Laplace transform of t: L{t}

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:03.330
Let's try to fill in our Laplace
transform table a

00:00:03.330 --> 00:00:03.960
little bit more.

00:00:03.960 --> 00:00:06.000
And a good place to start is
just to write our definition

00:00:06.000 --> 00:00:07.820
of the Laplace transform.

00:00:07.820 --> 00:00:14.240
The Laplace transform of some
function f of t is equal to

00:00:14.240 --> 00:00:19.340
the integral from 0 to infinity,
of e to the minus

00:00:19.340 --> 00:00:23.540
st, times our function,
f of t dt.

00:00:23.540 --> 00:00:24.860
That's our definition.

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The very first one we solved
for-- we could even do it on

00:00:28.330 --> 00:00:32.630
the side right here-- was the
Laplace transform of 1.

00:00:32.630 --> 00:00:35.550
You know, we could almost view
that as t to the 0, and that

00:00:35.550 --> 00:00:38.220
was equal to the integral
from 0 to infinity.

00:00:38.220 --> 00:00:44.490
f of t was just 1, so it's e to
the minus st, dt, which is

00:00:44.490 --> 00:00:47.080
equal to the antiderivative of
e to the minus st, which is

00:00:47.080 --> 00:00:53.910
minus 1 over s, e to the minus
st. And then you have to

00:00:53.910 --> 00:00:56.870
evaluate that from
0 to infinity.

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When you take the limit as this
term approaches infinity,

00:01:00.380 --> 00:01:02.740
this e to the minus, this
becomes e to the minus

00:01:02.740 --> 00:01:05.529
infinity, if we assume
s is greater than 0.

00:01:05.529 --> 00:01:08.160
So if we assume s is greater
than 0, this whole

00:01:08.160 --> 00:01:10.400
term goes to 0.

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So you end up with a 0 minus
this thing evaluated at 0.

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So when you evaluate t is equal
to 0, this term right

00:01:19.330 --> 00:01:24.790
here becomes 1, e to the 0
becomes 1, so it's minus minus

00:01:24.790 --> 00:01:29.585
1/s, which is the same
thing as plus 1/s.

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the?

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Laplace transform of 1,
of just the constant

00:01:32.960 --> 00:01:34.940
function 1, is 1/s.

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We already solved that.

00:01:36.660 --> 00:01:38.700
Now, let's increment
it a little bit.

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Let's see if we can figure out
the Laplace transform of t.

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So we can view this
as t to the 0.

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Now this is t to the 1.

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This is going to be equal to
the integral from 0 to

00:01:51.130 --> 00:01:58.870
infinity, of e to the
minus st times t dt.

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Now.

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I can tell you right now
that I don't have the

00:02:02.210 --> 00:02:03.870
antiderivative of
this memorized.

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I don't know what it is.

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But there's a sense that the
integration by parts could be

00:02:07.680 --> 00:02:10.269
useful, because integration by
parts kind of decomposes into

00:02:10.269 --> 00:02:11.680
a simpler problem.

00:02:11.680 --> 00:02:14.230
And I always forget integration
by parts, so I'll

00:02:14.230 --> 00:02:17.240
rederive it here in
this purple color.

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So if we have u times v, if we
take the derivative with

00:02:21.960 --> 00:02:25.620
respect to t of that, that's
equal to the derivative of the

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first times the second function
plus the first

00:02:29.180 --> 00:02:31.190
function times the derivative
of the second function.

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It's just the product rule.

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Now, if we take the integral
of both sides of this

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equation, we get uv is equal
to the antiderivative of u

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prime v plus the antiderivative
of uv prime.

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Now, since we want to apply
this to an integral, maybe

00:02:49.680 --> 00:02:53.080
let's make this what we want
to solve for, so we can get

00:02:53.080 --> 00:02:57.120
the integral of uv prime is
equal to-- we can just

00:02:57.120 --> 00:03:00.310
subtract this from that side
of the equation, so it's

00:03:00.310 --> 00:03:03.290
equal-- I'm just swapping the
sides, so I'm just solving for

00:03:03.290 --> 00:03:06.330
this, and to solve for this, I
just subtract this from that,

00:03:06.330 --> 00:03:11.390
so it's equal to uv minus the
integral of u prime v.

00:03:11.390 --> 00:03:12.290
So there you go.

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Even though I have trouble
memorizing this formula, it's

00:03:15.040 --> 00:03:18.660
not too hard to rederive as
long as you remember the

00:03:18.660 --> 00:03:21.350
product rule right there.

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So if we're going to do
integration by parts, it's

00:03:23.880 --> 00:03:27.740
good to define our v prime to
be something that's easy to

00:03:27.740 --> 00:03:29.460
take the antiderivative of,
because we're going to have to

00:03:29.460 --> 00:03:32.480
figure out v later on, and
it's good to take u to be

00:03:32.480 --> 00:03:34.690
something that's easy take
the derivative of.

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So let's make t is equal to our
u and let's make e to the

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minus st as being our v prime.

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If that's the case,
then what is v?

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Well, v's just the
antiderivative of that.

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In fact, we've done it before.

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It's minus 1/s, e to the
minus st. That's v.

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And then if we want to figure
out u prime, because we're

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going to have to figure out that
later anyway, u prime's

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just the derivative of t.

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That's just equal to 1.

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So let's apply this.

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Let's see, so the Laplace
transform of t is equal to uv.

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u is t, v is this right here.

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It's times minus 1/s, e to the
minus st. e to the minus st,

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that's the uv term
right there.

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And this is a definite
integral, right?

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So we're going to evaluate
this term right

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here from 0 to infinity.

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And then it's minus the integral
from 0 to infinity of

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u prime, which is
just 1 times v.

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v, we just figured out here, is
minus-- let me write it in

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v's color-- times minus 1/s--
this is my v right here--

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minus 1/s, e to the
minus st, dt.

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All of that is dt.

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So let's see if we can
simplify this.

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So this is equal to minus t/s,
e to the minus st, evaluated

00:05:31.640 --> 00:05:33.740
from 0 to infinity.

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And let's see, we could take--
well, this is just 1.

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1 times anything is 1.

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We can just not write that.

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And then bring the
minus 1/s out.

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So if we bring the minus 1/s
out, this becomes plus 1/s

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times the integral from
0 to infinity of e to

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the minus st, dt.

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And this should look
familiar to you.

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This is exactly what we
solved for right here.

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It was the Laplace
transform of 1.

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So let's keep that in mind.

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So this right here is the
Laplace transform of 1.

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And I want to write it that way,
because we're going to

00:06:10.500 --> 00:06:12.700
see a pattern of this
in the next video.

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I'm going to write that as
a Laplace transform of 1.

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But what is this equal to?

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So we're going to evaluate this
as it adds to infinity

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and then subtract from
that evaluated at 0.

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You can kind of view it as a
substitution, so this is equal

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to-- well, let me write
it this way.

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It's the limit as A approaches
infinity, of minus A/s, e to

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the minus sA.

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So that's this evaluated
at infinity.

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And then from that, we're
going to subtract this

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evaluated at 0.

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So minus all of this, but we
already have a minus sign

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here, so we could
write a plus.

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Plus 0/s times e to the
minus s times 0.

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And then, of course, we have
this term right here.

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So let me write that term.

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I'll do it in yellow or
let me do it in blue.

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Plus 1/s-- that's this right
there-- times the Laplace

00:07:09.040 --> 00:07:12.650
transform of 1.

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And what do we get?

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So what's the limit of this
as A approaches infinity?

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You might say, wow, you know, as
A approaches infinity right

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here, this becomes a
really big number.

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There's a minus sign in there,
so it would be a really big

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negative number.

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But this is an exponent.

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A is an exponent right here.

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So e to the minus infinity is
going to go to zero much

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faster than this is going
to go to infinity.

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This term right here is a much
stronger function, I guess is

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the way you could see it.

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And you could try it out on your
calculator, if you don't

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believe me.

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This term is going to overpower
this term and so

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this whole thing is going
to go to zero.

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Likewise, e to the minus-- e
to the 0, this is 1, but

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you're multiplying it times a
zero, so this is also going to

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go to zero, which is convenient
because all of this

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stuff just disappears.

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And we're left with the Laplace
transform of t is

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equal to 1/s times the Laplace
transform of 1.

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And we know what the Laplace
transform of 1 is.

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The Laplace transform of 1-- we
just did it at beginning of

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the video-- was equal to 1/s,
if we assume that s is

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greater than zero.

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In fact, we have to assume that
s was greater than zero

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here in order to assume that
this goes to zero.

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Only if s is greater than zero,
when you get a minus

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infinity here does this
approach zero.

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So fair enough.

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So the Laplace transform of t
is equal to 1/s times 1/s,

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which is equal to 1/s squared,
where s is greater than zero.

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So we have one more entry
in our table, and

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then we can use this.

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What we're going to do in the
next video is build up to the

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Laplace transform of t to
any arbitrary exponent.

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And we'll do this in
the next video.


Using the Laplace transform to solve a nonhomogeneous eq

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:03.880
It's been over a year since I
last did a video with the

00:00:03.880 --> 00:00:05.420
differential equations playlist,
and I thought I

00:00:05.420 --> 00:00:07.370
would start kicking up, making
a couple of videos.

00:00:07.370 --> 00:00:10.840
And I think where I left,
I said that I would do a

00:00:10.840 --> 00:00:15.340
non-homogenous linear equation
using the Laplace Transform.

00:00:15.340 --> 00:00:18.040
So let's do one, as a bit of a
warm-up, now that we've had

00:00:18.040 --> 00:00:20.050
a-- or at least I've had
a one-year hiatus.

00:00:20.050 --> 00:00:21.660
Maybe you're watching these
continuously, so you're

00:00:21.660 --> 00:00:24.040
probably more warmed
up than I am.

00:00:24.040 --> 00:00:31.500
So if we have the equation the
second derivative of y plus y

00:00:31.500 --> 00:00:36.820
is equal to sine of 2t.

00:00:36.820 --> 00:00:39.750
And we're given some initial
conditions here.

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The initial conditions are y
of 0 is equal to 2, and y

00:00:46.190 --> 00:00:50.880
prime of 0 is equal to 1.

00:00:50.880 --> 00:00:52.250
And where we left off--
and now you

00:00:52.250 --> 00:00:53.090
probably remember this.

00:00:53.090 --> 00:00:55.570
You probably recently watched
the last video.

00:00:55.570 --> 00:00:56.940
To solve these, we just
take the Laplace

00:00:56.940 --> 00:00:58.480
Transforms of all the sides.

00:00:58.480 --> 00:01:02.540
We solve for the Laplace
Transform of the function.

00:01:02.540 --> 00:01:03.990
Then we take the inverse
Laplace Transform.

00:01:03.990 --> 00:01:06.530
If that doesn't make sense, then
let's just do it in this

00:01:06.530 --> 00:01:08.390
video, and hopefully
the example

00:01:08.390 --> 00:01:11.190
will clarify all confusion.

00:01:11.190 --> 00:01:13.650
So in the last video-- it was
either the last one or the

00:01:13.650 --> 00:01:18.460
previous one-- I showed you that
the Laplace Transform of

00:01:18.460 --> 00:01:23.360
the second derivative of y is
equal to s squared times the

00:01:23.360 --> 00:01:28.210
Laplace Transform of y-- and we
keep lowering the degree on

00:01:28.210 --> 00:01:32.540
s-- so minus s times y of 0.

00:01:32.540 --> 00:01:34.530
You can kind of think of it
as taking the derivative.

00:01:34.530 --> 00:01:35.510
This is an integral.

00:01:35.510 --> 00:01:38.260
It's not exactly the
anti-derivative of this.

00:01:38.260 --> 00:01:40.330
But the Laplace Transform
it is an integral.

00:01:40.330 --> 00:01:42.290
The Transform is an integral.

00:01:42.290 --> 00:01:47.530
So y of 0 is kind of one
derivative away from that.

00:01:47.530 --> 00:01:52.400
And then minus y prime of 0.

00:01:52.400 --> 00:01:53.940
And then we could also
rewrite this.

00:01:53.940 --> 00:01:56.270
And this is just a purely
notational issue.

00:01:56.270 --> 00:01:59.300
I could write this, instead of
writing the Laplace Transform

00:01:59.300 --> 00:02:02.600
of y all the time, I could write
this as s squared times

00:02:02.600 --> 00:02:05.600
capital Y of s-- because this
is going to be a function of

00:02:05.600 --> 00:02:12.700
s, not a function of y-- minus
s times y of 0 minus

00:02:12.700 --> 00:02:14.670
y prime y of 0.

00:02:14.670 --> 00:02:16.510
These are going to be
numbers, right?

00:02:16.510 --> 00:02:17.420
These aren't functions.

00:02:17.420 --> 00:02:22.500
These are the function evaluated
at 0, or the

00:02:22.500 --> 00:02:24.190
derivative of the function
evaluated at 0.

00:02:24.190 --> 00:02:25.690
And we know what these
values are.

00:02:25.690 --> 00:02:31.910
y of 0, right here, is 2,
and y prime of 0 is 1.

00:02:31.910 --> 00:02:33.340
It was given to us.

00:02:33.340 --> 00:02:35.740
So if we take the Laplace
Transforms of both sides of

00:02:35.740 --> 00:02:37.940
this equation, first we're
going to want to take the

00:02:37.940 --> 00:02:39.950
Laplace Transform of this term
right there, which we've

00:02:39.950 --> 00:02:40.700
really just done.

00:02:40.700 --> 00:02:46.960
The Laplace Transform of the
second derivative is s squared

00:02:46.960 --> 00:02:51.620
times the Laplace Transform of
the function, which we write

00:02:51.620 --> 00:02:56.690
as capital Y of s, minus this,
minus 2s-- they gave us that

00:02:56.690 --> 00:03:01.900
initial condition--
and then minus 1.

00:03:01.900 --> 00:03:02.660
Right?

00:03:02.660 --> 00:03:05.770
This term right here is
just 1, so minus 1.

00:03:05.770 --> 00:03:07.560
So that's term right there.

00:03:07.560 --> 00:03:08.490
Then we want to take
the Laplace

00:03:08.490 --> 00:03:11.380
Transform of y by itself.

00:03:11.380 --> 00:03:16.100
So this is just plus Y of
s, right, the Laplace

00:03:16.100 --> 00:03:17.150
Transform of y.

00:03:17.150 --> 00:03:20.110
So I'll just rewrite Laplace
Transform of y.

00:03:20.110 --> 00:03:22.500
I'm just rewriting it
in this notation.

00:03:22.500 --> 00:03:23.200
Y of S.

00:03:23.200 --> 00:03:26.230
It's good to get used
to either one.

00:03:26.230 --> 00:03:28.060
This is going to be equal
to the Laplace

00:03:28.060 --> 00:03:31.470
Transform of sine of 2t.

00:03:31.470 --> 00:03:34.930
And I showed you in a video last
year that we showed what

00:03:34.930 --> 00:03:37.740
the Laplace Transform of sine
of at is, but I'll write it

00:03:37.740 --> 00:03:39.690
down here just so
you remember it.

00:03:39.690 --> 00:03:49.170
Laplace Transform of the sine
of at is equal to a over s

00:03:49.170 --> 00:03:52.325
squared plus a squared.

00:03:56.040 --> 00:04:02.220
So the Laplace Transform
of sine of 2t.

00:04:02.220 --> 00:04:03.380
Here, a is 2.

00:04:03.380 --> 00:04:07.140
This is going to be 2 over
s squared plus 4.

00:04:07.140 --> 00:04:09.080
So if we take the Laplace
Transform of both sides of

00:04:09.080 --> 00:04:13.630
this, the right-hand side
is going to be 2 over s

00:04:13.630 --> 00:04:16.589
squared plus 4.

00:04:16.589 --> 00:04:19.390
Now what we can do is we can
separate out all the Y of s

00:04:19.390 --> 00:04:23.240
terms. And so we can factor,
well I guess we could say,

00:04:23.240 --> 00:04:25.270
factor out their coefficients,
so that's a Y of s term,

00:04:25.270 --> 00:04:26.920
that's a Y of s term.

00:04:26.920 --> 00:04:30.170
And so we could write the
left-hand side here as s

00:04:30.170 --> 00:04:34.610
squared-- that's that term--
plus 1-- the coefficient on

00:04:34.610 --> 00:04:39.685
that term-- s squared plus
1, times Y of s.

00:04:39.685 --> 00:04:41.600
Let me do it in green.

00:04:41.600 --> 00:04:47.450
So this is Y of s, and this Y
of s, times Y of s, and then

00:04:47.450 --> 00:04:51.900
we have the non-Y of s terms.
These two right here.

00:04:51.900 --> 00:04:58.770
So minus 2s, minus 1, is equal
to 2 over s squared plus 4.

00:04:58.770 --> 00:05:01.490
We can add 2s plus 1 to both
sides, to essentially move

00:05:01.490 --> 00:05:08.320
this to the right-hand side, and
we're left with s squared

00:05:08.320 --> 00:05:20.390
plus 1, times Y of s, is equal
to 2 over s squared plus 4,

00:05:20.390 --> 00:05:24.750
plus 2s, plus 1.

00:05:24.750 --> 00:05:28.010
Now we can divide both sides of
this equation by s squared

00:05:28.010 --> 00:05:32.630
plus 1, and we get the Laplace
Transform of Y.

00:05:32.630 --> 00:05:39.130
Y of s is equal to-- let me
switch colors-- it's equal to

00:05:39.130 --> 00:05:45.880
2 over s squared plus 4 times
this thing right here.

00:05:45.880 --> 00:05:48.890
I'm dividing both sides of this
equation by this term

00:05:48.890 --> 00:05:49.540
right there.

00:05:49.540 --> 00:05:53.160
So times s squared plus 1-- it's
in the denominator so I'm

00:05:53.160 --> 00:05:59.530
dividing by it-- plus 2s plus
1-- I have to divide both of

00:05:59.530 --> 00:06:03.810
those terms by the s squared
plus 1-- divided by s squared

00:06:03.810 --> 00:06:09.290
plus 1, divided by
s squared plus 1.

00:06:09.290 --> 00:06:11.830
Now, in order to be able to
take the inverse Laplace

00:06:11.830 --> 00:06:15.350
Transform of this, I need to get
it in some type of simple

00:06:15.350 --> 00:06:16.310
fraction form.

00:06:16.310 --> 00:06:18.070
These are actually easier to
do, but this was one's a

00:06:18.070 --> 00:06:18.810
little bit difficult.

00:06:18.810 --> 00:06:21.790
I want to do some partial
fraction decomposition to

00:06:21.790 --> 00:06:25.380
break this up into maybe
simpler fractions.

00:06:25.380 --> 00:06:27.120
So what I want to do, I'm going
to do a little bit of an

00:06:27.120 --> 00:06:28.100
aside here.

00:06:28.100 --> 00:06:31.890
And this really is the hardest
part of these problems. The

00:06:31.890 --> 00:06:34.140
algebra, breaking
this thing up.

00:06:34.140 --> 00:06:36.780
So I'm going to break this up.

00:06:36.780 --> 00:06:37.940
So let me write this this way.

00:06:37.940 --> 00:06:45.080
2 over s squared plus 4 times
s squared plus 1.

00:06:45.080 --> 00:06:47.660
I'm going to break this
up into two fractions.

00:06:47.660 --> 00:06:50.410
This is the partial fraction
decomposition.

00:06:50.410 --> 00:06:52.660
One fraction is s
squared plus 4.

00:06:52.660 --> 00:06:56.420
And the other fraction
is s squared plus 1.

00:06:56.420 --> 00:07:02.110
And since both of these
denominators are of degree 2,

00:07:02.110 --> 00:07:06.080
the numerators are going
to be of degree 1.

00:07:06.080 --> 00:07:09.690
So they're going to be some--
let me write it this way--

00:07:09.690 --> 00:07:12.630
this one will be As plus B.

00:07:12.630 --> 00:07:16.520
And then this one will
be Cs plus D.

00:07:16.520 --> 00:07:17.970
This is just pure algebra.

00:07:17.970 --> 00:07:20.130
This is just partial fraction
decomposition.

00:07:20.130 --> 00:07:22.320
I've made a couple
of videos on it.

00:07:22.320 --> 00:07:25.760
And I'm saying that I'm assuming
that this expression

00:07:25.760 --> 00:07:27.330
right here can be broken
up into two

00:07:27.330 --> 00:07:28.790
expressions of this form.

00:07:28.790 --> 00:07:31.860
And I need to solve for
A, B, C, and D.

00:07:31.860 --> 00:07:32.940
So let's see how
we can do that.

00:07:32.940 --> 00:07:36.310
So if I were to start with these
two and add them up,

00:07:36.310 --> 00:07:37.470
what do I get?

00:07:37.470 --> 00:07:40.820
I would have to multiply these
times-- so my denominator, my

00:07:40.820 --> 00:07:46.040
common denominator, would be
this thing again-- it would be

00:07:46.040 --> 00:07:53.530
s squared plus 4 times
s squared plus 1.

00:07:53.530 --> 00:08:00.180
And now I'm going to have to
multiply the As plus B times

00:08:00.180 --> 00:08:01.623
this s squared plus 1.

00:08:05.040 --> 00:08:06.870
This, as it is right now,
these two terms

00:08:06.870 --> 00:08:07.420
would cancel out.

00:08:07.420 --> 00:08:09.070
You'll just get this term,
but I need to add it

00:08:09.070 --> 00:08:10.940
to this right here.

00:08:10.940 --> 00:08:24.970
So you get plus Cs plus D
times s squared plus 4.

00:08:24.970 --> 00:08:27.660
And now let's see what we could
do to match up the terms

00:08:27.660 --> 00:08:29.810
here with this number
2 right here.

00:08:29.810 --> 00:08:31.650
So let's multiply
all of this out.

00:08:31.650 --> 00:08:39.780
So As times s squared
is As to the 3rd.

00:08:39.780 --> 00:08:43.630
As times 1 is plus As.

00:08:43.630 --> 00:08:48.620
B times s squared, so plus Bs
squared, and then you have B

00:08:48.620 --> 00:08:51.690
times 1 is plus B.

00:08:51.690 --> 00:08:54.525
And then you have Cs times s
squared, that's Cs to the 3rd.

00:08:58.130 --> 00:09:02.025
And then Cs times 4,
so it's plus 4Cs.

00:09:05.130 --> 00:09:08.240
These problems are tiring.

00:09:08.240 --> 00:09:11.560
And I also have a cold, so this
is especially tiring, but

00:09:11.560 --> 00:09:13.410
I'll soldier forward.

00:09:13.410 --> 00:09:14.150
Where was I?

00:09:14.150 --> 00:09:16.200
So I multiplied the C's times
each of these, now I have to

00:09:16.200 --> 00:09:17.040
multiply the D's.

00:09:17.040 --> 00:09:22.210
So plus Ds squared-- that's D
times that one-- plus D times

00:09:22.210 --> 00:09:24.910
4, so plus 4D.

00:09:24.910 --> 00:09:25.950
So that's all of them.

00:09:25.950 --> 00:09:28.600
And I just wrote it this way
so I have the common degree

00:09:28.600 --> 00:09:30.460
terms under each other.

00:09:30.460 --> 00:09:35.140
So if I were to add the entire
numerator, I get-- and I'll

00:09:35.140 --> 00:09:42.100
just switch colors, somewhat
arbitrarily-- I get A plus C

00:09:42.100 --> 00:09:46.580
times s to the 3rd plus-- let
me write the s squared term

00:09:46.580 --> 00:09:51.510
next-- plus B plus D times s
squared-- now I'll write this

00:09:51.510 --> 00:10:01.760
s term-- plus A plus 4C times
s plus B plus 4D.

00:10:01.760 --> 00:10:02.870
This is just the numerator.

00:10:02.870 --> 00:10:04.710
This is when I just added
these two things up.

00:10:04.710 --> 00:10:10.300
This whole thing up here
simplifies to this.

00:10:10.300 --> 00:10:13.030
I don't know if the word
simplify is appropriate.

00:10:13.030 --> 00:10:15.730
But it becomes this expression
right here.

00:10:15.730 --> 00:10:16.960
And that's just the numerator.

00:10:16.960 --> 00:10:18.880
The denominator is still what
we had written before.

00:10:18.880 --> 00:10:23.240
The denominator is still the s
squared plus 4, times the s

00:10:23.240 --> 00:10:25.130
squared plus 1.

00:10:25.130 --> 00:10:28.720
Of course, I have to show
that this is a fraction.

00:10:28.720 --> 00:10:31.100
And this is going to be equal
to this thing over here.

00:10:31.100 --> 00:10:39.030
2 over s squared plus 4 times
s squared plus 1.

00:10:39.030 --> 00:10:43.380
Now, why did I go through this
whole mess right here?

00:10:43.380 --> 00:10:45.110
Well, the reason why I went
through it is because we

00:10:45.110 --> 00:10:48.100
should be able to solve
for A, B, C, and D.

00:10:48.100 --> 00:10:50.600
So let's see, A plus C.

00:10:50.600 --> 00:10:52.490
This is the coefficient
on the s cubed term.

00:10:52.490 --> 00:10:54.850
Do we see any s cubed
terms here?

00:10:54.850 --> 00:10:56.820
No, we see no s cubed
terms here.

00:10:56.820 --> 00:11:02.180
So A plus C-- let me write this
down-- A plus C must be

00:11:02.180 --> 00:11:03.900
equal to 0, because we
see nothing here that

00:11:03.900 --> 00:11:05.990
has an s to the third.

00:11:05.990 --> 00:11:08.690
B plus D is a coefficient
on the s squared term.

00:11:08.690 --> 00:11:10.720
Do we see any s squared
terms here?

00:11:10.720 --> 00:11:13.120
No, so B plus D must
be equal to 0.

00:11:16.350 --> 00:11:23.260
A plus 4C are the coefficient
of the s term.

00:11:23.260 --> 00:11:25.500
I see no s term over here.

00:11:25.500 --> 00:11:35.310
So A plus 4C must also
be equal to 0.

00:11:35.310 --> 00:11:40.000
And then finally, we look at
just the constant terms. And

00:11:40.000 --> 00:11:41.930
we do have a constant term
on the left-hand

00:11:41.930 --> 00:11:42.890
side of this equation.

00:11:42.890 --> 00:11:44.040
We have 2.

00:11:44.040 --> 00:11:48.700
so B plus 4D-- I didn't want to
make it that thick-- B plus

00:11:48.700 --> 00:11:53.295
4D must be equal to 2.

00:11:58.380 --> 00:12:00.420
This just seems like these
linear equations are pretty

00:12:00.420 --> 00:12:01.720
easy to solve for.

00:12:01.720 --> 00:12:06.150
Let's subtract this from this.

00:12:06.150 --> 00:12:08.780
So A-- or let me subtract the
bottom one from the top one--

00:12:08.780 --> 00:12:11.700
so A minus A, that's 0A.

00:12:11.700 --> 00:12:17.290
And then C minus 4C minus
3C is equal to 0.

00:12:17.290 --> 00:12:20.210
And so you get C
is equal to 0.

00:12:20.210 --> 00:12:23.130
If C is equal to 0, A plus C
is equals to 0, A must be

00:12:23.130 --> 00:12:25.810
equal to 0.

00:12:25.810 --> 00:12:27.460
And let's do the same
thing here.

00:12:27.460 --> 00:12:29.890
Let's subtract this from that.

00:12:29.890 --> 00:12:35.610
So you get B minus B is 0, and
then minus 3D-- that's just D

00:12:35.610 --> 00:12:39.900
minus 4D-- and then 0 minus
2 is equal to minus 2.

00:12:39.900 --> 00:12:48.080
And then you get D is equal to--
what do we get?-- D is

00:12:48.080 --> 00:12:50.080
equal to 2/3.

00:12:53.190 --> 00:12:58.670
Minus 2 divided by minus 3 is
2/3, and then-- this isn't a

00:12:58.670 --> 00:13:01.100
minus here, I wrote that there
later-- we said B plus D is

00:13:01.100 --> 00:13:01.900
equal to 0.

00:13:01.900 --> 00:13:04.370
So B must be the opposite
of D, right?

00:13:04.370 --> 00:13:08.580
We could write B is equal
to minus D, or B is

00:13:08.580 --> 00:13:11.150
equal to minus 2/3.

00:13:11.150 --> 00:13:14.510
Let's remember all of this and
go back to our original

00:13:14.510 --> 00:13:17.510
problem, because we've
kind of-- actually

00:13:17.510 --> 00:13:18.660
let me just be clear.

00:13:18.660 --> 00:13:26.110
We can rewrite 2 over s
squared plus 4 times s

00:13:26.110 --> 00:13:28.150
squared plus 1.

00:13:28.150 --> 00:13:32.750
We can rewrite this as, well,
A is 0, B is minus 2/3.

00:13:32.750 --> 00:13:39.710
So this is equal to minus 2/3
over s squared plus 4.

00:13:39.710 --> 00:13:42.450
And then C is 0, we
figured that out.

00:13:42.450 --> 00:13:44.590
And then D is 2/3.

00:13:44.590 --> 00:13:51.530
So plus 2/3 over s
squared plus 1.

00:13:51.530 --> 00:13:55.300
So all of that work that I just
did, that was just to

00:13:55.300 --> 00:13:57.890
break up this piece
right here.

00:13:57.890 --> 00:13:59.640
That was just to break up
that piece right there.

00:13:59.640 --> 00:14:01.320
And then, of course, we have
these other two pieces here

00:14:01.320 --> 00:14:02.820
that we can't forget about.

00:14:02.820 --> 00:14:05.550
So after all of this work,
what do we have?

00:14:05.550 --> 00:14:06.470
And I'm going to make
sure I don't make a

00:14:06.470 --> 00:14:08.390
careless mistake here.

00:14:08.390 --> 00:14:14.390
We get the Laplace Transform
of Y-- as you can see, the

00:14:14.390 --> 00:14:20.670
algebra is the hardest part
here-- is equal to this first

00:14:20.670 --> 00:14:23.930
term-- I'm just going back--
this first term, which I've

00:14:23.930 --> 00:14:27.880
now decomposed into this.

00:14:27.880 --> 00:14:32.850
So it's minus-- let me write it
this way-- minus 1/3-- and

00:14:32.850 --> 00:14:34.820
I think you're going to see in
a second why I'm writing this

00:14:34.820 --> 00:14:45.270
way-- minus 1/3 times 2 over s
squared plus 4, and then plus

00:14:45.270 --> 00:14:51.600
2/3 times 1 over s
squared plus 1.

00:14:51.600 --> 00:14:52.400
And you're probably saying,
Sal, why are you

00:14:52.400 --> 00:14:53.470
writing it this way?

00:14:53.470 --> 00:14:56.250
Well you can already immediately
see that this is

00:14:56.250 --> 00:14:59.030
the Laplace Transform
of sine of 2t.

00:14:59.030 --> 00:15:01.530
This is the Laplace Transform
of sine of t.

00:15:01.530 --> 00:15:03.750
So I wanted to write this 2
here, because this is 2, this

00:15:03.750 --> 00:15:04.370
is 2 squared.

00:15:04.370 --> 00:15:05.860
This is 1, this is 1 squared.

00:15:05.860 --> 00:15:07.760
So I wanted to write
it in this form.

00:15:07.760 --> 00:15:08.910
This was just the first term.

00:15:08.910 --> 00:15:11.400
We have two more terms
to worry about.

00:15:11.400 --> 00:15:12.770
I don't want to make
a careless mistake.

00:15:12.770 --> 00:15:15.710
I have 2s over s
squared plus 1.

00:15:15.710 --> 00:15:18.420
So let me write that down.

00:15:18.420 --> 00:15:31.350
So plus 2 times s over s squared
plus 1, plus-- last

00:15:31.350 --> 00:15:34.730
one-- plus 1 over s
squared plus 1.

00:15:42.050 --> 00:15:45.670
Now we just take the inverse
Laplace Transform of the whole

00:15:45.670 --> 00:15:49.330
thing, and then we'll
know what Y is.

00:15:52.810 --> 00:15:56.080
So, you know, just to remember
the Laplace Transform.

00:15:56.080 --> 00:15:57.345
So this is going to be
a little inverse.

00:15:57.345 --> 00:15:59.210
This is going to
be sine of 2t.

00:15:59.210 --> 00:16:01.410
Let me just write, just so we
have it here, so you know I'm

00:16:01.410 --> 00:16:03.090
not doing some type of voodoo.

00:16:03.090 --> 00:16:09.210
The Laplace Transform of sine
of at is equal to a over s

00:16:09.210 --> 00:16:11.220
squared plus a squared.

00:16:11.220 --> 00:16:17.910
And the Laplace Transform of
cosine of at is equal to s

00:16:17.910 --> 00:16:19.720
over s squared plus a squared.

00:16:19.720 --> 00:16:21.540
Let's just remember those two
things when we take the

00:16:21.540 --> 00:16:25.680
inverse Laplace Transform of
both sides of this equation.

00:16:25.680 --> 00:16:28.140
The inverse Laplace Transform of
the Laplace Transform of y,

00:16:28.140 --> 00:16:30.540
well that's just y.

00:16:30.540 --> 00:16:35.010
y-- maybe I'll write it as a
function of t-- is equal to--

00:16:35.010 --> 00:16:38.210
well this is the Laplace
Transform of sine of 2t.

00:16:38.210 --> 00:16:40.730
You can just do some pattern
matching right here.

00:16:40.730 --> 00:16:43.450
If a is equal to 2, then this
would be the Laplace Transform

00:16:43.450 --> 00:16:44.520
of sine of 2t.

00:16:44.520 --> 00:16:55.800
So it's minus 1/3 times sine of
2t plus 2/3 times-- this is

00:16:55.800 --> 00:16:57.310
the Laplace Transform
of sine of t.

00:16:57.310 --> 00:17:00.100
If you just make a is equal
to 1, sine of t's Laplace

00:17:00.100 --> 00:17:02.440
Transform is 1 over
s squared plus 1.

00:17:02.440 --> 00:17:08.140
So plus 2/3 times the sine of
t-- let me do the next one in

00:17:08.140 --> 00:17:12.430
blue, just because it was
already written in blue-- plus

00:17:12.430 --> 00:17:15.780
2 times-- this is the Laplace
Transform of cosine of t.

00:17:15.780 --> 00:17:19.079
If you make a is equal to 1,
then the cosine t Laplace

00:17:19.079 --> 00:17:21.140
Transform is s over
s squared plus 1.

00:17:21.140 --> 00:17:27.980
So 2 times cosine of t-- and
then one last term-- plus--

00:17:27.980 --> 00:17:30.150
this is just like this one over
here, this is just the

00:17:30.150 --> 00:17:35.470
Laplace Transform of sine
of t-- plus sine of t.

00:17:35.470 --> 00:17:36.570
And we're almost done.

00:17:36.570 --> 00:17:38.270
We're essentially done, but
there's a little bit more

00:17:38.270 --> 00:17:40.290
simplification we can do.

00:17:40.290 --> 00:17:43.820
I have 2/3 times the sign of t
here, and then I have another

00:17:43.820 --> 00:17:48.810
1 sine of t here, so I can
add the 2/3 to the 1.

00:17:48.810 --> 00:17:51.610
What's 2/3 plus 1, or 3/3?

00:17:51.610 --> 00:17:52.610
It's 5/3.

00:17:52.610 --> 00:18:01.270
So I can write y of t is equal
to minus 1/3 sine of 2t plus--

00:18:01.270 --> 00:18:03.430
these two terms I'm just
going to add up--

00:18:03.430 --> 00:18:07.720
plus 5/3 sine of t.

00:18:07.720 --> 00:18:13.350
And then I have this last term
here, plus 2cosine of t.

00:18:13.350 --> 00:18:15.950
So this was a hairy problem,
a lot of work.

00:18:15.950 --> 00:18:18.340
And we saw that the hardest
part really was just the

00:18:18.340 --> 00:18:21.700
partial fraction decomposition
that we did up here, not

00:18:21.700 --> 00:18:23.110
making any careless mistakes.

00:18:23.110 --> 00:18:25.900
But at the end, we got a pretty
neat answer that's not

00:18:25.900 --> 00:18:33.060
too complicated, that satisfies
this non-homogenous

00:18:33.060 --> 00:18:34.420
differential equation.

00:18:34.420 --> 00:18:36.540
We were able to incorporate
the boundary

00:18:36.540 --> 00:18:38.450
conditions as we did it.

00:18:38.450 --> 00:18:43.185
Anyway, hopefully you found
that vaguely satisfying.

00:18:43.185 --> 00:18:45.770
This is a good warm-up
after a year of no

00:18:45.770 --> 00:18:47.020
differential equations.


Viruses

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:03.210
Considering that I have a cold
right now, I can't imagine a

00:00:03.210 --> 00:00:08.109
more appropriate topic to make
a video on than a virus.

00:00:08.109 --> 00:00:09.930
And I didn't want to
make it that thick.

00:00:09.930 --> 00:00:11.380
A virus, or viruses.

00:00:14.760 --> 00:00:19.080
And in my opinion, viruses are,
on some level, the most

00:00:19.080 --> 00:00:21.450
fascinating thing in
all of biology.

00:00:21.450 --> 00:00:25.560
Because they really blur the
boundary between what is an

00:00:25.560 --> 00:00:27.440
inanimate object and
what is life?

00:00:30.080 --> 00:00:33.410
I mean if we look at ourselves,
or life as one of

00:00:33.410 --> 00:00:35.210
those things that you know
it when you see it.

00:00:35.210 --> 00:00:39.300
If you see something that,
it's born, it grows, it's

00:00:39.300 --> 00:00:40.460
constantly changing.

00:00:40.460 --> 00:00:41.510
Maybe it moves around.

00:00:41.510 --> 00:00:42.260
Maybe it doesn't.

00:00:42.260 --> 00:00:44.400
But it's metabolizing things
around itself.

00:00:44.400 --> 00:00:46.310
It reproduces and
then it dies.

00:00:46.310 --> 00:00:48.300
You say, hey, that's
probably life.

00:00:48.300 --> 00:00:50.960
And in this, we throw most
things that we see-- or we

00:00:50.960 --> 00:00:52.815
throw in, us.

00:00:52.815 --> 00:00:55.360
We throw in bacteria.

00:00:55.360 --> 00:00:57.120
We throw in plants.

00:00:57.120 --> 00:01:00.830
I mean, I could-- I'm kind of
butchering the taxonomy system

00:01:00.830 --> 00:01:03.275
here, but we tend to know
life when we see it.

00:01:03.275 --> 00:01:07.480
But all viruses are, they're
just a bunch of genetic

00:01:07.480 --> 00:01:09.740
information inside
of a protein.

00:01:09.740 --> 00:01:11.230
Inside of a protein capsule.

00:01:11.230 --> 00:01:13.490
So let me draw.

00:01:13.490 --> 00:01:16.340
And the genetic information
can come in any form.

00:01:16.340 --> 00:01:22.240
So it can be an RNA, it could
be DNA, it could be

00:01:22.240 --> 00:01:24.670
single-stranded RNA,
double-stranded RNA.

00:01:24.670 --> 00:01:27.260
Sometimes for single stranded
they'll write these two little

00:01:27.260 --> 00:01:29.250
S's in front of it.

00:01:29.250 --> 00:01:30.950
Let's say they are talking about
double stranded DNA,

00:01:30.950 --> 00:01:32.910
they'll put a ds
in front of it.

00:01:32.910 --> 00:01:35.330
But the general idea-- and
viruses can come in all of

00:01:35.330 --> 00:01:38.110
these forms-- is that they have
some genetic information,

00:01:38.110 --> 00:01:40.530
some chain of nucleic acids.

00:01:40.530 --> 00:01:43.730
Either as single or double
stranded RNA or single or

00:01:43.730 --> 00:01:44.560
double stranded DNA.

00:01:44.560 --> 00:01:47.970
And it's just contained inside
some type of protein

00:01:47.970 --> 00:01:49.880
structure, which is
called the capsid.

00:01:49.880 --> 00:01:53.240
And kind of the classic
drawing is kind of an

00:01:53.240 --> 00:01:56.650
icosahedron type
looking thing.

00:01:56.650 --> 00:01:58.200
Let me see if I can
do justice to it.

00:01:58.200 --> 00:02:00.610
It looks something like this.

00:02:00.610 --> 00:02:02.800
And not all viruses have to
look exactly like this.

00:02:02.800 --> 00:02:05.160
There's thousands of
types of viruses.

00:02:05.160 --> 00:02:06.850
And we're really just scratching
the surface and

00:02:06.850 --> 00:02:09.630
understanding even what viruses
are out there and all

00:02:09.630 --> 00:02:14.800
of the different ways that they
can essentially replicate

00:02:14.800 --> 00:02:15.220
themselves.

00:02:15.220 --> 00:02:16.620
We'll talk more about
that in the future.

00:02:16.620 --> 00:02:19.110
And I would suspect that pretty
much any possible way

00:02:19.110 --> 00:02:21.530
of replication probably
does somehow exist

00:02:21.530 --> 00:02:22.750
in the virus world.

00:02:22.750 --> 00:02:25.260
But they really are just these
proteins, these protein

00:02:25.260 --> 00:02:27.450
capsids, are just made up
of a bunch of little

00:02:27.450 --> 00:02:29.070
proteins put together.

00:02:29.070 --> 00:02:31.250
And inside they have some
genetic material, which might

00:02:31.250 --> 00:02:33.700
be DNA or it might be RNA.

00:02:33.700 --> 00:02:35.870
So let me draw their
genetic material.

00:02:35.870 --> 00:02:38.420
The protein is not necessarily
transparent, but if it was,

00:02:38.420 --> 00:02:42.960
you would see some genetic
material inside of there.

00:02:42.960 --> 00:02:48.440
So the question is, is
this thing life?

00:02:48.440 --> 00:02:49.580
It seems pretty inanimate.

00:02:49.580 --> 00:02:50.540
It doesn't grow.

00:02:50.540 --> 00:02:51.510
It doesn't change.

00:02:51.510 --> 00:02:53.200
It doesn't metabolize things.

00:02:53.200 --> 00:02:55.200
This thing, left to its
own devices, is just

00:02:55.200 --> 00:02:55.850
going to sit there.

00:02:55.850 --> 00:02:59.600
It's just going to sit there the
way a book on a table just

00:02:59.600 --> 00:03:00.690
sits there.

00:03:00.690 --> 00:03:02.280
It won't change anything.

00:03:02.280 --> 00:03:06.560
But what happens is,
the debate arises.

00:03:06.560 --> 00:03:08.470
I mean you might say, hey Sal,
when you define it that way,

00:03:08.470 --> 00:03:10.690
just looks like a bunch of
molecules put together.

00:03:10.690 --> 00:03:11.740
That isn't life.

00:03:11.740 --> 00:03:14.350
But it starts to seem like life
all of a sudden when it

00:03:14.350 --> 00:03:17.100
comes in contact with the
things that we normally

00:03:17.100 --> 00:03:18.220
consider life.

00:03:18.220 --> 00:03:22.480
So what viruses do, the classic
example is, a virus

00:03:22.480 --> 00:03:24.680
will attach itself to a cell.

00:03:24.680 --> 00:03:28.590
So let me draw this thing
a little bit smaller.

00:03:28.590 --> 00:03:30.260
So let's say that this
is my virus.

00:03:30.260 --> 00:03:32.562
I'll draw it as a
little hexagon.

00:03:32.562 --> 00:03:34.980
And what it does is, it'll
attach itself to a cell.

00:03:34.980 --> 00:03:35.795
And it could be any
type of cell.

00:03:35.795 --> 00:03:39.155
It could be a bacteria cell, it
could be a plant cell, it

00:03:39.155 --> 00:03:41.190
could be a human cell.

00:03:41.190 --> 00:03:43.820
Let me draw the cell here.

00:03:43.820 --> 00:03:48.540
Cells are usually far larger
than the virus.

00:03:48.540 --> 00:03:51.610
In the case of cells that have
soft membranes, the virus

00:03:51.610 --> 00:03:53.850
figures out some way
to enter it.

00:03:53.850 --> 00:03:58.780
Sometimes it can essentially
fuse-- I don't want to

00:03:58.780 --> 00:04:01.390
complicate the issue-- but
sometimes viruses have their

00:04:01.390 --> 00:04:02.480
own little membranes.

00:04:02.480 --> 00:04:03.740
And we'll talk about
in a second where

00:04:03.740 --> 00:04:04.760
it gets their membranes.

00:04:04.760 --> 00:04:08.270
So a virus might have its
own membrane like that.

00:04:08.270 --> 00:04:10.520
That's around its capsid.

00:04:10.520 --> 00:04:13.470
And then these membranes
will fuse.

00:04:13.470 --> 00:04:16.785
And then the virus will be able
to enter into the cell.

00:04:16.785 --> 00:04:18.000
Now, that's one method.

00:04:18.000 --> 00:04:19.670
And another method,
and they're seldom

00:04:19.670 --> 00:04:20.589
all the same way.

00:04:20.589 --> 00:04:24.020
But let's say another method
would be, the virus

00:04:24.020 --> 00:04:26.910
convinces-- just based on some
protein receptors on it, or

00:04:26.910 --> 00:04:29.810
protein receptors on the cells--
and obviously this has

00:04:29.810 --> 00:04:31.650
to be kind of a Trojan
horse type of thing.

00:04:31.650 --> 00:04:33.600
The cell doesn't want viruses.

00:04:33.600 --> 00:04:36.390
So the virus has to somehow
convince the cell that it's a

00:04:36.390 --> 00:04:38.050
non-foreign particle.

00:04:38.050 --> 00:04:41.770
We could do hundreds of videos
on how viruses work and it's a

00:04:41.770 --> 00:04:43.430
continuing field of research.

00:04:43.430 --> 00:04:47.580
But sometimes you might have a
virus that just gets consumed

00:04:47.580 --> 00:04:48.120
by the cell.

00:04:48.120 --> 00:04:49.930
Maybe the cell just thinks it's
something that it needs

00:04:49.930 --> 00:04:50.960
to consume.

00:04:50.960 --> 00:04:53.130
So the cell wraps around
it like this.

00:04:56.780 --> 00:04:58.930
And these sides will
eventually merge.

00:04:58.930 --> 00:05:01.670
And then the cell and the
virus will go into it.

00:05:01.670 --> 00:05:03.160
This is called endocytosis.

00:05:03.160 --> 00:05:04.020
I'll just talk about that.

00:05:04.020 --> 00:05:06.640
It just brings it into
its cytoplasm.

00:05:06.640 --> 00:05:11.020
It doesn't happen
just to viruses.

00:05:11.020 --> 00:05:12.890
But this is one mechanism
that can enter.

00:05:12.890 --> 00:05:17.320
And then in cases where the cell
in question-- for example

00:05:17.320 --> 00:05:20.280
in the situation with bacteria--
if the cell has a

00:05:20.280 --> 00:05:25.520
very hard shell-- let me
do it in a good color.

00:05:25.520 --> 00:05:29.040
So let's say that this is
a bacteria right here.

00:05:29.040 --> 00:05:30.510
And it has a hard shell.

00:05:30.510 --> 00:05:32.830
The viruses don't even
enter the cell.

00:05:32.830 --> 00:05:36.440
They just hang out outside
of the cell like this.

00:05:36.440 --> 00:05:37.740
Not drawing to scale.

00:05:37.740 --> 00:05:40.690
And they actually inject
their genetic material.

00:05:40.690 --> 00:05:43.790
So there's obviously a huge--
there's a wide variety of ways

00:05:43.790 --> 00:05:45.680
of how the viruses
get into cells.

00:05:45.680 --> 00:05:46.780
But that's beside the point.

00:05:46.780 --> 00:05:49.420
The interesting thing is that
they do get into the cell.

00:05:49.420 --> 00:05:52.240
And once they do get into the
cell, they release their

00:05:52.240 --> 00:05:54.570
genetic material
into the cell.

00:05:54.570 --> 00:05:56.970
So their genetic material
will float around.

00:05:56.970 --> 00:06:01.390
If their genetic material is
already in the form of RNA--

00:06:01.390 --> 00:06:05.700
and I could imagine almost every
possibility of different

00:06:05.700 --> 00:06:08.160
ways for viruses to work
probably do exist in nature.

00:06:08.160 --> 00:06:09.400
We just haven't found them.

00:06:09.400 --> 00:06:11.630
But the ones that we've already
found really do kind

00:06:11.630 --> 00:06:13.880
of do it in every
possible way.

00:06:13.880 --> 00:06:20.720
So if they have RNA, this RNA
can immediately start being

00:06:20.720 --> 00:06:23.170
used to essentially-- let's
say this is the

00:06:23.170 --> 00:06:25.540
nucleus of the cell.

00:06:25.540 --> 00:06:28.850
That's the nucleus of the cell
and it normally has the DNA in

00:06:28.850 --> 00:06:29.480
it like that.

00:06:29.480 --> 00:06:30.720
Maybe I'll do the DNA in
a different color.

00:06:30.720 --> 00:06:37.260
But DNA gets transcribed
into RNA, normally.

00:06:37.260 --> 00:06:39.940
So normally, the cell, this a
normal working cell, the RNA

00:06:39.940 --> 00:06:43.540
exits the nucleus, it goes to
the ribosomes, and then you

00:06:43.540 --> 00:06:46.690
have the RNA in conjunction with
the tRNA and it produces

00:06:46.690 --> 00:06:49.260
these proteins.

00:06:49.260 --> 00:06:51.030
The RNA codes for different
proteins.

00:06:51.030 --> 00:06:53.670
And I talk about that in
a different video.

00:06:53.670 --> 00:06:57.000
So these proteins get formed and
eventually, they can form

00:06:57.000 --> 00:06:58.800
the different structures
in a cell.

00:06:58.800 --> 00:07:03.190
But what a virus does is it
hijacks this process here.

00:07:03.190 --> 00:07:04.490
Hijacks this mechanism.

00:07:04.490 --> 00:07:08.550
This RNA will essentially go and
do what the cell's own RNA

00:07:08.550 --> 00:07:09.230
would have done.

00:07:09.230 --> 00:07:11.140
And it starts coding for
its own proteins.

00:07:11.140 --> 00:07:12.250
Obviously it's not
going to code for

00:07:12.250 --> 00:07:13.260
the same things there.

00:07:13.260 --> 00:07:15.790
And actually some of the first
proteins it codes for often

00:07:15.790 --> 00:07:19.670
start killing the DNA and the
RNA that might otherwise

00:07:19.670 --> 00:07:20.380
compete with it.

00:07:20.380 --> 00:07:22.390
So it codes its own proteins.

00:07:22.390 --> 00:07:26.860
And then those proteins start
making more viral shells.

00:07:26.860 --> 00:07:31.520
So those proteins just start
constructing more and more

00:07:31.520 --> 00:07:33.840
viral shells.

00:07:33.840 --> 00:07:36.000
At the same time, this
RNA is replicating.

00:07:36.000 --> 00:07:39.730
It's using the cell's own
mechanisms. Left to its own

00:07:39.730 --> 00:07:40.820
devices it would
just sit there.

00:07:40.820 --> 00:07:43.800
But once it enters into a cell
it can use all of the nice

00:07:43.800 --> 00:07:47.000
machinery that a cell has around
to replicate itself.

00:07:47.000 --> 00:07:51.360
And it's kind of amazing, just
the biochemistry of it.

00:07:51.360 --> 00:07:54.810
That these RNA molecules
then find themselves

00:07:54.810 --> 00:07:57.470
back in these capsids.

00:07:57.470 --> 00:08:00.290
And then once there's enough
of these and the cell has

00:08:00.290 --> 00:08:05.620
essentially all of its resources
have been depleted,

00:08:05.620 --> 00:08:09.990
the viruses, these individual
new viruses that have

00:08:09.990 --> 00:08:15.410
replicated themselves using all
of the cell's mechanisms,

00:08:15.410 --> 00:08:17.370
will find some way
to exit the cell.

00:08:17.370 --> 00:08:20.860
The most-- I don't want to
say, typical, because we

00:08:20.860 --> 00:08:23.110
haven't even discovered all the
different types of viruses

00:08:23.110 --> 00:08:26.750
there are-- but one that's, I
guess, talked about the most,

00:08:26.750 --> 00:08:28.760
is when there's enough of
these, they'll release

00:08:28.760 --> 00:08:30.880
proteins or they'll construct
proteins.

00:08:30.880 --> 00:08:32.390
Because they don't
make their own.

00:08:32.390 --> 00:08:35.860
That essentially cause the cell
to either kill itself or

00:08:35.860 --> 00:08:37.140
its membrane to dissolve.

00:08:37.140 --> 00:08:39.260
So the membrane dissolves.

00:08:39.260 --> 00:08:41.480
And essentially the
cell lyses.

00:08:41.480 --> 00:08:42.409
Let me write that down.

00:08:42.409 --> 00:08:43.840
The cell lyses.

00:08:43.840 --> 00:08:46.110
And lyses just means that
the cell's membrane just

00:08:46.110 --> 00:08:46.900
disappears.

00:08:46.900 --> 00:08:50.480
And then all of these guys can
emerge for themselves.

00:08:50.480 --> 00:08:53.470
Now I talked about before that
have some of these guys, that

00:08:53.470 --> 00:08:54.750
they have their own membrane.

00:08:54.750 --> 00:08:56.710
So how did they get
there, these

00:08:56.710 --> 00:08:58.400
kind of bilipid membranes?

00:08:58.400 --> 00:09:00.900
Well some of them, what they
do is, once they replicate

00:09:00.900 --> 00:09:07.380
inside of a cell, they exit
maybe not even killing-- they

00:09:07.380 --> 00:09:08.650
don't have to lyse.

00:09:08.650 --> 00:09:11.620
Everything I talk about, these
are specific ways that a virus

00:09:11.620 --> 00:09:12.150
might work.

00:09:12.150 --> 00:09:15.630
But viruses really kind of
explore-- well different types

00:09:15.630 --> 00:09:17.620
of viruses do almost every
different combination you

00:09:17.620 --> 00:09:22.610
could imagine of replicating
and coding for proteins and

00:09:22.610 --> 00:09:23.400
escaping from cells.

00:09:23.400 --> 00:09:24.720
Some of them just bud.

00:09:24.720 --> 00:09:26.560
And when they bud, they
essentially, you can kind of

00:09:26.560 --> 00:09:28.070
imagine that they push
against the cell

00:09:28.070 --> 00:09:29.670
wall, or the membrane.

00:09:29.670 --> 00:09:30.740
I shouldn't say cell wall.

00:09:30.740 --> 00:09:32.630
The cell's outer membrane.

00:09:32.630 --> 00:09:36.130
And then when they push against
it, they take some of

00:09:36.130 --> 00:09:38.070
the membrane with them.

00:09:38.070 --> 00:09:41.260
Until eventually the cell
will-- when this goes up

00:09:41.260 --> 00:09:43.810
enough, this'll pop together
and it'll take some of the

00:09:43.810 --> 00:09:44.565
membrane with it.

00:09:44.565 --> 00:09:47.280
And you could imagine why that
would be useful thing

00:09:47.280 --> 00:09:49.030
to have with you.

00:09:49.030 --> 00:09:51.260
Because now that you have this
membrane, you kind of look

00:09:51.260 --> 00:09:52.520
like this cell.

00:09:52.520 --> 00:09:55.300
So when you want to go infect
another cell like this, you're

00:09:55.300 --> 00:09:58.530
not going to necessarily look
like a foreign particle.

00:09:58.530 --> 00:10:02.780
So it's a very useful way to
look like something that

00:10:02.780 --> 00:10:03.470
you're not.

00:10:03.470 --> 00:10:06.940
And if you don't think that this
is creepy-crawly enough,

00:10:06.940 --> 00:10:10.780
that you're hijacking the DNA
of an organism, viruses can

00:10:10.780 --> 00:10:14.740
actually change the
DNA an organism.

00:10:14.740 --> 00:10:18.830
And actually one of the most
common examples is HIV virus.

00:10:18.830 --> 00:10:21.480
Let me write that down.

00:10:21.480 --> 00:10:27.480
HIV, which is a type of
retrovirus, which is

00:10:27.480 --> 00:10:28.280
fascinating.

00:10:28.280 --> 00:10:31.930
Because what they do is, so
they have RNA in them.

00:10:35.810 --> 00:10:38.030
And when they enter into a cell,
let's say that they got

00:10:38.030 --> 00:10:39.430
into the cell.

00:10:39.430 --> 00:10:42.950
So it's inside of the
cell like this.

00:10:42.950 --> 00:10:46.570
They actually bring along
with them a protein.

00:10:46.570 --> 00:10:49.600
And every time you say, where
do they get this protein?

00:10:49.600 --> 00:10:51.390
All of this stuff came from
a different cell.

00:10:51.390 --> 00:10:54.490
They use some other cell's amino
acids and ribosomes and

00:10:54.490 --> 00:10:56.800
nucleic acids and everything
to build themselves.

00:10:56.800 --> 00:10:58.780
So any proteins that they
have in them came

00:10:58.780 --> 00:11:00.080
from another cell.

00:11:00.080 --> 00:11:04.610
But they bring with them, this
protein reverse transcriptase.

00:11:04.610 --> 00:11:07.850
And the reverse transcriptase
takes their RNA and

00:11:07.850 --> 00:11:11.490
codes it into DNA.

00:11:11.490 --> 00:11:14.790
So its RNA to DNA.

00:11:14.790 --> 00:11:16.810
Which when it was first
discovered was, kind of,

00:11:16.810 --> 00:11:19.950
people always thought that you
always went from DNA to RNA,

00:11:19.950 --> 00:11:22.110
but this kind of broke
that paradigm.

00:11:22.110 --> 00:11:23.850
But it codes from RNA to DNA.

00:11:23.850 --> 00:11:29.020
And if that's not bad enough,
it'll incorporate that DNA

00:11:29.020 --> 00:11:31.350
into the DNA of the host cell.

00:11:31.350 --> 00:11:34.500
So that DNA will incorporate
itself into the

00:11:34.500 --> 00:11:35.200
DNA of the host cell.

00:11:35.200 --> 00:11:37.690
Let's say the yellow is the
DNA of the host cell.

00:11:37.690 --> 00:11:41.000
And this is its nucleus.

00:11:41.000 --> 00:11:44.070
So it actually messes with
the genetic makeup

00:11:44.070 --> 00:11:46.030
of what it's infecting.

00:11:46.030 --> 00:11:51.570
And when I made the videos on
bacteria I said, hey for every

00:11:51.570 --> 00:11:53.790
one human cell we have twenty
bacteria cells.

00:11:53.790 --> 00:11:56.420
And they live with us and
they're useful and they're

00:11:56.420 --> 00:12:00.100
part of us and they're 10% of
our dry mass and all of that.

00:12:00.100 --> 00:12:02.170
But bacteria are kind of
along for the ride.

00:12:02.170 --> 00:12:03.860
They don't change who we are.

00:12:03.860 --> 00:12:07.010
But these retroviruses, they're
actually changing our

00:12:07.010 --> 00:12:07.940
genetic makeup.

00:12:07.940 --> 00:12:10.650
I mean, my genes, I take
very personally.

00:12:10.650 --> 00:12:12.380
They define who I am.

00:12:12.380 --> 00:12:14.630
But these guys will
actually go in and

00:12:14.630 --> 00:12:16.240
change my genetic makeup.

00:12:16.240 --> 00:12:19.300
And then once they're part of
the DNA, then just the natural

00:12:19.300 --> 00:12:25.330
DNA to RNA to protein
process will code

00:12:25.330 --> 00:12:27.180
their actual proteins.

00:12:27.180 --> 00:12:30.160
Or their-- what they need to--
so sometimes they'll lay

00:12:30.160 --> 00:12:31.420
dormant and do nothing.

00:12:31.420 --> 00:12:33.730
And sometimes-- let's say
sometimes in some type of

00:12:33.730 --> 00:12:35.790
environmental trigger,
they'll start coding

00:12:35.790 --> 00:12:37.110
for themselves again.

00:12:37.110 --> 00:12:38.620
And they'll start
producing more.

00:12:38.620 --> 00:12:41.980
But they're producing it
directly from the organism's

00:12:41.980 --> 00:12:43.110
cell's DNA.

00:12:43.110 --> 00:12:45.120
They become part of
the organism.

00:12:45.120 --> 00:12:49.280
I mean I can't imagine a more
intimate way to become part of

00:12:49.280 --> 00:12:51.530
an organism than to become
part of its DNA.

00:12:51.530 --> 00:12:52.870
I can't imagine any
other way to

00:12:52.870 --> 00:12:55.680
actually define an organism.

00:12:55.680 --> 00:13:00.910
And if this by itself is not
eerie enough, and just so you

00:13:00.910 --> 00:13:06.770
know, this notion right here,
when a virus becomes part of

00:13:06.770 --> 00:13:09.430
an organism's DNA, this
is called a provirus.

00:13:13.180 --> 00:13:16.890
But if this isn't eerie enough,
they estimate-- so if

00:13:16.890 --> 00:13:23.500
this infects a cell in my nose
or in my arm, as this cell

00:13:23.500 --> 00:13:27.250
experiences mitosis, all of
its offspring-- but its

00:13:27.250 --> 00:13:29.030
offspring are genetically
identical-- are going to have

00:13:29.030 --> 00:13:30.600
this viral DNA.

00:13:30.600 --> 00:13:32.060
And that might be fine,
but at least my

00:13:32.060 --> 00:13:33.120
children won't get it.

00:13:33.120 --> 00:13:35.980
You know, at least it won't
become part of my species.

00:13:35.980 --> 00:13:38.710
But it doesn't have to just
infect somatic cells, it could

00:13:38.710 --> 00:13:41.170
infect a germ cell.

00:13:41.170 --> 00:13:44.510
So it could go into
a germ cell.

00:13:44.510 --> 00:13:46.850
And the germ cells, we've
learned already, these are the

00:13:46.850 --> 00:13:49.380
ones that produce gametes.

00:13:49.380 --> 00:13:52.930
For men, that's sperm and
for women it's eggs.

00:13:52.930 --> 00:13:55.180
But you could imagine, once
you've infected a germ cell,

00:13:55.180 --> 00:13:59.010
once you become part of a germ
cell's DNA, then I'm passing

00:13:59.010 --> 00:14:02.600
on that viral DNA to my
son or my daughter.

00:14:02.600 --> 00:14:05.540
And they are going to pass
it on to their children.

00:14:05.540 --> 00:14:10.260
And just that idea by itself
is, at least to my mind.

00:14:10.260 --> 00:14:12.130
vaguely creepy.

00:14:12.130 --> 00:14:16.050
And people estimate that 5-8%--
and this kind of really

00:14:16.050 --> 00:14:19.480
blurs, it makes you think about
what we as humans really

00:14:19.480 --> 00:14:26.800
are-- but the estimate is 5-8%
of the human genome-- so when

00:14:26.800 --> 00:14:29.500
I talked about bacteria I just
talked about things that were

00:14:29.500 --> 00:14:30.850
along for the ride.

00:14:30.850 --> 00:14:33.240
But the current estimate, and
I looked up this a lot.

00:14:33.240 --> 00:14:35.210
I found 8% someplace,
5% someplace.

00:14:35.210 --> 00:14:35.990
It's all a guess.

00:14:35.990 --> 00:14:37.650
I mean people are doing it based
on just looking at the

00:14:37.650 --> 00:14:40.690
DNA and how similar it is to
DNA in other organisms. But

00:14:40.690 --> 00:14:47.390
the estimate is 5-8% of the
human genome is from viruses,

00:14:47.390 --> 00:14:51.760
is from ancient retroviruses
that incorporated themselves

00:14:51.760 --> 00:14:53.530
into the human germ line.

00:14:53.530 --> 00:14:55.725
So into the human DNA.

00:14:55.725 --> 00:14:57.860
So these are called endogenous
retroviruses.

00:15:04.800 --> 00:15:07.150
Which is mind blowing to me,
because it's not just saying

00:15:07.150 --> 00:15:09.290
these things are along for the
ride or that they might help

00:15:09.290 --> 00:15:10.070
us or hurt us.

00:15:10.070 --> 00:15:14.000
It's saying that we are--
5-8% of our DNA

00:15:14.000 --> 00:15:16.390
actually comes from viruses.

00:15:16.390 --> 00:15:17.850
And this is another thing
that speaks to

00:15:17.850 --> 00:15:19.720
just genetic variation.

00:15:19.720 --> 00:15:24.210
Because viruses do something--
I mean this is called

00:15:24.210 --> 00:15:27.760
horizontal transfer of DNA.

00:15:27.760 --> 00:15:30.480
And you could imagine, as a
virus goes from one species to

00:15:30.480 --> 00:15:37.220
the next, as it goes from
Species A to B, if it mutates

00:15:37.220 --> 00:15:41.310
to be able to infiltrate these
cells, it might take some--

00:15:41.310 --> 00:15:43.810
it'll take the DNA that
it already has, that

00:15:43.810 --> 00:15:45.490
makes it, it with it.

00:15:45.490 --> 00:15:48.180
But sometimes, when it starts
coding for some of these other

00:15:48.180 --> 00:15:52.830
guys, so let's say that this
is a provirus right here.

00:15:52.830 --> 00:15:55.860
Where the blue part is
the original virus.

00:15:55.860 --> 00:16:00.960
The yellow is the organism's
historic DNA.

00:16:00.960 --> 00:16:04.230
Sometimes when it codes, it
takes up little sections of

00:16:04.230 --> 00:16:07.200
the other organism's DNA.

00:16:07.200 --> 00:16:11.810
So maybe most of it was the
viral DNA, but it might have,

00:16:11.810 --> 00:16:14.070
when it transcribed and
translated itself, it might

00:16:14.070 --> 00:16:16.710
have taken a little bit-- or at
least when it translated or

00:16:16.710 --> 00:16:18.460
replicated itself-- it might
take a little bit of the

00:16:18.460 --> 00:16:20.540
organism's previous DNA.

00:16:20.540 --> 00:16:23.690
So it's actually cutting parts
of DNA from one organism and

00:16:23.690 --> 00:16:24.950
bringing it to another
organism.

00:16:24.950 --> 00:16:27.490
Taking it from one member of a
species to another member of

00:16:27.490 --> 00:16:27.810
the species.

00:16:27.810 --> 00:16:29.750
But it can definitely
go cross-species.

00:16:29.750 --> 00:16:32.860
So you have this idea all of
a sudden that DNA can jump

00:16:32.860 --> 00:16:34.090
between species.

00:16:34.090 --> 00:16:36.410
It really kind of-- I don't
know, for me it makes me

00:16:36.410 --> 00:16:39.930
appreciate how interconnected--
as a species,

00:16:39.930 --> 00:16:41.950
we kind of imagine that we're
by ourselves and can only

00:16:41.950 --> 00:16:45.110
reproduce with each other and
have genetic variation within

00:16:45.110 --> 00:16:45.860
a population.

00:16:45.860 --> 00:16:50.190
But viruses introduce this
notion of horizontal transfer

00:16:50.190 --> 00:16:53.090
via transduction.

00:16:53.090 --> 00:16:57.640
Horizontal transduction is just
the idea of, look when I

00:16:57.640 --> 00:17:00.660
replicate this virus, I might
take a little bit of the

00:17:00.660 --> 00:17:04.349
organism that I'm freeloading
off of, I might take a little

00:17:04.349 --> 00:17:06.010
bit of their DNA with me.

00:17:06.010 --> 00:17:09.160
And infect that DNA into
the next organism.

00:17:09.160 --> 00:17:11.329
So you actually have this
DNA, this jumping,

00:17:11.329 --> 00:17:13.119
from organism to organism.

00:17:13.119 --> 00:17:17.300
So it kind of unifies
all DNA-based life.

00:17:17.300 --> 00:17:20.609
Which is all the life that
we know on the planet.

00:17:20.609 --> 00:17:24.900
And if all of this isn't creepy
enough-- and actually

00:17:24.900 --> 00:17:28.540
maybe I'll save the creepiest
part for the end.

00:17:28.540 --> 00:17:30.800
But there's a whole-- we could
talk all about the different

00:17:30.800 --> 00:17:31.670
classes of viruses.

00:17:31.670 --> 00:17:33.810
But just so you're familiar with
some of the terminology,

00:17:33.810 --> 00:17:37.895
when a virus attacks bacteria,
which they often do.

00:17:37.895 --> 00:17:41.150
And we study these the most
because this might be a good

00:17:41.150 --> 00:17:42.630
alternative to antibiotics.

00:17:42.630 --> 00:17:45.430
Because viruses that attack
bacteria might-- sometimes the

00:17:45.430 --> 00:17:47.820
bacteria is far worse for the
virus-- but these are called

00:17:47.820 --> 00:17:49.070
bacteriaphages.

00:17:52.610 --> 00:17:55.320
And I've already talked to you
about how they have their DNA.

00:17:55.320 --> 00:17:57.890
But since bacteria have hard
walls, they will just inject

00:17:57.890 --> 00:18:00.940
the DNA inside of
the bacteria.

00:18:00.940 --> 00:18:06.630
And when you talk about DNA,
this idea of a provirus.

00:18:06.630 --> 00:18:09.170
So when a virus lyses it
like this, this is

00:18:09.170 --> 00:18:11.090
called the lytic cycle.

00:18:11.090 --> 00:18:13.760
This is just some terminology
that's good to know if you're

00:18:13.760 --> 00:18:16.250
going to take a biology
exam about this stuff.

00:18:16.250 --> 00:18:18.600
And when the virus incorporates
it into the DNA

00:18:18.600 --> 00:18:22.350
and lays dormant, incorporates
into the DNA of the host

00:18:22.350 --> 00:18:25.420
organism and lays dormant for
awhile, this is called the

00:18:25.420 --> 00:18:28.030
lysogenic cycle.

00:18:28.030 --> 00:18:32.030
And normally, a provirus is
essentially experiencing a

00:18:32.030 --> 00:18:35.910
lysogenic cycle in eurkaryotes,
in organisms that

00:18:35.910 --> 00:18:38.250
have a nuclear membrane.

00:18:38.250 --> 00:18:40.890
Normally when people talk about
the lysogenic cycle,

00:18:40.890 --> 00:18:44.750
they're talking about viral DNA
laying dormant in the DNA

00:18:44.750 --> 00:18:45.600
of bacteria.

00:18:45.600 --> 00:18:48.600
Or bacteriophage DNA
laying dormant

00:18:48.600 --> 00:18:51.050
in the DNA of bacteria.

00:18:51.050 --> 00:18:53.380
But just to kind of give you
an idea of what this, quote

00:18:53.380 --> 00:18:59.480
unquote, looks like,
right here.

00:18:59.480 --> 00:19:01.550
I got these two pictures
from Wikipedia.

00:19:01.550 --> 00:19:02.800
One is from the CDC.

00:19:11.360 --> 00:19:14.200
These little green dots you see
right here all over the

00:19:14.200 --> 00:19:17.540
surface, this big thing you
see here, this is a white

00:19:17.540 --> 00:19:20.330
blood cell.

00:19:20.330 --> 00:19:21.760
Part of the human
immune system.

00:19:21.760 --> 00:19:23.050
This is a white blood cell.

00:19:27.540 --> 00:19:31.060
And what you see emerging from
the surface, essentially

00:19:31.060 --> 00:19:34.510
budding from the surface of this
white blood cell-- and

00:19:34.510 --> 00:19:36.300
this gives you a sense
of scale too--

00:19:36.300 --> 00:19:39.700
these are HIV-1 viruses.

00:19:39.700 --> 00:19:46.970
And so you're familiar with the
terminology, the HIV is a

00:19:46.970 --> 00:19:51.810
virus that infects white
blood cells.

00:19:51.810 --> 00:19:54.890
AIDS is the syndrome you get
once your immune system is

00:19:54.890 --> 00:19:55.900
weakened to the point.

00:19:55.900 --> 00:19:58.860
And then many people suffer
infections that people with a

00:19:58.860 --> 00:20:01.450
strong immune system normally
won't suffer from.

00:20:01.450 --> 00:20:03.050
But this is creepy.

00:20:03.050 --> 00:20:06.360
These things went inside this
huge cell, they used the

00:20:06.360 --> 00:20:11.555
cell's own mechanism to
reproduce its own DNA or its

00:20:11.555 --> 00:20:14.300
own RNA and these
protein capsids.

00:20:14.300 --> 00:20:16.270
And then they bud from the cell
and take a little bit of

00:20:16.270 --> 00:20:17.490
the membrane with it.

00:20:17.490 --> 00:20:20.990
And they can even leave some
of their DNA behind in this

00:20:20.990 --> 00:20:22.340
cell's own DNA.

00:20:22.340 --> 00:20:25.280
So they really change what
the cell is all about.

00:20:25.280 --> 00:20:26.890
This is another creepy
picture.

00:20:26.890 --> 00:20:28.140
These are bacteriaphages.

00:20:34.500 --> 00:20:36.110
And these show you what
I said before.

00:20:36.110 --> 00:20:38.450
This is a bacteria right here.

00:20:38.450 --> 00:20:41.360
This is its cell wall.

00:20:41.360 --> 00:20:42.100
And it's hard.

00:20:42.100 --> 00:20:45.020
So it's hard to just
emerge into it.

00:20:45.020 --> 00:20:47.040
Or you can't just merge,
fuse membranes with it.

00:20:47.040 --> 00:20:52.290
So they hang out on the outside
of this bacteria.

00:20:52.290 --> 00:20:55.410
And they are essentially
injecting their genetic

00:20:55.410 --> 00:20:58.280
material into the
bacteria itself.

00:20:58.280 --> 00:20:59.890
And you could imagine,
just looking at the

00:20:59.890 --> 00:21:01.000
size of these things.

00:21:01.000 --> 00:21:02.090
I mean, this is a cell.

00:21:02.090 --> 00:21:04.350
And it looks like a whole
planet or something.

00:21:04.350 --> 00:21:05.650
Or this is a bacteria and these

00:21:05.650 --> 00:21:06.620
things are so much smaller.

00:21:06.620 --> 00:21:08.170
Roughly 1/100 of a bacteria.

00:21:08.170 --> 00:21:10.570
And these are much less than
1/100 of this cell we're

00:21:10.570 --> 00:21:11.110
talking about.

00:21:11.110 --> 00:21:15.510
And they're extremely
hard to filter for.

00:21:15.510 --> 00:21:16.520
To kind of keep out.

00:21:16.520 --> 00:21:20.780
Because they are such,
such small particles.

00:21:20.780 --> 00:21:23.290
If you think that these are
exotic things that exist for

00:21:23.290 --> 00:21:31.360
things like HIV or Ebola , which
they do cause, or SARS,

00:21:31.360 --> 00:21:32.340
you're right.

00:21:32.340 --> 00:21:33.730
But they're also
common things.

00:21:33.730 --> 00:21:35.490
I mean, I said at the beginning
of this video that I

00:21:35.490 --> 00:21:36.300
have a cold.

00:21:36.300 --> 00:21:40.040
And I have a cold because some
viruses have infected the

00:21:40.040 --> 00:21:42.610
tissue in my nasal passage.

00:21:42.610 --> 00:21:46.190
And they're causing me to have
a runny nose and whatnot.

00:21:46.190 --> 00:21:48.540
And viruses also cause
the chicken pox.

00:21:48.540 --> 00:21:50.550
They cause the herpes
simplex virus.

00:21:50.550 --> 00:21:51.540
Causes cold sores.

00:21:51.540 --> 00:21:53.430
So they're with us all around.

00:21:53.430 --> 00:21:56.000
I can almost guarantee
you have some virus

00:21:56.000 --> 00:21:57.740
with you as you speak.

00:21:57.740 --> 00:21:58.630
They're all around you.

00:21:58.630 --> 00:22:04.320
But it's a very

00:22:04.320 --> 00:22:06.480
philosophically puzzling question.

00:22:06.480 --> 00:22:09.090
Because I started with, at the
beginning, are these life?

00:22:09.090 --> 00:22:11.495
And at first when I just showed
it to you, look they

00:22:11.495 --> 00:22:13.740
are just this protein
with some nucleic

00:22:13.740 --> 00:22:15.280
acid molecule in it.

00:22:15.280 --> 00:22:16.340
And it's not doing anything.

00:22:16.340 --> 00:22:17.780
And that doesn't look
like life to me.

00:22:17.780 --> 00:22:18.610
It's not moving around.

00:22:18.610 --> 00:22:19.660
It doesn't have a metabolism.

00:22:19.660 --> 00:22:20.390
It's not eating.

00:22:20.390 --> 00:22:21.560
It's not reproducing.

00:22:21.560 --> 00:22:23.120
But then all of a sudden, when
you think about what it's

00:22:23.120 --> 00:22:27.130
doing to cells and how it uses
cells to kind of reproduce.

00:22:27.130 --> 00:22:29.700
It kind of like-- in business
terms it's asset light.

00:22:29.700 --> 00:22:31.920
It doesn't need all of the
machinery because it can use

00:22:31.920 --> 00:22:35.850
other people's machinery
to replicate itself.

00:22:35.850 --> 00:22:38.080
You almost kind of want
to view it as a

00:22:38.080 --> 00:22:39.520
smarter form of life.

00:22:39.520 --> 00:22:42.480
Because it doesn't go through
all of the trouble of what

00:22:42.480 --> 00:22:44.110
every other form of life has.

00:22:44.110 --> 00:22:47.570
It makes you question what life
is, or even what we are.

00:22:47.570 --> 00:22:51.510
Are we these things that contain
DNA or are we just

00:22:51.510 --> 00:22:54.170
transport mechanisms
for the DNA?

00:22:54.170 --> 00:22:56.260
And these are kind of the
more important things.

00:22:56.260 --> 00:22:59.710
And these viral infections are
just battles between different

00:22:59.710 --> 00:23:02.340
forms of DNA and RNA
and whatnot.

00:23:02.340 --> 00:23:04.330
Anyway I don't want to get
too philosophical on you.

00:23:04.330 --> 00:23:08.840
But hopefully this gives you a
good idea of what viruses are

00:23:08.840 --> 00:23:12.940
and why they really are, in my
mind, the most fascinating

00:23:12.940 --> 00:23:15.870
pseudo organism in
all of biology.


Proofs of derivatives of ln(x) and e^x

WEBVTT
Kind: captions
Language: en

00:00:00.370 --> 00:00:03.450
In the first version of the
video of the proof of the

00:00:03.450 --> 00:00:10.860
derivative of the natural
log of x, where the first

00:00:10.860 --> 00:00:13.180
time I proved this is
a couple of years ago.

00:00:13.180 --> 00:00:16.960
And the very next video I
proved that the derivative

00:00:16.960 --> 00:00:19.640
of e to the x is
equal to e to the x.

00:00:19.640 --> 00:00:23.410
I've been charged with some of
making a circular proof, and

00:00:23.410 --> 00:00:26.160
I'm pretty convinced that
my proof wasn't circular.

00:00:26.160 --> 00:00:28.660
So what I want to do in this
video, now that I have a little

00:00:28.660 --> 00:00:31.480
bit more space to work with, a
little bit more sophisticated

00:00:31.480 --> 00:00:34.280
tools, I'm going to redo the
proof and I'm going to do these

00:00:34.280 --> 00:00:38.910
in the same video to show you
at no point do I assume this

00:00:38.910 --> 00:00:40.920
before I actually show it.

00:00:40.920 --> 00:00:42.330
So let's start with the proof.

00:00:42.330 --> 00:00:45.410
So the first thing I need to do
is prove this thing up here.

00:00:45.410 --> 00:00:46.320
I want to keep track of this.

00:00:46.320 --> 00:00:49.730
I don't assume this until
I actually show it.

00:00:49.730 --> 00:00:52.370
So let's start with the
proof, the derivative of

00:00:52.370 --> 00:00:54.310
the natural log of x.

00:00:54.310 --> 00:00:57.730
So the derivative of the
natural log of x, we can

00:00:57.730 --> 00:01:01.580
just to go to the basic
definition of a derivative.

00:01:01.580 --> 00:01:07.940
It's equal to the limit as
delta x approaches 0 of the

00:01:07.940 --> 00:01:15.170
natural log of x plus delta x
minus the natural log of x.

00:01:15.170 --> 00:01:19.880
All of that over delta x.

00:01:19.880 --> 00:01:22.150
Now we can just use the
property of logarithms.

00:01:22.150 --> 00:01:25.100
If I have the log of a minus
the log of b, that's the same

00:01:25.100 --> 00:01:27.900
thing as a log of a over b.

00:01:27.900 --> 00:01:29.800
So let me re-write it that way.

00:01:29.800 --> 00:01:36.490
So this is going to be equal
to the limit as delta

00:01:36.490 --> 00:01:38.900
x approaches 0.

00:01:38.900 --> 00:01:40.900
I could take this 1 over
delta x right here.

00:01:40.900 --> 00:01:48.370
1 over delta x times the
natural log of x plus delta

00:01:48.370 --> 00:01:50.590
x divided by this x.

00:01:50.590 --> 00:01:53.810
Just doing the logarithm
properties right there.

00:01:53.810 --> 00:01:55.830
Then I can re-write this --
first of all, when I have this

00:01:55.830 --> 00:01:58.160
coefficient in front of a
logarithm I can make

00:01:58.160 --> 00:01:59.480
this the exponent.

00:01:59.480 --> 00:02:01.190
And then I can simplify
this in here.

00:02:01.190 --> 00:02:06.940
So this is going to be equal to
the limit as delta x approaches

00:02:06.940 --> 00:02:11.870
0 of the natural log -- let
me do this in a new color.

00:02:11.870 --> 00:02:14.380
Let me do it in a
completely new color.

00:02:14.380 --> 00:02:18.660
The natural log of -- the
inside here I'll just

00:02:18.660 --> 00:02:20.020
divide everything by x.

00:02:20.020 --> 00:02:22.390
So x divided by x is 1.

00:02:22.390 --> 00:02:26.600
Then plus delta x over x.

00:02:26.600 --> 00:02:30.310
Then I had this 1 over delta x
sitting out here, and I can

00:02:30.310 --> 00:02:31.490
make that the exponent.

00:02:31.490 --> 00:02:34.030
That's just an exponent
rule right there, or

00:02:34.030 --> 00:02:35.440
a logarithm property.

00:02:35.440 --> 00:02:38.810
1 over delta x.

00:02:38.810 --> 00:02:40.100
Now I'm going to make
a substitution.

00:02:40.100 --> 00:02:43.910
Remember, all of this, this was
all just from my definition

00:02:43.910 --> 00:02:44.500
of a derivative.

00:02:44.500 --> 00:02:46.130
This was all equal to
the derivative of the

00:02:46.130 --> 00:02:47.740
natural log of x.

00:02:47.740 --> 00:02:51.250
I have still yet to
in any way use this.

00:02:51.250 --> 00:02:54.550
And I won't use that until
I actually show it to you.

00:02:54.550 --> 00:02:57.600
I've become very defensive
about these claims

00:02:57.600 --> 00:03:00.180
of circularity.

00:03:00.180 --> 00:03:03.040
They're my fault because that
shows that I wasn't clear

00:03:03.040 --> 00:03:05.950
enough in my earlier versions
of these proofs, so I'll try

00:03:05.950 --> 00:03:07.140
to be more clear this time.

00:03:07.140 --> 00:03:09.120
So let's see if we can
simplify this into terms

00:03:09.120 --> 00:03:10.840
that we recognize.

00:03:10.840 --> 00:03:15.860
Let's make the substitution so
that we can get e in maybe

00:03:15.860 --> 00:03:17.790
terms that we recognize.

00:03:17.790 --> 00:03:26.830
Let's make the substitute delta
x over x is equal to 1 over n.

00:03:26.830 --> 00:03:28.490
If we multiply -- this
is the same thing.

00:03:28.490 --> 00:03:29.890
This is the equivalent
to substitution.

00:03:29.890 --> 00:03:32.910
If we multiply both sides of
this by x, as saying that

00:03:32.910 --> 00:03:36.160
delta x is equal to x over n.

00:03:36.160 --> 00:03:37.200
These are equivalent
statements.

00:03:37.200 --> 00:03:40.490
I just multiplied both
sides by x here.

00:03:40.490 --> 00:03:46.480
Now if we take the limit as n
approaches infinity of this

00:03:46.480 --> 00:03:49.780
term right here, that's
equivalent -- that's completely

00:03:49.780 --> 00:03:53.670
equivalent to taking the limit
as delta x approaches 0.

00:03:56.680 --> 00:03:59.890
If we're defining delta x to be
this thing, and we take the

00:03:59.890 --> 00:04:02.590
limit as its denominator
approaches 0, we're going

00:04:02.590 --> 00:04:05.120
to make delta x go to 0.

00:04:05.120 --> 00:04:06.520
So let's make that
substitution.

00:04:06.520 --> 00:04:12.810
So all of this is going to be
equal to the limit as -- now

00:04:12.810 --> 00:04:14.275
we've gotten rid
of our delta x.

00:04:14.275 --> 00:04:19.000
We're going to say the limit as
an approaches infinity of the

00:04:19.000 --> 00:04:23.380
natural log -- I'll go back to
that mauve color -- the natural

00:04:23.380 --> 00:04:28.900
log of 1 plus -- now, I said
that instead of delta x over x,

00:04:28.900 --> 00:04:31.580
I made the substitution that
that is equal to 1 over n.

00:04:31.580 --> 00:04:35.010
So that's 1 plus 1 over n.

00:04:35.010 --> 00:04:37.860
And then what's 1 over delta x?

00:04:37.860 --> 00:04:41.690
Well delta x is equal to x over
n, so 1 over delta x is going

00:04:41.690 --> 00:04:42.875
to be the inverse of this.

00:04:42.875 --> 00:04:47.140
It's going to be n over x.

00:04:47.140 --> 00:04:50.700
And then we can re-write this
expression right here --

00:04:50.700 --> 00:04:52.670
let me re-write it again.

00:04:52.670 --> 00:04:58.600
This is equal to the limit as
n approaches infinity of the

00:04:58.600 --> 00:05:02.730
natural log of 1 plus 1 over n.

00:05:02.730 --> 00:05:05.610
What I can do is I can separate
out this n from the 1 over x.

00:05:05.610 --> 00:05:08.130
I could say this is to
the n, and then all of

00:05:08.130 --> 00:05:11.200
this to the 1 over x.

00:05:11.200 --> 00:05:13.730
Once again, this is just
an exponent property.

00:05:13.730 --> 00:05:16.290
If I raise something to the n
and then to the 1 over x, I

00:05:16.290 --> 00:05:18.110
could just multiply
the exponent to get

00:05:18.110 --> 00:05:19.660
to the n over x.

00:05:19.660 --> 00:05:21.660
So these two statements
are equivalent.

00:05:21.660 --> 00:05:24.720
But now we can use logarithm
properties to say hey, if this

00:05:24.720 --> 00:05:26.730
is the exponent, I can just
stick it out in front of the

00:05:26.730 --> 00:05:29.100
coefficient right here.

00:05:29.100 --> 00:05:32.300
I could put it out right there.

00:05:32.300 --> 00:05:34.830
And just remember, this was all
of the derivative with respect

00:05:34.830 --> 00:05:37.710
to x of the natural log of x.

00:05:37.710 --> 00:05:38.680
So what is that equal to?

00:05:38.680 --> 00:05:41.340
We could put this 1 out
of x in the front here.

00:05:41.340 --> 00:05:44.040
In fact, that 1 out of x term,
it has nothing to do with n.

00:05:44.040 --> 00:05:45.550
It's kind of a constant
term when you think

00:05:45.550 --> 00:05:46.290
of it in terms of n.

00:05:46.290 --> 00:05:49.000
So we can actually put it
all the way out here.

00:05:49.000 --> 00:05:50.060
We could put it either place.

00:05:50.060 --> 00:05:55.490
So we could say 1 over x times
all that stuff in mauve.

00:05:55.490 --> 00:06:03.400
The limit as n approaches
infinity of the natural log

00:06:03.400 --> 00:06:08.680
of 1 plus 1 over n to the n.

00:06:08.680 --> 00:06:10.560
The natural log of
all of that stuff.

00:06:10.560 --> 00:06:14.670
Or, just to make the point
clear, we can re-write this

00:06:14.670 --> 00:06:19.550
part -- let me make a salmon
color -- equal to 1 over x

00:06:19.550 --> 00:06:27.420
times the natural log of the
limit as n approaches infinity.

00:06:27.420 --> 00:06:30.015
I'm just switching places here,
because obviously what we care

00:06:30.015 --> 00:06:33.830
is what happens to this term as
it approaches infinity, of

00:06:33.830 --> 00:06:37.550
1 plus 1 over n to the n.

00:06:37.550 --> 00:06:39.910
Well what is -- this should
look a little familiar to you

00:06:39.910 --> 00:06:42.760
on some of the first videos
where we talked about e -- this

00:06:42.760 --> 00:06:45.020
is one of the definitions of e.

00:06:45.020 --> 00:06:46.030
e is defined.

00:06:46.030 --> 00:06:46.880
I'm just being clear here.

00:06:46.880 --> 00:06:50.650
I'm still not using
this at all.

00:06:50.650 --> 00:06:55.020
I'm just stating that the
definition of e, e is equal

00:06:55.020 --> 00:07:00.760
to the limit as n approaches
infinity of 1 plus

00:07:00.760 --> 00:07:02.750
1 over n to the n.

00:07:02.750 --> 00:07:04.490
This is just the
definition of e.

00:07:04.490 --> 00:07:07.330
And natural log is defined
to be the logarithm

00:07:07.330 --> 00:07:09.670
of base, this thing.

00:07:09.670 --> 00:07:12.890
So this thing is e.

00:07:12.890 --> 00:07:14.510
So I'm saying that the
derivative of the natural log

00:07:14.510 --> 00:07:18.680
of x is equal to 1 over x
times the natural log.

00:07:18.680 --> 00:07:19.860
This thing right here is e.

00:07:19.860 --> 00:07:22.050
That's what the
definition of e is.

00:07:22.050 --> 00:07:24.800
I'm not using the definition
of the derivative e, or

00:07:24.800 --> 00:07:28.360
the definition of the
derivative of e to the x.

00:07:28.360 --> 00:07:30.450
I'm just using the
definition of e.

00:07:30.450 --> 00:07:35.120
And the definition of
natural log is log base e.

00:07:35.120 --> 00:07:38.370
This says the power that you
have to raise e to to get to e,

00:07:38.370 --> 00:07:40.870
well this is just equal to 1.

00:07:40.870 --> 00:07:43.620
There we get that the
derivative of the natural log

00:07:43.620 --> 00:07:47.400
of x is equal to 1 over x.

00:07:47.400 --> 00:07:52.110
So, so far I think you'll be
satisfied that we've proven

00:07:52.110 --> 00:07:56.450
this first statement up here,
and in no way did we use

00:07:56.450 --> 00:07:58.020
this statement right here.

00:07:58.020 --> 00:08:01.540
I just used the definition
of e, but that's fine.

00:08:01.540 --> 00:08:04.420
I mean we assumed we know the
definition of e, even when we

00:08:04.420 --> 00:08:07.070
just talk about natural log,
we assume that it's base e.

00:08:07.070 --> 00:08:10.820
In no way did I assume
this to begin with.

00:08:10.820 --> 00:08:13.760
Now, given that we've shown
this and we didn't assume

00:08:13.760 --> 00:08:16.750
this at all, let's see
if we can show this.

00:08:16.750 --> 00:08:22.270
So the derivative -- let's do a
little bit of an exercise here.

00:08:26.610 --> 00:08:30.170
Actually, I could probably
do it in the margins.

00:08:30.170 --> 00:08:34.220
Let's take the derivative
of this function.

00:08:34.220 --> 00:08:37.246
The natural log of e to the x.

00:08:40.250 --> 00:08:42.140
So there's two ways we
can approach this.

00:08:42.140 --> 00:08:45.210
The first way we could simplify
this and we could say this is

00:08:45.210 --> 00:08:47.520
the exact same thing
as the derivative.

00:08:47.520 --> 00:08:51.740
We could put this x out
front of x times the

00:08:51.740 --> 00:08:53.820
natural log of e.

00:08:53.820 --> 00:08:55.910
And what's the
natural log of e?

00:08:55.910 --> 00:09:01.460
The natural log of e we
already know is equal to 1.

00:09:01.460 --> 00:09:03.930
So this is just the
derivative of x.

00:09:03.930 --> 00:09:08.030
And the derivative
of x is equal to 1.

00:09:08.030 --> 00:09:09.190
So that's pretty
straightforward.

00:09:09.190 --> 00:09:13.540
The derivative, in no way did
we assume this to begin with.

00:09:13.540 --> 00:09:16.140
We just simplified this
expression to just this is the

00:09:16.140 --> 00:09:18.260
same thing as the derivative
of x, because this

00:09:18.260 --> 00:09:19.660
term cancels out.

00:09:19.660 --> 00:09:21.600
And the derivative
of x is just 1.

00:09:21.600 --> 00:09:23.210
Or we could do it
the other way.

00:09:23.210 --> 00:09:25.380
We could do the chain rule.

00:09:25.380 --> 00:09:29.640
We could say that this could be
viewed as the derivative of

00:09:29.640 --> 00:09:33.960
this inner function, of this
inner expression, so the

00:09:33.960 --> 00:09:35.180
derivative of the inner
expression, I don't

00:09:35.180 --> 00:09:35.970
know what that is.

00:09:35.970 --> 00:09:37.660
I'm not assuming
anything about it.

00:09:37.660 --> 00:09:38.720
I just don't know what it is.

00:09:38.720 --> 00:09:40.410
So I'll write it in
yellow right here.

00:09:40.410 --> 00:09:42.650
So it's equal to the
derivative with respect

00:09:42.650 --> 00:09:45.080
to x of e to the x.

00:09:45.080 --> 00:09:46.080
I don't know what this is.

00:09:46.080 --> 00:09:48.350
I have no clue what this
is, and I haven't assumed

00:09:48.350 --> 00:09:49.430
anything about what it is.

00:09:49.430 --> 00:09:51.150
I'm just using the chain rule.

00:09:51.150 --> 00:09:54.070
If the derivative of this
inside function with respect to

00:09:54.070 --> 00:09:57.920
x, which is this right here,
times the derivative of this

00:09:57.920 --> 00:10:01.410
outside function with respect
to the inside function.

00:10:01.410 --> 00:10:04.080
So the derivative of natural
log of x with respect

00:10:04.080 --> 00:10:05.730
to x is 1 over x.

00:10:05.730 --> 00:10:08.160
So the derivative of natural
log of anything with

00:10:08.160 --> 00:10:10.310
respect to anything is
1 over that anything.

00:10:10.310 --> 00:10:13.710
So it's going to be equal to --
so the derivative of natural

00:10:13.710 --> 00:10:15.850
log of x with respect to e to
the x is equal to 1

00:10:15.850 --> 00:10:19.190
over e to the x.

00:10:19.190 --> 00:10:23.860
Once again, I in no way
assumed this right here.

00:10:23.860 --> 00:10:26.920
So far in anything we've done,
we haven't assumed that.

00:10:26.920 --> 00:10:30.580
But clearly, my derivatives,
either way I solve it -- one

00:10:30.580 --> 00:10:32.210
way I solve it I got 1.

00:10:32.210 --> 00:10:34.650
The other way, I kind
of didn't solve it.

00:10:34.650 --> 00:10:37.030
I got this expression
right here.

00:10:37.030 --> 00:10:39.226
They must be equal
to each other.

00:10:39.226 --> 00:10:40.760
So let me write that down.

00:10:40.760 --> 00:10:42.330
This must be equal to that.

00:10:42.330 --> 00:10:44.260
It's just we just looked at it
two different ways and got

00:10:44.260 --> 00:10:45.620
two different results.

00:10:45.620 --> 00:10:47.340
But I still don't know
what this thing is.

00:10:47.340 --> 00:10:49.400
I just left it kind of open.

00:10:49.400 --> 00:10:51.010
I just said whatever
the derivative of e to

00:10:51.010 --> 00:10:52.700
the x happens to be.

00:10:52.700 --> 00:10:56.480
But we know, since these two
expressions are equal, we know

00:10:56.480 --> 00:11:01.150
that the derivative with
respect to x of whatever e to

00:11:01.150 --> 00:11:03.635
the x -- so whatever the
derivative with respect to x of

00:11:03.635 --> 00:11:07.140
e to the x happens to be, we
know that when we multiply that

00:11:07.140 --> 00:11:11.340
times 1 over e to the x --
that's when we just did the

00:11:11.340 --> 00:11:13.960
chain rule -- that we should
get the same result as when we

00:11:13.960 --> 00:11:15.930
approached the problem
the other way.

00:11:15.930 --> 00:11:18.840
That should be equal to this
approach because they're both

00:11:18.840 --> 00:11:21.070
different ways of looking
at the derivative of the

00:11:21.070 --> 00:11:22.530
natural log of e to the x.

00:11:22.530 --> 00:11:24.990
So that should be equal to 1.

00:11:24.990 --> 00:11:27.090
Well, we're almost there.

00:11:27.090 --> 00:11:29.650
We could just simplify this
and solve for our mystery

00:11:29.650 --> 00:11:31.250
derivative of e to the x.

00:11:31.250 --> 00:11:34.520
Multiply both sides of this
equation by e to the x, and you

00:11:34.520 --> 00:11:38.890
get the derivative with respect
to x of e to the x is

00:11:38.890 --> 00:11:41.630
equal to e to the x.

00:11:41.630 --> 00:11:43.790
And I want to clarify this.

00:11:43.790 --> 00:11:47.650
At no point in this entire
proof, at no point

00:11:47.650 --> 00:11:50.300
did I assume this.

00:11:50.300 --> 00:11:52.090
In fact, this is the
first time that I'm even

00:11:52.090 --> 00:11:54.880
making the statement.

00:11:54.880 --> 00:11:58.080
I didn't have to assume this
when I showed you that the

00:11:58.080 --> 00:12:00.850
derivative of the natural
log of x is 1 over x.

00:12:00.850 --> 00:12:04.580
And I didn't have to assume
this to kind of get to it.

00:12:04.580 --> 00:12:07.320
So in no way is this
proof circular.

00:12:07.320 --> 00:12:10.520
So anyway, I didn't want to
appear defensive, but I

00:12:10.520 --> 00:12:12.080
wanted to clarify this up.

00:12:12.080 --> 00:12:17.080
Because I don't want to in any
way blame those who think that

00:12:17.080 --> 00:12:18.910
my original proof was circular.

00:12:18.910 --> 00:12:21.370
It's my fault because I
didn't explain it properly.

00:12:21.370 --> 00:12:23.580
So hopefully this should
provide a little bit of

00:12:23.580 --> 00:12:25.910
clarity on the issue.