05:58.798
where it came from, and they may have made

00:05:58.798 --> 00:06:00.832
a mistake in some way on the form.

00:06:00.832 --> 00:06:02.499
Sal: I see, but then that Estimated Family

00:06:02.499 --> 00:06:05.166
Contribution, then that gets
sent to the universities

00:06:05.166 --> 00:06:08.115
that you have, usually at
this point, have gotten

00:06:08.115 --> 00:06:10.293
admission to.

00:06:10.293 --> 00:06:13.395
Sean: It'll go to every
university ahead of time,

00:06:13.395 --> 00:06:15.594
so if the student
applies to seven schools,

00:06:15.594 --> 00:06:18.029
you're gonna want it there
well before decisions

00:06:18.029 --> 00:06:20.244
come out, there are usually
very strict deadlines

00:06:20.244 --> 00:06:23.278
to get things there in time, but then,

00:06:23.278 --> 00:06:24.895
usually what happens in
the admissions office

00:06:24.895 --> 00:06:28.359
is if Sal, if I'm reading your application

00:06:28.359 --> 00:06:29.645
in the admissions office, and I say great,

00:06:29.645 --> 00:06:31.229
Sal Khan is gonna be
admitted, and I call down

00:06:31.229 --> 00:06:33.530
to our financial aid
office, this is all done

00:06:33.530 --> 00:06:35.715
electronically now, but
the financial officer,

00:06:35.715 --> 00:06:37.765
what he will do is open
up your electronic file,

00:06:37.765 --> 00:06:40.015
and he will see the
FAFSA sitting in there,

00:06:40.015 --> 00:06:42.016
and then he will see
that, and it will say, ok,

00:06:42.016 --> 00:06:45.549
the Khan family should be able to pay

00:06:45.549 --> 00:06:47.897
1000 dollars towards Sal's education,

00:06:47.897 --> 00:06:50.481
we cost 50 thousand
dollars, so now we're gonna

00:06:50.481 --> 00:06:51.997
have to create a financial aid package for

00:06:51.997 --> 00:06:54.267
49 thousand dollars to make it possible

00:06:54.267 --> 00:06:55.697
for you to come here.

00:06:55.697 --> 00:06:57.431
Sal: And so that's before you've been

00:06:57.431 --> 00:06:59.480
officially admitted?

00:06:59.480 --> 00:07:02.099
Sean: It's typically
when you get admitted,

00:07:02.099 --> 00:07:03.814
then the admissions
office will say, ok now

00:07:03.814 --> 00:07:05.581
calculate a financial aid package.

00:07:05.581 --> 00:07:08.116
Sal: And are these, I can imagine a lot of

00:07:08.116 --> 00:07:09.796
students would worry about this, are these

00:07:09.796 --> 00:07:13.270
kind of separate processes,
you know, or will

00:07:13.270 --> 00:07:15.717
the admissions officers
be able to see the kind of

00:07:15.717 --> 00:07:17.570
Estimated Family Contribution,
and then does that

00:07:17.570 --> 00:07:19.984
have the potential to kind of affect your

00:07:19.984 --> 00:07:22.926
chances of admissions?

00:07:22.926 --> 00:07:25.951
Jim: Oh absolutely, I
think colleges are mindful

00:07:25.951 --> 00:07:29.200
of the programs that they
have in place to support

00:07:29.200 --> 00:07:32.999
these students, and so the
college's overall mission

00:07:32.999 --> 00:07:36.551
plays a role in how they're
distributing financial aid,

00:07:36.551 --> 00:07:39.119
and the way they view
students that they are going

00:07:39.119 --> 00:07:44.368
to support, through what
they uncover in the FAFSA.

00:07:44.368 --> 00:07:46.502
Sean: And Sal, let me add to that, there's

00:07:46.502 --> 00:07:48.467
two important definitions here.

00:07:48.467 --> 00:07:51.568
One is institutions
that are need blind, and

00:07:51.568 --> 00:07:53.857
institutions that are need sensitive.

00:07:53.857 --> 00:07:56.133
And that's an important
question for students to ask

00:07:56.133 --> 00:07:58.368
as they are starting to research colleges.

00:07:58.368 --> 00:08:03.017
Typically need blind
schools mean, whether or not

00:08:03.017 --> 00:08:04.822
you are applying for aid has no impact on

00:08:04.822 --> 00:08:06.534
an admissions decision.

00:08:06.534 --> 00:08:09.917
And, we will fund you
to 100 percent of your

00:08:09.917 --> 00:08:13.318
demonstrated need, based on the FAFSA.

00:08:13.318 --> 00:08:15.050
Ok, so that's an important one.

00:08:15.050 --> 00:08:17.201
Needs sensitive, and this
is getting to your point,

00:08:17.201 --> 00:08:18.952
there are schools, and
again, the great majority of

00:08:18.952 --> 00:08:20.901
schools are need sensitive,
because they have limited

00:08:20.901 --> 00:08:23.601
budgets, so they might
take into account at

00:08:23.601 --> 00:08:26.750
some point in the process, financial aid.

00:08:26.750 --> 00:08:29.500
If we're sort of talking
to really high-achieving,

00:08:29.500 --> 00:08:31.600
low-income students right now, typically

00:08:31.600 --> 00:08:33.701
they are still gonna make out very well,

00:08:33.701 --> 00:08:35.533
because even though the school might take

00:08:35.533 --> 00:08:37.285
into account need, and
the student may need

00:08:37.285 --> 00:08:39.667
as much as a full ride,
if they were really

00:08:39.667 --> 00:08:41.617
top, top students, that's
still somebody they are

00:08:41.617 --> 00:08:43.451
going to support and probably pay for.

00:08:43.451 --> 00:08:45.701
If you are a student that's not as strong,

00:08:45.701 --> 00:08:47.338
that's when you might get
caught in exactly what

00:08:47.338 --> 00:08:49.001
you are asking about, is though you need

00:08:49.001 --> 00:08:50.969
x number of dollars,
you're sort of right in

00:08:50.969 --> 00:08:53.670
a range where we have
a lot of choices, that

00:08:53.670 --> 00:08:55.470
may make you less attractive.

00:08:55.470 --> 00:08:58.754
So, again the story I
always tell students is,

00:08:58.754 --> 00:09:00.685
the better student you
are, not only the more

00:09:00.685 --> 00:09:02.003
opportunities you'll have for admission,

00:09:02.003 --> 00:09:03.353
but the better financial aid packages

00:09:03.353 --> 00:09:04.370
you'll be eligible for.

00:09:04.370 --> 00:09:05.135
Jim: That's right.

00:09:05.135 --> 00:09:06.586
Sal: And is it possible it might even go

00:09:06.586 --> 00:09:09.135
the other way, where when
they look at your need,

00:09:09.135 --> 00:09:12.144
and they say wow, he or
she comes from a family

00:09:12.144 --> 00:09:14.470
that didn't have all the resources of many

00:09:14.470 --> 00:09:15.836
of our other applicants,
that that actually

00:09:15.836 --> 00:09:20.026
could be a positive, when you apply?

00:09:20.026 --> 00:09:22.969
Jim: Oh absolutely, I
think the, for example

00:09:22.969 --> 00:09:25.336
Andover operates on a need blind admission

00:09:25.336 --> 00:09:28.370
initiative, and we're
looking for students who meet

00:09:28.370 --> 00:09:32.170
the demands of our
program, and when you see

00:09:32.170 --> 00:09:34.503
students who may not have
had all the preparation

00:09:34.503 --> 00:09:36.812
in terms of leading up
to coming to Andover,

00:09:36.812 --> 00:09:39.303
it's very compelling to
find students who could

00:09:39.303 --> 00:09:43.132
manage the rigor of our
program, and not have

00:09:43.132 --> 00:09:44.828
had that preparation, and so we would

00:09:44.828 --> 00:09:47.736
be very attracted to
them, and want to provide

00:09:47.736 --> 00:09:49.602
a financial aid package that supports them

00:09:49.602 --> 00:09:50.852
at our school.

00:09:50.852 --> 00:09:52.536
Sal: Right.
Jim: Colleges operate the same way.

00:09:52.536 --> 00:09:53.421
Sal: And Phillips Andover, obviously

00:09:53.421 --> 00:09:57.503
a well-recognized high
school, and colleges,

00:09:57.503 --> 00:09:58.363
you're saying, are operating

00:09:58.363 --> 00:10:00.004
in a very similar way, or many do.

00:10:00.004 --> 00:10:01.136
Jim: Exactly, yes they do.

00:10:01.136 --> 00:10:04.903
Sal: Very cool.


Introduction to complex numbers

WEBVTT
Kind: captions
Language: en

00:00:00.743 --> 00:00:02.728
Voiceover:Most of your mathematical lives

00:00:02.728 --> 00:00:04.583
you've been studying real numbers.

00:00:04.583 --> 00:00:08.136
Real numbers include
things like zero, and one,

00:00:08.136 --> 00:00:15.308
and zero point three
repeating, and pi, and e,

00:00:15.308 --> 00:00:17.535
and I could keep listing real numbers.

00:00:17.535 --> 00:00:18.641
These are the numbers that you're

00:00:18.641 --> 00:00:20.555
kind of familiar with.

00:00:20.555 --> 00:00:22.910
Then we explored something interesting.

00:00:22.910 --> 00:00:24.468
We explored the notion of what if

00:00:24.468 --> 00:00:26.309
there was a number that if I squared it

00:00:26.309 --> 00:00:28.535
I would get negative one.

00:00:28.535 --> 00:00:31.103
We defined that thing
that if we squared it

00:00:31.103 --> 00:00:34.560
we got negative one, we
defined that thing as i.

00:00:35.160 --> 00:00:37.749
So we defined a whole new class of numbers

00:00:37.749 --> 00:00:40.222
which you could really view as multiples

00:00:40.222 --> 00:00:42.259
of the imaginary unit.

00:00:42.259 --> 00:00:47.357
So imaginary numbers
would be i and negative i,

00:00:47.357 --> 00:00:52.839
and pi times i, and e times i.

00:00:52.839 --> 00:00:55.754
This might raise another
interesting question.

00:00:55.754 --> 00:00:58.783
What if I combined
imaginary and real numbers?

00:00:58.783 --> 00:01:00.422
What if I had numbers
that were essentially

00:01:00.422 --> 00:01:02.848
sums or differences of
real or imaginary numbers?

00:01:02.848 --> 00:01:06.504
For example, let's say
that I had the number.

00:01:06.504 --> 00:01:08.612
Let's say I call it z,

00:01:08.612 --> 00:01:11.793
and z tends to be the most used variable

00:01:11.793 --> 00:01:12.942
when we're talking about

00:01:12.942 --> 00:01:15.921
what I'm about to talk
about, complex numbers.

00:01:15.921 --> 00:01:19.253
Let's say that z is equal to,

00:01:19.253 --> 00:01:23.258
is equal to the real number five plus

00:01:23.258 --> 00:01:28.130
the imaginary number three times i.

00:01:28.130 --> 00:01:29.354
So this thing right over here

00:01:29.354 --> 00:01:31.667
we have a real number
plus an imaginary number.

00:01:31.667 --> 00:01:33.319
You might be tempted to
add these two things,

00:01:33.319 --> 00:01:34.512
but you can't.

00:01:34.512 --> 00:01:35.890
They won't make any sense.

00:01:35.890 --> 00:01:37.417
These are kind of going in different,

00:01:37.417 --> 00:01:39.704
we'll think about it visually in a second,

00:01:39.704 --> 00:01:41.505
but you can't simplify this anymore.

00:01:41.505 --> 00:01:43.174
You can't add this real number

00:01:43.174 --> 00:01:44.951
to this imaginary number.

00:01:44.951 --> 00:01:46.457
A number like this, let me make it clear,

00:01:46.457 --> 00:01:51.669
that's real and this is
imaginary, imaginary.

00:01:51.669 --> 00:01:56.541
A number like this we
call a complex number,

00:01:56.541 --> 00:02:00.107
a complex number.

00:02:00.107 --> 00:02:02.524
It has a real part and an imaginary part.

00:02:02.524 --> 00:02:04.664
Sometimes you'll see notation like this,

00:02:04.664 --> 00:02:07.201
or someone will say what's the real part?

00:02:07.201 --> 00:02:10.115
What's the real part of
our complex number, z?

00:02:10.115 --> 00:02:13.879
Well, that would be the
five right over there.

00:02:13.879 --> 00:02:14.876
Then they might say,

00:02:14.876 --> 00:02:16.840
"Well, what's the imaginary part?

00:02:16.840 --> 00:02:20.751
"What's the imaginary part
of our complex number, z?

00:02:20.751 --> 00:02:23.883
And then typically the
way that this function

00:02:23.883 --> 00:02:25.296
is defined they really want to know

00:02:25.296 --> 00:02:28.799
what multiple of i is this imaginary part

00:02:28.799 --> 00:02:29.818
right over here.

00:02:29.818 --> 00:02:34.414
In this case it is going to
be, it is going to be three.

00:02:34.414 --> 00:02:35.953
We can visualize this.

00:02:35.953 --> 00:02:38.368
We can visualize this in two dimensions.

00:02:38.368 --> 00:02:39.690
Instead of having the traditional

00:02:39.690 --> 00:02:42.496
two-dimensional Cartesian plane

00:02:42.496 --> 00:02:44.590
with real numbers on the horizontal

00:02:44.590 --> 00:02:46.155
and the vertical axis,

00:02:46.155 --> 00:02:48.960
what we do to plot complex numbers

00:02:48.960 --> 00:02:52.798
is we on the vertical axis we plot

00:02:52.798 --> 00:02:56.290
the imaginary part, so
that's the imaginary part.

00:02:56.290 --> 00:02:59.625
On the horizontal axis
we plot the real part.

00:02:59.625 --> 00:03:04.352
We plot the real part just like that.

00:03:04.352 --> 00:03:06.051
We plot the real part.

00:03:06.051 --> 00:03:08.230
For example, z right over here

00:03:08.230 --> 00:03:09.706
which is five plus three i,

00:03:09.706 --> 00:03:12.767
the real part is five so we would go

00:03:12.767 --> 00:03:17.238
one, two, three, four, five.

00:03:17.238 --> 00:03:18.321
That's five right over there.

00:03:18.321 --> 00:03:20.107
The imaginary part is three.

00:03:20.107 --> 00:03:25.540
One, two, three, and so
on the complex plane,

00:03:25.540 --> 00:03:28.491
on the complex plane we would visualize

00:03:28.491 --> 00:03:31.744
that number right over here.

00:03:31.744 --> 00:03:33.377
This right over here is how we

00:03:33.377 --> 00:03:36.239
would visualize z on the complex plane.

00:03:36.239 --> 00:03:38.793
It's five, positive five
in the real direction,

00:03:38.793 --> 00:03:41.173
positive three in the imaginary direction.

00:03:41.173 --> 00:03:43.284
We could plot other complex numbers.

00:03:43.284 --> 00:03:46.106
Let's say we have the complex number a

00:03:46.106 --> 00:03:50.145
which is equal to let's
say it's negative two

00:03:50.145 --> 00:03:51.388
plus i.

00:03:51.388 --> 00:03:53.254
Where would I plot that?

00:03:53.254 --> 00:03:55.613
Well, the real part is negative two,

00:03:55.613 --> 00:03:56.932
negative two,

00:03:56.932 --> 00:03:59.163
and the imaginary part is going to be

00:03:59.163 --> 00:04:01.424
you could imagine this as plus one i

00:04:01.424 --> 00:04:02.563
so we go one up.

00:04:02.563 --> 00:04:04.196
It's going to be right over there.

00:04:04.196 --> 00:04:07.220
That right over there
is our complex number.

00:04:07.220 --> 00:04:10.130
Our complex number a
would be at that point

00:04:10.130 --> 00:04:12.149
of the complex,

00:04:12.149 --> 00:04:14.188
complex, let me write that,

00:04:14.188 --> 00:04:17.909
that point of the complex plane.

00:04:17.909 --> 00:04:19.426
Let me just do one more.

00:04:19.426 --> 00:04:22.148
Let's say you had a complex number b

00:04:22.148 --> 00:04:24.645
which is going to be,

00:04:24.645 --> 00:04:29.494
let's say it is, let's say
it's four minus three i.

00:04:29.494 --> 00:04:30.809
Where would we plot that?

00:04:30.809 --> 00:04:32.886
Well, one, two, three, four,

00:04:32.886 --> 00:04:36.740
and then let's see minus one, two, three.

00:04:36.740 --> 00:04:39.358
Our negative three gets
us right over there.

00:04:39.358 --> 00:04:40.499
That right over there would be

00:04:40.499 --> 00:04:43.045
the complex number b.


Saltatory conduction in neurons

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:04.720
Now that we know how a signal
can spread through a neuron,

00:00:04.720 --> 00:00:07.310
through an electrotonic
potential and action

00:00:07.310 --> 00:00:09.950
potential and combinations
of the two, let's

00:00:09.950 --> 00:00:11.520
put it all together
by looking again

00:00:11.520 --> 00:00:14.270
at the structure of a neuron,
the anatomy of a neuron,

00:00:14.270 --> 00:00:16.020
and thinking about why
it has that anatomy

00:00:16.020 --> 00:00:18.600
and how it all can work.

00:00:18.600 --> 00:00:20.460
So we've already talked
about the dendrites

00:00:20.460 --> 00:00:23.570
as being where the
neuron can be stimulated

00:00:23.570 --> 00:00:24.860
from multiple inputs.

00:00:24.860 --> 00:00:27.700
If we're in the
brain, these dendrites

00:00:27.700 --> 00:00:32.409
might be near the terminal
ends of axons of other neurons.

00:00:32.409 --> 00:00:34.240
If we're some type
of sensory cell,

00:00:34.240 --> 00:00:36.130
these dendrites
could be stimulated

00:00:36.130 --> 00:00:37.985
by some type of sensory input.

00:00:37.985 --> 00:00:39.860
But let's just say, for
the sake of argument,

00:00:39.860 --> 00:00:41.459
they are stimulated in some way.

00:00:41.459 --> 00:00:43.250
And because they're
stimulated in some way,

00:00:43.250 --> 00:00:53.510
it allows positive ions
to flood into the neuron

00:00:53.510 --> 00:00:54.329
from the outside.

00:00:54.329 --> 00:00:56.120
As we know, there's a
potential difference.

00:00:56.120 --> 00:00:59.470
It's more negative
inside of the neuron

00:00:59.470 --> 00:01:00.960
than outside of the neuron.

00:01:00.960 --> 00:01:04.140
And so if a channel
gets opened up

00:01:04.140 --> 00:01:06.090
because of some
stimulus, that would

00:01:06.090 --> 00:01:10.690
allow positive ions to flow in.

00:01:10.690 --> 00:01:13.370
And the primary positive
ions we've been talking about

00:01:13.370 --> 00:01:14.400
are the sodium ions.

00:01:14.400 --> 00:01:17.910
Maybe this is some
type of sodium gate

00:01:17.910 --> 00:01:22.310
that gets opened up
because of this stimulus.

00:01:22.310 --> 00:01:26.120
So when that happens, you
will have electrotonic spread.

00:01:26.120 --> 00:01:30.440
You will have an electrotonic
potential being spread.

00:01:30.440 --> 00:01:32.570
So let's say that
we had a voltmeter

00:01:32.570 --> 00:01:35.320
right here on the axon hillock.

00:01:35.320 --> 00:01:39.490
It's kind of the hill that leads
to the axon right over here.

00:01:39.490 --> 00:01:42.450
So what you might see happening
after some amount of time--

00:01:42.450 --> 00:01:43.950
so let me draw.

00:01:43.950 --> 00:01:46.800
So let's say this is our
voltage in millivolts

00:01:46.800 --> 00:01:49.179
across the membrane--
our voltage difference,

00:01:49.179 --> 00:01:49.720
I should say.

00:01:49.720 --> 00:01:51.340
This is the passage of time.

00:01:51.340 --> 00:01:53.910
Let's say the stimulus
happens at time 0.

00:01:53.910 --> 00:01:55.940
But right at time
0, we haven't really

00:01:55.940 --> 00:01:57.750
noticed it with our voltmeter.

00:01:57.750 --> 00:02:00.290
Our voltage right across the
membrane right over there

00:02:00.290 --> 00:02:04.370
is at that equilibrium,
negative 70 millivolts.

00:02:04.370 --> 00:02:07.350
But after some small
amount of time,

00:02:07.350 --> 00:02:10.919
this electrotonic potential
has gotten to this point,

00:02:10.919 --> 00:02:12.640
because all of these
positive charges

00:02:12.640 --> 00:02:14.840
are trying to get
away from each other.

00:02:14.840 --> 00:02:16.000
It's gotten to that point.

00:02:16.000 --> 00:02:21.975
And you might see a
bump in the voltage--

00:02:21.975 --> 00:02:24.960
in the voltage difference,
I guess I should say.

00:02:24.960 --> 00:02:26.480
This thing might go up.

00:02:26.480 --> 00:02:28.620
So it might look
something like that.

00:02:28.620 --> 00:02:30.400
Now, that by itself
might not be--

00:02:30.400 --> 00:02:34.180
we might have gotten the
voltage difference low enough,

00:02:34.180 --> 00:02:35.260
I guess we could say.

00:02:35.260 --> 00:02:41.600
Or we might not have gotten
the voltage inside of the cell

00:02:41.600 --> 00:02:45.730
positive enough in order to
trigger the voltage-gated ion

00:02:45.730 --> 00:02:46.610
channels.

00:02:46.610 --> 00:02:48.260
And so maybe nothing happens.

00:02:48.260 --> 00:02:54.390
Maybe this right over here,
this is negative 55 millivolts.

00:02:54.390 --> 00:02:57.530
And so that's what you have
to get the voltage up to,

00:02:57.530 --> 00:02:59.400
the voltage difference
up to, in order

00:02:59.400 --> 00:03:06.620
to trigger the ion
channels right over there.

00:03:06.620 --> 00:03:10.590
So those are the sodium channels
to get positive charge in.

00:03:10.590 --> 00:03:14.040
Here's the potassium channels
to get the positive charge out.

00:03:14.040 --> 00:03:16.710
The axon hillock
has a ton of these,

00:03:16.710 --> 00:03:18.010
because these are really there.

00:03:18.010 --> 00:03:20.690
Once they get triggered,
they can trigger an impulse

00:03:20.690 --> 00:03:22.870
that can then go
down the entire axon,

00:03:22.870 --> 00:03:26.510
and maybe stimulate other
things, maybe in the brain

00:03:26.510 --> 00:03:29.550
or whatever else this neuron
might be connected to.

00:03:29.550 --> 00:03:31.790
So maybe that stimulus by
itself didn't trigger it.

00:03:31.790 --> 00:03:33.789
But let's say that there's
another stimulus that

00:03:33.789 --> 00:03:37.260
happens right at the same
time, or around the same time.

00:03:37.260 --> 00:03:38.060
And that happens.

00:03:38.060 --> 00:03:40.000
And on its own, that
might have caused

00:03:40.000 --> 00:03:44.356
a similar type of
bump right over here.

00:03:44.356 --> 00:03:45.730
But when you add
the two together

00:03:45.730 --> 00:03:47.450
and they're happening
at the same time,

00:03:47.450 --> 00:03:51.852
their combined bumps
are enough to trigger

00:03:51.852 --> 00:03:53.310
an action potential
in the hillock,

00:03:53.310 --> 00:03:55.650
or a series of action
potentials in the hillock.

00:03:55.650 --> 00:04:00.620
And so then, you really have,
essentially, fired the neuron.

00:04:00.620 --> 00:04:03.170
So now all sorts
of positive charge

00:04:03.170 --> 00:04:07.360
gets flushed into the neuron.

00:04:07.360 --> 00:04:10.210
And then purely through
electrotonic spread,

00:04:10.210 --> 00:04:15.740
you will have this electrotonic
potential spread down the axon.

00:04:15.740 --> 00:04:17.300
Now, this is the
interesting part,

00:04:17.300 --> 00:04:18.966
because we can think
a little bit about,

00:04:18.966 --> 00:04:21.420
what is the best way for
an axon to be designed?

00:04:21.420 --> 00:04:24.570
In general, if you're trying
to transfer a current,

00:04:24.570 --> 00:04:28.080
the ideal thing to do is, the
thing that you're transferring

00:04:28.080 --> 00:04:32.126
the current down should
conduct really well.

00:04:32.126 --> 00:04:33.750
Or you could say it
has low resistance.

00:04:37.430 --> 00:04:40.250
But you want it to be
surrounded by an insulator.

00:04:40.250 --> 00:04:41.490
You want it to be surrounded.

00:04:41.490 --> 00:04:43.310
So if this was a
cross section, you

00:04:43.310 --> 00:04:46.340
want it to be surrounded
by an insulator that

00:04:46.340 --> 00:04:49.050
has high resistance.

00:04:49.050 --> 00:05:00.410
And the reason is because
you don't want the potential

00:05:00.410 --> 00:05:08.100
to leak across your
membrane-- high resistance

00:05:08.100 --> 00:05:08.790
right over here.

00:05:08.790 --> 00:05:13.910
If you didn't have something
high resistance around it,

00:05:13.910 --> 00:05:15.642
your current would
actually go slower.

00:05:15.642 --> 00:05:17.850
This is true if you're just
dealing with electronics.

00:05:17.850 --> 00:05:21.420
If you just had a bunch of
copper wires on one side,

00:05:21.420 --> 00:05:22.950
and you had some
copper wires that

00:05:22.950 --> 00:05:25.690
were surrounded by a
really good insulator,

00:05:25.690 --> 00:05:29.850
a really good resistor-- for
example, plastic or rubber

00:05:29.850 --> 00:05:30.630
of some kind.

00:05:30.630 --> 00:05:35.690
The current is actually going
to have less energy loss.

00:05:35.690 --> 00:05:37.490
It's going to travel
faster when it's

00:05:37.490 --> 00:05:39.399
surrounded by an insulator.

00:05:39.399 --> 00:05:40.690
So you might say, OK, well gee.

00:05:40.690 --> 00:05:44.400
The best thing to do would be
to surround this entire axon

00:05:44.400 --> 00:05:46.030
with a good insulator.

00:05:46.030 --> 00:05:47.890
And for the most
part, that is true.

00:05:47.890 --> 00:05:50.520
It is surrounded by
a good insulator.

00:05:50.520 --> 00:05:52.270
That is what the
myelin sheath is.

00:05:52.270 --> 00:05:55.560
So let's say we want to surround
this whole thing with just one

00:05:55.560 --> 00:06:00.392
big grouping of Schwann's
cells, so one big myelin

00:06:00.392 --> 00:06:01.850
sheath-- which is
a good insulator.

00:06:01.850 --> 00:06:04.430
It does not conduct
current well.

00:06:04.430 --> 00:06:13.640
So this right over here is just
one big myelin sheath right

00:06:13.640 --> 00:06:14.650
over here.

00:06:14.650 --> 00:06:16.790
Now, what's the
problem with this?

00:06:16.790 --> 00:06:19.270
Well, if this axon is
really long-- and let's say,

00:06:19.270 --> 00:06:21.280
you know, you're a
dinosaur or something.

00:06:21.280 --> 00:06:23.390
And you're trying
to go up your neck,

00:06:23.390 --> 00:06:24.980
and your neck is 20 feet long.

00:06:24.980 --> 00:06:27.750
Or even a human being,
we're a reasonable size.

00:06:27.750 --> 00:06:30.640
And you're going several
feet, or even whatever,

00:06:30.640 --> 00:06:32.980
you want to go a
reasonable distance

00:06:32.980 --> 00:06:38.220
purely with electrotonic
spread, your signal, remember,

00:06:38.220 --> 00:06:38.990
it dissipates.

00:06:38.990 --> 00:06:42.390
Your signal is going to be
really weak right over here.

00:06:42.390 --> 00:06:47.104
You're going to have a weak
signal on the other end.

00:06:47.104 --> 00:06:48.520
It might not be
even strong enough

00:06:48.520 --> 00:06:50.130
to make anything
interesting happen

00:06:50.130 --> 00:06:52.700
at these terminals, which
wouldn't be strong enough

00:06:52.700 --> 00:06:54.790
to trigger, maybe,
other neurons,

00:06:54.790 --> 00:06:57.440
or whatever else might need
to happen at this other end.

00:06:57.440 --> 00:07:00.520
So then you say,
OK, well then why

00:07:00.520 --> 00:07:02.749
don't we try to
boost the signal?

00:07:02.749 --> 00:07:04.290
Well, how would you
boost the signal?

00:07:04.290 --> 00:07:04.790
You say, OK.

00:07:04.790 --> 00:07:06.510
I like having this
myelin sheath.

00:07:06.510 --> 00:07:10.480
But why don't we put gaps in the
myelin sheath every so often?

00:07:10.480 --> 00:07:13.110
And then those gaps
would allow the membrane

00:07:13.110 --> 00:07:15.280
to interface with the outside.

00:07:15.280 --> 00:07:18.950
And in those areas, we could
put some voltage-gated channels

00:07:18.950 --> 00:07:22.300
that can release
action potentials,

00:07:22.300 --> 00:07:24.590
in order to essentially
boost the signal.

00:07:24.590 --> 00:07:27.940
And that's is exactly what the
anatomy of a typical neuron

00:07:27.940 --> 00:07:28.970
is like.

00:07:28.970 --> 00:07:33.310
So instead of just one big
insulating sheath like this,

00:07:33.310 --> 00:07:36.280
it would-- let me
make some gaps here.

00:07:36.280 --> 00:07:38.630
Whoops, I'm going
to do that in black.

00:07:38.630 --> 00:07:40.570
So actually, let me
just draw it like this.

00:07:40.570 --> 00:07:43.495
Let me just erase this.

00:07:43.495 --> 00:07:47.310
So clear, and let me clear this.

00:07:47.310 --> 00:07:49.920
That's good enough.

00:07:49.920 --> 00:07:55.100
And so what we could do
is we could put gaps in it

00:07:55.100 --> 00:07:59.120
right over here where
the axon, the axonal

00:07:59.120 --> 00:08:04.100
membrane itself can interface
with its surroundings.

00:08:04.100 --> 00:08:05.840
And of course, we
know we call those

00:08:05.840 --> 00:08:09.760
gaps the nodes of
Ranvier, or Ran-Veer.

00:08:09.760 --> 00:08:12.130
I'm not really sure
how to pronounce it.

00:08:12.130 --> 00:08:15.200
So let me put
those gaps in here.

00:08:15.200 --> 00:08:18.210
So you put those gaps in here,
so these are the myelin sheath.

00:08:18.210 --> 00:08:20.960
And this right over here
is a node of Ranvier.

00:08:23.890 --> 00:08:29.960
These are nodes of
Ran-Veer, or Ranvier.

00:08:29.960 --> 00:08:34.590
And right in those little nodes,
right in those nodes, right

00:08:34.590 --> 00:08:36.890
where the myelin
sheath isn't, we

00:08:36.890 --> 00:08:43.789
can put these voltage-gated
channels to essentially boost

00:08:43.789 --> 00:08:44.350
the signal.

00:08:44.350 --> 00:08:46.500
If the signal had to
go electrotonically

00:08:46.500 --> 00:08:48.270
all the way over here,
it'd be very weak.

00:08:48.270 --> 00:08:49.960
It's going to dissipate
as it goes down,

00:08:49.960 --> 00:08:52.210
but it could be just strong
enough right at this point

00:08:52.210 --> 00:08:54.840
in order to trigger these
voltage-gated channels,

00:08:54.840 --> 00:08:57.280
in order to essentially boost
the signal again, in order

00:08:57.280 --> 00:09:00.271
to trigger an action
potential, boost the signal.

00:09:00.271 --> 00:09:01.770
And now the signal
is boosted, it'll

00:09:01.770 --> 00:09:03.960
dissipate, dissipate,
dissipate, boost.

00:09:03.960 --> 00:09:07.800
And it'll boost right
over here again.

00:09:07.800 --> 00:09:10.350
And then it'll dissipate,
dissipate, dissipate,

00:09:10.350 --> 00:09:12.340
and boost.

00:09:12.340 --> 00:09:14.350
Dissipate, dissipate, boost.

00:09:14.350 --> 00:09:16.540
And so by having
this combination,

00:09:16.540 --> 00:09:17.980
you want the myelin sheath.

00:09:17.980 --> 00:09:22.970
You want the insulator in
order to keep the transmission

00:09:22.970 --> 00:09:27.940
of the current to fast, in order
to have minimal energy loss.

00:09:27.940 --> 00:09:33.120
But you do need these areas
where the myelin sheath isn't

00:09:33.120 --> 00:09:37.780
in order to boost the signal, in
order for the action potentials

00:09:37.780 --> 00:09:41.640
to get triggered, and so your
signal can keep being-- well,

00:09:41.640 --> 00:09:43.550
I guess keep being
amplified, if we

00:09:43.550 --> 00:09:47.017
wanted to talk in kind of
electrical engineering speak.

00:09:47.017 --> 00:09:49.100
And this type of conduction,
where the signal just

00:09:49.100 --> 00:09:52.940
keeps boosting, and if you were
just to superficially observe

00:09:52.940 --> 00:09:55.350
it, it looks like the
signal is almost jumping.

00:09:55.350 --> 00:09:57.660
It gets triggered here,
then it gets triggered,

00:09:57.660 --> 00:09:58.680
here then it gets
triggered here,

00:09:58.680 --> 00:10:00.920
then it gets triggered here,
then it gets triggered here.

00:10:00.920 --> 00:10:02.470
This is called
saltatory conduction.

00:10:13.470 --> 00:10:17.570
It comes from the Latin
word saltare-- once again,

00:10:17.570 --> 00:10:18.820
I don't know how to pronounce.

00:10:18.820 --> 00:10:20.460
My Latin isn't too good.

00:10:20.460 --> 00:10:22.570
But it comes from the
Latin word saltare,

00:10:22.570 --> 00:10:25.730
which means to jump
around or to hop around.

00:10:25.730 --> 00:10:28.790
And that's because it looks like
the signal is hopping around.

00:10:28.790 --> 00:10:30.590
But that's not exactly
what's happening.

00:10:30.590 --> 00:10:34.080
The signal is traveling
passively through.

00:10:34.080 --> 00:10:36.380
It gets triggered here
in the axon hillock.

00:10:36.380 --> 00:10:41.099
Then it travels passively
through electrotonic spread.

00:10:41.099 --> 00:10:42.140
And then it gets boosted.

00:10:42.140 --> 00:10:43.890
And you have the
myelin sheath around it

00:10:43.890 --> 00:10:45.600
to make sure it goes
as fast as possible,

00:10:45.600 --> 00:10:47.790
and you get very
little loss of signal.

00:10:47.790 --> 00:10:50.140
And then it gets boosted
at the nodes of Ranvier,

00:10:50.140 --> 00:10:54.570
because it triggers these
voltage-gated channels again.

00:10:54.570 --> 00:10:56.774
That triggers an
action potential.

00:10:56.774 --> 00:10:58.190
And then your
signal gets boosted,

00:10:58.190 --> 00:11:00.439
and then it dissipates--
boosted, dissipates, boosted,

00:11:00.439 --> 00:11:02.290
dissipates, boosted, dissipates.

00:11:02.290 --> 00:11:04.010
Maybe it could even
get boosted again.

00:11:04.010 --> 00:11:07.440
And then it can trigger
whatever else it has to trigger.


Electrotonic and action potentials

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.340
We've already seen that when a
neuron is in its resting state

00:00:03.340 --> 00:00:05.900
there's a voltage difference
across the membrane.

00:00:05.900 --> 00:00:08.529
And so in these diagrams right
over here, this right over here

00:00:08.529 --> 00:00:09.195
is the membrane.

00:00:13.450 --> 00:00:16.840
This right over here is
the inside of the neuron,

00:00:16.840 --> 00:00:19.490
and this right over
here is the outside.

00:00:19.490 --> 00:00:22.500
That's the outside and of
course this is the outside.

00:00:22.500 --> 00:00:24.230
This is the outside as well.

00:00:24.230 --> 00:00:26.190
So if you had a
voltmeter measuring

00:00:26.190 --> 00:00:28.070
the potential difference
across the membrane,

00:00:28.070 --> 00:00:32.659
so if you took this voltage
minus this voltage right

00:00:32.659 --> 00:00:35.450
over here, the voltage
between this and that,

00:00:35.450 --> 00:00:38.435
you would get negative-- let's
say for the sake of argument,

00:00:38.435 --> 00:00:40.140
let's say it would
measure, it would

00:00:40.140 --> 00:00:42.450
average about negative
70 millivolts.

00:00:42.450 --> 00:00:44.150
So this is in
millivolts, negative 70.

00:00:48.359 --> 00:00:50.400
And I'll do it actually
for both of these graphs.

00:00:50.400 --> 00:00:52.620
We're going to use both
of these to describe

00:00:52.620 --> 00:00:55.817
slightly different, or actually
quite different, scenarios.

00:00:55.817 --> 00:00:58.150
And you could have another
voltmeter out here in yellow,

00:00:58.150 --> 00:01:02.190
and that's a little further
out, but that's also

00:01:02.190 --> 00:01:05.610
going to register
negative 70 millivolts.

00:01:05.610 --> 00:01:08.620
Now let's make something
interesting happen.

00:01:08.620 --> 00:01:11.950
Let's say that, for
some reason, let's

00:01:11.950 --> 00:01:17.230
say that the membrane
becomes permeable to sodium.

00:01:17.230 --> 00:01:19.710
So sodium just starts
flooding through.

00:01:19.710 --> 00:01:22.220
It's going to flood
through for two reasons.

00:01:22.220 --> 00:01:23.820
One, it is a positive ion.

00:01:23.820 --> 00:01:25.270
It's more positive
on the outside

00:01:25.270 --> 00:01:29.212
than the inside, so positive
charge will want to flood in.

00:01:29.212 --> 00:01:31.170
And the other reason why
it'll want to flood in

00:01:31.170 --> 00:01:33.500
is because there's a higher
concentration of sodium

00:01:33.500 --> 00:01:34.960
on the outside
than on the inside.

00:01:34.960 --> 00:01:38.274
So it'll just go down its
concentration gradient.

00:01:38.274 --> 00:01:40.690
And the reason why we have a
higher concentration gradient

00:01:40.690 --> 00:01:42.680
on the sodium on the
outside than the inside,

00:01:42.680 --> 00:01:46.280
we've already seen, is because
of the sodium potassium pump.

00:01:46.280 --> 00:01:48.906
But anyway, so you're going
to have this increase.

00:01:48.906 --> 00:01:51.280
You're going to really have
this spike in positive charge

00:01:51.280 --> 00:01:51.780
flowing.

00:01:51.780 --> 00:01:55.470
And then what's going to be the
dynamic then inside the neuron?

00:01:55.470 --> 00:01:58.146
Well, if you have all this
positive charge right over here

00:01:58.146 --> 00:01:59.770
the other positive
charge in the neuron

00:01:59.770 --> 00:02:01.311
is going to want to
get away from it.

00:02:08.461 --> 00:02:10.460
And this is not just in
the rightward direction.

00:02:10.460 --> 00:02:12.210
It's really going to
be in all directions.

00:02:12.210 --> 00:02:13.797
In all directions
the positive charge,

00:02:13.797 --> 00:02:15.880
they're going to want to
get away from each other.

00:02:15.880 --> 00:02:17.430
So this one's going
to move that way,

00:02:17.430 --> 00:02:18.390
and then that's going
to make that one

00:02:18.390 --> 00:02:19.931
want to move that
way, which is going

00:02:19.931 --> 00:02:21.680
to make that one want
to move that way.

00:02:21.680 --> 00:02:23.870
So if we let some
time pass, what's

00:02:23.870 --> 00:02:27.590
the voltage going to look
like on this blue voltmeter?

00:02:27.590 --> 00:02:30.574
Well after some time, because
more and more positive

00:02:30.574 --> 00:02:32.740
charges are trying to get
away from these other ones

00:02:32.740 --> 00:02:34.450
right over here as
the concentration

00:02:34.450 --> 00:02:36.482
of these positive
charges spread out,

00:02:36.482 --> 00:02:38.565
you're going to see the
voltage start to increase.

00:02:41.851 --> 00:02:43.600
And then as they fully
get spread out then

00:02:43.600 --> 00:02:48.100
it might return to
something of an equilibrium.

00:02:48.100 --> 00:02:52.946
And then if we go a little
bit further down the neuron

00:02:52.946 --> 00:02:54.570
a little more time
will pass before you

00:02:54.570 --> 00:02:57.450
see a voltage increase, but
because this thing is just

00:02:57.450 --> 00:03:00.670
getting spread out across
more and more distance,

00:03:00.670 --> 00:03:03.740
the effect is going
to be more limited.

00:03:03.740 --> 00:03:06.430
You're not going to
see as much of a bump

00:03:06.430 --> 00:03:10.320
in the voltage over here
than you saw over here.

00:03:10.320 --> 00:03:14.880
And this type of spread of, I
guess you could say a signal,

00:03:14.880 --> 00:03:17.790
is called electrotonic spread.

00:03:17.790 --> 00:03:19.275
Let me write that down.

00:03:26.060 --> 00:03:28.870
Or this is the spread of
an electrotonic potential.

00:03:32.067 --> 00:03:33.900
So there's a couple of
characteristics here.

00:03:33.900 --> 00:03:35.144
One, it's passive.

00:03:35.144 --> 00:03:36.560
This part that we
drew right here,

00:03:36.560 --> 00:03:39.040
this isn't the
electrotonic spread.

00:03:39.040 --> 00:03:41.190
The electrotonic spread is
what happens after that.

00:03:41.190 --> 00:03:43.290
Once you have this high
concentration here,

00:03:43.290 --> 00:03:45.480
the fact that a few
moments later you're

00:03:45.480 --> 00:03:46.980
going to have a
higher concentration

00:03:46.980 --> 00:03:49.514
of positive charge here, and
a few moments later a higher

00:03:49.514 --> 00:03:50.680
positive concentration here.

00:03:50.680 --> 00:03:53.550
This is a passive phenomenon.

00:03:53.550 --> 00:03:58.200
So this thing right over
here, it is passive.

00:03:58.200 --> 00:03:59.640
And it also dissipates.

00:03:59.640 --> 00:04:02.015
The signal gets weaker and
weaker the further and further

00:04:02.015 --> 00:04:04.514
you get out because this stuff
just gets further and further

00:04:04.514 --> 00:04:05.160
spread out.

00:04:05.160 --> 00:04:07.240
So it's passive
and it dissipates.

00:04:11.440 --> 00:04:13.750
Now let's play out
this scenario again,

00:04:13.750 --> 00:04:19.380
but let's also throw in some
voltage-gated ion channels

00:04:19.380 --> 00:04:20.265
right over here.

00:04:20.265 --> 00:04:22.390
So let's say this right
over here that I'm drawing,

00:04:22.390 --> 00:04:27.455
let's say this is a
voltage-gated sodium channel.

00:04:34.510 --> 00:04:42.220
Let's say it opens at
negative 55 millivolts.

00:04:42.220 --> 00:04:45.910
So that would be
right around there.

00:04:45.910 --> 00:04:49.220
So that is when it opens
at negative 55 millivolts.

00:04:49.220 --> 00:04:51.480
Let me draw that
threshold there.

00:04:51.480 --> 00:05:02.280
And let's say it closes at
positive 40 millivolts, right

00:05:02.280 --> 00:05:02.780
over there.

00:05:02.780 --> 00:05:04.530
I'm just trying to
show the threshold.

00:05:04.530 --> 00:05:09.140
And let's say we also have
a potassium channel too,

00:05:09.140 --> 00:05:09.910
right over here.

00:05:12.820 --> 00:05:16.980
So this is a potassium channel,
the infamous leaky potassium

00:05:16.980 --> 00:05:19.190
channels, which are
the true reason why

00:05:19.190 --> 00:05:23.590
we have this voltage
difference across the membrane.

00:05:23.590 --> 00:05:25.120
But this potassium
channel, let's

00:05:25.120 --> 00:05:27.200
say it opens when
this one closes.

00:05:27.200 --> 00:05:29.154
So it opens, just for
the sake of argument,

00:05:29.154 --> 00:05:31.570
these aren't going to be the
exact numbers but to give you

00:05:31.570 --> 00:05:35.840
the idea, at positive
40 millivolts.

00:05:35.840 --> 00:05:41.710
And let's say it closes
at negative 80 millivolts.

00:05:41.710 --> 00:05:50.080
So that one opens up here,
and then it closes down here.

00:05:50.080 --> 00:05:52.500
Now what is going to happen?

00:05:52.500 --> 00:05:58.320
Well just like we saw before--
Let's let our positive charge

00:05:58.320 --> 00:06:05.040
flood in here at the
left side of this neuron,

00:06:05.040 --> 00:06:09.815
I guess we could say, and then
because of electrotonic spread,

00:06:09.815 --> 00:06:11.440
a little bit later
you're going to have

00:06:11.440 --> 00:06:15.420
the potential across the
membrane at this point

00:06:15.420 --> 00:06:17.575
is going to start to
become less negative.

00:06:17.575 --> 00:06:18.950
The potential
difference is going

00:06:18.950 --> 00:06:22.140
to become less negative, just
like we saw right over here.

00:06:22.140 --> 00:06:23.810
So it's going to
become less negative.

00:06:23.810 --> 00:06:25.810
But it's not just going
to be just a little bump

00:06:25.810 --> 00:06:27.851
and then go back down,
because what happens right

00:06:27.851 --> 00:06:31.630
when the potential hits
negative 55 millivolts?

00:06:31.630 --> 00:06:36.320
Well then it's going to trigger
the opening of this sodium

00:06:36.320 --> 00:06:37.170
channel.

00:06:37.170 --> 00:06:40.580
So the sodium channel is going
to open because the voltage got

00:06:40.580 --> 00:06:47.295
high enough, and so you're going
to have sodium flood in again.

00:06:47.295 --> 00:06:48.420
So what's that going to do?

00:06:48.420 --> 00:06:54.050
Well that's going to
spike up the voltage.

00:06:54.050 --> 00:06:56.000
So it's going to look
something like that.

00:06:56.000 --> 00:06:58.530
It's going to keep flowing
in, keep flowing in.

00:06:58.530 --> 00:07:00.040
The voltage is going to
get more and more positive.

00:07:00.040 --> 00:07:02.040
Because remember, this
is going to be flowing in

00:07:02.040 --> 00:07:03.290
for two reasons.

00:07:03.290 --> 00:07:04.945
One, there's just more charge.

00:07:04.945 --> 00:07:07.000
It's more positive
outside than the inside

00:07:07.000 --> 00:07:08.990
so it's going to go
across a voltage gradient,

00:07:08.990 --> 00:07:13.060
or go down the voltage gradient,
or the electro potential

00:07:13.060 --> 00:07:16.590
gradient, but also there's a
higher concentration of sodium

00:07:16.590 --> 00:07:19.310
out here than there is in here
because of the sodium potassium

00:07:19.310 --> 00:07:22.100
pump, and so it'll also want
to go down its concentration

00:07:22.100 --> 00:07:22.600
gradient.

00:07:22.600 --> 00:07:25.530
So it's just going to keep
flowing in even past the point

00:07:25.530 --> 00:07:29.752
at which you have
no voltage gradient,

00:07:29.752 --> 00:07:31.460
but because of the
concentration gradient

00:07:31.460 --> 00:07:32.610
it's going to keep going.

00:07:32.610 --> 00:07:35.430
But then, as you get to
positive 40 millivolts,

00:07:35.430 --> 00:07:37.300
this channel is going to close.

00:07:37.300 --> 00:07:39.260
So that's going to
stop flooding in.

00:07:39.260 --> 00:07:42.289
And you also have the
potassium channel opening.

00:07:42.289 --> 00:07:43.830
And the potassium
channel, now you're

00:07:43.830 --> 00:07:46.690
more positive on the inside than
the outside, at least locally

00:07:46.690 --> 00:07:47.750
right over here.

00:07:47.750 --> 00:07:51.770
And so now you're going to
have this positively-charged

00:07:51.770 --> 00:07:53.340
potassium ions want
to get out, want

00:07:53.340 --> 00:07:56.620
to get out from this
positive environment.

00:07:56.620 --> 00:07:59.730
And so the voltage is going
to get more and more negative,

00:07:59.730 --> 00:08:02.340
and it's going to go beyond
neutral because potassium

00:08:02.340 --> 00:08:05.829
is going to want to go down,
not just its voltage gradient,

00:08:05.829 --> 00:08:08.120
it's going to do that while
it's positive on the inside

00:08:08.120 --> 00:08:11.410
and negative on the outside,
or more positive on the inside

00:08:11.410 --> 00:08:14.030
than it is on the
outside, but it'll also

00:08:14.030 --> 00:08:15.970
want to go down its
concentration gradient.

00:08:15.970 --> 00:08:18.359
There's a higher
concentration of potassium

00:08:18.359 --> 00:08:20.650
on the inside than on the
outside because of the sodium

00:08:20.650 --> 00:08:21.910
potassium pump.

00:08:21.910 --> 00:08:24.670
So the potassium will
just keep going out,

00:08:24.670 --> 00:08:27.410
and out, and out, and out,
and then at negative 80

00:08:27.410 --> 00:08:31.690
millivolts the potassium
channel closes, and then we

00:08:31.690 --> 00:08:37.730
can get back to our
equilibrium state.

00:08:37.730 --> 00:08:40.169
Now why is this interesting?

00:08:40.169 --> 00:08:43.245
Well we had the electrotonic
spread up to this point.

00:08:43.245 --> 00:08:45.370
But the signal would just
keep dissipating and keep

00:08:45.370 --> 00:08:47.990
dissipating, and if
you get far enough

00:08:47.990 --> 00:08:51.039
it would be very hard
to notice that signal.

00:08:51.039 --> 00:08:52.580
And so what this
essentially just did

00:08:52.580 --> 00:08:54.490
is it just boosted
the signal again.

00:08:54.490 --> 00:08:58.420
It just boosted the signal,
and now, a few moments later,

00:08:58.420 --> 00:09:03.022
if you were to measure
the potential difference--

00:09:03.022 --> 00:09:05.480
because these things are trying
to get away from each other

00:09:05.480 --> 00:09:08.540
again, once again you have
electrotonic spread-- if you

00:09:08.540 --> 00:09:12.400
were to measure the potential
difference across the membrane

00:09:12.400 --> 00:09:16.390
where this yellow voltmeter
is, then you're going to have--

00:09:16.390 --> 00:09:20.990
So where that yellow one is,
before it had just a little

00:09:20.990 --> 00:09:22.630
dissipated bump
here, but now it's

00:09:22.630 --> 00:09:27.210
going to have quite a nice bump.

00:09:27.210 --> 00:09:30.350
And if you actually had
another voltage-gated channel

00:09:30.350 --> 00:09:33.590
right over here, then
that would boost it again.

00:09:33.590 --> 00:09:41.850
And so this kind of very
active boosting of the voltage,

00:09:41.850 --> 00:09:43.705
this is called an
action potential.

00:09:47.880 --> 00:09:50.210
You could view this as the
boosting of the signal.

00:09:50.210 --> 00:09:53.700
The signal is spreading,
electrotonic spread, then you

00:09:53.700 --> 00:09:56.770
trigger a channel, a
voltage-gated channel,

00:09:56.770 --> 00:09:59.740
then that boosts
the signal again.

00:09:59.740 --> 00:10:02.740
And as we'll see, the neuron
uses a combination, just

00:10:02.740 --> 00:10:05.760
the way we described it here,
in order to spread a signal,

00:10:05.760 --> 00:10:09.810
in order for it to have
the signal spread, in order

00:10:09.810 --> 00:10:12.140
to obviously to spread
passively, but then

00:10:12.140 --> 00:10:17.011
to boost it so that the signal
can cover over long distances.


Derivative of triple composition

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:03.780
Consider the functions f and
g with the graphs shown below.

00:00:03.780 --> 00:00:08.210
If capital G of x is equal to g
of f of x whole thing squared,

00:00:08.210 --> 00:00:12.700
what is the value of G
prime, capital G prime, of 5?

00:00:12.700 --> 00:00:14.560
And I encourage you to
now pause this video

00:00:14.560 --> 00:00:17.330
and try to solve it on your own.

00:00:17.330 --> 00:00:20.280
So let's try to think
through this somewhat

00:00:20.280 --> 00:00:23.930
complicated-looking function
definition right over here.

00:00:23.930 --> 00:00:27.070
So we have capital G of x.

00:00:27.070 --> 00:00:29.312
And actually, let me
write it this way.

00:00:29.312 --> 00:00:31.270
Let me write it this way,
I'll do it in yellow.

00:00:31.270 --> 00:00:37.850
We have capital G of x is
equal to this quantity squared.

00:00:37.850 --> 00:00:45.970
What we're squaring
is g of f of x.

00:00:45.970 --> 00:00:47.810
g of f of x is what
we're squaring.

00:00:47.810 --> 00:00:51.950
Or another way to
write G of x, If h of x

00:00:51.950 --> 00:00:54.880
were to be equal
to x squared, we

00:00:54.880 --> 00:01:05.690
could write G of x is equal
to h of this business, h

00:01:05.690 --> 00:01:06.952
of g of f of x.

00:01:06.952 --> 00:01:08.452
Let me just copy
and paste that so I

00:01:08.452 --> 00:01:10.180
don't have to keep
switching colors.

00:01:10.180 --> 00:01:15.550
So copy and paste, there we go.

00:01:15.550 --> 00:01:18.110
So this is another
way of writing

00:01:18.110 --> 00:01:22.000
G of x, where
whatever g of f of x,

00:01:22.000 --> 00:01:26.470
we input then to h of x, which
is really just squaring it.

00:01:26.470 --> 00:01:28.600
So there's a couple of
ways that we can write out

00:01:28.600 --> 00:01:30.882
the derivative of capital
G with respect to x.

00:01:30.882 --> 00:01:32.340
And you could
imagine this is going

00:01:32.340 --> 00:01:33.280
to involve the chain rule.

00:01:33.280 --> 00:01:35.080
But I like to write it
out, just to clarify

00:01:35.080 --> 00:01:37.520
in my head what's going
on and to make sure

00:01:37.520 --> 00:01:40.010
that it actually
makes some sense.

00:01:40.010 --> 00:01:41.730
So one thing that
we could write,

00:01:41.730 --> 00:01:45.310
we could write the
derivative of G

00:01:45.310 --> 00:01:49.340
with respect-- I'll mix
notations a little bit--

00:01:49.340 --> 00:01:53.610
but I'll write the derivative
of G of x with respect

00:01:53.610 --> 00:02:05.770
to x is equal to the
derivative of this whole thing.

00:02:05.770 --> 00:02:08.560
So let me copy and paste
it, copy and paste.

00:02:08.560 --> 00:02:15.590
It's equal to this derivative
of this whole thing with respect

00:02:15.590 --> 00:02:18.350
to what's inside of
that whole thing.

00:02:18.350 --> 00:02:22.880
So if you wanted to treat
g of f of x as a variable,

00:02:22.880 --> 00:02:25.210
so with respect to that.

00:02:25.210 --> 00:02:29.770
So copy and paste.

00:02:29.770 --> 00:02:32.730
So it's going to be the
derivative of this whole thing

00:02:32.730 --> 00:02:38.680
with respect to g of f of
x times the derivative of g

00:02:38.680 --> 00:02:47.970
of f of x with respect to f
of x, with respect to-- I'll

00:02:47.970 --> 00:02:52.030
just copy and paste this
part, whoops-- with respect

00:02:52.030 --> 00:02:52.830
to f of x.

00:02:52.830 --> 00:02:54.490
And I like to write this out.

00:02:54.490 --> 00:02:55.820
It feels good.

00:02:55.820 --> 00:03:00.600
It looks like these are rational
expressions with differentials.

00:03:00.600 --> 00:03:03.560
It's really a notation more
than to be taken literally.

00:03:03.560 --> 00:03:06.120
But it feels good, or
at least in my mind

00:03:06.120 --> 00:03:08.620
it's a little bit more intuitive
why all of this works out.

00:03:08.620 --> 00:03:14.680
So with respect to f of x times
the derivative of-- and I'm

00:03:14.680 --> 00:03:16.080
using non-standard
notation here,

00:03:16.080 --> 00:03:19.070
but it helps me really
conceptualize this-- times

00:03:19.070 --> 00:03:28.040
the derivative of f of
x with respect to x.

00:03:28.040 --> 00:03:29.930
Or another way we
could write this

00:03:29.930 --> 00:03:37.930
is G prime of x is
equal to h prime

00:03:37.930 --> 00:03:42.100
of g of f of x, h
prime of-- actually,

00:03:42.100 --> 00:03:44.580
let me do it here--
h prime of this.

00:03:48.020 --> 00:03:53.330
So copy and paste,
h prime of that,

00:03:53.330 --> 00:04:05.186
times g prime of f of x,
times g prime of this.

00:04:05.186 --> 00:04:09.530
So copy and then paste.

00:04:09.530 --> 00:04:12.600
So times g prime of that.

00:04:12.600 --> 00:04:14.190
Put some parentheses there.

00:04:14.190 --> 00:04:19.969
Times f prime of x.

00:04:22.840 --> 00:04:25.800
And I like writing it this
way, because you notice

00:04:25.800 --> 00:04:28.370
if these were-- and once
again, this is more notation,

00:04:28.370 --> 00:04:30.070
but it gives a sense
of what's going on.

00:04:30.070 --> 00:04:31.580
If you did view
these as fractions,

00:04:31.580 --> 00:04:32.880
that would cancel with that.

00:04:32.880 --> 00:04:34.410
That would cancel with that.

00:04:34.410 --> 00:04:36.940
You're taking the derivative
of everything with respect

00:04:36.940 --> 00:04:40.000
to x, which is exactly
what you wanted to do.

00:04:40.000 --> 00:04:41.820
And let me put some
parentheses here

00:04:41.820 --> 00:04:43.950
so it makes a little bit
clearer what's going on.

00:04:43.950 --> 00:04:45.590
But this thing, in
my brain, I like

00:04:45.590 --> 00:04:47.006
to translate that
as, well, that's

00:04:47.006 --> 00:04:48.720
just h prime of g of f of x.

00:04:48.720 --> 00:04:50.420
This is g prime of f of x.

00:04:50.420 --> 00:04:52.370
This is f prime of x.

00:04:52.370 --> 00:04:54.152
And going from this
to try to answer

00:04:54.152 --> 00:04:56.610
your question, the question
that they're asking us actually

00:04:56.610 --> 00:04:58.420
isn't too bad.

00:04:58.420 --> 00:05:01.110
So we want to know,
what's G prime of 5?

00:05:01.110 --> 00:05:06.770
So everywhere we see an
x, let's change it to a 5.

00:05:06.770 --> 00:05:10.080
So we're going to say, we need
to figure out what G prime of 5

00:05:10.080 --> 00:05:10.840
is.

00:05:10.840 --> 00:05:14.480
G prime of 5 is equal
to-- and actually,

00:05:14.480 --> 00:05:17.770
let me just copy and
paste this whole thing.

00:05:17.770 --> 00:05:21.950
So copy and paste.

00:05:21.950 --> 00:05:25.620
And so let me, everywhere
where I see an x,

00:05:25.620 --> 00:05:27.315
I'm going to
replace it with a 5.

00:05:27.315 --> 00:05:29.340
So let me get rid of that.

00:05:29.340 --> 00:05:31.680
Let me get rid of that.

00:05:31.680 --> 00:05:34.660
And let me get rid of that.

00:05:34.660 --> 00:05:40.890
And so I have a 5, a 5, and a 5.

00:05:40.890 --> 00:05:43.021
So what is f of 5?

00:05:43.021 --> 00:05:47.110
f of 5 is equal to negative 1.

00:05:47.110 --> 00:05:52.340
So this right over here
simplifies to negative 1.

00:05:52.340 --> 00:05:55.410
This right over here
simplifies to negative 1.

00:05:55.410 --> 00:05:57.860
And what's f prime
of negative 5?

00:05:57.860 --> 00:05:59.960
Well, that's the slope
of the tangent line

00:05:59.960 --> 00:06:01.370
at this point right over here.

00:06:01.370 --> 00:06:03.244
And we see that the
derivative, or the slope,

00:06:03.244 --> 00:06:04.960
of the tangent line here is 0.

00:06:04.960 --> 00:06:09.820
So this right over here
is going to be equal to 0.

00:06:09.820 --> 00:06:11.540
Now that's really interesting.

00:06:11.540 --> 00:06:15.910
So we could keep trying to,
well, what's g of negative 1?

00:06:15.910 --> 00:06:17.340
What's g prime of negative 1?

00:06:17.340 --> 00:06:20.707
You could see g of
negative 1, g of negative 1

00:06:20.707 --> 00:06:23.290
we see is negative 1. g prime
of negative 1 is the slope here,

00:06:23.290 --> 00:06:24.750
which is also negative 1.

00:06:24.750 --> 00:06:27.023
Then we could calculate
h prime of these values,

00:06:27.023 --> 00:06:28.190
et cetera, et cetera.

00:06:28.190 --> 00:06:29.750
But we don't even
have to do that.

00:06:29.750 --> 00:06:31.583
Because this is the
product of three things,

00:06:31.583 --> 00:06:33.760
and one of these things
right over here is a 0.

00:06:33.760 --> 00:06:35.390
So 0 times anything
times anything

00:06:35.390 --> 00:06:40.470
is going to be equal to 0.

00:06:40.470 --> 00:06:42.880
Another way of thinking
about it is, f of x is

00:06:42.880 --> 00:06:45.100
isn't changing when
x is equal to 5.

00:06:45.100 --> 00:06:48.480
If f of x isn't changing
when x is equal to 5,

00:06:48.480 --> 00:06:51.060
then the input into the g
isn't going to be changing.

00:06:51.060 --> 00:06:53.980
So the g function isn't going
to be-- in the composition

00:06:53.980 --> 00:06:56.110
g of f of x-- isn't
going to be changing.

00:06:56.110 --> 00:06:58.470
And so h of g of f of x
isn't going to be changing.

00:06:58.470 --> 00:07:00.440
So g of x isn't
going to be changing.

00:07:00.440 --> 00:07:04.150
And so the derivative of
capital G of x at x equals 5

00:07:04.150 --> 00:07:07.630
is going to be equal to 0.


Total distance traveled by a particle

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:02.590
The position of a
particle moving along

00:00:02.590 --> 00:00:04.730
a number line is
given by s of t is

00:00:04.730 --> 00:00:08.810
equal to 2/3 t to the third
minus 6t squared plus 10t,

00:00:08.810 --> 00:00:10.600
for t is greater
than or equal to 0,

00:00:10.600 --> 00:00:12.850
where t is time in seconds.

00:00:12.850 --> 00:00:16.340
The particle moves both left and
right in the first 6 seconds.

00:00:16.340 --> 00:00:19.640
What is the total distance
traveled by the particle for 0

00:00:19.640 --> 00:00:23.042
is less than or equal to t
is less than or equal to 6?

00:00:23.042 --> 00:00:24.500
So let's just remind
ourselves what

00:00:24.500 --> 00:00:26.990
they mean by total distance.

00:00:26.990 --> 00:00:29.280
If I were to say
start there, and if I

00:00:29.280 --> 00:00:32.689
were to move 3
units to the right

00:00:32.689 --> 00:00:37.270
and then I were to move
4 units to the left,

00:00:37.270 --> 00:00:40.370
and I'll say negative 4 to show
that I'm moving to the left,

00:00:40.370 --> 00:00:46.002
then my total distance
right over here is 7.

00:00:46.002 --> 00:00:47.720
3 to the right
and 4 to the left.

00:00:47.720 --> 00:00:49.560
Even though my position
right over here

00:00:49.560 --> 00:00:51.075
is going to be negative 1.

00:00:53.630 --> 00:00:56.240
Or you could say
my net distance,

00:00:56.240 --> 00:00:58.490
or you could say my
displacement is negative 1.

00:00:58.490 --> 00:01:00.990
I'm 1 to the left
of where I started.

00:01:00.990 --> 00:01:03.030
The total distance is 7.

00:01:03.030 --> 00:01:04.370
So now we've clarified that.

00:01:04.370 --> 00:01:06.379
I encourage you to
now pause this video

00:01:06.379 --> 00:01:07.670
and try to answer the question.

00:01:07.670 --> 00:01:09.260
What is the total
distance traveled

00:01:09.260 --> 00:01:13.930
by the particle in
these first 6 seconds?

00:01:13.930 --> 00:01:16.244
So the easiest way I
can think of addressing

00:01:16.244 --> 00:01:17.660
this is to think
about, well, when

00:01:17.660 --> 00:01:19.034
is this thing
moving to the right

00:01:19.034 --> 00:01:20.590
and when is it
moving to the left?

00:01:20.590 --> 00:01:22.320
And it's going to be
moving to the right

00:01:22.320 --> 00:01:24.020
when the velocity is
positive, and it's

00:01:24.020 --> 00:01:26.639
going to be moving to the left
when the velocity is negative.

00:01:26.639 --> 00:01:28.180
So this really boils
down to thinking

00:01:28.180 --> 00:01:30.265
about when is the velocity
positive or negative.

00:01:30.265 --> 00:01:31.890
And to think about
that, let's actually

00:01:31.890 --> 00:01:34.996
graph the velocity function
or make a rough sketch of it.

00:01:34.996 --> 00:01:36.370
So this is the
position function.

00:01:36.370 --> 00:01:37.869
The velocity function
is going to be

00:01:37.869 --> 00:01:41.510
the derivative of the position
function with respect to time.

00:01:41.510 --> 00:01:44.110
So the derivative of
2/3 t to the third

00:01:44.110 --> 00:01:48.100
is going to be 2t squared.

00:01:48.100 --> 00:01:54.100
And then we have
minus 12t plus 10.

00:01:54.100 --> 00:01:57.580
And so let's just
try to graph this.

00:01:57.580 --> 00:01:59.840
This is going to be an
upward opening parabola.

00:01:59.840 --> 00:02:01.520
This is clearly a quadratic.

00:02:01.520 --> 00:02:04.720
And the coefficient on
the second degree term,

00:02:04.720 --> 00:02:06.580
on the t squared term,
is a positive number,

00:02:06.580 --> 00:02:08.538
so it's going to be an
upward opening parabola.

00:02:08.538 --> 00:02:10.630
It's going to look
something like this.

00:02:10.630 --> 00:02:12.636
And we're assuming that
it switches direction.

00:02:12.636 --> 00:02:14.510
So it's going to be
positive some of the time

00:02:14.510 --> 00:02:16.350
and negative for
some of the time.

00:02:16.350 --> 00:02:22.370
So it should intersect the
t-axis where it's negative.

00:02:22.370 --> 00:02:24.620
The function is going to be
negative in that interval,

00:02:24.620 --> 00:02:27.630
and it's going to be positive
outside of that interval.

00:02:27.630 --> 00:02:29.550
So the easiest thing
I could think of doing

00:02:29.550 --> 00:02:31.480
is let's try to find
what the 0's are.

00:02:31.480 --> 00:02:34.300
Then we can draw this
upward opening parabola.

00:02:34.300 --> 00:02:37.700
So to find its 0's, let's
just set this thing equal to 0

00:02:37.700 --> 00:02:43.620
so we get 2t squared minus
12t plus 10 is equal to 0.

00:02:43.620 --> 00:02:46.234
Divide both sides by 2
just to get rid of this 2,

00:02:46.234 --> 00:02:47.650
make this leading
coefficient a 1.

00:02:47.650 --> 00:02:51.890
We get t squared minus
6t plus 5 is equal to 0.

00:02:51.890 --> 00:02:53.440
That made it a lot
easier to factor.

00:02:53.440 --> 00:02:59.920
This can be factored into
t minus 1 times t minus 5.

00:02:59.920 --> 00:03:02.110
Negative 1 times
negative 5 is 5.

00:03:02.110 --> 00:03:05.190
Negative 1 plus negative
5 is negative 6.

00:03:05.190 --> 00:03:06.730
This is equal to 0.

00:03:06.730 --> 00:03:09.640
So this left hand
side of the equation

00:03:09.640 --> 00:03:12.150
is going to be equal to 0 if
either one of these things

00:03:12.150 --> 00:03:13.110
is equal to 0.

00:03:13.110 --> 00:03:15.160
Take the product of
two things equaling 0,

00:03:15.160 --> 00:03:17.440
well, you get 0 if
either one of them is 0.

00:03:17.440 --> 00:03:22.160
So either t is equal to
1 or t is equal to 5.

00:03:22.160 --> 00:03:23.860
So now let's graph it.

00:03:23.860 --> 00:03:25.960
So let's draw our axes.

00:03:25.960 --> 00:03:29.340
So I could say that's
my velocity axis.

00:03:29.340 --> 00:03:31.450
And let me draw
the-- we only care

00:03:31.450 --> 00:03:33.910
for positive values of time.

00:03:33.910 --> 00:03:37.300
So let's draw
something like this.

00:03:37.300 --> 00:03:38.975
Positive time.

00:03:38.975 --> 00:03:39.800
And let's see.

00:03:39.800 --> 00:03:44.750
Let's take that 1, 2, 3, 4, 5.

00:03:44.750 --> 00:03:46.010
We could keep going.

00:03:46.010 --> 00:03:47.530
So this is t equals 1.

00:03:47.530 --> 00:03:49.130
This is t is equal to 5.

00:03:49.130 --> 00:03:50.960
This is our t-axis.

00:03:50.960 --> 00:03:51.972
And let's graph it.

00:03:51.972 --> 00:03:53.930
So it's going to be an
upward opening parabola.

00:03:53.930 --> 00:03:56.609
It's going to intersect
both of these points.

00:03:56.609 --> 00:03:58.150
And so its vertex
is going to be when

00:03:58.150 --> 00:04:00.220
t is equal to 3 right
in between those points.

00:04:00.220 --> 00:04:03.452
So it's going to look
something like this.

00:04:03.452 --> 00:04:05.910
That's the only way to make an
upward opening parabola that

00:04:05.910 --> 00:04:10.140
intersects the t-axis
at both of these points.

00:04:10.140 --> 00:04:13.200
So it'll go like that,
and it'll go like this.

00:04:13.200 --> 00:04:14.316
It'll intersect.

00:04:14.316 --> 00:04:16.149
When t equals 0, we
actually can figure out.

00:04:16.149 --> 00:04:17.950
When t equals 0
our velocity is 10.

00:04:17.950 --> 00:04:21.930
So the v-intercept, we could
say, is 10 right over here.

00:04:21.930 --> 00:04:23.660
So that's what it looks like.

00:04:23.660 --> 00:04:30.840
So we see that the velocity
is positive for time

00:04:30.840 --> 00:04:32.430
between 0 and 1.

00:04:32.430 --> 00:04:37.250
And it's also positive for
time is greater than 5 seconds.

00:04:37.250 --> 00:04:39.450
And we see that our
velocity is negative,

00:04:39.450 --> 00:04:46.000
or that we're moving to the
left, between 1 and 5 seconds.

00:04:46.000 --> 00:04:48.720
The velocity is below the
t-axis right over here.

00:04:48.720 --> 00:04:50.644
It is negative.

00:04:50.644 --> 00:04:52.560
So let's just think about
what our position is

00:04:52.560 --> 00:04:56.580
at each of these points, at
time 0, at time 1, at time 5,

00:04:56.580 --> 00:04:58.100
and what we care about time 6.

00:05:00.970 --> 00:05:03.150
And then think about
what the distance

00:05:03.150 --> 00:05:06.290
it would have had to
travel between those times.

00:05:06.290 --> 00:05:07.780
So let's think about it.

00:05:07.780 --> 00:05:10.660
So let's make a
little table here.

00:05:10.660 --> 00:05:12.240
Let's make a little table.

00:05:12.240 --> 00:05:15.430
So this is time, and this is
our position at that time.

00:05:15.430 --> 00:05:20.470
So we care about time 0,
time 1, time 5 seconds,

00:05:20.470 --> 00:05:22.730
and time 6 seconds.

00:05:22.730 --> 00:05:26.910
So at 0 seconds, we know
that our position is 0.

00:05:26.910 --> 00:05:29.310
S of 0 is 0.

00:05:29.310 --> 00:05:35.490
At 1 second, this is going
to be 2/3 minus 6 plus 10.

00:05:35.490 --> 00:05:37.720
So it's going to be 4 and 2/3.

00:05:37.720 --> 00:05:41.530
So I'll write down 4 and 2/3.

00:05:41.530 --> 00:05:44.865
At 5 seconds, let's
see, it's 2/3 times--

00:05:44.865 --> 00:05:46.240
I'm going to write
this one down.

00:05:46.240 --> 00:05:50.720
So it's going to
be 2/3 times 125.

00:05:50.720 --> 00:05:58.690
That's the same
thing as 250 over 3,

00:05:58.690 --> 00:06:00.520
which is the same thing.

00:06:00.520 --> 00:06:02.270
Let's see, 250 over 3.

00:06:02.270 --> 00:06:02.960
That's the same.

00:06:02.960 --> 00:06:09.100
83 times 3 is 249, so
this is 83 and 1/3.

00:06:12.940 --> 00:06:14.100
That's this first term.

00:06:17.160 --> 00:06:20.360
Minus 6 times 25.

00:06:23.780 --> 00:06:29.850
So that's going to be
minus 150 plus 10 times 5.

00:06:29.850 --> 00:06:31.800
So plus 50.

00:06:31.800 --> 00:06:33.410
So this is going to simplify.

00:06:33.410 --> 00:06:38.125
Minus 150 plus 50, that's
going to be minus 100.

00:06:38.125 --> 00:06:42.400
83 and 1/3 minus 100.

00:06:42.400 --> 00:06:46.730
That's going to be
negative 16 and 2/3.

00:06:46.730 --> 00:06:52.920
So negative 16 and 2/3 is
its position after 5 seconds.

00:06:52.920 --> 00:06:55.010
And then at the 6
seconds, it's going

00:06:55.010 --> 00:06:57.570
to be 2/3 times 6 to the third.

00:06:57.570 --> 00:06:58.870
I have to write this one down.

00:06:58.870 --> 00:07:05.900
2/3 times 6 to the third
minus 6 times 6 squared.

00:07:05.900 --> 00:07:11.460
Well, that's just going to be
minus 6 to the third again.

00:07:11.460 --> 00:07:15.107
6 times 6 squared plus 60.

00:07:15.107 --> 00:07:15.690
And let's see.

00:07:15.690 --> 00:07:18.950
How can we simplify
this right over here?

00:07:18.950 --> 00:07:20.820
Well, this part
right over here we

00:07:20.820 --> 00:07:23.170
can rewrite as-- we could
factor out as 6 to the third.

00:07:23.170 --> 00:07:28.389
This is 6 to the third
times 2/3 minus 1 plus 60.

00:07:28.389 --> 00:07:30.430
Scroll down a little bit
and get some more space.

00:07:30.430 --> 00:07:34.600
So it's going to be 6 to
the third times negative 1/3

00:07:34.600 --> 00:07:36.130
plus 60.

00:07:36.130 --> 00:07:37.601
And let's see.

00:07:37.601 --> 00:07:38.600
Let's write it this way.

00:07:38.600 --> 00:07:45.510
This is going to be 6 squared
times 6 times negative 1/3

00:07:45.510 --> 00:07:47.290
plus 60.

00:07:47.290 --> 00:07:49.130
This right over
here is negative 2.

00:07:49.130 --> 00:07:50.630
So it's negative 2 times 36.

00:07:50.630 --> 00:07:53.675
This is negative 72 plus 60.

00:07:57.580 --> 00:08:01.154
So this is going to be
negative 12 right over there.

00:08:01.154 --> 00:08:03.320
So now we just have to
think, how far did it travel?

00:08:03.320 --> 00:08:05.150
Well, it starts
traveling to the right.

00:08:05.150 --> 00:08:07.699
It's going to travel
to the right 4 and 2/3.

00:08:07.699 --> 00:08:08.740
So let's write this down.

00:08:08.740 --> 00:08:12.090
So we're going to
have 4 and 2/3.

00:08:12.090 --> 00:08:14.320
And then it's going
to travel to the left.

00:08:14.320 --> 00:08:19.510
Let's see, to go from 4 and
2/3 to negative 16 and 2/3,

00:08:19.510 --> 00:08:22.050
that means you traveled
4 and 2/3 again.

00:08:22.050 --> 00:08:24.240
You traveled 4 and
2/3 to the left,

00:08:24.240 --> 00:08:29.110
and then you traveled another
16 and 2/3 to the left.

00:08:29.110 --> 00:08:31.709
Just a reminder, we're 4
and 2/3 to the right now.

00:08:31.709 --> 00:08:34.000
We have to go 4 and 2/3 to
the left back to the origin,

00:08:34.000 --> 00:08:36.840
and then we have to go 16
and 2/3 again to the left.

00:08:36.840 --> 00:08:39.070
So that's why this
move from here to here

00:08:39.070 --> 00:08:43.100
is going to be 4 and 2/3 to
the left followed by 16 and 2/3

00:08:43.100 --> 00:08:44.035
to the left.

00:08:44.035 --> 00:08:45.910
Another way to think
about it, the difference

00:08:45.910 --> 00:08:49.120
between these two
points is what?

00:08:49.120 --> 00:08:52.500
It's going to be 4 and
2/3 plus 16 and 2/3.

00:08:52.500 --> 00:08:55.597
If you do 4 and 2/3 minus
negative 16 and 2/3,

00:08:55.597 --> 00:08:57.930
you're going to have, that's
the same thing as 4 and 2/3

00:08:57.930 --> 00:09:01.230
plus 16 and 2/3.

00:09:01.230 --> 00:09:10.060
And then to go from negative
16 and 2/3 to negative 12,

00:09:10.060 --> 00:09:15.150
that means you went another
4 and 2/3 now to the right.

00:09:15.150 --> 00:09:17.750
So now this is 4 and 2/3.

00:09:17.750 --> 00:09:21.370
Now you're moving 4
and 2/3 to the right.

00:09:21.370 --> 00:09:24.850
And so we just have to
add up all of these.

00:09:24.850 --> 00:09:27.580
We just have to add up
all of these values.

00:09:27.580 --> 00:09:29.070
So what is this going to be?

00:09:29.070 --> 00:09:34.352
So this is going to be 2/3
times 4, so this part of it

00:09:34.352 --> 00:09:36.060
right over here, the
fraction part of it.

00:09:36.060 --> 00:09:39.860
2/3 times 4 is 8 over 3.

00:09:39.860 --> 00:09:44.840
And let's see, 4 plus
4 plus 16 plus 4 is 28.

00:09:44.840 --> 00:09:47.390
So 28 and 8/3, that's a very
strange way to write it.

00:09:47.390 --> 00:09:54.320
Because 8/3 is the same
thing as 2 and 2/3.

00:09:54.320 --> 00:09:58.790
So 28 plus 2 and
2/3 is 30 and 2/3.

00:09:58.790 --> 00:10:02.190
So the total distance
traveled over those 6 seconds

00:10:02.190 --> 00:10:07.931
is 30 and 2/3 units.


Analyzing particle movement based on graphs

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:04.290
A particle moves along a
number line not shown for t

00:00:04.290 --> 00:00:06.250
is greater than or equal to 0.

00:00:06.250 --> 00:00:09.180
Its position function, s
of t, is shown in blue.

00:00:09.180 --> 00:00:11.990
So this is its position
as a function of time.

00:00:11.990 --> 00:00:15.030
Its velocity function,
v of t, is in red.

00:00:15.030 --> 00:00:16.210
That's velocity.

00:00:16.210 --> 00:00:20.150
And its acceleration
function, a if t, is in green.

00:00:20.150 --> 00:00:23.421
All are graphed with respect
to time t in seconds.

00:00:23.421 --> 00:00:25.670
With the graphs as an aid,
answer the questions below.

00:00:25.670 --> 00:00:26.750
So that's what's going on here.

00:00:26.750 --> 00:00:28.387
So it's position as
a function of time.

00:00:28.387 --> 00:00:30.220
Actually, let me just
draw their number line

00:00:30.220 --> 00:00:34.170
that they did not depict just so
we can really think about this.

00:00:34.170 --> 00:00:38.400
So let's say that's
our number line.

00:00:38.400 --> 00:00:43.240
Let's say this right
over here is 0.

00:00:43.240 --> 00:00:44.720
That's 1.

00:00:44.720 --> 00:00:46.630
That is 2.

00:00:46.630 --> 00:00:49.480
This is negative 1.

00:00:49.480 --> 00:00:51.200
So we're defining
going to the right

00:00:51.200 --> 00:00:53.600
as the positive direction.

00:00:53.600 --> 00:00:54.910
So what's happening here?

00:00:54.910 --> 00:01:02.750
So at time equals 0 right
over here, s of 0 is 0.

00:01:02.750 --> 00:01:05.430
And then as time
increases, our position

00:01:05.430 --> 00:01:09.140
increases all the way
until time equals 1.

00:01:09.140 --> 00:01:12.050
At time equals 1,
our position is 2.

00:01:12.050 --> 00:01:14.860
So at time equals 1,
our position is 2.

00:01:14.860 --> 00:01:19.640
And then our position, s
of t, starts decreasing.

00:01:19.640 --> 00:01:23.420
So one way to think about it
is, and you see we move up,

00:01:23.420 --> 00:01:25.000
we move to the
right really fast.

00:01:25.000 --> 00:01:25.770
We get to 2.

00:01:25.770 --> 00:01:29.240
We stop at 2, and then we
start moving to the left.

00:01:29.240 --> 00:01:31.970
So at time equals 0, the
first second looks like this.

00:01:31.970 --> 00:01:35.660
We go zoom, oh,
slow down and stop.

00:01:35.660 --> 00:01:38.380
And then we start
moving the other way.

00:01:38.380 --> 00:01:40.180
And then we start drifting.

00:01:40.180 --> 00:01:43.470
Notice our position
is decreasing.

00:01:43.470 --> 00:01:46.140
So our position is
decreasing, but it's

00:01:46.140 --> 00:01:48.720
decreasing at ever slower,
slower, and slower and slower

00:01:48.720 --> 00:01:49.220
rates.

00:01:49.220 --> 00:01:53.160
It's not clear if we'll
ever get back to the origin.

00:01:53.160 --> 00:01:54.519
So that's what's going on here.

00:01:54.519 --> 00:01:56.560
And we see that no matter
which graph we look at.

00:01:56.560 --> 00:01:59.130
Our position function is
definitely telling that story.

00:01:59.130 --> 00:02:02.070
Our velocity function, which is
the derivative of the position

00:02:02.070 --> 00:02:03.460
function, is telling that story.

00:02:03.460 --> 00:02:06.430
Out the gate, we have a
high positive velocity,

00:02:06.430 --> 00:02:08.690
but we decelerate quickly.

00:02:08.690 --> 00:02:10.966
And at 1 second,
our velocity is 0,

00:02:10.966 --> 00:02:13.090
and then we start having
a negative velocity, which

00:02:13.090 --> 00:02:14.970
means we're moving to the left.

00:02:14.970 --> 00:02:20.160
So fast rightward velocity,
but we decelerate quickly,

00:02:20.160 --> 00:02:22.420
stop at time equals
1 second, and then

00:02:22.420 --> 00:02:25.550
we start drifting to the left.

00:02:25.550 --> 00:02:29.630
And the acceleration also
shows that same narrative.

00:02:29.630 --> 00:02:31.980
But anyway, let's actually
answer the questions.

00:02:31.980 --> 00:02:33.600
The initial velocity
of the particle

00:02:33.600 --> 00:02:35.030
is blank units per second.

00:02:35.030 --> 00:02:38.150
I encourage you to pause
this video and answer that.

00:02:38.150 --> 00:02:41.040
Well, we just said the velocity,
let's see, at time equals 0,

00:02:41.040 --> 00:02:42.570
we're at 8 units per second.

00:02:42.570 --> 00:02:44.470
So we'll just put
8 right over there.

00:02:44.470 --> 00:02:48.750
The particles moving
to the right when

00:02:48.750 --> 00:02:51.627
t is in the interval, and
since they're doing this

00:02:51.627 --> 00:02:53.960
as a member of, they really
want this kind of in the set

00:02:53.960 --> 00:02:56.530
notation t is a member
of the interval.

00:02:56.530 --> 00:02:58.120
Well, when are we
moving to the right?

00:02:58.120 --> 00:02:59.080
We already went over that.

00:02:59.080 --> 00:03:00.600
We're moving it to
the right-- there's

00:03:00.600 --> 00:03:02.058
a couple of ways
to think about it.

00:03:02.058 --> 00:03:04.960
When our velocity
is greater than-- so

00:03:04.960 --> 00:03:09.470
we're moving to the right
when v of t is greater than 0.

00:03:09.470 --> 00:03:12.150
When v of t is less than 0,
we're moving to the left.

00:03:12.150 --> 00:03:14.110
When v of t is equal
to 0, we're stationary.

00:03:14.110 --> 00:03:16.240
So when is v of
t greater than 0?

00:03:16.240 --> 00:03:22.460
Well, it's between t being
0, velocity is definitely

00:03:22.460 --> 00:03:25.800
positive, all the way to t is
1, but not including t is 1.

00:03:25.800 --> 00:03:27.496
So I'll put a parentheses there.

00:03:27.496 --> 00:03:28.870
So this is equivalent
to saying--

00:03:28.870 --> 00:03:31.530
so t is a member
of that interval

00:03:31.530 --> 00:03:35.540
is equivalent to saying that
0 is less than or equal to t

00:03:35.540 --> 00:03:38.000
is less than 1.

00:03:38.000 --> 00:03:42.150
Once again, the first
second, at time 0,

00:03:42.150 --> 00:03:46.470
we're going fast, slow
down, and then stop

00:03:46.470 --> 00:03:49.570
for an infinitesimal moment,
and then we start drifting back.

00:03:49.570 --> 00:03:51.220
That happens at time equals 1.

00:03:51.220 --> 00:03:53.090
We start drifting back.

00:03:53.090 --> 00:03:57.000
The total distance traveled
by the particle for t

00:03:57.000 --> 00:03:59.309
in the interval between
0 and 3 is blank units.

00:03:59.309 --> 00:04:01.350
So once again, I encourage
you to pause the video

00:04:01.350 --> 00:04:04.510
and try to answer that,
the total distance.

00:04:04.510 --> 00:04:05.470
So this is interesting.

00:04:05.470 --> 00:04:09.030
Don't get distance
confused with displacement.

00:04:09.030 --> 00:04:13.860
If I were to move
three to the right

00:04:13.860 --> 00:04:16.740
and then I were to move
back one to the left,

00:04:16.740 --> 00:04:19.730
the total distance
I've traveled is four.

00:04:19.730 --> 00:04:24.600
The distance I traveled is
four, while the displacement

00:04:24.600 --> 00:04:27.172
would be a positive.

00:04:27.172 --> 00:04:28.630
We could maybe put
a minus 1 there.

00:04:28.630 --> 00:04:30.096
We moved one to the left.

00:04:30.096 --> 00:04:31.720
So three to the right,
one to the left.

00:04:31.720 --> 00:04:37.670
Well, our displacement would
be a net of positive 2.

00:04:37.670 --> 00:04:40.385
So they're asking, what's
the total distance traveled?

00:04:40.385 --> 00:04:49.650
So between time 0 and time 1,
we have moved two to the right.

00:04:53.880 --> 00:04:58.020
And then between time 1
and time 3, we move back,

00:04:58.020 --> 00:05:00.930
or to the left, we move half.

00:05:00.930 --> 00:05:03.982
So to the left, we move half.

00:05:03.982 --> 00:05:05.190
So what's our total distance?

00:05:05.190 --> 00:05:07.360
It's going to be
two to the right

00:05:07.360 --> 00:05:11.745
plus half to the left, which
is going to be 2.5 units.

00:05:18.250 --> 00:05:20.080
And we're done.