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PROFESSOR: Today
we're going to keep
00:00:24.510 --> 00:00:30.660
on going with related rates.
00:00:30.660 --> 00:00:32.710
And you may recall
that last time we
00:00:32.710 --> 00:00:39.000
were in the middle of a
problem with this geometry.
00:00:39.000 --> 00:00:41.260
There was a right triangle.
00:00:41.260 --> 00:00:43.190
There was a road.
00:00:43.190 --> 00:00:47.000
Which was going this
way, from right to left.
00:00:47.000 --> 00:00:53.830
And the police were up here,
monitoring the situation.
00:00:53.830 --> 00:00:55.990
30 feet from the road.
00:00:55.990 --> 00:00:59.990
And you're here.
00:00:59.990 --> 00:01:04.214
And you're heading this way.
00:01:04.214 --> 00:01:06.130
Maybe it's a two lane
highway, but anyway it's
00:01:06.130 --> 00:01:07.680
only going this direction.
00:01:07.680 --> 00:01:11.670
And this distance was 50 feet.
00:01:11.670 --> 00:01:17.070
So, because you're moving,
this distance is varying
00:01:17.070 --> 00:01:19.120
and so we gave it a letter.
00:01:19.120 --> 00:01:22.460
And, similarly, your distance
to the foot of the perpendicular
00:01:22.460 --> 00:01:25.050
with the road is also varying.
00:01:25.050 --> 00:01:28.410
At this instant it's
40, because this is a 3,
00:01:28.410 --> 00:01:31.570
4, 5 right triangle.
00:01:31.570 --> 00:01:35.830
So this was the situation
that we were in last time.
00:01:35.830 --> 00:01:38.090
And we're going to pick
up where we left off.
00:01:38.090 --> 00:01:48.210
The question is,
are you speeding
00:01:48.210 --> 00:01:54.170
if the rate of change
of D with respect to t
00:01:54.170 --> 00:01:56.510
is 80 feet per second.
00:01:56.510 --> 00:01:59.290
Now, technically
that would be -80,
00:01:59.290 --> 00:02:03.030
because you're going
towards the policemen.
00:02:03.030 --> 00:02:10.550
Alright, so D is shrinking at
a rate of -80 feet per second.
00:02:10.550 --> 00:02:15.620
And I remind you that
95 feet per second
00:02:15.620 --> 00:02:17.660
is approximately
the speed limit.
00:02:17.660 --> 00:02:21.370
Which is 65 miles per hour.
00:02:21.370 --> 00:02:24.850
So, again, this is
where we were last time.
00:02:24.850 --> 00:02:30.150
And, got a little
question mark there.
00:02:30.150 --> 00:02:34.140
And so let's solve this problem.
00:02:34.140 --> 00:02:37.480
So, this is the setup.
00:02:37.480 --> 00:02:38.880
There's a right triangle.
00:02:38.880 --> 00:02:42.290
So there's a relationship
between these lengths.
00:02:42.290 --> 00:02:49.800
And the relationship is
that x^2 + 30^2 = D^2.
00:02:49.800 --> 00:02:53.730
So that's the first
relationship that we have.
00:02:53.730 --> 00:02:55.540
And the second
relationship that we have,
00:02:55.540 --> 00:02:57.380
we've already written down.
00:02:57.380 --> 00:03:01.540
Which is dx/dt - oops, sorry.
00:03:01.540 --> 00:03:05.540
dD/dt = -80.
00:03:08.620 --> 00:03:14.920
Now, the idea here is
relatively straightforward.
00:03:14.920 --> 00:03:17.790
We just want to use
differentiation.
00:03:17.790 --> 00:03:25.810
Now, you could solve for x.
00:03:25.810 --> 00:03:28.960
Alright, x is the square
root of D^2 - 30^2.
00:03:31.580 --> 00:03:32.820
That's one possibility.
00:03:32.820 --> 00:03:36.060
But this is basically
a waste of time.
00:03:36.060 --> 00:03:37.810
It's a waste of your time.
00:03:37.810 --> 00:03:44.340
So it's easier, or
easiest, to follow
00:03:44.340 --> 00:03:46.350
this method of implicit
differentiation,
00:03:46.350 --> 00:03:49.080
which I want to encourage
you to get used to.
00:03:49.080 --> 00:03:51.490
Namely, we just
differentiate this equation
00:03:51.490 --> 00:03:53.460
with respect to time.
00:03:53.460 --> 00:03:56.480
Now, when you do that, you have
to remember that you are not
00:03:56.480 --> 00:03:59.120
allowed to plug in a constant.
00:03:59.120 --> 00:04:00.857
Namely 40, for t.
00:04:00.857 --> 00:04:02.440
You have to keep in
mind what's really
00:04:02.440 --> 00:04:04.565
going on in this problem
which is that x is moving,
00:04:04.565 --> 00:04:05.670
it's changing.
00:04:05.670 --> 00:04:07.250
And D is also changing.
00:04:07.250 --> 00:04:10.090
So you have to
differentiate first
00:04:10.090 --> 00:04:12.030
before you plug in the values.
00:04:12.030 --> 00:04:15.160
So the easiest thing is
to use, in this case,
00:04:15.160 --> 00:04:20.130
implicit differentiation.
00:04:20.130 --> 00:04:33.000
And if I do that, I get 2x
dx/dt is equal to 2D dD/dt.
00:04:33.000 --> 00:04:34.910
No more DDT left.
00:04:34.910 --> 00:04:36.640
We hope.
00:04:36.640 --> 00:04:39.160
Except in this blackboard.
00:04:39.160 --> 00:04:40.690
So there's our situation.
00:04:40.690 --> 00:04:50.670
Now, if I just plug in,
now I can plug in values.
00:04:50.670 --> 00:04:52.820
So this is after
taking the derivative.
00:04:52.820 --> 00:04:54.820
And, indeed, we
have here 2 times
00:04:54.820 --> 00:04:57.500
the value for x which
is 40 at this instant.
00:04:57.500 --> 00:05:01.070
And then we have dx/dt.
00:05:01.070 --> 00:05:07.510
And that's equal to 2
times D, which is 50.
00:05:07.510 --> 00:05:12.900
And then dD/dt is -80.
00:05:12.900 --> 00:05:20.130
So the 80's cancel and we
see that dx/dt = -100 feet
00:05:20.130 --> 00:05:21.660
per second.
00:05:21.660 --> 00:05:26.985
And so the answer to
the question is yes.
00:05:26.985 --> 00:05:28.360
Although you
probably wouldn't be
00:05:28.360 --> 00:05:32.540
pulled over for this
much of a violation.
00:05:32.540 --> 00:05:36.240
So that's-- right, it's
more than 65 miles an hour,
00:05:36.240 --> 00:05:41.190
by a little bit.
00:05:41.190 --> 00:05:43.380
So that's the end
of this question.
00:05:43.380 --> 00:05:46.460
And usually in these rate
of change or related rates
00:05:46.460 --> 00:05:49.100
questions, this is considered to
be the answer to the question,
00:05:49.100 --> 00:05:53.530
if you like.
00:05:53.530 --> 00:05:55.600
So that's one example.
00:05:55.600 --> 00:05:57.890
I'm going to give
one more example
00:05:57.890 --> 00:06:01.285
before we go on to
some other applications
00:06:01.285 --> 00:06:04.560
of implicit differentiation.
00:06:04.560 --> 00:06:17.250
So my second example is going
to be, you have a conical tank.
00:06:17.250 --> 00:06:26.640
With top of radius
4 feet, let's say.
00:06:26.640 --> 00:06:31.770
And depth 10 feet.
00:06:31.770 --> 00:06:34.710
So that's our situation.
00:06:34.710 --> 00:06:37.730
And then it's being
filled with water.
00:06:37.730 --> 00:06:42.830
So, is being filled with water.
00:06:42.830 --> 00:06:49.900
At 2 cubic feet per minute.
00:06:49.900 --> 00:06:52.980
So there is our situation.
00:06:52.980 --> 00:07:05.750
And then the question is,
how fast is the water rising
00:07:05.750 --> 00:07:15.040
when it is at depth 5 feet?
00:07:15.040 --> 00:07:19.020
So if this thing is
half-full in the sense--
00:07:19.020 --> 00:07:21.770
well not half-full in
terms of total volume,
00:07:21.770 --> 00:07:25.640
but half-full in
terms of height.
00:07:25.640 --> 00:07:36.540
What's the speed at which
the water is rising.
00:07:36.540 --> 00:07:39.970
So, how do we set up
problems like this?
00:07:39.970 --> 00:07:42.240
Well, we talked
about this last time.
00:07:42.240 --> 00:07:51.780
The first step is to set
up a diagram and variables.
00:07:51.780 --> 00:07:54.450
And I'm just going
to draw the picture.
00:07:54.450 --> 00:07:58.320
And I'm actually going to
draw the picture twice.
00:07:58.320 --> 00:08:01.590
So here's the conical tank.
00:08:01.590 --> 00:08:05.040
And we have this
radius, which is 4.
00:08:05.040 --> 00:08:07.970
And we have this
height, which is 10.
00:08:07.970 --> 00:08:10.950
So that's just to allow me
to think about this problem.
00:08:10.950 --> 00:08:18.270
Now, it turns out because
we have a varying depth
00:08:18.270 --> 00:08:21.730
and so on, and this
is just the top.
00:08:21.730 --> 00:08:24.530
That I'd better depict also
the level at which the water
00:08:24.530 --> 00:08:26.700
actually is at present.
00:08:26.700 --> 00:08:30.710
And furthermore, it's better
to do this schematically,
00:08:30.710 --> 00:08:31.830
as you'll see.
00:08:31.830 --> 00:08:38.200
So the key point here is
to draw this triangle here.
00:08:38.200 --> 00:08:45.230
Which shows me the 10 and
shows me the 4, over here.
00:08:45.230 --> 00:08:48.290
And then imagine that
I'm filling it partway.
00:08:48.290 --> 00:08:52.400
So maybe we'll put that water
level in in another color here.
00:08:52.400 --> 00:08:54.370
So here's the water level.
00:08:54.370 --> 00:08:56.130
And the water level,
I'm going to depict
00:08:56.130 --> 00:08:59.659
that horizontal
distance here, as r.
00:08:59.659 --> 00:09:00.950
That's going to be my variable.
00:09:00.950 --> 00:09:04.150
That's the radius of
the top of the water.
00:09:04.150 --> 00:09:06.359
So I'm taking a
cross-section of this,
00:09:06.359 --> 00:09:08.150
because that geometrically
the only thing I
00:09:08.150 --> 00:09:10.230
have to keep track of.
00:09:10.230 --> 00:09:11.820
At least initially.
00:09:11.820 --> 00:09:14.630
So this is our water level.
00:09:14.630 --> 00:09:18.540
And it's really rotated around.
00:09:18.540 --> 00:09:23.130
But I'm depicting just this
one half-slice of the picture.
00:09:23.130 --> 00:09:26.300
And then similarly,
I have the height.
00:09:26.300 --> 00:09:30.390
Which is this dimension there.
00:09:30.390 --> 00:09:32.570
Or, if you like, the
depth of the water.
00:09:32.570 --> 00:09:42.140
So the water has filled us
up, up to this point here.
00:09:42.140 --> 00:09:46.050
So I set it up this
way so that it's clear
00:09:46.050 --> 00:09:48.830
that we have two triangles
here, and that one
00:09:48.830 --> 00:09:51.210
piece of information we
can get from the geometry
00:09:51.210 --> 00:09:53.880
is the similar
triangles information.
00:09:53.880 --> 00:09:59.870
Namely, that r / h = 4 / 10.
00:09:59.870 --> 00:10:09.760
So this is by far the most
typical geometric fact
00:10:09.760 --> 00:10:13.450
that you'll have to
glean from a picture.
00:10:13.450 --> 00:10:16.150
So that's one piece
of information
00:10:16.150 --> 00:10:18.920
that we get from this picture.
00:10:18.920 --> 00:10:20.420
Now, the second
thing we have to do
00:10:20.420 --> 00:10:25.622
is set up formulas for
the volume of water,
00:10:25.622 --> 00:10:27.330
and then figure out
what's going on here.
00:10:27.330 --> 00:10:33.460
So the volume of
water is-- of a cone.
00:10:33.460 --> 00:10:36.950
So again, you have to know
something about geometry
00:10:36.950 --> 00:10:38.540
to do many of these problems.
00:10:38.540 --> 00:10:41.190
So you have to know that
the volume of a cone
00:10:41.190 --> 00:10:44.500
is 1/3 base times height.
00:10:44.500 --> 00:10:45.900
Now, this one is upside down.
00:10:45.900 --> 00:10:49.130
The base is on the top
and it's going down
00:10:49.130 --> 00:10:50.720
but it works the same way.
00:10:50.720 --> 00:10:52.270
That doesn't affect volume.
00:10:52.270 --> 00:10:57.450
So it's 1/3, and the base
is pi r^2, that's the base,
00:10:57.450 --> 00:11:01.890
and times h, which
is the height.
00:11:01.890 --> 00:11:06.290
So this is the setup
for this problem.
00:11:06.290 --> 00:11:11.480
And now, having
our relationship,
00:11:11.480 --> 00:11:14.580
we have one relationship left
that we have to remember.
00:11:14.580 --> 00:11:17.630
Because we have one more piece
of information in this problem.
00:11:17.630 --> 00:11:20.990
Namely, how fast the
volume is changing.
00:11:20.990 --> 00:11:22.960
It's going at 2 cubic
feet per minute.
00:11:22.960 --> 00:11:29.740
It's increasing, so that
means that dV/dt = 2.
00:11:29.740 --> 00:11:33.970
So I've now gotten
rid of all the words
00:11:33.970 --> 00:11:38.010
and I have only formulas left.
00:11:38.010 --> 00:11:43.574
I started here with the
words, I drew some pictures,
00:11:43.574 --> 00:11:44.740
and I derived some formulas.
00:11:44.740 --> 00:11:46.240
Actually, there's
one thing missing.
00:11:46.240 --> 00:11:49.930
What's missing?
00:11:49.930 --> 00:11:52.570
Yeah.
00:11:52.570 --> 00:11:56.949
STUDENT: [INAUDIBLE]
00:11:56.949 --> 00:11:57.740
PROFESSOR: Exactly.
00:11:57.740 --> 00:11:58.656
What you want to find.
00:11:58.656 --> 00:12:00.690
What I left out is the question.
00:12:00.690 --> 00:12:15.110
So the question is, what
is dh(dt when h = 5?
00:12:15.110 --> 00:12:20.486
So that's the question here.
00:12:20.486 --> 00:12:22.360
Now, we've got the whole
problem and we never
00:12:22.360 --> 00:12:25.510
have to look at it
again if you like.
00:12:25.510 --> 00:12:32.610
We just have to pay
attention to this piece here.
00:12:32.610 --> 00:12:34.780
So let's carry it out.
00:12:34.780 --> 00:12:36.870
So what happens here.
00:12:36.870 --> 00:12:39.190
So look, you could do this
by implicit differentiation.
00:12:39.190 --> 00:12:42.830
But it's so easy to express
r as a function of h
00:12:42.830 --> 00:12:44.560
that that seems kind of foolish.
00:12:44.560 --> 00:12:48.900
So let's write r as 2/5 h.
00:12:48.900 --> 00:12:51.280
That's coming from this
first equation here.
00:12:51.280 --> 00:12:53.800
And then let's
substitute that in.
00:12:53.800 --> 00:12:58.800
That means that V =
1/3 pi (2/5 h)^2 h.
00:13:04.810 --> 00:13:06.630
And now I have to
differentiate that.
00:13:06.630 --> 00:13:08.850
So now I will use
implicit differentiation.
00:13:08.850 --> 00:13:12.030
It's very foolish at this
point to take cube roots
00:13:12.030 --> 00:13:13.320
to solve for h.
00:13:13.320 --> 00:13:16.080
You just get yourself
into a bunch of junk.
00:13:16.080 --> 00:13:18.180
So there is a stage at
which we're still using
00:13:18.180 --> 00:13:19.490
implicit differentiation here.
00:13:19.490 --> 00:13:24.772
I'm not going to try to solve
for h as a function of V.
00:13:24.772 --> 00:13:26.480
Instead I'm just
differentiating straight
00:13:26.480 --> 00:13:28.230
out from this formula,
which is relatively
00:13:28.230 --> 00:13:29.530
easy to differentiate.
00:13:29.530 --> 00:13:33.795
So this is dV/dt,
which of course is 2,
00:13:33.795 --> 00:13:37.440
is equal to, and if I
differentiate it I just get
00:13:37.440 --> 00:13:42.390
this constant, pi/3, this
other constant, (2/5)^2.
00:13:42.390 --> 00:13:45.200
And then I have to
differentiate h^3.
00:13:45.200 --> 00:13:47.290
(h^2)h.
00:13:47.290 --> 00:13:50.050
So that's 3h^2 dh/dt.
00:13:53.140 --> 00:13:59.760
That's the chain rule.
00:13:59.760 --> 00:14:02.080
So now let's plug in
all of our numbers.
00:14:02.080 --> 00:14:04.810
Again, the other
theme is, you don't
00:14:04.810 --> 00:14:08.300
plug in numbers, fixed
numbers, until everything
00:14:08.300 --> 00:14:09.600
has stopped moving.
00:14:09.600 --> 00:14:12.790
At this point, we've already
calculated our rates of change.
00:14:12.790 --> 00:14:14.140
So now I can put in the numbers.
00:14:14.140 --> 00:14:23.790
So, 2 is equal to pi/3 (2/5)^2
times 3, and then h was 5,
00:14:23.790 --> 00:14:26.010
so this is 5^2.
00:14:26.010 --> 00:14:27.870
And then I have dh/dt.
00:14:27.870 --> 00:14:31.840
There's only one unknown thing
left in this problem, which
00:14:31.840 --> 00:14:32.560
is dh/dt.
00:14:32.560 --> 00:14:33.990
Everything else is a number.
00:14:33.990 --> 00:14:35.650
And if you do all
the cancellations,
00:14:35.650 --> 00:14:38.210
you see that this cancels this.
00:14:38.210 --> 00:14:41.149
One of the 2's cancels -
well, this cancels this.
00:14:41.149 --> 00:14:42.690
And then one of the
2's cancels that.
00:14:42.690 --> 00:14:49.600
So all told what we have
here is that dh/dt = 1/2 pi.
00:14:54.080 --> 00:14:58.700
And so that happens
to be feet per second.
00:14:58.700 --> 00:15:04.597
This is the whole story.
00:15:04.597 --> 00:15:05.680
Questions, way back there.
00:15:05.680 --> 00:15:09.297
STUDENT: [INAUDIBLE]
00:15:09.297 --> 00:15:11.630
PROFESSOR: Where did I get--
the question is where did I
00:15:11.630 --> 00:15:14.590
get r = 2/5 h from.
00:15:14.590 --> 00:15:17.680
The answer was, it came
from similar triangles,
00:15:17.680 --> 00:15:18.964
which is over here.
00:15:18.964 --> 00:15:20.380
I did this similar
triangle thing.
00:15:20.380 --> 00:15:23.530
And I got this
relationship here.
00:15:23.530 --> 00:15:29.620
r/h = 4/10, but then
I canceled it, got 2/5
00:15:29.620 --> 00:15:31.724
and brought the h over.
00:15:31.724 --> 00:15:32.890
Another question, over here.
00:15:32.890 --> 00:15:45.470
STUDENT: [INAUDIBLE]
00:15:45.470 --> 00:15:53.100
PROFESSOR: The question
was the following.
00:15:53.100 --> 00:15:56.815
Suppose you're at this stage,
can you write from here dV/dh
00:15:56.815 --> 00:16:01.390
- so, suppose you're here -
and work out what this is.
00:16:01.390 --> 00:16:05.030
It's going to end up being
some constant times h^2.
00:16:05.030 --> 00:16:14.990
And then also use
dV/dt = dV/dh dh/dt.
00:16:14.990 --> 00:16:17.520
Which the chain rule.
00:16:17.520 --> 00:16:20.830
And the answer is yes.
00:16:20.830 --> 00:16:26.310
We can do that, and indeed that
is what my next sentence is.
00:16:26.310 --> 00:16:27.750
That's exactly what I'm saying.
00:16:27.750 --> 00:16:32.780
So when I said this-- sorry,
when you said this, I did that.
00:16:32.780 --> 00:16:34.080
That's exactly what I did.
00:16:34.080 --> 00:16:41.110
This chunk is exactly dV/dh.
00:16:41.110 --> 00:16:43.390
That's just what I'm doing.
00:16:43.390 --> 00:16:44.730
OK.
00:16:44.730 --> 00:16:48.750
So, definitely, that's
what I had in mind.
00:16:48.750 --> 00:16:49.760
Yeah, another question.
00:16:49.760 --> 00:16:55.224
STUDENT: [INAUDIBLE]
00:16:55.224 --> 00:16:57.390
PROFESSOR: You're asking
me whether my arithmetic is
00:16:57.390 --> 00:16:58.040
right or not?
00:16:58.040 --> 00:17:03.890
STUDENT: [INAUDIBLE]
00:17:03.890 --> 00:17:06.850
PROFESSOR: Pi - per second.
00:17:06.850 --> 00:17:07.350
Oh.
00:17:07.350 --> 00:17:09.480
This should - no, OK, right.
00:17:09.480 --> 00:17:11.474
I guess it's per minute.
00:17:11.474 --> 00:17:12.890
Since the other
one is per minute.
00:17:12.890 --> 00:17:14.400
Thank you.
00:17:14.400 --> 00:17:16.756
Yes.
00:17:16.756 --> 00:17:17.880
Was there another question?
00:17:17.880 --> 00:17:21.810
Probably also fixing
my seconds to minutes.
00:17:21.810 --> 00:17:23.150
Way back there.
00:17:23.150 --> 00:17:27.490
STUDENT: I don't understand,
why did you have to do all that.
00:17:27.490 --> 00:17:30.930
Isn't the speed 2
cubic feet per minute?
00:17:30.930 --> 00:17:33.000
PROFESSOR: The speed
at which you're
00:17:33.000 --> 00:17:35.920
filling it is 2 cubic feet,
but the water level is rising
00:17:35.920 --> 00:17:38.750
at a different rate, depending
on whether you're low
00:17:38.750 --> 00:17:40.350
down or high up.
00:17:40.350 --> 00:17:44.220
It depends on how wide
the pond, the surface, is.
00:17:44.220 --> 00:17:45.310
So in fact it's not.
00:17:45.310 --> 00:17:48.950
In fact, the answer wasn't
2 cubic-- it wasn't.
00:17:48.950 --> 00:17:49.900
There's a rate there.
00:17:49.900 --> 00:17:52.090
That is, that's how much
volume is being added.
00:17:52.090 --> 00:17:54.650
But then there's another number
that we're keeping track of,
00:17:54.650 --> 00:17:55.780
which is the height.
00:17:55.780 --> 00:17:58.790
Or, if you like, the
depth of the water.
00:17:58.790 --> 00:17:59.792
OK.
00:17:59.792 --> 00:18:01.750
So this is the whole
point about related rates.
00:18:01.750 --> 00:18:03.670
Is you have one
variable, which is V,
00:18:03.670 --> 00:18:04.882
which is volume of something.
00:18:04.882 --> 00:18:06.340
You have another
variable, which is
00:18:06.340 --> 00:18:10.675
h, which is the height of
the cone of water there.
00:18:10.675 --> 00:18:13.300
And you're keeping track of one
variable in terms of the other.
00:18:13.300 --> 00:18:15.990
And the relationship will
always be a chain rule
00:18:15.990 --> 00:18:16.866
type of relationship.
00:18:16.866 --> 00:18:18.573
So, therefore, you'll
be able to-- if you
00:18:18.573 --> 00:18:20.810
know one you'll be able
to figure out the other.
00:18:20.810 --> 00:18:22.864
Provided you get all
of the related rates.
00:18:22.864 --> 00:18:24.530
These are what are
called related rates.
00:18:24.530 --> 00:18:28.050
This is a rate of something with
respect to something, etc. etc.
00:18:28.050 --> 00:18:30.540
So it's really all
about the chain rule.
00:18:30.540 --> 00:18:38.540
And just fitting it
to word problems.
00:18:38.540 --> 00:18:41.900
So there's a couple of examples.
00:18:41.900 --> 00:18:44.430
And I'll let you
work out some more.
00:18:44.430 --> 00:18:50.100
So now, the next thing
that I want to do
00:18:50.100 --> 00:18:54.080
is to give you one
more max-min problem.
00:18:54.080 --> 00:18:58.670
Which has to do with
this device, which
00:18:58.670 --> 00:19:00.400
I brought with me.
00:19:00.400 --> 00:19:02.480
So this is a ring.
00:19:02.480 --> 00:19:04.150
Happens to be a
napkin ring, and this
00:19:04.150 --> 00:19:09.450
is some parachute string, which
I use when I go backpacking.
00:19:09.450 --> 00:19:15.382
And the question is
if you have a-- you
00:19:15.382 --> 00:19:17.090
think of this is a
weight, it's flexible.
00:19:17.090 --> 00:19:19.040
It's allowed to move along here.
00:19:19.040 --> 00:19:23.480
And the question is, if I fix
these two ends where my two
00:19:23.480 --> 00:19:25.290
hands are, where my
right hand is here
00:19:25.290 --> 00:19:28.410
and where my left
hand is over there.
00:19:28.410 --> 00:19:32.500
And the question is, where
does this ring settle down.
00:19:32.500 --> 00:19:35.130
Now, it's obvious, or
should be maybe obvious,
00:19:35.130 --> 00:19:38.020
is that if my two hands
are at equal heights,
00:19:38.020 --> 00:19:41.232
it should settle in the middle.
00:19:41.232 --> 00:19:42.940
The question that
we're trying to resolve
00:19:42.940 --> 00:19:47.290
is what if one hand is a
little higher than the other.
00:19:47.290 --> 00:19:49.040
What happens, or
if the other way.
00:19:49.040 --> 00:19:52.370
Where does it settle down?
00:19:52.370 --> 00:19:55.210
So in order to depict
this problem properly,
00:19:55.210 --> 00:19:57.480
I need two volunteers
to help me out.
00:19:57.480 --> 00:20:00.400
Can I have some help?
00:20:00.400 --> 00:20:03.370
OK.
00:20:03.370 --> 00:20:06.320
So I need one of you to hold
the right side, and one of you
00:20:06.320 --> 00:20:08.460
to hold the left side.
00:20:08.460 --> 00:20:09.260
OK.
00:20:09.260 --> 00:20:11.180
And just hold it
against the blackboard.
00:20:11.180 --> 00:20:12.240
We're going to trace.
00:20:12.240 --> 00:20:15.335
So stick it about here,
in the middle somewhere.
00:20:15.335 --> 00:20:17.210
And now we want to make
sure that this one is
00:20:17.210 --> 00:20:18.960
a little higher, all right?
00:20:18.960 --> 00:20:22.430
So we'll have to-- yeah,
let's get a little higher up.
00:20:22.430 --> 00:20:25.730
That's probably good enough.
00:20:25.730 --> 00:20:29.770
So the experiment has been done.
00:20:29.770 --> 00:20:31.100
We now see where this thing is.
00:20:31.100 --> 00:20:33.810
But, so now hold on tight.
00:20:33.810 --> 00:20:35.960
This thing stretches.
00:20:35.960 --> 00:20:38.050
So we want to get it
stretched so that we
00:20:38.050 --> 00:20:40.667
can see what it is properly.
00:20:40.667 --> 00:20:43.000
So this thing isn't heavy
enough for this demonstration.
00:20:43.000 --> 00:20:46.480
I should've had a ten-ton
brick attached there.
00:20:46.480 --> 00:20:49.830
But if I did that,
than I would tax my,
00:20:49.830 --> 00:20:52.310
right, I would tax your
abilities to-- right,
00:20:52.310 --> 00:20:55.400
so we're going to try to trace
out what the possibilities are
00:20:55.400 --> 00:21:03.810
here.
00:21:03.810 --> 00:21:06.560
So this is, roughly
speaking, where this thing--
00:21:06.560 --> 00:21:07.960
and so now where does it settle.
00:21:07.960 --> 00:21:10.550
Well, it settles about here.
00:21:10.550 --> 00:21:12.630
So we're going to put
that as X marks the spot.
00:21:12.630 --> 00:21:15.900
OK, thank you very much.
00:21:15.900 --> 00:21:20.960
Got to remember where
those dots-- OK, all right.
00:21:20.960 --> 00:21:24.040
Sorry, I forgot
to mark the spots.
00:21:24.040 --> 00:21:27.410
OK, so here's the situation.
00:21:27.410 --> 00:21:29.860
We have a problem.
00:21:29.860 --> 00:21:33.170
And we've hung a string.
00:21:33.170 --> 00:21:39.630
And it went down like this
and then it went like that.
00:21:39.630 --> 00:21:43.710
And we discovered
where it settled.
00:21:43.710 --> 00:21:44.750
Physically.
00:21:44.750 --> 00:21:47.820
So we want to explain this
mathematically, and see what's
00:21:47.820 --> 00:21:49.280
going on with this problem.
00:21:49.280 --> 00:21:50.940
So this is a real-life problem.
00:21:50.940 --> 00:21:52.850
It honestly is the
problem you have to solve
00:21:52.850 --> 00:21:54.100
if you want to build a bridge.
00:21:54.100 --> 00:21:55.570
Like, a suspension bridge.
00:21:55.570 --> 00:21:57.010
This, among many problems.
00:21:57.010 --> 00:21:59.350
It's a very serious
and important problem.
00:21:59.350 --> 00:22:02.940
And this is the simplest
one of this type.
00:22:02.940 --> 00:22:04.990
So we've got our shape here.
00:22:04.990 --> 00:22:08.255
This should be a straight
line, maybe not quite as angled
00:22:08.255 --> 00:22:10.920
as that.
00:22:10.920 --> 00:22:13.030
The first, we've already
drawn the diagram
00:22:13.030 --> 00:22:14.670
and we've more or
less visualized
00:22:14.670 --> 00:22:16.010
what's going on here.
00:22:16.010 --> 00:22:21.460
But the first step after the
diagram is to give letters.
00:22:21.460 --> 00:22:24.630
Is to label things
appropriately.
00:22:24.630 --> 00:22:29.170
And I do not expect you to be
able to do this, at this stage.
00:22:29.170 --> 00:22:31.390
Because it requires
a lot of experience.
00:22:31.390 --> 00:22:33.040
But I'm going to do it for you.
00:22:33.040 --> 00:22:34.630
We're going to just do that.
00:22:34.630 --> 00:22:37.300
So the simplest
thing to do is to use
00:22:37.300 --> 00:22:39.800
the coordinates of the plane.
00:22:39.800 --> 00:22:43.750
And if you do that, it's also
easiest to use the origin.
00:22:43.750 --> 00:22:46.590
My favorite number is 0 and
it should be yours, too.
00:22:46.590 --> 00:22:51.420
So we're going to make
this point be (0, 0).
00:22:51.420 --> 00:22:54.550
Now, there's another fixed
point in this problem.
00:22:54.550 --> 00:22:57.089
And it's this point over here.
00:22:57.089 --> 00:22:58.880
And we don't know what
its coordinates are.
00:22:58.880 --> 00:23:00.850
So we're just going to
give them letters, a and b.
00:23:00.850 --> 00:23:02.225
But those letters
are going to be
00:23:02.225 --> 00:23:06.110
fixed numbers in this problem.
00:23:06.110 --> 00:23:08.840
And we want to solve it for
all possible a's and b's.
00:23:08.840 --> 00:23:10.470
Now, the interesting
thing, remember,
00:23:10.470 --> 00:23:12.630
is what happens when b Is not 0.
00:23:12.630 --> 00:23:15.825
If b = 0, we already have
a clue that the point
00:23:15.825 --> 00:23:16.950
should be the center point.
00:23:16.950 --> 00:23:19.190
It should be exactly
that X, the middle point,
00:23:19.190 --> 00:23:22.730
which I'm going to label in a
second, is halfway in between.
00:23:22.730 --> 00:23:25.930
So now the variable point that
I'm going to use is down here.
00:23:25.930 --> 00:23:30.220
I'm going to call
this point (x, y).
00:23:30.220 --> 00:23:31.600
So here's my setup.
00:23:31.600 --> 00:23:37.800
I've now given labels to all the
things on the diagram so far.
00:23:37.800 --> 00:23:42.260
Most of the things
on the diagram.
00:23:42.260 --> 00:23:47.700
So now, what else
do I have to do?
00:23:47.700 --> 00:23:55.920
Well, I have to explain to you
that this is a minimization
00:23:55.920 --> 00:23:56.850
problem.
00:23:56.850 --> 00:23:59.010
What happens,
actually, physically
00:23:59.010 --> 00:24:03.050
is that the weight settles
to the lowest point.
00:24:03.050 --> 00:24:05.820
That's the thing that has
the lowest potential energy.
00:24:05.820 --> 00:24:09.370
So we're minimizing a function.
00:24:09.370 --> 00:24:13.260
And it's this curve here.
00:24:13.260 --> 00:24:17.000
The constraint is that we're
restricted to this curve.
00:24:17.000 --> 00:24:18.660
So this is a constraint curve.
00:24:18.660 --> 00:24:24.590
And we want the lowest
point of this curve.
00:24:24.590 --> 00:24:29.040
So now, we need a little
bit more language in order
00:24:29.040 --> 00:24:31.180
to describe what it
is that we've got.
00:24:31.180 --> 00:24:36.500
And the constraint curve, we
got it in a particular way.
00:24:36.500 --> 00:24:39.320
Namely, we strung some
string from here to there.
00:24:39.320 --> 00:24:41.090
And what happens
at all these points
00:24:41.090 --> 00:24:45.920
is that the total length
of the string is the same.
00:24:45.920 --> 00:24:48.530
So one way of expressing
the constraint
00:24:48.530 --> 00:24:52.270
is that the length of
the string is constant.
00:24:52.270 --> 00:24:54.520
And so in order to figure
out what the constraint is,
00:24:54.520 --> 00:24:58.120
what this curve is, I have to
describe that analytically.
00:24:58.120 --> 00:25:01.830
And I'm going to do that by
drawing in some helping lines.
00:25:01.830 --> 00:25:05.400
Namely, some right triangles to
figure out what this length is.
00:25:05.400 --> 00:25:07.290
And what the other length is.
00:25:07.290 --> 00:25:11.410
So this length is pretty easy
if I draw a right triangle here.
00:25:11.410 --> 00:25:14.410
Because we go over
x and we go down y.
00:25:14.410 --> 00:25:19.150
So this length is the
square root of x^2 + y^2.
00:25:19.150 --> 00:25:22.420
That's the Pythagorean theorem.
00:25:22.420 --> 00:25:26.200
Similarly, over here, I'm
going to get another length.
00:25:26.200 --> 00:25:28.360
Which is a little bit of a mess.
00:25:28.360 --> 00:25:30.670
It's the vertical.
00:25:30.670 --> 00:25:35.290
So I'm just going to label one
half of it, so that you see.
00:25:35.290 --> 00:25:38.600
So this horizontal
distance is x.
00:25:38.600 --> 00:25:41.180
And this horizontal
distance from this top
00:25:41.180 --> 00:25:44.000
point with this right
angle, over there.
00:25:44.000 --> 00:25:47.660
It starts at x and ends at a.
00:25:47.660 --> 00:25:50.820
The right-most point is
a in the x-coordinate.
00:25:50.820 --> 00:25:56.150
So the whole distance is a - x.
00:25:56.150 --> 00:26:00.700
So that's this leg of
this right triangle.
00:26:00.700 --> 00:26:05.770
And, similarly, the vertical
distance will be b - y.
00:26:05.770 --> 00:26:08.300
And so, the formula here,
which is a little complicated
00:26:08.300 --> 00:26:13.770
for this length, is the square
root of (a-x)^2 + (b-y)^2.
00:26:17.160 --> 00:26:20.502
So here are the
two formulas that
00:26:20.502 --> 00:26:22.460
are going to allow me to
set up my problem now.
00:26:22.460 --> 00:26:25.240
So, my goal is to set it
up the way I did here, just
00:26:25.240 --> 00:26:26.940
with formulas.
00:26:26.940 --> 00:26:40.590
And not with diagrams
and not with names.
00:26:40.590 --> 00:26:42.910
So here's what I'd like to do.
00:26:42.910 --> 00:26:46.840
I claim that what's constrained,
if I'm along that curve,
00:26:46.840 --> 00:26:48.910
is that the total
length is constant.
00:26:48.910 --> 00:26:50.730
So that's this statement here.
00:26:50.730 --> 00:26:55.830
The square root of x^2 + y^2
plus this other square root.
00:26:55.830 --> 00:27:02.360
These are the two
lengths of string.
00:27:02.360 --> 00:27:08.210
Is equal to some number,
L, which is constant.
00:27:08.210 --> 00:27:15.790
And this, as I said, is what
I'm calling my constraint.
00:27:15.790 --> 00:27:16.290
Yeah.
00:27:16.290 --> 00:27:20.480
STUDENT: [INAUDIBLE]
00:27:20.480 --> 00:27:24.750
PROFESSOR: So the question
is, shouldn't it be b+y.
00:27:24.750 --> 00:27:28.210
No, and the reason is that
y is a negative number.
00:27:28.210 --> 00:27:30.300
It's below 0.
00:27:30.300 --> 00:27:39.210
So it's actually the sum,
-y is a positive number.
00:27:39.210 --> 00:27:43.840
All right, so
here's the formula.
00:27:43.840 --> 00:27:53.000
And then, we want to find
the minimum of something.
00:27:53.000 --> 00:27:56.250
So what is it that we're
finding the minimum of?
00:27:56.250 --> 00:27:58.370
This is actually the
hardest part of the problem,
00:27:58.370 --> 00:27:59.030
conceptually.
00:27:59.030 --> 00:28:02.650
I tried to prepare it, but it's
very hard to figure this out.
00:28:02.650 --> 00:28:07.540
We're finding the least what?
00:28:07.540 --> 00:28:10.740
It's y.
00:28:10.740 --> 00:28:11.830
We got a name for that.
00:28:11.830 --> 00:28:17.220
So we want to find the lowest y.
00:28:17.220 --> 00:28:19.690
Now, the reason why it
seems a little weird
00:28:19.690 --> 00:28:21.770
is you might think of y
as just being a variable.
00:28:21.770 --> 00:28:25.180
But really, y is a function.
00:28:25.180 --> 00:28:29.920
It's really y = y(x)
is defined implicitly
00:28:29.920 --> 00:28:34.930
by the constraint equation.
00:28:34.930 --> 00:28:36.420
That's what that curve is.
00:28:36.420 --> 00:28:40.250
And notice the bottom
point is exactly
00:28:40.250 --> 00:28:46.160
the place where the tangent
line will be horizontal.
00:28:46.160 --> 00:28:48.180
Which is just what we want.
00:28:48.180 --> 00:29:03.600
So from the diagram, the
bottom point is where y' = 0.
00:29:03.600 --> 00:29:15.195
So this is the critical point.
00:29:15.195 --> 00:29:25.720
Yeah?
00:29:25.720 --> 00:29:27.540
STUDENT: [INAUDIBLE]
00:29:27.540 --> 00:29:28.440
PROFESSOR: Exactly.
00:29:28.440 --> 00:29:31.780
So I'm deriving for
you-- so the question
00:29:31.780 --> 00:29:35.590
is, could I have just tried
to find y' = 0 to begin with.
00:29:35.590 --> 00:29:37.320
The answer is yes, absolutely.
00:29:37.320 --> 00:29:39.070
And in fact I'm leading
in that direction.
00:29:39.070 --> 00:29:41.820
I'm just showing
you, so I'm trying
00:29:41.820 --> 00:29:43.930
to make the following,
very subtle, point.
00:29:43.930 --> 00:29:48.500
Which is in
maximum-minimum problems,
00:29:48.500 --> 00:29:51.130
we always have to keep
track of two things.
00:29:51.130 --> 00:29:54.924
Often the interesting point
is the critical point.
00:29:54.924 --> 00:29:56.840
And that indeed turns
out to be the case here.
00:29:56.840 --> 00:29:59.840
But we always have
to check the ends.
00:29:59.840 --> 00:30:02.125
And so there are several
ways of checking the ends.
00:30:02.125 --> 00:30:03.666
One is, we did this
physical problem.
00:30:03.666 --> 00:30:05.410
We can see that
it's coming up here.
00:30:05.410 --> 00:30:06.910
We can see that
it's coming up here.
00:30:06.910 --> 00:30:10.480
Therefore the bottom must
be at this critical point.
00:30:10.480 --> 00:30:14.140
So that's OK, so that's
one way of checking it.
00:30:14.140 --> 00:30:17.872
Another way of checking it is
the reasoning that I just gave.
00:30:17.872 --> 00:30:19.330
But it's really
the same reasoning.
00:30:19.330 --> 00:30:20.650
I'm pointing to
this thing and I'm
00:30:20.650 --> 00:30:23.010
showing you that the bottom
is somewhere in the middle.
00:30:23.010 --> 00:30:26.470
So, therefore, it is a place
of horizontal tangency.
00:30:26.470 --> 00:30:28.970
That's the reasoning
that I'm using.
00:30:28.970 --> 00:30:30.510
So, again, this
is to avoid having
00:30:30.510 --> 00:30:33.775
to evaluate a limit
of an end or to use
00:30:33.775 --> 00:30:35.150
the second derivative
test, which
00:30:35.150 --> 00:30:42.310
is a total catastrophe
in this case.
00:30:42.310 --> 00:30:45.470
OK, now.
00:30:45.470 --> 00:30:47.530
There's one other
thing that you might
00:30:47.530 --> 00:30:51.081
know about this if you've seen
this geometric construction
00:30:51.081 --> 00:30:51.580
before.
00:30:51.580 --> 00:30:54.300
With a string and chalk.
00:30:54.300 --> 00:30:57.917
Which is that this
curve is an eclipse.
00:30:57.917 --> 00:30:59.750
It turns out, this is
a piece of an eclipse.
00:30:59.750 --> 00:31:00.870
It's a huge ellipse.
00:31:00.870 --> 00:31:03.580
These two points turn out
to be the so-called foci
00:31:03.580 --> 00:31:05.500
of the ellipse.
00:31:05.500 --> 00:31:09.530
However, that geometric fact
is totally useless for solving
00:31:09.530 --> 00:31:10.860
this problem.
00:31:10.860 --> 00:31:12.390
Completely useless.
00:31:12.390 --> 00:31:14.850
If you actually write out
the formulas for the ellipse,
00:31:14.850 --> 00:31:17.130
you'll discover that you
have a much harder problem
00:31:17.130 --> 00:31:18.090
on your hands.
00:31:18.090 --> 00:31:20.150
And it will take you
twice or ten times
00:31:20.150 --> 00:31:23.720
as long, so, it's true
that it's an ellipse,
00:31:23.720 --> 00:31:25.670
but it doesn't help.
00:31:25.670 --> 00:31:29.510
OK, so what we're going to
do instead is much simpler.
00:31:29.510 --> 00:31:32.390
We're going to leave
this expression alone
00:31:32.390 --> 00:31:35.480
and we're just going to
differentiate implicitly.
00:31:35.480 --> 00:31:42.430
So again, we use
implicit differentiation
00:31:42.430 --> 00:31:49.430
on the constraint equation.
00:31:49.430 --> 00:31:52.380
So that's the equation which
is directly above me there,
00:31:52.380 --> 00:31:54.310
at the top.
00:31:54.310 --> 00:31:57.470
And I have to differentiate
it with respect to x.
00:31:57.470 --> 00:31:59.190
So that's a little
ugly, but we've
00:31:59.190 --> 00:32:01.080
done this a few times before.
00:32:01.080 --> 00:32:02.940
When you differentiate
a square root,
00:32:02.940 --> 00:32:08.230
the square root goes
into the denominator.
00:32:08.230 --> 00:32:11.500
And there's a factor of 1/2,
so there's a 2x which cancels.
00:32:11.500 --> 00:32:12.750
So I claim it's this.
00:32:12.750 --> 00:32:17.180
Now, because y depends on
x, there's also a y y' here.
00:32:17.180 --> 00:32:22.170
So technically speaking,
it's twice this with a half.
00:32:22.170 --> 00:32:24.310
2/2 times that.
00:32:24.310 --> 00:32:30.790
So that's the differentiation
of the first piece of this guy.
00:32:30.790 --> 00:32:32.360
Now I'm going to
do this second one,
00:32:32.360 --> 00:32:34.060
and it's also the chain rule.
00:32:34.060 --> 00:32:39.150
But you're just going to
have to let me do it for you.
00:32:39.150 --> 00:32:41.690
Because it's just a little
bit too long for you
00:32:41.690 --> 00:32:43.847
to pay attention to.
00:32:43.847 --> 00:32:45.430
It turns out there's
a minus sign that
00:32:45.430 --> 00:32:49.710
comes out, because there's
a - x and a - y there.
00:32:49.710 --> 00:32:51.940
And then the numerator,
the denominator
00:32:51.940 --> 00:32:55.100
is the same massive square root.
00:32:55.100 --> 00:32:59.920
So it's (a - (a-x)^2, (b-y)^2.
00:32:59.920 --> 00:33:05.670
And the numerator
is a - x, which is
00:33:05.670 --> 00:33:07.750
what replaces the x over here.
00:33:07.750 --> 00:33:13.660
And then another term,
which is (b-y)y'.
00:33:13.660 --> 00:33:17.140
I claim that that's analogous
to what I did in the first term.
00:33:17.140 --> 00:33:20.086
And you'll just have to
check this on your own.
00:33:20.086 --> 00:33:21.460
Because I did it
too fast for you
00:33:21.460 --> 00:33:23.607
to be able to check yourself.
00:33:23.607 --> 00:33:26.190
Now, that's going to be equal
to what, on the right-hand side?
00:33:26.190 --> 00:33:29.510
What's the derivative
of L with respect to x?
00:33:29.510 --> 00:33:30.540
It's 0.
00:33:30.540 --> 00:33:32.300
It's not changing
in the problem.
00:33:32.300 --> 00:33:37.630
Although my parachute
stuff stretches.
00:33:37.630 --> 00:33:40.700
We tried to stretch it
to its fullest extent.
00:33:40.700 --> 00:33:45.100
So that we kept it fixed,
that was the goal here.
00:33:45.100 --> 00:33:50.380
So now, this looks
like a total mess.
00:33:50.380 --> 00:33:52.940
But, it's not.
00:33:52.940 --> 00:33:54.440
And let me show you why.
00:33:54.440 --> 00:33:57.110
It simplifies a great deal.
00:33:57.110 --> 00:34:01.140
And let me show you exactly how.
00:34:01.140 --> 00:34:03.400
So, first of all,
the whole point
00:34:03.400 --> 00:34:06.860
is we're looking for
the place where y' = 0.
00:34:06.860 --> 00:34:14.080
So that means that these
terms go away. y' = 0.
00:34:14.080 --> 00:34:16.690
So they're gone.
00:34:16.690 --> 00:34:21.070
And now what we have is
the following equation.
00:34:21.070 --> 00:34:27.810
It's x divided by square root of
x^2 + y^2 is equal to, if I put
00:34:27.810 --> 00:34:32.450
it on the other side the minus
sign is changed to a plus sign,
00:34:32.450 --> 00:34:37.470
a - x divided by this
other massive object,
00:34:37.470 --> 00:34:39.020
(a-x)^2 + (b-y)^2.
00:34:43.300 --> 00:34:45.510
So this is what
it simplifies to.
00:34:45.510 --> 00:34:51.020
Now again, that also might
look complicated to you.
00:34:51.020 --> 00:34:55.890
But I claim that
this is something,
00:34:55.890 --> 00:34:59.620
this is a kind of
equilibrium equation
00:34:59.620 --> 00:35:01.980
that can be interpreted
geometrically,
00:35:01.980 --> 00:35:05.360
in a way that is very
meaningful and important.
00:35:05.360 --> 00:35:09.330
So first of all, let me
observe for you that this x is
00:35:09.330 --> 00:35:10.730
something on our picture.
00:35:10.730 --> 00:35:14.160
And the square root of x^2 +
y^2 is something on our picture.
00:35:14.160 --> 00:35:18.870
Namely, if I go over to
the picture, here was x
00:35:18.870 --> 00:35:20.760
and this was a right triangle.
00:35:20.760 --> 00:35:25.700
And this hypotenuse was
square root of x^2 + y^2.
00:35:25.700 --> 00:35:34.200
So, if I call this angle alpha,
then this is the sine of alpha.
00:35:34.200 --> 00:35:34.700
Right?
00:35:34.700 --> 00:35:36.950
It's a right triangle,
that's the opposite leg.
00:35:36.950 --> 00:35:44.940
So this guy is
the sine of alpha.
00:35:44.940 --> 00:35:48.510
Similarly, the other side
has an interpretation
00:35:48.510 --> 00:35:50.800
for the other right triangle.
00:35:50.800 --> 00:35:56.150
If this angle is beta, then
the opposite side is a-x,
00:35:56.150 --> 00:36:00.850
and the hypotenuse is what was
in the denominator over there.
00:36:00.850 --> 00:36:10.130
So this side is sine of beta.
00:36:10.130 --> 00:36:14.080
And so what this
condition is telling us
00:36:14.080 --> 00:36:19.230
is that alpha = beta.
00:36:19.230 --> 00:36:22.620
Which is the hidden
symmetry in the problem.
00:36:22.620 --> 00:36:24.940
I don't know if you
can actually see it
00:36:24.940 --> 00:36:28.490
when I show you this thing.
00:36:28.490 --> 00:36:32.920
But, no matter how I tilt
it, actually the two angles
00:36:32.920 --> 00:36:39.330
from the horizontal
are the same.
00:36:39.330 --> 00:36:40.920
In the middle it's
kind of obvious
00:36:40.920 --> 00:36:42.540
that that should be the case.
00:36:42.540 --> 00:36:44.360
But on the sides
it's not obvious
00:36:44.360 --> 00:36:46.820
that that's what's happening.
00:36:46.820 --> 00:36:49.882
Now, this has even-- so that's
a symmetry, if you like,
00:36:49.882 --> 00:36:50.590
of the situation.
00:36:50.590 --> 00:36:52.590
These two angles are equal.
00:36:52.590 --> 00:36:55.440
But there's something
more to be said.
00:36:55.440 --> 00:36:58.290
If you do a force diagram for
this, what you'll discover
00:36:58.290 --> 00:37:03.680
is that the tension on
the two lines is the same.
00:37:03.680 --> 00:37:05.690
Which means that when
you build something
00:37:05.690 --> 00:37:11.350
which is hanging like this, it
will involve the least stress.
00:37:11.350 --> 00:37:13.260
If you hang
something very heavy,
00:37:13.260 --> 00:37:15.790
and one side carries twice
as much load as the other,
00:37:15.790 --> 00:37:18.060
then you have twice as
much chance of its falling
00:37:18.060 --> 00:37:19.260
and breaking.
00:37:19.260 --> 00:37:22.530
If they each hold
the same strength,
00:37:22.530 --> 00:37:26.320
then you've distributed the load
in a much more balanced way.
00:37:26.320 --> 00:37:28.660
So this is a kind of
a balance condition,
00:37:28.660 --> 00:37:31.740
and it's very typical of
minimization problems.
00:37:31.740 --> 00:37:34.040
And fortunately, there
are nice solutions
00:37:34.040 --> 00:37:37.020
which distribute the
weight reasonably well.
00:37:37.020 --> 00:37:41.290
That's certainly the principle
of suspension bridges.
00:37:41.290 --> 00:37:42.820
So yeah, one more question.
00:37:42.820 --> 00:37:44.760
STUDENT: [INAUDIBLE]
00:37:44.760 --> 00:37:48.767
PROFESSOR: OK, so
the question where
00:37:48.767 --> 00:37:50.600
it hangs, that is, what
the formula for x is
00:37:50.600 --> 00:37:53.030
and what the formula for y is.
00:37:53.030 --> 00:37:55.900
And other things, like the
equation for the ellipse
00:37:55.900 --> 00:37:58.310
and lots of other
stuff like that.
00:37:58.310 --> 00:38:01.950
Those are things that I
will answer for you in a set
00:38:01.950 --> 00:38:03.700
of notes which I will hand out.
00:38:03.700 --> 00:38:06.310
And they're just a mess.
00:38:06.310 --> 00:38:08.400
You see they're, just
as in that other problem
00:38:08.400 --> 00:38:11.604
that we did last time, there's
some illuminating things you
00:38:11.604 --> 00:38:13.020
can say about the
problem and then
00:38:13.020 --> 00:38:15.890
there's some messy formulas.
00:38:15.890 --> 00:38:19.222
You know, you want to try to
pick out the simple things
00:38:19.222 --> 00:38:19.930
that you can say.
00:38:19.930 --> 00:38:21.740
In fact, that's a
property of math,
00:38:21.740 --> 00:38:24.500
you want to focus on the
more comprehensible things.
00:38:24.500 --> 00:38:26.770
On the other hand,
it can be done.
00:38:26.770 --> 00:38:30.290
It just takes a little
bit of computation.
00:38:30.290 --> 00:38:33.630
So I didn't answer the question
of what the lowest y was.
00:38:33.630 --> 00:38:39.020
But I'll do that for you.
00:38:39.020 --> 00:38:42.284
Maybe I'll just mention one
more thing about this problem.
00:38:42.284 --> 00:38:44.700
This is a very amusing problem
from a completely different
00:38:44.700 --> 00:38:45.890
point of view.
00:38:45.890 --> 00:38:50.280
If you sort of roll
the ellipse around,
00:38:50.280 --> 00:38:55.580
you get the same phenomenon
from each place here.
00:38:55.580 --> 00:38:58.170
So it doesn't matter
where a and 0 are.
00:38:58.170 --> 00:38:59.490
You'll get the same phenomenon.
00:38:59.490 --> 00:39:02.040
That is, the tangent line.
00:39:02.040 --> 00:39:06.030
So, this angle and that
angle will be equal.
00:39:06.030 --> 00:39:09.934
So you can also read that
as being the angle over here
00:39:09.934 --> 00:39:11.350
and the angle over
here are equal.
00:39:11.350 --> 00:39:15.730
That is, beta-- that is
the complementary angles
00:39:15.730 --> 00:39:17.290
are also equal.
00:39:17.290 --> 00:39:21.170
And if you interpret
this as a ray of light,
00:39:21.170 --> 00:39:23.540
and this as a mirror,
then this would
00:39:23.540 --> 00:39:27.040
be saying that if you start at
one focus, every ray of light
00:39:27.040 --> 00:39:29.310
will bounce and go
to the other focus.
00:39:29.310 --> 00:39:32.690
So that's a property
that an ellipse has.
00:39:32.690 --> 00:39:36.460
More precisely, a property
that this kind of curve has.
00:39:36.460 --> 00:39:39.080
And in fact, a few
years ago there
00:39:39.080 --> 00:39:43.590
was this great piece
of art at something
00:39:43.590 --> 00:39:46.330
called the DeCordova Museum,
which I recommended very highly
00:39:46.330 --> 00:39:52.460
to you go sometime to visit
in your four years here.
00:39:52.460 --> 00:39:59.000
There was a collection
of miniature golf holes.
00:39:59.000 --> 00:40:02.070
So they had a bunch of
mini golf pieces of art.
00:40:02.070 --> 00:40:04.320
And every one was completely
different from the other.
00:40:04.320 --> 00:40:07.910
And one of them was
called hole in one.
00:40:07.910 --> 00:40:11.110
And the tee was at one
focus of the ellipse.
00:40:11.110 --> 00:40:15.390
And the hole was at the
other focus of the ellipse.
00:40:15.390 --> 00:40:18.660
So, no matter how you
hit the golf ball,
00:40:18.660 --> 00:40:21.790
it always goes into the hole.
00:40:21.790 --> 00:40:24.595
No matter where it bounces,
it just, one bounce
00:40:24.595 --> 00:40:26.820
and it's in the hole.
00:40:26.820 --> 00:40:30.550
So that's actually a
consequence of the computation
00:40:30.550 --> 00:40:40.260
that we just did.
00:40:40.260 --> 00:40:42.170
Time to go on.
00:40:42.170 --> 00:41:06.150
We're going to now talk
about something else.
00:41:06.150 --> 00:41:10.990
So our next topic
is Newton's method.
00:41:10.990 --> 00:41:24.020
Which is one of the greatest
applications of calculus.
00:41:24.020 --> 00:41:27.610
And I'm going to
describe it here for you.
00:41:27.610 --> 00:41:34.870
And we'll illustrate
it on an example,
00:41:34.870 --> 00:41:39.970
which is solving the
equation x^2 = 5.
00:41:39.970 --> 00:41:42.000
We're going to find
the square root of 5.
00:41:42.000 --> 00:41:45.520
Now, you can actually solve
any equation this way.
00:41:45.520 --> 00:41:46.910
Any equation that
you understand,
00:41:46.910 --> 00:41:49.822
you can solve this
way, essentially.
00:41:49.822 --> 00:41:52.280
So even though I'm doing it
for the square root of 5, which
00:41:52.280 --> 00:41:56.990
is something that you can
figure out on your calculator,
00:41:56.990 --> 00:41:59.000
in fact this is
really at the heart
00:41:59.000 --> 00:42:02.210
of many of the ways in
which calculators work.
00:42:02.210 --> 00:42:04.880
So, the first thing is to make
this problem a little bit more
00:42:04.880 --> 00:42:06.420
abstract.
00:42:06.420 --> 00:42:13.830
We're going to set
f(x) = x^2 - 5.
00:42:13.830 --> 00:42:19.840
And then we're going
to solve f(x) = 0.
00:42:19.840 --> 00:42:24.120
So this is the sort of standard
form for solving such a--
00:42:24.120 --> 00:42:27.360
So you take some either
complicated or simple function
00:42:27.360 --> 00:42:29.490
of x, linear
functions are boring,
00:42:29.490 --> 00:42:32.550
quadratic functions are
already interesting.
00:42:32.550 --> 00:42:36.272
And cubic functions, as
I've said a few times,
00:42:36.272 --> 00:42:37.980
you don't even have
formulas for solving.
00:42:37.980 --> 00:42:39.688
So this would be the
only method you have
00:42:39.688 --> 00:42:43.750
for solving them numerically.
00:42:43.750 --> 00:42:45.810
So here's how it works.
00:42:45.810 --> 00:42:50.010
So the idea, I'll
plot this function.
00:42:50.010 --> 00:42:55.810
Here's the function, it's
a parabola, y = x^2 - 5.
00:42:55.810 --> 00:43:01.090
It dips below 0, sorry, it
should be centered, but anyway.
00:43:01.090 --> 00:43:06.530
And now the idea here is
to start with a guess.
00:43:06.530 --> 00:43:11.380
And square root of 5 is pretty
close to the square root of 4,
00:43:11.380 --> 00:43:12.360
which is 2.
00:43:12.360 --> 00:43:17.030
So my first guess is
going to be 2, here.
00:43:17.030 --> 00:43:30.510
So start with initial guess.
00:43:30.510 --> 00:43:32.320
So that's our first guess.
00:43:32.320 --> 00:43:35.860
And now, what we're
going to do is
00:43:35.860 --> 00:43:40.080
we're going to pretend that
the function is linear.
00:43:40.080 --> 00:43:41.300
That's all we're going to do.
00:43:41.300 --> 00:43:44.530
And then if the
function were linear,
00:43:44.530 --> 00:43:47.432
we're going to try to
find where the 0 is.
00:43:47.432 --> 00:43:49.265
So if the function is
linear, what we'll use
00:43:49.265 --> 00:43:54.990
is we'll plot the point
where 2 is on, that is,
00:43:54.990 --> 00:43:59.040
the point (2, f(2)),
and then we're
00:43:59.040 --> 00:44:06.660
going to draw the
tangent line here.
00:44:06.660 --> 00:44:14.560
And this is going
to be our new guess.
00:44:14.560 --> 00:44:22.420
x equals x, which I'll call x_1.
00:44:22.420 --> 00:44:26.610
So the idea here is that this
point may be somewhat far
00:44:26.610 --> 00:44:31.250
from where it crosses, but this
point will be a little closer.
00:44:31.250 --> 00:44:33.830
And now we're going to do
this over and over again.
00:44:33.830 --> 00:44:38.340
And see how fast it gets to
the place we're aiming for.
00:44:38.340 --> 00:44:42.020
So we have to work out
what the formulas are.
00:44:42.020 --> 00:44:53.700
And that's the strategy.
00:44:53.700 --> 00:45:01.360
So now, the first step
here is, we have our guess,
00:45:01.360 --> 00:45:06.020
we have our tangent line.
00:45:06.020 --> 00:45:13.540
Which has the formula
y - y_0 = m(x - x_0).
00:45:13.540 --> 00:45:16.220
So that's the general
form for a tangent line.
00:45:16.220 --> 00:45:20.140
And now, I have to
tell you what x_1 is.
00:45:20.140 --> 00:45:29.970
In terms of this tangent
line. x_1 is the x-intercept.
00:45:29.970 --> 00:45:32.810
The tangent line, if you look
over here at the diagram,
00:45:32.810 --> 00:45:36.220
the tangent line
is the orange line.
00:45:36.220 --> 00:45:41.340
Where that crosses the axis,
that's where I want to put x_1.
00:45:41.340 --> 00:45:43.530
So that's the x-intercept.
00:45:43.530 --> 00:45:46.670
Now, how do you find
the x-intercept?
00:45:46.670 --> 00:45:49.760
You find it by setting y = 0.
00:45:49.760 --> 00:45:52.270
That horizontal line is y = 0.
00:45:52.270 --> 00:45:59.790
So I set y = 0, and I get
0 - y_0 = m (x_1 - x_0).
00:45:59.790 --> 00:46:02.300
So I changed two things
in this equation.
00:46:02.300 --> 00:46:04.970
I plugged in 0 here, for y.
00:46:04.970 --> 00:46:07.020
And I said that the
place where that happens
00:46:07.020 --> 00:46:13.050
is going to be where x is x_1.
00:46:13.050 --> 00:46:15.530
So now let's solve.
00:46:15.530 --> 00:46:19.700
And what we get here is
-.y divided by-- sorry,
00:46:19.700 --> 00:46:26.390
-y_0 / m = x_1 - x_0.
00:46:26.390 --> 00:46:30.800
And now I can get
a formula for x_1.
00:46:30.800 --> 00:46:46.600
So x_1 = x_0 - y_0 / m.
00:46:46.600 --> 00:46:53.140
I now need to tell you
what this formula means,
00:46:53.140 --> 00:46:56.070
in terms of the function f.
00:46:56.070 --> 00:46:59.230
So first of all, x_0 is
x_0, whatever x_0 is.
00:46:59.230 --> 00:47:06.230
And y_0, I claim, is f(x_0).
00:47:06.230 --> 00:47:11.080
And m is the slope
at that same place.
00:47:11.080 --> 00:47:17.230
So it's f'(x_0).
00:47:17.230 --> 00:47:19.650
And this is the whole story.
00:47:19.650 --> 00:47:25.440
This is the formula which
will enable us to calculate
00:47:25.440 --> 00:47:34.520
basically any root.
00:47:34.520 --> 00:47:36.072
I'm going to repeat
this formula,
00:47:36.072 --> 00:47:37.530
so I'm going to
tell you again what
00:47:37.530 --> 00:47:40.620
Newton's method is,
and put a little more
00:47:40.620 --> 00:47:43.080
colorful box around it.
00:47:43.080 --> 00:47:53.210
So Newton's method involves
the following formula.
00:47:53.210 --> 00:47:57.470
In order to get
the (n+1)st point,
00:47:57.470 --> 00:48:00.250
that's our better
and better guess,
00:48:00.250 --> 00:48:03.530
I'm going to take the nth one
and then I'm going to plug
00:48:03.530 --> 00:48:04.330
in this formula.
00:48:04.330 --> 00:48:09.870
So f(x_n) / f'(x_n).
00:48:09.870 --> 00:48:13.630
So this is the basic
formula, and if you like,
00:48:13.630 --> 00:48:18.820
this is the idea of just
repeating what I had before.
00:48:18.820 --> 00:48:21.900
Now, we've gone from
geometry, from just pictures,
00:48:21.900 --> 00:48:23.520
to an honest to
goodness formula which
00:48:23.520 --> 00:48:25.690
is completely
implementable and very easy
00:48:25.690 --> 00:48:29.450
to implement in any case.
00:48:29.450 --> 00:48:33.540
So let's see how it works
in the case that we've got.
00:48:33.540 --> 00:48:41.160
Which is x_0 = 2.
f(x) = x^2 - 5.
00:48:41.160 --> 00:48:44.850
Let's see how to
implement this formula.
00:48:44.850 --> 00:48:49.920
So first of all, I have to
calculate for you, f'(x).
00:48:49.920 --> 00:48:55.200
That's 2x.
00:48:55.200 --> 00:49:05.420
And so, x_1 is equal to
x_0 minus, so f', sorry,
00:49:05.420 --> 00:49:08.690
f(x) would be x_0^2 - 5.
00:49:08.690 --> 00:49:10.550
That's what's in the numerator.
00:49:10.550 --> 00:49:13.220
And in the denominator
I have the derivative,
00:49:13.220 --> 00:49:18.820
so that's 2 x_0.
00:49:18.820 --> 00:49:23.920
And so all told, well, let's
work it out in two steps here.
00:49:23.920 --> 00:49:27.190
This is -1/2 x_0
for the first term,
00:49:27.190 --> 00:49:31.750
and then plus (5/2) /
x_0 for the second term.
00:49:31.750 --> 00:49:40.450
And these two terms combine, so
we have here 1/2 x_0 plus 5/2
00:49:40.450 --> 00:49:43.010
with an x_0 in the denominator.
00:49:43.010 --> 00:49:53.310
So here's the formula
for x_1. in this case.
00:49:53.310 --> 00:50:03.620
Now I'd like to show
you how well this works.
00:50:03.620 --> 00:50:10.790
So first of all, we have x_1
is 1/2 * 2, if I plug in x1,
00:50:10.790 --> 00:50:16.630
plus 5/4, which is 9/4.
00:50:16.630 --> 00:50:25.620
And x_2, I have 1/2
* 9/4 + + 5/2 * 4/9.
00:50:28.550 --> 00:50:30.360
That's the next one.
00:50:30.360 --> 00:50:37.610
And if you work this
out, it's 161/72.
00:50:37.610 --> 00:50:41.302
And then x_3 is kind of long.
00:50:41.302 --> 00:50:43.760
But I will just write down what
it is, so that you can see.
00:50:43.760 --> 00:50:49.270
It's 1/2 * 161/72
plus 5/2 and then
00:50:49.270 --> 00:50:50.600
I do the reciprocal of that.
00:50:50.600 --> 00:50:56.620
So 72/161.
00:50:56.620 --> 00:51:00.770
So let's see how good these are.
00:51:00.770 --> 00:51:05.850
I carefully calculated
how far off they are.
00:51:05.850 --> 00:51:07.510
Somewhere on my notes.
00:51:07.510 --> 00:51:11.660
So I'll just take a look
and see what I said.
00:51:11.660 --> 00:51:13.930
Oh yeah, I did do it.
00:51:13.930 --> 00:51:20.460
So, what's the square root
of 5 minus-- so here's n,
00:51:20.460 --> 00:51:24.420
here's the square root of
5 minus x_n, if you like.
00:51:24.420 --> 00:51:26.800
Or the other way around.
00:51:26.800 --> 00:51:28.900
The size of this.
00:51:28.900 --> 00:51:31.540
You'll have to decide
on your homework
00:51:31.540 --> 00:51:33.960
whether it comes out positive
or negative, to the right
00:51:33.960 --> 00:51:35.660
or to the left of the answer.
00:51:35.660 --> 00:51:37.460
But let's do this.
00:51:37.460 --> 00:51:41.070
So when n = 0, the guess was 2.
00:51:41.070 --> 00:51:45.330
And we're off by
about 2 * 10^(-1).
00:51:45.330 --> 00:51:48.690
And if n = 1, so
that's this 9/4,
00:51:48.690 --> 00:51:53.120
that's off by about 10^(-2).
00:51:53.120 --> 00:51:58.240
And then n = 2, that's
this number here, right?
00:51:58.240 --> 00:52:03.540
And that's off by
about 4 * 10^(-5).
00:52:03.540 --> 00:52:05.840
That's already as
good an approximation
00:52:05.840 --> 00:52:10.450
to the square root of 5 as
you'll ever need in your life.
00:52:10.450 --> 00:52:17.050
If you do 3, this number
here turns out to be accurate
00:52:17.050 --> 00:52:20.140
to 10^(-10) or so.
00:52:20.140 --> 00:52:23.391
This goes right off to the
edge of my calculator, this one
00:52:23.391 --> 00:52:23.890
here.
00:52:23.890 --> 00:52:26.500
So already, with
the third iterate.
00:52:26.500 --> 00:52:31.340
you're of way past the accuracy
that you need for most things.
00:52:31.340 --> 00:52:32.070
Yep, question.
00:52:32.070 --> 00:52:32.903
STUDENT: [INAUDIBLE]
00:52:32.903 --> 00:52:38.660
PROFESSOR: How come
the x_0 disappears?
00:52:38.660 --> 00:52:55.360
STUDENT: [INAUDIBLE] PROFESSOR:
So, from here to here
00:52:55.360 --> 00:52:56.010
[INAUDIBLE]
00:52:56.010 --> 00:52:56.380
STUDENT: [INAUDIBLE]
00:52:56.380 --> 00:52:58.296
PROFESSOR: Hang on,
folks, we have a question.
00:52:58.296 --> 00:52:59.390
Let's just answer it.
00:52:59.390 --> 00:53:02.870
So here we have an x_0,
and here we have -1/2,
00:53:02.870 --> 00:53:08.050
there's an x_0^2 and
an x which cancel.
00:53:08.050 --> 00:53:11.480
And here we have a
minus, and a -5/2 x0.
00:53:11.480 --> 00:53:16.150
So I combine the 1 - 1/2,
I got +1/2, that's all.
00:53:16.150 --> 00:53:17.680
OK?
00:53:17.680 --> 00:53:18.890
All right, thanks.
00:53:18.890 --> 00:53:21.910
We'll have to ask other
questions after class.