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PROF.
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JERISON: So, we're ready
to begin Lecture 10,
00:00:24.660 --> 00:00:28.860
and what I'm going to begin
with is by finishing up
00:00:28.860 --> 00:00:33.880
some things from last time.
00:00:33.880 --> 00:00:42.460
We'll talk about
approximations, and I
00:00:42.460 --> 00:00:47.650
want to fill in a
number of comments
00:00:47.650 --> 00:00:52.440
and get you a little bit more
oriented in the point of view
00:00:52.440 --> 00:00:55.980
that I'm trying to express
about approximations.
00:00:55.980 --> 00:00:58.790
So, first of all, I
want to remind you
00:00:58.790 --> 00:01:03.800
of the actual applied example
that I wrote down last time.
00:01:03.800 --> 00:01:08.730
So that was this business here.
00:01:08.730 --> 00:01:11.590
There was something
from special relativity.
00:01:11.590 --> 00:01:15.990
And the approximation that
we used was the linear
00:01:15.990 --> 00:01:22.170
approximation, with a -1/2 power
that comes out to be T(1 + 1/2
00:01:22.170 --> 00:01:22.670
v^2/c^2).
00:01:27.340 --> 00:01:30.850
I want to reiterate why this
is a useful way of thinking
00:01:30.850 --> 00:01:31.930
of things.
00:01:31.930 --> 00:01:34.830
And why this is that this
comes up in real life.
00:01:34.830 --> 00:01:37.090
Why this is maybe more
important than everything
00:01:37.090 --> 00:01:39.820
that I've taught you
about technically so far.
00:01:39.820 --> 00:01:46.750
So, first of all, what this is
telling us is the change in T
00:01:46.750 --> 00:01:50.430
divided by T, if you do the
arithmetic here and subtract T,
00:01:50.430 --> 00:01:55.970
that's using the change
in T is T' - T here.
00:01:55.970 --> 00:02:01.440
If you work that out, this is
approximately the same as 1/2
00:02:01.440 --> 00:02:04.140
v^2/c^2.
00:02:04.140 --> 00:02:06.270
So what is this saying?
00:02:06.270 --> 00:02:08.580
This is saying that if you
have this satellite, which
00:02:08.580 --> 00:02:12.610
is going at speed v, and
little c is the speed of light,
00:02:12.610 --> 00:02:17.060
then the change in the
watch down here on earth,
00:02:17.060 --> 00:02:20.770
relative to the time
on the satellite,
00:02:20.770 --> 00:02:24.050
is going to be proportional
to this ratio here.
00:02:24.050 --> 00:02:25.710
So, physically,
this makes sense.
00:02:25.710 --> 00:02:28.120
This is time divided by time.
00:02:28.120 --> 00:02:30.670
And this is velocity squared
divided by velocity squared.
00:02:30.670 --> 00:02:32.620
So, in each case,
the units divide out.
00:02:32.620 --> 00:02:34.880
So this is a
dimensionless quantity.
00:02:34.880 --> 00:02:36.960
And this is a
dimensionless quantity.
00:02:36.960 --> 00:02:40.840
And the only point here
that we're trying to make
00:02:40.840 --> 00:02:43.440
is just this notion
of proportionality.
00:02:43.440 --> 00:02:45.470
So I want to write this down.
00:02:45.470 --> 00:02:48.010
Just-- in summary.
00:02:48.010 --> 00:02:50.210
So the error
fraction, if you like,
00:02:50.210 --> 00:02:53.630
which is sort of the number
of significant digits
00:02:53.630 --> 00:03:03.090
that we have in our measurement,
is proportional, in this case,
00:03:03.090 --> 00:03:04.860
to this quantity.
00:03:04.860 --> 00:03:09.030
It happens to be proportional
to this quantity here.
00:03:09.030 --> 00:03:16.980
And the factor is,
happens to be, 1/2.
00:03:16.980 --> 00:03:21.350
So these proportionality factors
are what we're looking for.
00:03:21.350 --> 00:03:22.404
Their rates of change.
00:03:22.404 --> 00:03:23.820
Their rates of
change of something
00:03:23.820 --> 00:03:25.550
with respect to something else.
00:03:25.550 --> 00:03:28.800
Now, on your homework,
you have something
00:03:28.800 --> 00:03:30.740
rather similar to this.
00:03:30.740 --> 00:03:38.460
So in Problem, on Part
2B, Part II, Problem 1,
00:03:38.460 --> 00:03:41.580
there's the speed
of a pitch, right?
00:03:41.580 --> 00:03:44.100
And the speed of the pitch
is changing depending
00:03:44.100 --> 00:03:45.600
on how high the mound is.
00:03:45.600 --> 00:03:47.460
And the point here
is that that's
00:03:47.460 --> 00:03:50.150
approximately
proportional to the change
00:03:50.150 --> 00:03:52.320
in the height of the mound.
00:03:52.320 --> 00:03:54.290
In that problem, we
had this delta h,
00:03:54.290 --> 00:03:56.830
that was the x variable
in that problem.
00:03:56.830 --> 00:03:58.330
And what you're
trying to figure out
00:03:58.330 --> 00:04:02.904
is what the constant
of proportionality is.
00:04:02.904 --> 00:04:04.820
That's what you're aiming
for in this problem.
00:04:04.820 --> 00:04:08.070
So there's a linear
relationship, approximately,
00:04:08.070 --> 00:04:11.890
to all intents and purposes
this is an equality.
00:04:11.890 --> 00:04:13.290
Because the lower
order terms are
00:04:13.290 --> 00:04:14.720
unimportant for the problem.
00:04:14.720 --> 00:04:18.020
Just as over here, this function
is a little bit complicated.
00:04:18.020 --> 00:04:19.800
This function is a
little more simple.
00:04:19.800 --> 00:04:22.800
For the purposes of this
problem, they are the same.
00:04:22.800 --> 00:04:27.370
Because the errors
are negligible
00:04:27.370 --> 00:04:29.760
for the particular problem
that we're working on.
00:04:29.760 --> 00:04:34.210
So we might as well work with
the simpler relationship.
00:04:34.210 --> 00:04:37.621
And similarly, over here,
so you could do this with,
00:04:37.621 --> 00:04:39.120
in this case with
square roots, it's
00:04:39.120 --> 00:04:41.980
not so hard here with
reciprocals of square roots.
00:04:41.980 --> 00:04:45.300
It's also not terribly
hard to do it numerically.
00:04:45.300 --> 00:04:47.850
And the reason why we're
not doing it numerically
00:04:47.850 --> 00:04:51.260
is that, as I say,
this is something that
00:04:51.260 --> 00:04:53.560
happens all across engineering.
00:04:53.560 --> 00:04:55.640
People are looking for
these linear relationships
00:04:55.640 --> 00:05:00.240
between the change in some input
and the change in the output.
00:05:00.240 --> 00:05:03.210
And if you don't make
these simplifications,
00:05:03.210 --> 00:05:06.760
then when you get, say,
a dozen of them together,
00:05:06.760 --> 00:05:09.540
you can't figure
out what's going on.
00:05:09.540 --> 00:05:12.640
In this case the design of the
satellite, it's very important.
00:05:12.640 --> 00:05:15.160
The speed actually
isn't just one speed.
00:05:15.160 --> 00:05:19.070
Because it's the relative
speed of you to the satellite.
00:05:19.070 --> 00:05:21.349
And you might be-- it depends
on your angle of sight
00:05:21.349 --> 00:05:22.890
with the satellite
what the speed is.
00:05:22.890 --> 00:05:24.300
So it varies quite a bit.
00:05:24.300 --> 00:05:26.280
So you really need
this rule of thumb.
00:05:26.280 --> 00:05:28.372
Then there are all kinds
of other considerations
00:05:28.372 --> 00:05:29.080
in this question.
00:05:29.080 --> 00:05:30.852
Like, for example,
there's the fact
00:05:30.852 --> 00:05:33.530
that we're sitting on Earth
and so we're rotating around
00:05:33.530 --> 00:05:36.250
on what's called a
non-inertial frame.
00:05:36.250 --> 00:05:38.721
So there's the question
of that acceleration.
00:05:38.721 --> 00:05:41.220
There's the question that the
gravity that I experience here
00:05:41.220 --> 00:05:44.830
on Earth is not the same
as up at the satellite.
00:05:44.830 --> 00:05:47.870
And that also creates
a difference in time,
00:05:47.870 --> 00:05:49.580
as a result of
general relativity.
00:05:49.580 --> 00:05:53.400
So all of these
considerations come down
00:05:53.400 --> 00:05:56.030
to formulas which are
this complicated or maybe
00:05:56.030 --> 00:05:56.910
a tiny bit more.
00:05:56.910 --> 00:05:58.330
Not really that much.
00:05:58.330 --> 00:06:00.700
And then people simplify
them enormously to these very
00:06:00.700 --> 00:06:02.090
simple-minded rules.
00:06:02.090 --> 00:06:05.130
And they don't keep
track of what's going on.
00:06:05.130 --> 00:06:06.830
So in order to
design the system,
00:06:06.830 --> 00:06:08.960
you must make these
simplifications,
00:06:08.960 --> 00:06:12.630
otherwise you can't even
think about what's going on.
00:06:12.630 --> 00:06:13.860
This comes up in everything.
00:06:13.860 --> 00:06:17.422
In weather forecasting,
economic forecasting.
00:06:17.422 --> 00:06:19.880
Figuring out whether there's
going to be an asteroid that's
00:06:19.880 --> 00:06:22.140
going to hit the Earth.
00:06:22.140 --> 00:06:23.970
Every single one of
these things involves
00:06:23.970 --> 00:06:27.900
dozens of these considerations.
00:06:27.900 --> 00:06:29.951
OK, there was a question
that I saw, here.
00:06:29.951 --> 00:06:30.450
Yes.
00:06:30.450 --> 00:06:38.821
STUDENT: [INAUDIBLE]
00:06:38.821 --> 00:06:39.320
PROF.
00:06:39.320 --> 00:06:40.120
JERISON: Yeah.
00:06:40.120 --> 00:06:41.950
Basically, any
problem where you have
00:06:41.950 --> 00:06:43.840
a derivative, the
rate of change also
00:06:43.840 --> 00:06:46.017
depends upon what
the base point is.
00:06:46.017 --> 00:06:46.850
That's the question.
00:06:46.850 --> 00:06:49.070
You're saying, doesn't
this delta v also
00:06:49.070 --> 00:06:51.050
depend, I had a base
point in that problem.
00:06:51.050 --> 00:06:53.290
I happened to decide that
pitchers pitch on average
00:06:53.290 --> 00:06:55.350
about 90 miles an hour.
00:06:55.350 --> 00:06:58.327
Whereas, in fact, some pitchers
pitch at 100 miles an hour,
00:06:58.327 --> 00:07:00.410
some pitch at 80 miles an
hour, and of course they
00:07:00.410 --> 00:07:02.040
vary the speed of the pitch.
00:07:02.040 --> 00:07:03.510
And so, this varies
a little bit.
00:07:03.510 --> 00:07:05.426
In fact, that's sort of
a second order effect.
00:07:05.426 --> 00:07:08.220
It does change the constant
of proportionality.
00:07:08.220 --> 00:07:11.790
It's a rate of change at
a different base point.
00:07:11.790 --> 00:07:14.230
Which we're considering fixed.
00:07:14.230 --> 00:07:17.060
In fact, that's sort of
a second order effect.
00:07:17.060 --> 00:07:19.700
When you actually do the
computations, what you discover
00:07:19.700 --> 00:07:21.940
is that it doesn't make
that much difference.
00:07:21.940 --> 00:07:22.630
To the a.
00:07:22.630 --> 00:07:25.040
And that's something that
you get from experience.
00:07:25.040 --> 00:07:28.980
That it turns out, which things
matter and which things don't.
00:07:28.980 --> 00:07:31.820
And yet again, that's exactly
the same sort of consideration
00:07:31.820 --> 00:07:34.472
but at the next order of what
I'm talking about here is.
00:07:34.472 --> 00:07:36.430
You have to have enough
experience with numbers
00:07:36.430 --> 00:07:38.992
to know that if you take, if
you vary something a little bit
00:07:38.992 --> 00:07:40.950
it's not going to change
the answer that you're
00:07:40.950 --> 00:07:42.680
looking for very much.
00:07:42.680 --> 00:07:45.530
And that's exactly the
point that I'm making.
00:07:45.530 --> 00:07:51.830
So I can't make them all
at once, all such points.
00:07:51.830 --> 00:07:55.290
So that's my pitch for
understanding things
00:07:55.290 --> 00:07:56.470
from this point of view.
00:07:56.470 --> 00:08:02.000
Now, we're going to go on, now,
to quadratic approximations,
00:08:02.000 --> 00:08:13.290
which are a little
more complicated.
00:08:13.290 --> 00:08:15.920
So, we talked a little
bit about this last time
00:08:15.920 --> 00:08:16.850
but I didn't finish.
00:08:16.850 --> 00:08:19.930
So I want to finish this up.
00:08:19.930 --> 00:08:22.320
And the first thing
that I should say
00:08:22.320 --> 00:08:30.740
is that you use these when the
linear approximation is not
00:08:30.740 --> 00:08:35.680
enough.
00:08:35.680 --> 00:08:38.270
OK, so, that's something
that you really
00:08:38.270 --> 00:08:40.390
need to get a little
experience with.
00:08:40.390 --> 00:08:43.810
In economics, I told
you they use logarithms.
00:08:43.810 --> 00:08:46.030
So sometimes they use
log-linear functions.
00:08:46.030 --> 00:08:48.010
Sometimes they use
log-quadratic functions
00:08:48.010 --> 00:08:49.860
when the log-linear
ones don't work.
00:08:49.860 --> 00:08:53.472
So most modeling in economics
is with log-quadratic functions.
00:08:53.472 --> 00:08:55.680
And if you've made it any
more complicated than that,
00:08:55.680 --> 00:08:56.740
it's useless.
00:08:56.740 --> 00:08:57.850
And it's a mess.
00:08:57.850 --> 00:08:58.810
And people don't do it.
00:08:58.810 --> 00:09:02.790
So they stick with the
quadratic ones, typically.
00:09:02.790 --> 00:09:07.010
So the basic formula here,
and I'm going to take the base
00:09:07.010 --> 00:09:14.380
point to be 0, is that f(x) is
approximately f(0) + f'(0) x.
00:09:14.380 --> 00:09:16.100
That's the linear part.
00:09:16.100 --> 00:09:17.174
Plus this extra term.
00:09:17.174 --> 00:09:18.090
Which is f''(0)/2 x^2.
00:09:21.440 --> 00:09:27.740
And this is supposed
to work for x near 0.
00:09:27.740 --> 00:09:34.310
So I've chosen the base
point as simply as possible.
00:09:34.310 --> 00:09:38.060
So here's more or less
where we left off last time.
00:09:38.060 --> 00:09:41.710
And one thing that I said
I was going to explain,
00:09:41.710 --> 00:09:48.390
which I will now, is
why it's 1/2 f''(0).
00:09:48.390 --> 00:09:51.610
So we need to know that.
00:09:51.610 --> 00:09:54.770
So let's work that
out here first of all.
00:09:54.770 --> 00:09:57.290
So I'm just going
to do it by example.
00:09:57.290 --> 00:09:59.860
So if you like, the
answer is just, well,
00:09:59.860 --> 00:10:03.600
what happens when
you have a parabola?
00:10:03.600 --> 00:10:06.640
A parabola's a quadratic.
00:10:06.640 --> 00:10:09.250
It had better-- its quadratic
approximation had better
00:10:09.250 --> 00:10:10.314
be itself.
00:10:10.314 --> 00:10:11.480
It's got to be the best one.
00:10:11.480 --> 00:10:13.010
So it's got to be itself.
00:10:13.010 --> 00:10:15.240
So this formula, if
it's going to work,
00:10:15.240 --> 00:10:21.890
has to work on the nose,
for quadratic functions.
00:10:21.890 --> 00:10:23.770
So, let's take a look.
00:10:23.770 --> 00:10:28.130
If I differentiate,
I get b + 2cx.
00:10:28.130 --> 00:10:32.230
If I differentiate a
second time, I get 2c.
00:10:32.230 --> 00:10:34.580
And now let's plug it in.
00:10:34.580 --> 00:10:39.450
Well, we can recover, what is
it that we want to recover?
00:10:39.450 --> 00:10:42.150
We want to recover
these numbers a, b and c
00:10:42.150 --> 00:10:45.930
using the derivatives
evaluated at 0.
00:10:45.930 --> 00:10:49.500
So let's see.
00:10:49.500 --> 00:10:51.210
It's pretty easy, actually.
00:10:51.210 --> 00:10:52.569
f(0) = a.
00:10:52.569 --> 00:10:53.360
That's on the nose.
00:10:53.360 --> 00:10:58.070
If you plug in x = 0 here, these
terms drop out and you get a.
00:10:58.070 --> 00:11:02.220
And now, f'(0),
whoops that was wrong.
00:11:02.220 --> 00:11:05.100
So I wrote f' but
what I meant was f.
00:11:05.100 --> 00:11:07.950
So f(0) is a.
00:11:07.950 --> 00:11:09.200
Let's back up.
00:11:09.200 --> 00:11:13.130
f(0) is a, so if I
plug in x = 0 I get a.
00:11:13.130 --> 00:11:18.220
Now, f'(0), that's
this next formula here,
00:11:18.220 --> 00:11:21.180
f'(0), I plug in 0
here, and I get b.
00:11:21.180 --> 00:11:22.220
That's also good.
00:11:22.220 --> 00:11:24.220
And that's exactly what
the linear approximation
00:11:24.220 --> 00:11:25.540
is supposed to be.
00:11:25.540 --> 00:11:28.480
But now you notice, f'' is 2c.
00:11:28.480 --> 00:11:34.440
So to recover c, I
better take half of it.
00:11:34.440 --> 00:11:35.120
And that's it.
00:11:35.120 --> 00:11:38.010
That's the reason.
00:11:38.010 --> 00:11:40.690
There's no chance that any
other formula could work.
00:11:40.690 --> 00:11:45.050
And this one does.
00:11:45.050 --> 00:11:50.310
So that's the explanation
for the formula.
00:11:50.310 --> 00:11:54.215
And now I remind you that I had
a collection of basic formulas
00:11:54.215 --> 00:11:55.090
written on the board.
00:11:55.090 --> 00:12:00.700
And I want to just make sure
we know all of them again.
00:12:00.700 --> 00:12:07.580
So, first of all, there was
sine x is approximately x.
00:12:07.580 --> 00:12:12.610
Cosine x is approximately
1 - 1/2 x^2.
00:12:12.610 --> 00:12:19.230
And e^x is approximately
1 + x + 1/2 x^2.
00:12:19.230 --> 00:12:22.230
So those were three that
I mentioned last time.
00:12:22.230 --> 00:12:28.980
And, again, this is
all for x near 0.
00:12:28.980 --> 00:12:31.160
All for x near 0 only.
00:12:31.160 --> 00:12:35.360
These are wildly wrong far
away, but near 0 they're
00:12:35.360 --> 00:12:37.730
nice, good, quadratic
approximations.
00:12:37.730 --> 00:12:39.300
Now, the other
two approximations
00:12:39.300 --> 00:12:44.960
that I want to mention
are the logarithm,
00:12:44.960 --> 00:12:46.960
and we use the
base point shifted.
00:12:46.960 --> 00:12:50.500
So we can put it at x near 0.
00:12:50.500 --> 00:12:54.545
And this one - sorry, this is an
approximately equals sign there
00:12:54.545 --> 00:12:59.410
- turns out to be x - 1/2 x^2.
00:12:59.410 --> 00:13:08.979
And the last one is (1+x)^r,
which turns out to be 1 + rx +
00:13:08.979 --> 00:13:09.520
r(r-1)/2 x^2.
00:13:14.480 --> 00:13:19.980
Now, eventually, your mind
will converge on all of these
00:13:19.980 --> 00:13:23.380
and you'll find them
relatively easy to memorize.
00:13:23.380 --> 00:13:25.820
But it'll take some
getting used to.
00:13:25.820 --> 00:13:29.670
And I'm not claiming that
you should recognize them
00:13:29.670 --> 00:13:32.370
and understand them all now.
00:13:32.370 --> 00:13:35.630
But I'm going to put a
giant box around this.
00:13:35.630 --> 00:13:41.793
STUDENT: [INAUDIBLE]
00:13:41.793 --> 00:13:42.293
PROF.
00:13:42.293 --> 00:13:42.834
JERISON: Yes.
00:13:42.834 --> 00:13:44.543
So the question was,
you get all of these
00:13:44.543 --> 00:13:45.834
if you use that equation there.
00:13:45.834 --> 00:13:47.500
That's exactly what
I'm going to do.
00:13:47.500 --> 00:13:52.350
So I already did it actually
for these three, last time.
00:13:52.350 --> 00:13:55.880
But I didn't do it
yet for these two.
00:13:55.880 --> 00:13:58.490
But I will do it in
about two minutes.
00:13:58.490 --> 00:14:00.405
Well, maybe five minutes.
00:14:00.405 --> 00:14:10.410
But first I want to explain
just a few things about these.
00:14:10.410 --> 00:14:12.850
They all follow from
the basic formula.
00:14:12.850 --> 00:14:16.780
In fact, that one deserves
a pink box too, doesn't it.
00:14:16.780 --> 00:14:18.020
That one's pretty important.
00:14:18.020 --> 00:14:19.770
Alright.
00:14:19.770 --> 00:14:23.020
Yeah.
00:14:23.020 --> 00:14:25.100
Maybe even some little sparkles.
00:14:25.100 --> 00:14:32.830
Alright.
00:14:32.830 --> 00:14:33.430
OK.
00:14:33.430 --> 00:14:36.390
So that's pretty important.
00:14:36.390 --> 00:14:39.900
Almost as important as the more
basic one without this term
00:14:39.900 --> 00:14:41.190
here.
00:14:41.190 --> 00:14:47.480
So now, let me just tell
you a little bit more
00:14:47.480 --> 00:14:54.740
about the significance.
00:14:54.740 --> 00:14:56.641
Again, this is just
to reinforce something
00:14:56.641 --> 00:14:57.640
that we've already done.
00:14:57.640 --> 00:14:59.264
But it's closely
related to what you're
00:14:59.264 --> 00:15:00.880
doing on your problem set.
00:15:00.880 --> 00:15:05.230
So it's worth your
while to recall this.
00:15:05.230 --> 00:15:10.870
So, there's this expression
that we were dealing with.
00:15:10.870 --> 00:15:13.370
And we talked about
it in lecture.
00:15:13.370 --> 00:15:19.610
And we showed that this tends
to e as k goes to infinity.
00:15:19.610 --> 00:15:21.110
So that's what we
showed in lecture.
00:15:21.110 --> 00:15:25.310
And the way that we did that
was, we took the logarithm
00:15:25.310 --> 00:15:27.320
and we wrote it
as k times, sorry,
00:15:27.320 --> 00:15:29.640
the logarithm of 1 + 1/k.
00:15:32.530 --> 00:15:35.780
And then we evaluated
the limit of this.
00:15:35.780 --> 00:15:38.000
And I want to do
this limit again,
00:15:38.000 --> 00:15:40.220
using linear approximation.
00:15:40.220 --> 00:15:42.660
To show you how easy it
is if you just remember
00:15:42.660 --> 00:15:44.820
the linear approximation.
00:15:44.820 --> 00:15:47.720
And then we'll explain where the
quadratic approximation comes
00:15:47.720 --> 00:15:48.390
in.
00:15:48.390 --> 00:15:51.770
So I claim that this
is approximately
00:15:51.770 --> 00:15:56.470
equal to k times 1/k.
00:15:59.300 --> 00:16:00.750
Now, why is that?
00:16:00.750 --> 00:16:04.350
Well, that's just this
linear approximation.
00:16:04.350 --> 00:16:05.500
So what did I use here?
00:16:05.500 --> 00:16:14.760
I used log of 1+x is
approximately x, for x = 1/k.
00:16:14.760 --> 00:16:18.600
That's what I used in
this approximation here.
00:16:18.600 --> 00:16:20.330
And that's the
linear approximation
00:16:20.330 --> 00:16:24.010
to the natural logarithm.
00:16:24.010 --> 00:16:27.250
And this number is
relatively easy to evaluate.
00:16:27.250 --> 00:16:28.150
I know how to do it.
00:16:28.150 --> 00:16:31.010
It's equal to 1.
00:16:31.010 --> 00:16:34.680
That's the same, well,
so where does this work?
00:16:34.680 --> 00:16:37.450
This works where
this thing is near 0.
00:16:37.450 --> 00:16:41.610
Which is when k is
going to infinity.
00:16:41.610 --> 00:16:44.170
This thing is working only
when k is going to infinity.
00:16:44.170 --> 00:16:46.660
So what it's really saying,
this approximation formula,
00:16:46.660 --> 00:16:50.690
it's really saying that as
we go to infinity, in k,
00:16:50.690 --> 00:16:54.720
this thing is going to 1.
00:16:54.720 --> 00:16:58.890
As k goes to infinity.
00:16:58.890 --> 00:17:00.410
So that's what it's saying.
00:17:00.410 --> 00:17:01.740
That's the substance there.
00:17:01.740 --> 00:17:05.200
And that's how we want to
use it, in many instances.
00:17:05.200 --> 00:17:06.580
Just to evaluate limits.
00:17:06.580 --> 00:17:09.180
We also want to realize
that it's nearby
00:17:09.180 --> 00:17:12.800
when k is pretty large, like
100 or something like that.
00:17:12.800 --> 00:17:16.660
Now, so that's the idea of
the linear approximation.
00:17:16.660 --> 00:17:21.920
Now, if you want to get the
rate of convergence here,
00:17:21.920 --> 00:17:27.950
so the rate of what's
called convergence.
00:17:27.950 --> 00:17:34.260
So convergence means how fast
this is going towards that.
00:17:34.260 --> 00:17:36.220
What I have to do is
take the difference.
00:17:36.220 --> 00:17:40.670
I have to take ln a_k, and I
have to subtract 1 from it.
00:17:40.670 --> 00:17:42.420
And I know that
this is going to 0,
00:17:42.420 --> 00:17:47.870
and the question
is how big is this.
00:17:47.870 --> 00:17:51.760
We want it to be very small.
00:17:51.760 --> 00:17:54.490
And the answer
we're going to get,
00:17:54.490 --> 00:18:03.570
so the answer just uses the
quadratic approximation.
00:18:03.570 --> 00:18:06.520
So if I just have a little
bit more detail, then
00:18:06.520 --> 00:18:08.720
this expression
here, in other words,
00:18:08.720 --> 00:18:11.210
I have the next
higher-order term.
00:18:11.210 --> 00:18:15.420
This is like 1/k,
this is like 1 / k^2.
00:18:15.420 --> 00:18:18.890
Then I can understand
how big the difference
00:18:18.890 --> 00:18:24.260
is between the expression
that I've got and its limit.
00:18:24.260 --> 00:18:26.200
And so that's what's
on your homework.
00:18:26.200 --> 00:18:31.610
This is on your problem set.
00:18:31.610 --> 00:18:35.050
OK, so that is more or
less an explanation for one
00:18:35.050 --> 00:18:38.860
of the things that quadratic
approximations are good for.
00:18:38.860 --> 00:18:42.900
And I'm going to give you
one more illustration.
00:18:42.900 --> 00:18:45.360
One more illustration.
00:18:45.360 --> 00:18:47.472
And then we'll actually
check these formulas.
00:18:47.472 --> 00:18:48.430
Yeah, another question.
00:18:48.430 --> 00:18:55.520
STUDENT: [INAUDIBLE]
00:18:55.520 --> 00:18:56.020
PROF.
00:18:56.020 --> 00:18:58.470
JERISON: That's a very
good question here.
00:18:58.470 --> 00:19:01.260
When they, which in
this case means maybe,
00:19:01.260 --> 00:19:07.510
me, when I give you
a question, does one
00:19:07.510 --> 00:19:13.030
specify whether you want to
use a linear or a quadratic
00:19:13.030 --> 00:19:14.430
approximation.
00:19:14.430 --> 00:19:17.910
The answer is, in
real life when you're
00:19:17.910 --> 00:19:22.470
faced with a problem like
this, where some satellite is
00:19:22.470 --> 00:19:24.700
orbiting and you want to
know the effects of gravity
00:19:24.700 --> 00:19:28.304
or something like that, nobody
is going to tell you anything.
00:19:28.304 --> 00:19:29.720
They're not even
going to tell you
00:19:29.720 --> 00:19:32.480
whether a linear approximation
is relevant, or a quadratic
00:19:32.480 --> 00:19:33.740
or anything.
00:19:33.740 --> 00:19:36.170
So you're on your own.
00:19:36.170 --> 00:19:39.830
When I give you a question,
at least for right now,
00:19:39.830 --> 00:19:42.640
I'm always going to tell you.
00:19:42.640 --> 00:19:44.640
But as time goes on
I'd like you to get
00:19:44.640 --> 00:19:48.360
used to when it's
enough to get away
00:19:48.360 --> 00:19:49.840
with a linear approximation.
00:19:49.840 --> 00:19:53.540
And you should only use
a quadratic approximation
00:19:53.540 --> 00:19:55.690
if somebody forces you to.
00:19:55.690 --> 00:19:57.960
You should always start
trying with a linear one.
00:19:57.960 --> 00:20:00.610
Because the quadratic ones are
much more complicated as you'll
00:20:00.610 --> 00:20:03.390
see in this next example.
00:20:03.390 --> 00:20:05.320
OK, so the example
that I want to use
00:20:05.320 --> 00:20:07.740
is, you're going to be
stuck with it because I'm
00:20:07.740 --> 00:20:09.110
asking for the quadratic.
00:20:09.110 --> 00:20:16.190
So we're going to find the
quadratic approximation
00:20:16.190 --> 00:20:23.740
near-- for x near 0.
00:20:23.740 --> 00:20:24.525
To what?
00:20:24.525 --> 00:20:26.360
Well, this is the
same function that we
00:20:26.360 --> 00:20:30.840
used in the last lecture.
00:20:30.840 --> 00:20:32.880
I think this was it.
00:20:32.880 --> 00:20:34.090
e^(-3x) (1+x)^(-1/2).
00:20:37.460 --> 00:20:40.670
OK.
00:20:40.670 --> 00:20:45.060
So, unfortunately, I stuck
it in the wrong place
00:20:45.060 --> 00:20:47.620
to be able to fit this
very long formula here.
00:20:47.620 --> 00:20:51.100
So I'm going to switch it.
00:20:51.100 --> 00:20:57.070
I'm just going to write it here.
00:20:57.070 --> 00:21:00.140
And we're going to just
do the approximation.
00:21:00.140 --> 00:21:03.830
So we're going to say
quadratic, in parentheses.
00:21:03.830 --> 00:21:08.450
And we'll say x near 0.
00:21:08.450 --> 00:21:12.160
So that's what I want.
00:21:12.160 --> 00:21:15.424
So now, here's
what I have to do.
00:21:15.424 --> 00:21:17.590
Well, I have to write in
the quadratic approximation
00:21:17.590 --> 00:21:25.710
for e^(-3x), and I'm going to
use this formula right here.
00:21:25.710 --> 00:21:33.970
And so that's 1 +
(-3x) + (-3x)^2 / 2.
00:21:33.970 --> 00:21:36.530
And the other factor,
I'm going to have
00:21:36.530 --> 00:21:40.910
to use this formula down here.
00:21:40.910 --> 00:21:43.460
Because r is - 1/2.
00:21:43.460 --> 00:21:50.300
And so that's 1 - 1/2 x
+ 1/2 (-1/2)(-3/2)x^2.
00:21:59.090 --> 00:22:09.970
So this is the r term, and
this is the r - 1 term.
00:22:09.970 --> 00:22:11.630
And now I'm going
to do something
00:22:11.630 --> 00:22:15.550
which is the only good thing
about quadratic approximations.
00:22:15.550 --> 00:22:17.660
They're messy, they're
long, there's nothing
00:22:17.660 --> 00:22:19.450
particularly good about them.
00:22:19.450 --> 00:22:21.670
But there is one good
thing about them.
00:22:21.670 --> 00:22:25.610
Which is that you always get to
ignore the higher order terms.
00:22:25.610 --> 00:22:29.420
So even though this looks like
a very ugly multiplication,
00:22:29.420 --> 00:22:31.590
I can do it in my head.
00:22:31.590 --> 00:22:33.940
Just watching it.
00:22:33.940 --> 00:22:38.710
Because I get a 1 * 1, I'm
forced with that term here.
00:22:38.710 --> 00:22:41.200
And then I get the cross
terms which are linear,
00:22:41.200 --> 00:22:44.150
which is -3x - 1/2 x.
00:22:44.150 --> 00:22:46.460
We already did that
when we calculated
00:22:46.460 --> 00:22:48.550
the linear
approximation, so that's
00:22:48.550 --> 00:22:51.670
this times the 1 and
this times that 1.
00:22:51.670 --> 00:22:55.230
And now I have three cross
terms which are quadratic.
00:22:55.230 --> 00:22:58.880
So one of them is these two
linear terms are multiplying.
00:22:58.880 --> 00:23:02.160
So that's plus 3/2 x^2.
00:23:02.160 --> 00:23:05.210
That's -3 times -1/2.
00:23:05.210 --> 00:23:08.160
And then there's this
term, multiplying the 1,
00:23:08.160 --> 00:23:11.730
that's plus 9/2 x^2.
00:23:11.730 --> 00:23:15.080
And then there's one last term,
which is this monster here.
00:23:15.080 --> 00:23:23.260
Multiplying 1, and that is -3/8.
00:23:23.260 --> 00:23:31.430
So the great thing is, we
drop x^3, x^4, etc., terms.
00:23:31.430 --> 00:23:37.930
Yeah?
00:23:37.930 --> 00:23:39.656
STUDENT: [INAUDIBLE]
00:23:39.656 --> 00:23:40.156
PROF.
00:23:40.156 --> 00:23:42.500
JERISON: OK, well
so copy it down.
00:23:42.500 --> 00:23:45.620
And you work it out
as I'm doing it now.
00:23:45.620 --> 00:23:47.930
So what I did is, I
multiplied 1 by 1.
00:23:47.930 --> 00:23:50.290
I'm using the
distributive law here.
00:23:50.290 --> 00:23:51.560
That was this one.
00:23:51.560 --> 00:23:55.120
I multiplied this 3x by this
one, that was that term.
00:23:55.120 --> 00:23:58.550
I multiplied this by
this, that's that term.
00:23:58.550 --> 00:24:01.120
And then I multiplied
this by this.
00:24:01.120 --> 00:24:03.860
In other words, two x
terms that gave me an x^2
00:24:03.860 --> 00:24:06.940
and a (-3)(-1/2).
00:24:06.940 --> 00:24:08.499
And I'm going to
stop at that point.
00:24:08.499 --> 00:24:10.290
Because the point is
it's just all the rest
00:24:10.290 --> 00:24:12.230
of the terms that come up.
00:24:12.230 --> 00:24:14.320
Now, the reason, the only
reason why it's easy,
00:24:14.320 --> 00:24:16.319
is that I only have to
go up to x squared terms.
00:24:16.319 --> 00:24:21.462
I don't have to do
the higher ones.
00:24:21.462 --> 00:24:22.795
Another question, way back here.
00:24:22.795 --> 00:24:23.545
Yeah, right there.
00:24:23.545 --> 00:24:28.696
STUDENT: [INAUDIBLE]
00:24:28.696 --> 00:24:29.196
PROF.
00:24:29.196 --> 00:24:29.696
JERISON: OK.
00:24:29.696 --> 00:24:38.690
So somebody can check
my arithmetic, too.
00:24:38.690 --> 00:24:39.470
Good.
00:24:39.470 --> 00:24:40.361
STUDENT: [INAUDIBLE]
00:24:40.361 --> 00:24:40.861
PROF.
00:24:40.861 --> 00:24:43.466
JERISON: Why do I get to drop
all the higher-order terms.
00:24:43.466 --> 00:24:46.210
So, that's because
the situation where
00:24:46.210 --> 00:24:50.200
I'm going to apply this is
the situation in which x is,
00:24:50.200 --> 00:24:52.610
say, 1/100.
00:24:52.610 --> 00:24:55.030
So here's about 1/100.
00:24:55.030 --> 00:24:57.210
Here's something which
is on the order of 1/100.
00:24:57.210 --> 00:25:00.130
This is on the order of 1/100^2.
00:25:00.130 --> 00:25:02.450
1/100^2, all of these terms.
00:25:02.450 --> 00:25:07.400
Now, these cubic and quartic
terms are of the order
00:25:07.400 --> 00:25:10.520
of 1/100^3.
00:25:10.520 --> 00:25:11.420
That's 10^(-6).
00:25:11.420 --> 00:25:12.420
6.
00:25:12.420 --> 00:25:14.730
And the point is
that I'm not claiming
00:25:14.730 --> 00:25:16.230
that I have an exact answer.
00:25:16.230 --> 00:25:19.950
And I'm going to drop things
of that order of magnitude.
00:25:19.950 --> 00:25:22.890
So I'm saving everything
up to 4 decimal places.
00:25:22.890 --> 00:25:29.300
I'm throwing away things which
are 6 decimal places out.
00:25:29.300 --> 00:25:30.870
Does that answer your question?
00:25:30.870 --> 00:25:35.211
STUDENT: [INAUDIBLE]
00:25:35.211 --> 00:25:35.710
PROF.
00:25:35.710 --> 00:25:36.370
JERISON: So.
00:25:36.370 --> 00:25:39.770
That's the situation, and now
you can combine the terms.
00:25:39.770 --> 00:25:43.780
I mean, it's not
very impressive here.
00:25:43.780 --> 00:25:54.300
This is equal to 1 - 7/2
x, maybe, plus 51/8 x^2.
00:25:54.300 --> 00:25:57.530
If I've made that-- if those
minus signs hadn't canceled,
00:25:57.530 --> 00:25:59.560
I would have gotten
the wrong answer here.
00:25:59.560 --> 00:26:00.140
Anyway.
00:26:00.140 --> 00:26:03.430
So, this is a 2 here, sorry.
00:26:03.430 --> 00:26:04.920
7/2.
00:26:04.920 --> 00:26:07.864
This is the linear
approximation we got last time
00:26:07.864 --> 00:26:09.530
and here's the extra
information that we
00:26:09.530 --> 00:26:11.540
got from this calculation.
00:26:11.540 --> 00:26:17.650
Which is this 51/8 term.
00:26:17.650 --> 00:26:19.620
Right, you have to
accept that there's
00:26:19.620 --> 00:26:21.890
a certain degree of
complexity to this problem
00:26:21.890 --> 00:26:23.840
and the answer is
sufficiently complicated
00:26:23.840 --> 00:26:25.740
so it can't be less
arithmetic because we
00:26:25.740 --> 00:26:29.380
get this peculiar
51/8 there, right.
00:26:29.380 --> 00:26:31.950
So one of the
things to realize is
00:26:31.950 --> 00:26:35.120
that these kinds of problems,
because they involve
00:26:35.120 --> 00:26:37.135
many, many terms
are always going
00:26:37.135 --> 00:26:43.960
to involve a little bit
of complicated arithmetic.
00:26:43.960 --> 00:26:46.910
Last little bit,
I did promise you
00:26:46.910 --> 00:26:51.360
that I was going to derive
these two relations, as I said.
00:26:51.360 --> 00:26:53.020
Did the ones in the left column.
00:26:53.020 --> 00:26:56.640
So let's carry that out.
00:26:56.640 --> 00:27:01.400
And as someone just pointed out,
it all comes from this formula
00:27:01.400 --> 00:27:01.900
here.
00:27:01.900 --> 00:27:07.710
So let's just check it.
00:27:07.710 --> 00:27:12.160
So we'll start with
the log function.
00:27:12.160 --> 00:27:18.810
This is the function, f,
and then f' is 1/(1+x).
00:27:18.810 --> 00:27:24.620
And f'', so this is f', this
is f'', is - -1 / (1+x)^2.
00:27:28.160 --> 00:27:31.730
And now I have to plug in x = 0.
00:27:31.730 --> 00:27:35.180
So at x = 0 this is
ln 1, which is 0.
00:27:35.180 --> 00:27:37.860
So this is at x = 0.
00:27:37.860 --> 00:27:41.850
I'm getting 0 here, I
plug in 0 and I get 1.
00:27:41.850 --> 00:27:45.720
And here, I plug
in 0 and I get -1.
00:27:45.720 --> 00:27:48.050
So now I go and I look
up at that formula,
00:27:48.050 --> 00:27:50.270
which is way in that
upper corner there.
00:27:50.270 --> 00:27:53.420
And I see that the coefficient
on the constant is 0.
00:27:53.420 --> 00:27:55.260
The coefficient on x is 1.
00:27:55.260 --> 00:27:59.290
And then the other coefficient,
the very last one, is -1/2.
00:27:59.290 --> 00:28:01.010
So this is the -1 here.
00:28:01.010 --> 00:28:05.340
And then in the formula,
there's a 2 in the denominator.
00:28:05.340 --> 00:28:08.400
So it's half of whatever I get
for this second derivative,
00:28:08.400 --> 00:28:10.590
at 0.
00:28:10.590 --> 00:28:13.100
So this is the
approximation formula, which
00:28:13.100 --> 00:28:17.260
is way up in that corner there.
00:28:17.260 --> 00:28:21.290
Similarly, if I
do it for (1+x)^r,
00:28:21.290 --> 00:28:25.990
I have to differentiate
that, I get r(1+x)^(r-1),
00:28:25.990 --> 00:28:27.300
and then r(r-1)(x+1)^(r-2).
00:28:31.270 --> 00:28:33.830
So here are the derivatives.
00:28:33.830 --> 00:28:41.110
And so if I evaluate
them at x = 0, I get 1.
00:28:41.110 --> 00:28:43.240
That's 1^r is 1.
00:28:43.240 --> 00:28:46.910
And here I get r.
00:28:46.910 --> 00:28:49.580
1^(r-1) times r.
00:28:49.580 --> 00:28:54.010
And here, I plug in x
= 0 and I get r(r-1).
00:28:59.790 --> 00:29:02.880
So again, the pattern
is right above it here.
00:29:02.880 --> 00:29:05.300
The 1 is there, the r is there.
00:29:05.300 --> 00:29:08.920
And then instead of
r(r-1), I have half that.
00:29:08.920 --> 00:29:21.480
For the coefficient.
00:29:21.480 --> 00:29:22.660
So these are just examples.
00:29:22.660 --> 00:29:24.700
Obviously if we had a
more complicated function,
00:29:24.700 --> 00:29:26.050
we might carry this out.
00:29:26.050 --> 00:29:28.210
But as a practical
matter, we try
00:29:28.210 --> 00:29:30.810
to stick with the
ones in the pink box
00:29:30.810 --> 00:29:42.460
and just use algebra
to get other formulas.
00:29:42.460 --> 00:29:46.430
So I want to shift gears now
and treat the subject that was
00:29:46.430 --> 00:29:48.910
supposed to be this lecture.
00:29:48.910 --> 00:29:51.230
And we're not quite
caught up, but we
00:29:51.230 --> 00:29:54.890
will try to do our best to
do as much as we can today.
00:29:54.890 --> 00:29:59.880
So the next topic
is curve sketching.
00:29:59.880 --> 00:30:18.120
And so let's get
started with that.
00:30:18.120 --> 00:30:24.950
So now, happily in this
subject, there are more pictures
00:30:24.950 --> 00:30:26.880
and it's a little
bit more geometric.
00:30:26.880 --> 00:30:30.420
And there's relatively
little computation.
00:30:30.420 --> 00:30:33.600
So let's hope we can do this.
00:30:33.600 --> 00:30:36.890
So I want to-- so
here we go, we'll
00:30:36.890 --> 00:30:44.930
start with curve sketching.
00:30:44.930 --> 00:30:47.520
And the goal here--
00:30:47.520 --> 00:30:56.410
STUDENT: [INAUDIBLE]
00:30:56.410 --> 00:30:56.910
PROF.
00:30:56.910 --> 00:31:00.880
JERISON: So that's like
'liner', the last time.
00:31:00.880 --> 00:31:09.600
That's kind of sketchy
spelling, isn't it?
00:31:09.600 --> 00:31:12.800
Yeah, there are certain kinds
of things which I can't spell.
00:31:12.800 --> 00:31:16.640
But, all right.
00:31:16.640 --> 00:31:18.741
Sketching.
00:31:18.741 --> 00:31:19.240
Alright.
00:31:19.240 --> 00:31:21.350
So here's our goal.
00:31:21.350 --> 00:31:38.280
Our goal is to draw the
graph of f, using f' and f''.
00:31:42.380 --> 00:31:47.190
Whether they're
positive or negative.
00:31:47.190 --> 00:31:48.440
So that's it.
00:31:48.440 --> 00:31:52.400
This is the goal here.
00:31:52.400 --> 00:31:57.020
However, there is a big warning
that I want to give you.
00:31:57.020 --> 00:32:04.000
And this is one
that unfortunately
00:32:04.000 --> 00:32:07.005
I now have to make you
unlearn, especially
00:32:07.005 --> 00:32:08.380
those that you
that have actually
00:32:08.380 --> 00:32:10.700
had a little bit of
calculus before, I
00:32:10.700 --> 00:32:12.929
want to make you unlearn
some of your instincts
00:32:12.929 --> 00:32:13.720
that you developed.
00:32:13.720 --> 00:32:15.511
So this will be harder
for those of you who
00:32:15.511 --> 00:32:19.840
have actually done this before.
00:32:19.840 --> 00:32:22.650
But for the rest of you,
it will be relatively easy.
00:32:22.650 --> 00:32:35.490
Which is, don't abandon
your precalculus skills.
00:32:35.490 --> 00:32:42.380
And common sense.
00:32:42.380 --> 00:32:46.760
So there's a great deal
of common sense in this.
00:32:46.760 --> 00:32:50.555
And it actually trumps
some of the calculus.
00:32:50.555 --> 00:32:56.190
The calculus just fills in
what you didn't quite know yet.
00:32:56.190 --> 00:32:59.660
So I will try to
illustrate this.
00:32:59.660 --> 00:33:01.390
And because we're
running a bit late,
00:33:01.390 --> 00:33:03.910
I won't get to the some
of the main punchlines
00:33:03.910 --> 00:33:05.970
until next lecture.
00:33:05.970 --> 00:33:07.380
But I want you to do it.
00:33:07.380 --> 00:33:09.412
So for now, I'm just
going to tell you
00:33:09.412 --> 00:33:10.620
about the general principles.
00:33:10.620 --> 00:33:14.100
And in the process I'm going
to introduce the terminology.
00:33:14.100 --> 00:33:17.167
Just, the words that we
need to use to describe what
00:33:17.167 --> 00:33:18.000
is that we're doing.
00:33:18.000 --> 00:33:19.375
And there's also
a certain amount
00:33:19.375 --> 00:33:22.160
of carelessness with that
in many of the treatments
00:33:22.160 --> 00:33:23.070
that you'll see.
00:33:23.070 --> 00:33:24.750
And a lot of hastiness.
00:33:24.750 --> 00:33:29.350
So just be a little patient
and we will do this.
00:33:29.350 --> 00:33:33.740
So, the first principle
is the following.
00:33:33.740 --> 00:33:40.670
If f' is positive,
then f is increasing.
00:33:40.670 --> 00:33:44.530
That's a straightforward
idea, and it's closely related
00:33:44.530 --> 00:33:46.730
to this tangent
line approximation
00:33:46.730 --> 00:33:49.220
or the linear approximation
that I just did.
00:33:49.220 --> 00:33:50.230
You can just imagine.
00:33:50.230 --> 00:33:53.050
Here's the tangent line,
here's the function.
00:33:53.050 --> 00:33:55.680
And if the tangent
line is pointing up,
00:33:55.680 --> 00:33:58.490
then the function is
also going up, too.
00:33:58.490 --> 00:34:00.310
So that's all that's
going on here.
00:34:00.310 --> 00:34:10.370
Similarly, if f' is negative,
then f is decreasing.
00:34:10.370 --> 00:34:12.220
And that's the basic idea.
00:34:12.220 --> 00:34:17.170
Now, the second step is
also fairly straightforward.
00:34:17.170 --> 00:34:21.130
It's just a second-order
effect of the same type.
00:34:21.130 --> 00:34:29.190
If you have f'' as positive,
then that means that f' is
00:34:29.190 --> 00:34:33.150
increasing.
00:34:33.150 --> 00:34:36.740
That's the same principle
applied one step up.
00:34:36.740 --> 00:34:37.320
Right?
00:34:37.320 --> 00:34:41.880
Because if f'' is positive,
that means it's the derivative
00:34:41.880 --> 00:34:42.710
of f'.
00:34:42.710 --> 00:34:45.970
So it's the same
principle just repeated.
00:34:45.970 --> 00:34:49.710
And now I just want to
draw a picture of this.
00:34:49.710 --> 00:34:52.470
Here's a picture of it, I claim.
00:34:52.470 --> 00:34:54.790
And it looks like
something's going down.
00:34:54.790 --> 00:34:56.430
And I did that on purpose.
00:34:56.430 --> 00:34:58.650
But there is something
that's increasing here.
00:34:58.650 --> 00:35:02.470
Which is, the slope is
very steep negative here.
00:35:02.470 --> 00:35:06.330
And it's less steep
negative over here.
00:35:06.330 --> 00:35:09.790
So we have the slope which is
some negative number, say, -4.
00:35:09.790 --> 00:35:13.500
And here it's -3.
00:35:13.500 --> 00:35:14.800
So it's increasing.
00:35:14.800 --> 00:35:18.250
It's getting less negative,
and maybe eventually it'll
00:35:18.250 --> 00:35:19.680
curve up the other way.
00:35:19.680 --> 00:35:23.180
And this is a picture of
what I'm talking about here.
00:35:23.180 --> 00:35:25.630
That's what it means to
say that f' is increasing.
00:35:25.630 --> 00:35:28.020
The slope is getting larger.
00:35:28.020 --> 00:35:34.930
And the way to describe a curve
like this is that it's concave.
00:35:34.930 --> 00:35:41.620
So f is concave up.
00:35:41.620 --> 00:35:49.320
And similarly, f'' negative is
going to be the same thing as f
00:35:49.320 --> 00:35:59.770
concave-- or implies
f concave down.
00:35:59.770 --> 00:36:04.850
So those are the ways in
which derivatives will help us
00:36:04.850 --> 00:36:08.270
qualitatively to draw graphs.
00:36:08.270 --> 00:36:09.880
But as I said
before, we still have
00:36:09.880 --> 00:36:13.360
to use a little bit of common
sense when we draw the graphs.
00:36:13.360 --> 00:36:15.710
These are just the
additional bits of help
00:36:15.710 --> 00:36:17.850
that we have from calculus.
00:36:17.850 --> 00:36:25.410
In drawing pictures.
00:36:25.410 --> 00:36:30.650
So I'm going to go
through one example
00:36:30.650 --> 00:36:36.190
to introduce all the notations.
00:36:36.190 --> 00:36:42.210
And then eventually, so probably
at the beginning of next time,
00:36:42.210 --> 00:36:45.340
I'll give you a
systematic strategy
00:36:45.340 --> 00:36:48.600
that's going to work when
what I'm describing now
00:36:48.600 --> 00:36:52.360
goes wrong, or a
little bit wrong.
00:36:52.360 --> 00:36:58.730
So let's begin with a
straightforward example.
00:36:58.730 --> 00:37:03.280
So, the first example that I'll
give you is the function f(x) =
00:37:03.280 --> 00:37:04.690
3x - x^3.
00:37:07.520 --> 00:37:11.130
Just, as I said, to be able to
introduce all the notations.
00:37:11.130 --> 00:37:17.640
Now, if you differentiate
it, you get 3 - 3x^2.
00:37:17.640 --> 00:37:20.884
And I can factor that.
00:37:20.884 --> 00:37:21.800
This is 3 (1-x) (1+x).
00:37:26.440 --> 00:37:27.750
OK?
00:37:27.750 --> 00:37:33.140
And so, I can decide
whether the derivative
00:37:33.140 --> 00:37:37.390
is positive or negative.
00:37:37.390 --> 00:37:38.830
Easily enough.
00:37:38.830 --> 00:37:50.120
Namely, just staring at this,
I can see that when -1 < x < 1,
00:37:50.120 --> 00:37:53.550
in that range there, both these
numbers, both these factors,
00:37:53.550 --> 00:37:55.550
are positive.
00:37:55.550 --> 00:37:59.290
1-x is a positive number and
1 1+x is a positive number.
00:37:59.290 --> 00:38:04.710
So, in this range,
f'(x) is positive.
00:38:04.710 --> 00:38:09.700
So this thing is,
so f is increasing.
00:38:09.700 --> 00:38:13.310
And similarly, in
the other ranges,
00:38:13.310 --> 00:38:15.770
if x is very, very
large, this becomes,
00:38:15.770 --> 00:38:18.190
if it crosses 1, in
fact, this becomes,
00:38:18.190 --> 00:38:21.310
this factor becomes negative
and this one stays positive.
00:38:21.310 --> 00:38:29.680
So when x > 1, we have
that f'(x) is negative.
00:38:29.680 --> 00:38:35.510
And so f is decreasing.
00:38:35.510 --> 00:38:39.540
And the same thing goes
for the other side.
00:38:39.540 --> 00:38:42.500
When it's less than -1,
that also works this way.
00:38:42.500 --> 00:38:46.420
Because when it's less than
-1, this factor is positive.
00:38:46.420 --> 00:38:50.300
But the other one is negative.
00:38:50.300 --> 00:38:56.290
So in both of these cases,
we get that it's decreasing.
00:38:56.290 --> 00:39:07.300
So now, here's the schematic
picture of this function.
00:39:07.300 --> 00:39:12.950
So here's -1, here's 1.
00:39:12.950 --> 00:39:19.840
It's going to go down, up, down.
00:39:19.840 --> 00:39:21.590
That's what it's doing.
00:39:21.590 --> 00:39:23.710
Maybe I'll just leave
it alone like this.
00:39:23.710 --> 00:39:28.120
That's what it looks like.
00:39:28.120 --> 00:39:31.400
So, this is the
kind of information
00:39:31.400 --> 00:39:32.980
we can get right off the bat.
00:39:32.980 --> 00:39:38.089
And you notice immediately
that it's very important,
00:39:38.089 --> 00:39:39.880
from the features of
the function, the sort
00:39:39.880 --> 00:39:42.060
of key features of the
function that we see here,
00:39:42.060 --> 00:39:44.690
are these two places.
00:39:44.690 --> 00:39:49.270
Maybe I'll even mark
them in a, like this.
00:39:49.270 --> 00:39:59.090
And these things
are turning points.
00:39:59.090 --> 00:40:00.430
So what are they?
00:40:00.430 --> 00:40:03.920
Well, they're just
the points where
00:40:03.920 --> 00:40:05.749
the derivative changes sign.
00:40:05.749 --> 00:40:07.790
Where it's negative here
and it's positive there,
00:40:07.790 --> 00:40:09.850
so there it must be 0.
00:40:09.850 --> 00:40:12.830
So we have a definition, and
this is the most important
00:40:12.830 --> 00:40:21.110
definition in this subject,
which is that is if f'(x_0) =
00:40:21.110 --> 00:40:33.370
0, we call x_0 a critical point.
00:40:33.370 --> 00:40:36.620
The word 'turning point' is not
used just because, in fact, it
00:40:36.620 --> 00:40:38.880
doesn't have to turn
around at those points.
00:40:38.880 --> 00:40:42.710
But certainly, if it turns
around then this will happen.
00:40:42.710 --> 00:40:44.980
And we also have
another notation,
00:40:44.980 --> 00:40:50.580
which is the number
y_0 which is f(x_0) is
00:40:50.580 --> 00:40:59.100
called a critical value.
00:40:59.100 --> 00:41:02.230
So these are the key
numbers that we're
00:41:02.230 --> 00:41:05.340
going to have to work out
in order to understand
00:41:05.340 --> 00:41:18.750
what the function looks like.
00:41:18.750 --> 00:41:27.990
So what I'm going to
do is just plot them.
00:41:27.990 --> 00:41:30.440
We're going to plot the
critical points and the values.
00:41:30.440 --> 00:41:34.650
Well, we found the critical
points relatively easily.
00:41:34.650 --> 00:41:37.240
I didn't write it down here
but it's pretty obvious.
00:41:37.240 --> 00:41:42.353
If you set f(x) = 0,
that implies that (1 -
00:41:42.353 --> 00:41:50.570
x)(1 + x) = 0, which implies
that x is plus or minus 1.
00:41:50.570 --> 00:41:52.710
So those are known as
the critical points.
00:41:52.710 --> 00:41:56.420
And now, in order to get
the critical values here,
00:41:56.420 --> 00:42:02.280
I have to plug in f(1), for
instance, the function is 3x -
00:42:02.280 --> 00:42:07.480
x^2, so there's this 3
* 1 - 1^3, which is 2.
00:42:07.480 --> 00:42:17.180
And f(-1), which is 3(-1)
- (-1)^3, which is -2.
00:42:17.180 --> 00:42:21.780
And so I can plot
the function here.
00:42:21.780 --> 00:42:26.160
So here's the point -1
and here's, up here, is 2.
00:42:26.160 --> 00:42:28.620
So this is-- whoops,
which one is it?
00:42:28.620 --> 00:42:29.160
Yeah.
00:42:29.160 --> 00:42:32.560
This is -1, so it's down here.
00:42:32.560 --> 00:42:35.160
So it's (-1, -2).
00:42:35.160 --> 00:42:41.570
And then over here, I
have the point (1, 2).
00:42:41.570 --> 00:42:44.570
Alright, now, what
information do
00:42:44.570 --> 00:42:48.930
I get from - so I've now
plotted two, I claim,
00:42:48.930 --> 00:42:50.630
very interesting points.
00:42:50.630 --> 00:42:55.000
What information
do I get from this?
00:42:55.000 --> 00:42:59.680
The answer is, I know
something very nearby.
00:42:59.680 --> 00:43:01.830
Because I've already
checked that the thing
00:43:01.830 --> 00:43:04.770
is coming down from the
left, and coming back up.
00:43:04.770 --> 00:43:07.370
And so it must be
shaped like this.
00:43:07.370 --> 00:43:08.330
Over here.
00:43:08.330 --> 00:43:11.620
The tangent line is 0, it's
going to be level there.
00:43:11.620 --> 00:43:14.490
And similarly over here,
it's going to do that.
00:43:14.490 --> 00:43:22.211
So this is what we know so far,
based on what we've computed.
00:43:22.211 --> 00:43:22.710
Question.
00:43:22.710 --> 00:43:36.916
STUDENT: [INAUDIBLE]
00:43:36.916 --> 00:43:37.416
PROF.
00:43:37.416 --> 00:43:38.150
JERISON: The question
is, what happens
00:43:38.150 --> 00:43:39.233
if there's a sharp corner.
00:43:39.233 --> 00:43:43.350
The answer is,
calculus is-- it's not
00:43:43.350 --> 00:43:45.050
called a critical point.
00:43:45.050 --> 00:43:47.090
It's a something else.
00:43:47.090 --> 00:43:50.440
And it's a very
important point, too.
00:43:50.440 --> 00:43:52.570
And we will be discussing
those kinds of points.
00:43:52.570 --> 00:43:54.540
There are much more
dramatic instances of that.
00:43:54.540 --> 00:43:56.640
That's part of what
we're going to say.
00:43:56.640 --> 00:43:58.670
But I just want to
save that, all right.
00:43:58.670 --> 00:44:02.490
We will be discussing.
00:44:02.490 --> 00:44:03.010
Yeah.
00:44:03.010 --> 00:44:03.540
Question.
00:44:03.540 --> 00:44:08.516
STUDENT: [INAUDIBLE] PROF.
00:44:08.516 --> 00:44:10.316
JERISON: The question
that was asked
00:44:10.316 --> 00:44:12.070
was, how did I know
at the critical point
00:44:12.070 --> 00:44:17.190
that it's concave down over
here and concave up over here.
00:44:17.190 --> 00:44:21.590
The answer is that
I actually did not
00:44:21.590 --> 00:44:24.300
use the second derivative yet.
00:44:24.300 --> 00:44:26.510
What I used is another
piece of information.
00:44:26.510 --> 00:44:28.760
I used the information
that I derived over here.
00:44:28.760 --> 00:44:32.500
That f' is positive,
where f' is positive
00:44:32.500 --> 00:44:33.670
and where it's negative.
00:44:33.670 --> 00:44:36.980
So what I know is that
the graph is going down
00:44:36.980 --> 00:44:39.470
to the left of -1.
00:44:39.470 --> 00:44:41.530
It's going up to
the right, here.
00:44:41.530 --> 00:44:44.920
It's going up here and
it's going down there.
00:44:44.920 --> 00:44:47.060
I did not use the
second derivative.
00:44:47.060 --> 00:44:49.960
I used the first derivative.
00:44:49.960 --> 00:44:51.840
OK, but I didn't
just use the fact
00:44:51.840 --> 00:44:55.834
that there was a
turning point here.
00:44:55.834 --> 00:44:57.250
So, actually, I
was using the fact
00:44:57.250 --> 00:44:58.416
that it was a turning point.
00:44:58.416 --> 00:45:01.080
I wasn't using the fact that
it had the second derivative,
00:45:01.080 --> 00:45:01.730
though.
00:45:01.730 --> 00:45:02.230
OK.
00:45:02.230 --> 00:45:03.120
For now.
00:45:03.120 --> 00:45:09.500
You can also see it by the
second derivative as well.
00:45:09.500 --> 00:45:13.710
So now, the next thing
that I'd like to do,
00:45:13.710 --> 00:45:16.170
I need to finish off this graph.
00:45:16.170 --> 00:45:19.950
And I just want to do it a
little bit carefully here.
00:45:19.950 --> 00:45:22.710
In the order that is reasonable.
00:45:22.710 --> 00:45:26.830
Now, you might happen
to notice, and there's
00:45:26.830 --> 00:45:31.660
nothing wrong with this.
00:45:31.660 --> 00:45:34.826
So let's even fill in a guess.
00:45:34.826 --> 00:45:36.700
In order to fill in a
guess, though, and have
00:45:36.700 --> 00:45:38.190
it be even vaguely
right, I do have
00:45:38.190 --> 00:45:39.606
to notice that
this thing crosses,
00:45:39.606 --> 00:45:41.800
this function
crosses the origin.
00:45:41.800 --> 00:45:48.120
The function f(x) = 3x -
x^3 happens to also have
00:45:48.120 --> 00:45:50.780
the property that f(0) = 0.
00:45:50.780 --> 00:45:52.560
Again, common sense.
00:45:52.560 --> 00:45:54.590
You're allowed to use
your common sense.
00:45:54.590 --> 00:45:58.320
You're allowed to notice a value
of the function and put it in.
00:45:58.320 --> 00:46:00.270
So there's nothing
wrong with that.
00:46:00.270 --> 00:46:03.960
If you happen to
have such a value.
00:46:03.960 --> 00:46:06.650
So, now we can guess what our
function is going to look like.
00:46:06.650 --> 00:46:10.010
It's going to maybe
come down like this.
00:46:10.010 --> 00:46:11.670
Come up like this.
00:46:11.670 --> 00:46:13.130
And come down like this.
00:46:13.130 --> 00:46:15.230
That could be what
it looks like.
00:46:15.230 --> 00:46:16.800
But, you know,
another possibility
00:46:16.800 --> 00:46:19.820
is it sort of comes along
here and goes out that way.
00:46:19.820 --> 00:46:22.260
Comes along here and goes
out that way, who knows?
00:46:22.260 --> 00:46:25.070
It happens, by the way,
that it's an odd function.
00:46:25.070 --> 00:46:25.570
Right?
00:46:25.570 --> 00:46:26.640
Those are all odd powers.
00:46:26.640 --> 00:46:28.710
So, actually, it's
symmetric on the right half
00:46:28.710 --> 00:46:29.860
and the left half.
00:46:29.860 --> 00:46:31.289
And crosses at 0.
00:46:31.289 --> 00:46:32.830
So everything that
we do on the right
00:46:32.830 --> 00:46:34.996
is going to be the same as
what happens on the left.
00:46:34.996 --> 00:46:36.620
That's another piece
of common sense.
00:46:36.620 --> 00:46:38.120
You want to make
use of that as much
00:46:38.120 --> 00:46:40.060
as possible, whenever
you're drawing anything.
00:46:40.060 --> 00:46:42.340
Don't want to throw
out information.
00:46:42.340 --> 00:46:46.540
So this function
happens to be odd.
00:46:46.540 --> 00:46:48.700
Odd, and f(0) = 0.
00:46:48.700 --> 00:46:52.520
I'm considering those to be
kinds of precalculus skills
00:46:52.520 --> 00:47:00.610
that I want you to use
as much as you can.
00:47:00.610 --> 00:47:02.880
So now, here's the
first feature which
00:47:02.880 --> 00:47:08.880
is unfortunately ignored in
most discussions of functions.
00:47:08.880 --> 00:47:11.340
And it's strange,
because nowadays we
00:47:11.340 --> 00:47:13.380
have graphing things.
00:47:13.380 --> 00:47:19.080
And it's really the only
part of the exercise
00:47:19.080 --> 00:47:22.480
that you couldn't do, at least
on this relatively simpleminded
00:47:22.480 --> 00:47:28.180
level, with a
graphing calculator.
00:47:28.180 --> 00:47:33.790
And that is what I would
call the ends of the problem.
00:47:33.790 --> 00:47:37.000
So what happens off the
screen, is the question.
00:47:37.000 --> 00:47:39.880
And that basically is the
theoretical part of the problem
00:47:39.880 --> 00:47:41.430
that you have to address.
00:47:41.430 --> 00:47:42.920
You can program this.
00:47:42.920 --> 00:47:45.410
You can draw all the
pictures that you want.
00:47:45.410 --> 00:47:48.190
But what you won't see
is what's off the screen.
00:47:48.190 --> 00:47:50.260
You need to know
something to figure out
00:47:50.260 --> 00:47:51.780
what's off the screen.
00:47:51.780 --> 00:47:54.560
So, in this case, I'm talking
about what's off the screen
00:47:54.560 --> 00:48:01.930
going to the right,
or going to the left.
00:48:01.930 --> 00:48:06.100
So let's check the ends.
00:48:06.100 --> 00:48:07.680
So here, let's just take a look.
00:48:07.680 --> 00:48:12.280
We have the function f(x),
which is, sorry, 3x - x^3.
00:48:12.280 --> 00:48:14.620
Again this is a
precalculus sort of thing.
00:48:14.620 --> 00:48:16.230
And we're imagining
now, let's just
00:48:16.230 --> 00:48:18.360
do x goes to plus infinity.
00:48:18.360 --> 00:48:19.670
So what happens here.
00:48:19.670 --> 00:48:25.460
When x is gigantic, this term
is completely negligible.
00:48:25.460 --> 00:48:30.480
And it just behaves like -x^3,
which goes to minus infinity
00:48:30.480 --> 00:48:32.600
as x goes to plus infinity.
00:48:32.600 --> 00:48:40.330
And similarly, f(x)
goes to plus infinity
00:48:40.330 --> 00:48:46.150
if x goes to minus infinity.
00:48:46.150 --> 00:48:50.060
Now let me pull down this
picture again, and show you
00:48:50.060 --> 00:48:53.140
what piece of the
information we've got.
00:48:53.140 --> 00:48:55.610
We now know that it is
heading up this way.
00:48:55.610 --> 00:48:58.560
It doesn't go like this,
it goes up like that.
00:48:58.560 --> 00:49:00.220
And I'm going to
put an arrow for it,
00:49:00.220 --> 00:49:02.580
And it's going down like this.
00:49:02.580 --> 00:49:06.500
Heading down to minus
infinity as x goes out farther
00:49:06.500 --> 00:49:07.310
to the right.
00:49:07.310 --> 00:49:15.120
And going out to plus infinity
as x goes farther to the left.
00:49:15.120 --> 00:49:19.070
So now there's
hardly anything left
00:49:19.070 --> 00:49:21.910
of this function to describe.
00:49:21.910 --> 00:49:27.120
There's really nothing left
except maybe decoration.
00:49:27.120 --> 00:49:29.650
And we kind of like
that decoration,
00:49:29.650 --> 00:49:32.560
so we will pay attention to it.
00:49:32.560 --> 00:49:35.580
And to do that, we'll have to
check the second derivative.
00:49:35.580 --> 00:49:40.150
So if we differentiate a second
time, the first derivative was,
00:49:40.150 --> 00:49:42.680
remember, 3 - 3x^2.
00:49:42.680 --> 00:49:53.540
So the second derivative is -6x.
00:49:53.540 --> 00:50:02.470
So now we notice that f''(x)
is negative if x is positive.
00:50:02.470 --> 00:50:08.310
And f''(x) is positive
if x is negative.
00:50:08.310 --> 00:50:13.170
And so in this part
it's concave down.
00:50:13.170 --> 00:50:18.990
And in this part
it's concave up.
00:50:18.990 --> 00:50:22.390
And now I'm going to switch the
boards so that you'll, and draw
00:50:22.390 --> 00:50:24.860
it.
00:50:24.860 --> 00:50:30.630
And you see that it was
begging to be this way.
00:50:30.630 --> 00:50:32.960
So we'll fill in
the rest of it here.
00:50:32.960 --> 00:50:35.560
Maybe in a nice color here.
00:50:35.560 --> 00:50:38.100
So this is the whole graph
and this is the correct graph.
00:50:38.100 --> 00:50:41.450
It comes down in one swoop
down here, and comes up here.
00:50:41.450 --> 00:50:46.620
And then it changes to concave
down right at the origin.
00:50:46.620 --> 00:50:49.250
So this point is of
interest, not only
00:50:49.250 --> 00:50:51.750
because it's the place
where it crosses the axis,
00:50:51.750 --> 00:51:00.690
but it's also what's
called an inflection point.
00:51:00.690 --> 00:51:02.970
Inflection point,
that's a point where--
00:51:02.970 --> 00:51:07.620
because f'' at that
place is equal to 0.
00:51:07.620 --> 00:51:10.970
So it's a place where the
second derivative is 0.
00:51:10.970 --> 00:51:15.160
We also consider those
to be interesting points.
00:51:15.160 --> 00:51:21.010
Now, so let me just making
one closing remark here.
00:51:21.010 --> 00:51:26.590
Which is that all of this
information fits together.
00:51:26.590 --> 00:51:29.610
And we're going to have much,
much harder examples of this
00:51:29.610 --> 00:51:32.570
where you'll actually have to
think about what's going on.
00:51:32.570 --> 00:51:35.190
But there's a lot of
stuff protecting you.
00:51:35.190 --> 00:51:39.050
And functions will
behave themselves
00:51:39.050 --> 00:51:40.920
and turn around appropriately.
00:51:40.920 --> 00:51:43.473
Anyway, we'll talk
about it next time.