Fundamental theorem of calculus (Part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:03.140
Let's say I have
some function f that

00:00:03.140 --> 00:00:11.160
is continuous on an
interval between a and b.

00:00:11.160 --> 00:00:13.840
And I have these brackets here,
so it also includes a and b

00:00:13.840 --> 00:00:15.010
in the interval.

00:00:15.010 --> 00:00:16.540
So let me graph
this just so we get

00:00:16.540 --> 00:00:18.540
a sense of what
I'm talking about.

00:00:18.540 --> 00:00:20.700
So that's my vertical axis.

00:00:20.700 --> 00:00:22.310
This is my horizontal axis.

00:00:22.310 --> 00:00:24.170
I'm going to label
my horizontal axis

00:00:24.170 --> 00:00:26.610
t so we can save x for later.

00:00:26.610 --> 00:00:28.860
I can still make this
y right over there.

00:00:28.860 --> 00:00:30.310
And let me graph.

00:00:30.310 --> 00:00:35.340
This right over here is the
graph of y is equal to f of t.

00:00:35.340 --> 00:00:41.780
Now our lower endpoint is a,
so that's a right over there.

00:00:41.780 --> 00:00:43.485
Our upper boundary is b.

00:00:46.190 --> 00:00:47.209
Let me make that clear.

00:00:47.209 --> 00:00:49.750
And actually just to show that
we're including that endpoint,

00:00:49.750 --> 00:00:52.690
let me make them bold
lines, filled in lines.

00:00:52.690 --> 00:00:54.662
So lower boundary,
a, upper boundary, b.

00:00:54.662 --> 00:00:56.120
We're just saying
and I've drawn it

00:00:56.120 --> 00:00:58.720
this way that f is
continuous on that.

00:00:58.720 --> 00:01:01.950
Now let's define
some new function.

00:01:01.950 --> 00:01:03.600
Let's define some
new function that's

00:01:03.600 --> 00:01:09.050
the area under the curve
between a and some point

00:01:09.050 --> 00:01:10.330
that's in our interval.

00:01:10.330 --> 00:01:13.560
Let me pick this
right over here, x.

00:01:13.560 --> 00:01:17.180
So let's define
some new function

00:01:17.180 --> 00:01:22.690
to capture the area under
the curve between a and x.

00:01:22.690 --> 00:01:25.680
Well, how do we denote
the area under the curve

00:01:25.680 --> 00:01:27.424
between two endpoints?

00:01:27.424 --> 00:01:29.090
Well, we just use our
definite integral.

00:01:29.090 --> 00:01:30.290
That's our Riemann integral.

00:01:30.290 --> 00:01:33.110
It's really that right
now before we come up

00:01:33.110 --> 00:01:34.624
with the conclusion
of this video,

00:01:34.624 --> 00:01:36.040
it really just
represents the area

00:01:36.040 --> 00:01:38.100
under the curve
between two endpoints.

00:01:38.100 --> 00:01:40.220
So this right over
here, we can say

00:01:40.220 --> 00:01:55.886
is the definite integral
from a to x of f of t dt.

00:01:59.540 --> 00:02:02.230
Now this right over here is
going to be a function of x--

00:02:02.230 --> 00:02:06.660
and let me make
it clear-- where x

00:02:06.660 --> 00:02:10.740
is in the interval
between a and b.

00:02:10.740 --> 00:02:12.260
This thing right
over here is going

00:02:12.260 --> 00:02:14.330
to be another function of x.

00:02:14.330 --> 00:02:18.810
This value is going to depend
on what x we actually choose.

00:02:18.810 --> 00:02:21.630
So let's define this
as a function of x.

00:02:21.630 --> 00:02:28.376
So I'm going to say that this
is equal to uppercase F of x.

00:02:28.376 --> 00:02:29.250
So all fair and good.

00:02:29.250 --> 00:02:31.390
Uppercase F of x is a function.

00:02:31.390 --> 00:02:34.770
If you give me an x value
that's between a and b,

00:02:34.770 --> 00:02:37.910
it'll tell you the
area under lowercase f

00:02:37.910 --> 00:02:40.780
of t between a and x.

00:02:40.780 --> 00:02:43.950
Now the cool part, the
fundamental theorem

00:02:43.950 --> 00:02:45.160
of calculus.

00:02:45.160 --> 00:02:47.060
The fundamental
theorem of calculus

00:02:47.060 --> 00:02:48.870
tells us-- let me
write this down

00:02:48.870 --> 00:02:50.450
because this is a big deal.

00:02:50.450 --> 00:02:57.580
Fundamental theorem-- that's
not an abbreviation-- theorem

00:02:57.580 --> 00:03:01.230
of calculus tells
us that if we were

00:03:01.230 --> 00:03:03.847
to take the derivative
of our capital F,

00:03:03.847 --> 00:03:06.430
so the derivative-- let me make
sure I have enough space here.

00:03:06.430 --> 00:03:08.480
So if I were to take the
derivative of capital

00:03:08.480 --> 00:03:11.420
F with respect to x, which
is the same thing as taking

00:03:11.420 --> 00:03:13.070
the derivative of
this with respect

00:03:13.070 --> 00:03:15.840
to x, which is equal to
the derivative of all

00:03:15.840 --> 00:03:18.520
of this business--
let me copy this.

00:03:18.520 --> 00:03:24.940
So copy and then paste,
which is the same thing.

00:03:24.940 --> 00:03:26.880
I've defined capital
F as this stuff.

00:03:26.880 --> 00:03:28.550
So if I'm taking the derivative
of the left hand side,

00:03:28.550 --> 00:03:29.930
it's the same thing as
taking the derivative

00:03:29.930 --> 00:03:31.260
of the right hand side.

00:03:31.260 --> 00:03:33.320
The fundamental
theorem of calculus

00:03:33.320 --> 00:03:44.420
tells us that this is going to
be equal to lowercase f of x.

00:03:44.420 --> 00:03:46.470
Now why is this a big deal?

00:03:46.470 --> 00:03:48.130
Why does it get such
an important title

00:03:48.130 --> 00:03:54.920
as the fundamental
theorem of calculus?

00:03:54.920 --> 00:03:58.340
Well, it tells us that for
any continuous function

00:03:58.340 --> 00:04:01.130
f, if I define a
function, that is,

00:04:01.130 --> 00:04:05.320
the area under the curve
between a and x right over here,

00:04:05.320 --> 00:04:08.580
that the derivative of that
function is going to be f.

00:04:08.580 --> 00:04:09.580
So let me make it clear.

00:04:09.580 --> 00:04:15.370
Every continuous function,
every continuous f,

00:04:15.370 --> 00:04:26.760
has an antiderivative
capital F of x.

00:04:26.760 --> 00:04:28.747
That by itself is a cool thing.

00:04:28.747 --> 00:04:30.330
But the other really
cool thing-- or I

00:04:30.330 --> 00:04:32.110
guess these are
somewhat related.

00:04:32.110 --> 00:04:34.550
Remember, coming into
this, all we did,

00:04:34.550 --> 00:04:37.170
we just viewed the
definite integral

00:04:37.170 --> 00:04:41.890
as symbolizing as the area under
the curve between two points.

00:04:41.890 --> 00:04:44.060
That's where that Riemann
definition of integration

00:04:44.060 --> 00:04:45.290
comes from.

00:04:45.290 --> 00:04:49.032
But now we see a connection
between that and derivatives.

00:04:49.032 --> 00:04:50.740
When you're taking
the definite integral,

00:04:50.740 --> 00:04:52.448
one way of thinking,
especially if you're

00:04:52.448 --> 00:04:54.350
taking a definite
integral between a lower

00:04:54.350 --> 00:04:56.730
boundary and an x, one way
to think about it is you're

00:04:56.730 --> 00:04:58.970
essentially taking
an antiderivative.

00:04:58.970 --> 00:05:01.900
So we now see a
connection-- and this

00:05:01.900 --> 00:05:04.280
is why it is the fundamental
theorem of calculus.

00:05:04.280 --> 00:05:05.850
It connects
differential calculus

00:05:05.850 --> 00:05:14.330
and integral calculus--
connection between derivatives,

00:05:14.330 --> 00:05:17.010
or maybe I should say
antiderivatives, derivatives

00:05:17.010 --> 00:05:18.040
and integration.

00:05:18.040 --> 00:05:21.160
Which before this video, we
just viewed integration as area

00:05:21.160 --> 00:05:21.710
under curve.

00:05:21.710 --> 00:05:24.720
Now we see it has a
connection to derivatives.

00:05:24.720 --> 00:05:27.120
Well, how would you actually
use the fundamental theorem

00:05:27.120 --> 00:05:27.760
of calculus?

00:05:27.760 --> 00:05:30.520
Well, maybe in the context
of a calculus class.

00:05:30.520 --> 00:05:32.740
And we'll do the intuition
for why this happens

00:05:32.740 --> 00:05:35.220
or why this is true and maybe
a proof in later videos.

00:05:35.220 --> 00:05:38.310
But how would you actually
apply this right over here?

00:05:38.310 --> 00:05:41.910
Well, let's say someone
told you that they

00:05:41.910 --> 00:05:43.530
want to find the derivative.

00:05:43.530 --> 00:05:45.030
Let me do this in
a new color just

00:05:45.030 --> 00:05:46.850
to show this is an example.

00:05:46.850 --> 00:05:50.090
Let's say someone wanted to
find the derivative with respect

00:05:50.090 --> 00:05:53.074
to x of the integral
from-- I don't know.

00:05:53.074 --> 00:05:54.490
I'll pick some
random number here.

00:05:54.490 --> 00:05:59.900
So pi to x -- I'll put
something crazy here --

00:05:59.900 --> 00:06:06.120
cosine squared of t
over the natural log

00:06:06.120 --> 00:06:10.910
of t minus the
square root of t dt.

00:06:10.910 --> 00:06:12.980
So they want you take the
derivative with respect

00:06:12.980 --> 00:06:14.021
to x of this crazy thing.

00:06:14.021 --> 00:06:18.790
Remember, this thing in the
parentheses is a function of x.

00:06:18.790 --> 00:06:21.332
Its value, it's going to have
a value that is dependent on x.

00:06:21.332 --> 00:06:22.748
If you give it a
different x, it's

00:06:22.748 --> 00:06:24.140
going to have a different value.

00:06:24.140 --> 00:06:26.830
So what's the derivative
of this with respect to x?

00:06:26.830 --> 00:06:29.030
Well, the fundamental
theorem of calculus tells us

00:06:29.030 --> 00:06:31.720
it can be very simple.

00:06:31.720 --> 00:06:34.310
We essentially-- and you can
even pattern match up here.

00:06:34.310 --> 00:06:35.768
And we'll get more
intuition of why

00:06:35.768 --> 00:06:37.065
this is true in future videos.

00:06:37.065 --> 00:06:38.440
But essentially,
everywhere where

00:06:38.440 --> 00:06:41.307
you see this right
over here is an f of t.

00:06:41.307 --> 00:06:43.140
Everywhere you see a
t, replace it with an x

00:06:43.140 --> 00:06:44.490
and it becomes an f of x.

00:06:44.490 --> 00:06:47.850
So this is going to be
equal to cosine squared

00:06:47.850 --> 00:06:53.980
of x over the natural log of
x minus the square root of x.

00:06:53.980 --> 00:06:56.700
You take the derivative of
the indefinite integral where

00:06:56.700 --> 00:06:58.410
the upper boundary
is x right over here.

00:06:58.410 --> 00:07:02.050
It just becomes whatever you
were taking the integral of,

00:07:02.050 --> 00:07:07.770
that as a function instead of
t, that is now a function x.

00:07:07.770 --> 00:07:10.210
So it can really simplify
sometimes taking a derivative.

00:07:10.210 --> 00:07:12.840
And sometimes you'll see on
exams these trick problems

00:07:12.840 --> 00:07:14.320
where you had this really
hairy thing that you

00:07:14.320 --> 00:07:15.910
need to take a definite
integral of and then take

00:07:15.910 --> 00:07:17.285
the derivative,
and you just have

00:07:17.285 --> 00:07:19.290
to remember the fundamental
theorem of calculus,

00:07:19.290 --> 00:07:20.790
the thing that ties
it all together,

00:07:20.790 --> 00:07:22.740
connects derivatives
and integration,

00:07:22.740 --> 00:07:26.350
that you can just simplify it
by realizing that this is just

00:07:26.350 --> 00:07:29.900
going to be instead of a
function lowercase f of t,

00:07:29.900 --> 00:07:31.570
it's going to be
lowercase f of x.

00:07:31.570 --> 00:07:32.770
Let me make it clear.

00:07:32.770 --> 00:07:35.550
In this example right over
here, this right over here

00:07:35.550 --> 00:07:37.730
was lowercase f of t.

00:07:37.730 --> 00:07:41.610
And now it became
lowercase f of x.

00:07:41.610 --> 00:07:45.044
This right over here was our a.

00:07:45.044 --> 00:07:46.460
And notice, it
doesn't matter what

00:07:46.460 --> 00:07:48.592
the lower boundary
of a actually is.

00:07:48.592 --> 00:07:50.300
You don't have anything
on the right hand

00:07:50.300 --> 00:07:52.691
side that is in some
way dependent on a.

00:07:52.691 --> 00:07:53.940
Anyway, hope you enjoyed that.

00:07:53.940 --> 00:07:56.470
And in the next few videos,
we'll think about the intuition

00:07:56.470 --> 00:08:00.420
and do more examples making
use of the fundamental theorem

00:08:00.420 --> 00:08:01.870
of calculus.


Related rates: shadow

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:02.320
It's late at night,
and some type

00:00:02.320 --> 00:00:05.860
of nocturnal predatory
bird, maybe this is an owl,

00:00:05.860 --> 00:00:08.390
is diving for its dinner.

00:00:08.390 --> 00:00:13.500
So this right over
here is a mouse.

00:00:13.500 --> 00:00:18.360
And it's diving straight
down near a street light.

00:00:18.360 --> 00:00:20.720
And let's get some information
about what's going on.

00:00:20.720 --> 00:00:25.610
So the street light right
over here is 20 feet high.

00:00:25.610 --> 00:00:31.140
So this is a 20 foot
high street lamp.

00:00:31.140 --> 00:00:33.710
And right at this
moment, the-- and I

00:00:33.710 --> 00:00:35.360
haven't drawn it
completely to scale--

00:00:35.360 --> 00:00:39.200
the owl is 15 feet
above the mouse.

00:00:39.200 --> 00:00:44.540
So this distance right
over here is 15 feet.

00:00:44.540 --> 00:00:48.840
And the mouse itself is 10
feet from the base of the lamp.

00:00:48.840 --> 00:00:49.640
Let me draw that.

00:00:49.640 --> 00:00:56.780
So the mouse is 10 feet
from the base of the lamp.

00:00:56.780 --> 00:00:58.850
And we also know-- we
have our little radar

00:00:58.850 --> 00:01:02.500
gun out-- we know that this
owl is diving straight down.

00:01:05.090 --> 00:01:09.090
And right now, it is
going 20 feet per second.

00:01:09.090 --> 00:01:17.290
So right now, this is going
down at 20 feet per second.

00:01:17.290 --> 00:01:20.210
Now what we're curious about
is we have the light over here.

00:01:20.210 --> 00:01:23.490
Light is coming from the
street lamp in every direction.

00:01:23.490 --> 00:01:26.080
And it creates a
shadow of the owl.

00:01:26.080 --> 00:01:28.130
So right now the
shadow is out here.

00:01:28.130 --> 00:01:30.440
And as the owl goes
further and further down,

00:01:30.440 --> 00:01:35.510
the shadow's going to move
to the left like that.

00:01:35.510 --> 00:01:38.040
And so, given everything that
we've set up right over here,

00:01:38.040 --> 00:01:44.300
the question is, at what
rate is the shadow moving?

00:01:44.300 --> 00:01:46.770
So let's think about what we
know and what we don't know.

00:01:46.770 --> 00:01:49.620
And to do that, let's
set up some variables.

00:01:49.620 --> 00:01:52.910
So let me draw the same thing a
little bit more geometrically.

00:01:52.910 --> 00:01:58.130
So let's say that this right
over here is the street light.

00:01:58.130 --> 00:02:00.350
That is 20 feet tall.

00:02:00.350 --> 00:02:06.770
And then this right over
here is the height of the owl

00:02:06.770 --> 00:02:09.780
right at this moment,
so this is 15 feet.

00:02:09.780 --> 00:02:11.520
The distance between
the base of the lamp

00:02:11.520 --> 00:02:14.650
and where the owl is going,
where that mouse is right now.

00:02:14.650 --> 00:02:16.760
This is 10 feet.

00:02:16.760 --> 00:02:18.880
And if I were to think
about where the shadow is,

00:02:18.880 --> 00:02:22.970
well, the light's
from right over here.

00:02:22.970 --> 00:02:25.790
And so the owl blocks the
light right over there,

00:02:25.790 --> 00:02:27.880
so the shadow is going
to be right over there.

00:02:27.880 --> 00:02:31.330
So if you just draw a straight
line from the source of light

00:02:31.330 --> 00:02:33.639
through the owl and
you just keep going

00:02:33.639 --> 00:02:35.180
and you hit the
ground, you are going

00:02:35.180 --> 00:02:36.596
to figure out where
the shadow is.

00:02:36.596 --> 00:02:44.500
So the shadow is going
to be right over here.

00:02:44.500 --> 00:02:46.110
It's going to be
right over there.

00:02:46.110 --> 00:02:49.140
And we need to figure out
how quickly is that moving.

00:02:49.140 --> 00:02:54.190
And it's going to be moving
in the leftward direction.

00:02:54.190 --> 00:02:57.110
So let's set up some
variables over here.

00:02:57.110 --> 00:02:59.090
So let's say-- so
what's changing?

00:02:59.090 --> 00:03:01.410
Well, we know that the height
of the owl is changing.

00:03:01.410 --> 00:03:03.000
So let's call that y.

00:03:03.000 --> 00:03:05.220
Right at this moment,
it's equal to 15,

00:03:05.220 --> 00:03:07.290
but it is actually changing.

00:03:07.290 --> 00:03:14.345
And let's call the
distance between the shadow

00:03:14.345 --> 00:03:19.790
and the mouse x.

00:03:19.790 --> 00:03:22.040
Now given this set
up, can we come up

00:03:22.040 --> 00:03:24.185
with a relationship
between x and y?

00:03:24.185 --> 00:03:26.310
And then using that
relationship, what we're really

00:03:26.310 --> 00:03:29.440
trying to come up with is,
what is the rate at which x

00:03:29.440 --> 00:03:31.780
is changing with
respect to time?

00:03:31.780 --> 00:03:33.880
We know what y is
right at this moment.

00:03:33.880 --> 00:03:37.800
We know what dy/dt is
right at this moment.

00:03:37.800 --> 00:03:39.790
Can we come up the
relationship between x and y

00:03:39.790 --> 00:03:41.540
and maybe take the
derivative with respect

00:03:41.540 --> 00:03:44.970
to t so we can
figure out what dx/dt

00:03:44.970 --> 00:03:48.990
is at a given moment in time?

00:03:48.990 --> 00:03:51.094
Well, both of these
triangles-- and when

00:03:51.094 --> 00:03:52.510
I say both of these
triangles, let

00:03:52.510 --> 00:03:54.720
me be clear what
I'm talking about.

00:03:54.720 --> 00:03:58.590
This triangle right over here,
the smaller triangle in green,

00:03:58.590 --> 00:04:01.600
is a similar triangle
to the larger triangle.

00:04:01.600 --> 00:04:04.630
This is a similar triangle
to this larger triangle

00:04:04.630 --> 00:04:06.640
that I am tracing in blue.

00:04:06.640 --> 00:04:08.770
It's similar to this larger one.

00:04:08.770 --> 00:04:10.250
How do I know that?

00:04:10.250 --> 00:04:13.430
Well, they both have a right
angle, right over here.

00:04:13.430 --> 00:04:15.190
They both share this angle.

00:04:15.190 --> 00:04:16.820
So all three-- they
have two angles

00:04:16.820 --> 00:04:22.280
in common-- then all three
angles must be in common.

00:04:22.280 --> 00:04:23.780
So they are similar
triangles, which

00:04:23.780 --> 00:04:26.390
means the ratio between
corresponding sides

00:04:26.390 --> 00:04:28.410
must be the same.

00:04:28.410 --> 00:04:33.790
So we know that
the ratio of x to y

00:04:33.790 --> 00:04:38.370
must be the ratio of this
entire base, which is x plus 10,

00:04:38.370 --> 00:04:42.860
to the height of the
larger triangle, to 20.

00:04:42.860 --> 00:04:45.280
And right there, we have a
relationship between x and y.

00:04:45.280 --> 00:04:48.150
And if we take the derivative
of both sides with respect to t,

00:04:48.150 --> 00:04:50.875
we're probably
doing pretty well.

00:04:50.875 --> 00:04:52.750
Now before taking the
derivative with respect

00:04:52.750 --> 00:04:54.660
to t-- I could do
it right over here--

00:04:54.660 --> 00:04:56.400
just to simplify
things a little bit,

00:04:56.400 --> 00:04:57.780
let me just cross multiply.

00:04:57.780 --> 00:05:01.030
So let me multiply both
sides of this equation by 20

00:05:01.030 --> 00:05:04.190
and y, just so that I
don't have as many things

00:05:04.190 --> 00:05:05.600
in the denominator.

00:05:05.600 --> 00:05:08.700
So on the left hand side
it simplifies to 20x.

00:05:08.700 --> 00:05:11.570
I don't want to write over it,
well, I'll just write-- 20x.

00:05:11.570 --> 00:05:15.580
And on the left
hand side is 20 x.

00:05:15.580 --> 00:05:17.200
And then on the
right hand side--

00:05:17.200 --> 00:05:26.254
let's see this cancels with
that-- we have xy plus 10y.

00:05:26.254 --> 00:05:27.670
And now let me
take the derivative

00:05:27.670 --> 00:05:30.990
of both sides with
respect to time.

00:05:30.990 --> 00:05:34.274
So the derivative of 20 times
something with respect to time

00:05:34.274 --> 00:05:36.940
is going to be the derivative of
20 times something with respect

00:05:36.940 --> 00:05:38.852
to the something,
which is just 20.

00:05:38.852 --> 00:05:40.560
That's the derivative
of 20x with respect

00:05:40.560 --> 00:05:45.030
to x, times dx or the derivative
of x with respect to t,

00:05:45.030 --> 00:05:46.669
is equal to.

00:05:46.669 --> 00:05:48.460
Now over here we're
going have to break out

00:05:48.460 --> 00:05:50.166
a little bit of
the product rule.

00:05:50.166 --> 00:05:52.040
So first we want to
figure out the derivative

00:05:52.040 --> 00:05:53.770
of x with respect to time.

00:05:56.900 --> 00:06:00.430
So the derivative of the first
thing times the second thing,

00:06:00.430 --> 00:06:01.590
times y.

00:06:01.590 --> 00:06:03.720
Plus just the first thing
times the derivative

00:06:03.720 --> 00:06:04.585
of the second thing.

00:06:04.585 --> 00:06:06.710
So derivative of y with
respect to t is just dy/dt.

00:06:09.690 --> 00:06:12.500
And then finally,
right over here.

00:06:12.500 --> 00:06:14.830
The derivative of
10y with respect to t

00:06:14.830 --> 00:06:18.320
is the derivative of 10y with
respect to y, which is just 10,

00:06:18.320 --> 00:06:21.247
times the derivative of y with
respect to t, which is dy/dt.

00:06:23.544 --> 00:06:24.460
And there you have it.

00:06:24.460 --> 00:06:29.400
You have your relationship
between dx/dt, dy/dt, and x

00:06:29.400 --> 00:06:30.689
and y.

00:06:30.689 --> 00:06:32.480
So let's just make sure
we have everything.

00:06:32.480 --> 00:06:35.682
This is what we're trying
to solve for, dx/dt.

00:06:35.682 --> 00:06:37.640
And let's see here, we
have another dx/dt here.

00:06:37.640 --> 00:06:39.300
We're going to try
to solve for that.

00:06:39.300 --> 00:06:42.380
We know what y is.
y is equal to 15.

00:06:42.380 --> 00:06:45.690
We know what dy/dt is, dy/dt.

00:06:45.690 --> 00:06:48.610
If we make the convention
since y is decreasing,

00:06:48.610 --> 00:06:50.340
we can say it's -20.

00:06:50.340 --> 00:06:52.840
So we know what this is.

00:06:52.840 --> 00:06:56.210
And so if we just know what,
so we know what this is.

00:06:56.210 --> 00:07:01.380
So if we just know what x
is we can solve for dx/dt.

00:07:01.380 --> 00:07:04.310
So what is x right
at this moment?

00:07:04.310 --> 00:07:06.505
Well, we can use
this first equation--

00:07:06.505 --> 00:07:08.130
we could actually
use this one up here,

00:07:08.130 --> 00:07:11.040
but this one is simplified a
little bit-- to actually solve

00:07:11.040 --> 00:07:11.680
for x.

00:07:11.680 --> 00:07:14.346
So let's do that, and then we'll
substitute back into this thing

00:07:14.346 --> 00:07:15.870
where we've taken
the derivative.

00:07:15.870 --> 00:07:22.400
So we get 20 times x
is equal to x times y.

00:07:22.400 --> 00:07:24.055
y is 15.

00:07:24.055 --> 00:07:26.180
And just remember, I could
have used this equation,

00:07:26.180 --> 00:07:27.970
but this is just
one step further.

00:07:27.970 --> 00:07:29.660
We've already
crossed multiplied.

00:07:29.660 --> 00:07:32.240
So it's x times y. y is 15.

00:07:32.240 --> 00:07:35.680
So it's x times 15,
plus 10 times y.

00:07:35.680 --> 00:07:39.047
Plus 10 times 15.

00:07:39.047 --> 00:07:39.880
Did I do that right?

00:07:39.880 --> 00:07:45.490
20x is equal to x times
15 plus 10 times 15.

00:07:45.490 --> 00:07:47.080
So let's see if you subtract.

00:07:47.080 --> 00:07:52.450
So just this is 20x is
equal to 15x, plus 150.

00:07:52.450 --> 00:07:55.020
Subtract 15x from
both sides, you

00:07:55.020 --> 00:07:59.900
get 5x is equal to 30--
sorry, 5x is equal to 150,

00:07:59.900 --> 00:08:01.420
my brain is getting ahead.

00:08:01.420 --> 00:08:03.330
5x is equal to 150.

00:08:03.330 --> 00:08:04.690
Divide both sides by 5.

00:08:04.690 --> 00:08:08.560
You get x is equal to 30 feet.

00:08:08.560 --> 00:08:11.760
x is equal to 30 feet
right at this moment.

00:08:11.760 --> 00:08:15.240
So this distance, just going
back to our original diagram.

00:08:15.240 --> 00:08:17.782
This distance right
over here is 30 feet.

00:08:17.782 --> 00:08:19.740
So let's substitute all
the values we know back

00:08:19.740 --> 00:08:23.580
into this equation to
actually solve for dx/dt.

00:08:23.580 --> 00:08:25.800
So we have-- so let me
do it right over here--

00:08:25.800 --> 00:08:29.860
we have 20 times dx/dt.

00:08:29.860 --> 00:08:31.830
I'll do that in orange,
we'll solve for that.

00:08:31.830 --> 00:08:33.288
Actually no, I
already used orange.

00:08:33.288 --> 00:08:36.260
So let's say dx/dt-- I'll
use this pink-- 20 times

00:08:36.260 --> 00:08:41.860
dx/dt is equal to dx/dt times y.

00:08:41.860 --> 00:08:44.400
y right now is 15 feet.

00:08:44.400 --> 00:08:50.070
So times 15, times-- I
didn't want to do that color.

00:08:50.070 --> 00:08:56.220
Times 15 plus x, we
already know that x is 30.

00:08:56.220 --> 00:08:59.160
Plus 30, times dy/dt.

00:08:59.160 --> 00:09:00.280
What is dy/dt?

00:09:00.280 --> 00:09:07.400
dy/dt, we could say is -20 feet
per second. y is decreasing,

00:09:07.400 --> 00:09:11.150
the bird is the bird is
diving down to get its dinner.

00:09:11.150 --> 00:09:16.530
So times negative-- well,
times 20 feet per second.

00:09:16.530 --> 00:09:18.090
So that's that right over there.

00:09:18.090 --> 00:09:20.640
Plus 10 times dy/dt.

00:09:20.640 --> 00:09:25.840
So plus 10 times
-20 feet per second.

00:09:25.840 --> 00:09:27.910
And now we just solve for dx/dt.

00:09:27.910 --> 00:09:29.160
So let's see, what do we have?

00:09:29.160 --> 00:09:34.280
We have 20 times-- let's see,
let me subtract 15 dx/dt from

00:09:34.280 --> 00:09:37.310
both sides of this equation--
and we get 5dx/dt's.

00:09:40.880 --> 00:09:44.040
I just subtracted this from
both sides of the equation.

00:09:44.040 --> 00:09:45.920
This is 15dx/dt this is 20.

00:09:45.920 --> 00:09:50.046
So that we have 5dx/dt's,
is equal to this

00:09:50.046 --> 00:09:55.250
is, this part right
over here is -600.

00:09:55.250 --> 00:09:59.120
And this part right
over here is -200.

00:09:59.120 --> 00:10:05.720
So it's equal to -800
feet per second-- or -800,

00:10:05.720 --> 00:10:08.350
and actually this will
actually be in feet per second.

00:10:08.350 --> 00:10:12.660
And so dx/dt is equal to,
dividing both sides by 5,

00:10:12.660 --> 00:10:15.720
5 times 16 is 80.

00:10:15.720 --> 00:10:20.640
So this is -160 feet per second.

00:10:20.640 --> 00:10:21.390
And we're done.

00:10:21.390 --> 00:10:25.760
And we see the shadow is
moving very, very, very, very

00:10:25.760 --> 00:10:26.780
fast to the left.

00:10:26.780 --> 00:10:28.869
x is decreasing,
and we see that.

00:10:28.869 --> 00:10:30.660
That's why we have this
negative sign here.

00:10:30.660 --> 00:10:32.300
The value of x is decreasing.

00:10:32.300 --> 00:10:38.140
It is moving to the left
at quite a nice speed here.


Related rates: Approaching cars

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:03.900
So this car right over here
is approaching an intersection

00:00:03.900 --> 00:00:06.230
at 60 miles per hour.

00:00:06.230 --> 00:00:08.590
And right now, right
at this moment,

00:00:08.590 --> 00:00:14.420
it is 0.8 miles from
the intersection.

00:00:14.420 --> 00:00:16.450
Now we have this
truck over here,

00:00:16.450 --> 00:00:18.280
it's approaching the
same intersection

00:00:18.280 --> 00:00:20.370
on a street that is
perpendicular to the street

00:00:20.370 --> 00:00:21.980
that the car is on.

00:00:21.980 --> 00:00:30.090
And right now it is 0.6
miles from the intersection.

00:00:30.090 --> 00:00:31.720
And it is approaching
the intersection

00:00:31.720 --> 00:00:34.000
at 30 miles per hour.

00:00:34.000 --> 00:00:36.820
Now my question to you
is, what is the rate

00:00:36.820 --> 00:00:40.350
at which the distance
between the car and the truck

00:00:40.350 --> 00:00:42.730
is changing?

00:00:42.730 --> 00:00:44.390
Well to think about
that, let's first

00:00:44.390 --> 00:00:45.848
just think about
what we're asking.

00:00:45.848 --> 00:00:47.500
So we're asking
about the distance

00:00:47.500 --> 00:00:49.230
between the car and the truck.

00:00:49.230 --> 00:00:51.480
So right at this
moment, when the car

00:00:51.480 --> 00:00:53.240
is 0.8 miles from
the intersection,

00:00:53.240 --> 00:00:55.510
the truck is 0.6 miles
from the intersection.

00:00:55.510 --> 00:00:57.700
The truck is traveling at
30 miles per hour towards

00:00:57.700 --> 00:01:00.250
intersection, the car is
travelling at 60 miles per hour

00:01:00.250 --> 00:01:02.220
towards the intersection
right at this moment.

00:01:02.220 --> 00:01:04.940
What is the rate at which
this distance right over here

00:01:04.940 --> 00:01:06.610
is changing?

00:01:06.610 --> 00:01:08.790
And just so that we have
some variables in place,

00:01:08.790 --> 00:01:11.610
let's call this distance s.

00:01:11.610 --> 00:01:13.540
So what we really are
trying to figure out

00:01:13.540 --> 00:01:20.360
is right at this moment, what
is ds dt going to be equal to?

00:01:20.360 --> 00:01:21.960
Let's think about
what we know that we

00:01:21.960 --> 00:01:24.000
could use to somehow
come to terms

00:01:24.000 --> 00:01:26.890
or figure out what ds dt is.

00:01:26.890 --> 00:01:32.100
Well we know the distance of
the car and the intersection.

00:01:32.100 --> 00:01:33.600
And let's just
call that distance,

00:01:33.600 --> 00:01:36.280
let's call that-- I don't know,
let's call that distance y.

00:01:36.280 --> 00:01:38.610
So y is equal to 0.8 miles.

00:01:38.610 --> 00:01:40.310
We also know that--
so let me write

00:01:40.310 --> 00:01:43.760
this-- we know that y
is 0.8 miles right now.

00:01:47.460 --> 00:01:51.060
We also know the dy
dt, the rate at which y

00:01:51.060 --> 00:01:54.590
is changing with
respect to time is what?

00:01:54.590 --> 00:01:58.520
Well y is decreasing
by 60 miles per hour.

00:01:58.520 --> 00:02:06.240
So let me write it as
negative 60 miles per hour.

00:02:06.240 --> 00:02:11.090
Now similarly, let's say that
this distance right over here

00:02:11.090 --> 00:02:14.700
is x. x is 0.6 miles
right at this moment.

00:02:14.700 --> 00:02:19.410
So we know that x is
equal to 0.6 miles.

00:02:19.410 --> 00:02:24.070
What is the rate at which x is
changing with respect to time?

00:02:24.070 --> 00:02:25.982
Well, we know it's
30 miles per hour

00:02:25.982 --> 00:02:27.940
is how fast we're
approaching the intersection,

00:02:27.940 --> 00:02:31.770
but x is decreasing by
30 miles every hour.

00:02:31.770 --> 00:02:38.590
So we should say it's
negative 30 miles per hour.

00:02:38.590 --> 00:02:39.780
So we know what y is.

00:02:39.780 --> 00:02:40.530
We know what x is.

00:02:40.530 --> 00:02:42.710
We know how fast y is
changing, how fast x

00:02:42.710 --> 00:02:44.860
is changing with
respect to time.

00:02:44.860 --> 00:02:46.710
So what we could try
to do here is come up

00:02:46.710 --> 00:02:50.640
with a relationship
between x, y, and s.

00:02:50.640 --> 00:02:52.580
And then differentiate
that relationship

00:02:52.580 --> 00:02:54.290
with respect to time.

00:02:54.290 --> 00:02:56.460
And it seems like we have
pretty much everything

00:02:56.460 --> 00:02:58.580
we need to solve for this.

00:02:58.580 --> 00:03:03.460
So what's a relationship
between x, y, and s?

00:03:03.460 --> 00:03:05.380
Well we know that this
is a right triangle.

00:03:05.380 --> 00:03:07.240
The streets are
perpendicular to each other.

00:03:07.240 --> 00:03:09.270
So we can use the
Pythagorean theorem.

00:03:09.270 --> 00:03:14.690
We know that x squared
plus y squared is

00:03:14.690 --> 00:03:17.540
going to be equal to s squared.

00:03:17.540 --> 00:03:19.000
And then we can
take the derivative

00:03:19.000 --> 00:03:20.440
of both sides of
this with respect

00:03:20.440 --> 00:03:22.606
to time to get a relationship
between all the things

00:03:22.606 --> 00:03:24.070
that we care about.

00:03:24.070 --> 00:03:27.275
So what's the derivative of x
squared with respect to time?

00:03:27.275 --> 00:03:29.650
Well, so you're going to need
the derivative of x squared

00:03:29.650 --> 00:03:32.770
with respect to x,
which is just 2x,

00:03:32.770 --> 00:03:36.950
times the derivative of x with
respect to time, times dx dt.

00:03:36.950 --> 00:03:38.620
Once again, just the chain rule.

00:03:38.620 --> 00:03:41.970
Derivative of something squared
with respect to the something,

00:03:41.970 --> 00:03:45.110
times the derivative of the
something with respect to time.

00:03:45.110 --> 00:03:48.304
And we use similar
logic right over here

00:03:48.304 --> 00:03:49.970
when we want to take
the derivative of y

00:03:49.970 --> 00:03:51.480
squared with respect to time.

00:03:51.480 --> 00:03:53.670
Derivative of y squared
with respect to y,

00:03:53.670 --> 00:03:57.030
times the derivative of
y with respect to time.

00:03:57.030 --> 00:03:59.264
Now on the right-hand
side of this equation,

00:03:59.264 --> 00:04:01.180
we once again take the
derivative with respect

00:04:01.180 --> 00:04:01.535
to time.

00:04:01.535 --> 00:04:03.900
So it's the derivative of s
squared with respect to s,

00:04:03.900 --> 00:04:05.180
which is just 2s.

00:04:05.180 --> 00:04:07.580
Times the derivative of
s with respect to time.

00:04:07.580 --> 00:04:12.146
Once again, this is all just an
application of the chain rule.

00:04:12.146 --> 00:04:13.520
So now it looks
like we know what

00:04:13.520 --> 00:04:15.840
x is, we know what
dx dt is, we know

00:04:15.840 --> 00:04:18.339
what y is, we know
what dy dt is.

00:04:18.339 --> 00:04:19.910
All we need to
figure out is what s

00:04:19.910 --> 00:04:22.860
and then what ds dt is, the
rate at which this distance is

00:04:22.860 --> 00:04:24.540
changing with respect to time.

00:04:24.540 --> 00:04:27.010
Well what's s right now?

00:04:27.010 --> 00:04:29.310
Well we can actually use
the Pythagorean theorem

00:04:29.310 --> 00:04:30.810
at this exact moment.

00:04:30.810 --> 00:04:35.280
We know that x
squared-- so x is 0.6--

00:04:35.280 --> 00:04:40.470
we know that 0.6 squared
plus y squared, 0.8 squared

00:04:40.470 --> 00:04:42.500
is equal to s squared.

00:04:42.500 --> 00:04:48.270
Well this is 0.36, plus
0.64 is equal to s squared.

00:04:48.270 --> 00:04:50.980
This is 1, it is
equal to s squared.

00:04:50.980 --> 00:04:52.730
And we only care about
positive distances,

00:04:52.730 --> 00:04:54.940
so we have s is
equal to 1 right now.

00:04:54.940 --> 00:04:56.719
So we also know what s is.

00:04:56.719 --> 00:04:58.510
So let's substitute
all of these numbers in

00:04:58.510 --> 00:05:01.140
and then try to solve for
what we came here to do.

00:05:01.140 --> 00:05:03.210
Solve for ds dt.

00:05:03.210 --> 00:05:07.240
So the rate at which, so
2 times x-- maybe I'll

00:05:07.240 --> 00:05:10.225
do that in yellow--
2 times x, x is 0.6

00:05:10.225 --> 00:05:13.660
so it's going to
be 1.2 times dx dt,

00:05:13.660 --> 00:05:15.600
so that's negative
30 miles per hour.

00:05:15.600 --> 00:05:17.770
So times negative
30 miles per hour,

00:05:17.770 --> 00:05:24.420
plus 2 times y is
1.6, times dy dt

00:05:24.420 --> 00:05:27.902
is negative 60 miles per hour.

00:05:27.902 --> 00:05:29.360
And I'm not writing
the units here.

00:05:29.360 --> 00:05:30.270
But if you were to
write the units,

00:05:30.270 --> 00:05:32.510
you will see that all of
our distances are in miles.

00:05:32.510 --> 00:05:34.850
And all of our time
is within hours,

00:05:34.850 --> 00:05:36.350
so we're going to
get an answer when

00:05:36.350 --> 00:05:38.604
we solve for ds dt
that's miles per hour.

00:05:38.604 --> 00:05:41.020
But I encourage you, if you
want to, to actually write out

00:05:41.020 --> 00:05:43.210
the units and see
how they work out.

00:05:43.210 --> 00:05:46.530
And so this is going to
be equal to 2 times s.

00:05:46.530 --> 00:05:50.190
Well, s is 1 mile, so it's
just going to be 2 times

00:05:50.190 --> 00:05:53.460
ds dt, which is what
we're trying to solve for.

00:05:53.460 --> 00:05:56.050
So what do we get here
on the left-hand side?

00:05:56.050 --> 00:06:01.230
So 1.2 times negative
30, that's negative 36.

00:06:01.230 --> 00:06:01.730
Right?

00:06:01.730 --> 00:06:04.440
1/5 of 30 is 6,
yep that's right.

00:06:04.440 --> 00:06:06.710
And then 1.6 times
negative sixty,

00:06:06.710 --> 00:06:12.420
that's going to be negative
96, is equal to 2 times ds dt,

00:06:12.420 --> 00:06:14.930
is equal to 2 times
the rate at which

00:06:14.930 --> 00:06:17.210
our distance is changing
with respect to time.

00:06:17.210 --> 00:06:21.840
On the left-hand side right
over here, this is negative 132.

00:06:21.840 --> 00:06:26.400
Negative 132 is equal
to 2 times ds dt.

00:06:26.400 --> 00:06:30.760
Divide both sides by
2, we get negative 66.

00:06:30.760 --> 00:06:35.640
And now we could put our units
if we want, miles per hour

00:06:35.640 --> 00:06:37.650
is the rate at
which our distance

00:06:37.650 --> 00:06:39.310
is changing with
respect to time.

00:06:39.310 --> 00:06:42.930
So ds dt is negative
66 miles per hour.

00:06:42.930 --> 00:06:45.785
Does it make sense that we
got a negative number here?

00:06:45.785 --> 00:06:48.720
Well sure, this distance
is decreasing right

00:06:48.720 --> 00:06:52.211
at this moment, as they
approach the intersection.


Related rates: balloon

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.830
You're watching some type
of hot air balloon show,

00:00:03.830 --> 00:00:07.430
and you're curious about
how quickly one hot air

00:00:07.430 --> 00:00:10.030
balloon in particular is rising.

00:00:10.030 --> 00:00:12.080
And you have some
information at your disposal.

00:00:12.080 --> 00:00:15.170
You know the spot on the
ground that is directly

00:00:15.170 --> 00:00:16.267
below the hot air balloon.

00:00:16.267 --> 00:00:17.850
Let's say it took
off from that point,

00:00:17.850 --> 00:00:20.130
it's just been going
straight up ever since.

00:00:20.130 --> 00:00:21.700
And you know, you've
measured it out,

00:00:21.700 --> 00:00:24.310
that you're 500 meters
away from there.

00:00:24.310 --> 00:00:29.992
So you know that you are
500 meters away from that.

00:00:29.992 --> 00:00:31.700
And you're also able
to measure the angle

00:00:31.700 --> 00:00:33.700
between the horizontal
and the hot air balloon.

00:00:33.700 --> 00:00:35.241
You could do that
with, I don't know,

00:00:35.241 --> 00:00:36.890
I'm not exactly
a surveyor, but I

00:00:36.890 --> 00:00:38.767
guess a viewfinder or
something like that.

00:00:38.767 --> 00:00:41.350
So you're able to-- and I'm not
sure if that's the right tool,

00:00:41.350 --> 00:00:43.016
but there are tools
that you can measure

00:00:43.016 --> 00:00:47.270
the angles between the
horizontal and something that's

00:00:47.270 --> 00:00:48.680
not on the horizontal.

00:00:48.680 --> 00:00:51.090
So you know that this
angle right over here

00:00:51.090 --> 00:00:55.630
is pi over 4 radians,
or 45 degrees.

00:00:55.630 --> 00:00:57.090
We're going to
keep it pi over 4,

00:00:57.090 --> 00:00:59.256
because when you take
derivatives of trig functions,

00:00:59.256 --> 00:01:02.000
you assume that you're
dealing with radians.

00:01:02.000 --> 00:01:05.269
So right over here, this
is pi over 4 radians.

00:01:05.269 --> 00:01:08.530
And you also are able to measure
the rate at which this angle is

00:01:08.530 --> 00:01:09.400
changing.

00:01:09.400 --> 00:01:20.390
So this is changing at
0.2 radians per minute.

00:01:20.390 --> 00:01:22.124
Now my question to
you, or the question

00:01:22.124 --> 00:01:24.540
that you're trying to figure
out as you watch this hot air

00:01:24.540 --> 00:01:27.820
balloon, is how fast
is it rising right now?

00:01:27.820 --> 00:01:30.500
How fast is it rising
just as the angle

00:01:30.500 --> 00:01:33.750
between the horizontal and
kind of the line between you

00:01:33.750 --> 00:01:36.250
and the hot air balloon
is pi over 4 radians,

00:01:36.250 --> 00:01:41.330
and that angle is changing
at 0.2 radians per minute?

00:01:41.330 --> 00:01:42.760
So let's think
about what we know

00:01:42.760 --> 00:01:44.590
and what we're
trying to figure out.

00:01:44.590 --> 00:01:45.920
So we know a couple of things.

00:01:45.920 --> 00:01:49.210
We know that theta
is equal to pi over 4

00:01:49.210 --> 00:01:52.210
if we call theta the
angle right over here.

00:01:52.210 --> 00:01:54.420
So this is theta.

00:01:54.420 --> 00:01:56.610
We also know the rate at
which data is changing.

00:01:56.610 --> 00:01:58.210
We know d theta.

00:01:58.210 --> 00:01:59.270
Let me do this in yellow.

00:01:59.270 --> 00:02:07.848
We know d theta dt is equal
to 0.2 radians per minute.

00:02:07.848 --> 00:02:09.389
Now what are we
trying to figure out?

00:02:09.389 --> 00:02:11.472
Well, we're trying to
figure out the rate at which

00:02:11.472 --> 00:02:13.420
the height of the
balloon is changing.

00:02:13.420 --> 00:02:15.320
So if you call
this distance right

00:02:15.320 --> 00:02:23.470
over here h, what we want
to figure out is dh dt.

00:02:23.470 --> 00:02:26.090
That's what we don't know.

00:02:26.090 --> 00:02:27.560
So what we'd want
to come up with

00:02:27.560 --> 00:02:32.550
is a relationship between dh
dt, d theta dt, and maybe theta,

00:02:32.550 --> 00:02:33.695
if we need it.

00:02:33.695 --> 00:02:35.070
Or another way to
think about it,

00:02:35.070 --> 00:02:37.540
if we can come up with
the relationship between h

00:02:37.540 --> 00:02:40.700
and theta, then we could take
the derivative with respect

00:02:40.700 --> 00:02:43.320
to t, and we'll probably get
a relationship between all

00:02:43.320 --> 00:02:44.530
of this stuff.

00:02:44.530 --> 00:02:48.980
So what's the relationship
between theta and h?

00:02:48.980 --> 00:02:50.950
Well, it's a little
bit of trigonometry.

00:02:50.950 --> 00:02:52.970
We know we're trying
to figure out h.

00:02:52.970 --> 00:02:55.740
We already know what this
length is right over here.

00:02:55.740 --> 00:02:58.050
We know opposite over adjacent.

00:02:58.050 --> 00:03:00.050
That's the definition
of tangent.

00:03:00.050 --> 00:03:01.740
So let's write that down.

00:03:01.740 --> 00:03:07.680
So we know that the
tangent of theta

00:03:07.680 --> 00:03:11.220
is equal to the opposite
side-- the opposite side is

00:03:11.220 --> 00:03:16.230
equal to h-- over the
adjacent side, which

00:03:16.230 --> 00:03:20.860
we know is going
to be a fixed 500.

00:03:20.860 --> 00:03:23.670
So there you have a relationship
between theta and h.

00:03:23.670 --> 00:03:30.560
And then to figure out a
relationship between d theta

00:03:30.560 --> 00:03:32.950
dt, the rate at which theta
changes with respect to t,

00:03:32.950 --> 00:03:36.500
and the rate at which h
changes with respect to t,

00:03:36.500 --> 00:03:38.270
we just have to
take the derivative

00:03:38.270 --> 00:03:41.590
of both sides of this with
respect to t implicitly.

00:03:41.590 --> 00:03:42.680
So let's do that.

00:03:42.680 --> 00:03:46.640
And actually, let me move over
this h over 500 a little bit.

00:03:46.640 --> 00:03:48.520
So let me move it
over a little bit

00:03:48.520 --> 00:03:51.360
so I have space to show
the derivative operator.

00:03:51.360 --> 00:03:53.270
So let's write it like that.

00:03:53.270 --> 00:03:56.390
And now let's take the
derivative with respect to t.

00:03:56.390 --> 00:03:58.185
So d dt.

00:03:58.185 --> 00:04:00.810
I'm going to take the derivative
with respect to t on the left.

00:04:00.810 --> 00:04:02.768
We're going to take the
derivative with respect

00:04:02.768 --> 00:04:04.429
to t on the right.

00:04:04.429 --> 00:04:05.970
So what's the
derivative with respect

00:04:05.970 --> 00:04:08.130
to t of tangent of theta?

00:04:08.130 --> 00:04:11.270
Well, we're just going to
apply the chain rule here.

00:04:11.270 --> 00:04:12.870
It's going to be
first the derivative

00:04:12.870 --> 00:04:17.610
of the tangent of theta with
respect to theta, which is just

00:04:17.610 --> 00:04:21.829
secant squared of theta,
times the derivative of theta

00:04:21.829 --> 00:04:26.080
with respect to t,
times d theta dt.

00:04:26.080 --> 00:04:32.470
Once again, this is
just the derivative

00:04:32.470 --> 00:04:37.950
of the tangent, the tangent
of something with respect

00:04:37.950 --> 00:04:40.660
to that something
times the derivative

00:04:40.660 --> 00:04:42.924
of the something
with respect to t.

00:04:42.924 --> 00:04:44.590
Derivative of tangent
theta with respect

00:04:44.590 --> 00:04:47.230
to theta times the derivative
theta with respect to t

00:04:47.230 --> 00:04:50.790
gives us the derivative of
tangent of theta with respect

00:04:50.790 --> 00:04:54.110
to t, which is what
we want when we

00:04:54.110 --> 00:04:55.900
use this type of a
derivative operator.

00:04:55.900 --> 00:04:57.816
We're taking the derivative
with respect to t,

00:04:57.816 --> 00:05:00.250
not just the derivative
with respect to theta.

00:05:00.250 --> 00:05:00.750
Fair enough.

00:05:00.750 --> 00:05:02.400
So this is the left hand side.

00:05:02.400 --> 00:05:05.460
And then the right hand
side becomes, well,

00:05:05.460 --> 00:05:09.370
it's just going to
be 1 over 500 dh dt.

00:05:09.370 --> 00:05:13.740
So 1 over 500 dh dt.

00:05:13.740 --> 00:05:16.130
We're literally saying
it's just 1 over 500 times

00:05:16.130 --> 00:05:18.850
the derivative of h
with respect to t.

00:05:18.850 --> 00:05:20.730
But now we have
our relationship.

00:05:20.730 --> 00:05:24.870
We have the relationship
that we actually care about.

00:05:24.870 --> 00:05:27.960
We have a relationship between
the rate at which the height is

00:05:27.960 --> 00:05:30.125
changing with respect
to time and the rate

00:05:30.125 --> 00:05:32.280
at which the angle is
changing with respect to time

00:05:32.280 --> 00:05:34.210
and our angle at any moment.

00:05:34.210 --> 00:05:36.850
So we can just take these values
up here, throw it in here,

00:05:36.850 --> 00:05:38.690
and then solve for the unknown.

00:05:38.690 --> 00:05:39.470
So let's do that.

00:05:39.470 --> 00:05:41.270
Let's do that right over here.

00:05:41.270 --> 00:05:43.730
So we get secant
squared of theta.

00:05:43.730 --> 00:05:46.440
So we get secant squared.

00:05:46.440 --> 00:05:49.090
Right now our
theta is pi over 4.

00:05:49.090 --> 00:05:50.534
Secant squared of pi over 4.

00:05:50.534 --> 00:05:52.200
Let me write those
colors in to show you

00:05:52.200 --> 00:05:54.420
that I'm putting
these values in.

00:05:54.420 --> 00:05:59.050
Secant squared of pi over 4.

00:05:59.050 --> 00:06:01.150
Secant times d theta dt.

00:06:01.150 --> 00:06:03.970
Well, that is just 0.2.

00:06:03.970 --> 00:06:08.120
So times 0.2.

00:06:08.120 --> 00:06:14.419
And then this is going to
be equal to 1 over 500.

00:06:14.419 --> 00:06:15.460
And we want to make sure.

00:06:15.460 --> 00:06:17.700
Since this is in
radians per minute,

00:06:17.700 --> 00:06:19.170
we're going to get meters per.

00:06:19.170 --> 00:06:20.692
And this is meters
right over here.

00:06:20.692 --> 00:06:22.900
We're going to get meters
per minute right over here.

00:06:22.900 --> 00:06:26.140
We just want to make sure we
know what our units are doing.

00:06:26.140 --> 00:06:29.160
I haven't written the units
here to save some space.

00:06:29.160 --> 00:06:33.196
But we get 1 over
500 times dh dt.

00:06:33.196 --> 00:06:34.570
So if we want to
solve for dh dt,

00:06:34.570 --> 00:06:36.790
we can multiply
both sides by 500,

00:06:36.790 --> 00:06:39.530
and you get the rate at
which our height is changing

00:06:39.530 --> 00:06:42.240
is equal to 500
times, let's see,

00:06:42.240 --> 00:06:44.280
secant squared of pi over 4.

00:06:44.280 --> 00:06:49.924
That is 1 over cosine
squared of pi over 4.

00:06:49.924 --> 00:06:51.090
Let me write this over here.

00:06:51.090 --> 00:06:57.810
Cosine of pi over 4 is
square root of 2 over 2.

00:06:57.810 --> 00:07:06.490
Cosine squared of pi over 4 is
going to be equal to 2 over 4,

00:07:06.490 --> 00:07:08.450
which is equal to 1/2.

00:07:08.450 --> 00:07:12.850
And so secant squared of pi
over 4 is just 1 over that,

00:07:12.850 --> 00:07:14.810
is equal to 2.

00:07:14.810 --> 00:07:17.830
So this is going to be
equal to-- let me rewrite

00:07:17.830 --> 00:07:20.790
this instead of-- so the
secant squared of pi over 4--

00:07:20.790 --> 00:07:23.290
let me erase this
right over here.

00:07:23.290 --> 00:07:26.770
Secant squared of pi over 4,
all of this business right

00:07:26.770 --> 00:07:28.700
over here simplifies to 2.

00:07:28.700 --> 00:07:34.594
So times 2 times 0.2 times 0.2.

00:07:34.594 --> 00:07:35.760
So what is this going to be?

00:07:35.760 --> 00:07:39.570
This is going to
be 500 times 0.4.

00:07:39.570 --> 00:07:43.650
So this is equal to
500 times-- let me just

00:07:43.650 --> 00:07:48.360
write a dot instead--
times 0.4, which

00:07:48.360 --> 00:07:51.520
is equal to-- let me make
sure I get this right.

00:07:51.520 --> 00:07:54.755
This would be with two 0's
and one behind the decimal.

00:07:54.755 --> 00:07:55.730
Yep, there you go.

00:07:55.730 --> 00:07:57.930
It would be 200.

00:07:57.930 --> 00:08:00.770
So the rate at which our
height is changing with respect

00:08:00.770 --> 00:08:09.920
to time right at that moment
is 200 meters per minute.

00:08:09.920 --> 00:08:16.810
dh dt is equal to 200
meters per minute.


Related rates: Falling ladder

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.920
So I've got a 10 foot ladder
that's leaning against a wall.

00:00:03.920 --> 00:00:07.587
But it's on very slick ground,
and it starts to slide outward.

00:00:07.587 --> 00:00:09.670
And right when it's-- and
right at the moment that

00:00:09.670 --> 00:00:12.170
we're looking at this ladder,
the base of the ladder is 8

00:00:12.170 --> 00:00:14.020
feet away from the
base of the wall.

00:00:14.020 --> 00:00:16.610
And it's sliding outward
at 4 feet per second.

00:00:16.610 --> 00:00:19.340
And we'll assume that the top
of the latter kind of glides

00:00:19.340 --> 00:00:22.050
along the side of the wall,
it stays kind of in contact

00:00:22.050 --> 00:00:24.320
with the wall and
moves straight down.

00:00:24.320 --> 00:00:27.130
And we see right over here, the
arrow is moving straight down.

00:00:27.130 --> 00:00:29.290
And our question
is, how fast is it

00:00:29.290 --> 00:00:33.296
moving straight
down at that moment?

00:00:33.296 --> 00:00:34.920
So let's think about
this a little bit.

00:00:34.920 --> 00:00:37.300
What do we know and
what do we not know?

00:00:37.300 --> 00:00:39.900
So if we call the distance
between-- let's call

00:00:39.900 --> 00:00:42.460
the distance between the
base of the wall and the base

00:00:42.460 --> 00:00:44.280
the ladder, let's call that x.

00:00:44.280 --> 00:00:47.600
We know right now x
is equal to 8 feet.

00:00:47.600 --> 00:00:49.120
We also know the
rate at which x is

00:00:49.120 --> 00:00:51.120
changing with respect to time.

00:00:51.120 --> 00:00:53.350
The rate at which x is
changing with respect to time

00:00:53.350 --> 00:00:54.800
is 4 feet per second.

00:00:54.800 --> 00:00:58.700
So we could call this dx dt.

00:00:58.700 --> 00:01:03.480
Now let's call the distance
between the top of the ladder

00:01:03.480 --> 00:01:05.820
and the base of the latter h.

00:01:05.820 --> 00:01:08.042
Let's call that h.

00:01:08.042 --> 00:01:09.750
So what we're really
trying to figure out

00:01:09.750 --> 00:01:15.720
is what dh dt is,
given that we know

00:01:15.720 --> 00:01:17.850
all of this other information.

00:01:17.850 --> 00:01:21.060
So let's see if we can come up
with the relationship between x

00:01:21.060 --> 00:01:23.980
and h and then take the
derivative with respect

00:01:23.980 --> 00:01:25.920
to time, maybe using
the chain rule.

00:01:25.920 --> 00:01:28.330
And see if we can
solve for dh dt knowing

00:01:28.330 --> 00:01:30.274
all of this information.

00:01:30.274 --> 00:01:32.190
Well, we know the
relationship between x and h

00:01:32.190 --> 00:01:34.760
at any time because of
the Pythagorean theorem.

00:01:34.760 --> 00:01:37.010
We can assume this
is a right angle.

00:01:37.010 --> 00:01:43.230
So we know that x
squared plus h squared is

00:01:43.230 --> 00:01:45.480
going to be equal to the
length of the ladder squared,

00:01:45.480 --> 00:01:48.404
is going to be equal to 100.

00:01:48.404 --> 00:01:49.820
And what we care
about is the rate

00:01:49.820 --> 00:01:52.580
at which these things
change with respect to time.

00:01:52.580 --> 00:01:54.920
So let's take the
derivative with respect

00:01:54.920 --> 00:01:57.260
to time of both sides of this.

00:01:57.260 --> 00:02:02.060
We're doing a little bit of
implicit differentiation.

00:02:02.060 --> 00:02:05.470
So what's the derivative with
respect to time of x squared?

00:02:05.470 --> 00:02:09.342
Well the derivative of x
squared with respect to x is 2x.

00:02:09.342 --> 00:02:10.800
And we're going to
have to multiply

00:02:10.800 --> 00:02:15.940
that times the derivative of
x with respect to t, dx dt.

00:02:15.940 --> 00:02:17.670
Just to be clear, this
is the chain rule.

00:02:17.670 --> 00:02:22.040
This is the derivative of x
squared with respect to x,

00:02:22.040 --> 00:02:26.540
which is 2x, times dx
dt to get the derivative

00:02:26.540 --> 00:02:28.690
of x squared with
respect to time.

00:02:28.690 --> 00:02:30.362
Just the chain rule.

00:02:30.362 --> 00:02:32.070
Now similarly, what's
the derivative of h

00:02:32.070 --> 00:02:34.200
squared with respect to time?

00:02:34.200 --> 00:02:37.150
Well that's just going
to be 2h, the derivative

00:02:37.150 --> 00:02:39.540
of h squared with
respect to h is 2h times

00:02:39.540 --> 00:02:43.550
the derivative of h
with respect to time.

00:02:43.550 --> 00:02:46.680
Once again, this right
over here is the derivative

00:02:46.680 --> 00:02:50.130
of h squared with respect
to h, times the derivative

00:02:50.130 --> 00:02:53.210
of h with respect to time, which
gives us the derivative of h

00:02:53.210 --> 00:02:55.142
squared with respect to time.

00:02:55.142 --> 00:02:56.850
And what do we get on
the right-hand side

00:02:56.850 --> 00:02:57.805
of our equation?

00:02:57.805 --> 00:02:59.680
Well the length of our
ladder isn't changing.

00:02:59.680 --> 00:03:02.290
This 100 isn't going to
change with respect to time.

00:03:02.290 --> 00:03:05.780
Derivative of a constant
is just equal 0.

00:03:05.780 --> 00:03:07.870
So now we have
it, a relationship

00:03:07.870 --> 00:03:10.210
between the rate of change
of h with respect to time.

00:03:10.210 --> 00:03:12.830
The rate of change of
x with respect to time.

00:03:12.830 --> 00:03:14.970
And then at a given
point in time,

00:03:14.970 --> 00:03:17.920
when the length of
x is x and h is h.

00:03:17.920 --> 00:03:21.740
But do we know what h is
when x is equal to 8 feet?

00:03:21.740 --> 00:03:23.360
Well, we can figure it out.

00:03:23.360 --> 00:03:26.591
When x is equal to 8 feet, we
can use the Pythagorean theorem

00:03:26.591 --> 00:03:27.090
again.

00:03:27.090 --> 00:03:31.650
We get 8 feet squared,
plus h squared

00:03:31.650 --> 00:03:34.640
is going to be equal to 100.

00:03:34.640 --> 00:03:36.730
So 8 squared is 64.

00:03:36.730 --> 00:03:41.780
Subtract it from both sides, you
get h squared is equal to 36.

00:03:41.780 --> 00:03:44.080
Take the positive square
root, a negative square root

00:03:44.080 --> 00:03:46.080
doesn't make sense because
then the ladder would

00:03:46.080 --> 00:03:48.410
be below the ground, it
would be somehow underground.

00:03:48.410 --> 00:03:51.220
So we get h is equal to 6.

00:03:51.220 --> 00:03:53.140
So this is something
that was essentially

00:03:53.140 --> 00:03:54.890
given by the problem.

00:03:54.890 --> 00:03:55.570
So now we know.

00:03:55.570 --> 00:03:57.320
We can look at this original
thing right over here,

00:03:57.320 --> 00:03:58.900
we know what x is,
that was given.

00:03:58.900 --> 00:04:00.690
Right now x is 8 feet.

00:04:00.690 --> 00:04:03.030
We know the rate of change
of x with respect to time.

00:04:03.030 --> 00:04:04.680
It's 4 feet per second.

00:04:04.680 --> 00:04:08.090
We know what h is
right now, it is 6.

00:04:08.090 --> 00:04:12.150
So then we can solve for the
rate of h with respect to time.

00:04:12.150 --> 00:04:13.380
So let's do that.

00:04:13.380 --> 00:04:19.690
So we get 2 times 8
feet, times 4 feet

00:04:19.690 --> 00:04:35.040
per second, so times
4, plus 2h, is going

00:04:35.040 --> 00:04:38.200
to be plus 2 times, our
height right now is 6,

00:04:38.200 --> 00:04:40.250
times the rate at which
our height is changing

00:04:40.250 --> 00:04:43.050
with respect to t is equal to 0.

00:04:43.050 --> 00:04:47.140
And so we get 2 times
8 times 4 is 64.

00:04:47.140 --> 00:04:53.350
Plus 12 dh dt is equal to 0.

00:04:53.350 --> 00:04:57.320
We can subtract 64 from
both sides, we get 12.

00:04:57.320 --> 00:04:59.540
12 times the derivative
of h with respect to time

00:04:59.540 --> 00:05:02.800
is equal to negative 64.

00:05:02.800 --> 00:05:05.520
And then we just have to
divide both sides by 12.

00:05:05.520 --> 00:05:07.550
And so now we get a
little bit of a drum roll.

00:05:07.550 --> 00:05:09.600
The derivative,
the rate of change

00:05:09.600 --> 00:05:13.280
of h with respect to time is
equal to negative 64 divided

00:05:13.280 --> 00:05:14.870
by 12.

00:05:14.870 --> 00:05:18.240
It's equal to
negative 64 over 12,

00:05:18.240 --> 00:05:20.620
which is the same
thing as negative 16

00:05:20.620 --> 00:05:22.730
over 3, yeah that's right.

00:05:22.730 --> 00:05:24.530
Which is equal to--
let me scroll over

00:05:24.530 --> 00:05:27.070
to the right a little
bit-- negative 5

00:05:27.070 --> 00:05:30.877
and 1/3 feet per second.

00:05:30.877 --> 00:05:31.460
So we're done.

00:05:31.460 --> 00:05:32.490
But let's just do
a reality check,

00:05:32.490 --> 00:05:34.989
does that make sense that we
got a negative value right over

00:05:34.989 --> 00:05:35.650
here?

00:05:35.650 --> 00:05:39.150
Well our height is decreasing.

00:05:39.150 --> 00:05:42.220
So it completely makes sense
that its rate of change

00:05:42.220 --> 00:05:46.620
is indeed negative.

00:05:46.620 --> 00:05:48.440
And we're done.


Related rates intro

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:03.860
So let's say that we've
got a pool of water

00:00:03.860 --> 00:00:07.665
and I drop a rock into the
middle of that pool of water.

00:00:11.170 --> 00:00:14.840
And a little while later,
a little wave, a ripple

00:00:14.840 --> 00:00:17.720
has formed that is moving
radially outward from where

00:00:17.720 --> 00:00:18.840
I dropped the rock.

00:00:18.840 --> 00:00:20.680
So let's see how
well I can draw that.

00:00:20.680 --> 00:00:22.480
So it's moving
radially outwards.

00:00:22.480 --> 00:00:24.920
So that is the
ripple that is formed

00:00:24.920 --> 00:00:28.320
from me dropping the
rock into the water.

00:00:28.320 --> 00:00:31.160
So it's a circle centered
at where the rock initially

00:00:31.160 --> 00:00:32.259
hit the water.

00:00:32.259 --> 00:00:38.310
And let's say right at this
moment the radius of the circle

00:00:38.310 --> 00:00:40.073
is equal to 3 centimeters.

00:00:43.250 --> 00:00:48.380
And we also know that
the radius is increasing

00:00:48.380 --> 00:00:52.370
at a rate of 1
centimeter per second.

00:00:52.370 --> 00:01:04.920
So radius growing at rate
of 1 centimeter per second.

00:01:04.920 --> 00:01:07.120
So given this, right
now our circle,

00:01:07.120 --> 00:01:10.210
our ripple circle has a
radius of 3 centimeters.

00:01:10.210 --> 00:01:12.010
And we know that the
radius is growing

00:01:12.010 --> 00:01:13.360
at 1 centimeter per second.

00:01:13.360 --> 00:01:16.920
Given that, at what rate
is the area growing?

00:01:16.920 --> 00:01:29.260
At what rate is area
of circle growing?

00:01:29.260 --> 00:01:30.760
Interesting.

00:01:30.760 --> 00:01:32.245
So let's think
about what we know

00:01:32.245 --> 00:01:33.870
and then what we
don't know, what we're

00:01:33.870 --> 00:01:35.590
trying to figure out.

00:01:35.590 --> 00:01:40.890
So if we call this radius
r, we know that right now r

00:01:40.890 --> 00:01:43.880
is equal to 3 centimeters.

00:01:43.880 --> 00:01:47.240
We also know the
rate at which r is

00:01:47.240 --> 00:01:49.450
changing with respect to time.

00:01:49.450 --> 00:01:53.790
We also know this information
right over here. dr dt,

00:01:53.790 --> 00:01:56.630
the rate at which the radius is
changing with respect to time,

00:01:56.630 --> 00:01:58.350
is 1 centimeter per second.

00:02:02.900 --> 00:02:05.060
Now what do we
need to figure out?

00:02:05.060 --> 00:02:08.410
Well, they say at what rate is
the area of the circle growing?

00:02:08.410 --> 00:02:10.080
So we need to figure
out at what rate

00:02:10.080 --> 00:02:11.560
is the area of
the circle-- where

00:02:11.560 --> 00:02:14.890
a is the area of the circle--
at what rate is this growing?

00:02:14.890 --> 00:02:18.370
This is what we
need to figure out.

00:02:18.370 --> 00:02:20.200
So what might be
useful here is if we

00:02:20.200 --> 00:02:23.200
can come up with a relationship
between the area of the circle

00:02:23.200 --> 00:02:25.170
and the radius of the
circle and maybe take

00:02:25.170 --> 00:02:26.924
the derivative with
respect to time.

00:02:26.924 --> 00:02:29.090
And we'll have to use a
little bit of the chain rule

00:02:29.090 --> 00:02:29.984
to do that.

00:02:29.984 --> 00:02:31.900
So what is the relationship
at any given point

00:02:31.900 --> 00:02:33.750
in time between the
area of the circle

00:02:33.750 --> 00:02:35.770
and the radius of the circle?

00:02:35.770 --> 00:02:38.390
Well, this is
elementary geometry.

00:02:38.390 --> 00:02:40.160
The area of a circle
is going to be

00:02:40.160 --> 00:02:47.070
equal to pi times the radius
of the circle squared.

00:02:47.070 --> 00:02:48.820
Now what we want
to do is figure out

00:02:48.820 --> 00:02:51.321
the rate at which the area is
changing with respect to time.

00:02:51.321 --> 00:02:52.778
So why don't we
take the derivative

00:02:52.778 --> 00:02:54.630
of both sides of this
with respect to time?

00:02:54.630 --> 00:02:56.350
And let me give myself a
little more real estate.

00:02:56.350 --> 00:02:58.340
Actually, let me just
rewrite what I just had.

00:02:58.340 --> 00:03:03.570
So pi r squared.

00:03:03.570 --> 00:03:04.980
Area is equal to pi r squared.

00:03:04.980 --> 00:03:07.229
I'm going to take the
derivative of both sides of this

00:03:07.229 --> 00:03:08.980
with respect to time.

00:03:08.980 --> 00:03:11.006
So the derivative
with respect to time.

00:03:11.006 --> 00:03:13.006
I'm not taking the
derivative with respect to r,

00:03:13.006 --> 00:03:16.732
I'm taking the derivative
with respect to time.

00:03:16.732 --> 00:03:18.440
So on the left-hand
side right over here,

00:03:18.440 --> 00:03:21.865
I'm going to have the
derivative of our area.

00:03:21.865 --> 00:03:23.990
Actually, let me just write
it in that green color.

00:03:28.780 --> 00:03:33.420
I'm going to have the
derivative of our area

00:03:33.420 --> 00:03:36.310
with respect to time
on the left-hand side.

00:03:36.310 --> 00:03:38.730
And on the right-hand
side, what do I have?

00:03:38.730 --> 00:03:41.100
Well, if I'm taking the
derivative of a constant times

00:03:41.100 --> 00:03:42.580
something, I can take
the constant out.

00:03:42.580 --> 00:03:43.550
So let me just do that.

00:03:43.550 --> 00:03:50.625
Pi times the derivative with
respect to time of r squared.

00:03:50.625 --> 00:03:53.000
And to make it a little bit
clearer what I'm about to do,

00:03:53.000 --> 00:03:54.458
why I'm using the
chain rule, we're

00:03:54.458 --> 00:03:56.732
assuming that r is
a function of time.

00:03:56.732 --> 00:03:58.190
If r wasn't a
function of time then

00:03:58.190 --> 00:03:59.690
area wouldn't be a
function of time.

00:03:59.690 --> 00:04:02.240
So instead of just writing
r, let me make it explicit

00:04:02.240 --> 00:04:03.490
that it is a function of time.

00:04:03.490 --> 00:04:05.290
I'll write r of t.

00:04:05.290 --> 00:04:07.319
So it's r of t,
which we're squaring.

00:04:07.319 --> 00:04:09.610
And we want to find the
derivative of this with respect

00:04:09.610 --> 00:04:10.880
to time.

00:04:10.880 --> 00:04:13.270
And here we just have
to apply the chain rule.

00:04:13.270 --> 00:04:16.990
We're taking the derivative of
something squared with respect

00:04:16.990 --> 00:04:17.756
to that something.

00:04:17.756 --> 00:04:20.089
So the derivative of that
something squared with respect

00:04:20.089 --> 00:04:25.810
to the something is going
to be 2 times that something

00:04:25.810 --> 00:04:27.415
to the first power.

00:04:27.415 --> 00:04:28.290
Let me make it clear.

00:04:28.290 --> 00:04:37.420
This is the derivative of r of t
squared with respect to r of t.

00:04:37.420 --> 00:04:39.630
The derivative of something
squared with respect

00:04:39.630 --> 00:04:40.461
to that something.

00:04:40.461 --> 00:04:42.710
If it was a derivative of x
squared with respect to x,

00:04:42.710 --> 00:04:43.500
we'd have 2x.

00:04:43.500 --> 00:04:46.990
If it was the derivative of r
of t squared with respect to r

00:04:46.990 --> 00:04:48.247
of t, it's 2r of t.

00:04:48.247 --> 00:04:50.580
But this doesn't get us just
the derivative with respect

00:04:50.580 --> 00:04:51.080
to time.

00:04:51.080 --> 00:04:54.450
This is just the derivative
with respect to r of t.

00:04:54.450 --> 00:04:57.270
The derivative at which this
changes with respect to time,

00:04:57.270 --> 00:05:04.830
we have to multiply this
times the rate at which r of t

00:05:04.830 --> 00:05:07.400
changes with respect to time.

00:05:07.400 --> 00:05:09.960
So the rate at which r of t
changes with respect to time?

00:05:09.960 --> 00:05:13.270
Well, we could just
write that as dr dt.

00:05:13.270 --> 00:05:15.240
These are equivalent
expressions.

00:05:15.240 --> 00:05:17.680
And of course, we
have our pi out front.

00:05:17.680 --> 00:05:19.930
And I just want to
emphasize this is just

00:05:19.930 --> 00:05:21.590
the chain rule right over here.

00:05:21.590 --> 00:05:25.060
The derivative of something
squared with respect to time

00:05:25.060 --> 00:05:27.310
is going to be the derivative
of the something squared

00:05:27.310 --> 00:05:28.559
with respect to the something.

00:05:28.559 --> 00:05:30.960
So that's 2 times the
something, times the derivative

00:05:30.960 --> 00:05:32.860
of that something
with respect to time.

00:05:32.860 --> 00:05:34.260
I can't emphasize enough.

00:05:34.260 --> 00:05:38.370
What we did right over here,
this is the chain rule.

00:05:38.370 --> 00:05:42.150
That is the chain rule.

00:05:42.150 --> 00:05:44.160
So we're left with
pi times this is

00:05:44.160 --> 00:05:48.290
equal to the derivative of
our area with respect to time.

00:05:48.290 --> 00:05:50.534
Now let me rewrite
all this again

00:05:50.534 --> 00:05:51.950
just so it cleans
up a little bit.

00:05:51.950 --> 00:05:55.050
So we have the derivative
of our area with respect

00:05:55.050 --> 00:05:58.594
to time is equal to
pi times-- actually

00:05:58.594 --> 00:05:59.760
let me put that 2 out front.

00:05:59.760 --> 00:06:04.510
Is equal to 2 times
pi times-- I can now

00:06:04.510 --> 00:06:05.990
switch back to just
calling this r.

00:06:05.990 --> 00:06:07.760
We know that r is
a function of t.

00:06:07.760 --> 00:06:11.970
So I'll just write 2
pi times r times dr dt.

00:06:11.970 --> 00:06:14.180
Actually, let me
make the r in blue.

00:06:14.180 --> 00:06:17.225
2 pi r dr dt.

00:06:22.320 --> 00:06:24.000
Now what do we know?

00:06:24.000 --> 00:06:25.150
We know what r is.

00:06:25.150 --> 00:06:28.390
We know that r, at this
moment right in time,

00:06:28.390 --> 00:06:30.220
is 3 centimeters.

00:06:30.220 --> 00:06:32.410
Right now r is 3 centimeters.

00:06:32.410 --> 00:06:37.420
We know dr dt right now is
1 centimeter per second.

00:06:37.420 --> 00:06:41.840
We know this is 1
centimeter per second.

00:06:41.840 --> 00:06:44.160
So what's da dt
going to be equal to?

00:06:44.160 --> 00:06:47.720
Well, it's going to be equal
to-- do that same green-- 2

00:06:47.720 --> 00:06:56.140
pi times 3 times 3
times 1 times-- that's

00:06:56.140 --> 00:07:00.240
purple-- times 1
centimeter per second.

00:07:00.240 --> 00:07:02.190
And let's make sure we
get the units right.

00:07:02.190 --> 00:07:04.880
So we have a centimeter
times a centimeter.

00:07:04.880 --> 00:07:06.836
So it's going to be centimeters.

00:07:06.836 --> 00:07:07.960
That's too dark of a color.

00:07:07.960 --> 00:07:10.540
It's going to be square
centimeters, centimeters

00:07:10.540 --> 00:07:13.250
times centimeters, square
centimeter per second, which

00:07:13.250 --> 00:07:15.960
is the exact units we
need for a change in area.

00:07:15.960 --> 00:07:19.730
So we have da dt is equal to
this. da, the rate at which

00:07:19.730 --> 00:07:24.400
area is changing with respect
to time, is equal to 6 pi.

00:07:24.400 --> 00:07:32.590
So it's going to be a little
bit over 18 centimeters squared

00:07:32.590 --> 00:07:33.220
per second.

00:07:33.220 --> 00:07:34.100
Right at that moment.

00:07:34.100 --> 00:07:35.150
Yep, 3 times 2 pi.

00:07:35.150 --> 00:07:36.970
So 6 pi centimeters
squared per second

00:07:36.970 --> 00:07:40.190
is how fast the
area is changing.

00:07:40.190 --> 00:07:42.830
And we are done.


Related rates: water pouring into a cone

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.240
So we have got a very
interesting scenario here.

00:00:03.240 --> 00:00:08.730
I have this conical thimble-like
cup that is 4 centimeters high.

00:00:08.730 --> 00:00:11.450
And also, the diameter
of the top of the cup

00:00:11.450 --> 00:00:13.450
is also 4 centimeters.

00:00:13.450 --> 00:00:16.480
And I'm pouring water
into this cup right now.

00:00:16.480 --> 00:00:20.560
And I'm pouring the water at
a rate of 1 cubic centimeter.

00:00:20.560 --> 00:00:22.800
1 cubic centimeter per second.

00:00:22.800 --> 00:00:26.130
And right at this
moment, there is a height

00:00:26.130 --> 00:00:28.930
of 2 centimeters of water
in the cup right now.

00:00:28.930 --> 00:00:32.070
So the height right now from the
bottom of the cup to this point

00:00:32.070 --> 00:00:34.230
right over here
is 2 centimeters.

00:00:34.230 --> 00:00:36.980
So my question to you
is, at what rate--

00:00:36.980 --> 00:00:39.700
we know the rate at which the
water is flowing into the cup,

00:00:39.700 --> 00:00:43.390
we're being given
a volume per time.

00:00:43.390 --> 00:00:45.550
My question to you is
right at this moment,

00:00:45.550 --> 00:00:47.270
right when we are
filling our cup

00:00:47.270 --> 00:00:49.660
at 1 cubic centimeter
per second.

00:00:49.660 --> 00:00:51.910
And we have exactly
2 centimeters

00:00:51.910 --> 00:00:55.200
of water in the cup,
2 centimeters deep

00:00:55.200 --> 00:00:57.580
of water in the cup,
what is the rate at which

00:00:57.580 --> 00:01:01.770
the height of the
water is changing?

00:01:01.770 --> 00:01:05.540
What is the rate at which
this height right over here

00:01:05.540 --> 00:01:06.440
is actually changing?

00:01:06.440 --> 00:01:10.902
We know it's 2 centimeters,
but how fast is it changing?

00:01:10.902 --> 00:01:12.610
Well let's think about
this a little bit.

00:01:12.610 --> 00:01:13.877
What are we being given?

00:01:13.877 --> 00:01:15.960
We're given-- we are being
given the rate at which

00:01:15.960 --> 00:01:19.530
the volume of the water is
changing with respect to time.

00:01:19.530 --> 00:01:20.740
So let's write that down.

00:01:20.740 --> 00:01:24.290
We are being given the rate at
which the volume of the water

00:01:24.290 --> 00:01:26.900
is changing with
respect to time,

00:01:26.900 --> 00:01:34.630
and we're told that this is 1
cubic centimeter per second.

00:01:34.630 --> 00:01:36.440
And what are we
trying to figure out?

00:01:36.440 --> 00:01:38.810
Well we're trying to
figure out how fast

00:01:38.810 --> 00:01:41.330
the height of the water is
changing with respect to time.

00:01:41.330 --> 00:01:43.550
We know that the height
right now is 2 centimeters.

00:01:43.550 --> 00:01:45.560
But what we want to
figure out is the rate

00:01:45.560 --> 00:01:50.300
at which the height is
changing with respect to time.

00:01:50.300 --> 00:01:52.060
If we can figure
out this, then we

00:01:52.060 --> 00:01:54.880
have essentially
answered the question.

00:01:54.880 --> 00:01:56.620
So one way that we
can do this is we

00:01:56.620 --> 00:01:59.530
can come up with the
relationship between the volume

00:01:59.530 --> 00:02:02.704
at any moment in time and the
height at any moment in time.

00:02:02.704 --> 00:02:04.120
And then maybe
take the derivative

00:02:04.120 --> 00:02:06.870
of that relationship,
possibly using the chain rule,

00:02:06.870 --> 00:02:09.684
to come up with a relationship
between the rate at which

00:02:09.684 --> 00:02:12.100
the volume is changing and the
rate at which the height is

00:02:12.100 --> 00:02:13.250
changing.

00:02:13.250 --> 00:02:14.900
So let's try to do
it step by step.

00:02:14.900 --> 00:02:17.024
So first of all, can we
come up with a relationship

00:02:17.024 --> 00:02:21.220
between the volume and the
height at any given moment?

00:02:21.220 --> 00:02:23.550
Well we have also
been given the formula

00:02:23.550 --> 00:02:25.690
for the volume of a
cone right over here.

00:02:25.690 --> 00:02:28.480
The volume of a cone
is 1/3 times the area

00:02:28.480 --> 00:02:30.485
of the base of the
cone, times the height.

00:02:30.485 --> 00:02:32.110
And we won't prove
it here, although we

00:02:32.110 --> 00:02:33.110
could prove it later on.

00:02:33.110 --> 00:02:36.330
Especially when we start
doing solids of revolutions

00:02:36.330 --> 00:02:38.700
within in integral calculus.

00:02:38.700 --> 00:02:40.450
But we'll just take
it on faith right now,

00:02:40.450 --> 00:02:44.340
that this is how we can figure
out the volume of a cone.

00:02:44.340 --> 00:02:47.280
So given this can we
figure out volume--

00:02:47.280 --> 00:02:49.710
can we figure out an
expression that relates volume

00:02:49.710 --> 00:02:51.470
to the height of the cone?

00:02:51.470 --> 00:02:53.200
Well we could say
that volume-- and I'll

00:02:53.200 --> 00:02:56.380
do it in this blue color--
the volume of water

00:02:56.380 --> 00:02:57.980
is what we really care about.

00:02:57.980 --> 00:03:00.740
The volume of water
is going to be

00:03:00.740 --> 00:03:07.950
equal to 1/3 times the area
of the surface of the water--

00:03:07.950 --> 00:03:16.890
area of water surface-- times
our height of the water.

00:03:16.890 --> 00:03:18.710
So times h.

00:03:18.710 --> 00:03:21.010
So how can we figure out the
area of the water surface,

00:03:21.010 --> 00:03:23.320
preferably in terms of h?

00:03:23.320 --> 00:03:25.290
Well we see right over
here, the diameter

00:03:25.290 --> 00:03:27.920
across the top of the
cone is 4 centimeters.

00:03:27.920 --> 00:03:31.840
And the height of the
whole cup is 4 centimeters.

00:03:31.840 --> 00:03:34.600
And so that ratio is
going to be true of any--

00:03:34.600 --> 00:03:36.100
at any depth of water.

00:03:36.100 --> 00:03:39.100
It's always going to have the
same ratio between the diameter

00:03:39.100 --> 00:03:41.140
across the top and the height.

00:03:41.140 --> 00:03:43.230
Because these are
lines right over here.

00:03:43.230 --> 00:03:47.290
So at any given point, the
ratio between this and this

00:03:47.290 --> 00:03:48.420
is going to be the same.

00:03:48.420 --> 00:03:50.530
So at any given
point, the diameter

00:03:50.530 --> 00:03:53.500
across the surface of the
water-- if the depth is h,

00:03:53.500 --> 00:03:55.370
the diameter across the
surface of the water

00:03:55.370 --> 00:03:57.690
is also going to be h.

00:03:57.690 --> 00:03:59.112
And so from that
we can figure out

00:03:59.112 --> 00:04:00.570
what are the radius
is going to be.

00:04:00.570 --> 00:04:02.730
The radius is going
to be h over 2.

00:04:02.730 --> 00:04:05.060
And so the area of
the water surface

00:04:05.060 --> 00:04:10.500
is going to be pi r squared,
pi times the radius squared.

00:04:10.500 --> 00:04:13.490
h over 2 squared.

00:04:13.490 --> 00:04:15.670
That's the area of the
surface of the water.

00:04:15.670 --> 00:04:17.810
And of course we still
have the 1/3 out here.

00:04:17.810 --> 00:04:21.980
And we're still multiplying
by this h over here.

00:04:21.980 --> 00:04:23.900
So let me see if I
can simplify this.

00:04:23.900 --> 00:04:33.530
So this gives us 1/3 times pi h
squared over 4 times another h,

00:04:33.530 --> 00:04:42.350
which is equal to-- we have pi,
h to the third power over 12.

00:04:42.350 --> 00:04:45.600
So that is our volume.

00:04:45.600 --> 00:04:48.930
Now what we want to do is relate
the volume, how fast the volume

00:04:48.930 --> 00:04:50.510
is changing with
respect to time,

00:04:50.510 --> 00:04:53.210
and how fast the height is
changing with respect to time.

00:04:53.210 --> 00:04:55.877
So we care with respect to
time, since we care so much

00:04:55.877 --> 00:04:57.710
about what's happening
with respect to time,

00:04:57.710 --> 00:05:00.290
let's take the derivative of
both sides of this equation

00:05:00.290 --> 00:05:01.529
with respect to time.

00:05:01.529 --> 00:05:03.820
To do that, and just so I
have enough space to do that,

00:05:03.820 --> 00:05:05.240
let me move this over.

00:05:05.240 --> 00:05:08.420
Let me move this over to
the right a little bit.

00:05:08.420 --> 00:05:10.580
So I just move this
over to the right.

00:05:10.580 --> 00:05:13.310
And so now we can take the
derivative with respect

00:05:13.310 --> 00:05:16.350
to time of both sides
of this business.

00:05:16.350 --> 00:05:21.010
So the derivative with
respect to time of our volume

00:05:21.010 --> 00:05:24.839
and the derivative with respect
to time of this business.

00:05:24.839 --> 00:05:27.130
Well the derivative with
respect to time of our volume,

00:05:27.130 --> 00:05:30.850
we could just rewrite that as dV
dt, this thing right over here.

00:05:30.850 --> 00:05:36.850
This is dV dt, and
this is going to be

00:05:36.850 --> 00:05:39.404
equal to-- well we could
take the constants out

00:05:39.404 --> 00:05:41.320
of this-- this is going
to be equal to pi over

00:05:41.320 --> 00:05:52.300
12 times the derivative
with respect to t of h, of h

00:05:52.300 --> 00:05:55.620
to the third power.

00:05:55.620 --> 00:05:57.930
And just so that the
next few things I do

00:05:57.930 --> 00:05:59.539
will appear a
little bit clearer,

00:05:59.539 --> 00:06:01.580
we're assuming that height
is a function of time.

00:06:01.580 --> 00:06:02.780
In fact, it's definitely
a function of time.

00:06:02.780 --> 00:06:04.750
As time goes on, the
height will change.

00:06:04.750 --> 00:06:06.960
Because we're pouring
more and more water here.

00:06:06.960 --> 00:06:09.210
So instead of just writing
h to the third power, which

00:06:09.210 --> 00:06:12.770
I could write over here,
let me write h of t

00:06:12.770 --> 00:06:13.620
to the third power.

00:06:13.620 --> 00:06:16.822
Just to make it clear that
this is a function of t.

00:06:16.822 --> 00:06:21.160
h of t to the third power.

00:06:21.160 --> 00:06:23.430
Now what is the derivative
with respect to t,

00:06:23.430 --> 00:06:26.300
of h of t to the third power.

00:06:26.300 --> 00:06:28.190
Now, you might be getting
a tingling feeling

00:06:28.190 --> 00:06:31.710
that the chain rule
might be applicable here.

00:06:31.710 --> 00:06:33.846
So let's think about
the chain rule.

00:06:33.846 --> 00:06:36.220
The chain rule tells us-- let
me rewrite everything else.

00:06:36.220 --> 00:06:43.070
dV with respect to t, is going
to be equal to pi over 12,

00:06:43.070 --> 00:06:45.320
times the derivative of
this with respect to t.

00:06:45.320 --> 00:06:47.570
If we want to take the
derivative of this with respect

00:06:47.570 --> 00:06:49.630
to t, we have something
to the third power.

00:06:49.630 --> 00:06:52.580
So we want to take the
derivative of something

00:06:52.580 --> 00:06:54.750
to the third power with
respect to something.

00:06:54.750 --> 00:06:56.670
So that's going to
be-- let me write

00:06:56.670 --> 00:06:59.350
this in a different color,
maybe in orange-- so that's

00:06:59.350 --> 00:07:06.460
going to be 3 times
our something squared,

00:07:06.460 --> 00:07:11.840
times the derivative of that
something, with respect to t.

00:07:11.840 --> 00:07:21.770
Times dh-- I've already used
that pink-- times dh dt.

00:07:21.770 --> 00:07:23.250
Let's just be very clear.

00:07:23.250 --> 00:07:25.610
This orange term right
over here-- and I'm

00:07:25.610 --> 00:07:32.760
just using the chain rule--
this is the derivative of h of t

00:07:32.760 --> 00:07:38.550
to the third power
with respect to h of t.

00:07:38.550 --> 00:07:41.990
And then we're going to multiply
that times the derivative of h

00:07:41.990 --> 00:07:46.520
of t with respect to t.

00:07:46.520 --> 00:07:48.290
And then that gives
us the derivative

00:07:48.290 --> 00:07:52.081
of this entire thing, h of t to
the third power, with respect

00:07:52.081 --> 00:07:52.580
to t.

00:07:52.580 --> 00:07:56.800
This will give us
the derivative of h

00:07:56.800 --> 00:08:03.630
of t to the third power with
respect to d with respect

00:08:03.630 --> 00:08:05.090
to t, which is
exactly what we want

00:08:05.090 --> 00:08:06.740
to do when we apply
this operator.

00:08:06.740 --> 00:08:08.480
How fast is this changing?

00:08:08.480 --> 00:08:11.510
How is this changing
with respect to time?

00:08:11.510 --> 00:08:13.172
So we can just
rewrite this, just

00:08:13.172 --> 00:08:14.380
so gets a little bit cleaner.

00:08:14.380 --> 00:08:15.879
Let me rewrite
everything I've done.

00:08:15.879 --> 00:08:18.640
So we've got dV, the rate
at which our volume is

00:08:18.640 --> 00:08:20.570
changing with respect to time.

00:08:20.570 --> 00:08:24.050
The rate at which our volume is
changing with respect to time

00:08:24.050 --> 00:08:31.460
is equal to pi over 12
times 3 h of t squared,

00:08:31.460 --> 00:08:36.289
or I could just write
that as 3h squared,

00:08:36.289 --> 00:08:38.929
times the rate at which
the height is changing

00:08:38.929 --> 00:08:41.960
with respect to
time, times dh dt.

00:08:47.222 --> 00:08:48.680
And you might be
a little confused.

00:08:48.680 --> 00:08:52.670
You might have been tempted to
take the derivative over here

00:08:52.670 --> 00:08:53.607
with respect to h.

00:08:53.607 --> 00:08:55.190
But remember, we're
thinking about how

00:08:55.190 --> 00:08:57.890
things are changing
with respect to time.

00:08:57.890 --> 00:09:00.010
So we're assuming--
we did express volume

00:09:00.010 --> 00:09:01.820
as a function of
height-- but we're

00:09:01.820 --> 00:09:03.820
saying that height itself
is a function of time.

00:09:03.820 --> 00:09:05.653
So we're taking the
derivative of everything

00:09:05.653 --> 00:09:06.790
with respect to time.

00:09:06.790 --> 00:09:08.900
So that's why the chain
rule came into play

00:09:08.900 --> 00:09:10.690
when we were taking
the derivative of h,

00:09:10.690 --> 00:09:12.740
or the derivative of
h of t, because we're

00:09:12.740 --> 00:09:15.230
assuming that h is
a function of time.

00:09:15.230 --> 00:09:18.740
Now what does this thing
right over here get us?

00:09:18.740 --> 00:09:20.660
Well we're telling us
at the exact moment

00:09:20.660 --> 00:09:22.880
that we set up this
problem, we know

00:09:22.880 --> 00:09:27.900
what dV dt is, we know that it's
1 centimeter cubed per second.

00:09:27.900 --> 00:09:29.530
We know that this
right over here

00:09:29.530 --> 00:09:33.620
is 1 centimeter
cubed per second.

00:09:33.620 --> 00:09:35.620
We know what our height
is right at this moment.

00:09:35.620 --> 00:09:37.090
We were told it
is 2 centimeters.

00:09:43.240 --> 00:09:45.200
So the only unknown
we have over here

00:09:45.200 --> 00:09:46.750
is the rate at
which our height is

00:09:46.750 --> 00:09:48.070
changing with respect to time.

00:09:48.070 --> 00:09:50.320
Which is exactly what
we needed to figure out

00:09:50.320 --> 00:09:51.834
in the first place.

00:09:51.834 --> 00:09:53.250
So we just have
to solve for that.

00:09:53.250 --> 00:09:55.870
So we get 1 cubic
centimeter-- let

00:09:55.870 --> 00:10:01.220
me make it clear-- we get 1
cubic centimeter per second--

00:10:01.220 --> 00:10:03.520
I won't write the units
to save some space-- is

00:10:03.520 --> 00:10:05.306
equal to pi over 2.

00:10:05.306 --> 00:10:06.930
And I'll write this
in a neutral color.

00:10:06.930 --> 00:10:08.638
Actually, let me write
in the same color.

00:10:08.638 --> 00:10:15.480
Is equal to pi over 2,
times 3, times h squared.

00:10:15.480 --> 00:10:18.616
h is 2 so you're going to get
4 squared centimeters if we

00:10:18.616 --> 00:10:19.420
kept the units.

00:10:19.420 --> 00:10:21.674
So 3 times 4.

00:10:21.674 --> 00:10:23.340
All right let me be
careful, that wasn't

00:10:23.340 --> 00:10:25.765
pi over 2, that was pi over 12.

00:10:25.765 --> 00:10:29.780
This is a pi over
12 right over here.

00:10:29.780 --> 00:10:39.350
So you get pi over 12, times 3
times 2 squared, times dh dt.

00:10:39.350 --> 00:10:41.020
All of this is equal to 1.

00:10:41.020 --> 00:10:42.920
So now I'll switch
to a neutral color.

00:10:42.920 --> 00:10:46.130
We get 1 is equal to,
well 3 times 4 is 12,

00:10:46.130 --> 00:10:47.710
cancels out with that 12.

00:10:47.710 --> 00:10:51.800
We get one is equal
to pi times dh dt.

00:10:51.800 --> 00:10:54.770
To solve for dh dt
divide both sides by pi.

00:10:54.770 --> 00:10:57.162
And we get our drum roll now.

00:10:57.162 --> 00:10:59.370
The rate at which our height
is changing with respect

00:10:59.370 --> 00:11:03.260
to time as we're putting 1 cubic
centimeter of water per second

00:11:03.260 --> 00:11:03.910
in.

00:11:03.910 --> 00:11:05.987
And right when our height
is 2 centimeters high,

00:11:05.987 --> 00:11:08.570
the rate at which this height
is changing with respect to time

00:11:08.570 --> 00:11:13.330
is 1 over pi.

00:11:13.330 --> 00:11:15.330
And I haven't done the
dimensional analysis

00:11:15.330 --> 00:11:19.486
but this is going to be
in centimeters per second.

00:11:19.486 --> 00:11:21.360
You can work through
the dimensional analysis

00:11:21.360 --> 00:11:24.250
if you like by putting in the
dimensions right over here.

00:11:24.250 --> 00:11:25.610
But there you have it.

00:11:25.610 --> 00:11:27.890
That's how fast
our height is going

00:11:27.890 --> 00:11:31.790
to be changing at
exactly that moment.


Optimization: sum of squares

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:04.300
We are being asked, what
is the smallest-- this

00:00:04.300 --> 00:00:07.680
is a little typo here--
what is the smallest

00:00:07.680 --> 00:00:10.480
possible sum of
squares of two numbers

00:00:10.480 --> 00:00:14.170
if their product is negative 16?

00:00:14.170 --> 00:00:19.670
So let's say that these
two numbers are x and y.

00:00:19.670 --> 00:00:22.950
So how could we define the
sum of the squares of the two

00:00:22.950 --> 00:00:23.710
numbers?

00:00:23.710 --> 00:00:25.940
So I'll just call that
the sum of the squares,

00:00:25.940 --> 00:00:28.840
s for sum of the squares,
and it would just

00:00:28.840 --> 00:00:32.910
be equal to x squared
plus y squared.

00:00:32.910 --> 00:00:35.470
And this is what we
want to minimize.

00:00:35.470 --> 00:00:40.900
We want to minimize s.

00:00:40.900 --> 00:00:43.690
Now, right now s is expressed
as a function of x and y.

00:00:43.690 --> 00:00:46.190
We don't know how to minimize
with respect to two variables,

00:00:46.190 --> 00:00:48.540
so we have to get this in
terms of only one variable.

00:00:48.540 --> 00:00:51.660
And lucky for us, they give us
another piece of information.

00:00:51.660 --> 00:00:54.850
Their product is negative 16.

00:00:54.850 --> 00:01:00.690
So x times y is
equal to negative 16.

00:01:00.690 --> 00:01:03.360
So let's say we
wanted this expression

00:01:03.360 --> 00:01:05.330
right over here
only in terms of x.

00:01:05.330 --> 00:01:07.690
Well, then we can
figure out what

00:01:07.690 --> 00:01:10.095
y is in terms of x
and then substitute.

00:01:10.095 --> 00:01:11.470
So let's do that
right over here.

00:01:11.470 --> 00:01:16.700
If we divide both sides by x,
we get y is equal to negative 16

00:01:16.700 --> 00:01:17.750
over x.

00:01:17.750 --> 00:01:20.740
And so let's replace
our y in this expression

00:01:20.740 --> 00:01:23.030
with negative 16 over x.

00:01:23.030 --> 00:01:27.150
So then we would get
our sum of squares

00:01:27.150 --> 00:01:29.140
as a function of
x is going to be

00:01:29.140 --> 00:01:32.390
equal to x squared
plus y squared.

00:01:32.390 --> 00:01:35.230
y is negative 16 over x.

00:01:38.210 --> 00:01:41.020
And then that's what
we will now square.

00:01:41.020 --> 00:01:46.900
So this is equal to x
squared plus, what is this?

00:01:46.900 --> 00:01:48.800
256 over x squared.

00:01:48.800 --> 00:01:54.870
Or we could write that as
256x to the negative 2 power.

00:01:54.870 --> 00:01:59.690
That is the sum of our squares
that we now want to minimize.

00:01:59.690 --> 00:02:02.196
Well, to minimize
this, we would want

00:02:02.196 --> 00:02:04.070
to look at the critical
points of this, which

00:02:04.070 --> 00:02:07.260
is where the derivative
is either 0 or undefined,

00:02:07.260 --> 00:02:09.972
and see whether those
critical points are possibly

00:02:09.972 --> 00:02:11.180
a minimum or a maximum point.

00:02:11.180 --> 00:02:12.680
They don't have to
be, but those are

00:02:12.680 --> 00:02:15.012
the ones if we have a
minimum or a maximum point,

00:02:15.012 --> 00:02:16.970
they're going to be one
of the critical points.

00:02:16.970 --> 00:02:18.640
So let's take the derivative.

00:02:18.640 --> 00:02:21.060
So the derivative
s prime-- let me

00:02:21.060 --> 00:02:24.630
do this in a different
color-- s prime of x.

00:02:24.630 --> 00:02:26.320
I'll do it right
over here, actually.

00:02:26.320 --> 00:02:28.970
The derivative s prime
of x with respect

00:02:28.970 --> 00:02:32.520
to x is going to be
equal to 2x times

00:02:32.520 --> 00:02:38.360
negative 2 times 2x plus
256 times negative 2.

00:02:38.360 --> 00:02:42.705
So that's minus 512x to
the negative 3 power.

00:02:45.610 --> 00:02:52.650
Now, this is going to be
undefined when x is equal to 0.

00:02:52.650 --> 00:02:55.620
But if x is equal to
0, then y is undefined.

00:02:55.620 --> 00:02:57.240
So this whole thing breaks down.

00:02:57.240 --> 00:03:01.330
So that isn't a useful
critical point, x equals 0.

00:03:01.330 --> 00:03:02.920
So let's think about
any other ones.

00:03:02.920 --> 00:03:04.777
Well, it's defined
everywhere else.

00:03:04.777 --> 00:03:07.110
So let's think about where
the derivative is equal to 0.

00:03:07.110 --> 00:03:09.570
So when does this thing equal 0?

00:03:09.570 --> 00:03:16.040
So when does 2x minus 512x
to the negative 3 equal 0?

00:03:16.040 --> 00:03:19.520
Well, we can add 512x to the
negative 3 to both sides.

00:03:19.520 --> 00:03:26.390
So you get 2x is equal to 512x
to the negative third power.

00:03:26.390 --> 00:03:32.010
We can multiply both sides
times x to the third power

00:03:32.010 --> 00:03:34.890
so all the x's go away
on the right-hand side.

00:03:34.890 --> 00:03:40.380
So you get 2x to the
fourth is equal to 512.

00:03:40.380 --> 00:03:42.730
We can divide both
sides by 2, and you

00:03:42.730 --> 00:03:49.940
get x to the fourth
power is equal to 256.

00:03:49.940 --> 00:03:52.650
And so what is the
fourth root of 256?

00:03:52.650 --> 00:03:54.900
Well, we could take the
square root of both sides just

00:03:54.900 --> 00:03:56.140
to help us here.

00:03:56.140 --> 00:03:57.990
So let's see.

00:03:57.990 --> 00:04:01.400
So it's going to be x squared
is going to be equal to 256

00:04:01.400 --> 00:04:03.160
is 16 squared.

00:04:03.160 --> 00:04:05.180
So this is 16.

00:04:05.180 --> 00:04:12.500
This is going to be x squared is
equal to 16 or x is equal to 4.

00:04:12.500 --> 00:04:15.000
Now that's our only
critical point we have,

00:04:15.000 --> 00:04:17.170
so that's probably
the x value that

00:04:17.170 --> 00:04:19.790
minimizes our sum of
squares right over here.

00:04:19.790 --> 00:04:21.529
But let's make sure
it's a minimum value.

00:04:21.529 --> 00:04:24.680
And to do that, we can just
do our second derivative test.

00:04:24.680 --> 00:04:25.700
So let's figure out.

00:04:25.700 --> 00:04:27.640
Let's take the
second derivative s

00:04:27.640 --> 00:04:31.260
prime prime of x and figure
out if we are concave upwards

00:04:31.260 --> 00:04:34.030
or downwards when
x is equal to 4.

00:04:34.030 --> 00:04:37.690
So s prime prime of x is
going to be equal to 2.

00:04:37.690 --> 00:04:41.430
And then we're going to have
negative 3 times negative 512.

00:04:41.430 --> 00:04:45.070
So I'll just write that
as plus 3 times 512.

00:04:45.070 --> 00:04:48.077
That's going to be 1,536.

00:04:48.077 --> 00:04:48.660
Is that right?

00:04:48.660 --> 00:04:51.190
Yeah, 3 times 500
is 1,500, 3 times 12

00:04:51.190 --> 00:04:54.080
is 36, x to the
negative 4 power.

00:04:57.521 --> 00:04:59.270
And this thing right
over here is actually

00:04:59.270 --> 00:05:00.900
going to be positive for any x.

00:05:04.815 --> 00:05:08.024
x to the negative 4, even
if the negative x value,

00:05:08.024 --> 00:05:09.190
that's going to be positive.

00:05:09.190 --> 00:05:10.650
Everything else is positive.

00:05:10.650 --> 00:05:12.590
This thing is always positive.

00:05:12.590 --> 00:05:18.840
So we are always in a
concave upwards situation.

00:05:18.840 --> 00:05:22.480
Concave upwards means that
our graph might look something

00:05:22.480 --> 00:05:23.270
like that.

00:05:23.270 --> 00:05:25.395
Actually, I don't want to
draw the little squiggle.

00:05:25.395 --> 00:05:26.857
It might look
something like that.

00:05:26.857 --> 00:05:29.440
And you see there's a reason why
the second derivative implies

00:05:29.440 --> 00:05:31.760
concave upwards, a second
derivative positive

00:05:31.760 --> 00:05:35.119
means that our derivative
is constantly increasing.

00:05:35.119 --> 00:05:36.910
So the derivative is
constantly increasing.

00:05:36.910 --> 00:05:39.920
It's negative, less
negative, even less negative.

00:05:39.920 --> 00:05:41.400
Let me do it in a
different color.

00:05:41.400 --> 00:05:45.320
You see it's negative, less
negative, even less negative,

00:05:45.320 --> 00:05:47.940
0, positive, more positive.

00:05:47.940 --> 00:05:50.230
So it's increasing
over the entire place.

00:05:50.230 --> 00:05:53.290
So if you have a
critical point where

00:05:53.290 --> 00:05:57.830
the derivative is equal to 0,
so the slope is equal to 0,

00:05:57.830 --> 00:06:00.110
and it's concave upwards,
you see pretty clearly

00:06:00.110 --> 00:06:04.170
that we have minimized
the function.

00:06:04.170 --> 00:06:06.662
So what is y going
to be equal to?

00:06:06.662 --> 00:06:08.370
We actually don't even
have to figure out

00:06:08.370 --> 00:06:10.440
what y has to be
equal to in order

00:06:10.440 --> 00:06:11.961
to minimize the sum of squares.

00:06:11.961 --> 00:06:13.460
We could just put
it back into this.

00:06:13.460 --> 00:06:17.010
But just for fun, we see that
y would be negative 16 over x.

00:06:17.010 --> 00:06:20.900
So y would be equal
to negative 4.

00:06:20.900 --> 00:06:23.690
And we could just figure out
now what our sum of squares is.

00:06:23.690 --> 00:06:26.460
Our minimum sum of
squares is going

00:06:26.460 --> 00:06:31.780
to be equal to 4 squared,
which is 16 plus negative 4

00:06:31.780 --> 00:06:35.524
squared plus another 16,
which is equal to 32.

00:06:35.524 --> 00:06:37.440
Now I know some of you
might be thinking, hey,

00:06:37.440 --> 00:06:39.106
I could have done
this without calculus.

00:06:39.106 --> 00:06:41.650
I could have just tried out
numbers whose product is

00:06:41.650 --> 00:06:45.560
negative 16 and I probably would
have tried out 4 and negative 4

00:06:45.560 --> 00:06:47.282
in not too much time
and then I would

00:06:47.282 --> 00:06:49.740
have been able to maybe figure
out it's lower than if I did

00:06:49.740 --> 00:06:53.552
2 and negative 8 or negative
2 and 8 or 1 and 16.

00:06:53.552 --> 00:06:55.010
And that's true,
you probably would

00:06:55.010 --> 00:06:56.197
have been able to do that.

00:06:56.197 --> 00:06:58.280
But you still wouldn't
have been able to feel good

00:06:58.280 --> 00:07:00.405
that that was a minimum
value, because you wouldn't

00:07:00.405 --> 00:07:03.130
have tried out 4.01 or 4.0011.

00:07:03.130 --> 00:07:06.030
In fact, you couldn't have tried
out all of the possible values.

00:07:06.030 --> 00:07:08.155
Remember, we didn't say
that this is only integers.

00:07:08.155 --> 00:07:12.700
It just happened to be that
our values just worked out

00:07:12.700 --> 00:07:14.940
to be integers in
this situation.

00:07:14.940 --> 00:07:17.330
You can imagine what would
happen if the problem wasn't

00:07:17.330 --> 00:07:19.630
if their product is
negative 16, but what

00:07:19.630 --> 00:07:23.080
if their product is negative 17?

00:07:23.080 --> 00:07:25.880
Or what if the product
is negative 16.5?

00:07:25.880 --> 00:07:28.204
Or what if their
product was pi squared?

00:07:28.204 --> 00:07:30.370
Then you wouldn't be able
to try everything else out

00:07:30.370 --> 00:07:32.370
and you would have
to resort to doing

00:07:32.370 --> 00:07:34.640
what we did in this video.