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Okay so I'd like to begin the
second lecture by reminding
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00:00:27,38 --> 00:00:30,77
you what we did last time.
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00:00:30,77 --> 00:00:54,55
So last time, we defined the
derivative as the slope
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00:00:54,55 --> 00:01:04,26
of a tangent line.
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00:01:04,26 --> 00:01:08,4
So that was our geometric point
of view and we also did a
14
00:01:08,4 --> 00:01:10,47
couple of computations.
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00:01:10,47 --> 00:01:14,5
We worked out that the
derivative of 1 /
16
00:01:14,5 --> 00:01:20,1
x was -1 / x^2.
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00:01:20,1 --> 00:01:27,1
And we also computed the
derivative of x ^ nth power for
18
00:01:27,1 --> 00:01:32,31
n = 1, 2, etc., and that turned
out to be x, I'm
19
00:01:32,31 --> 00:01:32,93
sorry, nx^(n-1).
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00:01:32,93 --> 00:01:36,97
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00:01:36,97 --> 00:01:46,53
So that's what we did last
time, and today I want to
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00:01:46,53 --> 00:01:53,43
finish up with other points of
view on what a derivative is.
23
00:01:53,43 --> 00:01:57,35
So this is extremely important,
it's almost the most important
24
00:01:57,35 --> 00:01:58,75
thing I'll be saying
in the class.
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00:01:58,75 --> 00:02:01,31
But you'll have to think about
it again when you start over
26
00:02:01,31 --> 00:02:04,6
and start using calculus
in the real world.
27
00:02:04,6 --> 00:02:14,71
So again we're talking about
what is a derivative and this
28
00:02:14,71 --> 00:02:19,66
is just a continuation
of last time.
29
00:02:19,66 --> 00:02:23,26
So, as I said last time, we
talked about geometric
30
00:02:23,26 --> 00:02:28,81
interpretations, and today what
we're gonna talk about is rate
31
00:02:28,81 --> 00:02:40
of change as an interpretation
of the derivative.
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00:02:40 --> 00:02:47,13
So remember we drew graphs of
functions, y = f(x) and we
33
00:02:47,13 --> 00:02:51,855
kept track of the change in x
and here the change
34
00:02:51,855 --> 00:02:56,14
in y, let's say.
35
00:02:56,14 --> 00:03:02,18
And then from this new point of
view a rate of change, keeping
36
00:03:02,18 --> 00:03:05,65
track of the rate of change of
x and the rate of change of y,
37
00:03:05,65 --> 00:03:08,71
it's the relative rate of
change we're interested in, and
38
00:03:08,71 --> 00:03:14,7
that's delta y / delta x and
that has another
39
00:03:14,7 --> 00:03:16,01
interpretation.
40
00:03:16,01 --> 00:03:21,65
This is the average change.
41
00:03:21,65 --> 00:03:26,88
Usually we would think of that,
if x were measuring time and so
42
00:03:26,88 --> 00:03:32,63
the average and that's when
this becomes a rate, and
43
00:03:32,63 --> 00:03:35,83
the average is over the
time interval delta x.
44
00:03:35,83 --> 00:03:45,47
And then the limiting value is
denoted dy/dx and so this one
45
00:03:45,47 --> 00:03:48,76
is the average rate of change
and this one is the
46
00:03:48,76 --> 00:03:59,86
instantaneous rate.
47
00:03:59,86 --> 00:04:02,67
Okay, so that's the point of
view that I'd like to discuss
48
00:04:02,67 --> 00:04:06,2
now and give you just
a couple of examples.
49
00:04:06,2 --> 00:04:12,98
So, let's see.
50
00:04:12,98 --> 00:04:19,62
Well, first of all, maybe some
examples from physics here.
51
00:04:19,62 --> 00:04:31,34
So q is usually the name for
a charge, and then dq/dt is
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00:04:31,34 --> 00:04:33,66
what's known as current.
53
00:04:33,66 --> 00:04:38,6
So that's one physical example.
54
00:04:38,6 --> 00:04:45,2
A second example, which is
probably the most tangible one,
55
00:04:45,2 --> 00:04:51,74
is we could denote know the
letter s by distance and
56
00:04:51,74 --> 00:04:58,52
then the rate of change is
that what we call speed.
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00:04:58,52 --> 00:05:03,85
So those are the two typical
examples and I just want to
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00:05:03,85 --> 00:05:08,35
illustrate the second example
in a little bit more detail
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00:05:08,35 --> 00:05:10,56
because I think it's important
to have some visceral
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00:05:10,56 --> 00:05:16,32
sense of this notion of
instantaneous speed.
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00:05:16,32 --> 00:05:22,57
And I get to use the example of
this very building to do that.
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00:05:22,57 --> 00:05:28,72
Probably you know, or maybe
you don't, that on Halloween
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there's an event that takes
place in this building or
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really from the top of this
building which is called
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the pumpkin drop.
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00:05:37 --> 00:05:44,45
So let's illustrates this
idea of rate of change
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with the pumpkin drop.
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So what happens is this
building, well let's see here's
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the building, and here's the
dot, that's the beautiful grass
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00:06:03,15 --> 00:06:07,47
out on this side of the
building, and then there's some
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00:06:07,47 --> 00:06:12,6
people up here and very small
objects, well they're not that
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00:06:12,6 --> 00:06:16,59
small when you're close to
them, that get dumped
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00:06:16,59 --> 00:06:19
over the side there.
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00:06:19 --> 00:06:21,51
And they fall down.
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00:06:21,51 --> 00:06:24,17
You know everything at MIT
or a lot of things at MIT
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are physics experiments.
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That's the pumpkin drop.
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So roughly speaking, the
building is about 300 feet
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high, we're down here on
the first usable floor.
80
00:06:39,57 --> 00:06:44,61
And so we're going to use
instead of 300 feet, just for
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00:06:44,61 --> 00:06:50,22
convenience purposes we'll use
80 meters because that makes
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00:06:50,22 --> 00:06:55,41
the numbers come out simply.
83
00:06:55,41 --> 00:07:05,52
So we have the height which
starts out at 80 meters at time
84
00:07:05,52 --> 00:07:10,3
0 and then the acceleration due
to gravity gives you this
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00:07:10,3 --> 00:07:13,33
formula for h, this
is the height.
86
00:07:13,33 --> 00:07:21,76
So at time t = 0, we're up
at the top, h is 80 meters,
87
00:07:21,76 --> 00:07:24,58
the units here are meters.
88
00:07:24,58 --> 00:07:32,2
And at time t = 4 you
notice, (5 * 4^2) is 80.
89
00:07:32,2 --> 00:07:34,24
I picked these numbers
conveniently so that we're
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00:07:34,24 --> 00:07:38,32
down at the bottom.
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00:07:38,32 --> 00:07:45,77
Okay, so this notion of average
change here, so the average
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00:07:45,77 --> 00:07:53,08
change, or the average speed
here, maybe we'll call it the
93
00:07:53,08 --> 00:08:02,29
average speed since that's,
over this time that it takes
94
00:08:02,29 --> 00:08:06,48
for the pumpkin to drop is
going to be the change
95
00:08:06,48 --> 00:08:10,17
in h / the change in t.
96
00:08:10,17 --> 00:08:18,35
Which starts out at, what
does it start out as?
97
00:08:18,35 --> 00:08:21,87
It starts out as 80, right?
98
00:08:21,87 --> 00:08:23,93
And it ends at 0.
99
00:08:23,93 --> 00:08:26,52
So actually we have
to do it backwards.
100
00:08:26,52 --> 00:08:33,29
We have to take 0 - 80 because
the first value is the final
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00:08:33,29 --> 00:08:37,39
position and the second value
is the initial position.
102
00:08:37,39 --> 00:08:41,47
And that's divided by 4
- 0; times 4 seconds
103
00:08:41,47 --> 00:08:43,68
minus times 0 seconds.
104
00:08:43,68 --> 00:08:49,2
And so that of course is
-20 meters per second.
105
00:08:49,2 --> 00:08:56,86
So the average speed of this
guy is 20 meters a second.
106
00:08:56,86 --> 00:09:00,97
Now, so why did I
pick this example?
107
00:09:00,97 --> 00:09:04,48
Because, of course, the
average, although interesting,
108
00:09:04,48 --> 00:09:06,86
is not really what anybody
cares about who actually
109
00:09:06,86 --> 00:09:08,67
goes to the event.
110
00:09:08,67 --> 00:09:12,37
All we really care about is the
instantaneous speed when it
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00:09:12,37 --> 00:09:22,95
hits the pavement and so that's
can be calculated
112
00:09:22,95 --> 00:09:23,61
at the bottom.
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00:09:23,61 --> 00:09:25,33
So what's the
instantaneous speed?
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00:09:25,33 --> 00:09:30,36
That's the derivative, or maybe
to be consistent with the
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00:09:30,36 --> 00:09:35,95
notation I've been using
so far, that's d/dt of h.
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00:09:35,95 --> 00:09:37,58
All right?
117
00:09:37,58 --> 00:09:39,09
So that's d/dt of h.
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00:09:39,09 --> 00:09:42,02
Now remember we have
formulas for these things.
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00:09:42,02 --> 00:09:43,85
We can differentiate
this function now.
120
00:09:43,85 --> 00:09:47,93
We did that yesterday.
121
00:09:47,93 --> 00:09:51,35
So we're gonna take the rate of
change and if you take a look
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00:09:51,35 --> 00:09:57,25
at it, it's just the rate of
change of 80 is 0, minus
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00:09:57,25 --> 00:10:02,86
the rate change for this
-5t^2, that's minus 10t.
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00:10:02,86 --> 00:10:09
So that's using the fact that
d/dt of 80 is equal to 0 and
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00:10:09 --> 00:10:12,35
d/dt of t^2 is equal to 2t.
126
00:10:12,35 --> 00:10:14,42
The special case...
127
00:10:14,42 --> 00:10:17,69
Well I'm cheating here,
but there's a special
128
00:10:17,69 --> 00:10:18,5
case that's obvious.
129
00:10:18,5 --> 00:10:19,85
I didn't throw it in over here.
130
00:10:19,85 --> 00:10:23,75
The case n = 2 is that
second case there.
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00:10:23,75 --> 00:10:30,38
But the case n = 0 also works.
132
00:10:30,38 --> 00:10:31,62
Because that's constants.
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00:10:31,62 --> 00:10:32,95
The derivative of
a constant is 0.
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00:10:32,95 --> 00:10:36,96
And then the factor n there's
0 and that's consistent.
135
00:10:36,96 --> 00:10:39,14
And actually if you look at the
formula above it you'll see
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00:10:39,14 --> 00:10:44,09
that it's the case of n = -1.
137
00:10:44,09 --> 00:10:49,82
So we'll get a larger pattern
soon enough with the powers.
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00:10:49,82 --> 00:10:50,45
Okay anyway.
139
00:10:50,45 --> 00:10:54,09
Back over here we have
our rate of change and
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00:10:54,09 --> 00:10:55,38
this is what it is.
141
00:10:55,38 --> 00:11:00,01
And at the bottom, at that
point of impact, we have t
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00:11:00,01 --> 00:11:07,75
= 4 and so h' which is the
derivative is equal to
143
00:11:07,75 --> 00:11:12,86
-40 meters per second.
144
00:11:12,86 --> 00:11:19,07
So twice as fast as the average
speed here, and if you need
145
00:11:19,07 --> 00:11:22,9
to convert that, that's
about 90 miles an hour.
146
00:11:22,9 --> 00:11:29,45
Which is why the police are
there at midnight on Halloween
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00:11:29,45 --> 00:11:33,41
to make sure you're all safe
and also why when you come you
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00:11:33,41 --> 00:11:37,33
have to be prepared to
clean up afterwards.
149
00:11:37,33 --> 00:11:40,26
So anyway that's what happens,
it's 90 miles an hour.
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00:11:40,26 --> 00:11:42,63
It's actually the buildings a
little taller, there's air
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00:11:42,63 --> 00:11:46,88
resistance and I'm sure you
can do a much more thorough
152
00:11:46,88 --> 00:11:50,35
study of this example.
153
00:11:50,35 --> 00:11:54,3
All right so now I want to give
you a couple of more examples
154
00:11:54,3 --> 00:11:58,81
because time and these kinds of
parameters and variables are
155
00:11:58,81 --> 00:12:02,57
not the only ones that are
important for calculus.
156
00:12:02,57 --> 00:12:06,3
If it were only this kind of
physics that was involved, then
157
00:12:06,3 --> 00:12:08,54
this would be a much more
specialized subject than it.
158
00:12:08,54 --> 00:12:13,34
Is And so I want to give you a
couple of examples that don't
159
00:12:13,34 --> 00:12:16,57
involved time as a variable.
160
00:12:16,57 --> 00:12:21,72
So the third example
I'll give here is.
161
00:12:21,72 --> 00:12:31,27
The letter t often denotes
temperature, and then dt/dx
162
00:12:31,27 --> 00:12:38,83
would be what is known as
the temperature gradient.
163
00:12:38,83 --> 00:12:43,94
Which we really care about a
lot when we're predicting the
164
00:12:43,94 --> 00:12:46,29
weather because it's that
temperature difference
165
00:12:46,29 --> 00:12:52,14
that causes air flows and
causes things to change.
166
00:12:52,14 --> 00:12:59,16
And then there's another theme
which is throughout the
167
00:12:59,16 --> 00:13:02,76
sciences and engineering which
I'm going to talk about under
168
00:13:02,76 --> 00:13:15,6
the heading of sensitivity
of measurements.
169
00:13:15,6 --> 00:13:18,47
So let me explain this.
170
00:13:18,47 --> 00:13:22,26
I don't want to belabor it
because I just am doing this in
171
00:13:22,26 --> 00:13:26,38
order to introduce you to the
ideas on your problem set which
172
00:13:26,38 --> 00:13:29,35
are the first case of this.
173
00:13:29,35 --> 00:13:35,26
So on problem set one you have
an example which is based on a
174
00:13:35,26 --> 00:13:39,55
simplified model of GPS, sort
of the Flat Earth Model.
175
00:13:39,55 --> 00:13:42,76
And in that situation, well, if
the Earth is flat it's just
176
00:13:42,76 --> 00:13:45,49
a horizontal line like this.
177
00:13:45,49 --> 00:13:54,02
And then you have a satellite,
which is over here, preferably
178
00:13:54,02 --> 00:14:03,06
above the earth, and the
satellite or the system knows
179
00:14:03,06 --> 00:14:05,2
exactly where the point
directly below the
180
00:14:05,2 --> 00:14:06,48
satellite is.
181
00:14:06,48 --> 00:14:12,17
So this point is
treated as known.
182
00:14:12,17 --> 00:14:24,43
And I'm sitting here with
my little GPS device and I
183
00:14:24,43 --> 00:14:26,44
want to know where I am.
184
00:14:26,44 --> 00:14:30,06
And the way I locate where I am
is I communicate with this
185
00:14:30,06 --> 00:14:36,74
satellite by radio signals and
I can measure this distance
186
00:14:36,74 --> 00:14:38,75
here which is called h.
187
00:14:38,75 --> 00:14:42,84
And then system will
computer this horizontal
188
00:14:42,84 --> 00:14:47,07
distance which is L.
189
00:14:47,07 --> 00:14:58,91
So in other words what is
measured, so h measured by
190
00:14:58,91 --> 00:15:04,91
radios, radio waves and a
clock, or various clocks.
191
00:15:04,91 --> 00:15:13,56
And then L is deduced from h.
192
00:15:13,56 --> 00:15:17,63
And what's critical in all of
these systems is that you
193
00:15:17,63 --> 00:15:20,32
don't know h exactly.
194
00:15:20,32 --> 00:15:26,33
There's an error in h which
will denote delta h.
195
00:15:26,33 --> 00:15:31,04
There's some degree
of uncertainty.
196
00:15:31,04 --> 00:15:35,55
The main uncertainty in GPS
is from the ionosphere.
197
00:15:35,55 --> 00:15:38,45
But there are lots of
corrections that are
198
00:15:38,45 --> 00:15:41,34
made of all kinds.
199
00:15:41,34 --> 00:15:43,65
And also if you're inside
a building it's a
200
00:15:43,65 --> 00:15:44,79
problem to measure it.
201
00:15:44,79 --> 00:15:48,22
But it's an extremely
important issue, as I'll
202
00:15:48,22 --> 00:15:49,73
explain in a second.
203
00:15:49,73 --> 00:16:01,44
So the idea is we then get at
delta L is estimated by
204
00:16:01,44 --> 00:16:06
considering this ratio delta
L/delta h which is going to be
205
00:16:06 --> 00:16:11,01
approximately the same as
the derivative of L
206
00:16:11,01 --> 00:16:13,91
with respect to h.
207
00:16:13,91 --> 00:16:16,32
So this is the thing
that's easy because of
208
00:16:16,32 --> 00:16:18,94
course it's calculus.
209
00:16:18,94 --> 00:16:22,85
Calculus is the easy part and
that allows us to deduce
210
00:16:22,85 --> 00:16:28,6
something about the real world
that's close by over here.
211
00:16:28,6 --> 00:16:31,98
So the reason why you should
care about this quite a bit
212
00:16:31,98 --> 00:16:34,87
is that it's used all the
time to land airplanes.
213
00:16:34,87 --> 00:16:38,22
So you really do care that they
actually know to within a few
214
00:16:38,22 --> 00:16:42,77
feet or even closer where
your plane is and how high
215
00:16:42,77 --> 00:16:48,57
up it is and so forth.
216
00:16:48,57 --> 00:16:48,65
All right.
217
00:16:48,65 --> 00:16:51,15
So that's it for the general
introduction of what
218
00:16:51,15 --> 00:16:52,01
a derivative is.
219
00:16:52,01 --> 00:16:54,26
I'm sure you'll be getting used
to this in a lot of different
220
00:16:54,26 --> 00:16:56,56
contexts throughout the course.
221
00:16:56,56 --> 00:17:04,51
And now we have to get back
down to some rigorous details.
222
00:17:04,51 --> 00:17:09,72
Ok, everybody happy with
what we've got so far?
223
00:17:09,72 --> 00:17:10,04
Yeah?
224
00:17:10,04 --> 00:17:13,4
Student: How did you get
the equation for height?
225
00:17:13,4 --> 00:17:14,98
Professor: Ah good
question question.
226
00:17:14,98 --> 00:17:18,56
The question was how did I get
this equation for height?
227
00:17:18,56 --> 00:17:25,38
I just made it up because it's
the formula from physics that
228
00:17:25,38 --> 00:17:30,06
you will learn when you take
8.01 and, in fact, it has to do
229
00:17:30,06 --> 00:17:35,87
with the fact that this is the
speed if you differentiate
230
00:17:35,87 --> 00:17:39,86
another time you get
acceleration and acceleration
231
00:17:39,86 --> 00:17:42,33
due to gravity is 10
meters per second.
232
00:17:42,33 --> 00:17:44,45
Which happens to be the
second derivative of this.
233
00:17:44,45 --> 00:17:46,96
But anyway I just pulled
it out of a hat from
234
00:17:46,96 --> 00:17:48,35
your physics class.
235
00:17:48,35 --> 00:17:55,51
So you can just say see 8.01 .
236
00:17:55,51 --> 00:18:02,84
All right, other questions?
237
00:18:02,84 --> 00:18:04,97
All right, so let's go on now.
238
00:18:04,97 --> 00:18:09,34
Now I have to be a little bit
more systematic about limits.
239
00:18:09,34 --> 00:18:20,13
So let's do that now.
240
00:18:20,13 --> 00:18:30,37
So now what I'd like to talk
about is limits and continuity.
241
00:18:30,37 --> 00:18:34,96
And this is a warm up for
deriving all the rest of the
242
00:18:34,96 --> 00:18:38,19
formulas, all the rest of
the formulas that I'm going
243
00:18:38,19 --> 00:18:41,6
to need to differentiate
every function you know.
244
00:18:41,6 --> 00:18:44,99
Remember, that's our goal and
we only have about a week left
245
00:18:44,99 --> 00:18:47,51
so we'd better get started.
246
00:18:47,51 --> 00:18:58,98
So first of all there is what
I will call easy limits.
247
00:18:58,98 --> 00:19:00,65
So what's an easy limit?
248
00:19:00,65 --> 00:19:07,55
An easy limit is something like
the limit as x goes to 4 of (x
249
00:19:07,55 --> 00:19:09,81
3 / x^2
250
00:19:09,81 --> 00:19:11,57
1).
251
00:19:11,57 --> 00:19:16,24
And with this kind of limit all
I have to do to evaluate it is
252
00:19:16,24 --> 00:19:21,48
to plug in x = 4 because,
so what I get here
253
00:19:21,48 --> 00:19:24,26
is 4 + 3 / (4^2
254
00:19:24,26 --> 00:19:27,9
1).
255
00:19:27,9 --> 00:19:31,56
And that's just 7 / 17.
256
00:19:31,56 --> 00:19:33,72
And that's the end of it.
257
00:19:33,72 --> 00:19:38,51
So those are the easy limits.
258
00:19:38,51 --> 00:19:43,07
The second kind of limit, well
so this isn't the only second
259
00:19:43,07 --> 00:19:45,29
kind of limit but I just want
to point this out, it's very
260
00:19:45,29 --> 00:19:55,68
important is that:
derivatives are are always
261
00:19:55,68 --> 00:19:59,37
harder than this.
262
00:19:59,37 --> 00:20:03,23
You can't get away
with nothing here.
263
00:20:03,23 --> 00:20:05,09
So, why is that?
264
00:20:05,09 --> 00:20:08,2
Well, when you take a
derivative, you're taking the
265
00:20:08,2 --> 00:20:20,27
limit as x goes to x0 of f(x),
well we'll write it all
266
00:20:20,27 --> 00:20:24,52
out in all its glory.
267
00:20:24,52 --> 00:20:28,79
Here's the formula
for the derivative.
268
00:20:28,79 --> 00:20:39,11
Now notice that if you plug in
x = x0, always gives 0 / 0.
269
00:20:39,11 --> 00:20:42,08
So it just basically
never works.
270
00:20:42,08 --> 00:20:51,22
So we always are going to need
some cancellation to make
271
00:20:51,22 --> 00:21:05,96
sense out of the limit.
272
00:21:05,96 --> 00:21:12,99
Now in order to make things a
little easier for myself to
273
00:21:12,99 --> 00:21:16,85
explain what's going on with
limits I need to introduce just
274
00:21:16,85 --> 00:21:18,66
one more piece of notation.
275
00:21:18,66 --> 00:21:21,3
What I'm gonna introduce
here is what's known as a
276
00:21:21,3 --> 00:21:23,38
left-hand and a right limit.
277
00:21:23,38 --> 00:21:27,295
If I take the limit as
x tends to x0 with a
278
00:21:27,295 --> 00:21:32,63
sign here of some function,
this is what's known as
279
00:21:32,63 --> 00:21:42,28
the right hand limit.
280
00:21:42,28 --> 00:21:44,87
And I can display it visually.
281
00:21:44,87 --> 00:21:45,95
So what does this mean?
282
00:21:45,95 --> 00:21:49,15
It means practically the same
thing as x tends to x0 except
283
00:21:49,15 --> 00:21:52,08
there is one more restriction
which has to do with this
284
00:21:52,08 --> 00:21:55,37
sign, which is we're going
from the plus side of x0.
285
00:21:55,37 --> 00:21:58,71
That means x is bigger than x0.
286
00:21:58,71 --> 00:22:01,77
And I say right-hand, so there
should be a hyphen here,
287
00:22:01,77 --> 00:22:07,64
right-hand limit because on the
number line, if x zero is over
288
00:22:07,64 --> 00:22:14,86
here the x is to the right.
289
00:22:14,86 --> 00:22:15,08
All right?
290
00:22:15,08 --> 00:22:16,75
So that's the right-hand limit.
291
00:22:16,75 --> 00:22:20,13
And then this being the left
side of the board, I'll put on
292
00:22:20,13 --> 00:22:23,454
the right side of the board
the left limit, just to
293
00:22:23,454 --> 00:22:24,56
make things confusing.
294
00:22:24,56 --> 00:22:30,52
So that one has the
minus sign here.
295
00:22:30,52 --> 00:22:33,94
I'm just a little dyslexic
and I hope you're not.
296
00:22:33,94 --> 00:22:38,2
So I may have
gotten that wrong.
297
00:22:38,2 --> 00:22:41,51
So this is the left-hand
limit, and I'll draw it.
298
00:22:41,51 --> 00:22:45,705
So of course that just means
x goes to x0 but x is
299
00:22:45,705 --> 00:22:48,26
to the left of x0 .
300
00:22:48,26 --> 00:22:54,27
And again, on the number line,
here's the x0 and the x is
301
00:22:54,27 --> 00:22:56,57
on the other side of it.
302
00:22:56,57 --> 00:22:59,66
Okay, so those two notations
are going to help us to
303
00:22:59,66 --> 00:23:01,83
clarify a bunch of things.
304
00:23:01,83 --> 00:23:06,22
It's much more convenient to
have this extra bit of
305
00:23:06,22 --> 00:23:10,37
description of limits than to
just consider limits
306
00:23:10,37 --> 00:23:15,88
from both sides.
307
00:23:15,88 --> 00:23:25,98
Okay so I want to give
an example of this.
308
00:23:25,98 --> 00:23:30,45
And also an example of how
you're going to think about
309
00:23:30,45 --> 00:23:32,11
these sorts of problems.
310
00:23:32,11 --> 00:23:38,02
So I'll take a function which
has two different definitions.
311
00:23:38,02 --> 00:23:44,085
Say it's x + 1, when
x > 0 and -x
312
00:23:44,085 --> 00:23:47,57
2, when x < 0.
313
00:23:47,57 --> 00:23:51,28
So maybe put commas there.
314
00:23:51,28 --> 00:23:58,54
So when x >
0, it's x + 1.
315
00:23:58,54 --> 00:24:01,03
Now I can draw a
picture of this.
316
00:24:01,03 --> 00:24:04,33
It's gonna be kind of little
small because I'm gonna try to
317
00:24:04,33 --> 00:24:07,67
fit it down in here, but maybe
I'll put the axis down below.
318
00:24:07,67 --> 00:24:14,5
So at height 1, I have to the
right something of slope 1
319
00:24:14,5 --> 00:24:16,89
so it goes up like this.
320
00:24:16,89 --> 00:24:18,24
All right?
321
00:24:18,24 --> 00:24:24,12
And then to the left of 0 I
have something which has slope
322
00:24:24,12 --> 00:24:30,72
-1, but it hits the axis
at 2 so it's up here.
323
00:24:30,72 --> 00:24:33,94
So I had this sort of
strange antennae figure
324
00:24:33,94 --> 00:24:35,15
here is my graph.
325
00:24:35,15 --> 00:24:43,71
Maybe I should draw these in
another color to depict that.
326
00:24:43,71 --> 00:24:50,72
And then if I calculate these
two limits here, what I see is
327
00:24:50,72 --> 00:25:00,04
that the limit as x goes to 0
from above of f(x), that's the
328
00:25:00,04 --> 00:25:07,99
same as the limit as x goes to
0 of the formula here, x + 1.
329
00:25:07,99 --> 00:25:10,43
Which turns out to be 1.
330
00:25:10,43 --> 00:25:15,36
And if I take the limit, so
that's the left-hand limit.
331
00:25:15,36 --> 00:25:20,7
Sorry, I told you
I was dyslexic.
332
00:25:20,7 --> 00:25:23,32
This is the right, so
it's that right-hand.
333
00:25:23,32 --> 00:25:25,08
Here we go.
334
00:25:25,08 --> 00:25:32,2
So now I'm going from the left,
and it's f(x) again, but now
335
00:25:32,2 --> 00:25:35,67
because I'm on that side the
thing I need to plug is the
336
00:25:35,67 --> 00:25:43,54
other formula, -x + 2, and
that's gonna give us 2.
337
00:25:43,54 --> 00:25:48,77
Now, notice that the left and
right limits, and this is one
338
00:25:48,77 --> 00:25:51,65
little tiny subtlety and it's
almost the only thing that I
339
00:25:51,65 --> 00:25:54,02
need you to really pay
attention to a little bit right
340
00:25:54,02 --> 00:26:06,21
now, is that this, we did
not need x = 0 value.
341
00:26:06,21 --> 00:26:11,86
In fact I never even told
you what f(0) was here.
342
00:26:11,86 --> 00:26:14,65
If we stick it in we
could stick it in.
343
00:26:14,65 --> 00:26:20,05
Okay let's say we stick
it in on this side.
344
00:26:20,05 --> 00:26:22,97
Let's make it be that
it's on this side.
345
00:26:22,97 --> 00:26:32,86
So that means that this point
is in and this point is out.
346
00:26:32,86 --> 00:26:37,68
So that's a typical notation:
this little open circle
347
00:26:37,68 --> 00:26:41,53
and this closed dot for
when you include the.
348
00:26:41,53 --> 00:26:46,36
So in that case the value of
f(x) happens to be the same as
349
00:26:46,36 --> 00:26:56,53
its right hand limit, namely
the value is 1 here and not 2.
350
00:26:56,53 --> 00:27:01,14
Okay, so that's the
first kind of example.
351
00:27:01,14 --> 00:27:06,61
Questions?
352
00:27:06,61 --> 00:27:13,5
Okay, so now our next
job is to introduce the
353
00:27:13,5 --> 00:27:17,27
definition of continuity.
354
00:27:17,27 --> 00:27:20,08
So that was the
other topic here.
355
00:27:20,08 --> 00:27:23,49
So we're going to define.
356
00:27:23,49 --> 00:27:39,3
So f is continuous at x0 means
that the limit of f(x) as
357
00:27:39,3 --> 00:27:44,44
x tends to x0 = f(x0) .
358
00:27:44,44 --> 00:27:47,09
Right?
359
00:27:47,09 --> 00:27:52,36
So the reason why I spend all
this time paying attention to
360
00:27:52,36 --> 00:27:55,04
the left and the right and so
on and so forth and focusing
361
00:27:55,04 --> 00:27:57,76
is that I want you to pay
attention for one moment to
362
00:27:57,76 --> 00:28:01,82
what the content of
this definition is.
363
00:28:01,82 --> 00:28:12,64
What it's saying is the
following: continuous at x0
364
00:28:12,64 --> 00:28:15,45
has various ingredients here.
365
00:28:15,45 --> 00:28:24,54
So the first one is that
this limit exists.
366
00:28:24,54 --> 00:28:28,23
And what that means is that
there's an honest limiting
367
00:28:28,23 --> 00:28:35,15
value both from the
left and right.
368
00:28:35,15 --> 00:28:39,25
And they also have
to be the same.
369
00:28:39,25 --> 00:28:41,98
All right, so that's
what's going on here.
370
00:28:41,98 --> 00:28:50,38
And the second property is
that f(x0) is defined.
371
00:28:50,38 --> 00:28:53,23
So I can't be in one of these
situations where I haven't
372
00:28:53,23 --> 00:29:05,22
even specified what f(x0)
is and they're equal.
373
00:29:05,22 --> 00:29:09,19
Okay, so that's the situation.
374
00:29:09,19 --> 00:29:13,56
Now again let me emphasize
a tricky part of the
375
00:29:13,56 --> 00:29:15,56
definition of a limit.
376
00:29:15,56 --> 00:29:20,71
This side, the left-hand side
is completely independent, is
377
00:29:20,71 --> 00:29:24,38
evaluated by a procedure which
does not involve the
378
00:29:24,38 --> 00:29:25,07
right-hand side.
379
00:29:25,07 --> 00:29:26,9
These are separate things.
380
00:29:26,9 --> 00:29:32,68
This one is, to evaluate
it, you always avoid
381
00:29:32,68 --> 00:29:34,31
the limit point.
382
00:29:34,31 --> 00:29:37,67
So that's if you like a
paradox, because it's exactly
383
00:29:37,67 --> 00:29:41,68
the question: is it true that
if you plug in x0 you get the
384
00:29:41,68 --> 00:29:44,3
same answer as if you
move in the limit?
385
00:29:44,3 --> 00:29:46,27
That's the issue that
we're considering here.
386
00:29:46,27 --> 00:29:49,01
We have to make that
distinction in order to say
387
00:29:49,01 --> 00:29:55,27
that these are two, otherwise
this is just tautalogical.
388
00:29:55,27 --> 00:29:56,63
It doesn't have any meaning.
389
00:29:56,63 --> 00:29:58,64
But in fact it does have a
meaning because one thing is
390
00:29:58,64 --> 00:30:01,25
evaluated separately with
reference to all the other
391
00:30:01,25 --> 00:30:05,85
points and the other is
evaluated right at the
392
00:30:05,85 --> 00:30:06,87
point in question.
393
00:30:06,87 --> 00:30:12,5
And indeed what these
things are, are exactly
394
00:30:12,5 --> 00:30:18,16
the easy limits.
395
00:30:18,16 --> 00:30:19,77
That's exactly what we're
talking about here.
396
00:30:19,77 --> 00:30:24,15
They're the ones you
can evaluate this way.
397
00:30:24,15 --> 00:30:25,64
So we have to make
the distinction.
398
00:30:25,64 --> 00:30:27,77
And these other ones are
gonna be the ones which we
399
00:30:27,77 --> 00:30:29,67
can't evaluate that way.
400
00:30:29,67 --> 00:30:32,47
So these are the nice ones and
that's why we care about them
401
00:30:32,47 --> 00:30:36,47
why we have a whole definitions
associated with them.
402
00:30:36,47 --> 00:30:38,7
All right?
403
00:30:38,7 --> 00:30:40,4
So now what's next?
404
00:30:40,4 --> 00:30:48,91
Well, I need to give you a a
little tour, very brief tour,
405
00:30:48,91 --> 00:30:54,09
of the zoo of what are known
as discontinuous functions.
406
00:30:54,09 --> 00:30:57,43
So sort of everything else
that's not continuous.
407
00:30:57,43 --> 00:31:04,55
So, the first example here, let
me just write it down here.
408
00:31:04,55 --> 00:31:13,67
It's jump discontinuities.
409
00:31:13,67 --> 00:31:15,3
So what would a jump
discontinuity be?
410
00:31:15,3 --> 00:31:18,73
Well we've actually
already seen it.
411
00:31:18,73 --> 00:31:21,89
The jump discontinuity
is the example that
412
00:31:21,89 --> 00:31:23,23
we had right there.
413
00:31:23,23 --> 00:31:35,26
This is when the limit from
the left and right exist,
414
00:31:35,26 --> 00:31:42,18
but are not equal.
415
00:31:42,18 --> 00:31:51,22
Okay, so that's as
in the example.
416
00:31:51,22 --> 00:31:51,44
Right?
417
00:31:51,44 --> 00:31:53,82
In this example, the two
limits, one of them was
418
00:31:53,82 --> 00:31:57,89
1 and of them was 2.
419
00:31:57,89 --> 00:32:02,15
So that's a jump discontinuity.
420
00:32:02,15 --> 00:32:09,87
And this kind of issue, of
whether something is continuous
421
00:32:09,87 --> 00:32:21,31
or not, may seem a little bit
technical but it is true that
422
00:32:21,31 --> 00:32:26,12
people have worried
about it a lot.
423
00:32:26,12 --> 00:32:30,08
Bob Merton, who was a professor
at MIT when he did his work for
424
00:32:30,08 --> 00:32:35,27
the Nobel prize in economics,
was interested in this very
425
00:32:35,27 --> 00:32:39,86
issue of whether stock prices
of various kinds are continuous
426
00:32:39,86 --> 00:32:42,54
from the left or right
in a certain model.
427
00:32:42,54 --> 00:32:46,53
And that was a very serious
issue in developing the model
428
00:32:46,53 --> 00:32:50,43
that priced things that
our hedge funds use
429
00:32:50,43 --> 00:32:51,84
all the time now.
430
00:32:51,84 --> 00:32:57,63
So left and right can really
mean something very different.
431
00:32:57,63 --> 00:33:01,96
In this case left is the past
and right is the future and it
432
00:33:01,96 --> 00:33:04,29
makes a big difference whether
things are continuous from the
433
00:33:04,29 --> 00:33:06,84
left or continuous
from the right.
434
00:33:06,84 --> 00:33:09,37
Right, is it true that the
point is here, here, somewhere
435
00:33:09,37 --> 00:33:11,72
in the middle, somewhere else.
436
00:33:11,72 --> 00:33:13,48
That's a serious issue.
437
00:33:13,48 --> 00:33:18,45
So the next example that
I want to give you is a
438
00:33:18,45 --> 00:33:22,72
little bit more subtle.
439
00:33:22,72 --> 00:33:32,14
It's what's known as a
removable discontinuity.
440
00:33:32,14 --> 00:33:39,65
And so what this means is
that the limit from left
441
00:33:39,65 --> 00:33:46,19
and right are equal.
442
00:33:46,19 --> 00:33:48,78
So a picture of that would be,
you have a function which is
443
00:33:48,78 --> 00:33:52,5
coming along like this and
there's a hole maybe where, who
444
00:33:52,5 --> 00:33:55,05
knows either the function is
undefined or maybe it's defined
445
00:33:55,05 --> 00:33:58,97
up here, and then it
just continues on.
446
00:33:58,97 --> 00:33:59,25
All right?
447
00:33:59,25 --> 00:34:01,21
So the two limits are the same.
448
00:34:01,21 --> 00:34:05,01
And then of course the function
in begging to be redefined
449
00:34:05,01 --> 00:34:07,37
so that we remove that hole.
450
00:34:07,37 --> 00:34:14,47
And that's why it's called
a removable discontinuity.
451
00:34:14,47 --> 00:34:18,14
Now let me give you an
example of this, or actually
452
00:34:18,14 --> 00:34:22,46
a couple of examples.
453
00:34:22,46 --> 00:34:28,47
So these are quite important
examples which you will be
454
00:34:28,47 --> 00:34:34,02
working with in a few minutes.
455
00:34:34,02 --> 00:34:41,79
So the first is the function
g(x), which is sin x / x, and
456
00:34:41,79 --> 00:34:50,52
the second will be the function
h(x), which is 1 - cos x / x.
457
00:34:50,52 --> 00:35:00,29
So we have a problem at
g(0) , g(0) is undefined.
458
00:35:00,29 --> 00:35:03,97
On the other hand it turns out
this function has what's called
459
00:35:03,97 --> 00:35:05,71
a removable singularity.
460
00:35:05,71 --> 00:35:14,63
Namely the limit as x goes to
0 of sin x / x does exist.
461
00:35:14,63 --> 00:35:17,05
In fact it's equal to 1.
462
00:35:17,05 --> 00:35:19,96
So that's a very important
limit that we will work out
463
00:35:19,96 --> 00:35:22,145
either at the end of this
lecture or the beginning
464
00:35:22,145 --> 00:35:23,42
of next lecture.
465
00:35:23,42 --> 00:35:35,37
And similarly, the limit of 1 -
cos x / x, as x goes to 0 is 0.
466
00:35:35,37 --> 00:35:38,44
Maybe I'll put that a
little farther away
467
00:35:38,44 --> 00:35:40,36
so you can read it.
468
00:35:40,36 --> 00:35:45,43
Okay, so these are very useful
facts that we're going
469
00:35:45,43 --> 00:35:47,8
to need later on.
470
00:35:47,8 --> 00:35:51,09
And what they say is that these
things have removable
471
00:35:51,09 --> 00:36:04,6
singularities, sorry removable
discontinuity at x = 0.
472
00:36:04,6 --> 00:36:13,03
All right so as I say, we'll
get to that in a few minutes.
473
00:36:13,03 --> 00:36:16,59
Okay so are there any
questions before I move on?
474
00:36:16,59 --> 00:36:16,9
Yeah?
475
00:36:16,9 --> 00:36:30,63
Student: [INAUDIBLE]
476
00:36:30,63 --> 00:36:38,3
Professor: The question
is: why is this true?
477
00:36:38,3 --> 00:36:40,3
Is that what your question is?
478
00:36:40,3 --> 00:36:44,963
The answer is it's very, very
unobvious, I haven't shown it
479
00:36:44,963 --> 00:36:49,67
to you yet, and if you were
not surprised by it then that
480
00:36:49,67 --> 00:36:51,56
would be very strange indeed.
481
00:36:51,56 --> 00:36:53,39
So we haven't done it yet.
482
00:36:53,39 --> 00:36:55,99
You have to stay
tuned until we do.
483
00:36:55,99 --> 00:36:57,21
Okay?
484
00:36:57,21 --> 00:36:59,25
We haven't shown it yet.
485
00:36:59,25 --> 00:37:02,49
And actually even this other
statement, which maybe seems
486
00:37:02,49 --> 00:37:05,76
stranger still, is also
not yet explained.
487
00:37:05,76 --> 00:37:09,33
Okay, so we are going to get
there, as I said, either at
488
00:37:09,33 --> 00:37:15,41
the end of this lecture or
at the beginning of next.
489
00:37:15,41 --> 00:37:22,56
Other questions?
490
00:37:22,56 --> 00:37:29,04
All right, so let me just
continue my tour of the
491
00:37:29,04 --> 00:37:34
zoo of discontinuities.
492
00:37:34 --> 00:37:38,24
And, I guess, I want to
illustrate something with the
493
00:37:38,24 --> 00:37:42,37
convenience of right and left
hand limits so I'll save this
494
00:37:42,37 --> 00:37:52,18
board about right and
left-hand limits.
495
00:37:52,18 --> 00:37:55,445
So a third type of
discontinuity is what's known
496
00:37:55,445 --> 00:38:07,32
as an infinite discontinuity.
497
00:38:07,32 --> 00:38:11,95
And we've already
encountered one of these.
498
00:38:11,95 --> 00:38:14,45
I'm going to draw
them over here.
499
00:38:14,45 --> 00:38:19,37
Remember the function
y is 1 / x.
500
00:38:19,37 --> 00:38:22,45
That's this function here.
501
00:38:22,45 --> 00:38:26,68
But now I'd like to draw also
the other branch of the
502
00:38:26,68 --> 00:38:31,14
hyperbola down here and
allow myself to consider
503
00:38:31,14 --> 00:38:32,32
negative values of x.
504
00:38:32,32 --> 00:38:35,91
So here's the graph of 1 / x.
505
00:38:35,91 --> 00:38:42,82
And the convenience here of
distinguishing the left and the
506
00:38:42,82 --> 00:38:46,62
right hand limits is very
important because here I
507
00:38:46,62 --> 00:38:49,39
can write down that the
limit as x goes to 0
508
00:38:49,39 --> 00:38:51,8
of 1 / x.
509
00:38:51,8 --> 00:38:57,3
Well that's coming from the
right and it's going up.
510
00:38:57,3 --> 00:39:00,58
So this limit is infinity.
511
00:39:00,58 --> 00:39:07,12
Whereas, the limit in the other
direction, from the left,
512
00:39:07,12 --> 00:39:10,63
that one is going down.
513
00:39:10,63 --> 00:39:16,51
And so it's quite different,
it's minus infinity.
514
00:39:16,51 --> 00:39:19,98
Now some people say that these
limits are undefined but
515
00:39:19,98 --> 00:39:22,94
actually they're going in some
very definite direction.
516
00:39:22,94 --> 00:39:26,12
So you should, whenever
possible, specify what
517
00:39:26,12 --> 00:39:26,64
these limits are.
518
00:39:26,64 --> 00:39:30,97
On the other hand, the
statement that the limit as
519
00:39:30,97 --> 00:39:37,25
x goes to 0 of 1 / x is
infinity is simply wrong.
520
00:39:37,25 --> 00:39:40,34
Okay, it's not that
people don't write this.
521
00:39:40,34 --> 00:39:41,68
It's just that it's wrong.
522
00:39:41,68 --> 00:39:43,47
It's not that they
don't write it down.
523
00:39:43,47 --> 00:39:45
In fact you'll probably see it.
524
00:39:45 --> 00:39:47,04
It's because people are
just thinking of the
525
00:39:47,04 --> 00:39:48,79
right hand branch.
526
00:39:48,79 --> 00:39:51,22
It's not that they're making
a mistake necessarily,
527
00:39:51,22 --> 00:39:53,19
but anyway, it's sloppy.
528
00:39:53,19 --> 00:39:56,2
And there's some sloppiness
that we'll endure and others
529
00:39:56,2 --> 00:39:57,08
that we'll try to avoid.
530
00:39:57,08 --> 00:40:00,12
So here, you want to say this,
and it does make a difference.
531
00:40:00,12 --> 00:40:04,99
You know, plus infinity is an
infinite number of dollars
532
00:40:04,99 --> 00:40:07,45
and minus infinity is and
infinite amount of debt.
533
00:40:07,45 --> 00:40:08,98
They're actually different.
534
00:40:08,98 --> 00:40:09,89
They're not the same.
535
00:40:09,89 --> 00:40:13,79
So, you know, this is sloppy
and this is actually
536
00:40:13,79 --> 00:40:15,54
more correct.
537
00:40:15,54 --> 00:40:18,38
Okay, so now in addition,
I just want to point
538
00:40:18,38 --> 00:40:21,35
out one more thing.
539
00:40:21,35 --> 00:40:24,34
Remember, we calculated
the derivative, and
540
00:40:24,34 --> 00:40:26,88
that was -1/x^2.
541
00:40:26,88 --> 00:40:31,71
But, I want to draw the graph
of that and make a few
542
00:40:31,71 --> 00:40:32,57
comments about it.
543
00:40:32,57 --> 00:40:36,91
So I'm going to draw the graph
directly underneath the
544
00:40:36,91 --> 00:40:38,82
graph of the function.
545
00:40:38,82 --> 00:40:41,29
And notice what this graphs is.
546
00:40:41,29 --> 00:40:48,53
It goes like this, it's always
negative, and it points down.
547
00:40:48,53 --> 00:40:52,3
So now this may look a little
strange, that the derivative
548
00:40:52,3 --> 00:40:57,1
of this thing is this guy,
but that's because of
549
00:40:57,1 --> 00:40:58,63
something very important.
550
00:40:58,63 --> 00:41:01,03
And you should always remember
this about derivatives.
551
00:41:01,03 --> 00:41:03,13
The derivative function
looks nothing like the
552
00:41:03,13 --> 00:41:04,86
function, necessarily.
553
00:41:04,86 --> 00:41:07,78
So you should just forget
about that as being an idea.
554
00:41:07,78 --> 00:41:10,33
Some people feel like if one
thing goes down, the other
555
00:41:10,33 --> 00:41:11,47
thing has to go down.
556
00:41:11,47 --> 00:41:13,03
Just forget that intuition.
557
00:41:13,03 --> 00:41:14,16
It's wrong.
558
00:41:14,16 --> 00:41:20,17
What we're dealing with here,
if you remember, is the slope.
559
00:41:20,17 --> 00:41:24,33
So if you have a slope here,
that corresponds to just a
560
00:41:24,33 --> 00:41:29,22
place over here and as the
slope gets a little bit less
561
00:41:29,22 --> 00:41:31,92
steep, that's why
we're approaching the
562
00:41:31,92 --> 00:41:33,32
horizontal axis.
563
00:41:33,32 --> 00:41:36,48
The number is getting a little
smaller as we close in.
564
00:41:36,48 --> 00:41:41,12
Now over here, the slope
is also negative.
565
00:41:41,12 --> 00:41:43,32
It is going down and as we
get down here it's getting
566
00:41:43,32 --> 00:41:44,58
more and more negative.
567
00:41:44,58 --> 00:41:48,27
As we go here the slope, this
function is going up, but
568
00:41:48,27 --> 00:41:50,05
its slope is going down.
569
00:41:50,05 --> 00:41:55,79
All right, so the slope is down
on both sides and the notation
570
00:41:55,79 --> 00:42:03,69
that we use for that is well
suited to this left
571
00:42:03,69 --> 00:42:09,41
and right business.
572
00:42:09,41 --> 00:42:16,45
Namely, the limit as x goes to
0 of -1 / x^2, that's going to
573
00:42:16,45 --> 00:42:18,14
be equal to minus infinity.
574
00:42:18,14 --> 00:42:21,41
And that applies
to x going to 0
575
00:42:21,41 --> 00:42:24,76
and x going to 0-.
576
00:42:24,76 --> 00:42:31,78
So both have this property.
577
00:42:31,78 --> 00:42:34,34
Finally let me just make
one last comment about
578
00:42:34,34 --> 00:42:37,66
these two graphs.
579
00:42:37,66 --> 00:42:43
This function here is an odd
function and when you take the
580
00:42:43 --> 00:42:45,33
derivative of an odd function
you always get an
581
00:42:45,33 --> 00:42:50,74
even function.
582
00:42:50,74 --> 00:42:54,38
That's closely related to the
fact that this 1 / x is an odd
583
00:42:54,38 --> 00:43:01,17
power and x^1 is an odd power
and x^2 is an even power.
584
00:43:01,17 --> 00:43:05,62
So all of this your intuition
should be reinforcing the fact
585
00:43:05,62 --> 00:43:11,07
that these pictures look right.
586
00:43:11,07 --> 00:43:16,74
Okay, now there's one last kind
of discontinuity that I want to
587
00:43:16,74 --> 00:43:27,46
mention briefly, which I will
call other ugly
588
00:43:27,46 --> 00:43:33,99
discontinuities.
589
00:43:33,99 --> 00:43:39,77
And there are lots
and lots of them.
590
00:43:39,77 --> 00:43:44,42
So one example would be
the function y = sin 1
591
00:43:44,42 --> 00:43:50,08
/ x, as x goes to 0.
592
00:43:50,08 --> 00:43:59,29
And that looks a
little bit like this.
593
00:43:59,29 --> 00:44:00,33
Back and forth and
back and forth.
594
00:44:00,33 --> 00:44:06,17
It oscillates infinitely
often as we tend to 0.
595
00:44:06,17 --> 00:44:19,26
There's no left or right
limit in this case.
596
00:44:19,26 --> 00:44:25,33
So there is a very large
quantity of things like that.
597
00:44:25,33 --> 00:44:29,35
Fortunately we're not gonna
deal with them in this course.
598
00:44:29,35 --> 00:44:32,62
A lot of times in real life
there are things that oscillate
599
00:44:32,62 --> 00:44:35,94
as time goes to infinity, but
we're not going to worry
600
00:44:35,94 --> 00:44:40,18
about that right now.
601
00:44:40,18 --> 00:44:49,4
Okay, so that's our final
mention of a discontinuity, and
602
00:44:49,4 --> 00:44:56,38
now I need to do just one more
piece of groundwork for
603
00:44:56,38 --> 00:44:59,36
our formulas next time.
604
00:44:59,36 --> 00:45:09,13
Namely, I want to check
for you one basic fact,
605
00:45:09,13 --> 00:45:10,28
one limiting tool.
606
00:45:10,28 --> 00:45:12,96
So this is going
to be a theorem.
607
00:45:12,96 --> 00:45:17,72
Fortunately it's a very
short theorem and has
608
00:45:17,72 --> 00:45:19,58
a very short proof.
609
00:45:19,58 --> 00:45:22,09
So the theorem goes under
the name differentiable
610
00:45:22,09 --> 00:45:28,21
implies continuous.
611
00:45:28,21 --> 00:45:33,89
And what it says is the
following: it says that if f is
612
00:45:33,89 --> 00:45:41,88
differentiable, in other words
its the derivative exists at
613
00:45:41,88 --> 00:45:59,84
x0, then f is continuous at x0.
614
00:45:59,84 --> 00:46:02,47
So, we're gonna need this is as
a tool, it's a key step in the
615
00:46:02,47 --> 00:46:05,75
product and quotient rules.
616
00:46:05,75 --> 00:46:12,38
So I'd like to prove
it right now for you.
617
00:46:12,38 --> 00:46:16,27
So here is the proof.
618
00:46:16,27 --> 00:46:20,43
Fortunately the proof
is just one line.
619
00:46:20,43 --> 00:46:24,12
So first of all, I want to
write in just the right
620
00:46:24,12 --> 00:46:27,41
way what it is that
we have to check.
621
00:46:27,41 --> 00:46:31,64
So what we have to check is
that the limit, as x goes
622
00:46:31,64 --> 00:46:41,39
to x0 of f(x) - f(x0) = 0.
623
00:46:41,39 --> 00:46:42,68
So this is what
we want to know.
624
00:46:42,68 --> 00:46:45,9
We don't know it yet, but
we're trying to check
625
00:46:45,9 --> 00:46:47,65
whether this is true or not.
626
00:46:47,65 --> 00:46:50,735
So that's the same as the
statement that the function is
627
00:46:50,735 --> 00:46:54,66
continuous because the limit of
f(x) is supposed to be f(x0)
628
00:46:54,66 --> 00:46:59,69
and so this difference
should have limit 0.
629
00:46:59,69 --> 00:47:03,97
And now, the way this is
proved is just by rewriting
630
00:47:03,97 --> 00:47:09,72
it by multiplying and
dividing by (x - x0).
631
00:47:09,72 --> 00:47:18,15
So I'll rewrite the limit as x
goes to x0 of (f(x) - f(x0)
632
00:47:18,15 --> 00:47:25,57
/ by x - x0) (x - x0).
633
00:47:25,57 --> 00:47:29,23
Okay, so I wrote down the same
expression that I had here.
634
00:47:29,23 --> 00:47:31,67
This is just the same limit,
but I multiplied and
635
00:47:31,67 --> 00:47:38,07
divided by (x - x0).
636
00:47:38,07 --> 00:47:45,25
And now when I take the limit
what happens is the limit of
637
00:47:45,25 --> 00:47:48,83
the first factor is f'(x0).
638
00:47:48,83 --> 00:47:53,94
That's the thing we know
exists by our assumption.
639
00:47:53,94 --> 00:48:00,37
And the limit of the second
factor is 0 because the
640
00:48:00,37 --> 00:48:06,7
limit as x goes to x0 of
(x - x0) is clearly 0 .
641
00:48:06,7 --> 00:48:09,21
So that's it.
642
00:48:09,21 --> 00:48:12,21
The answer is 0, which
is what we wanted.
643
00:48:12,21 --> 00:48:14,98
So that's the proof.
644
00:48:14,98 --> 00:48:19,9
Now there's something
exceedingly fishy looking about
645
00:48:19,9 --> 00:48:26,37
this proof and let me just
point to it before we proceed.
646
00:48:26,37 --> 00:48:33,05
Namely, you're used in limits
to setting x equal to 0.
647
00:48:33,05 --> 00:48:35,88
And this looks like we're
multiplying, dividing by 0,
648
00:48:35,88 --> 00:48:40,82
exactly the thing which makes
all proofs wrong in all kinds
649
00:48:40,82 --> 00:48:43,52
of algebraic situations
and so on and so forth.
650
00:48:43,52 --> 00:48:45,78
You've been taught that
that never works.
651
00:48:45,78 --> 00:48:47,75
Right?
652
00:48:47,75 --> 00:48:52,71
But somehow these limiting
tricks have found a way around
653
00:48:52,71 --> 00:48:55,88
this and let me just make
explicit what it is.
654
00:48:55,88 --> 00:49:03,5
In this limit we never
are using x = x0.
655
00:49:03,5 --> 00:49:06,11
That's exactly the one
value of x that we don't
656
00:49:06,11 --> 00:49:09,12
consider in this limit.
657
00:49:09,12 --> 00:49:11,91
That's how limits
are cooked up.
658
00:49:11,91 --> 00:49:15,91
And that's sort of been the
themes so far today, is that we
659
00:49:15,91 --> 00:49:18,96
don't have to consider that and
so this multiplication and
660
00:49:18,96 --> 00:49:21,45
division by this
number is legal.
661
00:49:21,45 --> 00:49:25,2
It may be small, this number,
but it's always non-zero.
662
00:49:25,2 --> 00:49:28,21
So this really works, and it's
really true, and we just
663
00:49:28,21 --> 00:49:32,56
checked that a differentiable
function is continuous.
664
00:49:32,56 --> 00:49:38,93
So I'm gonna have to carry out
these limits, which are very
665
00:49:38,93 --> 00:49:42,04
interesting 0 / 0
limits next time.
666
00:49:42,04 --> 00:49:46,03
But let's hang on for one
second to see if there any
667
00:49:46,03 --> 00:49:47,96
questions before we stop.
668
00:49:47,96 --> 00:49:48,99
Yeah, there is a question.
669
00:49:48,99 --> 00:49:53,45
Student: [INAUDIBLE]
670
00:49:53,45 --> 00:50:00,97
Professor: Repeat this
proof right here?
671
00:50:00,97 --> 00:50:02,83
Just say again.
672
00:50:02,83 --> 00:50:08,23
Student: [INAUDIBLE]
673
00:50:08,23 --> 00:50:13,21
Professor: Okay, so there are
two steps to the proof and
674
00:50:13,21 --> 00:50:17,87
the step that you're asking
about is the first step.
675
00:50:17,87 --> 00:50:18,58
Right?
676
00:50:18,58 --> 00:50:21,1
And what I'm saying is if
you have a number, and you
677
00:50:21,1 --> 00:50:24,64
multiply it by 10 / 10
it's the same number.
678
00:50:24,64 --> 00:50:26,92
If you multiply it by 3 /
3 it's the same number.
679
00:50:26,92 --> 00:50:30,11
2 / 2, 1 / 1, and so on.
680
00:50:30,11 --> 00:50:32,48
So it is okay to change
this to this, it's
681
00:50:32,48 --> 00:50:34,4
exactly the same thing.
682
00:50:34,4 --> 00:50:36,35
That's the first step.
683
00:50:36,35 --> 00:50:36,72
Yes?
684
00:50:36,72 --> 00:50:41,56
Student: [INAUDIBLE]
685
00:50:41,56 --> 00:50:45,01
Professor: Shhhh...
686
00:50:45,01 --> 00:50:52,51
The question was how does the
proof, how does this line, yeah
687
00:50:52,51 --> 00:50:53,96
where the question mark is.
688
00:50:53,96 --> 00:50:56,5
So what I checked was that this
number which is on the left
689
00:50:56,5 --> 00:51:02,58
hand side is equal to this very
long complicated number which
690
00:51:02,58 --> 00:51:06,27
is equal to this number which
is equal to this number.
691
00:51:06,27 --> 00:51:08,87
And so I've checked that this
number is equal to 0 because
692
00:51:08,87 --> 00:51:12,12
the last thing is 0.
693
00:51:12,12 --> 00:51:16,12
This is equal to that is
equal to that is equal to 0.
694
00:51:16,12 --> 00:51:17,6
And that's the proof.
695
00:51:17,6 --> 00:51:17,92
Yes?
696
00:51:17,92 --> 00:51:21,91
Student: [INAUDIBLE]
697
00:51:21,91 --> 00:51:30,58
Professor: So that's a
different question.
698
00:51:30,58 --> 00:51:36,04
Okay, so the hypothesis of
differentiability I use
699
00:51:36,04 --> 00:51:39,42
because this limit is
equal to this number.
700
00:51:39,42 --> 00:51:40,52
That that limit exits.
701
00:51:40,52 --> 00:51:44,17
That's how I use the
hypothesis of the theorem.
702
00:51:44,17 --> 00:51:47,84
The conclusion of the theorem
is the same as this because
703
00:51:47,84 --> 00:51:52,65
being continuous is the same
as limit as x goes to
704
00:51:52,65 --> 00:51:56,02
x0 of f(x) = f(x0).
705
00:51:56,02 --> 00:51:57,53
That's the definition
of continuity.
706
00:51:57,53 --> 00:52:02,15
And I subtracted f(x0) from
both sides to get this
707
00:52:02,15 --> 00:52:02,86
as being the same thing.
708
00:52:02,86 --> 00:52:06,46
So this claim is continuity
and it's the same as
709
00:52:06,46 --> 00:52:10,35
this question here.
710
00:52:10,35 --> 00:52:11,18
Last question.
711
00:52:11,18 --> 00:52:16,97
Student: How did you
get the 0 [INAUDIBLE]
712
00:52:16,97 --> 00:52:18,52
Professor: How did we
get the 0 from this?
713
00:52:18,52 --> 00:52:20,88
So the claim that is being
made, so the claim is why
714
00:52:20,88 --> 00:52:24,67
is this tending to that.
715
00:52:24,67 --> 00:52:27,41
So for example, I'm going
to have to erase something
716
00:52:27,41 --> 00:52:28,73
to explain that.
717
00:52:28,73 --> 00:52:35,24
So the claim is that the limit
as x goes to x0 of x - x0 = 0.
718
00:52:35,24 --> 00:52:37,16
That's what I'm claiming.
719
00:52:37,16 --> 00:52:39,49
Okay, does that answer
your question?
720
00:52:39,49 --> 00:52:40,99
Okay.
721
00:52:40,99 --> 00:52:42,42
All right.
722
00:52:42,42 --> 00:52:45,37
Ask me other stuff
after lecture.
723
00:52:45,37 --> 00:52:46,18