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PROFESSOR: In the
twelfth lecture,
00:00:24.390 --> 00:00:31.110
we're going to talk
about maxima and minima.
00:00:31.110 --> 00:00:33.110
Let's finish up what
we did last time.
00:00:33.110 --> 00:00:35.710
We really only just started
with maxima and minima.
00:00:35.710 --> 00:00:38.100
And then we're going to
talk about related rates.
00:00:38.100 --> 00:00:48.180
So, right now I want to
give you some examples
00:00:48.180 --> 00:00:51.430
of max-min problems.
00:00:51.430 --> 00:00:55.540
And we're going to start
with a fairly basic one.
00:00:55.540 --> 00:00:58.540
So what's the thing
about max-min problems?
00:00:58.540 --> 00:01:01.240
The main thing is
that we're asking
00:01:01.240 --> 00:01:05.560
you to do a little bit more
of the interpretation of word
00:01:05.560 --> 00:01:06.060
problems.
00:01:06.060 --> 00:01:09.970
So many of the problems are
expressed in terms of words.
00:01:09.970 --> 00:01:18.650
And so, in this case, we have
a wire which is length 1.
00:01:18.650 --> 00:01:29.610
Cut into two pieces.
00:01:29.610 --> 00:01:38.150
And then each piece
encloses a square.
00:01:38.150 --> 00:01:44.400
Sorry, encloses a square.
00:01:44.400 --> 00:01:47.170
And the problem - so
this is the setup.
00:01:47.170 --> 00:02:02.650
And the problem is to find
the largest area enclosed.
00:02:02.650 --> 00:02:03.910
So here's the problem.
00:02:03.910 --> 00:02:10.340
Now, in all of these
cases, in all these cases,
00:02:10.340 --> 00:02:11.600
there's a bunch of words.
00:02:11.600 --> 00:02:18.599
And your job is typically
to draw a diagram.
00:02:18.599 --> 00:02:20.890
So the first thing you want
to do is to draw a diagram.
00:02:20.890 --> 00:02:23.340
In this case, it can
be fairly schematic.
00:02:23.340 --> 00:02:25.400
Here's your unit length.
00:02:25.400 --> 00:02:27.440
And when you draw
the diagram, you're
00:02:27.440 --> 00:02:29.370
going to have to pick variables.
00:02:29.370 --> 00:02:34.700
So those are really
the two main tasks.
00:02:34.700 --> 00:02:35.665
To set up the problem.
00:02:35.665 --> 00:02:37.170
So you're drawing a diagram.
00:02:37.170 --> 00:02:40.860
This is like word problems
of old, in grade school
00:02:40.860 --> 00:02:42.410
through high school.
00:02:42.410 --> 00:02:50.020
Draw a diagram and
name the variables.
00:02:50.020 --> 00:02:52.900
So we'll be doing a
lot of that today.
00:02:52.900 --> 00:02:54.950
So here's my unit length.
00:02:54.950 --> 00:02:58.160
And I'm going to
choose the variable x
00:02:58.160 --> 00:03:00.280
to be the length of one
of the pieces of wire.
00:03:00.280 --> 00:03:03.240
And that makes the
other piece 1 - x.
00:03:03.240 --> 00:03:05.579
And that's pretty much
the whole diagram,
00:03:05.579 --> 00:03:07.870
except that there's something
that we did with the wire
00:03:07.870 --> 00:03:09.710
after we cut it in half.
00:03:09.710 --> 00:03:13.370
Namely, we built two
little boxes out of it.
00:03:13.370 --> 00:03:15.520
Like this, these
are our squares.
00:03:15.520 --> 00:03:19.900
And their side lengths
are x/4 and (1-x)/4.
00:03:25.020 --> 00:03:26.710
So, so far, so good.
00:03:26.710 --> 00:03:28.880
And now we have
to think, well, we
00:03:28.880 --> 00:03:30.450
want to find the largest area.
00:03:30.450 --> 00:03:33.050
So I need a formula for
area in terms of variables
00:03:33.050 --> 00:03:34.490
that I've described.
00:03:34.490 --> 00:03:35.790
And so that's the last thing.
00:03:35.790 --> 00:03:39.400
I'll give the letter A as
the label for the area.
00:03:39.400 --> 00:03:43.760
And then the area is
just the square of x/4
00:03:43.760 --> 00:03:51.805
plus the square of 1 minus x
- whoops, that strange 2 got
00:03:51.805 --> 00:03:57.780
in here - over 4.
00:03:57.780 --> 00:03:59.110
So far, so good.
00:03:59.110 --> 00:04:01.750
Now, the instinct
that you'll have,
00:04:01.750 --> 00:04:04.020
and I'm going to yield
to that instinct,
00:04:04.020 --> 00:04:07.370
is we should charge ahead
and just differentiate.
00:04:07.370 --> 00:04:07.870
All right?
00:04:07.870 --> 00:04:08.578
That's all right.
00:04:08.578 --> 00:04:10.430
We'll find the critical points.
00:04:10.430 --> 00:04:12.995
So we know that those
are important points.
00:04:12.995 --> 00:04:17.840
So we're going to find
the critical points.
00:04:17.840 --> 00:04:21.460
In other words, we
take the derivative,
00:04:21.460 --> 00:04:27.390
we set the derivative of
A with respect to x = 0.
00:04:27.390 --> 00:04:33.020
So if I do that differentiation,
I get the, well,
00:04:33.020 --> 00:04:37.240
so the first one,
x^2 / 16, that's 8.
00:04:37.240 --> 00:04:40.870
Sorry.
00:04:40.870 --> 00:04:44.060
That's x/8, right?
00:04:44.060 --> 00:04:45.570
That's the derivative of this.
00:04:45.570 --> 00:04:51.290
And if I differentiate this, I
get well, the derivative of 1 -
00:04:51.290 --> 00:04:56.930
x^2 is 2(1-x)(a-1).
00:04:56.930 --> 00:04:59.620
So it's -(1-x)/8.
00:05:03.170 --> 00:05:05.010
So there are two
minus signs in there,
00:05:05.010 --> 00:05:07.520
I'll let you ponder that
differentiation, which
00:05:07.520 --> 00:05:10.900
I did by the chain rule.
00:05:10.900 --> 00:05:12.730
Hang on a sec, OK?
00:05:12.730 --> 00:05:14.960
Just wait until we're done.
00:05:14.960 --> 00:05:17.480
So here's the derivative.
00:05:17.480 --> 00:05:19.890
Is there a problem?
00:05:19.890 --> 00:05:29.060
STUDENT: [INAUDIBLE]
00:05:29.060 --> 00:05:31.320
PROFESSOR: Right, so
there's a 1/16 here.
00:05:31.320 --> 00:05:32.850
This is x^2 / 16.
00:05:32.850 --> 00:05:36.870
And so it's 2x over
8, over 16, sorry.
00:05:36.870 --> 00:05:40.360
Which has an 8.
00:05:40.360 --> 00:05:41.790
That's OK.
00:05:41.790 --> 00:05:56.080
All right, so now, This is equal
to 0 if and only if x = 1 - x.
00:05:56.080 --> 00:06:02.170
That's 2x = 1, or in
other words x = 1/2.
00:06:02.170 --> 00:06:03.280
All right?
00:06:03.280 --> 00:06:06.530
So there's our critical point.
00:06:06.530 --> 00:06:11.710
So x = 1/2 is the
critical point.
00:06:11.710 --> 00:06:17.640
And the critical
value, which is what
00:06:17.640 --> 00:06:26.260
you get when you evaluate A at
1/2, is (1/2) / 4, that's 1/8.
00:06:26.260 --> 00:06:38.030
So that's (1/8)^2 +
(1/8)^2 which is 1/32.
00:06:38.030 --> 00:06:45.480
So, so far, so good.
00:06:45.480 --> 00:06:48.050
But we're not done yet.
00:06:48.050 --> 00:06:56.850
We're not done.
00:06:56.850 --> 00:06:59.070
So why aren't we done?
00:06:59.070 --> 00:07:04.590
Because we haven't
checked the end points.
00:07:04.590 --> 00:07:08.370
So let's check the end points.
00:07:08.370 --> 00:07:10.450
Now, in this problem,
the end points
00:07:10.450 --> 00:07:12.650
are really sort of excluded.
00:07:12.650 --> 00:07:17.530
The ends are between
0 and 1 here.
00:07:17.530 --> 00:07:22.740
That's the possible
lengths of the cut.
00:07:22.740 --> 00:07:25.410
And so what we should really
be doing is evaluating
00:07:25.410 --> 00:07:29.230
in the limit, so that would be
the right-hand limit as x goes
00:07:29.230 --> 00:07:33.780
to 0 of A. And if
you plug in x = 0,
00:07:33.780 --> 00:07:42.000
what you get here is 0 +
(1/4)^2, which is 1/16.
00:07:42.000 --> 00:07:49.710
And similarly, at the other
end, that's 1-, from the left,
00:07:49.710 --> 00:07:56.650
we get (1/4)^2 + 0,
which is also 1/16.
00:07:56.650 --> 00:08:01.750
So, what you see is that
the schematic picture
00:08:01.750 --> 00:08:05.090
of this function, and
this isn't even so far
00:08:05.090 --> 00:08:08.100
off from being the
right picture here,
00:08:08.100 --> 00:08:12.120
is that its level here
is 1/16 and then it
00:08:12.120 --> 00:08:14.520
dips down and goes up.
00:08:14.520 --> 00:08:15.020
Right?
00:08:15.020 --> 00:08:19.960
This is 1/2, this is 1, and
this level here is a half that.
00:08:19.960 --> 00:08:23.070
This is 1/32.
00:08:23.070 --> 00:08:25.900
So we did not
find, when we found
00:08:25.900 --> 00:08:29.740
the critical point we did not
find the largest area enclosed.
00:08:29.740 --> 00:08:33.060
We found the least
area enclosed.
00:08:33.060 --> 00:08:36.850
So if you don't pay attention
to what the function looks like,
00:08:36.850 --> 00:08:40.240
not only will you about half
the time get the wrong answer,
00:08:40.240 --> 00:08:43.980
you'll get the
absolute worst answer.
00:08:43.980 --> 00:08:47.770
You'll get the one which is
the polar opposite from what
00:08:47.770 --> 00:08:49.680
you want.
00:08:49.680 --> 00:08:52.250
So you have to pay a little bit
of attention to the function
00:08:52.250 --> 00:08:53.440
that you've got.
00:08:53.440 --> 00:08:55.530
And in this case it's
just very schematic.
00:08:55.530 --> 00:08:57.320
It dips down and
goes up, and that's
00:08:57.320 --> 00:08:59.514
true of pretty much
most functions.
00:08:59.514 --> 00:09:00.430
They're fairly simple.
00:09:00.430 --> 00:09:02.096
They maybe only have
one critical point.
00:09:02.096 --> 00:09:03.550
They only turn around once.
00:09:03.550 --> 00:09:06.310
But then, maybe the critical
point is the maximum
00:09:06.310 --> 00:09:07.760
or maybe it's the minimum.
00:09:07.760 --> 00:09:09.670
Or maybe it's neither, in fact.
00:09:09.670 --> 00:09:15.820
So we'll be discussing
that maybe some other time.
00:09:15.820 --> 00:09:26.260
So what we find here is that
we have the least area enclosed
00:09:26.260 --> 00:09:30.420
is 1/32.
00:09:30.420 --> 00:09:35.880
And this is true when x = 1/2.
00:09:35.880 --> 00:09:44.190
So these are equal squares.
00:09:44.190 --> 00:09:55.970
And most when there's
only one square.
00:09:55.970 --> 00:10:00.480
Which is more or less
the limiting situation,
00:10:00.480 --> 00:10:05.490
if one of the pieces disappears.
00:10:05.490 --> 00:10:09.840
Now, so that's the
first kind of example.
00:10:09.840 --> 00:10:14.560
And I just want to make one
more comment about terminology
00:10:14.560 --> 00:10:16.600
before we go on.
00:10:16.600 --> 00:10:20.690
And I will introduce it
with the following question.
00:10:20.690 --> 00:10:31.510
What is the minimum?
00:10:31.510 --> 00:10:39.620
So, what is the minimum?
00:10:39.620 --> 00:10:40.120
Yeah.
00:10:40.120 --> 00:10:44.032
STUDENT: [INAUDIBLE]
00:10:44.032 --> 00:10:44.740
PROFESSOR: Right.
00:10:44.740 --> 00:10:46.640
The lowest value
of the function.
00:10:46.640 --> 00:10:52.600
So the answer to that
question is 1/32.
00:10:52.600 --> 00:10:56.830
Now, the problem with
this question and you
00:10:56.830 --> 00:11:07.100
will-- so that refers
to the minimum value.
00:11:07.100 --> 00:11:08.900
But then there's
this other question
00:11:08.900 --> 00:11:15.950
which is where is the minimum.
00:11:15.950 --> 00:11:22.390
And the answer to
that is x = 1/2.
00:11:22.390 --> 00:11:30.310
So one of them is
the minimum point,
00:11:30.310 --> 00:11:33.640
and the other one is
the minimum value.
00:11:33.640 --> 00:11:35.840
So they're two separate things.
00:11:35.840 --> 00:11:39.210
Now, the problem is
that people are sloppy.
00:11:39.210 --> 00:11:43.550
And especially since you
usually find the critical point
00:11:43.550 --> 00:11:48.620
first, and the value,
that is plugging in for A,
00:11:48.620 --> 00:11:51.430
second, people will
stop short and they'll
00:11:51.430 --> 00:11:54.010
give the wrong answer to
the question, for instance.
00:11:54.010 --> 00:11:57.760
Now, both questions are
important to answer.
00:11:57.760 --> 00:12:01.040
You just need to have
a word to put there.
00:12:01.040 --> 00:12:02.490
So this is a little
bit careless.
00:12:02.490 --> 00:12:06.060
When we say what is the minimum,
some people will say 1/2.
00:12:06.060 --> 00:12:08.030
And that's literally wrong.
00:12:08.030 --> 00:12:09.800
They know what they mean.
00:12:09.800 --> 00:12:10.890
But it's just wrong.
00:12:10.890 --> 00:12:13.520
And when people ask this
question, they're being sloppy.
00:12:13.520 --> 00:12:14.210
Anyway.
00:12:14.210 --> 00:12:16.010
They should maybe
be a little clearer
00:12:16.010 --> 00:12:17.850
and say what's
the minimum value.
00:12:17.850 --> 00:12:20.320
Or, where is the value achieved.
00:12:20.320 --> 00:12:27.080
It's achieved at, or where is
the minimum value achieved.
00:12:27.080 --> 00:12:31.970
"Where is min achieved?", would
be a better way of phrasing
00:12:31.970 --> 00:12:34.050
this second question.
00:12:34.050 --> 00:12:37.900
So that it has an
unambiguous answer.
00:12:37.900 --> 00:12:41.670
And when people ask you
for the minimum point,
00:12:41.670 --> 00:12:45.490
they're also-- so why is it that
we call it the minimum point?
00:12:45.490 --> 00:12:48.510
We have this word, critical
point, which is what x = 1/2
00:12:48.510 --> 00:12:50.330
is, and critical value.
00:12:50.330 --> 00:12:52.840
And so I'm making those
same distinctions here.
00:12:52.840 --> 00:12:59.680
But there's another
notion of a minimum point,
00:12:59.680 --> 00:13:09.340
and this is an
alternative if you like.
00:13:09.340 --> 00:13:19.540
The minimum point is
the point (1/2, 1/32).
00:13:19.540 --> 00:13:25.350
Right, that's a
point on the graph.
00:13:25.350 --> 00:13:29.490
It's the point - well, so
that graph is way up there,
00:13:29.490 --> 00:13:30.740
but I'll just put it on there.
00:13:30.740 --> 00:13:33.780
That's this point.
00:13:33.780 --> 00:13:35.792
And you might say min there.
00:13:35.792 --> 00:13:38.166
And you might point to this
point, and you might say max.
00:13:38.166 --> 00:13:42.540
And similarly, this
one might be a max.
00:13:42.540 --> 00:13:44.840
So in other words,
what this means
00:13:44.840 --> 00:13:49.375
is simply that people
are a little sloppy.
00:13:49.375 --> 00:13:50.750
And sometimes they
mean one thing
00:13:50.750 --> 00:13:53.770
and sometimes they mean another.
00:13:53.770 --> 00:13:56.470
And you're just stuck with
this, because there'll
00:13:56.470 --> 00:13:58.200
be some authors who
will say one thing
00:13:58.200 --> 00:13:59.575
and some people
will mean another
00:13:59.575 --> 00:14:02.770
and you just have to live with
this little bit of annoying
00:14:02.770 --> 00:14:05.150
ambiguity.
00:14:05.150 --> 00:14:05.690
Yeah?
00:14:05.690 --> 00:14:08.280
STUDENT: [INAUDIBLE]
00:14:08.280 --> 00:14:12.735
PROFESSOR: OK, so that's
a good - very good.
00:14:12.735 --> 00:14:16.710
So here we go, find the
largest area enclosed.
00:14:16.710 --> 00:14:22.290
So that's sort of a
trick question, isn't it?
00:14:22.290 --> 00:14:28.410
So there are various -
that's a good thing to ask.
00:14:28.410 --> 00:14:30.530
That's sort of a
trick question, why?
00:14:30.530 --> 00:14:34.840
Because according
to the rules, we're
00:14:34.840 --> 00:14:38.280
trapped between the
two maxima at something
00:14:38.280 --> 00:14:40.320
which is strictly below.
00:14:40.320 --> 00:14:44.140
So in other words, one answer
to this question would be,
00:14:44.140 --> 00:14:48.000
and this is the answer that I
would probably give, is 1/16.
00:14:48.000 --> 00:14:50.640
But that's not really true.
00:14:50.640 --> 00:14:56.560
Because that's
only in the limit.
00:14:56.560 --> 00:15:01.346
As x goes to 0, or
as x goes to 1-.
00:15:01.346 --> 00:15:02.720
And if you like,
the most is when
00:15:02.720 --> 00:15:04.830
you've only got one square.
00:15:04.830 --> 00:15:07.550
Which breaks the
rules of the problem.
00:15:07.550 --> 00:15:11.290
So, essentially, it's
a trick question.
00:15:11.290 --> 00:15:13.670
But I would answer it this way.
00:15:13.670 --> 00:15:16.220
Because that's the most
interesting part of the answer,
00:15:16.220 --> 00:15:18.780
which is that it's 1/16
and it occurs really
00:15:18.780 --> 00:15:28.920
when one of the squares
disappears to nothing.
00:15:28.920 --> 00:15:34.180
So now, let's do
another example here.
00:15:34.180 --> 00:15:41.450
And I just want to
illustrate the second style,
00:15:41.450 --> 00:15:42.980
or the second type of question.
00:15:42.980 --> 00:15:43.480
Yeah.
00:15:43.480 --> 00:15:51.750
STUDENT: [INAUDIBLE]
00:15:51.750 --> 00:15:56.320
PROFESSOR: The question
is, since the question was,
00:15:56.320 --> 00:16:00.090
what was the largest area, why
did we find the least area.
00:16:00.090 --> 00:16:04.050
The reason is that when we go
about our procedure of looking
00:16:04.050 --> 00:16:10.116
for the least, or the most,
we'll automatically find both.
00:16:10.116 --> 00:16:12.490
Because we don't know which
one is which until we compare
00:16:12.490 --> 00:16:14.730
values.
00:16:14.730 --> 00:16:17.170
And actually, it's much
more to your advantage
00:16:17.170 --> 00:16:20.540
to figure out both the
maximum and minimum
00:16:20.540 --> 00:16:22.316
whenever you answer
such a question.
00:16:22.316 --> 00:16:24.440
Because otherwise you won't
understand the behavior
00:16:24.440 --> 00:16:26.590
of the function very well.
00:16:26.590 --> 00:16:27.850
So, the question.
00:16:27.850 --> 00:16:30.500
We started out with one
question, we answered both.
00:16:30.500 --> 00:16:31.750
We answered two questions.
00:16:31.750 --> 00:16:36.480
We answered the question of what
the largest and the smallest
00:16:36.480 --> 00:16:37.290
value was.
00:16:37.290 --> 00:16:39.505
STUDENT: Also, I'm
wondering if you
00:16:39.505 --> 00:16:42.680
can check both the minimum
[INAUDIBLE] approaches
00:16:42.680 --> 00:16:47.505
[INAUDIBLE].
00:16:47.505 --> 00:16:48.130
PROFESSOR: Yes.
00:16:48.130 --> 00:16:49.880
One can also use-- the
question is, can we
00:16:49.880 --> 00:16:51.490
use the second derivative test.
00:16:51.490 --> 00:16:53.480
And the answer is, yes we can.
00:16:53.480 --> 00:16:55.610
In fact, you can actually
also stare at this
00:16:55.610 --> 00:16:57.590
and see that it's
a sum of squares.
00:16:57.590 --> 00:17:00.850
So it's always curving up.
00:17:00.850 --> 00:17:04.710
It's a parabola with a
positive second coefficient.
00:17:04.710 --> 00:17:06.670
So you can differentiate
this twice.
00:17:06.670 --> 00:17:09.670
If you do you'll get
1/8 plus another 1/8
00:17:09.670 --> 00:17:11.450
and you'll get 1/16.
00:17:11.450 --> 00:17:14.540
So the second
derivative is 1/16.
00:17:14.540 --> 00:17:17.730
Is 1/4.
00:17:17.730 --> 00:17:21.444
And that's an acceptable
way of figuring it out.
00:17:21.444 --> 00:17:23.360
I'll mention the second
derivative test again,
00:17:23.360 --> 00:17:24.950
in this second example.
00:17:24.950 --> 00:17:32.580
So let me talk about
a second example.
00:17:32.580 --> 00:17:35.070
So again, this is going
to be another question.
00:17:35.070 --> 00:17:37.950
STUDENT: [INAUDIBLE]
00:17:37.950 --> 00:17:43.660
PROFESSOR: The question is,
when I say minimum or maximum
00:17:43.660 --> 00:17:44.880
point which will I mean.
00:17:44.880 --> 00:17:50.570
STUDENT: [INAUDIBLE]
00:17:50.570 --> 00:17:53.100
PROFESSOR: So I just
repeated the question.
00:17:53.100 --> 00:17:56.650
So the question is, when
I say minimum point,
00:17:56.650 --> 00:17:58.580
what will I mean?
00:17:58.580 --> 00:18:00.100
OK?
00:18:00.100 --> 00:18:05.490
And the answer is that for
the purposes of this class
00:18:05.490 --> 00:18:09.090
I will probably
avoid saying that.
00:18:09.090 --> 00:18:13.340
But I will say, probably,
where is the minimum achieved.
00:18:13.340 --> 00:18:14.680
Just in order to avoid that.
00:18:14.680 --> 00:18:17.980
If I actually said it, I often
am referring to the graph,
00:18:17.980 --> 00:18:19.300
and I mean this.
00:18:19.300 --> 00:18:21.280
And in fact, when you
get your little review
00:18:21.280 --> 00:18:23.090
for the second exam,
I'll say exactly
00:18:23.090 --> 00:18:25.860
that on the review sheet.
00:18:25.860 --> 00:18:28.700
And I'll make this very clear
when we were doing this.
00:18:28.700 --> 00:18:30.700
However, I just want to
prepare you for the fact
00:18:30.700 --> 00:18:35.480
that in real life, and even me
when I'm talking colloquially,
00:18:35.480 --> 00:18:37.750
when I say what's the
minimum point of something,
00:18:37.750 --> 00:18:48.730
I might actually be mixing it
up with this other notion here.
00:18:48.730 --> 00:18:54.380
So let's do another example.
00:18:54.380 --> 00:18:57.440
So this is an example
to get us used
00:18:57.440 --> 00:18:59.880
to the notion of constraints.
00:18:59.880 --> 00:19:08.250
So we have, so consider
a box without a top.
00:19:08.250 --> 00:19:16.860
Or, if you like, we're going
to find the box without a top.
00:19:16.860 --> 00:19:35.380
With least surface area
for a fixed volume.
00:19:35.380 --> 00:19:38.950
Find the box without
a top with least
00:19:38.950 --> 00:19:42.360
surface area for a fixed volume.
00:19:42.360 --> 00:19:47.940
The procedure for working
this out is the following.
00:19:47.940 --> 00:19:51.480
You make this diagram.
00:19:51.480 --> 00:19:56.220
And you set up the variables.
00:19:56.220 --> 00:20:00.390
In this case, we're going to
have four names of variables.
00:20:00.390 --> 00:20:02.890
We have four letters
that we have to choose.
00:20:02.890 --> 00:20:05.680
And we'll choose them in a kind
of a standard way, alright?
00:20:05.680 --> 00:20:08.260
So first I have to tell
you one more thing.
00:20:08.260 --> 00:20:12.880
Which is something that we
could calculate separately
00:20:12.880 --> 00:20:15.080
but I'm just going to
give it to you in advance.
00:20:15.080 --> 00:20:16.746
Which is that it turns
out that the best
00:20:16.746 --> 00:20:21.420
box has a square bottom.
00:20:21.420 --> 00:20:24.710
And that's going to get rid of
one of our variables for us.
00:20:24.710 --> 00:20:28.500
So it's got a square bottom, and
so let's draw a picture of it.
00:20:28.500 --> 00:20:36.860
So here's our box.
00:20:36.860 --> 00:20:40.680
Well, that goes down
like this, almost.
00:20:40.680 --> 00:20:49.620
Maybe I should get it
a little farther down.
00:20:49.620 --> 00:20:52.070
So here's our box.
00:20:52.070 --> 00:20:54.900
Let's correct that just a bit.
00:20:54.900 --> 00:20:57.760
So now, what about the
dimensions of this box?
00:20:57.760 --> 00:21:02.050
Well, this is going to be x,
and this is very foreshortened,
00:21:02.050 --> 00:21:03.180
but it's also x.
00:21:03.180 --> 00:21:06.130
The bottom is x by x,
it's the same dimensions.
00:21:06.130 --> 00:21:12.190
And then the vertical
dimension is y.
00:21:12.190 --> 00:21:13.750
So far, so good.
00:21:13.750 --> 00:21:16.620
Now, I promised you
two more letter names.
00:21:16.620 --> 00:21:21.040
I want to compute the volume.
00:21:21.040 --> 00:21:24.440
The volume is, the base is
x^2, and the height is y.
00:21:24.440 --> 00:21:26.780
So there's the volume.
00:21:26.780 --> 00:21:32.870
And then the area, the area
is the area of the bottom,
00:21:32.870 --> 00:21:37.940
which is x^2, that's the bottom.
00:21:37.940 --> 00:21:40.510
And then there are
the four sides.
00:21:40.510 --> 00:21:44.230
And the four sides are
rectangles of dimensions x, y.
00:21:44.230 --> 00:21:49.500
So it's 4xy.
00:21:49.500 --> 00:21:53.630
So these are the sides.
00:21:53.630 --> 00:21:56.680
And remember, there's no top.
00:21:56.680 --> 00:21:58.610
So that's our setup.
00:21:58.610 --> 00:22:02.670
So now, the difference
between this problem
00:22:02.670 --> 00:22:05.230
and the last problem
is that there
00:22:05.230 --> 00:22:06.690
are two variables
floating around,
00:22:06.690 --> 00:22:09.710
namely x and y, which
are not determined.
00:22:09.710 --> 00:22:17.670
But there's what's
called a constraint here.
00:22:17.670 --> 00:22:22.920
Namely, we've fixed the
relationship between x and y.
00:22:22.920 --> 00:22:30.880
And so, that means that we
can solve for y in terms of x.
00:22:30.880 --> 00:22:33.310
So y = V/x^2.
00:22:40.650 --> 00:22:43.660
And then, we can plug that
into the formula for A.
00:22:43.660 --> 00:22:50.280
So here we have A which
is x^2 + 4x v/x^2.
00:23:00.911 --> 00:23:01.410
Question.
00:23:01.410 --> 00:23:14.600
STUDENT: [INAUDIBLE]
00:23:14.600 --> 00:23:16.100
PROFESSOR: The
question is, will you
00:23:16.100 --> 00:23:17.350
need to know this intuitively?
00:23:17.350 --> 00:23:17.849
No.
00:23:17.849 --> 00:23:20.830
That's something that I
would have to give to you.
00:23:20.830 --> 00:23:24.870
I mean, it's actually
true that a lot
00:23:24.870 --> 00:23:28.040
of things, the correct answer
is something symmetric.
00:23:28.040 --> 00:23:30.240
In this last
problem, the minimum
00:23:30.240 --> 00:23:33.100
turned out to be exactly
halfway in between because there
00:23:33.100 --> 00:23:35.900
were sort of equal demands
from the two sides.
00:23:35.900 --> 00:23:37.480
And similarly,
here, what happens
00:23:37.480 --> 00:23:40.130
is if you elongate
one side, you get less
00:23:40.130 --> 00:23:46.050
- it actually is involved
with a two variable problem.
00:23:46.050 --> 00:23:49.440
Namely, if you have a rectangle
and you have a certain amount
00:23:49.440 --> 00:23:52.610
of length associated with
it, what's the optimal thing
00:23:52.610 --> 00:23:53.720
you can do with that.
00:23:53.720 --> 00:23:58.280
But I won't-- in other
words, the optimal rectangle,
00:23:58.280 --> 00:24:01.397
the least perimeter rectangle,
turns out to be a square.
00:24:01.397 --> 00:24:03.230
That's the little
sub-problem that leads you
00:24:03.230 --> 00:24:05.960
to this square bottom.
00:24:05.960 --> 00:24:09.270
But so that would have been
a separate max-min problem.
00:24:09.270 --> 00:24:11.900
Which I'm skipping, because I
want to do this slightly more
00:24:11.900 --> 00:24:16.240
interesting one.
00:24:16.240 --> 00:24:20.540
So now, here's
our formula for A,
00:24:20.540 --> 00:24:27.340
and now I want to follow the
same procedure as before.
00:24:27.340 --> 00:24:29.320
Namely, we look for
the critical point.
00:24:29.320 --> 00:24:35.530
Or points.
00:24:35.530 --> 00:24:37.260
So let's take a look.
00:24:37.260 --> 00:24:40.020
So again, A is (x^2 + 4v) / x.
00:24:43.430 --> 00:24:49.800
And A' is 2x - 4v/x^2.
00:24:49.800 --> 00:24:59.510
So if we set that equal
to 0, we get 2x = 2v/x^2.
00:24:59.510 --> 00:25:01.620
So 2x^3.
00:25:04.690 --> 00:25:07.270
How did that happen
to change into 2?
00:25:07.270 --> 00:25:09.590
Interesting, guess that's wrong.
00:25:09.590 --> 00:25:12.950
OK.
00:25:12.950 --> 00:25:20.480
So this is x^3 2v.
00:25:20.480 --> 00:25:23.730
And so x = 2^(1/3) v^(1/3).
00:25:28.010 --> 00:25:36.230
So this is the critical point.
00:25:36.230 --> 00:25:38.040
So we are not done.
00:25:38.040 --> 00:25:38.619
Right?
00:25:38.619 --> 00:25:40.160
We're not done,
because we don't even
00:25:40.160 --> 00:25:41.701
know whether this
is going to give us
00:25:41.701 --> 00:25:44.410
the worst box or the best
box, from this point of view.
00:25:44.410 --> 00:25:48.100
The one that uses the most
surface area or the least.
00:25:48.100 --> 00:25:51.990
So let's check the
ends, right away.
00:25:51.990 --> 00:25:54.030
To see what's happening.
00:25:54.030 --> 00:25:56.280
So can somebody tell
me what the ends, what
00:25:56.280 --> 00:25:58.060
the end values of x are?
00:25:58.060 --> 00:25:59.520
Where does x range from?
00:25:59.520 --> 00:26:05.124
STUDENT: [INAUDIBLE]
00:26:05.124 --> 00:26:07.040
PROFESSOR: What's the
smallest x can be, yeah.
00:26:07.040 --> 00:26:16.520
STUDENT: [INAUDIBLE]
00:26:16.520 --> 00:26:19.400
PROFESSOR: OK, the claim was
that the largest x could be
00:26:19.400 --> 00:26:24.040
root A, because somehow there's
this x^2 here and you can't get
00:26:24.040 --> 00:26:25.685
any further past than that.
00:26:25.685 --> 00:26:28.700
But there's a key feature
here of this problem.
00:26:28.700 --> 00:26:32.050
Which is that A is variable.
00:26:32.050 --> 00:26:34.390
The only thing that's
fixed in the problem
00:26:34.390 --> 00:26:47.440
is V. So if V is fixed,
what do you know about x?
00:26:47.440 --> 00:26:49.810
STUDENT: [INAUDIBLE]
00:26:49.810 --> 00:26:51.160
PROFESSOR: x > 0, yeah.
00:26:51.160 --> 00:26:53.060
The lower end
point, that's safe.
00:26:53.060 --> 00:26:55.440
Because that has to do
geometrically with the fact
00:26:55.440 --> 00:26:59.430
that we don't have any boxes
with negative dimensions.
00:26:59.430 --> 00:27:02.360
That would be refused by
the Post Office, definitely.
00:27:02.360 --> 00:27:04.190
Over and above the
empty top, which
00:27:04.190 --> 00:27:05.510
they wouldn't accept either.
00:27:05.510 --> 00:27:13.330
STUDENT: [INAUDIBLE]
00:27:13.330 --> 00:27:17.260
PROFESSOR: It's true that x is
less than square root of V / y.
00:27:17.260 --> 00:27:19.250
So that's using
this relationship.
00:27:19.250 --> 00:27:23.750
But notice that y = V/x^2.
00:27:26.850 --> 00:27:30.060
So 0 to infinity, I
just got a guess there
00:27:30.060 --> 00:27:32.080
over here, that's right.
00:27:32.080 --> 00:27:33.350
Here's the upper limit.
00:27:33.350 --> 00:27:35.750
So this is really
important to realize.
00:27:35.750 --> 00:27:37.420
This is most problems.
00:27:37.420 --> 00:27:40.340
Most problems, the variable, if
it doesn't have a limitation,
00:27:40.340 --> 00:27:42.370
usually just goes
out to infinity.
00:27:42.370 --> 00:27:45.790
And infinity is a very
important end for the problem.
00:27:45.790 --> 00:27:51.516
It's usually an easy
end to the problem, too.
00:27:51.516 --> 00:27:53.890
So there's a possibility that
if we push all the way down
00:27:53.890 --> 00:27:56.240
to x = 0, we'll
get a better box.
00:27:56.240 --> 00:27:58.230
It would be very strange box.
00:27:58.230 --> 00:28:01.620
A little bit like our
vanishing enclosure.
00:28:01.620 --> 00:28:04.920
And maybe an
infinitely long box,
00:28:04.920 --> 00:28:06.840
also very inconvenient one.
00:28:06.840 --> 00:28:07.990
Might be the best box.
00:28:07.990 --> 00:28:10.390
We'll have to see.
00:28:10.390 --> 00:28:12.810
So let's just take a
look at what happens.
00:28:12.810 --> 00:28:18.740
So we're looking at A, at 0+.
00:28:18.740 --> 00:28:25.760
And that's x^2 + 4V
/ x with x at 0+.
00:28:25.760 --> 00:28:26.760
So what happens to that?
00:28:26.760 --> 00:28:35.220
Notice right here, this
is going to infinity.
00:28:35.220 --> 00:28:38.500
So this is infinite.
00:28:38.500 --> 00:28:42.250
So that turns out
to be a bad box.
00:28:42.250 --> 00:28:45.730
Let's take a look
at the other end.
00:28:45.730 --> 00:28:51.930
So this is x^2 + 4V /
x, x going to infinity.
00:28:51.930 --> 00:28:59.930
And again, this term here means
that this thing is infinite.
00:28:59.930 --> 00:29:01.870
So the shape of this
thing, I'll draw
00:29:01.870 --> 00:29:05.000
this tiny little schematic
diagram over here.
00:29:05.000 --> 00:29:11.110
The shape of this thing
is like this, right?
00:29:11.110 --> 00:29:13.950
And so, when we find that
one turnaround point,
00:29:13.950 --> 00:29:19.000
which happened to be at this
strange point 2/3-- sorry,
00:29:19.000 --> 00:29:24.440
2^(1/3) V^(1/3), that is
going to be the minimum.
00:29:24.440 --> 00:29:30.090
So we've just discovered
that it's the minimum.
00:29:30.090 --> 00:29:31.690
Which is just what
we were hoping for.
00:29:31.690 --> 00:29:38.200
This is going to
be the optimal box.
00:29:38.200 --> 00:29:44.660
Now, since you asked
earlier and since it's worth
00:29:44.660 --> 00:29:49.360
checking this as
well, let's also
00:29:49.360 --> 00:29:51.740
check an alternative
justification.
00:29:51.740 --> 00:30:02.540
So an alternative
to checking ends
00:30:02.540 --> 00:30:11.540
is the second derivative test.
00:30:11.540 --> 00:30:14.200
I do not recommend the
second derivative test.
00:30:14.200 --> 00:30:16.230
I try my best, when
I give you problems,
00:30:16.230 --> 00:30:19.290
to make it really hard to apply
the second derivative test.
00:30:19.290 --> 00:30:21.320
But in this example,
the function
00:30:21.320 --> 00:30:24.570
is simple enough so
that it's perfectly OK.
00:30:24.570 --> 00:30:28.110
If you take the
derivative here, remember,
00:30:28.110 --> 00:30:34.760
this was whatever it
was, 2x - 4V / x^2.
00:30:34.760 --> 00:30:43.610
If I take the second
derivative, it's 2 + 8V / x^3.
00:30:43.610 --> 00:30:45.310
And that's positive.
00:30:45.310 --> 00:30:49.580
So this thing is concave up.
00:30:49.580 --> 00:30:52.180
And that's consistent
with its being--
00:30:52.180 --> 00:30:59.840
the critical point is a min.
00:30:59.840 --> 00:31:00.840
Is a minimum point.
00:31:00.840 --> 00:31:05.180
See how I almost said, is a min,
as opposed to minimum point.
00:31:05.180 --> 00:31:05.780
So watch out.
00:31:05.780 --> 00:31:06.280
Yes.
00:31:06.280 --> 00:31:12.235
STUDENT: [INAUDIBLE]
00:31:12.235 --> 00:31:13.860
PROFESSOR: You're
one step ahead of me.
00:31:13.860 --> 00:31:15.985
The question is, is this
the answer to the question
00:31:15.985 --> 00:31:20.090
or would we have to give y
and A and so on and so forth.
00:31:20.090 --> 00:31:23.120
So, again, this is
something that I
00:31:23.120 --> 00:31:26.220
want to emphasize and take
my time with right now.
00:31:26.220 --> 00:31:29.800
Because it depends, what
kind of real-life problem
00:31:29.800 --> 00:31:33.360
you're answering, what kind
of answer is appropriate.
00:31:33.360 --> 00:31:36.010
So, so far we've found
the critical point.
00:31:36.010 --> 00:31:38.080
We haven't found
the critical value.
00:31:38.080 --> 00:31:42.190
We haven't found the
dimensions of the box.
00:31:42.190 --> 00:31:46.400
So we're going to spend a little
bit more time on this, exactly
00:31:46.400 --> 00:31:48.920
in order to address
these questions.
00:31:48.920 --> 00:31:50.170
So, first of all.
00:31:50.170 --> 00:31:51.470
The value of y.
00:31:51.470 --> 00:31:55.120
So, so far we have
x = 2^(1/3) V^(1/3).
00:31:55.120 --> 00:31:57.170
And certainly if you're
going to build the box,
00:31:57.170 --> 00:32:00.410
you also want to know
what the y-value is.
00:32:00.410 --> 00:32:04.970
The y-value is going
to be, let's see.
00:32:04.970 --> 00:32:12.110
Well, it's V / x^2, so that's
V / (2^(1/3) V^(1/3) )^2,
00:32:12.110 --> 00:32:19.770
which comes out to
be 2^(2/3) V^(1/3).
00:32:19.770 --> 00:32:22.580
So there's the y-value.
00:32:22.580 --> 00:32:29.505
On top of that, we could
figure out the value of A.
00:32:29.505 --> 00:32:31.880
So that's also a perfectly
reasonable part of the answer.
00:32:31.880 --> 00:32:33.780
Depending on what
one is interested in,
00:32:33.780 --> 00:32:35.675
you might care how
much money it's going
00:32:35.675 --> 00:32:38.200
to cost you to build this box.
00:32:38.200 --> 00:32:39.330
This optimal box.
00:32:39.330 --> 00:32:43.370
And so you plug in the value of
A. So A, let's see, is up here.
00:32:43.370 --> 00:32:47.740
It's x x^2 + 4V/x.
00:32:47.740 --> 00:32:57.810
So that's going to be (2^(1/3)
V^(1/3) )^2 plus 4V / (2^(1/3)
00:32:57.810 --> 00:32:58.310
V^(1/3) ).
00:33:02.330 --> 00:33:07.300
And if you work that all out,
what you get turns out to be 3
00:33:07.300 --> 00:33:08.170
* 2^(1/3) V^(2/3).
00:33:13.730 --> 00:33:17.540
So if you like, one way
of answering this question
00:33:17.540 --> 00:33:23.160
is these three things.
00:33:23.160 --> 00:33:26.490
That would be the minimum point
corresponding to the graph.
00:33:26.490 --> 00:33:28.950
That would be the
answer to this question.
00:33:28.950 --> 00:33:31.710
But the reason why I'm
carrying it out in such detail
00:33:31.710 --> 00:33:35.370
is I want to show you that there
are much more meaningful ways
00:33:35.370 --> 00:33:37.150
of answering this question.
00:33:37.150 --> 00:34:02.810
So let me explain that.
00:34:02.810 --> 00:34:07.290
So let me go through some
more meaningful answers here.
00:34:07.290 --> 00:34:14.770
The first more meaningful
answer is the following idea.
00:34:14.770 --> 00:34:29.780
Simply, what are known as
dimensionless variables.
00:34:29.780 --> 00:34:33.050
So the first thing that you
notice is the scaling law.
00:34:33.050 --> 00:34:36.580
That A / V^(2/3) is the
thing that's a dimensionless
00:34:36.580 --> 00:34:37.260
quantity.
00:34:37.260 --> 00:34:39.360
That happens to be 3 * 2^(1/3).
00:34:42.150 --> 00:34:43.970
So that's one thing.
00:34:43.970 --> 00:34:45.470
If you want to
expand the volume,
00:34:45.470 --> 00:34:49.220
you'll have to expand the area
by the 2/3 power of the volume.
00:34:49.220 --> 00:34:55.430
And if you think of the area as
being in, say, square inches,
00:34:55.430 --> 00:35:00.000
and the volume of the box
as being in cubic inches,
00:35:00.000 --> 00:35:02.360
then you can see that this
is a dimensionless quantity
00:35:02.360 --> 00:35:04.070
and you have a
dimensionless number here,
00:35:04.070 --> 00:35:06.730
which is a characteristic
independent of what
00:35:06.730 --> 00:35:09.310
A and V were.
00:35:09.310 --> 00:35:15.710
The other dimensionless
quantity is the ratio of x to y.
00:35:15.710 --> 00:35:19.640
Or x to y.
00:35:19.640 --> 00:35:23.680
So, again, that's inches
divided by inches.
00:35:23.680 --> 00:35:32.950
And it's 2^(1/3) V^(1/3)
divided by 2^(-2/3) V^(1/3),
00:35:32.950 --> 00:35:36.150
which happens to be 2.
00:35:36.150 --> 00:35:41.390
So this is actually the
best answer to the question.
00:35:41.390 --> 00:35:46.120
And it shows you that
the box is a 2:1 box.
00:35:46.120 --> 00:35:50.200
If this is 2 and this is
1, that's the good box.
00:35:50.200 --> 00:35:56.630
And that is just the
shape, if you like,
00:35:56.630 --> 00:36:02.670
and it's the optimal shape.
00:36:02.670 --> 00:36:04.490
And certainly that,
aesthetically, that's
00:36:04.490 --> 00:36:12.187
the cleanest answer
to the question.
00:36:12.187 --> 00:36:13.520
There was a question right here.
00:36:13.520 --> 00:36:14.020
Yes.
00:36:14.020 --> 00:36:20.537
STUDENT: [INAUDIBLE]
00:36:20.537 --> 00:36:22.620
PROFESSOR: Could you repeat
that, I couldn't hear.
00:36:22.620 --> 00:36:24.036
STUDENT: I'm
wondering if you'd be
00:36:24.036 --> 00:36:29.850
able to get that answer
if you [INAUDIBLE] square.
00:36:29.850 --> 00:36:31.350
PROFESSOR: The
question is, could we
00:36:31.350 --> 00:36:33.058
have gotten the answer
if we weren't told
00:36:33.058 --> 00:36:34.380
that the bottom was square.
00:36:34.380 --> 00:36:39.014
The answer is, yes in
18.02 with multivariable.
00:36:39.014 --> 00:36:41.222
You would have to have three
letters here, an x, a y,
00:36:41.222 --> 00:36:43.470
and a z, if you like.
00:36:43.470 --> 00:36:48.170
And then you'd have to work
with all three of them.
00:36:48.170 --> 00:36:51.890
So I separated out
into one, there's
00:36:51.890 --> 00:36:55.550
a separate one variable problem
that you can do for the base.
00:36:55.550 --> 00:36:57.510
And then this is a second
one variable problem
00:36:57.510 --> 00:36:58.510
for this other thing.
00:36:58.510 --> 00:37:00.910
And it's just two consecutive
one variable problems
00:37:00.910 --> 00:37:03.170
that solve the
multivariable problem.
00:37:03.170 --> 00:37:05.650
Or, as I say in
multivariable calculus,
00:37:05.650 --> 00:37:08.740
you can just do it
all in one step.
00:37:08.740 --> 00:37:09.240
Yeah?
00:37:09.240 --> 00:37:11.820
STUDENT: [INAUDIBLE]
00:37:11.820 --> 00:37:14.550
PROFESSOR: Why did
I divide x by y,
00:37:14.550 --> 00:37:17.830
rather than y by x, or in any?
00:37:17.830 --> 00:37:22.320
So, again, what I was aiming for
was dimensionless quantities.
00:37:22.320 --> 00:37:26.640
So x and y are measured
in the same units.
00:37:26.640 --> 00:37:29.065
And also the
proportions of the box
00:37:29.065 --> 00:37:34.310
- so that's another word
for this is proportions -
00:37:34.310 --> 00:37:38.430
are something that's universal,
independent of the volume
00:37:38.430 --> 00:37:42.020
V. It's something you can say
about any box, at any scale.
00:37:42.020 --> 00:37:45.040
Whether it be, you
know, something
00:37:45.040 --> 00:37:48.781
by Cristo in the Common.
00:37:48.781 --> 00:37:50.530
Maybe we'll get in
here to do some fancy--
00:37:50.530 --> 00:37:55.540
STUDENT: [INAUDIBLE]
00:37:55.540 --> 00:37:58.630
PROFESSOR: The proportions
is - with geometric problems
00:37:58.630 --> 00:38:01.930
typically, when there's a
scaling to the problem, where
00:38:01.930 --> 00:38:05.020
the answer is the same at small
scales and at large scales
00:38:05.020 --> 00:38:07.000
- this is capturing that.
00:38:07.000 --> 00:38:10.070
So that's why, the ratios
are what's capturing that.
00:38:10.070 --> 00:38:12.410
And that's why it's
aesthetically the nicest thing
00:38:12.410 --> 00:38:13.860
to ask.
00:38:13.860 --> 00:38:17.905
STUDENT: So, what
exactly does the ratio
00:38:17.905 --> 00:38:21.639
of the area to the volume
ratio [INAUDIBLE] tell us?
00:38:21.639 --> 00:38:23.180
PROFESSOR: Unfortunately,
this number
00:38:23.180 --> 00:38:25.490
is a really obscure number.
00:38:25.490 --> 00:38:28.006
So the question is
what does this tell us.
00:38:28.006 --> 00:38:29.630
The only thing that
I want to emphasize
00:38:29.630 --> 00:38:31.800
is what's on the
left-hand side here.
00:38:31.800 --> 00:38:34.870
Which is, it's the area to
the 2/3 power over the volume,
00:38:34.870 --> 00:38:38.550
so it's a dimensionless quantity
that happens to be this.
00:38:38.550 --> 00:38:43.700
If you do this, for example, in
general with planar diagrams,
00:38:43.700 --> 00:38:47.200
circumference to area
is a bad ratio to take.
00:38:47.200 --> 00:38:48.810
What you want to
take is the square
00:38:48.810 --> 00:38:50.932
of circumference to area.
00:38:50.932 --> 00:38:52.390
Because the square
of circumference
00:38:52.390 --> 00:38:54.690
has the same dimensions,
that is, say,
00:38:54.690 --> 00:38:56.880
inches squared to area.
00:38:56.880 --> 00:38:58.840
Which is in square inches.
00:38:58.840 --> 00:39:01.360
So, again, it's these
dimensionless quantities
00:39:01.360 --> 00:39:03.440
that you want to cook up.
00:39:03.440 --> 00:39:06.810
And those are the ones that
will have universal properties.
00:39:06.810 --> 00:39:09.990
The most famous of
these is the circle
00:39:09.990 --> 00:39:13.880
that encloses the most
area for its circumference.
00:39:13.880 --> 00:39:17.876
And, again, that's only
true if you take the square
00:39:17.876 --> 00:39:18.880
of the circumference.
00:39:18.880 --> 00:39:25.620
You do the units correctly.
00:39:25.620 --> 00:39:26.290
Anyway.
00:39:26.290 --> 00:39:29.760
So we're here,
we've got a shape.
00:39:29.760 --> 00:39:31.560
We've got an answer
to this question.
00:39:31.560 --> 00:39:36.940
And I now want to
do this problem.
00:39:36.940 --> 00:39:38.267
Well, let's put it this way.
00:39:38.267 --> 00:39:40.350
I wanted to do this problem
by a different method.
00:39:40.350 --> 00:39:43.430
I think I'll take
the time to do it.
00:39:43.430 --> 00:39:46.790
So I want to do this problem
by a slightly different method
00:39:46.790 --> 00:39:48.880
here.
00:39:48.880 --> 00:39:59.160
So, here's Example 2 by
implicit differentiation.
00:39:59.160 --> 00:40:01.950
So the same example, but
now I'm going to do it
00:40:01.950 --> 00:40:03.720
by implicit differentiation.
00:40:03.720 --> 00:40:07.050
Well, I'll tell you the
advantages and disadvantages
00:40:07.050 --> 00:40:08.880
to this method here.
00:40:08.880 --> 00:40:20.130
So the situation is, you
have to start the same way.
00:40:20.130 --> 00:40:24.230
So here is the starting
place of the problem.
00:40:24.230 --> 00:40:34.980
And the goal was the minimum
of A with V constant.
00:40:34.980 --> 00:40:38.540
So this was the situation
that we were in.
00:40:38.540 --> 00:40:45.130
And now, what I want to
do is just differentiate.
00:40:45.130 --> 00:40:46.990
The function y is
implicitly a function
00:40:46.990 --> 00:40:54.550
of x, so I can differentiate
the first expression.
00:40:54.550 --> 00:41:00.380
And that yields
0 = 2xy + x^2 y'.
00:41:03.330 --> 00:41:11.590
So this is giving me my implicit
formula for y', So y' = -2xy /
00:41:11.590 --> 00:41:13.080
x^2.
00:41:13.080 --> 00:41:14.950
Or in other words, -2y/x.
00:41:19.660 --> 00:41:24.990
And then I also have the dA/dx.
00:41:24.990 --> 00:41:28.750
Now, you may notice I'm not
using primes quite as much.
00:41:28.750 --> 00:41:31.510
Because all of the
variables are varying,
00:41:31.510 --> 00:41:34.300
and so here I'm emphasizing
that it's a differentiation
00:41:34.300 --> 00:41:36.790
with respect to the variable x.
00:41:36.790 --> 00:41:48.130
And this becomes 2x + 4y + 4xy'.
00:41:48.130 --> 00:41:53.140
So again, this is
using the product rule.
00:41:53.140 --> 00:41:57.450
And now I can plug in for what
y' is, which is right above it.
00:41:57.450 --> 00:42:04.890
So this is 2x + 4y + 4x (-2y/x).
00:42:09.330 --> 00:42:14.150
And that's equal to 0.
00:42:14.150 --> 00:42:24.770
And so let's gather
that together.
00:42:24.770 --> 00:42:25.960
So what do we have?
00:42:25.960 --> 00:42:34.290
We have 2x + 4y, and then,
altogether, this is 8 - 8y,
00:42:34.290 --> 00:42:36.550
equals 0.
00:42:36.550 --> 00:42:41.600
So that's the same
thing as 2x = 4y.
00:42:41.600 --> 00:42:44.100
The - 4y goes to the other side.
00:42:44.100 --> 00:42:54.420
And so, x/y = 2.
00:42:54.420 --> 00:42:59.260
So this, I claim, so you
have to decide for yourself.
00:42:59.260 --> 00:43:03.630
But I claim that this is faster.
00:43:03.630 --> 00:43:07.350
It's faster, and also
it gets to the heart
00:43:07.350 --> 00:43:10.870
of the matter, which is this
scale-invariant proportions.
00:43:10.870 --> 00:43:13.360
Which is basically also nicer.
00:43:13.360 --> 00:43:16.720
So it gets to the
nicer answer, also.
00:43:16.720 --> 00:43:19.970
So those are the
advantages that this has.
00:43:19.970 --> 00:43:23.190
So it's faster, and
it gets to this,
00:43:23.190 --> 00:43:26.260
I'm going to call it nicer.
00:43:26.260 --> 00:43:38.640
And the disadvantage
is it did not
00:43:38.640 --> 00:43:55.620
check whether this critical
point is a max, min,
00:43:55.620 --> 00:43:59.540
or neither.
00:43:59.540 --> 00:44:02.280
So we didn't quite
finish the problem.
00:44:02.280 --> 00:44:10.820
But we got to the
answer very fast.
00:44:10.820 --> 00:44:11.640
Yeah, question.
00:44:11.640 --> 00:44:13.480
STUDENT: [INAUDIBLE]
00:44:13.480 --> 00:44:17.437
PROFESSOR: How
would you check it?
00:44:17.437 --> 00:44:18.270
STUDENT: [INAUDIBLE]
00:44:18.270 --> 00:44:20.510
PROFESSOR: Well, so it
gives you a candidate.
00:44:20.510 --> 00:44:23.390
The answer is-- so the question
is, how would you check it?
00:44:23.390 --> 00:44:26.720
The answer is that for
this particular problem,
00:44:26.720 --> 00:44:31.300
the only way to do it is
to do something like this.
00:44:31.300 --> 00:44:34.480
So in other words, it doesn't
save you that much time.
00:44:34.480 --> 00:44:37.730
But with many, many,
examples, you actually
00:44:37.730 --> 00:44:42.010
can tell immediately that if the
two ends, the thing is, say, 0,
00:44:42.010 --> 00:44:43.650
and inside it's positive.
00:44:43.650 --> 00:44:44.370
Things like that.
00:44:44.370 --> 00:44:53.640
So in many, many, cases
this is just as good.
00:44:53.640 --> 00:44:58.510
So now I'm going to
change subjects here.
00:44:58.510 --> 00:45:02.220
But the subject that I'm
going to talk about next
00:45:02.220 --> 00:45:07.550
is almost-- is very,
very closely linked.
00:45:07.550 --> 00:45:10.850
Namely, I talked about
implicit differentiation.
00:45:10.850 --> 00:45:12.770
Now, we're going to
just talk about dealing
00:45:12.770 --> 00:45:13.920
with lots of variables.
00:45:13.920 --> 00:45:15.360
And rates of change.
00:45:15.360 --> 00:45:16.960
So, essentially,
we're going to talk
00:45:16.960 --> 00:45:19.680
about the same type of thing.
00:45:19.680 --> 00:45:23.470
So, I'm going to tell
you about a subject which
00:45:23.470 --> 00:45:25.220
is called related rates.
00:45:25.220 --> 00:45:26.980
Which is really
just another excuse
00:45:26.980 --> 00:45:32.510
for getting used to setting
up variables and equations.
00:45:32.510 --> 00:45:34.320
So, here we go.
00:45:34.320 --> 00:45:36.000
Related rates.
00:45:36.000 --> 00:45:41.970
And I'm going to illustrate
this with one example today, one
00:45:41.970 --> 00:45:44.350
tomorrow.
00:45:44.350 --> 00:45:47.600
So here's my example for today.
00:45:47.600 --> 00:45:50.810
So, again, this is going
to be a police problem.
00:45:50.810 --> 00:45:53.630
But this is going to be a
word problem and-- sorry,
00:45:53.630 --> 00:45:56.910
I'm don't want to
scare you, no police.
00:45:56.910 --> 00:45:59.940
Well, there are police in the
story but they're not present.
00:45:59.940 --> 00:46:05.070
So, but I'm going to draw it
immediately with the diagram
00:46:05.070 --> 00:46:08.410
because I'm going to
save us the trouble.
00:46:08.410 --> 00:46:11.580
Although, you know, the point
here is to get from the words
00:46:11.580 --> 00:46:15.940
to the diagram.
00:46:15.940 --> 00:46:21.990
So you have the police, and
they're 30 feet from the road.
00:46:21.990 --> 00:46:25.750
And here's the road.
00:46:25.750 --> 00:46:36.610
And you're coming along,
here, in your, let's see,
00:46:36.610 --> 00:46:39.420
in your car going in
this direction here.
00:46:39.420 --> 00:46:43.760
And the police have radar.
00:46:43.760 --> 00:46:46.490
Which is bouncing
off of your car.
00:46:46.490 --> 00:46:53.880
And what they read off is
that you're 50 feet away.
00:46:53.880 --> 00:46:57.100
They also know that
you're approaching
00:46:57.100 --> 00:47:12.370
along the line of the radar at
a rate of 80 feet per second.
00:47:12.370 --> 00:47:19.260
Now, the question
is, are you speeding.
00:47:19.260 --> 00:47:20.730
That's the question.
00:47:20.730 --> 00:47:29.030
So when you're speeding, by
the way, 95 feet per second
00:47:29.030 --> 00:47:32.090
is about 65 miles per hour.
00:47:32.090 --> 00:47:35.890
So that's the threshold here.
00:47:35.890 --> 00:47:40.080
So what I want to
do now is show you
00:47:40.080 --> 00:47:43.860
how you set up a
problem like this.
00:47:43.860 --> 00:47:46.150
This distance is 50.
00:47:46.150 --> 00:47:50.460
This is 30, and because it's
the distance to a straight line
00:47:50.460 --> 00:47:52.030
you know that this
is a right angle.
00:47:52.030 --> 00:47:54.580
So we know that this
is a right triangle.
00:47:54.580 --> 00:47:56.810
And this is set out to be
a right triangle, which
00:47:56.810 --> 00:48:00.300
is an easy one, a 3,
4, 5 right triangle
00:48:00.300 --> 00:48:05.640
just so that we can do
the computations easily.
00:48:05.640 --> 00:48:09.180
So now, the question is,
how do we put the letters in
00:48:09.180 --> 00:48:10.970
to make this problem
work, to figure out
00:48:10.970 --> 00:48:12.310
what the rate of change is.
00:48:12.310 --> 00:48:15.930
So now, let me explain
that right now.
00:48:15.930 --> 00:48:18.590
And we will actually do
the computation next time.
00:48:18.590 --> 00:48:22.940
So the first thing is, you have
to understand what's changing
00:48:22.940 --> 00:48:24.850
and what's not.
00:48:24.850 --> 00:48:30.880
And we're going to use
t for time, in seconds.
00:48:30.880 --> 00:48:35.210
And now, an important
distance here
00:48:35.210 --> 00:48:37.820
is the distance to this
foot of this perpendicular.
00:48:37.820 --> 00:48:41.310
So I'm going to name that x.
00:48:41.310 --> 00:48:42.710
I'm going to give that letter x.
00:48:42.710 --> 00:48:44.480
Now, x is varying.
00:48:44.480 --> 00:48:47.359
The reason why I need a letter
for it as opposed to this 40
00:48:47.359 --> 00:48:49.150
is that it's going to
have a rate of change
00:48:49.150 --> 00:48:50.220
with respect to t.
00:48:50.220 --> 00:48:54.520
And, in fact, it's
related to-- the question
00:48:54.520 --> 00:49:00.432
is whether dx/dt is
faster or slower than 95.
00:49:00.432 --> 00:49:01.890
So that's the thing
that's varying.
00:49:01.890 --> 00:49:04.010
Now, there's something
else that's varying.
00:49:04.010 --> 00:49:06.000
This distance here
is also varying.
00:49:06.000 --> 00:49:08.760
So we need a letter for that.
00:49:08.760 --> 00:49:11.620
We do not need a
letter for this.
00:49:11.620 --> 00:49:12.920
Because it's never changing.
00:49:12.920 --> 00:49:15.010
We're assuming the
police are parked.
00:49:15.010 --> 00:49:17.382
They're not ready to roar
out and catch you just yet,
00:49:17.382 --> 00:49:18.840
and they're certainly
not in motion
00:49:18.840 --> 00:49:20.880
when they've got the
radar guns aimed at you.
00:49:20.880 --> 00:49:24.110
So you need to know
something about the sociology
00:49:24.110 --> 00:49:28.160
and style of police.
00:49:28.160 --> 00:49:30.950
So you need to know things
about the real world.
00:49:30.950 --> 00:49:36.145
Now, the last bit is,
what about this 80 here.
00:49:36.145 --> 00:49:37.770
So this is how fast
you're approaching.
00:49:37.770 --> 00:49:40.290
Now, that's measured
along the radar gun.
00:49:40.290 --> 00:49:44.950
I claim that that's d/dt
of this quantity here.
00:49:44.950 --> 00:49:46.910
So this is D is also changing.
00:49:46.910 --> 00:49:49.200
That's why we needed
a letter for it, too.
00:49:49.200 --> 00:49:51.730
So, next time, we'll just
put that all together
00:49:51.730 --> 00:49:55.010
and compute dx / dt.