WEBVTT
00:00:00.110 --> 00:00:02.490
The following content is
provided under a Creative
00:00:02.490 --> 00:00:03.156
Commons license.
00:00:03.156 --> 00:00:05.440
Your support will help
MIT OpenCourseWare
00:00:05.440 --> 00:00:09.960
continue to offer high quality
educational resources for free.
00:00:09.960 --> 00:00:12.630
To make a donation or to
view additional materials
00:00:12.630 --> 00:00:15.910
from hundreds of MIT courses
visit MIT OpenCourseWare
00:00:15.910 --> 00:00:22.600
at ocw.mit.edu.
00:00:22.600 --> 00:00:27.490
Okay so I'd like to begin the
second lecture by reminding you
00:00:27.490 --> 00:00:30.770
what we did last time.
00:00:30.770 --> 00:00:50.520
So last time, we
defined the derivative
00:00:50.520 --> 00:01:04.260
as the slope of a tangent line.
00:01:04.260 --> 00:01:07.080
So that was our
geometric point of view
00:01:07.080 --> 00:01:10.470
and we also did a
couple of computations.
00:01:10.470 --> 00:01:18.290
We worked out that the
derivative of 1 / x was -1 /
00:01:18.290 --> 00:01:20.100
x^2.
00:01:20.100 --> 00:01:24.870
And we also computed the
derivative of x to the nth
00:01:24.870 --> 00:01:32.097
power for n = 1, 2, etc.,
and that turned out to be x,
00:01:32.097 --> 00:01:32.930
I'm sorry, nx^(n-1).
00:01:36.970 --> 00:01:46.180
So that's what we did
last time, and today I
00:01:46.180 --> 00:01:51.580
want to finish up with
other points of view
00:01:51.580 --> 00:01:53.430
on what a derivative is.
00:01:53.430 --> 00:01:56.250
So this is extremely
important, it's
00:01:56.250 --> 00:01:58.750
almost the most important thing
I'll be saying in the class.
00:01:58.750 --> 00:02:01.310
But you'll have to think about
it again when you start over
00:02:01.310 --> 00:02:04.600
and start using calculus
in the real world.
00:02:04.600 --> 00:02:14.030
So again we're talking
about what is a derivative
00:02:14.030 --> 00:02:19.660
and this is just a
continuation of last time.
00:02:19.660 --> 00:02:23.260
So, as I said last time,
we talked about geometric
00:02:23.260 --> 00:02:28.130
interpretations, and today
what we're gonna talk about
00:02:28.130 --> 00:02:34.310
is rate of change
as an interpretation
00:02:34.310 --> 00:02:40.000
of the derivative.
00:02:40.000 --> 00:02:46.260
So remember we drew graphs
of functions, y = f(x)
00:02:46.260 --> 00:02:53.190
and we kept track of the change
in x and here the change in y,
00:02:53.190 --> 00:02:56.140
let's say.
00:02:56.140 --> 00:03:01.872
And then from this new point
of view a rate of change,
00:03:01.872 --> 00:03:04.080
keeping track of the rate
of change of x and the rate
00:03:04.080 --> 00:03:07.030
of change of y, it's the
relative rate of change
00:03:07.030 --> 00:03:12.590
we're interested in, and that's
delta y / delta x and that
00:03:12.590 --> 00:03:16.010
has another interpretation.
00:03:16.010 --> 00:03:21.650
This is the average change.
00:03:21.650 --> 00:03:26.880
Usually we would think of that,
if x were measuring time and so
00:03:26.880 --> 00:03:31.350
the average and that's
when this becomes a rate,
00:03:31.350 --> 00:03:35.830
and the average is over
the time interval delta x.
00:03:35.830 --> 00:03:42.670
And then the limiting
value is denoted dy/dx
00:03:42.670 --> 00:03:47.580
and so this one is the
average rate of change
00:03:47.580 --> 00:03:59.860
and this one is the
instantaneous rate.
00:03:59.860 --> 00:04:01.412
Okay, so that's
the point of view
00:04:01.412 --> 00:04:03.120
that I'd like to
discuss now and give you
00:04:03.120 --> 00:04:06.200
just a couple of examples.
00:04:06.200 --> 00:04:12.980
So, let's see.
00:04:12.980 --> 00:04:19.620
Well, first of all, maybe some
examples from physics here.
00:04:19.620 --> 00:04:26.450
So q is usually the
name for a charge,
00:04:26.450 --> 00:04:33.660
and then dq/dt is
what's known as current.
00:04:33.660 --> 00:04:38.600
So that's one physical example.
00:04:38.600 --> 00:04:45.200
A second example, which is
probably the most tangible one,
00:04:45.200 --> 00:04:51.500
is we could denote the
letter s by distance
00:04:51.500 --> 00:04:58.520
and then the rate of change
is what we call speed.
00:04:58.520 --> 00:05:02.110
So those are the
two typical examples
00:05:02.110 --> 00:05:06.550
and I just want to
illustrate the second example
00:05:06.550 --> 00:05:08.900
in a little bit more
detail because I think
00:05:08.900 --> 00:05:12.690
it's important to have some
visceral sense of this notion
00:05:12.690 --> 00:05:16.320
of instantaneous speed.
00:05:16.320 --> 00:05:22.570
And I get to use the example of
this very building to do that.
00:05:22.570 --> 00:05:25.860
Probably you know,
or maybe you don't,
00:05:25.860 --> 00:05:29.690
that on Halloween
there's an event that
00:05:29.690 --> 00:05:33.042
takes place in this
building or really
00:05:33.042 --> 00:05:34.500
from the top of
this building which
00:05:34.500 --> 00:05:37.000
is called the pumpkin drop.
00:05:37.000 --> 00:05:44.450
So let's illustrates this
idea of rate of change
00:05:44.450 --> 00:05:49.040
with the pumpkin drop.
00:05:49.040 --> 00:05:53.800
So what happens is,
this building-- well
00:05:53.800 --> 00:06:01.070
let's see here's the building,
and here's the dot, that's
00:06:01.070 --> 00:06:04.890
the beautiful grass out on
this side of the building,
00:06:04.890 --> 00:06:09.430
and then there's
some people up here
00:06:09.430 --> 00:06:12.310
and very small
objects, well they're
00:06:12.310 --> 00:06:15.240
not that small when
you're close to them, that
00:06:15.240 --> 00:06:19.000
get dumped over the side there.
00:06:19.000 --> 00:06:21.510
And they fall down.
00:06:21.510 --> 00:06:24.170
You know everything at MIT
or a lot of things at MIT
00:06:24.170 --> 00:06:28.430
are physics experiments.
00:06:28.430 --> 00:06:29.440
That's the pumpkin drop.
00:06:29.440 --> 00:06:32.555
So roughly speaking,
the building
00:06:32.555 --> 00:06:36.360
is about 300 feet
high, we're down here
00:06:36.360 --> 00:06:39.570
on the first usable floor.
00:06:39.570 --> 00:06:44.130
And so we're going to
use instead of 300 feet,
00:06:44.130 --> 00:06:46.330
just for convenience
purposes we'll
00:06:46.330 --> 00:06:55.410
use 80 meters because that makes
the numbers come out simply.
00:06:55.410 --> 00:07:04.380
So we have the height
which starts out
00:07:04.380 --> 00:07:09.590
at 80 meters at time 0 and then
the acceleration due to gravity
00:07:09.590 --> 00:07:13.330
gives you this formula
for h, this is the height.
00:07:13.330 --> 00:07:21.760
So at time t = 0, we're up
at the top, h is 80 meters,
00:07:21.760 --> 00:07:24.580
the units here are meters.
00:07:24.580 --> 00:07:32.200
And at time t = 4 you
notice, 5 * 4^2 is 80.
00:07:32.200 --> 00:07:34.030
I picked these numbers
conveniently so
00:07:34.030 --> 00:07:38.320
that we're down at the bottom.
00:07:38.320 --> 00:07:43.620
Okay, so this notion
of average change here,
00:07:43.620 --> 00:07:51.380
so the average change, or
the average speed here,
00:07:51.380 --> 00:07:55.636
maybe we'll call it
the average speed,
00:07:55.636 --> 00:08:02.910
since that's-- over this time
that it takes for the pumpkin
00:08:02.910 --> 00:08:07.385
to drop is going to be
the change in h divided
00:08:07.385 --> 00:08:10.170
by the change in t.
00:08:10.170 --> 00:08:18.350
Which starts out at, what
does it start out as?
00:08:18.350 --> 00:08:21.870
It starts out as 80, right?
00:08:21.870 --> 00:08:23.930
And it ends at 0.
00:08:23.930 --> 00:08:26.520
So actually we have
to do it backwards.
00:08:26.520 --> 00:08:32.110
We have to take 0 - 80
because the first value is
00:08:32.110 --> 00:08:35.500
the final position
and the second value
00:08:35.500 --> 00:08:37.390
is the initial position.
00:08:37.390 --> 00:08:41.470
And that's divided by
4 - 0; times 4 seconds
00:08:41.470 --> 00:08:43.680
minus times 0 seconds.
00:08:43.680 --> 00:08:49.200
And so that of course is
-20 meters per second.
00:08:49.200 --> 00:08:56.860
So the average speed of this
guy is 20 meters a second.
00:08:56.860 --> 00:09:00.970
Now, so why did I
pick this example?
00:09:00.970 --> 00:09:04.480
Because, of course, the
average, although interesting,
00:09:04.480 --> 00:09:06.540
is not really what
anybody cares about who
00:09:06.540 --> 00:09:08.670
actually goes to the event.
00:09:08.670 --> 00:09:11.460
All we really care about
is the instantaneous speed
00:09:11.460 --> 00:09:19.005
when it hits the pavement
and so that's can
00:09:19.005 --> 00:09:23.610
be calculated at the bottom.
00:09:23.610 --> 00:09:25.330
So what's the
instantaneous speed?
00:09:25.330 --> 00:09:29.530
That's the derivative,
or maybe to be
00:09:29.530 --> 00:09:31.720
consistent with the notation
I've been using so far,
00:09:31.720 --> 00:09:35.950
that's d/dt of h.
00:09:35.950 --> 00:09:37.580
All right?
00:09:37.580 --> 00:09:39.090
So that's d/dt of h.
00:09:39.090 --> 00:09:42.020
Now remember we have
formulas for these things.
00:09:42.020 --> 00:09:43.850
We can differentiate
this function now.
00:09:43.850 --> 00:09:47.930
We did that yesterday.
00:09:47.930 --> 00:09:51.350
So we're gonna take the rate of
change and if you take a look
00:09:51.350 --> 00:09:56.790
at it, it's just the rate
of change of 80 is 0,
00:09:56.790 --> 00:10:02.860
minus the rate change for
this -5t^2, that's minus 10t.
00:10:02.860 --> 00:10:08.750
So that's using the fact
that d/dt of 80 is equal to 0
00:10:08.750 --> 00:10:12.350
and d/dt of t^2 is equal to 2t.
00:10:12.350 --> 00:10:14.420
The special case...
00:10:14.420 --> 00:10:17.160
Well I'm cheating
here, but there's
00:10:17.160 --> 00:10:18.500
a special case that's obvious.
00:10:18.500 --> 00:10:19.850
I didn't throw it in over here.
00:10:19.850 --> 00:10:23.750
The case n = 2 is that
second case there.
00:10:23.750 --> 00:10:30.380
But the case n = 0 also works.
00:10:30.380 --> 00:10:31.534
Because that's constants.
00:10:31.534 --> 00:10:32.950
The derivative of
a constant is 0.
00:10:32.950 --> 00:10:36.781
And then the factor n there's
0 and that's consistent.
00:10:36.781 --> 00:10:38.780
And actually if you look
at the formula above it
00:10:38.780 --> 00:10:44.090
you'll see that it's
the case of n = -1.
00:10:44.090 --> 00:10:49.820
So we'll get a larger pattern
soon enough with the powers.
00:10:49.820 --> 00:10:50.450
Okay anyway.
00:10:50.450 --> 00:10:53.770
Back over here we have
our rate of change
00:10:53.770 --> 00:10:55.380
and this is what it is.
00:10:55.380 --> 00:10:59.350
And at the bottom, at
that point of impact,
00:10:59.350 --> 00:11:04.700
we have t = 4 and so h',
which is the derivative,
00:11:04.700 --> 00:11:12.860
is equal to -40
meters per second.
00:11:12.860 --> 00:11:16.540
So twice as fast as
the average speed here,
00:11:16.540 --> 00:11:22.900
and if you need to convert that,
that's about 90 miles an hour.
00:11:22.900 --> 00:11:29.450
Which is why the police are
there at midnight on Halloween
00:11:29.450 --> 00:11:33.310
to make sure you're all safe
and also why when you come
00:11:33.310 --> 00:11:37.330
you have to be prepared
to clean up afterwards.
00:11:37.330 --> 00:11:40.260
So anyway that's what happens,
it's 90 miles an hour.
00:11:40.260 --> 00:11:42.320
It's actually the
buildings a little taller,
00:11:42.320 --> 00:11:45.010
there's air resistance
and I'm sure you
00:11:45.010 --> 00:11:50.350
can do a much more thorough
study of this example.
00:11:50.350 --> 00:11:54.300
All right so now I want to give
you a couple of more examples
00:11:54.300 --> 00:11:58.700
because time and these kinds
of parameters and variables
00:11:58.700 --> 00:12:02.570
are not the only ones that
are important for calculus.
00:12:02.570 --> 00:12:05.610
If it were only this kind of
physics that was involved,
00:12:05.610 --> 00:12:09.710
then this would be a much more
specialized subject than it is.
00:12:09.710 --> 00:12:13.340
And so I want to give you a
couple of examples that don't
00:12:13.340 --> 00:12:16.570
involve time as a variable.
00:12:16.570 --> 00:12:20.175
So the third example
I'll give here
00:12:20.175 --> 00:12:27.260
is-- The letter T often
denotes temperature,
00:12:27.260 --> 00:12:35.660
and then dT/dx would be what
is known as the temperature
00:12:35.660 --> 00:12:38.830
gradient.
00:12:38.830 --> 00:12:43.310
Which we really care
about a lot when
00:12:43.310 --> 00:12:45.820
we're predicting the weather
because it's that temperature
00:12:45.820 --> 00:12:52.140
difference that causes air flows
and causes things to change.
00:12:52.140 --> 00:12:54.410
And then there's
another theme which
00:12:54.410 --> 00:13:01.110
is throughout the sciences
and engineering which
00:13:01.110 --> 00:13:07.700
I'm going to talk about under
the heading of sensitivity
00:13:07.700 --> 00:13:15.600
of measurements.
00:13:15.600 --> 00:13:18.470
So let me explain this.
00:13:18.470 --> 00:13:21.530
I don't want to belabor
it because I just
00:13:21.530 --> 00:13:23.810
am doing this in
order to introduce you
00:13:23.810 --> 00:13:26.380
to the ideas on your
problem set which
00:13:26.380 --> 00:13:29.350
are the first case of this.
00:13:29.350 --> 00:13:33.290
So on problem set one
you have an example
00:13:33.290 --> 00:13:37.360
which is based on a
simplified model of GPS,
00:13:37.360 --> 00:13:39.550
sort of the Flat Earth Model.
00:13:39.550 --> 00:13:42.400
And in that situation,
well, if the Earth is flat
00:13:42.400 --> 00:13:45.490
it's just a horizontal
line like this.
00:13:45.490 --> 00:13:54.020
And then you have a satellite,
which is over here, preferably
00:13:54.020 --> 00:14:02.440
above the earth, and the
satellite or the system
00:14:02.440 --> 00:14:05.810
knows exactly where the point
directly below the satellite
00:14:05.810 --> 00:14:06.480
is.
00:14:06.480 --> 00:14:12.170
So this point is
treated as known.
00:14:12.170 --> 00:14:22.160
And I'm sitting here
with my little GPS device
00:14:22.160 --> 00:14:26.440
and I want to know where I am.
00:14:26.440 --> 00:14:28.880
And the way I
locate where I am is
00:14:28.880 --> 00:14:34.750
I communicate with this
satellite by radio signals
00:14:34.750 --> 00:14:38.750
and I can measure this distance
here which is called h.
00:14:38.750 --> 00:14:45.060
And then system will compute
this horizontal distance which
00:14:45.060 --> 00:14:53.360
is L. So in other
words what is measured,
00:14:53.360 --> 00:15:03.020
so h measured by radios,
radio waves and a clock,
00:15:03.020 --> 00:15:04.910
or various clocks.
00:15:04.910 --> 00:15:13.560
And then L is deduced from h.
00:15:13.560 --> 00:15:16.730
And what's critical in
all of these systems
00:15:16.730 --> 00:15:20.320
is that you don't
know h exactly.
00:15:20.320 --> 00:15:26.330
There's an error in h
which will denote delta h.
00:15:26.330 --> 00:15:31.040
There's some degree
of uncertainty.
00:15:31.040 --> 00:15:35.550
The main uncertainty in
GPS is from the ionosphere.
00:15:35.550 --> 00:15:38.280
But there are lots
of corrections
00:15:38.280 --> 00:15:41.340
that are made of all kinds.
00:15:41.340 --> 00:15:43.570
And also if you're
inside a building
00:15:43.570 --> 00:15:44.790
it's a problem to measure it.
00:15:44.790 --> 00:15:47.970
But it's an extremely
important issue,
00:15:47.970 --> 00:15:49.730
as I'll explain in a second.
00:15:49.730 --> 00:15:54.040
So the idea is we
then get at delta
00:15:54.040 --> 00:16:04.246
L is estimated by considering
this ratio delta L/delta
00:16:04.246 --> 00:16:07.060
h which is going
to be approximately
00:16:07.060 --> 00:16:13.910
the same as the derivative
of L with respect to h.
00:16:13.910 --> 00:16:17.815
So this is the thing that's
easy because of course it's
00:16:17.815 --> 00:16:18.940
calculus.
00:16:18.940 --> 00:16:21.960
Calculus is the
easy part and that
00:16:21.960 --> 00:16:25.330
allows us to deduce something
about the real world that's
00:16:25.330 --> 00:16:28.600
close by over here.
00:16:28.600 --> 00:16:31.980
So the reason why you should
care about this quite a bit
00:16:31.980 --> 00:16:34.870
is that it's used all the
time to land airplanes.
00:16:34.870 --> 00:16:36.680
So you really do care
that they actually
00:16:36.680 --> 00:16:42.386
know to within a few feet or
even closer where your plane is
00:16:42.386 --> 00:16:48.150
and how high up it
is and so forth.
00:16:48.150 --> 00:16:48.650
All right.
00:16:48.650 --> 00:16:50.850
So that's it for the
general introduction
00:16:50.850 --> 00:16:52.010
of what a derivative is.
00:16:52.010 --> 00:16:53.670
I'm sure you'll be
getting used to this
00:16:53.670 --> 00:16:56.560
in a lot of different contexts
throughout the course.
00:16:56.560 --> 00:17:04.510
And now we have to get back
down to some rigorous details.
00:17:04.510 --> 00:17:09.540
Okay, everybody happy with
what we've got so far?
00:17:09.540 --> 00:17:10.040
Yeah?
00:17:10.040 --> 00:17:13.400
Student: How did you get
the equation for height?
00:17:13.400 --> 00:17:14.980
Professor: Ah good question.
00:17:14.980 --> 00:17:18.560
The question was how did I
get this equation for height?
00:17:18.560 --> 00:17:24.970
I just made it up because
it's the formula from physics
00:17:24.970 --> 00:17:29.592
that you will learn when
you take 8.01 and, in fact,
00:17:29.592 --> 00:17:35.060
it has to do with the fact
that this is the speed if you
00:17:35.060 --> 00:17:37.340
differentiate
another time you get
00:17:37.340 --> 00:17:40.550
acceleration and
acceleration due to gravity
00:17:40.550 --> 00:17:42.330
is 10 meters per second.
00:17:42.330 --> 00:17:44.450
Which happens to be the
second derivative of this.
00:17:44.450 --> 00:17:47.710
But anyway I just pulled it
out of a hat from your physics
00:17:47.710 --> 00:17:48.350
class.
00:17:48.350 --> 00:17:55.510
So you can just say see 8.01 .
00:17:55.510 --> 00:18:02.840
All right, other questions?
00:18:02.840 --> 00:18:04.970
All right, so let's go on now.
00:18:04.970 --> 00:18:09.340
Now I have to be a little bit
more systematic about limits.
00:18:09.340 --> 00:18:20.130
So let's do that now.
00:18:20.130 --> 00:18:30.370
So now what I'd like to talk
about is limits and continuity.
00:18:30.370 --> 00:18:34.200
And this is a warm
up for deriving
00:18:34.200 --> 00:18:37.900
all the rest of the formulas,
all the rest of the formulas
00:18:37.900 --> 00:18:40.430
that I'm going to
need to differentiate
00:18:40.430 --> 00:18:41.600
every function you know.
00:18:41.600 --> 00:18:44.120
Remember, that's our goal
and we only have about a week
00:18:44.120 --> 00:18:47.510
left so we'd better get started.
00:18:47.510 --> 00:18:58.980
So first of all there is
what I will call easy limits.
00:18:58.980 --> 00:19:00.650
So what's an easy limit?
00:19:00.650 --> 00:19:07.285
An easy limit is something like
the limit as x goes to 4 of x
00:19:07.285 --> 00:19:11.570
plus 3 over x^2 + 1.
00:19:11.570 --> 00:19:16.240
And with this kind of limit all
I have to do to evaluate it is
00:19:16.240 --> 00:19:23.770
to plug in x = 4 because,
so what I get here is 4 + 3
00:19:23.770 --> 00:19:27.900
divided by 4^2 + 1.
00:19:27.900 --> 00:19:31.560
And that's just 7 / 17.
00:19:31.560 --> 00:19:33.720
And that's the end of it.
00:19:33.720 --> 00:19:38.510
So those are the easy limits.
00:19:38.510 --> 00:19:42.669
The second kind of limit -
well so this isn't the only
00:19:42.669 --> 00:19:44.960
second kind of limit but I
just want to point this out,
00:19:44.960 --> 00:19:55.680
it's very important - is that:
derivatives are are always
00:19:55.680 --> 00:19:59.370
harder than this.
00:19:59.370 --> 00:20:03.230
You can't get away
with nothing here.
00:20:03.230 --> 00:20:05.090
So, why is that?
00:20:05.090 --> 00:20:07.620
Well, when you
take a derivative,
00:20:07.620 --> 00:20:13.160
you're taking the limit
as x goes to x_0 of f(x),
00:20:13.160 --> 00:20:24.520
well we'll write it all
out in all its glory.
00:20:24.520 --> 00:20:28.790
Here's the formula
for the derivative.
00:20:28.790 --> 00:20:39.110
Now notice that if you plug in
x = x:0, always gives 0 / 0.
00:20:39.110 --> 00:20:42.080
So it just basically
never works.
00:20:42.080 --> 00:20:50.940
So we always are going
to need some cancellation
00:20:50.940 --> 00:21:05.960
to make sense out of the limit.
00:21:05.960 --> 00:21:12.570
Now in order to make things
a little easier for myself
00:21:12.570 --> 00:21:15.700
to explain what's
going on with limits
00:21:15.700 --> 00:21:18.660
I need to introduce just
one more piece of notation.
00:21:18.660 --> 00:21:20.490
What I'm gonna
introduce here is what's
00:21:20.490 --> 00:21:23.380
known as a left-hand
and a right limit.
00:21:23.380 --> 00:21:29.500
If I take the limit as x tends
to x_0 with a plus sign here
00:21:29.500 --> 00:21:42.280
of some function, this is what's
known as the right-hand limit.
00:21:42.280 --> 00:21:44.870
And I can display it visually.
00:21:44.870 --> 00:21:45.950
So what does this mean?
00:21:45.950 --> 00:21:47.530
It means practically
the same thing
00:21:47.530 --> 00:21:51.160
as x tends to x_0 except there
is one more restriction which
00:21:51.160 --> 00:21:53.630
has to do with this plus
sign, which is we're going
00:21:53.630 --> 00:21:55.370
from the plus side of x_0.
00:21:55.370 --> 00:21:58.710
That means x is bigger than x_0.
00:21:58.710 --> 00:22:01.770
And I say right-hand, so
there should be a hyphen here,
00:22:01.770 --> 00:22:06.600
right-hand limit because
on the number line,
00:22:06.600 --> 00:22:14.580
if x_0 is over here
the x is to the right.
00:22:14.580 --> 00:22:15.080
All right?
00:22:15.080 --> 00:22:16.750
So that's the right-hand limit.
00:22:16.750 --> 00:22:19.550
And then this being the
left side of the board,
00:22:19.550 --> 00:22:22.716
I'll put on the right side
of the board the left limit,
00:22:22.716 --> 00:22:24.560
just to make things confusing.
00:22:24.560 --> 00:22:30.520
So that one has the
minus sign here.
00:22:30.520 --> 00:22:33.940
I'm just a little dyslexic
and I hope you're not.
00:22:33.940 --> 00:22:38.200
So I may have gotten that wrong.
00:22:38.200 --> 00:22:41.510
So this is the left-hand
limit, and I'll draw it.
00:22:41.510 --> 00:22:45.705
So of course that just
means x goes to x_0 but x is
00:22:45.705 --> 00:22:48.260
to the left of x_0 .
00:22:48.260 --> 00:22:52.290
And again, on the number
line, here's the x_0
00:22:52.290 --> 00:22:56.570
and the x is on the
other side of it.
00:22:56.570 --> 00:22:58.970
Okay, so those two
notations are going
00:22:58.970 --> 00:23:01.830
to help us to clarify
a bunch of things.
00:23:01.830 --> 00:23:04.580
It's much more
convenient to have
00:23:04.580 --> 00:23:08.520
this extra bit of
description of limits
00:23:08.520 --> 00:23:15.880
than to just consider
limits from both sides.
00:23:15.880 --> 00:23:25.980
Okay so I want to give
an example of this.
00:23:25.980 --> 00:23:29.310
And also an example
of how you're going to
00:23:29.310 --> 00:23:32.110
think about these
sorts of problems.
00:23:32.110 --> 00:23:38.020
So I'll take a function which
has two different definitions.
00:23:38.020 --> 00:23:47.570
Say it's x + 1, when x >
0 and -x + 2, when x < 0.
00:23:47.570 --> 00:23:51.280
So maybe put commas there.
00:23:51.280 --> 00:23:58.540
So when x > 0, it's x + 1.
00:23:58.540 --> 00:24:01.030
Now I can draw a
picture of this.
00:24:01.030 --> 00:24:02.955
It's gonna be kind
of a little small
00:24:02.955 --> 00:24:04.830
because I'm gonna try
to fit it down in here,
00:24:04.830 --> 00:24:07.670
but maybe I'll put
the axis down below.
00:24:07.670 --> 00:24:13.990
So at height 1, I have to
the right something of slope
00:24:13.990 --> 00:24:16.890
1 so it goes up like this.
00:24:16.890 --> 00:24:18.240
All right?
00:24:18.240 --> 00:24:26.500
And then to the left of 0 I have
something which has slope -1,
00:24:26.500 --> 00:24:30.720
but it hits the axis
at 2 so it's up here.
00:24:30.720 --> 00:24:34.175
So I had this sort of
strange antenna figure here,
00:24:34.175 --> 00:24:35.150
which is my graph.
00:24:35.150 --> 00:24:43.710
Maybe I should draw these in
another color to depict that.
00:24:43.710 --> 00:24:47.780
And then if I calculate
these two limits here,
00:24:47.780 --> 00:24:54.670
what I see is that
the limit as x
00:24:54.670 --> 00:25:00.860
goes to 0 from above of f(x),
that's the same as the limit
00:25:00.860 --> 00:25:07.990
as x goes to 0 of the
formula here, x + 1.
00:25:07.990 --> 00:25:10.430
Which turns out to be 1.
00:25:10.430 --> 00:25:15.360
And if I take the limit, so
that's the left-hand limit.
00:25:15.360 --> 00:25:20.700
Sorry, I told you
I was dyslexic.
00:25:20.700 --> 00:25:23.320
This is the right, so
it's that right-hand.
00:25:23.320 --> 00:25:25.080
Here we go.
00:25:25.080 --> 00:25:31.530
So now I'm going from the
left, and it's f(x) again,
00:25:31.530 --> 00:25:35.180
but now because I'm on that
side the thing I need to plug
00:25:35.180 --> 00:25:43.540
is the other formula, -x + 2,
and that's gonna give us 2.
00:25:43.540 --> 00:25:48.310
Now, notice that the left
and right limits, and this
00:25:48.310 --> 00:25:51.470
is one little tiny subtlety
and it's almost the only thing
00:25:51.470 --> 00:25:53.770
that I need you to really
pay attention to a little bit
00:25:53.770 --> 00:26:06.210
right now, is that this, we
did not need x = 0 value.
00:26:06.210 --> 00:26:11.860
In fact I never even told
you what f(0) was here.
00:26:11.860 --> 00:26:14.650
If we stick it in we
could stick it in.
00:26:14.650 --> 00:26:20.050
Okay let's say we stick
it in on this side.
00:26:20.050 --> 00:26:22.970
Let's make it be that
it's on this side.
00:26:22.970 --> 00:26:32.860
So that means that this point
is in and this point is out.
00:26:32.860 --> 00:26:37.680
So that's a typical notation:
this little open circle
00:26:37.680 --> 00:26:41.530
and this closed dot for
when you include the.
00:26:41.530 --> 00:26:44.830
So in that case
the value of f(x)
00:26:44.830 --> 00:26:48.360
happens to be the same
as its right-hand limit,
00:26:48.360 --> 00:26:56.530
namely the value is
1 here and not 2.
00:26:56.530 --> 00:27:01.140
Okay, so that's the
first kind of example.
00:27:01.140 --> 00:27:06.610
Questions?
00:27:06.610 --> 00:27:13.420
Okay, so now our next
job is to introduce
00:27:13.420 --> 00:27:17.270
the definition of continuity.
00:27:17.270 --> 00:27:20.080
So that was the
other topic here.
00:27:20.080 --> 00:27:23.490
So we're going to define.
00:27:23.490 --> 00:27:39.515
So f is continuous at x_0 means
that the limit of f(x) as x
00:27:39.515 --> 00:27:44.440
tends to x_0 is
equal to f(x_0) .
00:27:44.440 --> 00:27:47.090
Right?
00:27:47.090 --> 00:27:51.750
So the reason why I spend
all this time paying
00:27:51.750 --> 00:27:54.540
attention to the left and the
right and so on and so forth
00:27:54.540 --> 00:27:57.340
and focusing is that I want you
to pay attention for one moment
00:27:57.340 --> 00:28:01.820
to what the content
of this definition is.
00:28:01.820 --> 00:28:12.640
What it's saying is the
following: continuous at x_0
00:28:12.640 --> 00:28:15.450
has various ingredients here.
00:28:15.450 --> 00:28:24.540
So the first one is
that this limit exists.
00:28:24.540 --> 00:28:27.080
And what that means
is that there's
00:28:27.080 --> 00:28:35.150
an honest limiting value
both from the left and right.
00:28:35.150 --> 00:28:39.250
And they also have
to be the same.
00:28:39.250 --> 00:28:41.980
All right, so that's
what's going on here.
00:28:41.980 --> 00:28:50.380
And the second property
is that f(x_0) is defined.
00:28:50.380 --> 00:28:52.100
So I can't be in one
of these situations
00:28:52.100 --> 00:28:54.770
where I haven't
even specified what
00:28:54.770 --> 00:29:05.220
f(x_0) is and they're equal.
00:29:05.220 --> 00:29:09.190
Okay, so that's the situation.
00:29:09.190 --> 00:29:13.310
Now again let me
emphasize a tricky part
00:29:13.310 --> 00:29:15.560
of the definition of a limit.
00:29:15.560 --> 00:29:20.320
This side, the left-hand side
is completely independent,
00:29:20.320 --> 00:29:23.790
is evaluated by a
procedure which does not
00:29:23.790 --> 00:29:25.070
involve the right-hand side.
00:29:25.070 --> 00:29:26.900
These are separate things.
00:29:26.900 --> 00:29:34.310
This one is, to evaluate it, you
always avoid the limit point.
00:29:34.310 --> 00:29:37.670
So that's if you like a
paradox, because it's exactly
00:29:37.670 --> 00:29:41.290
the question: is it true
that if you plug in x_0
00:29:41.290 --> 00:29:44.300
you get the same answer as
if you move in the limit?
00:29:44.300 --> 00:29:46.270
That's the issue that
we're considering here.
00:29:46.270 --> 00:29:48.270
We have to make that
distinction in order
00:29:48.270 --> 00:29:50.880
to say that these
are two, otherwise
00:29:50.880 --> 00:29:55.270
this is just tautological.
00:29:55.270 --> 00:29:56.630
It doesn't have any meaning.
00:29:56.630 --> 00:29:58.046
But in fact it
does have a meaning
00:29:58.046 --> 00:30:00.620
because one thing is evaluated
separately with reference
00:30:00.620 --> 00:30:03.850
to all the other
points and the other
00:30:03.850 --> 00:30:06.870
is evaluated right at
the point in question.
00:30:06.870 --> 00:30:11.090
And indeed what
these things are,
00:30:11.090 --> 00:30:17.896
are exactly the easy limits.
00:30:17.896 --> 00:30:19.770
That's exactly what
we're talking about here.
00:30:19.770 --> 00:30:24.150
They're the ones you
can evaluate this way.
00:30:24.150 --> 00:30:25.640
So we have to make
the distinction.
00:30:25.640 --> 00:30:27.685
And these other ones are
gonna be the ones which
00:30:27.685 --> 00:30:29.670
we can't evaluate that way.
00:30:29.670 --> 00:30:31.540
So these are the
nice ones and that's
00:30:31.540 --> 00:30:33.830
why we care about them, why
we have a whole definition
00:30:33.830 --> 00:30:36.470
associated with them.
00:30:36.470 --> 00:30:38.700
All right?
00:30:38.700 --> 00:30:40.400
So now what's next?
00:30:40.400 --> 00:30:48.910
Well, I need to give you a a
little tour, very brief tour,
00:30:48.910 --> 00:30:54.090
of the zoo of what are known
as discontinuous functions.
00:30:54.090 --> 00:30:57.430
So sort of everything else
that's not continuous.
00:30:57.430 --> 00:31:04.550
So, the first example here,
let me just write it down here.
00:31:04.550 --> 00:31:13.670
It's jump discontinuities.
00:31:13.670 --> 00:31:15.300
So what would a jump
discontinuity be?
00:31:15.300 --> 00:31:18.730
Well we've actually
already seen it.
00:31:18.730 --> 00:31:21.790
The jump discontinuity
is the example
00:31:21.790 --> 00:31:23.230
that we had right there.
00:31:23.230 --> 00:31:32.490
This is when the limit
from the left and right
00:31:32.490 --> 00:31:42.180
exist, but are not equal.
00:31:42.180 --> 00:31:50.940
Okay, so that's
as in the example.
00:31:50.940 --> 00:31:51.440
Right?
00:31:51.440 --> 00:31:53.680
In this example, the
two limits, one of them
00:31:53.680 --> 00:31:57.890
was 1 and of them was 2.
00:31:57.890 --> 00:32:02.150
So that's a jump discontinuity.
00:32:02.150 --> 00:32:09.310
And this kind of issue,
of whether something
00:32:09.310 --> 00:32:14.940
is continuous or not, may
seem a little bit technical
00:32:14.940 --> 00:32:26.120
but it is true that people
have worried about it a lot.
00:32:26.120 --> 00:32:28.820
Bob Merton, who was a
professor at MIT when
00:32:28.820 --> 00:32:33.410
he did his work for the
Nobel prize in economics,
00:32:33.410 --> 00:32:36.180
was interested in
this very issue
00:32:36.180 --> 00:32:39.320
of whether stock
prices of various kinds
00:32:39.320 --> 00:32:42.540
are continuous from the left
or right in a certain model.
00:32:42.540 --> 00:32:44.580
And that was a
very serious issue
00:32:44.580 --> 00:32:49.150
in developing the model
that priced things
00:32:49.150 --> 00:32:51.840
that our hedge funds
use all the time now.
00:32:51.840 --> 00:32:57.630
So left and right can really
mean something very different.
00:32:57.630 --> 00:33:01.507
In this case left is the
past and right is the future
00:33:01.507 --> 00:33:03.340
and it makes a big
difference whether things
00:33:03.340 --> 00:33:06.840
are continuous from the left
or continuous from the right.
00:33:06.840 --> 00:33:09.120
Right, is it true that
the point is here,
00:33:09.120 --> 00:33:11.720
here, somewhere in the
middle, somewhere else.
00:33:11.720 --> 00:33:13.480
That's a serious issue.
00:33:13.480 --> 00:33:18.210
So the next example
that I want to give you
00:33:18.210 --> 00:33:22.720
is a little bit more subtle.
00:33:22.720 --> 00:33:32.140
It's what's known as a
removable discontinuity.
00:33:32.140 --> 00:33:43.010
And so what this means is that
the limit from left and right
00:33:43.010 --> 00:33:46.190
are equal.
00:33:46.190 --> 00:33:47.980
So a picture of
that would be, you
00:33:47.980 --> 00:33:50.480
have a function which is
coming along like this
00:33:50.480 --> 00:33:52.820
and there's a hole
maybe where, who knows
00:33:52.820 --> 00:33:56.270
either the function is undefined
or maybe it's defined up here,
00:33:56.270 --> 00:33:58.751
and then it just continues on.
00:33:58.751 --> 00:33:59.250
All right?
00:33:59.250 --> 00:34:01.210
So the two limits are the same.
00:34:01.210 --> 00:34:05.010
And then of course the function
is begging to be redefined
00:34:05.010 --> 00:34:07.370
so that we remove that hole.
00:34:07.370 --> 00:34:14.470
And that's why it's called
a removable discontinuity.
00:34:14.470 --> 00:34:17.710
Now let me give you
an example of this,
00:34:17.710 --> 00:34:22.460
or actually a
couple of examples.
00:34:22.460 --> 00:34:28.130
So these are quite
important examples
00:34:28.130 --> 00:34:34.020
which you will be working
with in a few minutes.
00:34:34.020 --> 00:34:41.660
So the first is the function
g(x), which is sin x / x,
00:34:41.660 --> 00:34:45.260
and the second will be the
function h(x), which is 1 -
00:34:45.260 --> 00:34:50.520
cos x over x.
00:34:50.520 --> 00:35:00.290
So we have a problem at
g(0), g(0) is undefined.
00:35:00.290 --> 00:35:03.760
On the other hand it turns
out this function has what's
00:35:03.760 --> 00:35:05.710
called a removable singularity.
00:35:05.710 --> 00:35:14.630
Namely the limit as x goes
to 0 of sin x / x does exist.
00:35:14.630 --> 00:35:17.050
In fact it's equal to 1.
00:35:17.050 --> 00:35:20.430
So that's a very important limit
that we will work out either
00:35:20.430 --> 00:35:23.420
at the end of this lecture or
the beginning of next lecture.
00:35:23.420 --> 00:35:30.940
And similarly, the
limit of 1 - cos x
00:35:30.940 --> 00:35:35.370
divided by x, as
x goes to 0, is 0.
00:35:35.370 --> 00:35:38.051
Maybe I'll put that
a little farther
00:35:38.051 --> 00:35:40.360
away so you can read it.
00:35:40.360 --> 00:35:44.940
Okay, so these are
very useful facts
00:35:44.940 --> 00:35:47.800
that we're going
to need later on.
00:35:47.800 --> 00:35:50.460
And what they say is
that these things have
00:35:50.460 --> 00:35:58.520
removable singularities, sorry
removable discontinuity at x
00:35:58.520 --> 00:36:04.600
= 0.
00:36:04.600 --> 00:36:13.030
All right so as I say, we'll
get to that in a few minutes.
00:36:13.030 --> 00:36:16.400
Okay so are there any
questions before I move on?
00:36:16.400 --> 00:36:16.900
Yeah?
00:36:16.900 --> 00:36:30.630
Student: [INAUDIBLE]
00:36:30.630 --> 00:36:38.300
Professor: The question
is: why is this true?
00:36:38.300 --> 00:36:40.300
Is that what your question is?
00:36:40.300 --> 00:36:44.070
The answer is it's
very, very unobvious,
00:36:44.070 --> 00:36:48.360
I haven't shown it to you
yet, and if you were not
00:36:48.360 --> 00:36:51.560
surprised by it then that
would be very strange indeed.
00:36:51.560 --> 00:36:53.390
So we haven't done it yet.
00:36:53.390 --> 00:36:55.990
You have to stay
tuned until we do.
00:36:55.990 --> 00:36:57.210
Okay?
00:36:57.210 --> 00:36:59.250
We haven't shown it yet.
00:36:59.250 --> 00:37:01.320
And actually even
this other statement,
00:37:01.320 --> 00:37:03.600
which maybe seems
stranger still,
00:37:03.600 --> 00:37:05.760
is also not yet explained.
00:37:05.760 --> 00:37:08.865
Okay, so we are going
to get there, as I said,
00:37:08.865 --> 00:37:10.240
either at the end
of this lecture
00:37:10.240 --> 00:37:15.410
or at the beginning of next.
00:37:15.410 --> 00:37:22.560
Other questions?
00:37:22.560 --> 00:37:28.180
All right, so let me
just continue my tour
00:37:28.180 --> 00:37:34.000
of the zoo of discontinuities.
00:37:34.000 --> 00:37:37.050
And, I guess, I want
to illustrate something
00:37:37.050 --> 00:37:41.440
with the convenience of
right and left hand limits
00:37:41.440 --> 00:37:52.180
so I'll save this board about
right and left-hand limits.
00:37:52.180 --> 00:37:54.970
So a third type of
discontinuity is
00:37:54.970 --> 00:38:07.320
what's known as an
infinite discontinuity.
00:38:07.320 --> 00:38:11.950
And we've already
encountered one of these.
00:38:11.950 --> 00:38:14.450
I'm going to draw
them over here.
00:38:14.450 --> 00:38:19.370
Remember the
function y is 1 / x.
00:38:19.370 --> 00:38:22.450
That's this function here.
00:38:22.450 --> 00:38:25.500
But now I'd like to draw
also the other branch
00:38:25.500 --> 00:38:31.140
of the hyperbola down here
and allow myself to consider
00:38:31.140 --> 00:38:32.320
negative values of x.
00:38:32.320 --> 00:38:35.910
So here's the graph of 1 / x.
00:38:35.910 --> 00:38:42.640
And the convenience here
of distinguishing the left
00:38:42.640 --> 00:38:46.620
and the right hand limits is
very important because here I
00:38:46.620 --> 00:38:51.800
can write down that the limit
as x goes to 0+ of 1 / x.
00:38:51.800 --> 00:38:57.300
Well that's coming from the
right and it's going up.
00:38:57.300 --> 00:39:00.580
So this limit is infinity.
00:39:00.580 --> 00:39:05.380
Whereas, the limit in
the other direction,
00:39:05.380 --> 00:39:10.630
from the left, that
one is going down.
00:39:10.630 --> 00:39:16.510
And so it's quite different,
it's minus infinity.
00:39:16.510 --> 00:39:19.860
Now some people say that
these limits are undefined
00:39:19.860 --> 00:39:22.940
but actually they're going in
some very definite direction.
00:39:22.940 --> 00:39:24.950
So you should,
whenever possible,
00:39:24.950 --> 00:39:26.640
specify what these limits are.
00:39:26.640 --> 00:39:30.860
On the other hand, the
statement that the limit
00:39:30.860 --> 00:39:37.250
as x goes to 0 of 1 / x is
infinity is simply wrong.
00:39:37.250 --> 00:39:40.340
Okay, it's not that
people don't write this.
00:39:40.340 --> 00:39:41.680
It's just that it's wrong.
00:39:41.680 --> 00:39:43.470
It's not that they
don't write it down.
00:39:43.470 --> 00:39:45.000
In fact you'll probably see it.
00:39:45.000 --> 00:39:48.055
It's because people are just
thinking of the right hand
00:39:48.055 --> 00:39:48.790
branch.
00:39:48.790 --> 00:39:51.220
It's not that they're making
a mistake necessarily,
00:39:51.220 --> 00:39:53.116
but anyway, it's sloppy.
00:39:53.116 --> 00:39:54.990
And there's some sloppiness
that we'll endure
00:39:54.990 --> 00:39:57.080
and others that
we'll try to avoid.
00:39:57.080 --> 00:40:00.120
So here, you want to say this,
and it does make a difference.
00:40:00.120 --> 00:40:04.990
You know, plus infinity is
an infinite number of dollars
00:40:04.990 --> 00:40:07.450
and minus infinity is and
infinite amount of debt.
00:40:07.450 --> 00:40:08.980
They're actually different.
00:40:08.980 --> 00:40:09.890
They're not the same.
00:40:09.890 --> 00:40:15.540
So, you know, this is sloppy and
this is actually more correct.
00:40:15.540 --> 00:40:17.885
Okay, so now in
addition, I just want
00:40:17.885 --> 00:40:21.350
to point out one more thing.
00:40:21.350 --> 00:40:24.210
Remember, we calculated
the derivative,
00:40:24.210 --> 00:40:26.880
and that was -1/x^2.
00:40:26.880 --> 00:40:31.196
But, I want to draw
the graph of that
00:40:31.196 --> 00:40:32.570
and make a few
comments about it.
00:40:32.570 --> 00:40:34.420
So I'm going to draw
the graph directly
00:40:34.420 --> 00:40:38.820
underneath the graph
of the function.
00:40:38.820 --> 00:40:41.290
And notice what this graphs is.
00:40:41.290 --> 00:40:48.530
It goes like this, it's always
negative, and it points down.
00:40:48.530 --> 00:40:51.480
So now this may look
a little strange,
00:40:51.480 --> 00:40:55.080
that the derivative of
this thing is this guy,
00:40:55.080 --> 00:40:58.630
but that's because of
something very important.
00:40:58.630 --> 00:41:01.030
And you should always remember
this about derivatives.
00:41:01.030 --> 00:41:03.995
The derivative function looks
nothing like the function,
00:41:03.995 --> 00:41:04.860
necessarily.
00:41:04.860 --> 00:41:07.780
So you should just forget
about that as being an idea.
00:41:07.780 --> 00:41:10.040
Some people feel like
if one thing goes down,
00:41:10.040 --> 00:41:11.470
the other thing has to go down.
00:41:11.470 --> 00:41:13.030
Just forget that intuition.
00:41:13.030 --> 00:41:14.160
It's wrong.
00:41:14.160 --> 00:41:20.170
What we're dealing with here,
if you remember, is the slope.
00:41:20.170 --> 00:41:23.870
So if you have a slope
here, that corresponds
00:41:23.870 --> 00:41:26.960
to just a place over
here and as the slope
00:41:26.960 --> 00:41:30.190
gets a little bit
less steep, that's
00:41:30.190 --> 00:41:33.320
why we're approaching
the horizontal axis.
00:41:33.320 --> 00:41:36.480
The number is getting a
little smaller as we close in.
00:41:36.480 --> 00:41:41.120
Now over here, the
slope is also negative.
00:41:41.120 --> 00:41:42.980
It is going down and
as we get down here
00:41:42.980 --> 00:41:44.580
it's getting more
and more negative.
00:41:44.580 --> 00:41:48.170
As we go here the slope,
this function is going up,
00:41:48.170 --> 00:41:50.050
but its slope is going down.
00:41:50.050 --> 00:41:55.790
All right, so the slope is down
on both sides and the notation
00:41:55.790 --> 00:42:03.690
that we use for that is
well suited to this left
00:42:03.690 --> 00:42:09.410
and right business.
00:42:09.410 --> 00:42:16.030
Namely, the limit as x
goes to 0 of -1 / x^2,
00:42:16.030 --> 00:42:18.140
that's going to be
equal to minus infinity.
00:42:18.140 --> 00:42:24.760
And that applies to x going
to 0+ and x going to 0-.
00:42:24.760 --> 00:42:31.780
So both have this property.
00:42:31.780 --> 00:42:34.040
Finally let me just
make one last comment
00:42:34.040 --> 00:42:37.660
about these two graphs.
00:42:37.660 --> 00:42:42.220
This function here
is an odd function
00:42:42.220 --> 00:42:44.620
and when you take the
derivative of an odd function
00:42:44.620 --> 00:42:50.740
you always get an even function.
00:42:50.740 --> 00:42:54.380
That's closely related to the
fact that this 1 / x is an odd
00:42:54.380 --> 00:43:01.170
power and-- x^1 is an odd
power and x^2 is an even power.
00:43:01.170 --> 00:43:05.620
So all of this your intuition
should be reinforcing the fact
00:43:05.620 --> 00:43:11.070
that these pictures look right.
00:43:11.070 --> 00:43:16.010
Okay, now there's one
last kind of discontinuity
00:43:16.010 --> 00:43:20.460
that I want to mention
briefly, which I will call
00:43:20.460 --> 00:43:33.990
other ugly discontinuities.
00:43:33.990 --> 00:43:39.770
And there are lots
and lots of them.
00:43:39.770 --> 00:43:44.220
So one example would
be the function y = sin
00:43:44.220 --> 00:43:50.080
1 / x, as x goes to 0.
00:43:50.080 --> 00:43:58.914
And that looks a
little bit like this.
00:43:58.914 --> 00:44:00.330
Back and forth and
back and forth.
00:44:00.330 --> 00:44:06.170
It oscillates infinitely
often as we tend to 0.
00:44:06.170 --> 00:44:19.260
There's no left or right
limit in this case.
00:44:19.260 --> 00:44:25.330
So there is a very large
quantity of things like that.
00:44:25.330 --> 00:44:29.350
Fortunately we're not gonna
deal with them in this course.
00:44:29.350 --> 00:44:31.500
A lot of times in
real life there
00:44:31.500 --> 00:44:34.800
are things that oscillate
as time goes to infinity,
00:44:34.800 --> 00:44:40.180
but we're not going to
worry about that right now.
00:44:40.180 --> 00:44:49.090
Okay, so that's our final
mention of a discontinuity,
00:44:49.090 --> 00:44:54.130
and now I need to do just
one more piece of groundwork
00:44:54.130 --> 00:44:59.360
for our formulas next time.
00:44:59.360 --> 00:45:09.130
Namely, I want to check
for you one basic fact,
00:45:09.130 --> 00:45:10.280
one limiting tool.
00:45:10.280 --> 00:45:12.960
So this is going
to be a theorem.
00:45:12.960 --> 00:45:17.450
Fortunately it's a
very short theorem
00:45:17.450 --> 00:45:19.580
and has a very short proof.
00:45:19.580 --> 00:45:22.090
So the theorem goes under
the name differentiable
00:45:22.090 --> 00:45:28.210
implies continuous.
00:45:28.210 --> 00:45:30.190
And what it says
is the following:
00:45:30.190 --> 00:45:35.600
it says that if f is
differentiable, in other words
00:45:35.600 --> 00:45:45.560
its-- the derivative
exists at x_0, then
00:45:45.560 --> 00:45:59.245
f is continuous at x_0.
00:45:59.245 --> 00:46:00.870
So, we're gonna need
this is as a tool,
00:46:00.870 --> 00:46:05.750
it's a key step in the
product and quotient rules.
00:46:05.750 --> 00:46:12.380
So I'd like to prove
it right now for you.
00:46:12.380 --> 00:46:16.270
So here is the proof.
00:46:16.270 --> 00:46:20.430
Fortunately the proof
is just one line.
00:46:20.430 --> 00:46:24.740
So first of all, I want to
write in just the right way what
00:46:24.740 --> 00:46:27.410
it is that we have to check.
00:46:27.410 --> 00:46:33.540
So what we have to check is that
the limit, as x goes to x_0,
00:46:33.540 --> 00:46:41.347
of f(x) - f(x_0) is equal to 0.
00:46:41.347 --> 00:46:42.680
So this is what we want to know.
00:46:42.680 --> 00:46:44.930
We don't know it
yet, but we're trying
00:46:44.930 --> 00:46:47.650
to check whether
this is true or not.
00:46:47.650 --> 00:46:49.790
So that's the same
as the statement
00:46:49.790 --> 00:46:52.180
that the function is continuous
because the limit of f(x)
00:46:52.180 --> 00:46:56.300
is supposed to be f(x_0) and
so this difference should
00:46:56.300 --> 00:46:59.690
have limit 0.
00:46:59.690 --> 00:47:02.730
And now, the way this
is proved is just
00:47:02.730 --> 00:47:09.720
by rewriting it by multiplying
and dividing by (x - x_0).
00:47:09.720 --> 00:47:17.381
So I'll rewrite the limit
as x goes to x_0 of f(x) -
00:47:17.381 --> 00:47:25.570
f(x_0) divided by x
- x_0 times x - x_0.
00:47:25.570 --> 00:47:29.230
Okay, so I wrote down the same
expression that I had here.
00:47:29.230 --> 00:47:32.080
This is just the same limit,
but I multiplied and divided
00:47:32.080 --> 00:47:38.070
by (x - x_0).
00:47:38.070 --> 00:47:45.150
And now when I take the limit
what happens is the limit
00:47:45.150 --> 00:47:48.830
of the first factor is f'(x_0).
00:47:48.830 --> 00:47:53.940
That's the thing we know
exists by our assumption.
00:47:53.940 --> 00:48:00.640
And the limit of the second
factor is 0 because the limit
00:48:00.640 --> 00:48:06.700
as x goes to x_0 of (x
- x_0) is clearly 0 .
00:48:06.700 --> 00:48:09.210
So that's it.
00:48:09.210 --> 00:48:12.210
The answer is 0, which
is what we wanted.
00:48:12.210 --> 00:48:14.980
So that's the proof.
00:48:14.980 --> 00:48:19.500
Now there's something
exceedingly fishy-looking
00:48:19.500 --> 00:48:26.370
about this proof and let me just
point to it before we proceed.
00:48:26.370 --> 00:48:33.050
Namely, you're used in limits
to setting x equal to 0.
00:48:33.050 --> 00:48:35.880
And this looks like we're
multiplying, dividing by 0,
00:48:35.880 --> 00:48:38.430
exactly the thing
which makes all proofs
00:48:38.430 --> 00:48:42.562
wrong in all kinds of
algebraic situations
00:48:42.562 --> 00:48:43.520
and so on and so forth.
00:48:43.520 --> 00:48:45.780
You've been taught
that that never works.
00:48:45.780 --> 00:48:47.750
Right?
00:48:47.750 --> 00:48:51.040
But somehow these
limiting tricks
00:48:51.040 --> 00:48:54.100
have found a way around
this and let me just
00:48:54.100 --> 00:48:55.880
make explicit what it is.
00:48:55.880 --> 00:49:03.500
In this limit we never
are using x = x_0.
00:49:03.500 --> 00:49:05.720
That's exactly the
one value of x that we
00:49:05.720 --> 00:49:09.120
don't consider in this limit.
00:49:09.120 --> 00:49:11.910
That's how limits are cooked up.
00:49:11.910 --> 00:49:14.840
And that's sort of been
the themes so far today,
00:49:14.840 --> 00:49:17.100
is that we don't
have to consider that
00:49:17.100 --> 00:49:19.990
and so this multiplication
and division by this number
00:49:19.990 --> 00:49:21.450
is legal.
00:49:21.450 --> 00:49:25.200
It may be small, this number,
but it's always non-zero.
00:49:25.200 --> 00:49:27.670
So this really works,
and it's really true,
00:49:27.670 --> 00:49:31.040
and we just checked that a
differentiable function is
00:49:31.040 --> 00:49:32.560
continuous.
00:49:32.560 --> 00:49:38.580
So I'm gonna have to carry
out these limits, which
00:49:38.580 --> 00:49:42.040
are very interesting 0
/ 0 limits next time.
00:49:42.040 --> 00:49:46.512
But let's hang on for one second
to see if there any questions
00:49:46.512 --> 00:49:47.907
before we stop.
00:49:47.907 --> 00:49:48.990
Yeah, there is a question.
00:49:48.990 --> 00:50:00.970
Student: [INAUDIBLE] Professor:
Repeat this proof right here?
00:50:00.970 --> 00:50:02.830
Just say again.
00:50:02.830 --> 00:50:08.230
Student: [INAUDIBLE]
00:50:08.230 --> 00:50:13.060
Professor: Okay, so there
are two steps to the proof
00:50:13.060 --> 00:50:17.870
and the step that you're
asking about is the first step.
00:50:17.870 --> 00:50:18.580
Right?
00:50:18.580 --> 00:50:20.890
And what I'm saying is
if you have a number,
00:50:20.890 --> 00:50:24.640
and you multiply it by 10
/ 10 it's the same number.
00:50:24.640 --> 00:50:26.920
If you multiply it by 3
/ 3 it's the same number.
00:50:26.920 --> 00:50:30.110
2 / 2, 1 / 1, and so on.
00:50:30.110 --> 00:50:32.385
So it is okay to
change this to this,
00:50:32.385 --> 00:50:34.400
it's exactly the same thing.
00:50:34.400 --> 00:50:36.220
That's the first step.
00:50:36.220 --> 00:50:36.720
Yes?
00:50:36.720 --> 00:50:41.560
Student: [INAUDIBLE]
00:50:41.560 --> 00:50:45.010
Professor: Shhhh...
00:50:45.010 --> 00:50:52.100
The question was how does the
proof, how does this line,
00:50:52.100 --> 00:50:53.960
yeah where the question mark is.
00:50:53.960 --> 00:50:55.910
So what I checked was
that this number which
00:50:55.910 --> 00:50:59.850
is on the left hand side
is equal to this very long
00:50:59.850 --> 00:51:04.800
complicated number which is
equal to this number which
00:51:04.800 --> 00:51:06.270
is equal to this number.
00:51:06.270 --> 00:51:08.600
And so I've checked that
this number is equal to 0
00:51:08.600 --> 00:51:12.120
because the last thing is 0.
00:51:12.120 --> 00:51:16.120
This is equal to that is
equal to that is equal to 0.
00:51:16.120 --> 00:51:17.420
And that's the proof.
00:51:17.420 --> 00:51:17.920
Yes?
00:51:17.920 --> 00:51:21.910
Student: [INAUDIBLE]
00:51:21.910 --> 00:51:30.580
Professor: So that's
a different question.
00:51:30.580 --> 00:51:35.750
Okay, so the hypothesis
of differentiability I
00:51:35.750 --> 00:51:39.420
use because this limit
is equal to this number.
00:51:39.420 --> 00:51:40.520
That that limit exits.
00:51:40.520 --> 00:51:44.170
That's how I use the
hypothesis of the theorem.
00:51:44.170 --> 00:51:46.560
The conclusion of the
theorem is the same
00:51:46.560 --> 00:51:51.300
as this because being
continuous is the same as limit
00:51:51.300 --> 00:51:56.020
as x goes to x_0 of
f(x) is equal to f(x_0).
00:51:56.020 --> 00:51:57.530
That's the definition
of continuity.
00:51:57.530 --> 00:52:00.990
And I subtracted
f(x_0) from both sides
00:52:00.990 --> 00:52:02.860
to get this as being
the same thing.
00:52:02.860 --> 00:52:08.100
So this claim is continuity and
it's the same as this question
00:52:08.100 --> 00:52:10.350
here.
00:52:10.350 --> 00:52:11.180
Last question.
00:52:11.180 --> 00:52:16.771
Student: How did you
get the 0 [INAUDIBLE]
00:52:16.771 --> 00:52:18.520
Professor: How did we
get the 0 from this?
00:52:18.520 --> 00:52:20.450
So the claim that is
being made, so the claim
00:52:20.450 --> 00:52:24.670
is why is this tending to that.
00:52:24.670 --> 00:52:27.410
So for example, I'm going
to have to erase something
00:52:27.410 --> 00:52:28.730
to explain that.
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
00:52:33.900 --> 00:52:35.240
is equal to 0.
00:52:35.240 --> 00:52:37.160
That's what I'm claiming.
00:52:37.160 --> 00:52:39.490
Okay, does that
answer your question?
00:52:39.490 --> 00:52:40.990
Okay.
00:52:40.990 --> 00:52:42.420
All right.
00:52:42.420 --> 00:52:45.320
Ask me other stuff
after lecture.