1
00:00:00,980 --> 00:00:06,370
GILBERT STRANG: OK well,
here we're at the beginning.
2
00:00:06,370 --> 00:00:12,200
And that I think it's worth
thinking about what we know.
3
00:00:12,200 --> 00:00:13,830
Calculus.
4
00:00:13,830 --> 00:00:16,950
Differential equations is the
big application of calculus,
5
00:00:16,950 --> 00:00:21,530
so it's kind of interesting to
see what part of calculus, what
6
00:00:21,530 --> 00:00:24,360
information and what
ideas from calculus,
7
00:00:24,360 --> 00:00:27,190
actually get used in
differential equations.
8
00:00:27,190 --> 00:00:31,440
And I'm going to
show you what I see,
9
00:00:31,440 --> 00:00:34,450
and it's not everything
by any means,
10
00:00:34,450 --> 00:00:41,420
it's some basic ideas, but not
all the details you learned.
11
00:00:41,420 --> 00:00:43,460
So I'm not saying
forget all those,
12
00:00:43,460 --> 00:00:48,930
but just focus on what matters.
13
00:00:48,930 --> 00:00:49,690
OK.
14
00:00:49,690 --> 00:00:54,500
So the calculus you
need is my topic.
15
00:00:54,500 --> 00:00:56,760
And the first thing
is, you really do
16
00:00:56,760 --> 00:01:00,790
need to know basic derivatives.
17
00:01:00,790 --> 00:01:04,150
The derivative of x to
the n, the derivative
18
00:01:04,150 --> 00:01:06,120
of sine and cosine.
19
00:01:06,120 --> 00:01:11,290
Above all, the derivative of e
to the x, which is e to the x.
20
00:01:11,290 --> 00:01:13,870
The derivative of e to
the x is e to the x.
21
00:01:13,870 --> 00:01:18,353
That's the wonderful equation
that is solved by e to the x.
22
00:01:18,353 --> 00:01:21,200
Dy dt equals y.
23
00:01:21,200 --> 00:01:23,070
We'll have to do more with that.
24
00:01:23,070 --> 00:01:28,240
And then the inverse function
related to the exponential
25
00:01:28,240 --> 00:01:29,440
is the logarithm.
26
00:01:29,440 --> 00:01:32,901
With that special
derivative of 1/x.
27
00:01:32,901 --> 00:01:33,400
OK.
28
00:01:33,400 --> 00:01:35,060
But you know those.
29
00:01:35,060 --> 00:01:41,440
Secondly, out of those
few specific facts,
30
00:01:41,440 --> 00:01:46,220
you can create the derivatives
of an enormous array
31
00:01:46,220 --> 00:01:50,020
of functions using
the key rules.
32
00:01:50,020 --> 00:01:54,100
The derivative of f
plus g is the derivative
33
00:01:54,100 --> 00:01:56,780
of f plus the derivative of g.
34
00:01:56,780 --> 00:01:59,730
Derivative is a
linear operation.
35
00:01:59,730 --> 00:02:05,300
The product rule fg
prime plus gf prime.
36
00:02:05,300 --> 00:02:06,600
The quotient rule.
37
00:02:06,600 --> 00:02:08,259
Who can remember that?
38
00:02:08,259 --> 00:02:12,090
And above all, the chain rule.
39
00:02:12,090 --> 00:02:15,880
The derivative of this--
of that chain of functions,
40
00:02:15,880 --> 00:02:23,080
that composite function is the
derivative of f with respect
41
00:02:23,080 --> 00:02:28,360
to g times the derivative
of g with respect to x.
42
00:02:28,360 --> 00:02:32,070
That's really-- that
it's chains of functions
43
00:02:32,070 --> 00:02:37,120
that really blow open the
functions or we can deal with.
44
00:02:37,120 --> 00:02:38,610
OK.
45
00:02:38,610 --> 00:02:42,890
And then the
fundamental theorem.
46
00:02:42,890 --> 00:02:45,440
So the fundamental theorem
involves the derivative
47
00:02:45,440 --> 00:02:47,040
and the integral.
48
00:02:47,040 --> 00:02:52,340
And it says that one is the
inverse operation to the other.
49
00:02:52,340 --> 00:03:03,600
The derivative of the integral
of a function is this.
50
00:03:03,600 --> 00:03:09,050
Here is y and the
integral goes from 0
51
00:03:09,050 --> 00:03:13,400
to x I don't care what
that dummy variable is.
52
00:03:13,400 --> 00:03:17,030
I can-- I'll change that
dummy variable to t.
53
00:03:17,030 --> 00:03:17,890
Whatever.
54
00:03:17,890 --> 00:03:20,830
I don't care.
55
00:03:20,830 --> 00:03:23,790
[? ET ?] to show
the dummy variable.
56
00:03:23,790 --> 00:03:27,740
The x is the limit
of integration.
57
00:03:27,740 --> 00:03:31,780
I won't discuss that
fundamental theorem,
58
00:03:31,780 --> 00:03:35,870
but it certainly is
fundamental and I'll use it.
59
00:03:35,870 --> 00:03:37,190
Maybe that's better.
60
00:03:37,190 --> 00:03:39,800
I'll use the fundamental
theorem right away.
61
00:03:39,800 --> 00:03:42,520
So-- but remember what it says.
62
00:03:42,520 --> 00:03:46,110
It says that if you take a
function, you integrate it,
63
00:03:46,110 --> 00:03:50,100
you take the derivative, you
get the function back again.
64
00:03:50,100 --> 00:03:54,260
OK can I apply
that to a really--
65
00:03:54,260 --> 00:04:02,140
I see this as a key example
in differential equations.
66
00:04:02,140 --> 00:04:04,980
And let me show you the
function I have in mind.
67
00:04:04,980 --> 00:04:08,570
The function I have in
mind, I'll call it y,
68
00:04:08,570 --> 00:04:13,670
is the interval from 0 to t.
69
00:04:13,670 --> 00:04:17,630
So it's a function
of t then, time, It's
70
00:04:17,630 --> 00:04:23,210
the integral of this,
e to the t minus s.
71
00:04:23,210 --> 00:04:24,015
Some function.
72
00:04:28,620 --> 00:04:35,650
That's a remarkable
formula for the solution
73
00:04:35,650 --> 00:04:38,110
to a basic
differential equation.
74
00:04:38,110 --> 00:04:43,770
So with this, that
solves the equation dy
75
00:04:43,770 --> 00:04:52,100
dt equals y plus q of t.
76
00:04:52,100 --> 00:04:54,680
So when I see that equation
and we'll see it again
77
00:04:54,680 --> 00:04:58,430
and we'll derive this
formula, but now I
78
00:04:58,430 --> 00:05:02,870
want to just use the
fundamental theorem of calculus
79
00:05:02,870 --> 00:05:05,300
to check the formula.
80
00:05:05,300 --> 00:05:09,570
What as we created-- as we
derive the formula-- well
81
00:05:09,570 --> 00:05:15,010
it won't be wrong because
our derivation will be good.
82
00:05:15,010 --> 00:05:18,270
But also, it would
be nice, I just
83
00:05:18,270 --> 00:05:21,880
think if you plug that in,
to that differential equation
84
00:05:21,880 --> 00:05:23,360
it's solved.
85
00:05:23,360 --> 00:05:25,710
OK so I want to take
the derivative of that.
86
00:05:25,710 --> 00:05:26,860
That's my job.
87
00:05:26,860 --> 00:05:31,260
And that's why I do it here
because it uses all the rules.
88
00:05:31,260 --> 00:05:34,400
OK to take that
derivative, I notice
89
00:05:34,400 --> 00:05:38,990
the t is appearing there
in the usual place,
90
00:05:38,990 --> 00:05:40,830
and it's also
inside the integral.
91
00:05:40,830 --> 00:05:43,310
But this is a simple function.
92
00:05:43,310 --> 00:05:47,140
I can take e to the
t-- I'm going to take e
93
00:05:47,140 --> 00:05:51,810
to the t out of the--
outside the integral.
94
00:05:51,810 --> 00:05:54,410
e to the t.
95
00:05:54,410 --> 00:06:01,410
So I have a function t
times another function of t.
96
00:06:01,410 --> 00:06:03,610
I'm going to use
the product rule
97
00:06:03,610 --> 00:06:07,730
and show that the
derivative of that product
98
00:06:07,730 --> 00:06:13,310
is one term will be y and
the other term will be q.
99
00:06:13,310 --> 00:06:18,065
Can I just apply the product
rule to this function
100
00:06:18,065 --> 00:06:21,660
that I've pulled out of a
hat, but you'll see it again.
101
00:06:21,660 --> 00:06:25,350
OK so it's a product
of this times this.
102
00:06:25,350 --> 00:06:33,480
So the derivative dy dt
is-- the product rule
103
00:06:33,480 --> 00:06:36,400
says take the derivative
of [INAUDIBLE] that
104
00:06:36,400 --> 00:06:38,350
is e to the [INAUDIBLE].
105
00:06:42,315 --> 00:06:50,916
Plus, the first thing times
the derivative of the second.
106
00:06:50,916 --> 00:06:53,110
Now I'm using the product rule.
107
00:06:53,110 --> 00:06:56,820
It just-- you have to notice
that e to the t came twice
108
00:06:56,820 --> 00:07:02,270
because it is there and
its derivative is the same.
109
00:07:02,270 --> 00:07:08,290
OK now, what's the
derivative of that?
110
00:07:08,290 --> 00:07:10,180
Fundamental theorem of calculus.
111
00:07:10,180 --> 00:07:13,910
We've integrated something, I
want to take its derivative,
112
00:07:13,910 --> 00:07:15,520
so I get that something.
113
00:07:15,520 --> 00:07:20,770
I get e to the minus tq of t.
114
00:07:20,770 --> 00:07:23,230
That's the fundamental theorem.
115
00:07:23,230 --> 00:07:25,240
Are you good with that?
116
00:07:25,240 --> 00:07:28,160
So let's just look
and see what we have.
117
00:07:28,160 --> 00:07:33,720
First term was exactly y.
118
00:07:33,720 --> 00:07:35,580
Exactly what is
above because when
119
00:07:35,580 --> 00:07:38,660
I took the derivative
of the first guy,
120
00:07:38,660 --> 00:07:42,690
the f it didn't change
it, so I still have y.
121
00:07:42,690 --> 00:07:45,010
What have I-- what
do I have here?
122
00:07:45,010 --> 00:07:49,650
E to the t times e to
the minus t is one.
123
00:07:49,650 --> 00:07:52,500
So e to the t cancels
e to the minus t
124
00:07:52,500 --> 00:07:55,930
and I'm left with q
of t Just what I want.
125
00:07:55,930 --> 00:07:57,880
So the two terms
from the product rule
126
00:07:57,880 --> 00:08:01,460
are the two terms in the
differential equation.
127
00:08:01,460 --> 00:08:05,940
I just think as you saw the
fundamental theorem was needed
128
00:08:05,940 --> 00:08:08,930
right there to find the
derivative of what's
129
00:08:08,930 --> 00:08:12,610
in that box, is what's
in those parentheses.
130
00:08:12,610 --> 00:08:15,820
I just like that the use
of the fundamental theorem.
131
00:08:15,820 --> 00:08:21,860
OK one more topic
of calculus we need.
132
00:08:21,860 --> 00:08:24,250
And here we go.
133
00:08:24,250 --> 00:08:31,570
So it involves the
tangent line to the graph.
134
00:08:31,570 --> 00:08:34,789
This tangent to the graph.
135
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.
136
00:08:44,140 --> 00:08:47,410
That's taking any
function, maybe
137
00:08:47,410 --> 00:08:49,600
you'd rather I just
called the function f.
138
00:08:52,600 --> 00:08:56,910
A function at a point
a little beyond t,
139
00:08:56,910 --> 00:09:01,860
is approximately
the function at t
140
00:09:01,860 --> 00:09:06,260
plus the correction because
it-- plus a delta f, right?
141
00:09:06,260 --> 00:09:07,870
A delta f.
142
00:09:07,870 --> 00:09:10,210
And what's the delta
f approximately?
143
00:09:10,210 --> 00:09:20,920
It's approximately delta t
times the derivative at t.
144
00:09:20,920 --> 00:09:25,720
That-- there's a lot of
symbols on that line,
145
00:09:25,720 --> 00:09:31,390
but it expresses the most basic
fact of differential calculus.
146
00:09:31,390 --> 00:09:35,590
If I put that f of t on
this side with a minus sign,
147
00:09:35,590 --> 00:09:38,230
then I have delta f.
148
00:09:38,230 --> 00:09:44,240
If I divide by that delta
t, then the same rule
149
00:09:44,240 --> 00:09:49,280
is saying that this is
approximately df dt.
150
00:09:49,280 --> 00:09:51,380
That's a fundamental
idea of calculus,
151
00:09:51,380 --> 00:09:55,070
that the derivative
is quite close.
152
00:09:55,070 --> 00:09:58,460
At the point t-- the
derivative at the point t
153
00:09:58,460 --> 00:10:01,930
is close to delta f
divided by delta t.
154
00:10:01,930 --> 00:10:05,100
It changes over a
short time interval.
155
00:10:05,100 --> 00:10:10,270
OK so that's the tangent line
because it starts with that's
156
00:10:10,270 --> 00:10:12,350
the constant term.
157
00:10:12,350 --> 00:10:16,710
It's a function of delta
t and that's the slope.
158
00:10:16,710 --> 00:10:18,436
Just draw a picture.
159
00:10:18,436 --> 00:10:20,940
So I'm drawing a picture here.
160
00:10:20,940 --> 00:10:24,680
So let me draw a
graph of-- oh there's
161
00:10:24,680 --> 00:10:27,090
the graph of e to the t.
162
00:10:27,090 --> 00:10:29,122
So it starts up with slope 1.
163
00:10:29,122 --> 00:10:30,580
Let me give it a
little slope here.
164
00:10:33,380 --> 00:10:36,250
OK the tangent line,
and of course it
165
00:10:36,250 --> 00:10:39,380
comes down here Not below.
166
00:10:39,380 --> 00:10:42,630
So the tangent
line is that line.
167
00:10:47,070 --> 00:10:48,510
That's the tangent line.
168
00:10:48,510 --> 00:10:51,080
That's this approximation to f.
169
00:10:51,080 --> 00:10:55,560
And you see as I-- here
is t equals 0 let's say.
170
00:10:55,560 --> 00:10:58,400
And here's t equal delta t.
171
00:10:58,400 --> 00:11:00,210
And you see if I
take a big step,
172
00:11:00,210 --> 00:11:03,070
my line is far from the curve.
173
00:11:03,070 --> 00:11:06,080
And we want to get closer.
174
00:11:06,080 --> 00:11:09,330
So the way to get
closer is we have
175
00:11:09,330 --> 00:11:11,130
to take into
account the bending.
176
00:11:11,130 --> 00:11:12,660
The curve is bending.
177
00:11:12,660 --> 00:11:17,060
What derivative tells
us about bending?
178
00:11:17,060 --> 00:11:24,571
That is delta t squared
times the second derivative.
179
00:11:27,320 --> 00:11:27,820
One half.
180
00:11:27,820 --> 00:11:31,450
It turns out a one
half shows in there.
181
00:11:31,450 --> 00:11:35,760
So this is the term that
changes the tangent line,
182
00:11:35,760 --> 00:11:38,820
to a tangent parabola.
183
00:11:38,820 --> 00:11:41,300
It notices the
bending at that point.
184
00:11:41,300 --> 00:11:43,620
The second derivative
at that point.
185
00:11:43,620 --> 00:11:45,210
So it curves up.
186
00:11:45,210 --> 00:11:49,400
It doesn't follow it perfectly,
but as well-- much better
187
00:11:49,400 --> 00:11:51,100
than the other.
188
00:11:51,100 --> 00:11:53,735
So this is the line.
189
00:11:53,735 --> 00:11:54,610
Here is the parabola.
190
00:11:57,264 --> 00:11:58,305
And here is the function.
191
00:12:02,320 --> 00:12:04,060
The real one.
192
00:12:04,060 --> 00:12:05,690
OK.
193
00:12:05,690 --> 00:12:10,580
I won't review the theory there
that it pulls out that one
194
00:12:10,580 --> 00:12:12,350
half, but you could check it.
195
00:12:12,350 --> 00:12:16,690
Now finally, what if we
want to do even better?
196
00:12:16,690 --> 00:12:19,010
Well we need to take into
account the third derivative
197
00:12:19,010 --> 00:12:21,480
and then the fourth
derivative and so on,
198
00:12:21,480 --> 00:12:24,970
and if we get all
those derivatives then,
199
00:12:24,970 --> 00:12:29,075
all of them that means,
we will be at the function
200
00:12:29,075 --> 00:12:32,100
because that's a nice
function, e to the t.
201
00:12:32,100 --> 00:12:37,110
We can recreate that
function from knowing
202
00:12:37,110 --> 00:12:42,565
its height, its
slope, its bending
203
00:12:42,565 --> 00:12:44,350
and all the rest of the terms.
204
00:12:44,350 --> 00:12:48,130
So there's a whole lot
more-- Infinitely many terms.
205
00:12:48,130 --> 00:12:51,010
That one over two-- the
good way to think of one
206
00:12:51,010 --> 00:12:55,830
over two, one half, is one over
two factorial, two times one.
207
00:12:55,830 --> 00:12:59,650
Because this is one
over n factorial,
208
00:12:59,650 --> 00:13:04,170
times t to the
nth, pretty small,
209
00:13:04,170 --> 00:13:07,175
times the nth derivative
of the function.
210
00:13:10,170 --> 00:13:13,020
And keep going.
211
00:13:13,020 --> 00:13:19,310
That's called the Taylor
series named after Taylor.
212
00:13:19,310 --> 00:13:25,550
Kind of frightening at first.
213
00:13:25,550 --> 00:13:28,910
It's frightening because it's
got infinitely many terms.
214
00:13:28,910 --> 00:13:31,731
And the terms are getting
a little more comp--
215
00:13:31,731 --> 00:13:33,740
For most functions,
you really don't want
216
00:13:33,740 --> 00:13:35,680
to compute the nth derivative.
217
00:13:35,680 --> 00:13:39,100
For e to the t, I don't mind
computing the nth derivative
218
00:13:39,100 --> 00:13:44,790
because it's still e to the
t, but usually that's-- this
219
00:13:44,790 --> 00:13:46,950
isn't so practical.
220
00:13:46,950 --> 00:13:48,570
[INAUDIBLE] very practical.
221
00:13:48,570 --> 00:13:51,380
Tangent parabola,
quite practical.
222
00:13:51,380 --> 00:13:55,150
Higher order terms, less--
much less practical.
223
00:13:55,150 --> 00:13:59,210
But the formula is
beautiful because you
224
00:13:59,210 --> 00:14:02,340
see the pattern, that's
really what mathematics
225
00:14:02,340 --> 00:14:04,300
is about patterns,
and here you're
226
00:14:04,300 --> 00:14:08,880
seeing the pattern in
the higher, higher terms.
227
00:14:08,880 --> 00:14:14,050
They all fit that pattern and
when you add up all the terms,
228
00:14:14,050 --> 00:14:18,280
if you have a nice function,
then the approximation
229
00:14:18,280 --> 00:14:21,560
becomes perfect and you
would have equality.
230
00:14:21,560 --> 00:14:27,800
So to end this lecture,
approximate to equal provided
231
00:14:27,800 --> 00:14:30,080
we have a nice function.
232
00:14:30,080 --> 00:14:34,510
And those are the best functions
of mathematics and exponential
233
00:14:34,510 --> 00:14:36,010
is of course one of them.
234
00:14:36,010 --> 00:14:39,030
OK that's calculus.
235
00:14:39,030 --> 00:14:40,990
Well, part of calculus.
236
00:14:40,990 --> 00:14:42,790
Thank you.