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Professor: So, again
welcome to 18.01.
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00:00:24,180 --> 00:00:27,200
We're getting started
today with what
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00:00:27,200 --> 00:00:36,000
we're calling Unit One, a
highly imaginative title.
11
00:00:36,000 --> 00:00:43,050
And it's differentiation.
12
00:00:43,050 --> 00:00:45,960
So, let me first
tell you, briefly,
13
00:00:45,960 --> 00:00:49,320
what's in store in the
next couple of weeks.
14
00:00:49,320 --> 00:01:02,170
The main topic today is
what is a derivative.
15
00:01:02,170 --> 00:01:09,840
And, we're going to look at this
from several different points
16
00:01:09,840 --> 00:01:18,820
of view, and the first one is
the geometric interpretation.
17
00:01:18,820 --> 00:01:21,710
That's what we'll
spend most of today on.
18
00:01:21,710 --> 00:01:32,330
And then, we'll also talk
about a physical interpretation
19
00:01:32,330 --> 00:01:37,240
of what a derivative is.
20
00:01:37,240 --> 00:01:44,910
And then there's going to
be something else which
21
00:01:44,910 --> 00:01:48,030
I guess is maybe the reason
why Calculus is so fundamental,
22
00:01:48,030 --> 00:01:53,240
and why we always start with it
in most science and engineering
23
00:01:53,240 --> 00:02:01,420
schools, which is the importance
of derivatives, of this,
24
00:02:01,420 --> 00:02:09,290
to all measurements.
25
00:02:09,290 --> 00:02:11,390
So that means pretty
much every place.
26
00:02:11,390 --> 00:02:15,140
That means in science,
in engineering,
27
00:02:15,140 --> 00:02:23,780
in economics, in
political science, etc.
28
00:02:23,780 --> 00:02:27,730
Polling, lots of
commercial applications,
29
00:02:27,730 --> 00:02:29,610
just about everything.
30
00:02:29,610 --> 00:02:33,040
Now, that's what we'll
be getting started with,
31
00:02:33,040 --> 00:02:35,480
and then there's
another thing that we're
32
00:02:35,480 --> 00:02:41,500
gonna do in this unit, which
is we're going to explain
33
00:02:41,500 --> 00:02:49,740
how to differentiate anything.
34
00:02:49,740 --> 00:03:01,490
So, how to differentiate
any function you know.
35
00:03:01,490 --> 00:03:04,260
And that's kind of a
tall order, but let
36
00:03:04,260 --> 00:03:05,656
me just give you an example.
37
00:03:05,656 --> 00:03:07,072
If you want to
take the derivative
38
00:03:07,072 --> 00:03:09,640
- this we'll see
today is the notation
39
00:03:09,640 --> 00:03:13,750
for the derivative of something
- of some messy function like e
40
00:03:13,750 --> 00:03:15,420
^ x arctan x.
41
00:03:19,730 --> 00:03:25,730
We'll work this out by
the end of this unit.
42
00:03:25,730 --> 00:03:26,390
All right?
43
00:03:26,390 --> 00:03:29,090
Anything you can think of,
anything you can write down,
44
00:03:29,090 --> 00:03:32,140
we can differentiate it.
45
00:03:32,140 --> 00:03:37,820
All right, so that's what we're
gonna do, and today, as I said,
46
00:03:37,820 --> 00:03:39,680
we're gonna spend
most of our time
47
00:03:39,680 --> 00:03:44,690
on this geometric
interpretation.
48
00:03:44,690 --> 00:03:50,080
So let's begin with that.
49
00:03:50,080 --> 00:04:01,570
So here we go with the geometric
interpretation of derivatives.
50
00:04:01,570 --> 00:04:11,160
And, what we're going to do is
just ask the geometric problem
51
00:04:11,160 --> 00:04:25,490
of finding the tangent
line to some graph
52
00:04:25,490 --> 00:04:31,580
of some function at some point.
53
00:04:31,580 --> 00:04:33,250
Which is to say (x_0, y_0).
54
00:04:33,250 --> 00:04:42,460
So that's the problem that
we're addressing here.
55
00:04:42,460 --> 00:04:46,480
Alright, so here's
our problem, and now
56
00:04:46,480 --> 00:04:49,440
let me show you the solution.
57
00:04:49,440 --> 00:04:58,530
So, well, let's
graph the function.
58
00:04:58,530 --> 00:05:00,510
Here's its graph.
59
00:05:00,510 --> 00:05:02,590
Here's some point.
60
00:05:02,590 --> 00:05:09,620
All right, maybe I should
draw it just a bit lower.
61
00:05:09,620 --> 00:05:13,950
So here's a point P.
Maybe it's above the point
62
00:05:13,950 --> 00:05:19,080
x_0. x_0, by the way, this
was supposed to be an x_0.
63
00:05:19,080 --> 00:05:26,870
That was some fixed
place on the x-axis.
64
00:05:26,870 --> 00:05:32,240
And now, in order to
perform this mighty feat,
65
00:05:32,240 --> 00:05:36,700
I will use another
color of chalk.
66
00:05:36,700 --> 00:05:37,710
How about red?
67
00:05:37,710 --> 00:05:38,900
OK.
68
00:05:38,900 --> 00:05:42,080
So here it is.
69
00:05:42,080 --> 00:05:44,440
There's the tangent line,
well, not quite straight.
70
00:05:44,440 --> 00:05:45,860
Close enough.
71
00:05:45,860 --> 00:05:46,820
All right?
72
00:05:46,820 --> 00:05:49,040
I did it.
73
00:05:49,040 --> 00:05:50,700
That's the geometric problem.
74
00:05:50,700 --> 00:05:56,490
I achieved what I
wanted to do, and it's
75
00:05:56,490 --> 00:05:58,810
kind of an interesting
question, which unfortunately I
76
00:05:58,810 --> 00:06:01,520
can't solve for you
in this class, which
77
00:06:01,520 --> 00:06:03,240
is, how did I do that?
78
00:06:03,240 --> 00:06:04,870
That is, how
physically did I manage
79
00:06:04,870 --> 00:06:07,890
to know what to do to
draw this tangent line?
80
00:06:07,890 --> 00:06:10,610
But that's what geometric
problems are like.
81
00:06:10,610 --> 00:06:12,000
We visualize it.
82
00:06:12,000 --> 00:06:14,050
We can figure it out
somewhere in our brains.
83
00:06:14,050 --> 00:06:15,420
It happens.
84
00:06:15,420 --> 00:06:18,850
And the task that we
have now is to figure out
85
00:06:18,850 --> 00:06:23,670
how to do it analytically,
to do it in a way
86
00:06:23,670 --> 00:06:28,636
that a machine could
just as well as I did
87
00:06:28,636 --> 00:06:32,230
in drawing this tangent line.
88
00:06:32,230 --> 00:06:39,620
So, what did we learn in high
school about what a tangent
89
00:06:39,620 --> 00:06:40,770
line is?
90
00:06:40,770 --> 00:06:42,740
Well, a tangent line
has an equation,
91
00:06:42,740 --> 00:06:45,700
and any line through a
point has the equation y
92
00:06:45,700 --> 00:06:52,190
- y_0 is equal to m, the
slope, times x - x_0.
93
00:06:52,190 --> 00:06:58,880
So here's the equation
for that line,
94
00:06:58,880 --> 00:07:02,200
and now there are two
pieces of information
95
00:07:02,200 --> 00:07:07,310
that we're going to need to
work out what the line is.
96
00:07:07,310 --> 00:07:10,860
The first one is the point.
97
00:07:10,860 --> 00:07:13,230
That's that point P there.
98
00:07:13,230 --> 00:07:16,670
And to specify P,
given x, we need
99
00:07:16,670 --> 00:07:23,020
to know the level of y, which
is of course just f(x_0).
100
00:07:23,020 --> 00:07:25,080
That's not a calculus
problem, but anyway that's
101
00:07:25,080 --> 00:07:28,350
a very important
part of the process.
102
00:07:28,350 --> 00:07:31,830
So that's the first
thing we need to know.
103
00:07:31,830 --> 00:07:39,490
And the second thing we
need to know is the slope.
104
00:07:39,490 --> 00:07:42,140
And that's this number m.
105
00:07:42,140 --> 00:07:45,340
And in calculus we have
another name for it.
106
00:07:45,340 --> 00:07:48,170
We call it f prime of x_0.
107
00:07:48,170 --> 00:07:51,520
Namely, the derivative of f.
108
00:07:51,520 --> 00:07:53,079
So that's the calculus part.
109
00:07:53,079 --> 00:07:54,870
That's the tricky part,
and that's the part
110
00:07:54,870 --> 00:07:57,760
that we have to discuss now.
111
00:07:57,760 --> 00:08:00,910
So just to make
that explicit here,
112
00:08:00,910 --> 00:08:05,940
I'm going to make a definition,
which is that f '(x_0) ,
113
00:08:05,940 --> 00:08:19,110
which is known as the
derivative, of f, at x_0,
114
00:08:19,110 --> 00:08:40,940
is the slope of the tangent
line to y = f(x) at the point,
115
00:08:40,940 --> 00:08:47,860
let's just call it P.
116
00:08:47,860 --> 00:08:50,000
All right?
117
00:08:50,000 --> 00:08:55,270
So, that's what
it is, but still I
118
00:08:55,270 --> 00:08:59,190
haven't made any progress in
figuring out any better how
119
00:08:59,190 --> 00:09:01,120
I drew that line.
120
00:09:01,120 --> 00:09:03,700
So I have to say
something that's
121
00:09:03,700 --> 00:09:06,210
more concrete, because I
want to be able to cook up
122
00:09:06,210 --> 00:09:07,410
what these numbers are.
123
00:09:07,410 --> 00:09:11,430
I have to figure out
what this number m is.
124
00:09:11,430 --> 00:09:16,870
And one way of thinking about
that, let me just try this,
125
00:09:16,870 --> 00:09:19,094
so I certainly am
taking for granted that
126
00:09:19,094 --> 00:09:20,760
in sort of non-calculus
part that I know
127
00:09:20,760 --> 00:09:22,820
what a line through a point is.
128
00:09:22,820 --> 00:09:24,440
So I know this equation.
129
00:09:24,440 --> 00:09:31,967
But another possibility
might be, this line here,
130
00:09:31,967 --> 00:09:34,050
how do I know - well,
unfortunately, I didn't draw
131
00:09:34,050 --> 00:09:34,170
it quite straight,
but there it is -
132
00:09:34,170 --> 00:09:37,770
how do I know that this orange
line is not a tangent line,
133
00:09:37,770 --> 00:09:45,070
but this other line
is a tangent line?
134
00:09:45,070 --> 00:09:53,010
Well, it's actually
not so obvious,
135
00:09:53,010 --> 00:09:56,200
but I'm gonna describe
it a little bit.
136
00:09:56,200 --> 00:09:58,540
It's not really the
fact-- this thing
137
00:09:58,540 --> 00:10:01,050
crosses at some
other place, which
138
00:10:01,050 --> 00:10:04,490
is this point Q. But
it's not really the fact
139
00:10:04,490 --> 00:10:06,510
that the thing
crosses at two place,
140
00:10:06,510 --> 00:10:07,885
because the line
could be wiggly,
141
00:10:07,885 --> 00:10:10,570
the curve could be wiggly,
and it could cross back
142
00:10:10,570 --> 00:10:11,990
and forth a number of times.
143
00:10:11,990 --> 00:10:17,120
That's not what distinguishes
the tangent line.
144
00:10:17,120 --> 00:10:19,830
So I'm gonna have to
somehow grasp this,
145
00:10:19,830 --> 00:10:23,560
and I'll first do
it in language.
146
00:10:23,560 --> 00:10:27,213
And it's the
following idea: it's
147
00:10:27,213 --> 00:10:31,050
that if you take this
orange line, which
148
00:10:31,050 --> 00:10:37,990
is called a secant line,
and you think of the point Q
149
00:10:37,990 --> 00:10:42,510
as getting closer and closer to
P, then the slope of that line
150
00:10:42,510 --> 00:10:47,860
will get closer and closer
to the slope of the red line.
151
00:10:47,860 --> 00:10:53,247
And if we draw it close
enough, then that's
152
00:10:53,247 --> 00:10:54,330
gonna be the correct line.
153
00:10:54,330 --> 00:10:57,030
So that's really what I did,
sort of in my brain when
154
00:10:57,030 --> 00:10:58,400
I drew that first line.
155
00:10:58,400 --> 00:11:01,010
And so that's the way I'm
going to articulate it first.
156
00:11:01,010 --> 00:11:13,890
Now, so the tangent line is
equal to the limit of so called
157
00:11:13,890 --> 00:11:24,040
secant lines PQ,
as Q tends to P.
158
00:11:24,040 --> 00:11:31,550
And here we're thinking of P as
being fixed and Q as variable.
159
00:11:31,550 --> 00:11:35,420
All right?
160
00:11:35,420 --> 00:11:38,320
Again, this is still the
geometric discussion,
161
00:11:38,320 --> 00:11:42,090
but now we're gonna be able
to put symbols and formulas
162
00:11:42,090 --> 00:11:43,570
to this computation.
163
00:11:43,570 --> 00:11:56,230
And we'll be able to work
out formulas in any example.
164
00:11:56,230 --> 00:11:58,700
So let's do that.
165
00:11:58,700 --> 00:12:05,420
So first of all, I'm gonna write
out these points P and Q again.
166
00:12:05,420 --> 00:12:10,979
So maybe we'll put
P here and Q here.
167
00:12:10,979 --> 00:12:12,770
And I'm thinking of
this line through them.
168
00:12:12,770 --> 00:12:16,190
I guess it was orange, so
we'll leave it as orange.
169
00:12:16,190 --> 00:12:19,570
All right.
170
00:12:19,570 --> 00:12:24,010
And now I want to
compute its slope.
171
00:12:24,010 --> 00:12:27,080
So this, gradually, we'll
do this in two steps.
172
00:12:27,080 --> 00:12:28,970
And these steps
will introduce us
173
00:12:28,970 --> 00:12:31,960
to the basic notations which
are used throughout calculus,
174
00:12:31,960 --> 00:12:35,350
including multi-variable
calculus, across the board.
175
00:12:35,350 --> 00:12:37,960
So the first
notation that's used
176
00:12:37,960 --> 00:12:42,440
is you imagine here's
the x-axis underneath,
177
00:12:42,440 --> 00:12:47,490
and here's the x_0, the location
directly below the point P.
178
00:12:47,490 --> 00:12:51,500
And we're traveling here a
horizontal distance which
179
00:12:51,500 --> 00:12:53,650
is denoted by delta x.
180
00:12:53,650 --> 00:12:58,980
So that's delta x, so called.
181
00:12:58,980 --> 00:13:06,960
And we could also call
it the change in x.
182
00:13:06,960 --> 00:13:09,920
So that's one thing we want
to measure in order to get
183
00:13:09,920 --> 00:13:12,290
the slope of this line PQ.
184
00:13:12,290 --> 00:13:14,530
And the other thing
is this height.
185
00:13:14,530 --> 00:13:18,050
So that's this distance here,
which we denote delta f,
186
00:13:18,050 --> 00:13:21,980
which is the change in f.
187
00:13:21,980 --> 00:13:29,310
And then, the slope is just
the ratio, delta f / delta x.
188
00:13:29,310 --> 00:13:39,780
So this is the
slope of the secant.
189
00:13:39,780 --> 00:13:44,280
And the process I just described
over here with this limit
190
00:13:44,280 --> 00:13:46,380
applies not just to
the whole line itself,
191
00:13:46,380 --> 00:13:48,920
but also in particular
to its slope.
192
00:13:48,920 --> 00:13:53,990
And the way we write that is
the limit as delta x goes to 0.
193
00:13:53,990 --> 00:13:56,970
And that's going
to be our slope.
194
00:13:56,970 --> 00:14:10,850
So this is the slope
of the tangent line.
195
00:14:10,850 --> 00:14:11,860
OK.
196
00:14:11,860 --> 00:14:19,460
Now, This is still
a little general,
197
00:14:19,460 --> 00:14:26,460
and I want to work out
a more usable form here,
198
00:14:26,460 --> 00:14:28,500
a better formula for this.
199
00:14:28,500 --> 00:14:30,710
And in order to
do that, I'm gonna
200
00:14:30,710 --> 00:14:36,360
write delta f, the numerator
more explicitly here.
201
00:14:36,360 --> 00:14:41,130
The change in f, so
remember that the point P
202
00:14:41,130 --> 00:14:43,450
is the point (x_0, f(x_0)).
203
00:14:43,450 --> 00:14:51,180
All right, that's what we got
for the formula for the point.
204
00:14:51,180 --> 00:14:54,955
And in order to
compute these distances
205
00:14:54,955 --> 00:14:57,480
and in particular the
vertical distance here,
206
00:14:57,480 --> 00:15:00,820
I'm gonna have to get a
formula for Q as well.
207
00:15:00,820 --> 00:15:05,300
So if this horizontal
distance is delta x,
208
00:15:05,300 --> 00:15:11,020
then this location
is x_0 + delta x.
209
00:15:11,020 --> 00:15:13,810
And so the point
above that point
210
00:15:13,810 --> 00:15:20,870
has a formula, which
is x_0 plus delta
211
00:15:20,870 --> 00:15:31,680
x, f of - and this is a
mouthful - x_0 plus delta x.
212
00:15:31,680 --> 00:15:33,950
All right, so there's the
formula for the point Q.
213
00:15:33,950 --> 00:15:36,416
Here's the formula
for the point P.
214
00:15:36,416 --> 00:15:47,560
And now I can write a different
formula for the derivative,
215
00:15:47,560 --> 00:15:52,200
which is the following:
so this f'(x_0) ,
216
00:15:52,200 --> 00:15:58,380
which is the same as m, is going
to be the limit as delta x goes
217
00:15:58,380 --> 00:16:05,400
to 0 of the change in f, well
the change in f is the value
218
00:16:05,400 --> 00:16:12,580
of f at the upper point
here, which is x_0 + delta x,
219
00:16:12,580 --> 00:16:19,880
and minus its value at the
lower point P, which is f(x_0),
220
00:16:19,880 --> 00:16:23,250
divided by delta x.
221
00:16:23,250 --> 00:16:24,670
All right, so this
is the formula.
222
00:16:24,670 --> 00:16:28,300
I'm going to put
this in a little box,
223
00:16:28,300 --> 00:16:32,830
because this is by far the
most important formula today,
224
00:16:32,830 --> 00:16:35,410
which we use to derive
pretty much everything else.
225
00:16:35,410 --> 00:16:37,160
And this is the way
that we're going to be
226
00:16:37,160 --> 00:16:46,260
able to compute these numbers.
227
00:16:46,260 --> 00:17:06,280
So let's do an example.
228
00:17:06,280 --> 00:17:13,360
This example, we'll
call this example one.
229
00:17:13,360 --> 00:17:19,790
We'll take the function
f(x) , which is 1/x .
230
00:17:19,790 --> 00:17:23,590
That's sufficiently complicated
to have an interesting answer,
231
00:17:23,590 --> 00:17:27,660
and sufficiently straightforward
that we can compute
232
00:17:27,660 --> 00:17:32,670
the derivative fairly quickly.
233
00:17:32,670 --> 00:17:36,130
So what is it that
we're gonna do here?
234
00:17:36,130 --> 00:17:42,550
All we're going to do is we're
going to plug in this formula
235
00:17:42,550 --> 00:17:44,570
here for that function.
236
00:17:44,570 --> 00:17:47,580
That's all we're going
to do, and visually
237
00:17:47,580 --> 00:17:52,010
what we're accomplishing is
somehow to take the hyperbola,
238
00:17:52,010 --> 00:17:55,050
and take a point
on the hyperbola,
239
00:17:55,050 --> 00:18:00,890
and figure out
some tangent line.
240
00:18:00,890 --> 00:18:03,132
That's what we're
accomplishing when we do that.
241
00:18:03,132 --> 00:18:04,840
So we're accomplishing
this geometrically
242
00:18:04,840 --> 00:18:07,060
but we'll be doing
it algebraically.
243
00:18:07,060 --> 00:18:14,860
So first, we consider this
difference delta f / delta x
244
00:18:14,860 --> 00:18:16,570
and write out its formula.
245
00:18:16,570 --> 00:18:18,190
So I have to have a place.
246
00:18:18,190 --> 00:18:21,670
So I'm gonna make it again
above this point x_0, which
247
00:18:21,670 --> 00:18:22,550
is the general point.
248
00:18:22,550 --> 00:18:25,940
We'll make the
general calculation.
249
00:18:25,940 --> 00:18:30,310
So the value of f at the top,
when we move to the right
250
00:18:30,310 --> 00:18:35,920
by f(x), so I just read off
from this, read off from here.
251
00:18:35,920 --> 00:18:40,900
The formula, the first
thing I get here is 1 /
252
00:18:40,900 --> 00:18:43,530
(x_0 + delta x).
253
00:18:43,530 --> 00:18:46,560
That's the left hand term.
254
00:18:46,560 --> 00:18:50,252
Minus 1 / x_0, that's
the right hand term.
255
00:18:50,252 --> 00:18:54,180
And then I have to
divide that by delta x.
256
00:18:54,180 --> 00:18:57,920
OK, so here's our expression.
257
00:18:57,920 --> 00:19:00,460
And by the way this has a name.
258
00:19:00,460 --> 00:19:10,240
This thing is called
a difference quotient.
259
00:19:10,240 --> 00:19:12,330
It's pretty complicated,
because there's always
260
00:19:12,330 --> 00:19:13,579
a difference in the numerator.
261
00:19:13,579 --> 00:19:16,100
And in disguise, the
denominator is a difference,
262
00:19:16,100 --> 00:19:18,400
because it's the difference
between the value
263
00:19:18,400 --> 00:19:26,310
on the right side and the
value on the left side here.
264
00:19:26,310 --> 00:19:34,740
OK, so now we're going to
simplify it by some algebra.
265
00:19:34,740 --> 00:19:35,860
So let's just take a look.
266
00:19:35,860 --> 00:19:40,260
So this is equal to, let's
continue on the next level
267
00:19:40,260 --> 00:19:41,170
here.
268
00:19:41,170 --> 00:19:45,100
This is equal to 1
/ delta x times...
269
00:19:45,100 --> 00:19:49,220
All I'm going to do is put
it over a common denominator.
270
00:19:49,220 --> 00:19:56,420
So the common denominator
is (x_0 + delta x) * x_0.
271
00:19:56,420 --> 00:20:00,720
And so in the numerator for the
first expressions I have x_0,
272
00:20:00,720 --> 00:20:05,290
and for the second expression
I have x_0 + delta x.
273
00:20:05,290 --> 00:20:08,790
So this is the same thing as
I had in the numerator before,
274
00:20:08,790 --> 00:20:11,450
factoring out this denominator.
275
00:20:11,450 --> 00:20:17,040
And here I put that numerator
into this more amenable form.
276
00:20:17,040 --> 00:20:20,360
And now there are two
basic cancellations.
277
00:20:20,360 --> 00:20:33,160
The first one is that x_0 and
x_0 cancel, so we have this.
278
00:20:33,160 --> 00:20:38,894
And then the second step is that
these two expressions cancel,
279
00:20:38,894 --> 00:20:40,310
the numerator and
the denominator.
280
00:20:40,310 --> 00:20:44,090
Now we have a cancellation
that we can make use of.
281
00:20:44,090 --> 00:20:48,680
So we'll write that under here.
282
00:20:48,680 --> 00:20:57,690
And this is equals -1 over
x_0 plus delta x times x_0.
283
00:20:57,690 --> 00:21:03,480
And then the very last step
is to take the limit as delta
284
00:21:03,480 --> 00:21:09,550
x tends to 0, and
now we can do it.
285
00:21:09,550 --> 00:21:10,690
Before we couldn't do it.
286
00:21:10,690 --> 00:21:11,560
Why?
287
00:21:11,560 --> 00:21:15,220
Because the numerator and the
denominator gave us 0 / 0.
288
00:21:15,220 --> 00:21:17,810
But now that I've made
this cancellation,
289
00:21:17,810 --> 00:21:19,430
I can pass to the limit.
290
00:21:19,430 --> 00:21:22,100
And all that happens is
I set this delta x to 0,
291
00:21:22,100 --> 00:21:22,910
and I get -1/x_0^2.
292
00:21:25,950 --> 00:21:31,728
So that's the answer.
293
00:21:31,728 --> 00:21:33,644
All right, so in other
words what I've shown -
294
00:21:33,644 --> 00:21:36,010
let me put it up here - is
that f'(x_0) = -1/x_0^2.
295
00:21:52,700 --> 00:21:55,910
Now, let's look at the
graph just a little
296
00:21:55,910 --> 00:22:01,610
bit to check this for
plausibility, all right?
297
00:22:01,610 --> 00:22:04,940
What's happening here is,
first of all it's negative.
298
00:22:04,940 --> 00:22:08,320
It's less than 0,
which is a good thing.
299
00:22:08,320 --> 00:22:16,510
You see that slope
there is negative.
300
00:22:16,510 --> 00:22:20,860
That's the simplest check
that you could make.
301
00:22:20,860 --> 00:22:24,530
And the second thing that I
would just like to point out
302
00:22:24,530 --> 00:22:29,380
is that as x goes to infinity,
that as we go farther
303
00:22:29,380 --> 00:22:32,710
to the right, it gets
less and less steep.
304
00:22:32,710 --> 00:22:46,050
So as x_0 goes to infinity,
less and less steep.
305
00:22:46,050 --> 00:22:48,660
So that's also
consistent here, when
306
00:22:48,660 --> 00:22:51,460
x_0 is very large, this is
a smaller and smaller number
307
00:22:51,460 --> 00:22:54,270
in magnitude, although
it's always negative.
308
00:22:54,270 --> 00:23:00,750
It's always sloping down.
309
00:23:00,750 --> 00:23:03,860
All right, so I've managed
to fill the boards.
310
00:23:03,860 --> 00:23:06,010
So maybe I should stop
for a question or two.
311
00:23:06,010 --> 00:23:06,510
Yes?
312
00:23:06,510 --> 00:23:11,430
Student: [INAUDIBLE]
313
00:23:11,430 --> 00:23:18,640
Professor: So the question
is to explain again
314
00:23:18,640 --> 00:23:22,320
this limiting process.
315
00:23:22,320 --> 00:23:26,710
So the formula here is we
have basically two numbers.
316
00:23:26,710 --> 00:23:29,030
So in other words, why is
it that this expression,
317
00:23:29,030 --> 00:23:33,920
when delta x tends to 0,
is equal to -1 / x_0^2 ?
318
00:23:33,920 --> 00:23:37,890
Let me illustrate it by
sticking in a number for x_0
319
00:23:37,890 --> 00:23:39,740
to make it more explicit.
320
00:23:39,740 --> 00:23:42,770
All right, so for
instance, let me stick
321
00:23:42,770 --> 00:23:46,110
in here for x_0 the number 3.
322
00:23:46,110 --> 00:23:52,450
Then it's -1 over 3
plus delta x times 3.
323
00:23:52,450 --> 00:23:54,420
That's the situation
that we've got.
324
00:23:54,420 --> 00:23:56,030
And now the question
is what happens
325
00:23:56,030 --> 00:23:58,680
as this number gets smaller
and smaller and smaller,
326
00:23:58,680 --> 00:24:01,690
and gets to be practically 0?
327
00:24:01,690 --> 00:24:04,535
Well, literally what we can
do is just plug in 0 there,
328
00:24:04,535 --> 00:24:07,410
and you get 3 plus 0 times
3 in the denominator.
329
00:24:07,410 --> 00:24:08,840
-1 in the numerator.
330
00:24:08,840 --> 00:24:16,030
So this tends to
-1/9 (over 3^2).
331
00:24:16,030 --> 00:24:20,871
And that's what I'm saying in
general with this extra number
332
00:24:20,871 --> 00:24:21,370
here.
333
00:24:21,370 --> 00:24:25,360
Other questions?
334
00:24:25,360 --> 00:24:25,860
Yes.
335
00:24:25,860 --> 00:24:34,680
Student: [INAUDIBLE]
336
00:24:34,680 --> 00:24:39,360
Professor: So the
question is what
337
00:24:39,360 --> 00:24:43,650
happened between this
step and this step, right?
338
00:24:43,650 --> 00:24:45,750
Explain this step here.
339
00:24:45,750 --> 00:24:48,150
Alright, so there were
two parts to that.
340
00:24:48,150 --> 00:24:53,440
The first is this delta x which
is sitting in the denominator,
341
00:24:53,440 --> 00:24:56,010
I factored all
the way out front.
342
00:24:56,010 --> 00:24:57,970
And so what's in
the parentheses is
343
00:24:57,970 --> 00:25:00,600
supposed to be
the same as what's
344
00:25:00,600 --> 00:25:03,120
in the numerator of
this other expression.
345
00:25:03,120 --> 00:25:05,570
And then, at the
same time as doing
346
00:25:05,570 --> 00:25:08,080
that, I put that
expression, which
347
00:25:08,080 --> 00:25:10,332
is the difference
of two fractions,
348
00:25:10,332 --> 00:25:12,040
I expressed it with
a common denominator.
349
00:25:12,040 --> 00:25:13,498
So in the denominator
here, you see
350
00:25:13,498 --> 00:25:17,040
the product of the denominators
of the two fractions.
351
00:25:17,040 --> 00:25:20,230
And then I just figured out what
the numerator had to be without
352
00:25:20,230 --> 00:25:22,790
really...
353
00:25:22,790 --> 00:25:27,660
Other questions?
354
00:25:27,660 --> 00:25:32,840
OK.
355
00:25:32,840 --> 00:25:39,670
So I claim that on
the whole, calculus
356
00:25:39,670 --> 00:25:43,100
gets a bad rap,
that it's actually
357
00:25:43,100 --> 00:25:47,220
easier than most things.
358
00:25:47,220 --> 00:25:52,040
But there's a perception
that it's harder.
359
00:25:52,040 --> 00:25:56,840
And so I really have a duty
to give you the calculus made
360
00:25:56,840 --> 00:25:59,210
harder story here.
361
00:25:59,210 --> 00:26:03,630
So we have to make things
harder, because that's our job.
362
00:26:03,630 --> 00:26:06,280
And this is actually what
most people do in calculus,
363
00:26:06,280 --> 00:26:09,560
and it's the reason why
calculus has a bad reputation.
364
00:26:09,560 --> 00:26:15,020
So the secret is
that when people
365
00:26:15,020 --> 00:26:19,360
ask problems in calculus, they
generally ask them in context.
366
00:26:19,360 --> 00:26:22,700
And there are many, many
other things going on.
367
00:26:22,700 --> 00:26:25,420
And so the little piece of
the problem which is calculus
368
00:26:25,420 --> 00:26:28,490
is actually fairly routine and
has to be isolated and gotten
369
00:26:28,490 --> 00:26:28,990
through.
370
00:26:28,990 --> 00:26:31,130
But all the rest of it,
relies on everything else
371
00:26:31,130 --> 00:26:35,987
you learned in mathematics up
to this stage, from grade school
372
00:26:35,987 --> 00:26:36,820
through high school.
373
00:26:36,820 --> 00:26:39,889
So that's the complication.
374
00:26:39,889 --> 00:26:41,930
So now we're going to do
a little bit of calculus
375
00:26:41,930 --> 00:26:49,080
made hard.
376
00:26:49,080 --> 00:26:53,940
By talking about a word problem.
377
00:26:53,940 --> 00:26:57,890
We only have one sort of word
problem that we can pose,
378
00:26:57,890 --> 00:27:03,090
because all we've talked about
is this geometry point of view.
379
00:27:03,090 --> 00:27:06,080
So far those are the only kinds
of word problems we can pose.
380
00:27:06,080 --> 00:27:08,870
So what we're gonna do is
just pose such a problem.
381
00:27:08,870 --> 00:27:23,660
So find the areas of
triangles, enclosed
382
00:27:23,660 --> 00:27:39,770
by the axes and the
tangent to y = 1/x.
383
00:27:39,770 --> 00:27:43,570
OK, so that's a
geometry problem.
384
00:27:43,570 --> 00:27:46,600
And let me draw a picture of it.
385
00:27:46,600 --> 00:27:52,590
It's practically the same as
the picture for example one.
386
00:27:52,590 --> 00:27:54,910
We only consider
the first quadrant.
387
00:27:54,910 --> 00:27:55,820
Here's our shape.
388
00:27:55,820 --> 00:28:00,020
All right, it's the hyperbola.
389
00:28:00,020 --> 00:28:02,640
And here's maybe one
of our tangent lines,
390
00:28:02,640 --> 00:28:05,090
which is coming in like this.
391
00:28:05,090 --> 00:28:12,470
And then we're trying
to find this area here.
392
00:28:12,470 --> 00:28:14,300
Right, so there's our problem.
393
00:28:14,300 --> 00:28:16,070
So why does it have
to do with calculus?
394
00:28:16,070 --> 00:28:17,819
It has to do with
calculus because there's
395
00:28:17,819 --> 00:28:19,790
a tangent line in
it, so we're gonna
396
00:28:19,790 --> 00:28:24,200
need to do some calculus
to answer this question.
397
00:28:24,200 --> 00:28:30,500
But as you'll see, the
calculus is the easy part.
398
00:28:30,500 --> 00:28:34,060
So let's get started
with this problem.
399
00:28:34,060 --> 00:28:37,150
First of all, I'm gonna
label a few things.
400
00:28:37,150 --> 00:28:39,770
And one important thing
to remember of course,
401
00:28:39,770 --> 00:28:42,770
is that the curve is y = 1/x.
402
00:28:42,770 --> 00:28:44,830
That's perfectly
reasonable to do.
403
00:28:44,830 --> 00:28:48,850
And also, we're gonna calculate
the areas of the triangles,
404
00:28:48,850 --> 00:28:51,530
and you could ask
yourself, in terms of what?
405
00:28:51,530 --> 00:28:54,296
Well, we're gonna have to pick
a point and give it a name.
406
00:28:54,296 --> 00:28:55,670
And since we need
a number, we're
407
00:28:55,670 --> 00:28:57,169
gonna have to do
more than geometry.
408
00:28:57,169 --> 00:28:59,290
We're gonna have to do
some of this analysis
409
00:28:59,290 --> 00:29:01,010
just as we've done before.
410
00:29:01,010 --> 00:29:04,130
So I'm gonna pick a point and,
consistent with the labeling
411
00:29:04,130 --> 00:29:08,320
we've done before, I'm
gonna to call it (x_0, y_0).
412
00:29:08,320 --> 00:29:13,370
So that's almost half the
battle, having notations, x
413
00:29:13,370 --> 00:29:16,220
and y for the variables,
and x_0 and y_0,
414
00:29:16,220 --> 00:29:18,960
for the specific point.
415
00:29:18,960 --> 00:29:24,310
Now, once you see that
you have these labelings,
416
00:29:24,310 --> 00:29:28,070
I hope it's reasonable
to do the following.
417
00:29:28,070 --> 00:29:31,380
So first of all, this
is the point x_0,
418
00:29:31,380 --> 00:29:33,630
and over here is the point y_0.
419
00:29:33,630 --> 00:29:37,730
That's something that
we're used to in graphs.
420
00:29:37,730 --> 00:29:40,200
And in order to figure out
the area of this triangle,
421
00:29:40,200 --> 00:29:41,950
it's pretty clear
that we should find
422
00:29:41,950 --> 00:29:45,740
the base, which is that we
should find this location here.
423
00:29:45,740 --> 00:29:47,950
And we should find
the height, so we
424
00:29:47,950 --> 00:29:55,590
need to find that value there.
425
00:29:55,590 --> 00:29:58,810
Let's go ahead and do it.
426
00:29:58,810 --> 00:30:02,390
So how are we going to do this?
427
00:30:02,390 --> 00:30:14,960
Well, so let's just take a look.
428
00:30:14,960 --> 00:30:17,150
So what is it that
we need to do?
429
00:30:17,150 --> 00:30:21,073
I claim that there's
only one calculus step,
430
00:30:21,073 --> 00:30:25,630
and I'm gonna put a star
here for this tangent line.
431
00:30:25,630 --> 00:30:28,472
I have to understand
what the tangent line is.
432
00:30:28,472 --> 00:30:30,430
Once I've figured out
what the tangent line is,
433
00:30:30,430 --> 00:30:33,230
the rest of the problem
is no longer calculus.
434
00:30:33,230 --> 00:30:35,890
It's just that
slope that we need.
435
00:30:35,890 --> 00:30:38,410
So what's the formula
for the tangent line?
436
00:30:38,410 --> 00:30:45,830
Put that over here. it's going
to be y - y_0 is equal to,
437
00:30:45,830 --> 00:30:48,180
and here's the magic number,
we already calculated it.
438
00:30:48,180 --> 00:30:50,970
It's in the box over there.
439
00:30:50,970 --> 00:30:58,270
It's -1/x_0^2 ( x - x_0).
440
00:30:58,270 --> 00:31:12,670
So this is the only bit of
calculus in this problem.
441
00:31:12,670 --> 00:31:15,500
But now we're not done.
442
00:31:15,500 --> 00:31:16,750
We have to finish it.
443
00:31:16,750 --> 00:31:19,170
We have to figure out all
the rest of these quantities
444
00:31:19,170 --> 00:31:27,300
so we can figure out the area.
445
00:31:27,300 --> 00:31:31,160
All right.
446
00:31:31,160 --> 00:31:40,920
So how do we do that?
447
00:31:40,920 --> 00:31:44,680
Well, to find this
point, this has a name.
448
00:31:44,680 --> 00:31:52,730
We're gonna find the
so called x-intercept.
449
00:31:52,730 --> 00:31:54,630
That's the first thing
we're going to do.
450
00:31:54,630 --> 00:31:57,800
So to do that,
what we need to do
451
00:31:57,800 --> 00:32:02,450
is to find where this horizontal
line meets that diagonal line.
452
00:32:02,450 --> 00:32:10,910
And the equation for the
x-intercept is y = 0.
453
00:32:10,910 --> 00:32:13,315
So we plug in y = 0, that's
this horizontal line,
454
00:32:13,315 --> 00:32:15,240
and we find this point.
455
00:32:15,240 --> 00:32:18,440
So let's do that into star.
456
00:32:18,440 --> 00:32:22,830
We get 0 minus, oh one
other thing we need to know.
457
00:32:22,830 --> 00:32:28,770
We know that y0 is f(x_0)
, and f(x) is 1/x ,
458
00:32:28,770 --> 00:32:31,060
so this thing is 1/x_0.
459
00:32:33,780 --> 00:32:38,600
And that's equal to -1/x_0^2.
460
00:32:38,600 --> 00:32:41,920
And here's x, and here's x_0.
461
00:32:41,920 --> 00:32:46,500
All right, so in order
to find this x value,
462
00:32:46,500 --> 00:32:53,800
I have to plug in one
equation into the other.
463
00:32:53,800 --> 00:32:59,170
So this simplifies a bit.
464
00:32:59,170 --> 00:33:03,250
This is -x/x_0^2.
465
00:33:03,250 --> 00:33:09,420
And this is plus 1/x_0 because
the x_0 and x0^2 cancel
466
00:33:09,420 --> 00:33:10,480
somewhat.
467
00:33:10,480 --> 00:33:12,830
And so if I put this
on the other side,
468
00:33:12,830 --> 00:33:20,810
I get x / x_0^2 is
equal to 2 / x_0.
469
00:33:20,810 --> 00:33:27,878
And if I then multiply through
- so that's what this implies -
470
00:33:27,878 --> 00:33:39,930
and if I multiply through
by x_0^2 I get x = 2x_0.
471
00:33:39,930 --> 00:33:42,270
OK, so I claim that this
point we've just calculated,
472
00:33:42,270 --> 00:33:51,840
it's 2x_0.
473
00:33:51,840 --> 00:33:57,320
Now, I'm almost done.
474
00:33:57,320 --> 00:34:00,210
I need to get the other one.
475
00:34:00,210 --> 00:34:03,280
I need to get this one up here.
476
00:34:03,280 --> 00:34:06,600
Now I'm gonna use a very
big shortcut to do that.
477
00:34:06,600 --> 00:34:27,490
So the shortcut to the
y-intercept is to use symmetry.
478
00:34:27,490 --> 00:34:30,900
All right, I claim I can stare
at this and I can look at that,
479
00:34:30,900 --> 00:34:33,540
and I know the formula
for the y-intercept.
480
00:34:33,540 --> 00:34:39,891
It's equal to 2y_0.
481
00:34:39,891 --> 00:34:40,390
All right.
482
00:34:40,390 --> 00:34:42,120
That's what that one is.
483
00:34:42,120 --> 00:34:44,320
So this one is 2y_0.
484
00:34:44,320 --> 00:34:48,060
And the reason I know this
is the following: so here's
485
00:34:48,060 --> 00:34:52,690
the symmetry of the situation,
which is not completely direct.
486
00:34:52,690 --> 00:34:56,350
It's a kind of mirror
symmetry around the diagonal.
487
00:34:56,350 --> 00:35:05,380
It involves the exchange
of (x, y) with (y, x);
488
00:35:05,380 --> 00:35:06,850
so trading the roles of x and y.
489
00:35:06,850 --> 00:35:08,720
So the symmetry
that I'm using is
490
00:35:08,720 --> 00:35:11,980
that any formula I get that
involves x's and y's, if I
491
00:35:11,980 --> 00:35:14,520
trade all the x's and
replace them by y's and trade
492
00:35:14,520 --> 00:35:16,645
all the y's and replace
them by x's, then
493
00:35:16,645 --> 00:35:18,720
I'll have a correct
formula on the other way.
494
00:35:18,720 --> 00:35:20,910
So if everywhere I see
a y I make it an x,
495
00:35:20,910 --> 00:35:22,876
and everywhere I see
an x I make it a y,
496
00:35:22,876 --> 00:35:24,000
the switch will take place.
497
00:35:24,000 --> 00:35:27,230
So why is that?
498
00:35:27,230 --> 00:35:30,450
That's just an accident
of this equation.
499
00:35:30,450 --> 00:35:46,070
That's because, so the
symmetry explained...
500
00:35:46,070 --> 00:35:48,160
is that the equation is y = 1/x.
501
00:35:48,160 --> 00:35:52,690
But that's the same
thing as xy = 1,
502
00:35:52,690 --> 00:35:54,710
if I multiply
through by x, which
503
00:35:54,710 --> 00:35:58,740
is the same thing as x = 1/y.
504
00:35:58,740 --> 00:36:05,450
So here's where the x
and the y get reversed.
505
00:36:05,450 --> 00:36:08,470
OK now if you don't
trust this explanation,
506
00:36:08,470 --> 00:36:23,190
you can also get the
y-intercept by plugging x = 0
507
00:36:23,190 --> 00:36:28,720
into the equation star.
508
00:36:28,720 --> 00:36:29,300
OK?
509
00:36:29,300 --> 00:36:34,060
We plugged y = 0 in
and we got the x-value.
510
00:36:34,060 --> 00:36:43,080
And you can do the same thing
analogously the other way.
511
00:36:43,080 --> 00:36:47,160
All right so I'm almost done
with the geometry problem,
512
00:36:47,160 --> 00:36:58,280
and let's finish it off now.
513
00:36:58,280 --> 00:37:00,930
Well, let me hold off for one
second before I finish it off.
514
00:37:00,930 --> 00:37:05,030
What I'd like to say is just
make one more tiny remark.
515
00:37:05,030 --> 00:37:09,200
And this is the hardest part
of calculus in my opinion.
516
00:37:09,200 --> 00:37:11,890
So the hardest
part of calculus is
517
00:37:11,890 --> 00:37:17,560
that we call it one
variable calculus,
518
00:37:17,560 --> 00:37:20,080
but we're perfectly
happy to deal
519
00:37:20,080 --> 00:37:25,500
with four variables at a
time or five, or any number.
520
00:37:25,500 --> 00:37:29,800
In this problem, I had an
x, a y, an x_0 and a y_0.
521
00:37:29,800 --> 00:37:32,080
That's already four
different things
522
00:37:32,080 --> 00:37:35,464
that have various
relationships between them.
523
00:37:35,464 --> 00:37:37,880
Of course the manipulations
we do with them are algebraic,
524
00:37:37,880 --> 00:37:39,820
and when we're doing
the derivatives
525
00:37:39,820 --> 00:37:43,216
we just consider what's known
as one variable calculus.
526
00:37:43,216 --> 00:37:45,590
But really there are millions
of variable floating around
527
00:37:45,590 --> 00:37:46,930
potentially.
528
00:37:46,930 --> 00:37:49,280
So that's what makes
things complicated,
529
00:37:49,280 --> 00:37:51,380
and that's something that
you have to get used to.
530
00:37:51,380 --> 00:37:53,580
Now there's something
else which is more subtle,
531
00:37:53,580 --> 00:37:57,360
and that I think many
people who teach the subject
532
00:37:57,360 --> 00:38:00,380
or use the subject aren't
aware, because they've already
533
00:38:00,380 --> 00:38:03,960
entered into the language and
they're so comfortable with it
534
00:38:03,960 --> 00:38:06,820
that they don't even
notice this confusion.
535
00:38:06,820 --> 00:38:10,180
There's something deliberately
sloppy about the way
536
00:38:10,180 --> 00:38:12,700
we deal with these variables.
537
00:38:12,700 --> 00:38:14,770
The reason is very simple.
538
00:38:14,770 --> 00:38:16,750
There are already
four variables here.
539
00:38:16,750 --> 00:38:20,680
I don't wanna create six names
for variables or eight names
540
00:38:20,680 --> 00:38:23,620
for variables.
541
00:38:23,620 --> 00:38:26,710
But really in this problem
there were about eight.
542
00:38:26,710 --> 00:38:29,280
I just slipped them by you.
543
00:38:29,280 --> 00:38:30,750
So why is that?
544
00:38:30,750 --> 00:38:35,910
Well notice that the first time
that I got a formula for y_0
545
00:38:35,910 --> 00:38:39,580
here, it was this point.
546
00:38:39,580 --> 00:38:44,680
And so the formula for y_0,
which I plugged in right here,
547
00:38:44,680 --> 00:38:50,280
was from the equation of
the curve. y_0 = 1 / x_0.
548
00:38:50,280 --> 00:38:55,320
The second time I did it,
I did not use y = 1/x.
549
00:38:55,320 --> 00:39:01,640
I used this equation here,
so this is not y = 1/x.
550
00:39:01,640 --> 00:39:03,480
That's the wrong thing to do.
551
00:39:03,480 --> 00:39:05,960
It's an easy mistake to
make if the formulas are
552
00:39:05,960 --> 00:39:08,810
all a blur to you and
you're not paying attention
553
00:39:08,810 --> 00:39:11,140
to where they are
on the diagram.
554
00:39:11,140 --> 00:39:16,740
You see that x-intercept
calculation there involved
555
00:39:16,740 --> 00:39:21,420
where this horizontal line met
this diagonal line, and y = 0
556
00:39:21,420 --> 00:39:25,890
represented this line here.
557
00:39:25,890 --> 00:39:31,520
So the sloppiness is that y
means two different things.
558
00:39:31,520 --> 00:39:34,450
And we do this constantly
because it's way, way more
559
00:39:34,450 --> 00:39:37,640
complicated not to do it.
560
00:39:37,640 --> 00:39:40,060
It's much more convenient
for us to allow ourselves
561
00:39:40,060 --> 00:39:42,660
the flexibility
to change the role
562
00:39:42,660 --> 00:39:47,730
that this letter plays in
the middle of a computation.
563
00:39:47,730 --> 00:39:50,110
And similarly, later
on, if I had done this
564
00:39:50,110 --> 00:39:54,110
by this more straightforward
method, for the y-intercept,
565
00:39:54,110 --> 00:39:55,360
I would have set x equal to 0.
566
00:39:55,360 --> 00:39:59,990
That would have been this
vertical line, which is x = 0.
567
00:39:59,990 --> 00:40:03,520
But I didn't change the letter
x when I did that, because that
568
00:40:03,520 --> 00:40:06,180
would be a waste for us.
569
00:40:06,180 --> 00:40:09,340
So this is one of the main
confusions that happens.
570
00:40:09,340 --> 00:40:12,460
If you can keep
yourself straight,
571
00:40:12,460 --> 00:40:15,310
you're a lot better
off, and as I
572
00:40:15,310 --> 00:40:21,720
say this is one of
the complexities.
573
00:40:21,720 --> 00:40:24,910
All right, so now let's
finish off the problem.
574
00:40:24,910 --> 00:40:30,880
Let me finally get
this area here.
575
00:40:30,880 --> 00:40:33,730
So, actually I'll just
finish it off right here.
576
00:40:33,730 --> 00:40:41,240
So the area of the
triangle is, well
577
00:40:41,240 --> 00:40:42,880
it's the base times the height.
578
00:40:42,880 --> 00:40:46,550
The base is 2x_0, the height
is 2y_0, and a half of that.
579
00:40:46,550 --> 00:40:54,720
So it's 1/2 (2x_0) * (2y_0) ,
which is 2x_0 y_0, which is,
580
00:40:54,720 --> 00:40:57,780
lo and behold, 2.
581
00:40:57,780 --> 00:40:59,370
So the amusing
thing in this case
582
00:40:59,370 --> 00:41:02,010
is that it actually didn't
matter what x_0 and y_0 are.
583
00:41:02,010 --> 00:41:05,870
We get the same
answer every time.
584
00:41:05,870 --> 00:41:10,220
That's just an accident
of the function 1 / x.
585
00:41:10,220 --> 00:41:19,740
It happens to be the
function with that property.
586
00:41:19,740 --> 00:41:23,790
All right, so we have
some more business today,
587
00:41:23,790 --> 00:41:24,960
some serious business.
588
00:41:24,960 --> 00:41:30,980
So let me continue.
589
00:41:30,980 --> 00:41:41,270
So, first of all, I want to
give you a few more notations.
590
00:41:41,270 --> 00:41:49,420
And these are just
other notations
591
00:41:49,420 --> 00:41:51,790
that people use to
refer to derivatives.
592
00:41:51,790 --> 00:41:53,920
And the first one
is the following:
593
00:41:53,920 --> 00:41:56,850
we already wrote y = f(x).
594
00:41:56,850 --> 00:41:59,630
And so when we write
delta y, that means
595
00:41:59,630 --> 00:42:01,960
the same thing as delta f.
596
00:42:01,960 --> 00:42:04,350
That's a typical notation.
597
00:42:04,350 --> 00:42:13,670
And previously we wrote f
prime for the derivative,
598
00:42:13,670 --> 00:42:20,530
so this is Newton's
notation for the derivative.
599
00:42:20,530 --> 00:42:22,520
But there are other notations.
600
00:42:22,520 --> 00:42:27,830
And one of them is df/dx,
and another one is dy/dx,
601
00:42:27,830 --> 00:42:29,840
meaning exactly the same thing.
602
00:42:29,840 --> 00:42:32,870
And sometimes we
let the function
603
00:42:32,870 --> 00:42:40,520
slip down below so that becomes
d/dx of f and d/dx of y.
604
00:42:40,520 --> 00:42:44,340
So these are all notations that
are used for the derivative,
605
00:42:44,340 --> 00:42:49,150
and these were
initiated by Leibniz.
606
00:42:49,150 --> 00:42:55,120
And these notations are used
interchangeably, sometimes
607
00:42:55,120 --> 00:42:56,740
practically together.
608
00:42:56,740 --> 00:42:59,750
They both turn out to
be extremely useful.
609
00:42:59,750 --> 00:43:03,314
This one omits - notice
that this thing omits
610
00:43:03,314 --> 00:43:07,140
- the underlying
base point, x_0.
611
00:43:07,140 --> 00:43:09,100
That's one of the nuisances.
612
00:43:09,100 --> 00:43:11,440
It doesn't give you
all the information.
613
00:43:11,440 --> 00:43:18,070
But there are lots of situations
like that where people leave
614
00:43:18,070 --> 00:43:20,090
out some of the
important information,
615
00:43:20,090 --> 00:43:23,150
and you have to fill
it in from context.
616
00:43:23,150 --> 00:43:28,360
So that's another
couple of notations.
617
00:43:28,360 --> 00:43:33,330
So now I have one more
calculation for you today.
618
00:43:33,330 --> 00:43:35,530
I carried out this
calculation of the derivative
619
00:43:35,530 --> 00:43:45,780
of the function 1 / x.
620
00:43:45,780 --> 00:43:48,640
I wanna take care of
some other powers.
621
00:43:48,640 --> 00:43:59,150
So let's do that.
622
00:43:59,150 --> 00:44:08,730
So Example 2 is going to
be the function f(x) = x^n.
623
00:44:08,730 --> 00:44:14,160
n = 1, 2, 3; one of these guys.
624
00:44:14,160 --> 00:44:18,270
And now what we're trying to
figure out is the derivative
625
00:44:18,270 --> 00:44:21,470
with respect to x of
x^n in our new notation,
626
00:44:21,470 --> 00:44:27,070
what this is equal to.
627
00:44:27,070 --> 00:44:33,450
So again, we're going to form
this expression, delta f /
628
00:44:33,450 --> 00:44:35,370
delta x.
629
00:44:35,370 --> 00:44:38,830
And we're going to make some
algebraic simplification.
630
00:44:38,830 --> 00:44:44,950
So what we plug in for
delta f is ((x delta x)^n -
631
00:44:44,950 --> 00:44:48,240
x^n)/delta x.
632
00:44:48,240 --> 00:44:50,460
Now before, let
me just stick this
633
00:44:50,460 --> 00:44:52,350
in then I'm gonna erase it.
634
00:44:52,350 --> 00:44:56,490
Before, I wrote x_0
here and x_0 there.
635
00:44:56,490 --> 00:44:59,390
But now I'm going
to get rid of it,
636
00:44:59,390 --> 00:45:01,980
because in this particular
calculation, it's a nuisance.
637
00:45:01,980 --> 00:45:03,540
I don't have an x
floating around,
638
00:45:03,540 --> 00:45:06,310
which means something
different from the x_0.
639
00:45:06,310 --> 00:45:08,110
And I just don't
wanna have to keep
640
00:45:08,110 --> 00:45:10,250
on writing all those symbols.
641
00:45:10,250 --> 00:45:13,850
It's a waste of
blackboard energy.
642
00:45:13,850 --> 00:45:15,500
There's a total
amount of energy,
643
00:45:15,500 --> 00:45:18,264
and I've already filled
up so many blackboards
644
00:45:18,264 --> 00:45:21,710
that, there's just
a limited amount.
645
00:45:21,710 --> 00:45:23,650
Plus, I'm trying
to conserve chalk.
646
00:45:23,650 --> 00:45:25,880
Anyway, no 0's.
647
00:45:25,880 --> 00:45:28,840
So think of x as fixed.
648
00:45:28,840 --> 00:45:40,030
In this case, delta x moves and
x is fixed in this calculation.
649
00:45:40,030 --> 00:45:42,380
All right now, in order
to simplify this, in order
650
00:45:42,380 --> 00:45:44,810
to understand algebraically
what's going on,
651
00:45:44,810 --> 00:45:48,430
I need to understand what
the nth power of a sum is.
652
00:45:48,430 --> 00:45:50,100
And that's a famous formula.
653
00:45:50,100 --> 00:45:52,550
We only need a little
tiny bit of it,
654
00:45:52,550 --> 00:45:56,040
called the binomial theorem.
655
00:45:56,040 --> 00:46:06,350
So, the binomial theorem
which is in your text
656
00:46:06,350 --> 00:46:12,820
and explained in
an appendix, says
657
00:46:12,820 --> 00:46:15,880
that if you take
the sum of two guys
658
00:46:15,880 --> 00:46:18,190
and you take them to the
nth power, that of course
659
00:46:18,190 --> 00:46:24,750
is (x + delta x) multiplied
by itself n times.
660
00:46:24,750 --> 00:46:29,900
And so the first term is
x^n, that's when all of the n
661
00:46:29,900 --> 00:46:31,690
factors come in.
662
00:46:31,690 --> 00:46:35,620
And then, you could have this
factor of delta x and all
663
00:46:35,620 --> 00:46:36,786
the rest x's.
664
00:46:36,786 --> 00:46:39,160
So at least one term of the
form (x^(n-1)) times delta x.
665
00:46:41,820 --> 00:46:43,650
And how many times
does that happen?
666
00:46:43,650 --> 00:46:45,930
Well, it happens when
there's a factor from here,
667
00:46:45,930 --> 00:46:48,230
from the next factor, and
so on, and so on, and so on.
668
00:46:48,230 --> 00:46:54,330
There's a total of n possible
times that that happens.
669
00:46:54,330 --> 00:46:59,110
And now the great thing
is that, with this alone,
670
00:46:59,110 --> 00:47:05,120
all the rest of the
terms are junk that we
671
00:47:05,120 --> 00:47:06,640
won't have to worry about.
672
00:47:06,640 --> 00:47:11,650
So to be more specific,
there's a very careful notation
673
00:47:11,650 --> 00:47:12,460
for the junk.
674
00:47:12,460 --> 00:47:14,840
The junk is what's called
big O of (delta x)^2.
675
00:47:18,070 --> 00:47:25,740
What that means is that
these are terms of order,
676
00:47:25,740 --> 00:47:33,676
so with (delta x)^2,
(delta x)^3 or higher.
677
00:47:33,676 --> 00:47:38,780
All right, that's how.
678
00:47:38,780 --> 00:47:42,430
Very exciting,
higher order terms.
679
00:47:42,430 --> 00:47:47,540
OK, so this is the only
algebra that we need to do,
680
00:47:47,540 --> 00:47:50,890
and now we just need to combine
it together to get our result.
681
00:47:50,890 --> 00:47:54,750
So, now I'm going to just
carry out the cancellations
682
00:47:54,750 --> 00:48:02,480
that we need.
683
00:48:02,480 --> 00:48:03,790
So here we go.
684
00:48:03,790 --> 00:48:11,990
We have delta f / delta x, which
remember was 1 / delta x times
685
00:48:11,990 --> 00:48:25,420
this, which is this times, now
this is x^n plus nx^(n-1) delta
686
00:48:25,420 --> 00:48:35,320
x plus this junk
term, minus x^n.
687
00:48:35,320 --> 00:48:38,380
So that's what we
have so far based
688
00:48:38,380 --> 00:48:41,670
on our previous calculations.
689
00:48:41,670 --> 00:48:48,110
Now, I'm going to do the main
cancellation, which is this.
690
00:48:48,110 --> 00:48:49,420
All right.
691
00:48:49,420 --> 00:48:56,730
So, that's 1/delta x times
nx^(n-1) delta x plus this term
692
00:48:56,730 --> 00:49:01,220
here.
693
00:49:01,220 --> 00:49:05,020
And now I can divide
in by delta x.
694
00:49:05,020 --> 00:49:10,530
So I get nx^(n-1) plus,
now it's O(delta x).
695
00:49:10,530 --> 00:49:12,230
There's at least
one factor of delta
696
00:49:12,230 --> 00:49:14,390
x not two factors of
delta x, because I
697
00:49:14,390 --> 00:49:17,610
have to cancel one of them.
698
00:49:17,610 --> 00:49:19,860
And now I can just
take the limit.
699
00:49:19,860 --> 00:49:22,410
In the limit this
term is gonna be 0.
700
00:49:22,410 --> 00:49:25,850
That's why I called
it junk originally,
701
00:49:25,850 --> 00:49:26,960
because it disappears.
702
00:49:26,960 --> 00:49:31,010
And in math, junk is
something that goes away.
703
00:49:31,010 --> 00:49:37,350
So this tends to, as delta
x goes to 0, nx^(n-1).
704
00:49:37,350 --> 00:49:43,260
And so what I've shown you is
that d/dx of x to the n minus--
705
00:49:43,260 --> 00:49:47,790
sorry, n, is equal to nx^(n-1).
706
00:49:51,180 --> 00:49:53,710
So now this is gonna be
super important to you
707
00:49:53,710 --> 00:49:56,520
right on your problem set
in every possible way,
708
00:49:56,520 --> 00:49:59,040
and I want to tell you one
thing, one way in which it's
709
00:49:59,040 --> 00:50:00,200
very important.
710
00:50:00,200 --> 00:50:02,240
One way that extends
it immediately.
711
00:50:02,240 --> 00:50:10,950
So this thing extends
to polynomials.
712
00:50:10,950 --> 00:50:13,970
We get quite a lot out
of this one calculation.
713
00:50:13,970 --> 00:50:21,960
Namely, if I take d/dx of
something like (x^3 + 5x^10)
714
00:50:21,960 --> 00:50:25,900
that's gonna be equal to 3x^2,
that's applying this rule
715
00:50:25,900 --> 00:50:27,240
to x^3.
716
00:50:27,240 --> 00:50:35,110
And then here, I'll
get 5*10 so 50x^9.
717
00:50:35,110 --> 00:50:37,760
So this is the type of
thing that we get out of it,
718
00:50:37,760 --> 00:50:49,727
and we're gonna make more
hay with that next time.
719
00:50:49,727 --> 00:50:50,227
Question.
720
00:50:50,227 --> 00:50:50,727
Yes.
721
00:50:50,727 --> 00:50:51,690
I turned myself off.
722
00:50:51,690 --> 00:50:52,190
Yes?
723
00:50:52,190 --> 00:50:56,030
Student: [INAUDIBLE]
724
00:50:56,030 --> 00:51:01,030
Professor: The question
was the binomial theorem
725
00:51:01,030 --> 00:51:04,370
only works when
delta x goes to 0.
726
00:51:04,370 --> 00:51:06,930
No, the binomial theorem
is a general formula
727
00:51:06,930 --> 00:51:10,030
which also specifies
exactly what the junk is.
728
00:51:10,030 --> 00:51:11,950
It's very much more detailed.
729
00:51:11,950 --> 00:51:13,730
But we only needed this part.
730
00:51:13,730 --> 00:51:18,550
We didn't care what all
these crazy terms were.
731
00:51:18,550 --> 00:51:23,430
It's junk for our
purposes now, because we
732
00:51:23,430 --> 00:51:27,650
don't happen to need any more
than those first two terms.
733
00:51:27,650 --> 00:51:29,910
Yes, because delta x goes to 0.
734
00:51:29,910 --> 00:51:32,380
OK, see you next time.