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Okay so I'd like to begin the
second lecture by reminding you
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00:00:27,490 --> 00:00:30,770
what we did last time.
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00:00:30,770 --> 00:00:50,520
So last time, we
defined the derivative
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00:00:50,520 --> 00:01:04,260
as the slope of a tangent line.
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00:01:04,260 --> 00:01:07,080
So that was our
geometric point of view
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00:01:07,080 --> 00:01:10,470
and we also did a
couple of computations.
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00:01:10,470 --> 00:01:18,290
We worked out that the
derivative of 1 / x was -1 /
15
00:01:18,290 --> 00:01:20,100
x^2.
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00:01:20,100 --> 00:01:24,870
And we also computed the
derivative of x to the nth
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00:01:24,870 --> 00:01:32,097
power for n = 1, 2, etc.,
and that turned out to be x,
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00:01:32,097 --> 00:01:32,930
I'm sorry, nx^(n-1).
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00:01:36,970 --> 00:01:46,180
So that's what we did
last time, and today I
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00:01:46,180 --> 00:01:51,580
want to finish up with
other points of view
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00:01:51,580 --> 00:01:53,430
on what a derivative is.
22
00:01:53,430 --> 00:01:56,250
So this is extremely
important, it's
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00:01:56,250 --> 00:01:58,750
almost the most important thing
I'll be saying in the class.
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00:01:58,750 --> 00:02:01,310
But you'll have to think about
it again when you start over
25
00:02:01,310 --> 00:02:04,600
and start using calculus
in the real world.
26
00:02:04,600 --> 00:02:14,030
So again we're talking
about what is a derivative
27
00:02:14,030 --> 00:02:19,660
and this is just a
continuation of last time.
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00:02:19,660 --> 00:02:23,260
So, as I said last time,
we talked about geometric
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00:02:23,260 --> 00:02:28,130
interpretations, and today
what we're gonna talk about
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00:02:28,130 --> 00:02:34,310
is rate of change
as an interpretation
31
00:02:34,310 --> 00:02:40,000
of the derivative.
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00:02:40,000 --> 00:02:46,260
So remember we drew graphs
of functions, y = f(x)
33
00:02:46,260 --> 00:02:53,190
and we kept track of the change
in x and here the change in y,
34
00:02:53,190 --> 00:02:56,140
let's say.
35
00:02:56,140 --> 00:03:01,872
And then from this new point
of view a rate of change,
36
00:03:01,872 --> 00:03:04,080
keeping track of the rate
of change of x and the rate
37
00:03:04,080 --> 00:03:07,030
of change of y, it's the
relative rate of change
38
00:03:07,030 --> 00:03:12,590
we're interested in, and that's
delta y / delta x and that
39
00:03:12,590 --> 00:03:16,010
has another interpretation.
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00:03:16,010 --> 00:03:21,650
This is the average change.
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00:03:21,650 --> 00:03:26,880
Usually we would think of that,
if x were measuring time and so
42
00:03:26,880 --> 00:03:31,350
the average and that's
when this becomes a rate,
43
00:03:31,350 --> 00:03:35,830
and the average is over
the time interval delta x.
44
00:03:35,830 --> 00:03:42,670
And then the limiting
value is denoted dy/dx
45
00:03:42,670 --> 00:03:47,580
and so this one is the
average rate of change
46
00:03:47,580 --> 00:03:59,860
and this one is the
instantaneous rate.
47
00:03:59,860 --> 00:04:01,412
Okay, so that's
the point of view
48
00:04:01,412 --> 00:04:03,120
that I'd like to
discuss now and give you
49
00:04:03,120 --> 00:04:06,200
just a couple of examples.
50
00:04:06,200 --> 00:04:12,980
So, let's see.
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00:04:12,980 --> 00:04:19,620
Well, first of all, maybe some
examples from physics here.
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00:04:19,620 --> 00:04:26,450
So q is usually the
name for a charge,
53
00:04:26,450 --> 00:04:33,660
and then dq/dt is
what's known as current.
54
00:04:33,660 --> 00:04:38,600
So that's one physical example.
55
00:04:38,600 --> 00:04:45,200
A second example, which is
probably the most tangible one,
56
00:04:45,200 --> 00:04:51,500
is we could denote the
letter s by distance
57
00:04:51,500 --> 00:04:58,520
and then the rate of change
is what we call speed.
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00:04:58,520 --> 00:05:02,110
So those are the
two typical examples
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00:05:02,110 --> 00:05:06,550
and I just want to
illustrate the second example
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00:05:06,550 --> 00:05:08,900
in a little bit more
detail because I think
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00:05:08,900 --> 00:05:12,690
it's important to have some
visceral sense of this notion
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00:05:12,690 --> 00:05:16,320
of instantaneous speed.
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00:05:16,320 --> 00:05:22,570
And I get to use the example of
this very building to do that.
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00:05:22,570 --> 00:05:25,860
Probably you know,
or maybe you don't,
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00:05:25,860 --> 00:05:29,690
that on Halloween
there's an event that
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00:05:29,690 --> 00:05:33,042
takes place in this
building or really
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00:05:33,042 --> 00:05:34,500
from the top of
this building which
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00:05:34,500 --> 00:05:37,000
is called the pumpkin drop.
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00:05:37,000 --> 00:05:44,450
So let's illustrates this
idea of rate of change
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00:05:44,450 --> 00:05:49,040
with the pumpkin drop.
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00:05:49,040 --> 00:05:53,800
So what happens is,
this building-- well
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00:05:53,800 --> 00:06:01,070
let's see here's the building,
and here's the dot, that's
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00:06:01,070 --> 00:06:04,890
the beautiful grass out on
this side of the building,
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00:06:04,890 --> 00:06:09,430
and then there's
some people up here
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00:06:09,430 --> 00:06:12,310
and very small
objects, well they're
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00:06:12,310 --> 00:06:15,240
not that small when
you're close to them, that
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00:06:15,240 --> 00:06:19,000
get dumped over the side there.
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00:06:19,000 --> 00:06:21,510
And they fall down.
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00:06:21,510 --> 00:06:24,170
You know everything at MIT
or a lot of things at MIT
80
00:06:24,170 --> 00:06:28,430
are physics experiments.
81
00:06:28,430 --> 00:06:29,440
That's the pumpkin drop.
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00:06:29,440 --> 00:06:32,555
So roughly speaking,
the building
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00:06:32,555 --> 00:06:36,360
is about 300 feet
high, we're down here
84
00:06:36,360 --> 00:06:39,570
on the first usable floor.
85
00:06:39,570 --> 00:06:44,130
And so we're going to
use instead of 300 feet,
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00:06:44,130 --> 00:06:46,330
just for convenience
purposes we'll
87
00:06:46,330 --> 00:06:55,410
use 80 meters because that makes
the numbers come out simply.
88
00:06:55,410 --> 00:07:04,380
So we have the height
which starts out
89
00:07:04,380 --> 00:07:09,590
at 80 meters at time 0 and then
the acceleration due to gravity
90
00:07:09,590 --> 00:07:13,330
gives you this formula
for h, this is the height.
91
00:07:13,330 --> 00:07:21,760
So at time t = 0, we're up
at the top, h is 80 meters,
92
00:07:21,760 --> 00:07:24,580
the units here are meters.
93
00:07:24,580 --> 00:07:32,200
And at time t = 4 you
notice, 5 * 4^2 is 80.
94
00:07:32,200 --> 00:07:34,030
I picked these numbers
conveniently so
95
00:07:34,030 --> 00:07:38,320
that we're down at the bottom.
96
00:07:38,320 --> 00:07:43,620
Okay, so this notion
of average change here,
97
00:07:43,620 --> 00:07:51,380
so the average change, or
the average speed here,
98
00:07:51,380 --> 00:07:55,636
maybe we'll call it
the average speed,
99
00:07:55,636 --> 00:08:02,910
since that's-- over this time
that it takes for the pumpkin
100
00:08:02,910 --> 00:08:07,385
to drop is going to be
the change in h divided
101
00:08:07,385 --> 00:08:10,170
by the change in t.
102
00:08:10,170 --> 00:08:18,350
Which starts out at, what
does it start out as?
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00:08:18,350 --> 00:08:21,870
It starts out as 80, right?
104
00:08:21,870 --> 00:08:23,930
And it ends at 0.
105
00:08:23,930 --> 00:08:26,520
So actually we have
to do it backwards.
106
00:08:26,520 --> 00:08:32,110
We have to take 0 - 80
because the first value is
107
00:08:32,110 --> 00:08:35,500
the final position
and the second value
108
00:08:35,500 --> 00:08:37,390
is the initial position.
109
00:08:37,390 --> 00:08:41,470
And that's divided by
4 - 0; times 4 seconds
110
00:08:41,470 --> 00:08:43,680
minus times 0 seconds.
111
00:08:43,680 --> 00:08:49,200
And so that of course is
-20 meters per second.
112
00:08:49,200 --> 00:08:56,860
So the average speed of this
guy is 20 meters a second.
113
00:08:56,860 --> 00:09:00,970
Now, so why did I
pick this example?
114
00:09:00,970 --> 00:09:04,480
Because, of course, the
average, although interesting,
115
00:09:04,480 --> 00:09:06,540
is not really what
anybody cares about who
116
00:09:06,540 --> 00:09:08,670
actually goes to the event.
117
00:09:08,670 --> 00:09:11,460
All we really care about
is the instantaneous speed
118
00:09:11,460 --> 00:09:19,005
when it hits the pavement
and so that's can
119
00:09:19,005 --> 00:09:23,610
be calculated at the bottom.
120
00:09:23,610 --> 00:09:25,330
So what's the
instantaneous speed?
121
00:09:25,330 --> 00:09:29,530
That's the derivative,
or maybe to be
122
00:09:29,530 --> 00:09:31,720
consistent with the notation
I've been using so far,
123
00:09:31,720 --> 00:09:35,950
that's d/dt of h.
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00:09:35,950 --> 00:09:37,580
All right?
125
00:09:37,580 --> 00:09:39,090
So that's d/dt of h.
126
00:09:39,090 --> 00:09:42,020
Now remember we have
formulas for these things.
127
00:09:42,020 --> 00:09:43,850
We can differentiate
this function now.
128
00:09:43,850 --> 00:09:47,930
We did that yesterday.
129
00:09:47,930 --> 00:09:51,350
So we're gonna take the rate of
change and if you take a look
130
00:09:51,350 --> 00:09:56,790
at it, it's just the rate
of change of 80 is 0,
131
00:09:56,790 --> 00:10:02,860
minus the rate change for
this -5t^2, that's minus 10t.
132
00:10:02,860 --> 00:10:08,750
So that's using the fact
that d/dt of 80 is equal to 0
133
00:10:08,750 --> 00:10:12,350
and d/dt of t^2 is equal to 2t.
134
00:10:12,350 --> 00:10:14,420
The special case...
135
00:10:14,420 --> 00:10:17,160
Well I'm cheating
here, but there's
136
00:10:17,160 --> 00:10:18,500
a special case that's obvious.
137
00:10:18,500 --> 00:10:19,850
I didn't throw it in over here.
138
00:10:19,850 --> 00:10:23,750
The case n = 2 is that
second case there.
139
00:10:23,750 --> 00:10:30,380
But the case n = 0 also works.
140
00:10:30,380 --> 00:10:31,534
Because that's constants.
141
00:10:31,534 --> 00:10:32,950
The derivative of
a constant is 0.
142
00:10:32,950 --> 00:10:36,781
And then the factor n there's
0 and that's consistent.
143
00:10:36,781 --> 00:10:38,780
And actually if you look
at the formula above it
144
00:10:38,780 --> 00:10:44,090
you'll see that it's
the case of n = -1.
145
00:10:44,090 --> 00:10:49,820
So we'll get a larger pattern
soon enough with the powers.
146
00:10:49,820 --> 00:10:50,450
Okay anyway.
147
00:10:50,450 --> 00:10:53,770
Back over here we have
our rate of change
148
00:10:53,770 --> 00:10:55,380
and this is what it is.
149
00:10:55,380 --> 00:10:59,350
And at the bottom, at
that point of impact,
150
00:10:59,350 --> 00:11:04,700
we have t = 4 and so h',
which is the derivative,
151
00:11:04,700 --> 00:11:12,860
is equal to -40
meters per second.
152
00:11:12,860 --> 00:11:16,540
So twice as fast as
the average speed here,
153
00:11:16,540 --> 00:11:22,900
and if you need to convert that,
that's about 90 miles an hour.
154
00:11:22,900 --> 00:11:29,450
Which is why the police are
there at midnight on Halloween
155
00:11:29,450 --> 00:11:33,310
to make sure you're all safe
and also why when you come
156
00:11:33,310 --> 00:11:37,330
you have to be prepared
to clean up afterwards.
157
00:11:37,330 --> 00:11:40,260
So anyway that's what happens,
it's 90 miles an hour.
158
00:11:40,260 --> 00:11:42,320
It's actually the
buildings a little taller,
159
00:11:42,320 --> 00:11:45,010
there's air resistance
and I'm sure you
160
00:11:45,010 --> 00:11:50,350
can do a much more thorough
study of this example.
161
00:11:50,350 --> 00:11:54,300
All right so now I want to give
you a couple of more examples
162
00:11:54,300 --> 00:11:58,700
because time and these kinds
of parameters and variables
163
00:11:58,700 --> 00:12:02,570
are not the only ones that
are important for calculus.
164
00:12:02,570 --> 00:12:05,610
If it were only this kind of
physics that was involved,
165
00:12:05,610 --> 00:12:09,710
then this would be a much more
specialized subject than it is.
166
00:12:09,710 --> 00:12:13,340
And so I want to give you a
couple of examples that don't
167
00:12:13,340 --> 00:12:16,570
involve time as a variable.
168
00:12:16,570 --> 00:12:20,175
So the third example
I'll give here
169
00:12:20,175 --> 00:12:27,260
is-- The letter T often
denotes temperature,
170
00:12:27,260 --> 00:12:35,660
and then dT/dx would be what
is known as the temperature
171
00:12:35,660 --> 00:12:38,830
gradient.
172
00:12:38,830 --> 00:12:43,310
Which we really care
about a lot when
173
00:12:43,310 --> 00:12:45,820
we're predicting the weather
because it's that temperature
174
00:12:45,820 --> 00:12:52,140
difference that causes air flows
and causes things to change.
175
00:12:52,140 --> 00:12:54,410
And then there's
another theme which
176
00:12:54,410 --> 00:13:01,110
is throughout the sciences
and engineering which
177
00:13:01,110 --> 00:13:07,700
I'm going to talk about under
the heading of sensitivity
178
00:13:07,700 --> 00:13:15,600
of measurements.
179
00:13:15,600 --> 00:13:18,470
So let me explain this.
180
00:13:18,470 --> 00:13:21,530
I don't want to belabor
it because I just
181
00:13:21,530 --> 00:13:23,810
am doing this in
order to introduce you
182
00:13:23,810 --> 00:13:26,380
to the ideas on your
problem set which
183
00:13:26,380 --> 00:13:29,350
are the first case of this.
184
00:13:29,350 --> 00:13:33,290
So on problem set one
you have an example
185
00:13:33,290 --> 00:13:37,360
which is based on a
simplified model of GPS,
186
00:13:37,360 --> 00:13:39,550
sort of the Flat Earth Model.
187
00:13:39,550 --> 00:13:42,400
And in that situation,
well, if the Earth is flat
188
00:13:42,400 --> 00:13:45,490
it's just a horizontal
line like this.
189
00:13:45,490 --> 00:13:54,020
And then you have a satellite,
which is over here, preferably
190
00:13:54,020 --> 00:14:02,440
above the earth, and the
satellite or the system
191
00:14:02,440 --> 00:14:05,810
knows exactly where the point
directly below the satellite
192
00:14:05,810 --> 00:14:06,480
is.
193
00:14:06,480 --> 00:14:12,170
So this point is
treated as known.
194
00:14:12,170 --> 00:14:22,160
And I'm sitting here
with my little GPS device
195
00:14:22,160 --> 00:14:26,440
and I want to know where I am.
196
00:14:26,440 --> 00:14:28,880
And the way I
locate where I am is
197
00:14:28,880 --> 00:14:34,750
I communicate with this
satellite by radio signals
198
00:14:34,750 --> 00:14:38,750
and I can measure this distance
here which is called h.
199
00:14:38,750 --> 00:14:45,060
And then system will compute
this horizontal distance which
200
00:14:45,060 --> 00:14:53,360
is L. So in other
words what is measured,
201
00:14:53,360 --> 00:15:03,020
so h measured by radios,
radio waves and a clock,
202
00:15:03,020 --> 00:15:04,910
or various clocks.
203
00:15:04,910 --> 00:15:13,560
And then L is deduced from h.
204
00:15:13,560 --> 00:15:16,730
And what's critical in
all of these systems
205
00:15:16,730 --> 00:15:20,320
is that you don't
know h exactly.
206
00:15:20,320 --> 00:15:26,330
There's an error in h
which will denote delta h.
207
00:15:26,330 --> 00:15:31,040
There's some degree
of uncertainty.
208
00:15:31,040 --> 00:15:35,550
The main uncertainty in
GPS is from the ionosphere.
209
00:15:35,550 --> 00:15:38,280
But there are lots
of corrections
210
00:15:38,280 --> 00:15:41,340
that are made of all kinds.
211
00:15:41,340 --> 00:15:43,570
And also if you're
inside a building
212
00:15:43,570 --> 00:15:44,790
it's a problem to measure it.
213
00:15:44,790 --> 00:15:47,970
But it's an extremely
important issue,
214
00:15:47,970 --> 00:15:49,730
as I'll explain in a second.
215
00:15:49,730 --> 00:15:54,040
So the idea is we
then get at delta
216
00:15:54,040 --> 00:16:04,246
L is estimated by considering
this ratio delta L/delta
217
00:16:04,246 --> 00:16:07,060
h which is going
to be approximately
218
00:16:07,060 --> 00:16:13,910
the same as the derivative
of L with respect to h.
219
00:16:13,910 --> 00:16:17,815
So this is the thing that's
easy because of course it's
220
00:16:17,815 --> 00:16:18,940
calculus.
221
00:16:18,940 --> 00:16:21,960
Calculus is the
easy part and that
222
00:16:21,960 --> 00:16:25,330
allows us to deduce something
about the real world that's
223
00:16:25,330 --> 00:16:28,600
close by over here.
224
00:16:28,600 --> 00:16:31,980
So the reason why you should
care about this quite a bit
225
00:16:31,980 --> 00:16:34,870
is that it's used all the
time to land airplanes.
226
00:16:34,870 --> 00:16:36,680
So you really do care
that they actually
227
00:16:36,680 --> 00:16:42,386
know to within a few feet or
even closer where your plane is
228
00:16:42,386 --> 00:16:48,150
and how high up it
is and so forth.
229
00:16:48,150 --> 00:16:48,650
All right.
230
00:16:48,650 --> 00:16:50,850
So that's it for the
general introduction
231
00:16:50,850 --> 00:16:52,010
of what a derivative is.
232
00:16:52,010 --> 00:16:53,670
I'm sure you'll be
getting used to this
233
00:16:53,670 --> 00:16:56,560
in a lot of different contexts
throughout the course.
234
00:16:56,560 --> 00:17:04,510
And now we have to get back
down to some rigorous details.
235
00:17:04,510 --> 00:17:09,540
Okay, everybody happy with
what we've got so far?
236
00:17:09,540 --> 00:17:10,040
Yeah?
237
00:17:10,040 --> 00:17:13,400
Student: How did you get
the equation for height?
238
00:17:13,400 --> 00:17:14,980
Professor: Ah good question.
239
00:17:14,980 --> 00:17:18,560
The question was how did I
get this equation for height?
240
00:17:18,560 --> 00:17:24,970
I just made it up because
it's the formula from physics
241
00:17:24,970 --> 00:17:29,592
that you will learn when
you take 8.01 and, in fact,
242
00:17:29,592 --> 00:17:35,060
it has to do with the fact
that this is the speed if you
243
00:17:35,060 --> 00:17:37,340
differentiate
another time you get
244
00:17:37,340 --> 00:17:40,550
acceleration and
acceleration due to gravity
245
00:17:40,550 --> 00:17:42,330
is 10 meters per second.
246
00:17:42,330 --> 00:17:44,450
Which happens to be the
second derivative of this.
247
00:17:44,450 --> 00:17:47,710
But anyway I just pulled it
out of a hat from your physics
248
00:17:47,710 --> 00:17:48,350
class.
249
00:17:48,350 --> 00:17:55,510
So you can just say see 8.01 .
250
00:17:55,510 --> 00:18:02,840
All right, other questions?
251
00:18:02,840 --> 00:18:04,970
All right, so let's go on now.
252
00:18:04,970 --> 00:18:09,340
Now I have to be a little bit
more systematic about limits.
253
00:18:09,340 --> 00:18:20,130
So let's do that now.
254
00:18:20,130 --> 00:18:30,370
So now what I'd like to talk
about is limits and continuity.
255
00:18:30,370 --> 00:18:34,200
And this is a warm
up for deriving
256
00:18:34,200 --> 00:18:37,900
all the rest of the formulas,
all the rest of the formulas
257
00:18:37,900 --> 00:18:40,430
that I'm going to
need to differentiate
258
00:18:40,430 --> 00:18:41,600
every function you know.
259
00:18:41,600 --> 00:18:44,120
Remember, that's our goal
and we only have about a week
260
00:18:44,120 --> 00:18:47,510
left so we'd better get started.
261
00:18:47,510 --> 00:18:58,980
So first of all there is
what I will call easy limits.
262
00:18:58,980 --> 00:19:00,650
So what's an easy limit?
263
00:19:00,650 --> 00:19:07,285
An easy limit is something like
the limit as x goes to 4 of x
264
00:19:07,285 --> 00:19:11,570
plus 3 over x^2 + 1.
265
00:19:11,570 --> 00:19:16,240
And with this kind of limit all
I have to do to evaluate it is
266
00:19:16,240 --> 00:19:23,770
to plug in x = 4 because,
so what I get here is 4 + 3
267
00:19:23,770 --> 00:19:27,900
divided by 4^2 + 1.
268
00:19:27,900 --> 00:19:31,560
And that's just 7 / 17.
269
00:19:31,560 --> 00:19:33,720
And that's the end of it.
270
00:19:33,720 --> 00:19:38,510
So those are the easy limits.
271
00:19:38,510 --> 00:19:42,669
The second kind of limit -
well so this isn't the only
272
00:19:42,669 --> 00:19:44,960
second kind of limit but I
just want to point this out,
273
00:19:44,960 --> 00:19:55,680
it's very important - is that:
derivatives are are always
274
00:19:55,680 --> 00:19:59,370
harder than this.
275
00:19:59,370 --> 00:20:03,230
You can't get away
with nothing here.
276
00:20:03,230 --> 00:20:05,090
So, why is that?
277
00:20:05,090 --> 00:20:07,620
Well, when you
take a derivative,
278
00:20:07,620 --> 00:20:13,160
you're taking the limit
as x goes to x_0 of f(x),
279
00:20:13,160 --> 00:20:24,520
well we'll write it all
out in all its glory.
280
00:20:24,520 --> 00:20:28,790
Here's the formula
for the derivative.
281
00:20:28,790 --> 00:20:39,110
Now notice that if you plug in
x = x:0, always gives 0 / 0.
282
00:20:39,110 --> 00:20:42,080
So it just basically
never works.
283
00:20:42,080 --> 00:20:50,940
So we always are going
to need some cancellation
284
00:20:50,940 --> 00:21:05,960
to make sense out of the limit.
285
00:21:05,960 --> 00:21:12,570
Now in order to make things
a little easier for myself
286
00:21:12,570 --> 00:21:15,700
to explain what's
going on with limits
287
00:21:15,700 --> 00:21:18,660
I need to introduce just
one more piece of notation.
288
00:21:18,660 --> 00:21:20,490
What I'm gonna
introduce here is what's
289
00:21:20,490 --> 00:21:23,380
known as a left-hand
and a right limit.
290
00:21:23,380 --> 00:21:29,500
If I take the limit as x tends
to x_0 with a plus sign here
291
00:21:29,500 --> 00:21:42,280
of some function, this is what's
known as the right-hand limit.
292
00:21:42,280 --> 00:21:44,870
And I can display it visually.
293
00:21:44,870 --> 00:21:45,950
So what does this mean?
294
00:21:45,950 --> 00:21:47,530
It means practically
the same thing
295
00:21:47,530 --> 00:21:51,160
as x tends to x_0 except there
is one more restriction which
296
00:21:51,160 --> 00:21:53,630
has to do with this plus
sign, which is we're going
297
00:21:53,630 --> 00:21:55,370
from the plus side of x_0.
298
00:21:55,370 --> 00:21:58,710
That means x is bigger than x_0.
299
00:21:58,710 --> 00:22:01,770
And I say right-hand, so
there should be a hyphen here,
300
00:22:01,770 --> 00:22:06,600
right-hand limit because
on the number line,
301
00:22:06,600 --> 00:22:14,580
if x_0 is over here
the x is to the right.
302
00:22:14,580 --> 00:22:15,080
All right?
303
00:22:15,080 --> 00:22:16,750
So that's the right-hand limit.
304
00:22:16,750 --> 00:22:19,550
And then this being the
left side of the board,
305
00:22:19,550 --> 00:22:22,716
I'll put on the right side
of the board the left limit,
306
00:22:22,716 --> 00:22:24,560
just to make things confusing.
307
00:22:24,560 --> 00:22:30,520
So that one has the
minus sign here.
308
00:22:30,520 --> 00:22:33,940
I'm just a little dyslexic
and I hope you're not.
309
00:22:33,940 --> 00:22:38,200
So I may have gotten that wrong.
310
00:22:38,200 --> 00:22:41,510
So this is the left-hand
limit, and I'll draw it.
311
00:22:41,510 --> 00:22:45,705
So of course that just
means x goes to x_0 but x is
312
00:22:45,705 --> 00:22:48,260
to the left of x_0 .
313
00:22:48,260 --> 00:22:52,290
And again, on the number
line, here's the x_0
314
00:22:52,290 --> 00:22:56,570
and the x is on the
other side of it.
315
00:22:56,570 --> 00:22:58,970
Okay, so those two
notations are going
316
00:22:58,970 --> 00:23:01,830
to help us to clarify
a bunch of things.
317
00:23:01,830 --> 00:23:04,580
It's much more
convenient to have
318
00:23:04,580 --> 00:23:08,520
this extra bit of
description of limits
319
00:23:08,520 --> 00:23:15,880
than to just consider
limits from both sides.
320
00:23:15,880 --> 00:23:25,980
Okay so I want to give
an example of this.
321
00:23:25,980 --> 00:23:29,310
And also an example
of how you're going to
322
00:23:29,310 --> 00:23:32,110
think about these
sorts of problems.
323
00:23:32,110 --> 00:23:38,020
So I'll take a function which
has two different definitions.
324
00:23:38,020 --> 00:23:47,570
Say it's x + 1, when x >
0 and -x + 2, when x < 0.
325
00:23:47,570 --> 00:23:51,280
So maybe put commas there.
326
00:23:51,280 --> 00:23:58,540
So when x > 0, it's x + 1.
327
00:23:58,540 --> 00:24:01,030
Now I can draw a
picture of this.
328
00:24:01,030 --> 00:24:02,955
It's gonna be kind
of a little small
329
00:24:02,955 --> 00:24:04,830
because I'm gonna try
to fit it down in here,
330
00:24:04,830 --> 00:24:07,670
but maybe I'll put
the axis down below.
331
00:24:07,670 --> 00:24:13,990
So at height 1, I have to
the right something of slope
332
00:24:13,990 --> 00:24:16,890
1 so it goes up like this.
333
00:24:16,890 --> 00:24:18,240
All right?
334
00:24:18,240 --> 00:24:26,500
And then to the left of 0 I have
something which has slope -1,
335
00:24:26,500 --> 00:24:30,720
but it hits the axis
at 2 so it's up here.
336
00:24:30,720 --> 00:24:34,175
So I had this sort of
strange antenna figure here,
337
00:24:34,175 --> 00:24:35,150
which is my graph.
338
00:24:35,150 --> 00:24:43,710
Maybe I should draw these in
another color to depict that.
339
00:24:43,710 --> 00:24:47,780
And then if I calculate
these two limits here,
340
00:24:47,780 --> 00:24:54,670
what I see is that
the limit as x
341
00:24:54,670 --> 00:25:00,860
goes to 0 from above of f(x),
that's the same as the limit
342
00:25:00,860 --> 00:25:07,990
as x goes to 0 of the
formula here, x + 1.
343
00:25:07,990 --> 00:25:10,430
Which turns out to be 1.
344
00:25:10,430 --> 00:25:15,360
And if I take the limit, so
that's the left-hand limit.
345
00:25:15,360 --> 00:25:20,700
Sorry, I told you
I was dyslexic.
346
00:25:20,700 --> 00:25:23,320
This is the right, so
it's that right-hand.
347
00:25:23,320 --> 00:25:25,080
Here we go.
348
00:25:25,080 --> 00:25:31,530
So now I'm going from the
left, and it's f(x) again,
349
00:25:31,530 --> 00:25:35,180
but now because I'm on that
side the thing I need to plug
350
00:25:35,180 --> 00:25:43,540
is the other formula, -x + 2,
and that's gonna give us 2.
351
00:25:43,540 --> 00:25:48,310
Now, notice that the left
and right limits, and this
352
00:25:48,310 --> 00:25:51,470
is one little tiny subtlety
and it's almost the only thing
353
00:25:51,470 --> 00:25:53,770
that I need you to really
pay attention to a little bit
354
00:25:53,770 --> 00:26:06,210
right now, is that this, we
did not need x = 0 value.
355
00:26:06,210 --> 00:26:11,860
In fact I never even told
you what f(0) was here.
356
00:26:11,860 --> 00:26:14,650
If we stick it in we
could stick it in.
357
00:26:14,650 --> 00:26:20,050
Okay let's say we stick
it in on this side.
358
00:26:20,050 --> 00:26:22,970
Let's make it be that
it's on this side.
359
00:26:22,970 --> 00:26:32,860
So that means that this point
is in and this point is out.
360
00:26:32,860 --> 00:26:37,680
So that's a typical notation:
this little open circle
361
00:26:37,680 --> 00:26:41,530
and this closed dot for
when you include the.
362
00:26:41,530 --> 00:26:44,830
So in that case
the value of f(x)
363
00:26:44,830 --> 00:26:48,360
happens to be the same
as its right-hand limit,
364
00:26:48,360 --> 00:26:56,530
namely the value is
1 here and not 2.
365
00:26:56,530 --> 00:27:01,140
Okay, so that's the
first kind of example.
366
00:27:01,140 --> 00:27:06,610
Questions?
367
00:27:06,610 --> 00:27:13,420
Okay, so now our next
job is to introduce
368
00:27:13,420 --> 00:27:17,270
the definition of continuity.
369
00:27:17,270 --> 00:27:20,080
So that was the
other topic here.
370
00:27:20,080 --> 00:27:23,490
So we're going to define.
371
00:27:23,490 --> 00:27:39,515
So f is continuous at x_0 means
that the limit of f(x) as x
372
00:27:39,515 --> 00:27:44,440
tends to x_0 is
equal to f(x_0) .
373
00:27:44,440 --> 00:27:47,090
Right?
374
00:27:47,090 --> 00:27:51,750
So the reason why I spend
all this time paying
375
00:27:51,750 --> 00:27:54,540
attention to the left and the
right and so on and so forth
376
00:27:54,540 --> 00:27:57,340
and focusing is that I want you
to pay attention for one moment
377
00:27:57,340 --> 00:28:01,820
to what the content
of this definition is.
378
00:28:01,820 --> 00:28:12,640
What it's saying is the
following: continuous at x_0
379
00:28:12,640 --> 00:28:15,450
has various ingredients here.
380
00:28:15,450 --> 00:28:24,540
So the first one is
that this limit exists.
381
00:28:24,540 --> 00:28:27,080
And what that means
is that there's
382
00:28:27,080 --> 00:28:35,150
an honest limiting value
both from the left and right.
383
00:28:35,150 --> 00:28:39,250
And they also have
to be the same.
384
00:28:39,250 --> 00:28:41,980
All right, so that's
what's going on here.
385
00:28:41,980 --> 00:28:50,380
And the second property
is that f(x_0) is defined.
386
00:28:50,380 --> 00:28:52,100
So I can't be in one
of these situations
387
00:28:52,100 --> 00:28:54,770
where I haven't
even specified what
388
00:28:54,770 --> 00:29:05,220
f(x_0) is and they're equal.
389
00:29:05,220 --> 00:29:09,190
Okay, so that's the situation.
390
00:29:09,190 --> 00:29:13,310
Now again let me
emphasize a tricky part
391
00:29:13,310 --> 00:29:15,560
of the definition of a limit.
392
00:29:15,560 --> 00:29:20,320
This side, the left-hand side
is completely independent,
393
00:29:20,320 --> 00:29:23,790
is evaluated by a
procedure which does not
394
00:29:23,790 --> 00:29:25,070
involve the right-hand side.
395
00:29:25,070 --> 00:29:26,900
These are separate things.
396
00:29:26,900 --> 00:29:34,310
This one is, to evaluate it, you
always avoid the limit point.
397
00:29:34,310 --> 00:29:37,670
So that's if you like a
paradox, because it's exactly
398
00:29:37,670 --> 00:29:41,290
the question: is it true
that if you plug in x_0
399
00:29:41,290 --> 00:29:44,300
you get the same answer as
if you move in the limit?
400
00:29:44,300 --> 00:29:46,270
That's the issue that
we're considering here.
401
00:29:46,270 --> 00:29:48,270
We have to make that
distinction in order
402
00:29:48,270 --> 00:29:50,880
to say that these
are two, otherwise
403
00:29:50,880 --> 00:29:55,270
this is just tautological.
404
00:29:55,270 --> 00:29:56,630
It doesn't have any meaning.
405
00:29:56,630 --> 00:29:58,046
But in fact it
does have a meaning
406
00:29:58,046 --> 00:30:00,620
because one thing is evaluated
separately with reference
407
00:30:00,620 --> 00:30:03,850
to all the other
points and the other
408
00:30:03,850 --> 00:30:06,870
is evaluated right at
the point in question.
409
00:30:06,870 --> 00:30:11,090
And indeed what
these things are,
410
00:30:11,090 --> 00:30:17,896
are exactly the easy limits.
411
00:30:17,896 --> 00:30:19,770
That's exactly what
we're talking about here.
412
00:30:19,770 --> 00:30:24,150
They're the ones you
can evaluate this way.
413
00:30:24,150 --> 00:30:25,640
So we have to make
the distinction.
414
00:30:25,640 --> 00:30:27,685
And these other ones are
gonna be the ones which
415
00:30:27,685 --> 00:30:29,670
we can't evaluate that way.
416
00:30:29,670 --> 00:30:31,540
So these are the
nice ones and that's
417
00:30:31,540 --> 00:30:33,830
why we care about them, why
we have a whole definition
418
00:30:33,830 --> 00:30:36,470
associated with them.
419
00:30:36,470 --> 00:30:38,700
All right?
420
00:30:38,700 --> 00:30:40,400
So now what's next?
421
00:30:40,400 --> 00:30:48,910
Well, I need to give you a a
little tour, very brief tour,
422
00:30:48,910 --> 00:30:54,090
of the zoo of what are known
as discontinuous functions.
423
00:30:54,090 --> 00:30:57,430
So sort of everything else
that's not continuous.
424
00:30:57,430 --> 00:31:04,550
So, the first example here,
let me just write it down here.
425
00:31:04,550 --> 00:31:13,670
It's jump discontinuities.
426
00:31:13,670 --> 00:31:15,300
So what would a jump
discontinuity be?
427
00:31:15,300 --> 00:31:18,730
Well we've actually
already seen it.
428
00:31:18,730 --> 00:31:21,790
The jump discontinuity
is the example
429
00:31:21,790 --> 00:31:23,230
that we had right there.
430
00:31:23,230 --> 00:31:32,490
This is when the limit
from the left and right
431
00:31:32,490 --> 00:31:42,180
exist, but are not equal.
432
00:31:42,180 --> 00:31:50,940
Okay, so that's
as in the example.
433
00:31:50,940 --> 00:31:51,440
Right?
434
00:31:51,440 --> 00:31:53,680
In this example, the
two limits, one of them
435
00:31:53,680 --> 00:31:57,890
was 1 and of them was 2.
436
00:31:57,890 --> 00:32:02,150
So that's a jump discontinuity.
437
00:32:02,150 --> 00:32:09,310
And this kind of issue,
of whether something
438
00:32:09,310 --> 00:32:14,940
is continuous or not, may
seem a little bit technical
439
00:32:14,940 --> 00:32:26,120
but it is true that people
have worried about it a lot.
440
00:32:26,120 --> 00:32:28,820
Bob Merton, who was a
professor at MIT when
441
00:32:28,820 --> 00:32:33,410
he did his work for the
Nobel prize in economics,
442
00:32:33,410 --> 00:32:36,180
was interested in
this very issue
443
00:32:36,180 --> 00:32:39,320
of whether stock
prices of various kinds
444
00:32:39,320 --> 00:32:42,540
are continuous from the left
or right in a certain model.
445
00:32:42,540 --> 00:32:44,580
And that was a
very serious issue
446
00:32:44,580 --> 00:32:49,150
in developing the model
that priced things
447
00:32:49,150 --> 00:32:51,840
that our hedge funds
use all the time now.
448
00:32:51,840 --> 00:32:57,630
So left and right can really
mean something very different.
449
00:32:57,630 --> 00:33:01,507
In this case left is the
past and right is the future
450
00:33:01,507 --> 00:33:03,340
and it makes a big
difference whether things
451
00:33:03,340 --> 00:33:06,840
are continuous from the left
or continuous from the right.
452
00:33:06,840 --> 00:33:09,120
Right, is it true that
the point is here,
453
00:33:09,120 --> 00:33:11,720
here, somewhere in the
middle, somewhere else.
454
00:33:11,720 --> 00:33:13,480
That's a serious issue.
455
00:33:13,480 --> 00:33:18,210
So the next example
that I want to give you
456
00:33:18,210 --> 00:33:22,720
is a little bit more subtle.
457
00:33:22,720 --> 00:33:32,140
It's what's known as a
removable discontinuity.
458
00:33:32,140 --> 00:33:43,010
And so what this means is that
the limit from left and right
459
00:33:43,010 --> 00:33:46,190
are equal.
460
00:33:46,190 --> 00:33:47,980
So a picture of
that would be, you
461
00:33:47,980 --> 00:33:50,480
have a function which is
coming along like this
462
00:33:50,480 --> 00:33:52,820
and there's a hole
maybe where, who knows
463
00:33:52,820 --> 00:33:56,270
either the function is undefined
or maybe it's defined up here,
464
00:33:56,270 --> 00:33:58,751
and then it just continues on.
465
00:33:58,751 --> 00:33:59,250
All right?
466
00:33:59,250 --> 00:34:01,210
So the two limits are the same.
467
00:34:01,210 --> 00:34:05,010
And then of course the function
is begging to be redefined
468
00:34:05,010 --> 00:34:07,370
so that we remove that hole.
469
00:34:07,370 --> 00:34:14,470
And that's why it's called
a removable discontinuity.
470
00:34:14,470 --> 00:34:17,710
Now let me give you
an example of this,
471
00:34:17,710 --> 00:34:22,460
or actually a
couple of examples.
472
00:34:22,460 --> 00:34:28,130
So these are quite
important examples
473
00:34:28,130 --> 00:34:34,020
which you will be working
with in a few minutes.
474
00:34:34,020 --> 00:34:41,660
So the first is the function
g(x), which is sin x / x,
475
00:34:41,660 --> 00:34:45,260
and the second will be the
function h(x), which is 1 -
476
00:34:45,260 --> 00:34:50,520
cos x over x.
477
00:34:50,520 --> 00:35:00,290
So we have a problem at
g(0), g(0) is undefined.
478
00:35:00,290 --> 00:35:03,760
On the other hand it turns
out this function has what's
479
00:35:03,760 --> 00:35:05,710
called a removable singularity.
480
00:35:05,710 --> 00:35:14,630
Namely the limit as x goes
to 0 of sin x / x does exist.
481
00:35:14,630 --> 00:35:17,050
In fact it's equal to 1.
482
00:35:17,050 --> 00:35:20,430
So that's a very important limit
that we will work out either
483
00:35:20,430 --> 00:35:23,420
at the end of this lecture or
the beginning of next lecture.
484
00:35:23,420 --> 00:35:30,940
And similarly, the
limit of 1 - cos x
485
00:35:30,940 --> 00:35:35,370
divided by x, as
x goes to 0, is 0.
486
00:35:35,370 --> 00:35:38,051
Maybe I'll put that
a little farther
487
00:35:38,051 --> 00:35:40,360
away so you can read it.
488
00:35:40,360 --> 00:35:44,940
Okay, so these are
very useful facts
489
00:35:44,940 --> 00:35:47,800
that we're going
to need later on.
490
00:35:47,800 --> 00:35:50,460
And what they say is
that these things have
491
00:35:50,460 --> 00:35:58,520
removable singularities, sorry
removable discontinuity at x
492
00:35:58,520 --> 00:36:04,600
= 0.
493
00:36:04,600 --> 00:36:13,030
All right so as I say, we'll
get to that in a few minutes.
494
00:36:13,030 --> 00:36:16,400
Okay so are there any
questions before I move on?
495
00:36:16,400 --> 00:36:16,900
Yeah?
496
00:36:16,900 --> 00:36:30,630
Student: [INAUDIBLE]
497
00:36:30,630 --> 00:36:38,300
Professor: The question
is: why is this true?
498
00:36:38,300 --> 00:36:40,300
Is that what your question is?
499
00:36:40,300 --> 00:36:44,070
The answer is it's
very, very unobvious,
500
00:36:44,070 --> 00:36:48,360
I haven't shown it to you
yet, and if you were not
501
00:36:48,360 --> 00:36:51,560
surprised by it then that
would be very strange indeed.
502
00:36:51,560 --> 00:36:53,390
So we haven't done it yet.
503
00:36:53,390 --> 00:36:55,990
You have to stay
tuned until we do.
504
00:36:55,990 --> 00:36:57,210
Okay?
505
00:36:57,210 --> 00:36:59,250
We haven't shown it yet.
506
00:36:59,250 --> 00:37:01,320
And actually even
this other statement,
507
00:37:01,320 --> 00:37:03,600
which maybe seems
stranger still,
508
00:37:03,600 --> 00:37:05,760
is also not yet explained.
509
00:37:05,760 --> 00:37:08,865
Okay, so we are going
to get there, as I said,
510
00:37:08,865 --> 00:37:10,240
either at the end
of this lecture
511
00:37:10,240 --> 00:37:15,410
or at the beginning of next.
512
00:37:15,410 --> 00:37:22,560
Other questions?
513
00:37:22,560 --> 00:37:28,180
All right, so let me
just continue my tour
514
00:37:28,180 --> 00:37:34,000
of the zoo of discontinuities.
515
00:37:34,000 --> 00:37:37,050
And, I guess, I want
to illustrate something
516
00:37:37,050 --> 00:37:41,440
with the convenience of
right and left hand limits
517
00:37:41,440 --> 00:37:52,180
so I'll save this board about
right and left-hand limits.
518
00:37:52,180 --> 00:37:54,970
So a third type of
discontinuity is
519
00:37:54,970 --> 00:38:07,320
what's known as an
infinite discontinuity.
520
00:38:07,320 --> 00:38:11,950
And we've already
encountered one of these.
521
00:38:11,950 --> 00:38:14,450
I'm going to draw
them over here.
522
00:38:14,450 --> 00:38:19,370
Remember the
function y is 1 / x.
523
00:38:19,370 --> 00:38:22,450
That's this function here.
524
00:38:22,450 --> 00:38:25,500
But now I'd like to draw
also the other branch
525
00:38:25,500 --> 00:38:31,140
of the hyperbola down here
and allow myself to consider
526
00:38:31,140 --> 00:38:32,320
negative values of x.
527
00:38:32,320 --> 00:38:35,910
So here's the graph of 1 / x.
528
00:38:35,910 --> 00:38:42,640
And the convenience here
of distinguishing the left
529
00:38:42,640 --> 00:38:46,620
and the right hand limits is
very important because here I
530
00:38:46,620 --> 00:38:51,800
can write down that the limit
as x goes to 0+ of 1 / x.
531
00:38:51,800 --> 00:38:57,300
Well that's coming from the
right and it's going up.
532
00:38:57,300 --> 00:39:00,580
So this limit is infinity.
533
00:39:00,580 --> 00:39:05,380
Whereas, the limit in
the other direction,
534
00:39:05,380 --> 00:39:10,630
from the left, that
one is going down.
535
00:39:10,630 --> 00:39:16,510
And so it's quite different,
it's minus infinity.
536
00:39:16,510 --> 00:39:19,860
Now some people say that
these limits are undefined
537
00:39:19,860 --> 00:39:22,940
but actually they're going in
some very definite direction.
538
00:39:22,940 --> 00:39:24,950
So you should,
whenever possible,
539
00:39:24,950 --> 00:39:26,640
specify what these limits are.
540
00:39:26,640 --> 00:39:30,860
On the other hand, the
statement that the limit
541
00:39:30,860 --> 00:39:37,250
as x goes to 0 of 1 / x is
infinity is simply wrong.
542
00:39:37,250 --> 00:39:40,340
Okay, it's not that
people don't write this.
543
00:39:40,340 --> 00:39:41,680
It's just that it's wrong.
544
00:39:41,680 --> 00:39:43,470
It's not that they
don't write it down.
545
00:39:43,470 --> 00:39:45,000
In fact you'll probably see it.
546
00:39:45,000 --> 00:39:48,055
It's because people are just
thinking of the right hand
547
00:39:48,055 --> 00:39:48,790
branch.
548
00:39:48,790 --> 00:39:51,220
It's not that they're making
a mistake necessarily,
549
00:39:51,220 --> 00:39:53,116
but anyway, it's sloppy.
550
00:39:53,116 --> 00:39:54,990
And there's some sloppiness
that we'll endure
551
00:39:54,990 --> 00:39:57,080
and others that
we'll try to avoid.
552
00:39:57,080 --> 00:40:00,120
So here, you want to say this,
and it does make a difference.
553
00:40:00,120 --> 00:40:04,990
You know, plus infinity is
an infinite number of dollars
554
00:40:04,990 --> 00:40:07,450
and minus infinity is and
infinite amount of debt.
555
00:40:07,450 --> 00:40:08,980
They're actually different.
556
00:40:08,980 --> 00:40:09,890
They're not the same.
557
00:40:09,890 --> 00:40:15,540
So, you know, this is sloppy and
this is actually more correct.
558
00:40:15,540 --> 00:40:17,885
Okay, so now in
addition, I just want
559
00:40:17,885 --> 00:40:21,350
to point out one more thing.
560
00:40:21,350 --> 00:40:24,210
Remember, we calculated
the derivative,
561
00:40:24,210 --> 00:40:26,880
and that was -1/x^2.
562
00:40:26,880 --> 00:40:31,196
But, I want to draw
the graph of that
563
00:40:31,196 --> 00:40:32,570
and make a few
comments about it.
564
00:40:32,570 --> 00:40:34,420
So I'm going to draw
the graph directly
565
00:40:34,420 --> 00:40:38,820
underneath the graph
of the function.
566
00:40:38,820 --> 00:40:41,290
And notice what this graphs is.
567
00:40:41,290 --> 00:40:48,530
It goes like this, it's always
negative, and it points down.
568
00:40:48,530 --> 00:40:51,480
So now this may look
a little strange,
569
00:40:51,480 --> 00:40:55,080
that the derivative of
this thing is this guy,
570
00:40:55,080 --> 00:40:58,630
but that's because of
something very important.
571
00:40:58,630 --> 00:41:01,030
And you should always remember
this about derivatives.
572
00:41:01,030 --> 00:41:03,995
The derivative function looks
nothing like the function,
573
00:41:03,995 --> 00:41:04,860
necessarily.
574
00:41:04,860 --> 00:41:07,780
So you should just forget
about that as being an idea.
575
00:41:07,780 --> 00:41:10,040
Some people feel like
if one thing goes down,
576
00:41:10,040 --> 00:41:11,470
the other thing has to go down.
577
00:41:11,470 --> 00:41:13,030
Just forget that intuition.
578
00:41:13,030 --> 00:41:14,160
It's wrong.
579
00:41:14,160 --> 00:41:20,170
What we're dealing with here,
if you remember, is the slope.
580
00:41:20,170 --> 00:41:23,870
So if you have a slope
here, that corresponds
581
00:41:23,870 --> 00:41:26,960
to just a place over
here and as the slope
582
00:41:26,960 --> 00:41:30,190
gets a little bit
less steep, that's
583
00:41:30,190 --> 00:41:33,320
why we're approaching
the horizontal axis.
584
00:41:33,320 --> 00:41:36,480
The number is getting a
little smaller as we close in.
585
00:41:36,480 --> 00:41:41,120
Now over here, the
slope is also negative.
586
00:41:41,120 --> 00:41:42,980
It is going down and
as we get down here
587
00:41:42,980 --> 00:41:44,580
it's getting more
and more negative.
588
00:41:44,580 --> 00:41:48,170
As we go here the slope,
this function is going up,
589
00:41:48,170 --> 00:41:50,050
but its slope is going down.
590
00:41:50,050 --> 00:41:55,790
All right, so the slope is down
on both sides and the notation
591
00:41:55,790 --> 00:42:03,690
that we use for that is
well suited to this left
592
00:42:03,690 --> 00:42:09,410
and right business.
593
00:42:09,410 --> 00:42:16,030
Namely, the limit as x
goes to 0 of -1 / x^2,
594
00:42:16,030 --> 00:42:18,140
that's going to be
equal to minus infinity.
595
00:42:18,140 --> 00:42:24,760
And that applies to x going
to 0+ and x going to 0-.
596
00:42:24,760 --> 00:42:31,780
So both have this property.
597
00:42:31,780 --> 00:42:34,040
Finally let me just
make one last comment
598
00:42:34,040 --> 00:42:37,660
about these two graphs.
599
00:42:37,660 --> 00:42:42,220
This function here
is an odd function
600
00:42:42,220 --> 00:42:44,620
and when you take the
derivative of an odd function
601
00:42:44,620 --> 00:42:50,740
you always get an even function.
602
00:42:50,740 --> 00:42:54,380
That's closely related to the
fact that this 1 / x is an odd
603
00:42:54,380 --> 00:43:01,170
power and-- x^1 is an odd
power and x^2 is an even power.
604
00:43:01,170 --> 00:43:05,620
So all of this your intuition
should be reinforcing the fact
605
00:43:05,620 --> 00:43:11,070
that these pictures look right.
606
00:43:11,070 --> 00:43:16,010
Okay, now there's one
last kind of discontinuity
607
00:43:16,010 --> 00:43:20,460
that I want to mention
briefly, which I will call
608
00:43:20,460 --> 00:43:33,990
other ugly discontinuities.
609
00:43:33,990 --> 00:43:39,770
And there are lots
and lots of them.
610
00:43:39,770 --> 00:43:44,220
So one example would
be the function y = sin
611
00:43:44,220 --> 00:43:50,080
1 / x, as x goes to 0.
612
00:43:50,080 --> 00:43:58,914
And that looks a
little bit like this.
613
00:43:58,914 --> 00:44:00,330
Back and forth and
back and forth.
614
00:44:00,330 --> 00:44:06,170
It oscillates infinitely
often as we tend to 0.
615
00:44:06,170 --> 00:44:19,260
There's no left or right
limit in this case.
616
00:44:19,260 --> 00:44:25,330
So there is a very large
quantity of things like that.
617
00:44:25,330 --> 00:44:29,350
Fortunately we're not gonna
deal with them in this course.
618
00:44:29,350 --> 00:44:31,500
A lot of times in
real life there
619
00:44:31,500 --> 00:44:34,800
are things that oscillate
as time goes to infinity,
620
00:44:34,800 --> 00:44:40,180
but we're not going to
worry about that right now.
621
00:44:40,180 --> 00:44:49,090
Okay, so that's our final
mention of a discontinuity,
622
00:44:49,090 --> 00:44:54,130
and now I need to do just
one more piece of groundwork
623
00:44:54,130 --> 00:44:59,360
for our formulas next time.
624
00:44:59,360 --> 00:45:09,130
Namely, I want to check
for you one basic fact,
625
00:45:09,130 --> 00:45:10,280
one limiting tool.
626
00:45:10,280 --> 00:45:12,960
So this is going
to be a theorem.
627
00:45:12,960 --> 00:45:17,450
Fortunately it's a
very short theorem
628
00:45:17,450 --> 00:45:19,580
and has a very short proof.
629
00:45:19,580 --> 00:45:22,090
So the theorem goes under
the name differentiable
630
00:45:22,090 --> 00:45:28,210
implies continuous.
631
00:45:28,210 --> 00:45:30,190
And what it says
is the following:
632
00:45:30,190 --> 00:45:35,600
it says that if f is
differentiable, in other words
633
00:45:35,600 --> 00:45:45,560
its-- the derivative
exists at x_0, then
634
00:45:45,560 --> 00:45:59,245
f is continuous at x_0.
635
00:45:59,245 --> 00:46:00,870
So, we're gonna need
this is as a tool,
636
00:46:00,870 --> 00:46:05,750
it's a key step in the
product and quotient rules.
637
00:46:05,750 --> 00:46:12,380
So I'd like to prove
it right now for you.
638
00:46:12,380 --> 00:46:16,270
So here is the proof.
639
00:46:16,270 --> 00:46:20,430
Fortunately the proof
is just one line.
640
00:46:20,430 --> 00:46:24,740
So first of all, I want to
write in just the right way what
641
00:46:24,740 --> 00:46:27,410
it is that we have to check.
642
00:46:27,410 --> 00:46:33,540
So what we have to check is that
the limit, as x goes to x_0,
643
00:46:33,540 --> 00:46:41,347
of f(x) - f(x_0) is equal to 0.
644
00:46:41,347 --> 00:46:42,680
So this is what we want to know.
645
00:46:42,680 --> 00:46:44,930
We don't know it
yet, but we're trying
646
00:46:44,930 --> 00:46:47,650
to check whether
this is true or not.
647
00:46:47,650 --> 00:46:49,790
So that's the same
as the statement
648
00:46:49,790 --> 00:46:52,180
that the function is continuous
because the limit of f(x)
649
00:46:52,180 --> 00:46:56,300
is supposed to be f(x_0) and
so this difference should
650
00:46:56,300 --> 00:46:59,690
have limit 0.
651
00:46:59,690 --> 00:47:02,730
And now, the way this
is proved is just
652
00:47:02,730 --> 00:47:09,720
by rewriting it by multiplying
and dividing by (x - x_0).
653
00:47:09,720 --> 00:47:17,381
So I'll rewrite the limit
as x goes to x_0 of f(x) -
654
00:47:17,381 --> 00:47:25,570
f(x_0) divided by x
- x_0 times x - x_0.
655
00:47:25,570 --> 00:47:29,230
Okay, so I wrote down the same
expression that I had here.
656
00:47:29,230 --> 00:47:32,080
This is just the same limit,
but I multiplied and divided
657
00:47:32,080 --> 00:47:38,070
by (x - x_0).
658
00:47:38,070 --> 00:47:45,150
And now when I take the limit
what happens is the limit
659
00:47:45,150 --> 00:47:48,830
of the first factor is f'(x_0).
660
00:47:48,830 --> 00:47:53,940
That's the thing we know
exists by our assumption.
661
00:47:53,940 --> 00:48:00,640
And the limit of the second
factor is 0 because the limit
662
00:48:00,640 --> 00:48:06,700
as x goes to x_0 of (x
- x_0) is clearly 0 .
663
00:48:06,700 --> 00:48:09,210
So that's it.
664
00:48:09,210 --> 00:48:12,210
The answer is 0, which
is what we wanted.
665
00:48:12,210 --> 00:48:14,980
So that's the proof.
666
00:48:14,980 --> 00:48:19,500
Now there's something
exceedingly fishy-looking
667
00:48:19,500 --> 00:48:26,370
about this proof and let me just
point to it before we proceed.
668
00:48:26,370 --> 00:48:33,050
Namely, you're used in limits
to setting x equal to 0.
669
00:48:33,050 --> 00:48:35,880
And this looks like we're
multiplying, dividing by 0,
670
00:48:35,880 --> 00:48:38,430
exactly the thing
which makes all proofs
671
00:48:38,430 --> 00:48:42,562
wrong in all kinds of
algebraic situations
672
00:48:42,562 --> 00:48:43,520
and so on and so forth.
673
00:48:43,520 --> 00:48:45,780
You've been taught
that that never works.
674
00:48:45,780 --> 00:48:47,750
Right?
675
00:48:47,750 --> 00:48:51,040
But somehow these
limiting tricks
676
00:48:51,040 --> 00:48:54,100
have found a way around
this and let me just
677
00:48:54,100 --> 00:48:55,880
make explicit what it is.
678
00:48:55,880 --> 00:49:03,500
In this limit we never
are using x = x_0.
679
00:49:03,500 --> 00:49:05,720
That's exactly the
one value of x that we
680
00:49:05,720 --> 00:49:09,120
don't consider in this limit.
681
00:49:09,120 --> 00:49:11,910
That's how limits are cooked up.
682
00:49:11,910 --> 00:49:14,840
And that's sort of been
the themes so far today,
683
00:49:14,840 --> 00:49:17,100
is that we don't
have to consider that
684
00:49:17,100 --> 00:49:19,990
and so this multiplication
and division by this number
685
00:49:19,990 --> 00:49:21,450
is legal.
686
00:49:21,450 --> 00:49:25,200
It may be small, this number,
but it's always non-zero.
687
00:49:25,200 --> 00:49:27,670
So this really works,
and it's really true,
688
00:49:27,670 --> 00:49:31,040
and we just checked that a
differentiable function is
689
00:49:31,040 --> 00:49:32,560
continuous.
690
00:49:32,560 --> 00:49:38,580
So I'm gonna have to carry
out these limits, which
691
00:49:38,580 --> 00:49:42,040
are very interesting 0
/ 0 limits next time.
692
00:49:42,040 --> 00:49:46,512
But let's hang on for one second
to see if there any questions
693
00:49:46,512 --> 00:49:47,907
before we stop.
694
00:49:47,907 --> 00:49:48,990
Yeah, there is a question.
695
00:49:48,990 --> 00:50:00,970
Student: [INAUDIBLE] Professor:
Repeat this proof right here?
696
00:50:00,970 --> 00:50:02,830
Just say again.
697
00:50:02,830 --> 00:50:08,230
Student: [INAUDIBLE]
698
00:50:08,230 --> 00:50:13,060
Professor: Okay, so there
are two steps to the proof
699
00:50:13,060 --> 00:50:17,870
and the step that you're
asking about is the first step.
700
00:50:17,870 --> 00:50:18,580
Right?
701
00:50:18,580 --> 00:50:20,890
And what I'm saying is
if you have a number,
702
00:50:20,890 --> 00:50:24,640
and you multiply it by 10
/ 10 it's the same number.
703
00:50:24,640 --> 00:50:26,920
If you multiply it by 3
/ 3 it's the same number.
704
00:50:26,920 --> 00:50:30,110
2 / 2, 1 / 1, and so on.
705
00:50:30,110 --> 00:50:32,385
So it is okay to
change this to this,
706
00:50:32,385 --> 00:50:34,400
it's exactly the same thing.
707
00:50:34,400 --> 00:50:36,220
That's the first step.
708
00:50:36,220 --> 00:50:36,720
Yes?
709
00:50:36,720 --> 00:50:41,560
Student: [INAUDIBLE]
710
00:50:41,560 --> 00:50:45,010
Professor: Shhhh...
711
00:50:45,010 --> 00:50:52,100
The question was how does the
proof, how does this line,
712
00:50:52,100 --> 00:50:53,960
yeah where the question mark is.
713
00:50:53,960 --> 00:50:55,910
So what I checked was
that this number which
714
00:50:55,910 --> 00:50:59,850
is on the left hand side
is equal to this very long
715
00:50:59,850 --> 00:51:04,800
complicated number which is
equal to this number which
716
00:51:04,800 --> 00:51:06,270
is equal to this number.
717
00:51:06,270 --> 00:51:08,600
And so I've checked that
this number is equal to 0
718
00:51:08,600 --> 00:51:12,120
because the last thing is 0.
719
00:51:12,120 --> 00:51:16,120
This is equal to that is
equal to that is equal to 0.
720
00:51:16,120 --> 00:51:17,420
And that's the proof.
721
00:51:17,420 --> 00:51:17,920
Yes?
722
00:51:17,920 --> 00:51:21,910
Student: [INAUDIBLE]
723
00:51:21,910 --> 00:51:30,580
Professor: So that's
a different question.
724
00:51:30,580 --> 00:51:35,750
Okay, so the hypothesis
of differentiability I
725
00:51:35,750 --> 00:51:39,420
use because this limit
is equal to this number.
726
00:51:39,420 --> 00:51:40,520
That that limit exits.
727
00:51:40,520 --> 00:51:44,170
That's how I use the
hypothesis of the theorem.
728
00:51:44,170 --> 00:51:46,560
The conclusion of the
theorem is the same
729
00:51:46,560 --> 00:51:51,300
as this because being
continuous is the same as limit
730
00:51:51,300 --> 00:51:56,020
as x goes to x_0 of
f(x) is equal to f(x_0).
731
00:51:56,020 --> 00:51:57,530
That's the definition
of continuity.
732
00:51:57,530 --> 00:52:00,990
And I subtracted
f(x_0) from both sides
733
00:52:00,990 --> 00:52:02,860
to get this as being
the same thing.
734
00:52:02,860 --> 00:52:08,100
So this claim is continuity and
it's the same as this question
735
00:52:08,100 --> 00:52:10,350
here.
736
00:52:10,350 --> 00:52:11,180
Last question.
737
00:52:11,180 --> 00:52:16,771
Student: How did you
get the 0 [INAUDIBLE]
738
00:52:16,771 --> 00:52:18,520
Professor: How did we
get the 0 from this?
739
00:52:18,520 --> 00:52:20,450
So the claim that is
being made, so the claim
740
00:52:20,450 --> 00:52:24,670
is why is this tending to that.
741
00:52:24,670 --> 00:52:27,410
So for example, I'm going
to have to erase something
742
00:52:27,410 --> 00:52:28,730
to explain that.
743
00:52:28,730 --> 00:52:33,900
So the claim is that the limit
as x goes to x_0 of x - x_0
744
00:52:33,900 --> 00:52:35,240
is equal to 0.
745
00:52:35,240 --> 00:52:37,160
That's what I'm claiming.
746
00:52:37,160 --> 00:52:39,490
Okay, does that
answer your question?
747
00:52:39,490 --> 00:52:40,990
Okay.
748
00:52:40,990 --> 00:52:42,420
All right.
749
00:52:42,420 --> 00:52:45,320
Ask me other stuff
after lecture.