WEBVTT

00:00:00.980 --> 00:00:06.370
GILBERT STRANG: OK well,
here we're at the beginning.

00:00:06.370 --> 00:00:12.200
And that I think it's worth
thinking about what we know.

00:00:12.200 --> 00:00:13.830
Calculus.

00:00:13.830 --> 00:00:16.950
Differential equations is the
big application of calculus,

00:00:16.950 --> 00:00:21.530
so it's kind of interesting to
see what part of calculus, what

00:00:21.530 --> 00:00:24.360
information and what
ideas from calculus,

00:00:24.360 --> 00:00:27.190
actually get used in
differential equations.

00:00:27.190 --> 00:00:31.440
And I'm going to
show you what I see,

00:00:31.440 --> 00:00:34.450
and it's not everything
by any means,

00:00:34.450 --> 00:00:41.420
it's some basic ideas, but not
all the details you learned.

00:00:41.420 --> 00:00:43.460
So I'm not saying
forget all those,

00:00:43.460 --> 00:00:48.930
but just focus on what matters.

00:00:48.930 --> 00:00:49.690
OK.

00:00:49.690 --> 00:00:54.500
So the calculus you
need is my topic.

00:00:54.500 --> 00:00:56.760
And the first thing
is, you really do

00:00:56.760 --> 00:01:00.790
need to know basic derivatives.

00:01:00.790 --> 00:01:04.150
The derivative of x to
the n, the derivative

00:01:04.150 --> 00:01:06.120
of sine and cosine.

00:01:06.120 --> 00:01:11.290
Above all, the derivative of e
to the x, which is e to the x.

00:01:11.290 --> 00:01:13.870
The derivative of e to
the x is e to the x.

00:01:13.870 --> 00:01:18.353
That's the wonderful equation
that is solved by e to the x.

00:01:18.353 --> 00:01:21.200
Dy dt equals y.

00:01:21.200 --> 00:01:23.070
We'll have to do more with that.

00:01:23.070 --> 00:01:28.240
And then the inverse function
related to the exponential

00:01:28.240 --> 00:01:29.440
is the logarithm.

00:01:29.440 --> 00:01:32.901
With that special
derivative of 1/x.

00:01:32.901 --> 00:01:33.400
OK.

00:01:33.400 --> 00:01:35.060
But you know those.

00:01:35.060 --> 00:01:41.440
Secondly, out of those
few specific facts,

00:01:41.440 --> 00:01:46.220
you can create the derivatives
of an enormous array

00:01:46.220 --> 00:01:50.020
of functions using
the key rules.

00:01:50.020 --> 00:01:54.100
The derivative of f
plus g is the derivative

00:01:54.100 --> 00:01:56.780
of f plus the derivative of g.

00:01:56.780 --> 00:01:59.730
Derivative is a
linear operation.

00:01:59.730 --> 00:02:05.300
The product rule fg
prime plus gf prime.

00:02:05.300 --> 00:02:06.600
The quotient rule.

00:02:06.600 --> 00:02:08.259
Who can remember that?

00:02:08.259 --> 00:02:12.090
And above all, the chain rule.

00:02:12.090 --> 00:02:15.880
The derivative of this--
of that chain of functions,

00:02:15.880 --> 00:02:23.080
that composite function is the
derivative of f with respect

00:02:23.080 --> 00:02:28.360
to g times the derivative
of g with respect to x.

00:02:28.360 --> 00:02:32.070
That's really-- that
it's chains of functions

00:02:32.070 --> 00:02:37.120
that really blow open the
functions or we can deal with.

00:02:37.120 --> 00:02:38.610
OK.

00:02:38.610 --> 00:02:42.890
And then the
fundamental theorem.

00:02:42.890 --> 00:02:45.440
So the fundamental theorem
involves the derivative

00:02:45.440 --> 00:02:47.040
and the integral.

00:02:47.040 --> 00:02:52.340
And it says that one is the
inverse operation to the other.

00:02:52.340 --> 00:03:03.600
The derivative of the integral
of a function is this.

00:03:03.600 --> 00:03:09.050
Here is y and the
integral goes from 0

00:03:09.050 --> 00:03:13.400
to x I don't care what
that dummy variable is.

00:03:13.400 --> 00:03:17.030
I can-- I'll change that
dummy variable to t.

00:03:17.030 --> 00:03:17.890
Whatever.

00:03:17.890 --> 00:03:20.830
I don't care.

00:03:20.830 --> 00:03:23.790
[? ET ?] to show
the dummy variable.

00:03:23.790 --> 00:03:27.740
The x is the limit
of integration.

00:03:27.740 --> 00:03:31.780
I won't discuss that
fundamental theorem,

00:03:31.780 --> 00:03:35.870
but it certainly is
fundamental and I'll use it.

00:03:35.870 --> 00:03:37.190
Maybe that's better.

00:03:37.190 --> 00:03:39.800
I'll use the fundamental
theorem right away.

00:03:39.800 --> 00:03:42.520
So-- but remember what it says.

00:03:42.520 --> 00:03:46.110
It says that if you take a
function, you integrate it,

00:03:46.110 --> 00:03:50.100
you take the derivative, you
get the function back again.

00:03:50.100 --> 00:03:54.260
OK can I apply
that to a really--

00:03:54.260 --> 00:04:02.140
I see this as a key example
in differential equations.

00:04:02.140 --> 00:04:04.980
And let me show you the
function I have in mind.

00:04:04.980 --> 00:04:08.570
The function I have in
mind, I'll call it y,

00:04:08.570 --> 00:04:13.670
is the interval from 0 to t.

00:04:13.670 --> 00:04:17.630
So it's a function
of t then, time, It's

00:04:17.630 --> 00:04:23.210
the integral of this,
e to the t minus s.

00:04:23.210 --> 00:04:24.015
Some function.

00:04:28.620 --> 00:04:35.650
That's a remarkable
formula for the solution

00:04:35.650 --> 00:04:38.110
to a basic
differential equation.

00:04:38.110 --> 00:04:43.770
So with this, that
solves the equation dy

00:04:43.770 --> 00:04:52.100
dt equals y plus q of t.

00:04:52.100 --> 00:04:54.680
So when I see that equation
and we'll see it again

00:04:54.680 --> 00:04:58.430
and we'll derive this
formula, but now I

00:04:58.430 --> 00:05:02.870
want to just use the
fundamental theorem of calculus

00:05:02.870 --> 00:05:05.300
to check the formula.

00:05:05.300 --> 00:05:09.570
What as we created-- as we
derive the formula-- well

00:05:09.570 --> 00:05:15.010
it won't be wrong because
our derivation will be good.

00:05:15.010 --> 00:05:18.270
But also, it would
be nice, I just

00:05:18.270 --> 00:05:21.880
think if you plug that in,
to that differential equation

00:05:21.880 --> 00:05:23.360
it's solved.

00:05:23.360 --> 00:05:25.710
OK so I want to take
the derivative of that.

00:05:25.710 --> 00:05:26.860
That's my job.

00:05:26.860 --> 00:05:31.260
And that's why I do it here
because it uses all the rules.

00:05:31.260 --> 00:05:34.400
OK to take that
derivative, I notice

00:05:34.400 --> 00:05:38.990
the t is appearing there
in the usual place,

00:05:38.990 --> 00:05:40.830
and it's also
inside the integral.

00:05:40.830 --> 00:05:43.310
But this is a simple function.

00:05:43.310 --> 00:05:47.140
I can take e to the
t-- I'm going to take e

00:05:47.140 --> 00:05:51.810
to the t out of the--
outside the integral.

00:05:51.810 --> 00:05:54.410
e to the t.

00:05:54.410 --> 00:06:01.410
So I have a function t
times another function of t.

00:06:01.410 --> 00:06:03.610
I'm going to use
the product rule

00:06:03.610 --> 00:06:07.730
and show that the
derivative of that product

00:06:07.730 --> 00:06:13.310
is one term will be y and
the other term will be q.

00:06:13.310 --> 00:06:18.065
Can I just apply the product
rule to this function

00:06:18.065 --> 00:06:21.660
that I've pulled out of a
hat, but you'll see it again.

00:06:21.660 --> 00:06:25.350
OK so it's a product
of this times this.

00:06:25.350 --> 00:06:33.480
So the derivative dy dt
is-- the product rule

00:06:33.480 --> 00:06:36.400
says take the derivative
of [INAUDIBLE] that

00:06:36.400 --> 00:06:38.350
is e to the [INAUDIBLE].

00:06:42.315 --> 00:06:50.916
Plus, the first thing times
the derivative of the second.

00:06:50.916 --> 00:06:53.110
Now I'm using the product rule.

00:06:53.110 --> 00:06:56.820
It just-- you have to notice
that e to the t came twice

00:06:56.820 --> 00:07:02.270
because it is there and
its derivative is the same.

00:07:02.270 --> 00:07:08.290
OK now, what's the
derivative of that?

00:07:08.290 --> 00:07:10.180
Fundamental theorem of calculus.

00:07:10.180 --> 00:07:13.910
We've integrated something, I
want to take its derivative,

00:07:13.910 --> 00:07:15.520
so I get that something.

00:07:15.520 --> 00:07:20.770
I get e to the minus tq of t.

00:07:20.770 --> 00:07:23.230
That's the fundamental theorem.

00:07:23.230 --> 00:07:25.240
Are you good with that?

00:07:25.240 --> 00:07:28.160
So let's just look
and see what we have.

00:07:28.160 --> 00:07:33.720
First term was exactly y.

00:07:33.720 --> 00:07:35.580
Exactly what is
above because when

00:07:35.580 --> 00:07:38.660
I took the derivative
of the first guy,

00:07:38.660 --> 00:07:42.690
the f it didn't change
it, so I still have y.

00:07:42.690 --> 00:07:45.010
What have I-- what
do I have here?

00:07:45.010 --> 00:07:49.650
E to the t times e to
the minus t is one.

00:07:49.650 --> 00:07:52.500
So e to the t cancels
e to the minus t

00:07:52.500 --> 00:07:55.930
and I'm left with q
of t Just what I want.

00:07:55.930 --> 00:07:57.880
So the two terms
from the product rule

00:07:57.880 --> 00:08:01.460
are the two terms in the
differential equation.

00:08:01.460 --> 00:08:05.940
I just think as you saw the
fundamental theorem was needed

00:08:05.940 --> 00:08:08.930
right there to find the
derivative of what's

00:08:08.930 --> 00:08:12.610
in that box, is what's
in those parentheses.

00:08:12.610 --> 00:08:15.820
I just like that the use
of the fundamental theorem.

00:08:15.820 --> 00:08:21.860
OK one more topic
of calculus we need.

00:08:21.860 --> 00:08:24.250
And here we go.

00:08:24.250 --> 00:08:31.570
So it involves the
tangent line to the graph.

00:08:31.570 --> 00:08:34.789
This tangent to the graph.

00:08:34.789 --> 00:08:44.140
So it's a straight line and what
we need is y of t plus delta t.

00:08:44.140 --> 00:08:47.410
That's taking any
function, maybe

00:08:47.410 --> 00:08:49.600
you'd rather I just
called the function f.

00:08:52.600 --> 00:08:56.910
A function at a point
a little beyond t,

00:08:56.910 --> 00:09:01.860
is approximately
the function at t

00:09:01.860 --> 00:09:06.260
plus the correction because
it-- plus a delta f, right?

00:09:06.260 --> 00:09:07.870
A delta f.

00:09:07.870 --> 00:09:10.210
And what's the delta
f approximately?

00:09:10.210 --> 00:09:20.920
It's approximately delta t
times the derivative at t.

00:09:20.920 --> 00:09:25.720
That-- there's a lot of
symbols on that line,

00:09:25.720 --> 00:09:31.390
but it expresses the most basic
fact of differential calculus.

00:09:31.390 --> 00:09:35.590
If I put that f of t on
this side with a minus sign,

00:09:35.590 --> 00:09:38.230
then I have delta f.

00:09:38.230 --> 00:09:44.240
If I divide by that delta
t, then the same rule

00:09:44.240 --> 00:09:49.280
is saying that this is
approximately df dt.

00:09:49.280 --> 00:09:51.380
That's a fundamental
idea of calculus,

00:09:51.380 --> 00:09:55.070
that the derivative
is quite close.

00:09:55.070 --> 00:09:58.460
At the point t-- the
derivative at the point t

00:09:58.460 --> 00:10:01.930
is close to delta f
divided by delta t.

00:10:01.930 --> 00:10:05.100
It changes over a
short time interval.

00:10:05.100 --> 00:10:10.270
OK so that's the tangent line
because it starts with that's

00:10:10.270 --> 00:10:12.350
the constant term.

00:10:12.350 --> 00:10:16.710
It's a function of delta
t and that's the slope.

00:10:16.710 --> 00:10:18.436
Just draw a picture.

00:10:18.436 --> 00:10:20.940
So I'm drawing a picture here.

00:10:20.940 --> 00:10:24.680
So let me draw a
graph of-- oh there's

00:10:24.680 --> 00:10:27.090
the graph of e to the t.

00:10:27.090 --> 00:10:29.122
So it starts up with slope 1.

00:10:29.122 --> 00:10:30.580
Let me give it a
little slope here.

00:10:33.380 --> 00:10:36.250
OK the tangent line,
and of course it

00:10:36.250 --> 00:10:39.380
comes down here Not below.

00:10:39.380 --> 00:10:42.630
So the tangent
line is that line.

00:10:47.070 --> 00:10:48.510
That's the tangent line.

00:10:48.510 --> 00:10:51.080
That's this approximation to f.

00:10:51.080 --> 00:10:55.560
And you see as I-- here
is t equals 0 let's say.

00:10:55.560 --> 00:10:58.400
And here's t equal delta t.

00:10:58.400 --> 00:11:00.210
And you see if I
take a big step,

00:11:00.210 --> 00:11:03.070
my line is far from the curve.

00:11:03.070 --> 00:11:06.080
And we want to get closer.

00:11:06.080 --> 00:11:09.330
So the way to get
closer is we have

00:11:09.330 --> 00:11:11.130
to take into
account the bending.

00:11:11.130 --> 00:11:12.660
The curve is bending.

00:11:12.660 --> 00:11:17.060
What derivative tells
us about bending?

00:11:17.060 --> 00:11:24.571
That is delta t squared
times the second derivative.

00:11:27.320 --> 00:11:27.820
One half.

00:11:27.820 --> 00:11:31.450
It turns out a one
half shows in there.

00:11:31.450 --> 00:11:35.760
So this is the term that
changes the tangent line,

00:11:35.760 --> 00:11:38.820
to a tangent parabola.

00:11:38.820 --> 00:11:41.300
It notices the
bending at that point.

00:11:41.300 --> 00:11:43.620
The second derivative
at that point.

00:11:43.620 --> 00:11:45.210
So it curves up.

00:11:45.210 --> 00:11:49.400
It doesn't follow it perfectly,
but as well-- much better

00:11:49.400 --> 00:11:51.100
than the other.

00:11:51.100 --> 00:11:53.735
So this is the line.

00:11:53.735 --> 00:11:54.610
Here is the parabola.

00:11:57.264 --> 00:11:58.305
And here is the function.

00:12:02.320 --> 00:12:04.060
The real one.

00:12:04.060 --> 00:12:05.690
OK.

00:12:05.690 --> 00:12:10.580
I won't review the theory there
that it pulls out that one

00:12:10.580 --> 00:12:12.350
half, but you could check it.

00:12:12.350 --> 00:12:16.690
Now finally, what if we
want to do even better?

00:12:16.690 --> 00:12:19.010
Well we need to take into
account the third derivative

00:12:19.010 --> 00:12:21.480
and then the fourth
derivative and so on,

00:12:21.480 --> 00:12:24.970
and if we get all
those derivatives then,

00:12:24.970 --> 00:12:29.075
all of them that means,
we will be at the function

00:12:29.075 --> 00:12:32.100
because that's a nice
function, e to the t.

00:12:32.100 --> 00:12:37.110
We can recreate that
function from knowing

00:12:37.110 --> 00:12:42.565
its height, its
slope, its bending

00:12:42.565 --> 00:12:44.350
and all the rest of the terms.

00:12:44.350 --> 00:12:48.130
So there's a whole lot
more-- Infinitely many terms.

00:12:48.130 --> 00:12:51.010
That one over two-- the
good way to think of one

00:12:51.010 --> 00:12:55.830
over two, one half, is one over
two factorial, two times one.

00:12:55.830 --> 00:12:59.650
Because this is one
over n factorial,

00:12:59.650 --> 00:13:04.170
times t to the
nth, pretty small,

00:13:04.170 --> 00:13:07.175
times the nth derivative
of the function.

00:13:10.170 --> 00:13:13.020
And keep going.

00:13:13.020 --> 00:13:19.310
That's called the Taylor
series named after Taylor.

00:13:19.310 --> 00:13:25.550
Kind of frightening at first.

00:13:25.550 --> 00:13:28.910
It's frightening because it's
got infinitely many terms.

00:13:28.910 --> 00:13:31.731
And the terms are getting
a little more comp--

00:13:31.731 --> 00:13:33.740
For most functions,
you really don't want

00:13:33.740 --> 00:13:35.680
to compute the nth derivative.

00:13:35.680 --> 00:13:39.100
For e to the t, I don't mind
computing the nth derivative

00:13:39.100 --> 00:13:44.790
because it's still e to the
t, but usually that's-- this

00:13:44.790 --> 00:13:46.950
isn't so practical.

00:13:46.950 --> 00:13:48.570
[INAUDIBLE] very practical.

00:13:48.570 --> 00:13:51.380
Tangent parabola,
quite practical.

00:13:51.380 --> 00:13:55.150
Higher order terms, less--
much less practical.

00:13:55.150 --> 00:13:59.210
But the formula is
beautiful because you

00:13:59.210 --> 00:14:02.340
see the pattern, that's
really what mathematics

00:14:02.340 --> 00:14:04.300
is about patterns,
and here you're

00:14:04.300 --> 00:14:08.880
seeing the pattern in
the higher, higher terms.

00:14:08.880 --> 00:14:14.050
They all fit that pattern and
when you add up all the terms,

00:14:14.050 --> 00:14:18.280
if you have a nice function,
then the approximation

00:14:18.280 --> 00:14:21.560
becomes perfect and you
would have equality.

00:14:21.560 --> 00:14:27.800
So to end this lecture,
approximate to equal provided

00:14:27.800 --> 00:14:30.080
we have a nice function.

00:14:30.080 --> 00:14:34.510
And those are the best functions
of mathematics and exponential

00:14:34.510 --> 00:14:36.010
is of course one of them.

00:14:36.010 --> 00:14:39.030
OK that's calculus.

00:14:39.030 --> 00:14:40.990
Well, part of calculus.

00:14:40.990 --> 00:14:42.790
Thank you.