Mean value theorem

WEBVTT
Kind: captions
Language: en

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I've gotten several requests
to explain or teach the

00:00:04.660 --> 00:00:05.590
mean value theorem.

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So let's do that in this video.

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So this is the mean
value theorem.

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And I have mixed feelings
about the mean value theorem.

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It's kind of neat, but what
you'll see is, it might not be

00:00:20.490 --> 00:00:22.820
obvious to prove, but the
intuition behind it's

00:00:22.820 --> 00:00:23.860
pretty obvious.

00:00:23.860 --> 00:00:27.010
And the reason I have mixed
feelings about it, is that even

00:00:27.010 --> 00:00:34.130
though, as you'll hopefully
see, the intuition is pretty

00:00:34.130 --> 00:00:36.770
obvious, but they stick it in
the math books, and people are

00:00:36.770 --> 00:00:39.890
just trying to learn calculus,
and get to what matters, and

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then they put the mean value
theorem in there, and they have

00:00:41.710 --> 00:00:43.700
all of this function notation,
and they have all of these

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words, and it just
confuses people.

00:00:45.570 --> 00:00:47.890
So hopefully this video will
clarify that little bit,

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and I'm curious to see
what you think of it.

00:00:51.360 --> 00:00:51.880
So let's see.

00:00:51.880 --> 00:00:53.820
What does the mean
value theorem say?

00:00:53.820 --> 00:00:54.890
Let me draw some axes.

00:00:54.890 --> 00:00:56.505
I'll do a visual
explanation first.

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I think this calls for magenta.

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So that's my x-axis.

00:01:07.050 --> 00:01:11.340
This is my y-axis.

00:01:11.340 --> 00:01:14.000
And let's say I have
some function f of x.

00:01:17.110 --> 00:01:22.040
So let me draw my f of x.

00:01:22.040 --> 00:01:24.100
That's as good as any.

00:01:24.100 --> 00:01:26.950
And this is some function f
of x, and I'm going to put

00:01:26.950 --> 00:01:28.750
a few conditions on f of x.

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f of x has to be continuous
and differentiable.

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And I know a lot of you
probably get intimidated

00:01:42.380 --> 00:01:43.270
when you hear these words.

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It sounds like what a
mathematician would say, and

00:01:45.830 --> 00:01:47.420
it sounds very abstract.

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All continuous means is that
the curve is connected to

00:01:51.940 --> 00:01:54.280
itself as you go along it.

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And here, the conditions are
over a closed interval.

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This is another very
mathy term you'll see.

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So you'll often say, on a
closed interval from a to b.

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All that means is an interval,
let's say a is the low point,

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let's say this is a, we don't
know what number that is.

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That could be minus
5, or who knows.

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And let's say this is b, right
here, let me b right here.

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Let's say that's be.

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So when people talk about a
closed interval, defined on a

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closed interval, that means
that the function needs to be

00:02:24.360 --> 00:02:28.460
defined at every number between
a and b, and the function needs

00:02:28.460 --> 00:02:31.250
to be defined at a and at b.

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If they said over an open
interval between a and b, that

00:02:33.930 --> 00:02:37.180
means that it's only defined at
every value between a and b,

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but not necessarily at a and b.

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So it has to be continuous,
differentiable, and let's say

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it's defined over the closed
interval, and this is just

00:02:47.790 --> 00:02:51.480
the notation for it, a b.

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So that means, it has to be
defined at all of the x values

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from a to b, including a and b.

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If it was an open interval,
you would write it like this.

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You'd write a and b.

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That means an interval for all
the numbers between a and

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b, but not including those.

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So let's ignore that for now.

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So back to the mean
value theorem.

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So you know, hopefully,
what continuous means.

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Let me draw here a function
that's not continuous.

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So a function that is
not continuous would

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look like this.

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It would go like this,
and it would start up

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here, go like that.

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Right?

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So this would be an example
of a function, let's say,

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same axes, let me draw
it in a different color.

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If that was our y-- no,
that's not a good.

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If that was our y-axis, and
that was our x-axis, just to

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give you the reference
for what I drew.

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So if the function is
continuous, continuous,

00:03:43.470 --> 00:03:47.560
continuous, and then it jumps,
that disconnect, that

00:03:47.560 --> 00:03:50.240
would make this function
discontinuous, or it would not

00:03:50.240 --> 00:03:51.720
be a continuous function.

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So a function just has
to be continuous.

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And now what does
differentiable mean?

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Differentiable means that at
every point over the interval

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that we care about, you have to
be able to find the derivative.

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That means you can take
the derivative of it.

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It's a differentiable.

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And what else does that mean?

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Well, that means that if you
were to graph the derivative

00:04:09.150 --> 00:04:12.460
of this function, that
it is also continuous.

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And I'll let you think
about that for a second.

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And actually, in this video I'm
going to show you an example of

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a function that is continuous,
but not differentiable and

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because of that, the mean
value theorem breaks down.

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But anyway, let's get back
to the mean value theorem.

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Most of the functions we
deal with satisfy all

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three of these things.

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Unless, you know, you're doing
limit problems, and they try to

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make these things break down.

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Anyway, back to the function.

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So this function meets all
of these requirements.

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So all it says is, if I were
take the average slope

00:04:43.830 --> 00:04:46.120
between point a and point b.

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So what is the slope, the
average slope between

00:04:48.800 --> 00:04:50.450
point a and point b?

00:04:50.450 --> 00:04:54.060
Well, slope is just
rise over run, right?

00:04:54.060 --> 00:04:55.610
So what is it?

00:04:55.610 --> 00:04:58.810
Let me see if I can draw
the average slope.

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So the run would
be this distance.

00:05:04.700 --> 00:05:07.565
That'd be the run, right,
and this would be the rise.

00:05:10.400 --> 00:05:12.630
So this is the point,
right here, that's

00:05:12.630 --> 00:05:17.270
the point a, f of a.

00:05:17.270 --> 00:05:23.450
Over here, this is
the point b, f of b.

00:05:23.450 --> 00:05:27.270
So what's the average
slope between a and b?

00:05:27.270 --> 00:05:29.240
Well, it's rise over run.

00:05:29.240 --> 00:05:30.190
So what's the rise?

00:05:30.190 --> 00:05:31.510
What's this distance?

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How much have we gone up
from f of a to f of b?

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Well, the rise will
just be f of b, this

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height, minus f of a.

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f of b minus f of a.

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And what's the run,
what's this distance?

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Well, it's just b minus a.

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And if I were to draw a line
that has that average slope, it

00:05:59.430 --> 00:06:00.890
would look something like this.

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We could make it go through
those two points, but it

00:06:03.690 --> 00:06:06.750
really doesn't have to.

00:06:06.750 --> 00:06:10.130
Let me do it in a blue.

00:06:14.570 --> 00:06:16.640
So that's the average
slope between those

00:06:16.640 --> 00:06:18.090
two points, right?

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So what does the mean
value theorem tell us?

00:06:20.030 --> 00:06:24.690
It says, if f of x is defined
over this closed interval from

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a to b, and f of x is
continuous, and it's

00:06:27.960 --> 00:06:29.750
differentiable, that you could
take the derivative at any

00:06:29.750 --> 00:06:45.560
point, that there must be some
points c f prime of c is

00:06:45.560 --> 00:06:47.350
equal to this thing.

00:06:47.350 --> 00:06:50.740
So is equal to f prime of c.

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I shouldn't have
written it here.

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So what is that telling us?

00:06:53.820 --> 00:06:57.140
So all that's telling us,
is if we're continuous,

00:06:57.140 --> 00:07:00.000
differentiable, defined over
the closed interval, that

00:07:00.000 --> 00:07:03.890
there's some point c, oh, and c
has to be between a and b,

00:07:03.890 --> 00:07:07.550
there's some point between a
and b, and it could be at one

00:07:07.550 --> 00:07:11.540
of the points, but there's some
point c where the derivative at

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c, or the slope at c, the
instantaneous slope at c, is

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exactly equal to the average
slope over that interval.

00:07:19.700 --> 00:07:21.090
So what does that mean?

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So we can look at it visually.

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Is there any point along this
curve where the slope looks

00:07:27.790 --> 00:07:31.850
very similar to this average
slope that we calculated?

00:07:31.850 --> 00:07:33.030
Well, sure, let's see.

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It looks like, maybe,
this point, right here?

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Just the way I drew it.

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This is very inexact.

00:07:40.320 --> 00:07:43.520
But that point looks like the
slope, you know, I could

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say the slope is something
like that, right there.

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So we don't know what,
analytically, this function is,

00:07:49.970 --> 00:07:53.280
but visually, you could see
that at this point c, the

00:07:53.280 --> 00:07:57.220
derivative, so I just
picked that point.

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So this could be our point c.

00:08:00.160 --> 00:08:01.220
And how do we just say that?

00:08:01.220 --> 00:08:05.360
Well, because f prime of c is
this slope, and it's equal

00:08:05.360 --> 00:08:06.610
to the average slope.

00:08:06.610 --> 00:08:10.190
So f prime of c is this thing,
and it's going to be equal to

00:08:10.190 --> 00:08:12.460
the average slope over
the whole thing.

00:08:12.460 --> 00:08:15.230
And this curve actually
probably has another point

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where the slope is equal
to the average slope.

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Let's see.

00:08:18.210 --> 00:08:21.360
This one looks, like,
right around there.

00:08:21.360 --> 00:08:23.710
Just the way I drew it.

00:08:23.710 --> 00:08:26.460
Looks like the slope there
could look something like,

00:08:26.460 --> 00:08:28.110
could be parallel as well.

00:08:28.110 --> 00:08:31.000
These lines should be parallel.

00:08:31.000 --> 00:08:33.040
The tangent lines
should be parallel.

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So hopefully that makes
a little sense to you.

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Another way to think about it
is that your average, actually,

00:08:41.170 --> 00:08:45.450
let me draw a graph just
to make sure that we

00:08:45.450 --> 00:08:47.620
hit the point home.

00:08:47.620 --> 00:08:50.870
Let's draw my position
as a function of time.

00:08:50.870 --> 00:08:53.630
So this is something,
this'll make it applicable

00:08:53.630 --> 00:08:55.000
to the real world.

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So that's my x-axis, or
the time axis, that's

00:08:57.610 --> 00:09:00.270
my position axis.

00:09:00.270 --> 00:09:03.140
This is going back to our
original intuition of what

00:09:03.140 --> 00:09:05.470
even a derivative is.

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So this is time, and I call
this position, or distance,

00:09:09.720 --> 00:09:11.050
or it doesn't matter.

00:09:11.050 --> 00:09:12.490
Position.

00:09:12.490 --> 00:09:15.910
And if I was moving at a
constant velocity, my position

00:09:15.910 --> 00:09:19.190
as a function of time would
just be a straight line, right?

00:09:19.190 --> 00:09:21.050
And the velocity is
actually your slope.

00:09:21.050 --> 00:09:22.590
But let's say I had
a varying velocity.

00:09:22.590 --> 00:09:25.910
And in reality, if you're
driving a car, you are always

00:09:25.910 --> 00:09:27.640
at a variable velocity.

00:09:27.640 --> 00:09:31.770
So let's say I start at a
standstill at time t equals 0,

00:09:31.770 --> 00:09:39.810
and then I accelerate, then I
decelerate a little bit,

00:09:39.810 --> 00:09:43.150
decelerate a little bit, I keep
decelerating, and then I come

00:09:43.150 --> 00:09:45.860
to a standstill, so my
position stays still.

00:09:45.860 --> 00:09:49.920
Then I accelerate
again, decelerate,

00:09:49.920 --> 00:09:52.450
accelerate, et cetera.

00:09:52.450 --> 00:09:53.080
Right?

00:09:53.080 --> 00:09:57.145
So this could be, you know, I
have a variable velocity, and

00:09:57.145 --> 00:09:59.500
this could be my position
as a function of time.

00:09:59.500 --> 00:10:04.390
So all this says, let's
say that after, this

00:10:04.390 --> 00:10:05.900
is time 0, position 0.

00:10:05.900 --> 00:10:10.920
Let's say after 1 hour, let's
say that is 1 hour, this time

00:10:10.920 --> 00:10:16.685
equals 1 hour, let's say
I have gone 60 miles.

00:10:20.660 --> 00:10:21.710
So what can you say?

00:10:21.710 --> 00:10:29.160
You could say that my average
velocity equals just change in

00:10:29.160 --> 00:10:31.370
distance divided by
change in times.

00:10:31.370 --> 00:10:33.700
It equals 60 miles per hour.

00:10:38.230 --> 00:10:40.950
So what the mean values
theorem says, is OK.

00:10:40.950 --> 00:10:43.780
Your average velocity, so you
could almost view it as the

00:10:43.780 --> 00:10:47.430
average slope between this
point and this point with 60,

00:10:47.430 --> 00:10:51.410
if your average velocity was 60
miles per hour, there was some

00:10:51.410 --> 00:10:54.950
point in time, maybe more, but
there was at least one point in

00:10:54.950 --> 00:10:58.290
time, where you were going
exactly sixty miles per hour.

00:10:58.290 --> 00:10:59.590
That make sense, right?

00:10:59.590 --> 00:11:02.030
If you average 60 miles per
hour, maybe you're going 40

00:11:02.030 --> 00:11:03.880
miles per hour some of the
point, but at some point you

00:11:03.880 --> 00:11:07.440
went 80, and in between you had
to be going 60 miles per hour.

00:11:07.440 --> 00:11:10.300
So let me see if I can
draw that graphically.

00:11:10.300 --> 00:11:14.683
So this slope is my average
velocity, and the way I drew

00:11:14.683 --> 00:11:17.110
it, there's probably two
points, let's see, probably

00:11:17.110 --> 00:11:21.050
right around here, I was
probably going 60 miles per

00:11:21.050 --> 00:11:23.380
hour, the slope is probably
60 there, the instantaneous

00:11:23.380 --> 00:11:29.100
velocity probably
there, as well.

00:11:29.100 --> 00:11:32.710
So before I leave, let's do
this analytically, just

00:11:32.710 --> 00:11:36.010
to work with numbers.

00:11:36.010 --> 00:11:38.280
And the reason why I have mixed
feelings about the mean value

00:11:38.280 --> 00:11:42.040
theorem, it's useful later on,
probably if you become a math

00:11:42.040 --> 00:11:46.670
major you'll maybe use it to
prove some theorems, or maybe

00:11:46.670 --> 00:11:48.290
you'll prove it, itself.

00:11:48.290 --> 00:11:50.390
But if you're just applying
calculus for the most part,

00:11:50.390 --> 00:11:51.900
you're not going to be
using the mean value

00:11:51.900 --> 00:11:52.610
theorem too much.

00:11:52.610 --> 00:11:54.200
But anyway, if you've got to
know it, you've got to know it,

00:11:54.200 --> 00:11:56.720
and it tells you something else
about the world, so it's

00:11:56.720 --> 00:11:58.450
interesting that way.

00:11:58.450 --> 00:12:04.680
So let's say we have the
function f of x is equal to x

00:12:04.680 --> 00:12:09.210
squared minus 4x, and the
interval that I care about

00:12:09.210 --> 00:12:12.880
here is between, is a closed
interval, so I'm including

00:12:12.880 --> 00:12:15.880
2, from 2 to 4.

00:12:15.880 --> 00:12:19.200
Now, the mean value theorem
tells us that if this

00:12:19.200 --> 00:12:22.050
function is defined on this
interval, and it is, right?

00:12:22.050 --> 00:12:23.380
We could put any number.

00:12:23.380 --> 00:12:26.180
The domain of this is actually
all real numbers, I could put

00:12:26.180 --> 00:12:29.050
any number here, so obviously
it's going to be defined

00:12:29.050 --> 00:12:30.990
over this interval.

00:12:30.990 --> 00:12:34.140
But so it's defined over the
interval, this is continuous,

00:12:34.140 --> 00:12:34.870
this is differentiable.

00:12:34.870 --> 00:12:36.160
You could take the derivative,
and the derivative

00:12:36.160 --> 00:12:37.380
is continuous.

00:12:37.380 --> 00:12:39.810
So the mean value theorem
should apply here.

00:12:39.810 --> 00:12:42.960
So let's see what value of
c is equal to the average

00:12:42.960 --> 00:12:45.400
slope between 2 and 4.

00:12:45.400 --> 00:12:48.700
So what's the average
slope between 2 and 4?

00:12:48.700 --> 00:12:52.780
Well, it's going to be f of 4,
so the change in the function,

00:12:52.780 --> 00:12:59.900
f of 4 minus f of 2 divided by
the change in x, so 4 minus 2.

00:12:59.900 --> 00:13:01.460
So this equal to
the average slope.

00:13:01.460 --> 00:13:06.800
So f of 4 is 16
minus 16, right?

00:13:06.800 --> 00:13:07.800
So that's 0.

00:13:07.800 --> 00:13:08.370
Let me make sure of that.

00:13:08.370 --> 00:13:12.740
4 times 4, 16, minus 4
times 4, 16, right.

00:13:12.740 --> 00:13:14.560
Minus f of 2.

00:13:14.560 --> 00:13:19.500
f of 2 is 2 squared,
is 4, right, and then

00:13:19.500 --> 00:13:21.730
minus 4 times 2.

00:13:21.730 --> 00:13:23.630
So minus 8.

00:13:27.810 --> 00:13:29.390
All of that over 2.

00:13:29.390 --> 00:13:31.060
And so this equals minus 4.

00:13:31.060 --> 00:13:33.120
So this equals 4 over 2.

00:13:33.120 --> 00:13:36.700
So the average slope from
x is equal to 2 to x

00:13:36.700 --> 00:13:39.420
is equal to 4 is 2.

00:13:39.420 --> 00:13:42.350
And now the mean value theorem
tells us, that there must be

00:13:42.350 --> 00:13:45.670
some point that's between these
two, maybe including one of

00:13:45.670 --> 00:13:50.340
those, where the slope at that
point is exactly equal to 2.

00:13:50.340 --> 00:13:51.900
Let's figure out
what point that is.

00:13:51.900 --> 00:13:53.390
That c.

00:13:53.390 --> 00:13:55.720
Let's take the derivative,
because the derivative at c

00:13:55.720 --> 00:13:56.620
is going to be equal to 2.

00:13:56.620 --> 00:13:57.690
So we just take the derivative.

00:13:57.690 --> 00:14:04.110
So let's say f prime of x
is equal to 2x minus four.

00:14:04.110 --> 00:14:08.990
And we want to figure out, at
what x value does this equal 2.

00:14:08.990 --> 00:14:09.600
So we say,

00:14:09.600 --> 00:14:13.460
2x minus 4 is equal to 2.

00:14:13.460 --> 00:14:16.030
Where does the slope equal 2?

00:14:16.030 --> 00:14:24.000
And you get 2x is equal
to 6, x is equal to 3.

00:14:24.000 --> 00:14:27.880
So if x is equal to 3, the
derivative is exactly equal

00:14:27.880 --> 00:14:28.630
to the average slope.

00:14:28.630 --> 00:14:30.990
But let me see if I can,
let me get the graphing

00:14:30.990 --> 00:14:33.540
calculator here.

00:14:33.540 --> 00:14:36.910
Let me what I can do.

00:14:36.910 --> 00:14:38.980
OK.

00:14:38.980 --> 00:14:43.090
So here's the graph of
x squared minus 4x.

00:14:43.090 --> 00:14:44.600
Let me see if I can make
it a little bit bigger.

00:14:47.920 --> 00:14:52.050
The interval that we care
about is from here to here.

00:14:52.050 --> 00:14:55.910
So the average slope over
that interval was 2.

00:14:55.910 --> 00:14:59.010
So if we were to draw the
slope, it was like that, the

00:14:59.010 --> 00:15:00.900
slope would look like that.

00:15:00.900 --> 00:15:04.930
And at the point 3, the
slope is exactly 2.

00:15:04.930 --> 00:15:07.540
So let me actually draw that.

00:15:07.540 --> 00:15:11.320
This isn't too hard
to draw, for myself.

00:15:11.320 --> 00:15:13.540
Let me see.

00:15:13.540 --> 00:15:21.270
So if that's the x-axis, I'll
want that graph out of the way.

00:15:21.270 --> 00:15:22.160
That's the y-axis.

00:15:24.960 --> 00:15:28.820
So the graph goes through
the point 0, 0 as

00:15:28.820 --> 00:15:31.010
neatly as possible.

00:15:31.010 --> 00:15:31.990
Nope, that's not neat.

00:15:35.680 --> 00:15:40.610
So the graph goes something
like this, it dips up, then it

00:15:40.610 --> 00:15:43.610
goes like that, and actually
it keeps going straight up,

00:15:43.610 --> 00:15:45.660
like that, it's a parabola.

00:15:45.660 --> 00:15:47.620
So this is point 4.

00:15:47.620 --> 00:15:51.210
The point 2 is here.

00:15:51.210 --> 00:15:56.150
And at 2 we're at negative
4, so the vertex is at

00:15:56.150 --> 00:15:57.710
the point 2, minus four.

00:15:57.710 --> 00:16:01.390
So what we said, the average
slope, so the closed interval

00:16:01.390 --> 00:16:04.280
that we care about, between 2
and 4, it's from 2

00:16:04.280 --> 00:16:05.960
here to 4 here.

00:16:05.960 --> 00:16:09.160
That's the interval, 2 to 4.

00:16:09.160 --> 00:16:12.850
The average slope is 2.

00:16:12.850 --> 00:16:14.570
Doesn't look like it, only
because I've kind of

00:16:14.570 --> 00:16:17.010
compressed the y-axis.

00:16:17.010 --> 00:16:20.090
And we're saying, at the point
x is equal to 3, the slope

00:16:20.090 --> 00:16:22.010
is equal to exactly that.

00:16:22.010 --> 00:16:25.180
So at x is equal to three,
the slope is equal

00:16:25.180 --> 00:16:26.860
to the same thing.

00:16:26.860 --> 00:16:28.620
That's all the mean
value theorem is.

00:16:28.620 --> 00:16:29.840
I know sounds complicated.

00:16:29.840 --> 00:16:32.810
People talk about continuity,
and differentiability, and f

00:16:32.810 --> 00:16:36.520
prime of c, and all this, but
all it says is, there's some

00:16:36.520 --> 00:16:41.020
point between these two points
where the instantaneous slope,

00:16:41.020 --> 00:16:43.690
or slope exactly at that
point, is equal to the slope

00:16:43.690 --> 00:16:45.510
between these two points.

00:16:45.510 --> 00:16:47.650
Hope I didn't confuse you.


More implicit differentiation

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:04.190
I've been asked to do a few
more implicit differentiation

00:00:04.190 --> 00:00:06.490
problems, and this is something
that I think confuses a lot of

00:00:06.490 --> 00:00:09.100
people, so let's do a
couple more problems.

00:00:09.100 --> 00:00:10.740
And I think there was a typo.

00:00:10.740 --> 00:00:14.660
I forgot the user that actually
sent this, but I'm guessing

00:00:14.660 --> 00:00:16.210
this is what he intended.

00:00:16.210 --> 00:00:18.430
And if this isn't what he
intended, this is something

00:00:18.430 --> 00:00:19.760
close, so it should
give the idea.

00:00:19.760 --> 00:00:32.030
So the implicit function they
gave was x squared minus 24xy

00:00:32.030 --> 00:00:45.560
plus 16y squared minus 400x
minus 300y is equal to 0.

00:00:45.560 --> 00:00:46.670
So, how do we do this?

00:00:46.670 --> 00:00:50.770
Well, we take the derivative of
both sides with respect to x.

00:00:50.770 --> 00:00:52.800
I could write the derivative
operator on both sides, but I

00:00:52.800 --> 00:00:55.460
think you know that at this
point, so that's the next step.

00:00:55.460 --> 00:00:58.330
So let's do that, and
I'll do it in blue.

00:00:58.330 --> 00:01:04.330
OK, derivative of x square
with respect to x is 2x.

00:01:04.330 --> 00:01:08.360
And now, let's see, it's
going to be minus 24.

00:01:08.360 --> 00:01:13.520
I'm going to take the constant
term out, so minus 24 times

00:01:13.520 --> 00:01:15.580
the derivative of xy.

00:01:15.580 --> 00:01:16.600
So what's the derivative of xy?

00:01:16.600 --> 00:01:18.450
We could use the
product rule here.

00:01:18.450 --> 00:01:21.470
The derivative of the first
term x, well, that's

00:01:21.470 --> 00:01:22.810
just 1, right?

00:01:22.810 --> 00:01:27.140
So 1 is the derivative of x
times the second expression,

00:01:27.140 --> 00:01:29.750
or you could say the second
mini-function or whatever,

00:01:29.750 --> 00:01:36.670
times y, and then plus the
first expression x times

00:01:36.670 --> 00:01:38.560
the derivative of the
second expression.

00:01:38.560 --> 00:01:41.260
Well, what's the derivative
of y with respect to x?

00:01:41.260 --> 00:01:44.660
And this sometimes confuses
people because it's so easy.

00:01:44.660 --> 00:01:46.810
Well, it's just dy/dx.

00:01:46.810 --> 00:01:49.240
Instead of writing dy/dx, I'm
just going to write y prime

00:01:49.240 --> 00:01:51.400
here just to get you used to a
different type of notation,

00:01:51.400 --> 00:01:53.430
although I prefer dy/dx
because you realize you're

00:01:53.430 --> 00:01:54.870
using the chain rule.

00:01:54.870 --> 00:02:03.360
But anyway, xy prime, that's
dy/dx, plus-- so we'll

00:02:03.360 --> 00:02:04.490
do the chain rule again.

00:02:04.490 --> 00:02:09.330
So let's take the derivative of
this expression with respect to

00:02:09.330 --> 00:02:13.590
y first, so that's just 32y.

00:02:13.590 --> 00:02:15.650
But we're taking the derivative
with respect to x, so we have

00:02:15.650 --> 00:02:19.000
to take the derivative of this
with respect y, which we did,

00:02:19.000 --> 00:02:22.380
and then multiply that times
the derivative of y with

00:02:22.380 --> 00:02:25.880
respect to x, which is
just dy/dx, or y prime.

00:02:29.660 --> 00:02:34.960
And then this with respect to
x, well, that's just minus 400.

00:02:34.960 --> 00:02:39.170
And then what's the derivative
of this with respect to x?

00:02:39.170 --> 00:02:46.520
Well, it's minus 300, and then
times the derivative of y with

00:02:46.520 --> 00:02:48.510
respect to x because we're
doing the chain rule, right?

00:02:48.510 --> 00:02:50.110
Minus 300 is just the
derivative of this

00:02:50.110 --> 00:02:51.800
with respect y.

00:02:51.800 --> 00:02:54.600
And then-- I want
to hit this point.

00:02:54.600 --> 00:02:56.840
You know, some people when they
learn this, they just say, oh,

00:02:56.840 --> 00:02:58.580
whenever I take a derivative of
y, I just take the derivative

00:02:58.580 --> 00:03:00.210
and I multiply by y prime.

00:03:00.210 --> 00:03:00.890
And that's right.

00:03:00.890 --> 00:03:01.820
That's essentially
what you're doing.

00:03:01.820 --> 00:03:04.350
But I want you understand
that you're doing the chain

00:03:04.350 --> 00:03:05.860
rule when you do that.

00:03:05.860 --> 00:03:07.730
So you're taking the derivative
with respect to y and then

00:03:07.730 --> 00:03:10.800
multiplying that times the
derivative with respect to x,

00:03:10.800 --> 00:03:13.690
which we'll write as y prime
now, and all of that

00:03:13.690 --> 00:03:16.140
is equal to 0.

00:03:16.140 --> 00:03:18.040
Now let's simplify this.

00:03:18.040 --> 00:03:38.210
Let's see, I get 2x minus 24y
minus 24xy prime plus 32yy

00:03:38.210 --> 00:03:49.650
prime minus 400 minus 300y
prime is equal to 0.

00:03:49.650 --> 00:03:51.760
And let's switch colors.

00:03:51.760 --> 00:03:54.060
So let's separate all
the y prime terms out.

00:03:54.060 --> 00:03:56.590
So let's see, what are
the y prime terms?

00:03:56.590 --> 00:04:02.420
We have this term has a y prime
in it, we have this term has a

00:04:02.420 --> 00:04:06.950
y prime in it, and we have this
term has a y prime in it.

00:04:06.950 --> 00:04:08.140
So we can write these.

00:04:08.140 --> 00:04:09.130
Let's do the positive
ones first.

00:04:09.130 --> 00:04:13.770
So 32y, so this is the
same thing as y prime.

00:04:13.770 --> 00:04:28.850
y prime times 32y
minus 24x minus 300.

00:04:28.850 --> 00:04:30.800
Those are all the y prime
terms, and then we're left

00:04:30.800 --> 00:04:32.150
with just the regular terms.

00:04:32.150 --> 00:04:44.920
So then plus 2x minus 24y
minus 400 is equal to 0.

00:04:44.920 --> 00:04:47.130
I just rearranged it and
took out the y prime out of

00:04:47.130 --> 00:04:49.380
this, this and this term.

00:04:49.380 --> 00:04:51.500
Let's take all of these to the
other side of the equation.

00:04:51.500 --> 00:04:53.430
Now , essentially what I want
to do is get all the terms that

00:04:53.430 --> 00:04:55.830
don't have the y prime onto the
right-hand side and then divide

00:04:55.830 --> 00:04:58.980
by this, and then we will
solve the equation.

00:04:58.980 --> 00:05:03.130
Let's see, so we get y prime
times-- and this is-- you could

00:05:03.130 --> 00:05:04.595
also write, you know the
derivative with respect

00:05:04.595 --> 00:05:06.860
to x or dy/dx.

00:05:06.860 --> 00:05:09.210
In general, even though it
takes more time, I like to

00:05:09.210 --> 00:05:11.300
normally write dy/dx, because,
you remember, I'm taking the

00:05:11.300 --> 00:05:13.370
derivative with respect
to x. y prime?

00:05:13.370 --> 00:05:13.850
I don't know.

00:05:13.850 --> 00:05:15.180
You might be taking the
derivative with respect

00:05:15.180 --> 00:05:16.960
to who knows what?

00:05:16.960 --> 00:05:26.280
Let's see, 32y minus 24x minus
300 is equal to-- put all these

00:05:26.280 --> 00:05:28.570
on the right-hand side so you
get-- I'll switch the order.

00:05:28.570 --> 00:05:30.760
I like to put the
positive numbers first.

00:05:30.760 --> 00:05:37.620
24y minus 2x plus 400, right?

00:05:37.620 --> 00:05:40.340
And then divide both sides by
this, and I'll do it in a

00:05:40.340 --> 00:05:42.570
different color because we're
almost at the answer, and we

00:05:42.570 --> 00:05:47.250
get y prime-- the derivative
with respect to x is equal to

00:05:47.250 --> 00:05:56.230
this thing-- 24y minus 2x plus
400, all of that over this

00:05:56.230 --> 00:06:03.930
thing: 32y minus 24x minus 300.

00:06:03.930 --> 00:06:08.270
And let's see, we could at
least-- it looks like-- well,

00:06:08.270 --> 00:06:11.160
everything's divisible by 2, so
we could divide the numerator

00:06:11.160 --> 00:06:19.410
and the denominator by 2, and
you get 12y minus x plus 200

00:06:19.410 --> 00:06:26.820
over 16y minus 12x minus 150.

00:06:26.820 --> 00:06:29.550
And if I didn't make a careless
mistake, that's the answer.

00:06:29.550 --> 00:06:32.780
The derivative of y with
respect to x any time is this

00:06:32.780 --> 00:06:35.280
expression that has y's and x
in it, and that's cool because

00:06:35.280 --> 00:06:39.060
this is a fairly convoluted
implicit function, but we can

00:06:39.060 --> 00:06:40.900
figure out the slope of it at
any point because we

00:06:40.900 --> 00:06:42.190
know the derivative.

00:06:42.190 --> 00:06:43.430
Let's do the next one.

00:06:49.130 --> 00:06:52.080
And this one is-- this
one looks simpler.

00:06:52.080 --> 00:07:04.300
x minus y squared is equal
to 8 times y minus 6.

00:07:04.300 --> 00:07:05.240
All right.

00:07:05.240 --> 00:07:08.290
So let's just take the
derivative with respect to x of

00:07:08.290 --> 00:07:12.250
both sides, and I'll use the
dy/dx notation this time.

00:07:12.250 --> 00:07:14.710
So let's first use
the chain rule.

00:07:14.710 --> 00:07:16.880
Let's take the derivative
of the inside.

00:07:16.880 --> 00:07:21.280
So what's the derivative
of the inside function?

00:07:21.280 --> 00:07:22.480
Well, we could do
it either way.

00:07:22.480 --> 00:07:23.000
Let's do it the other way.

00:07:23.000 --> 00:07:25.290
I always do inside and out, but
let's take the derivative of

00:07:25.290 --> 00:07:29.130
this entire expression with
respect to x minus y.

00:07:29.130 --> 00:07:35.250
Well, that is 2
times x minus y.

00:07:35.250 --> 00:07:38.900
And then it's going to be that
times the derivative of x

00:07:38.900 --> 00:07:41.010
minus y with respect to x.

00:07:41.010 --> 00:07:45.060
So what the derivative of x
minus y with respect to x?

00:07:45.060 --> 00:07:49.110
Well, the derivative of x
with respect to x is just 1.

00:07:49.110 --> 00:07:51.830
And what's the derivative of--
well, we could say minus

00:07:51.830 --> 00:07:53.020
y with respect to x.

00:07:53.020 --> 00:07:56.870
Well, it's just minus dy/dx.

00:07:56.870 --> 00:08:00.530
You could say the derivative of
y with respect to y is-- or the

00:08:00.530 --> 00:08:03.550
derivative of minus y with
respect to y is minus 1, and

00:08:03.550 --> 00:08:05.970
then by the chain rule,
multiply that times by

00:08:05.970 --> 00:08:07.620
the derivative of y
with respect to x.

00:08:07.620 --> 00:08:10.200
Either way, I think you
get the point, hopefully.

00:08:10.200 --> 00:08:13.210
So that's the derivative
of the left-hand side.

00:08:13.210 --> 00:08:16.540
And now what's the derivative
of the right-hand side?

00:08:16.540 --> 00:08:19.200
Well, we could do the chain
rule or whatever, but, you

00:08:19.200 --> 00:08:24.130
know, this is the same thing
as 8y minus 48, right?

00:08:24.130 --> 00:08:29.880
So we could rewrite
this is 8y minus 48.

00:08:29.880 --> 00:08:32.450
And so, if you take the
derivative of this with

00:08:32.450 --> 00:08:35.050
respect to x, the constant
doesn't contribute.

00:08:35.050 --> 00:08:37.955
So what's the derivative
of 8y with respect to x?

00:08:37.955 --> 00:08:41.590
Well, it's the derivative of 8y
with respect to y, which is

00:08:41.590 --> 00:08:46.970
just 8, and by the chain rule,
times the derivative of 8y with

00:08:46.970 --> 00:08:50.290
respect to y times
the derivative of y

00:08:50.290 --> 00:08:53.530
with respect to x.

00:08:53.530 --> 00:08:57.270
And now, we just
solve for dy/dx.

00:08:57.270 --> 00:09:00.710
Let's see, you get-- and I will
do a different color now.

00:09:00.710 --> 00:09:12.910
2x minus 2y times 1 minus
dy/dx is equal to 8 dy/dx.

00:09:15.920 --> 00:09:17.290
And then what can we do?

00:09:17.290 --> 00:09:18.960
We can distribute this out.

00:09:18.960 --> 00:09:21.540
So 2x times this whole thing.

00:09:21.540 --> 00:09:28.650
So 2x minus 2x dy/dx, right?

00:09:28.650 --> 00:09:32.280
That times that, and then
that times that, and

00:09:32.280 --> 00:09:33.320
then this times this.

00:09:33.320 --> 00:09:40.600
So minus 2y and then minus
minus, or plus, 2y dy/dx.

00:09:43.850 --> 00:09:45.680
And then let's take this
because I want to put all the

00:09:45.680 --> 00:09:48.300
dy/dx terms on the same side,
so let's subtract this from

00:09:48.300 --> 00:09:49.310
both sides of the equation.

00:09:49.310 --> 00:09:56.880
So then we get minus 8
dy/dx is equal to 0.

00:09:56.880 --> 00:09:59.870
And then the next step, well,
let's just separate out

00:09:59.870 --> 00:10:02.470
all the dy/dx terms.

00:10:02.470 --> 00:10:05.450
So that's a dy/dx term,
that's one, that's one.

00:10:05.450 --> 00:10:08.790
They're easier to spot than
when you write y prime.

00:10:08.790 --> 00:10:13.610
So if we take those together
and take out the dy/dx, we get

00:10:13.610 --> 00:10:18.140
dy, the derivative of y with
respect to x, times-- let's

00:10:18.140 --> 00:10:19.540
see, are any of them positive?

00:10:19.540 --> 00:10:20.940
I'll put that one first.

00:10:20.940 --> 00:10:24.190
2y minus 2x, right?

00:10:24.190 --> 00:10:25.420
That's this one.

00:10:25.420 --> 00:10:37.450
Minus 8 and then plus 2x
minus 2y is equal to 0.

00:10:37.450 --> 00:10:39.140
Let's put these onto
the right-hand side

00:10:39.140 --> 00:10:40.880
of the equation.

00:10:40.880 --> 00:10:42.610
We're almost there.

00:10:42.610 --> 00:10:53.250
dy/dx times 2y minus 2x minus 8
is equal to 2y minus 2x-- I

00:10:53.250 --> 00:10:55.540
just took those and put them on
the right-hand side-- and

00:10:55.540 --> 00:10:58.740
now divide both sides by
this, and we are done.

00:10:58.740 --> 00:11:10.610
dy/dx is equal to 2y minus
2x, all of that over

00:11:10.610 --> 00:11:13.640
2y minus 2x minus 8.

00:11:13.640 --> 00:11:16.030
And once again we can divide
the numerator and the

00:11:16.030 --> 00:11:17.390
denominator by 2.

00:11:17.390 --> 00:11:24.960
So we're left with y minus
x over y minus x minus 8.

00:11:24.960 --> 00:11:25.850
And we're done.

00:11:25.850 --> 00:11:28.840
Hopefully, you found
that helpful.


Optimization  Example 4

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:04.060
I've been sent another
maximization or minimization

00:00:04.060 --> 00:00:08.050
problem, and this seemed to be
an interesting twist on what

00:00:08.050 --> 00:00:10.030
we've already done, so
let's do this one.

00:00:10.030 --> 00:00:10.290
Let's see.

00:00:10.290 --> 00:00:13.190
A rectangular storage container
with an open top is to have

00:00:13.190 --> 00:00:15.450
a volume of 10 meters cubed.

00:00:15.450 --> 00:00:17.920
The length of its base
is twice its width.

00:00:17.920 --> 00:00:19.103
Let me start drawing
this thing.

00:00:19.103 --> 00:00:22.270
It's a rectangular storage
container with an open top.

00:00:22.270 --> 00:00:24.490
So let's see, let me do
it a different color.

00:00:27.290 --> 00:00:29.560
OK, so I'll do it over here.

00:00:29.560 --> 00:00:30.470
So that's its base.

00:00:32.980 --> 00:00:34.000
So let's see.

00:00:34.000 --> 00:00:36.830
The length of its base
is twice its width.

00:00:36.830 --> 00:00:41.530
So this is the base, and if
I call this the width, so

00:00:41.530 --> 00:00:46.990
this is x, the length is
twice that, this is 2x.

00:00:46.990 --> 00:00:50.420
The length of its base is
twice the width, right?

00:00:50.420 --> 00:00:55.560
So this is the length of the
base, this is the width.

00:00:55.560 --> 00:00:56.840
OK?

00:00:56.840 --> 00:01:09.250
Material and then the height,
for now, is unknown.

00:01:09.250 --> 00:01:11.550
I think we should be able to
find a constraint on this,

00:01:11.550 --> 00:01:14.790
because we know what the
volume has to be, right?

00:01:14.790 --> 00:01:18.330
It says, materials for
the sides of cost $6

00:01:18.330 --> 00:01:19.320
per square meter.

00:01:19.320 --> 00:01:23.490
Find the cost of materials for
the cheapest such container.

00:01:23.490 --> 00:01:25.660
OK, so let's do a
couple things.

00:01:25.660 --> 00:01:29.700
First, let's figure out what
the height is in terms of x,

00:01:29.700 --> 00:01:32.440
and then we can find an
equation for the cost in terms

00:01:32.440 --> 00:01:34.450
of x, and then take the
derivatives, and find the

00:01:34.450 --> 00:01:36.090
minimum, et cetera, et cetera.

00:01:36.090 --> 00:01:37.230
So what do we know about this?

00:01:37.230 --> 00:01:40.010
What's the volume of this box?

00:01:40.010 --> 00:01:44.020
The volume of this box is
equal to the, what did I

00:01:44.020 --> 00:01:48.480
say, this is the length, but
it doesn't really matter.

00:01:48.480 --> 00:01:50.170
Length times the width
times the height.

00:01:50.170 --> 00:01:53.110
So it's 2x times x,
2x squared times h.

00:01:53.110 --> 00:01:58.120
So it's 2x squared
times h, right?

00:01:58.120 --> 00:02:00.210
Base times height times
width, whatever.

00:02:00.210 --> 00:02:01.560
Length times width
times height.

00:02:01.560 --> 00:02:04.670
Just the volume of a rectangle.

00:02:04.670 --> 00:02:07.930
And they already said that the
volume is 10 meters cubed,

00:02:07.930 --> 00:02:09.380
so that gives us a
good constraint.

00:02:09.380 --> 00:02:12.780
So that has to equal 10.

00:02:12.780 --> 00:02:20.950
Let's divide both sides by 2x
squared, and so what do we get?

00:02:20.950 --> 00:02:23.540
If we take this equation, we
divide both sides by 2x

00:02:23.540 --> 00:02:31.090
squared, we get h is equal to
10 over 2x squared, which is

00:02:31.090 --> 00:02:37.870
equal to, we could say,
5x to the minus 2 power.

00:02:37.870 --> 00:02:42.880
So this is, so we could
write this as 5x to

00:02:42.880 --> 00:02:44.550
the minus 2 power.

00:02:44.550 --> 00:02:45.440
That is h.

00:02:45.440 --> 00:02:46.240
We have just solved that.

00:02:46.240 --> 00:02:48.660
We were able to solve that
because they gave us constraint

00:02:48.660 --> 00:02:50.350
on the volume of the box.

00:02:50.350 --> 00:02:51.320
Fair enough.

00:02:51.320 --> 00:02:54.560
So now let's see what the cost
of the material, the total cost

00:02:54.560 --> 00:02:56.550
of the material,
is going to be.

00:02:56.550 --> 00:02:59.090
So I'll do that in magenta.

00:02:59.090 --> 00:03:01.510
The cost of the material.

00:03:01.510 --> 00:03:04.680
And I can even say,
as a function of x.

00:03:04.680 --> 00:03:05.450
is equal to what?

00:03:05.450 --> 00:03:06.710
OK.

00:03:06.710 --> 00:03:13.360
Material for the sides
costs $6 per square meter.

00:03:13.360 --> 00:03:16.970
So what is the surface
area of the sides?

00:03:16.970 --> 00:03:19.330
Oh no, they tell the material
for the base first.

00:03:19.330 --> 00:03:23.720
So material for the base
costs $10 per square meter.

00:03:23.720 --> 00:03:27.010
So this base right here,
the material costs

00:03:27.010 --> 00:03:29.800
$10 per square meter.

00:03:29.800 --> 00:03:31.185
So what is the
cost of the base?

00:03:31.185 --> 00:03:33.370
It's going to be area
of the base times 10.

00:03:33.370 --> 00:03:37.980
So it's 10 times the
area of the base.

00:03:37.980 --> 00:03:39.180
What's the area of the base?

00:03:39.180 --> 00:03:41.320
It's 2x times x.

00:03:41.320 --> 00:03:43.600
So it's 2x squared.

00:03:43.600 --> 00:03:46.080
So this is the cost of the
material on the base.

00:03:46.080 --> 00:03:50.330
And now we want to add the cost
of the material on the sides.

00:03:50.330 --> 00:03:50.910
So.

00:03:50.910 --> 00:03:52.660
And this is where gets a
little bit interesting.

00:03:52.660 --> 00:04:00.580
So there's two sides that
are x, this side and this

00:04:00.580 --> 00:04:04.390
side are the same size.

00:04:04.390 --> 00:04:05.510
Right?

00:04:05.510 --> 00:04:06.470
And so what are they?

00:04:06.470 --> 00:04:08.410
What are their areas?

00:04:08.410 --> 00:04:10.810
So it's going to be $6 per
square meter for those

00:04:10.810 --> 00:04:12.040
two sides, right?

00:04:12.040 --> 00:04:15.560
For this green side, and that
one back there, as well.

00:04:15.560 --> 00:04:17.180
I don't want to make
this too messy.

00:04:17.180 --> 00:04:18.050
So $6.

00:04:18.050 --> 00:04:18.900
And what's the area?

00:04:18.900 --> 00:04:20.530
And there's 2 of these
sides, I'm going to

00:04:20.530 --> 00:04:22.750
multiply them by 2.

00:04:22.750 --> 00:04:25.040
And so what's the area
of each of these sides?

00:04:25.040 --> 00:04:28.510
It's x times 5x minus 2.

00:04:28.510 --> 00:04:32.650
So that's 5x minus 1.

00:04:32.650 --> 00:04:33.090
All right?

00:04:33.090 --> 00:04:34.820
So now we have to figure
out this side here and

00:04:34.820 --> 00:04:36.310
that side over there.

00:04:36.310 --> 00:04:38.570
And let me do it in
a brown, maybe.

00:04:38.570 --> 00:04:42.450
So now we care about this
side up in front, and then

00:04:42.450 --> 00:04:44.120
that side back there.

00:04:44.120 --> 00:04:46.420
But that's also going
to cost $6, right?

00:04:46.420 --> 00:04:49.050
So plus $6 per square meter.

00:04:49.050 --> 00:04:52.000
And then there's two sides,
and then what is it?

00:04:52.000 --> 00:04:58.110
It's 2x times 5x to
the negative 2.

00:04:58.110 --> 00:05:02.970
So that's 10, right, 2 times
5 is 10, x times, x to the

00:05:02.970 --> 00:05:05.600
negative 2, x to
the negative 1.

00:05:05.600 --> 00:05:07.390
Let's simplify this.

00:05:07.390 --> 00:05:13.820
So we have the cost, and it
said it was an open top, so

00:05:13.820 --> 00:05:16.960
there's no there's no fabric
we need for the top.

00:05:16.960 --> 00:05:21.060
So the cost of putting the
fabric, or the material, on

00:05:21.060 --> 00:05:28.330
this on this open rectangle,
is, see, 20x squared plus,

00:05:28.330 --> 00:05:36.270
what's this, 12 times 5 is 60x
to the minus 1, plus, let's

00:05:36.270 --> 00:05:41.610
see, 12 times 10, 120x to the
minus one, and so we can

00:05:41.610 --> 00:05:50.450
further simplify that as 20 x
squared plus 180 x to

00:05:50.450 --> 00:05:53.350
the minus 1 power.

00:05:53.350 --> 00:05:55.020
And now we're ready to just
take the derivative and

00:05:55.020 --> 00:05:58.290
find out what the minimum
point is of this curve.

00:05:58.290 --> 00:06:00.250
So let's do that.

00:06:00.250 --> 00:06:06.450
So c prime of x, and this is
a fun, non-hairy derivative.

00:06:06.450 --> 00:06:14.810
So it's 40, 2 times 20
x and then minus 180

00:06:14.810 --> 00:06:16.890
x to the minus 2.

00:06:16.890 --> 00:06:19.800
And the minimum or maximum
point are going to be where

00:06:19.800 --> 00:06:22.110
this is equal to 0, so
let's set that equal to 0.

00:06:22.110 --> 00:06:25.280
Let me clear up some space.

00:06:25.280 --> 00:06:26.800
I think you understand
all of this, so let

00:06:26.800 --> 00:06:29.910
me get rid of that.

00:06:35.070 --> 00:06:38.320
Actually, I like the drawing,
but it's got to go,

00:06:38.320 --> 00:06:41.540
because I need the room.

00:06:41.540 --> 00:06:44.370
If you need to review it,
just rewind the video.

00:06:44.370 --> 00:06:44.640
OK.

00:06:44.640 --> 00:06:47.700
So we want to set the
derivative, to find the minimum

00:06:47.700 --> 00:06:49.800
or maximum point, at minimum or
maximum point on the curve, the

00:06:49.800 --> 00:06:51.960
slope is 0, so the
derivative is 0.

00:06:51.960 --> 00:06:53.650
So let's figure that out.

00:06:53.650 --> 00:07:05.280
So 40x minus 180 x to the
minus 2 is equal to 0.

00:07:05.280 --> 00:07:07.980
And the first thing I would do
is just, you could divide both

00:07:07.980 --> 00:07:17.060
sides by 20, and you get 2x
minus 9x minus 2 is equal to

00:07:17.060 --> 00:07:19.320
0 divided by 20, which is 0.

00:07:19.320 --> 00:07:20.860
And then, let's
see what you get.

00:07:20.860 --> 00:07:26.890
If you add 9x to the minus 2 to
both sides, you get 2x is equal

00:07:26.890 --> 00:07:32.250
to 9x to the minus 2, or
9 over x squared, right?

00:07:32.250 --> 00:07:35.380
That's the same thing
is 9x to the minus 2.

00:07:35.380 --> 00:07:39.570
Now, I don't know, let's
multiply both sides of this

00:07:39.570 --> 00:07:49.480
equation by x squared, so you
get 2x to the third is equal to

00:07:49.480 --> 00:08:01.130
9, or x to the third is equal
to 9/2, or x is equal to the

00:08:01.130 --> 00:08:06.780
cube root, or we could just
say 9/2 to the 1/3 power,

00:08:06.780 --> 00:08:08.090
whatever that could be.

00:08:08.090 --> 00:08:10.020
4.5 to the 1/3 power.

00:08:10.020 --> 00:08:12.850
And do we know that this is a
minimum or a maximum point?

00:08:12.850 --> 00:08:18.340
Well, we could go and take the
second derivative, and then

00:08:18.340 --> 00:08:21.200
confirm that we're concave
upwards at this point, and then

00:08:21.200 --> 00:08:23.260
we would know that we are at a
minimum point, and this is

00:08:23.260 --> 00:08:24.780
the cheapest such container.

00:08:24.780 --> 00:08:25.930
But I'll tell you a
little, you know.

00:08:25.930 --> 00:08:29.030
If you see this in a class, or
maybe you want to prove that

00:08:29.030 --> 00:08:32.820
for yourself on a test, but an
easy way to test that is just,

00:08:32.820 --> 00:08:34.590
try some other x, right?

00:08:34.590 --> 00:08:36.460
We have the cost
function right here.

00:08:36.460 --> 00:08:37.525
Try some other x.

00:08:37.525 --> 00:08:41.230
And you know that this is
either going to be a maximum or

00:08:41.230 --> 00:08:43.490
a minimum, or at least a local
maximum or minimum, or if you

00:08:43.490 --> 00:08:45.115
try another x and if it's
greater, then you

00:08:45.115 --> 00:08:45.890
have a minimum.

00:08:45.890 --> 00:08:48.240
Or the other thing is, well,
you know, if there's only one

00:08:48.240 --> 00:08:50.170
critical point, and they ask
you to find the cheapest

00:08:50.170 --> 00:08:52.480
container, that critical
point is probably going

00:08:52.480 --> 00:08:54.420
to be the minimum point.

00:08:54.420 --> 00:08:54.750
But anyway.

00:08:54.750 --> 00:08:57.860
So just for fun, let's
figure out what that

00:08:57.860 --> 00:09:02.790
is actually equal to.

00:09:02.790 --> 00:09:10.660
I'll actually go to
google.com, and let's see.

00:09:13.590 --> 00:09:17.480
We could just say, we want
9/2 to the 1/3 power.

00:09:17.480 --> 00:09:29.530
So that is 4.5 to the 1
divided by third power.

00:09:29.530 --> 00:09:31.350
1.65.

00:09:31.350 --> 00:09:33.770
So that's our answer.

00:09:33.770 --> 00:09:38.210
If x is equal to 1.65, you
know, plus some other points,

00:09:38.210 --> 00:09:40.340
then we have the cheapest
container to produce.

00:09:40.340 --> 00:09:43.050
It's 1.65 something something.

00:09:43.050 --> 00:09:44.430
And there you have it.

00:09:44.430 --> 00:09:47.370
Hopefully you found
that vaguely useful.


More limits

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:02.790
I've been asked to do some
more limit problems.

00:00:02.790 --> 00:00:04.430
So let's do some more.

00:00:04.430 --> 00:00:04.860
So let's see.

00:00:04.860 --> 00:00:06.150
What's the first
problem I have?

00:00:06.150 --> 00:00:19.330
It says the limit as x
approaches 0 of x minus 2 times

00:00:19.330 --> 00:00:26.970
the absolute value of x over
the absolute value of x.

00:00:26.970 --> 00:00:29.730
Now what immediately confuses
most people is this

00:00:29.730 --> 00:00:30.780
absolute value thing.

00:00:30.780 --> 00:00:32.960
Because how do you
deal with it?

00:00:32.960 --> 00:00:34.490
Because you can't
just subtract.

00:00:34.490 --> 00:00:36.410
If this was an x, it would
be a simple problem.

00:00:36.410 --> 00:00:37.560
Because then you could
just simplify it.

00:00:37.560 --> 00:00:40.490
You could subtract this 2x from
this x and then divide by x.

00:00:40.490 --> 00:00:41.550
And you'd get an answer.

00:00:41.550 --> 00:00:43.740
But this has an absolute
value, so you can't just

00:00:43.740 --> 00:00:45.540
directly manipulate it.

00:00:45.540 --> 00:00:48.630
Well what if we're able to get
rid of this absolute value?

00:00:48.630 --> 00:00:49.620
How do we do that?

00:00:49.620 --> 00:00:53.340
Well, in order for this limit
to exist-- in order for

00:00:53.340 --> 00:00:54.250
this limit-- I don't know.

00:00:54.250 --> 00:00:56.870
Let's say it equals l, right?

00:00:56.870 --> 00:00:58.820
And l is what we
have to solve for.

00:00:58.820 --> 00:01:02.390
In order for that limit to
exist, then the limit to x

00:01:02.390 --> 00:01:05.230
approaches 0 also has to
exist from the positive

00:01:05.230 --> 00:01:06.180
and the negative sides.

00:01:06.180 --> 00:01:07.040
What do I mean?

00:01:07.040 --> 00:01:10.380
Well that means that
this must also be true.

00:01:10.380 --> 00:01:15.570
The limit as x approaches 0
from the right-hand side, from

00:01:15.570 --> 00:01:23.300
positive numbers, of x minus 2
times the absolute value of x

00:01:23.300 --> 00:01:27.210
over the absolute value of
x has to be equal to l.

00:01:27.210 --> 00:01:33.240
And then we also know that the
limit as x approaches 0 from

00:01:33.240 --> 00:01:38.060
the negative side of the same
function, 2 times absolute

00:01:38.060 --> 00:01:42.680
value of x over the absolute
value of x, is also going

00:01:42.680 --> 00:01:43.830
to be equal to l, right?

00:01:43.830 --> 00:01:47.360
If this-- If we-- As we
approach 0 from the left,

00:01:47.360 --> 00:01:48.255
we get some number.

00:01:48.255 --> 00:01:51.790
And as we approach 0 from the
right, we get the same number,

00:01:51.790 --> 00:01:54.540
then we know that no matter
what direction we approach 0

00:01:54.540 --> 00:01:58.140
from, which is this situation,
we approach that number.

00:01:58.140 --> 00:01:59.385
So let's figure out
these two limits.

00:01:59.385 --> 00:02:01.250
And if they have the same
answer, then we have the answer

00:02:01.250 --> 00:02:03.100
to our original problem.

00:02:03.100 --> 00:02:05.990
So how can we simplify this
as x approaches 0 from

00:02:05.990 --> 00:02:08.180
the positive side?

00:02:08.180 --> 00:02:13.590
Well that's the same thing as
the limit as x approaches

00:02:13.590 --> 00:02:14.840
0 from the positive side.

00:02:14.840 --> 00:02:17.750
Well if we know we're
approaching 0 from the positive

00:02:17.750 --> 00:02:19.210
side, what do we know about x?

00:02:19.210 --> 00:02:21.500
We know that x is
positive, right?

00:02:21.500 --> 00:02:23.530
It's approaching
from the right.

00:02:23.530 --> 00:02:25.440
So if x is positive, we
can get rid of these

00:02:25.440 --> 00:02:26.450
absolute value signs.

00:02:26.450 --> 00:02:32.470
So we just get x
minus 2x over x.

00:02:32.470 --> 00:02:38.070
And so that equals the limit
as x approaches 0 from

00:02:38.070 --> 00:02:39.270
the positive side.

00:02:39.270 --> 00:02:39.590
Let's see.

00:02:39.590 --> 00:02:42.760
That's x minus 2x.

00:02:42.760 --> 00:02:46.310
This is minus x, right?

00:02:46.310 --> 00:02:47.150
x minus x.

00:02:47.150 --> 00:02:53.430
And then minus x divided
by x is minus 1.

00:02:53.430 --> 00:02:55.590
So the limit as x approaches
0 from the positive

00:02:55.590 --> 00:02:56.420
side of minus 1.

00:02:56.420 --> 00:02:58.120
Well that's just going
to equal minus 1.

00:03:05.120 --> 00:03:05.480
OK.

00:03:05.480 --> 00:03:07.970
Now let's take this case down
here when we're approaching

00:03:07.970 --> 00:03:09.700
from the negative side.

00:03:12.830 --> 00:03:15.400
So how can we think about this?

00:03:15.400 --> 00:03:22.240
What is the absolute value of
x if x is a negative number?

00:03:22.240 --> 00:03:25.930
Well the absolute value of x,
if x is a negative number,

00:03:25.930 --> 00:03:29.440
is going to be the
negative of x, right?

00:03:29.440 --> 00:03:33.490
If x-- Think of it this way: if
x is negative 1, when we take

00:03:33.490 --> 00:03:35.680
the absolute value of x, we're
essentially just multiplying

00:03:35.680 --> 00:03:37.610
x times negative 1 again.

00:03:37.610 --> 00:03:45.160
So another way to rewrite this
is: this is equal to the limit

00:03:45.160 --> 00:03:52.060
as x approaches 0 from
the negative side of

00:03:52.060 --> 00:03:58.670
x minus 2 times what?

00:03:58.670 --> 00:04:01.650
The absolute value of x is the
same thing as negative x.

00:04:04.540 --> 00:04:06.000
Hopefully that makes
sense to you, right?

00:04:06.000 --> 00:04:08.070
If we're dealing with negative
numbers, then taking the

00:04:08.070 --> 00:04:11.160
absolute value of a negative
number is the same thing as

00:04:11.160 --> 00:04:13.710
multiplying that negative
number times the negative to

00:04:13.710 --> 00:04:17.690
essentially make it into a
positive number, right?

00:04:17.690 --> 00:04:19.626
And then, of course, the
absolute value of x in the

00:04:19.626 --> 00:04:21.450
denominator, since we're
dealing with a negative number,

00:04:21.450 --> 00:04:23.260
is also equal to negative x.

00:04:26.420 --> 00:04:28.040
And let's see if we
can simplify that.

00:04:28.040 --> 00:04:33.800
So that means that the limit
as x approaches 0 from the

00:04:33.800 --> 00:04:36.380
negative side-- Let's see.

00:04:36.380 --> 00:04:39.600
I get x minus 2
times negative x.

00:04:39.600 --> 00:04:41.230
So these negatives cancel out.

00:04:41.230 --> 00:04:45.380
So I get x plus 2x, right?

00:04:45.380 --> 00:04:48.495
So I get 3x over minus x.

00:04:56.680 --> 00:05:00.450
That equals minus 3.

00:05:00.450 --> 00:05:02.100
So this is interesting.

00:05:02.100 --> 00:05:05.200
I am approaching a different
number when I approach it

00:05:05.200 --> 00:05:06.950
from the left-hand side.

00:05:06.950 --> 00:05:10.500
When I approached this function
from the left-hand side or

00:05:10.500 --> 00:05:11.850
from the right-hand side.

00:05:11.850 --> 00:05:15.160
So it looks like this
limit doesn't exist.

00:05:15.160 --> 00:05:15.880
Let's confirm that.

00:05:15.880 --> 00:05:22.680
Let me actually-- Let me see if
I can-- Let me get the graphing

00:05:22.680 --> 00:05:25.250
calculator and confirm.

00:05:25.250 --> 00:05:26.640
Let me type it in.

00:05:26.640 --> 00:05:32.636
So x minus-- I'll show you
what I'm doing --x-- so you

00:05:32.636 --> 00:05:38.110
don't get bored --minus
absolute value of x.

00:05:38.110 --> 00:05:43.080
And what am I-- Divided by
the absolute value of x.

00:05:48.240 --> 00:05:49.130
Let's see.

00:05:49.130 --> 00:05:51.770
Graph.

00:05:51.770 --> 00:05:53.570
And what does it have here?

00:05:53.570 --> 00:05:54.170
Zoom out.

00:06:01.030 --> 00:06:03.530
So this is-- Is this right?

00:06:03.530 --> 00:06:08.250
x minus the absolute
value of x-- Oh, sorry.

00:06:08.250 --> 00:06:10.390
It's x minus 2 times the
absolute value of x.

00:06:15.050 --> 00:06:16.170
Graph.

00:06:16.170 --> 00:06:18.190
There you go.

00:06:18.190 --> 00:06:19.100
Even the-- oh.

00:06:19.100 --> 00:06:20.690
I don't think you
can see it yet.

00:06:20.690 --> 00:06:24.060
--the graphing calculator
confirms the work we did.

00:06:24.060 --> 00:06:25.900
Although it connects them and
it makes you think the it

00:06:25.900 --> 00:06:26.710
somehow approaches here.

00:06:26.710 --> 00:06:28.490
But that's just because
it picks points and

00:06:28.490 --> 00:06:29.880
just plots them.

00:06:29.880 --> 00:06:32.220
So as you see, as you approach
from the right-hand side, you

00:06:32.220 --> 00:06:36.020
approach negative 1, right?

00:06:36.020 --> 00:06:37.790
Well actually you're at
negative 1 the whole time

00:06:37.790 --> 00:06:38.770
on the right-hand side.

00:06:38.770 --> 00:06:40.210
And as you approach from
the left-hand side, you

00:06:40.210 --> 00:06:41.410
approach negative 3.

00:06:41.410 --> 00:06:44.090
So the limit does not
exist at x is equal 0.

00:06:44.090 --> 00:06:47.290
You would say l is undefined.

00:06:47.290 --> 00:06:50.700
You know, there's a little
dirty secret about limits.

00:06:50.700 --> 00:06:53.270
It's not a dirty secret, but
in theory you should never

00:06:53.270 --> 00:06:54.260
get a limit problem wrong.

00:06:54.260 --> 00:06:55.170
And why?

00:06:55.170 --> 00:06:57.510
Well you should be able to
solve it analytically.

00:06:57.510 --> 00:06:59.860
But if you don't know how to
solve it analytically, just

00:06:59.860 --> 00:07:02.080
put it really really
small numbers here.

00:07:02.080 --> 00:07:06.810
Try out-- If you put
in-- Try 0.0001.

00:07:06.810 --> 00:07:10.870
Try numbers that are just
slightly larger then whatever

00:07:10.870 --> 00:07:12.210
your limit number is.

00:07:12.210 --> 00:07:13.410
And then slightly smaller.

00:07:13.410 --> 00:07:15.630
And then, just numerically, if
you have a calculator, see

00:07:15.630 --> 00:07:16.370
what it's approaching.

00:07:16.370 --> 00:07:18.370
And sometimes you don't
even need a calculator.

00:07:18.370 --> 00:07:20.740
You could probably calculate
this in your head with 0.01

00:07:20.740 --> 00:07:22.160
or something like that.

00:07:22.160 --> 00:07:23.875
Anyway, let's do
another problem.

00:07:27.470 --> 00:07:29.390
Invert colors.

00:07:29.390 --> 00:07:30.140
OK.

00:07:30.140 --> 00:07:43.900
So let's do the limit
as x approaches 0 of

00:07:43.900 --> 00:07:50.800
sine of 5x over 2x.

00:07:50.800 --> 00:07:54.000
Now this looks a lot
like sine of x over x.

00:07:54.000 --> 00:07:55.750
And we know sine of x over x.

00:07:55.750 --> 00:07:57.280
What is that?

00:07:57.280 --> 00:07:59.630
And if you don't believe me,
watch the videos where we prove

00:07:59.630 --> 00:08:02.520
it using the squeeze theorem.

00:08:02.520 --> 00:08:05.130
We proved that the limit as
x approaches 0 of sine of

00:08:05.130 --> 00:08:09.440
x over x is equal to 1.

00:08:09.440 --> 00:08:11.065
Now this looks
almost like that.

00:08:11.065 --> 00:08:13.165
It would be great if we
could get it into that form

00:08:13.165 --> 00:08:14.040
and then we'd be done.

00:08:14.040 --> 00:08:16.350
So how can we do that?

00:08:16.350 --> 00:08:18.000
Well let's try to substitute.

00:08:18.000 --> 00:08:20.413
Let's try to get a single
variable here instead of

00:08:20.413 --> 00:08:21.560
a 5 times a variable.

00:08:21.560 --> 00:08:26.610
So let's make a
substitution in magenta.

00:08:26.610 --> 00:08:30.880
And say that a is equal to 5x.

00:08:30.880 --> 00:08:34.440
And then that also means that--
So divide both sides by 5 and

00:08:34.440 --> 00:08:37.640
we get x is equal to a over 5.

00:08:37.640 --> 00:08:40.050
So let's make that
substitution and do this.

00:08:40.050 --> 00:08:44.500
So this-- The limit as x
approaches 0 would be

00:08:44.500 --> 00:08:46.050
the same thing as what?

00:08:46.050 --> 00:08:50.080
If approaches-- As x approaches
0 here, a is also going

00:08:50.080 --> 00:08:51.720
to approach 0, right?

00:08:51.720 --> 00:08:55.450
So this is the same thing as
the limit as a approaches 0.

00:08:55.450 --> 00:08:57.675
Or you could view it this
way: as a approaches 0, x

00:08:57.675 --> 00:08:58.720
is still approaching 0.

00:08:58.720 --> 00:09:01.490
Maybe at 1/5 the pace,
but it's the same thing.

00:09:01.490 --> 00:09:09.730
[SIDE COMMENTS]

00:09:09.730 --> 00:09:10.150
OK.

00:09:10.150 --> 00:09:16.510
Anyway, this should say the
limit is a approaches 0, right?

00:09:16.510 --> 00:09:18.920
And hopefully you're satisfied
that the limit as a approaches

00:09:18.920 --> 00:09:21.790
0 is the same-- That x is
still approaching 0, right?

00:09:21.790 --> 00:09:24.180
Because if a approaches
0 here, that's going

00:09:24.180 --> 00:09:25.610
to make x approach 0.

00:09:25.610 --> 00:09:27.680
So it's the limit as a
approaches 0 of sine

00:09:27.680 --> 00:09:34.080
of a over 2 times x.

00:09:34.080 --> 00:09:35.770
But x is a over 5.

00:09:35.770 --> 00:09:39.490
So it's 2 times a over 5.

00:09:39.490 --> 00:09:49.000
And so that is equal to the
limit as a approaches 0

00:09:49.000 --> 00:09:56.600
of sine of a over 2/5 a.

00:09:56.600 --> 00:09:57.660
Well that's the same thing.

00:09:57.660 --> 00:09:59.730
We could take this out.

00:09:59.730 --> 00:10:01.870
We could take this 1 over
2/5-- right, it's in

00:10:01.870 --> 00:10:02.810
the denominator --out.

00:10:02.810 --> 00:10:04.240
This is just a constant
term, so we can take

00:10:04.240 --> 00:10:04.910
it out of the limit.

00:10:04.910 --> 00:10:06.410
And obviously if we're taking
it out of the denominator,

00:10:06.410 --> 00:10:07.190
it flips, right?

00:10:07.190 --> 00:10:11.970
Because this is just
1 over 2/5 or 5/2.

00:10:11.970 --> 00:10:16.520
So that equals 5/2-- right, I
just took this out of the

00:10:16.520 --> 00:10:20.230
denominator and it flips,
right, because it's 1 over 2/5

00:10:20.230 --> 00:10:26.040
--times the limit as
a approaches 0 of

00:10:26.040 --> 00:10:30.840
sine of a over a.

00:10:30.840 --> 00:10:33.980
And now that looks an
awful lot like this.

00:10:33.980 --> 00:10:35.810
It's just we have an a instead
of an x, but that doesn't

00:10:35.810 --> 00:10:36.730
make a difference.

00:10:36.730 --> 00:10:38.670
So this is equal to 1.

00:10:38.670 --> 00:10:45.570
So this whole thing, since this
is equal to 1, is equal to 5/2.

00:10:45.570 --> 00:10:48.430
And, once again, if you get
this answer and you're not so

00:10:48.430 --> 00:10:52.180
sure, take your calculator
out and try-- calculate.

00:10:52.180 --> 00:10:56.750
What is the sine
of 5 times 0.001?

00:10:56.750 --> 00:11:04.190
So sine of 0.005 divided by
2 times 0.001, so 0.002.

00:11:04.190 --> 00:11:06.290
If you take that, you're going
to get a number that's awfully

00:11:06.290 --> 00:11:07.360
close to this number.

00:11:07.360 --> 00:11:08.800
This is approximate.

00:11:08.800 --> 00:11:10.180
This is exactly 2.5.

00:11:10.180 --> 00:11:12.490
You're probably going
to get like 2.49999 or

00:11:12.490 --> 00:11:14.690
something like that.

00:11:14.690 --> 00:11:15.600
So let's do another one.

00:11:20.770 --> 00:11:22.440
OK.

00:11:22.440 --> 00:11:30.970
So I have the limit as x
approaches 0-- this one looks a

00:11:30.970 --> 00:11:33.590
lot like the previous one,
although we have some exponents

00:11:33.590 --> 00:11:40.820
here --of sine squared
of x over x squared.

00:11:40.820 --> 00:11:43.040
And that almost looks like
sine of x over x, but I

00:11:43.040 --> 00:11:44.060
have these squared terms.

00:11:44.060 --> 00:11:45.720
What can I do?

00:11:45.720 --> 00:11:46.740
Well this is the same thing.

00:11:46.740 --> 00:11:49.620
This is sine of x
squared over x squared.

00:11:49.620 --> 00:11:55.720
So this is the same thing as
the limit as x approaches 0

00:11:55.720 --> 00:12:03.380
of sine of x over
x squared, right?

00:12:03.380 --> 00:12:06.230
If you were to take this out,
you'd square the numerator,

00:12:06.230 --> 00:12:07.480
square the denominator,
and you'd get this.

00:12:07.480 --> 00:12:10.050
If you took-- If you
wanted to change this.

00:12:10.050 --> 00:12:11.950
I won't say simplify.

00:12:11.950 --> 00:12:13.770
Well now this is interesting.

00:12:13.770 --> 00:12:17.180
Well this 2 is-- This
is a constant term.

00:12:17.180 --> 00:12:18.770
This isn't going to
change with the limit.

00:12:18.770 --> 00:12:21.130
So essentially, we can say,
well this is going to be equal

00:12:21.130 --> 00:12:30.050
to the limit as x approaches 0
of sine of x over x, the

00:12:30.050 --> 00:12:31.240
whole thing squared.

00:12:31.240 --> 00:12:31.770
And how is that?

00:12:31.770 --> 00:12:34.280
I couldn't have done this if
there was an x here, right,

00:12:34.280 --> 00:12:36.000
because that would have
changed the limit.

00:12:36.000 --> 00:12:37.790
But this is a constant term.

00:12:37.790 --> 00:12:42.790
So this-- The base is the only
thing that's going to change as

00:12:42.790 --> 00:12:44.610
I take-- as x approaches 0.

00:12:44.610 --> 00:12:47.870
So I can take the limit
within the base, I guess you

00:12:47.870 --> 00:12:49.040
could say, and get here.

00:12:49.040 --> 00:12:52.300
And now we're back to what we
proved in a previous video.

00:12:52.300 --> 00:12:54.280
What is the limit as
x approaches 0 of

00:12:54.280 --> 00:12:55.430
sine of x over x?

00:12:55.430 --> 00:12:55.950
Right.

00:12:55.950 --> 00:12:59.000
It's 1.

00:12:59.000 --> 00:13:00.810
So what's 1 squared?

00:13:00.810 --> 00:13:02.350
It is 1.

00:13:02.350 --> 00:13:03.740
There you go.

00:13:03.740 --> 00:13:05.820
And I've taken 13
minutes of your time.

00:13:05.820 --> 00:13:08.330
Hopefully you've found
it vaguely Useful.


Exponential Growth

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:03.060
Let's do a problem on
exponential growth.

00:00:03.060 --> 00:00:06.150
And if you think about it, what
we're going to do here with a

00:00:06.150 --> 00:00:09.820
bacteria, it's no different
than what we did with compound

00:00:09.820 --> 00:00:16.190
interest when we learned about
interest and the number e.

00:00:16.190 --> 00:00:18.340
Because if you think about it--
well, I'll draw the analogy

00:00:18.340 --> 00:00:19.680
as we do this problem.

00:00:19.680 --> 00:00:22.740
Anyway, this is a compound
growth problem, or an

00:00:22.740 --> 00:00:24.240
exponential growth problem.

00:00:24.240 --> 00:00:27.300
And it says: a bacteria culture
initially contains 100

00:00:27.300 --> 00:00:31.010
cells and grows at a rate
proportional to its size.

00:00:31.010 --> 00:00:34.770
After an hour, the population
has increased to 420.

00:00:34.770 --> 00:00:37.070
So this first says find an
expression for the number

00:00:37.070 --> 00:00:38.490
of bacteria after t hours.

00:00:38.490 --> 00:00:40.910
So in general, anything that--
When you're dealing with

00:00:40.910 --> 00:00:43.140
exponential growth or
exponential decay.

00:00:43.140 --> 00:00:46.130
Or continuously compounding
growth or continuously

00:00:46.130 --> 00:00:49.730
compounding decay, though
you'll very seldom

00:00:49.730 --> 00:00:52.480
hear that term.

00:00:52.480 --> 00:00:55.440
The amount of whatever you have
as a function of time will

00:00:55.440 --> 00:00:56.690
be this type of equation.

00:00:56.690 --> 00:00:58.780
I'll say bacteria as
a function of time.

00:00:58.780 --> 00:00:59.940
So I'll say b of t.

00:00:59.940 --> 00:01:02.310
And I'll do it here.

00:01:02.310 --> 00:01:06.180
So b of-- Well, let me
do it here. b of t.

00:01:06.180 --> 00:01:10.440
Bacteria as a function of time
is going to be equal to the

00:01:10.440 --> 00:01:13.260
initial bacteria, or the
initial of whatever we have for

00:01:13.260 --> 00:01:15.080
doing compound interest is
the amount we start

00:01:15.080 --> 00:01:20.980
with, e to the kt.

00:01:20.980 --> 00:01:23.370
Where k is kind of our
compounding rate.

00:01:23.370 --> 00:01:27.790
And so in general, if you
have this I naught, the

00:01:27.790 --> 00:01:28.400
initial amount you have.

00:01:28.400 --> 00:01:29.180
And that makes sense, right?

00:01:29.180 --> 00:01:31.960
Because if t equals 0, this
whole expression is 1.

00:01:31.960 --> 00:01:35.490
So b of 0 should
just be I naught.

00:01:35.490 --> 00:01:37.790
So if you know what you start
with, and then you have another

00:01:37.790 --> 00:01:39.120
point, you can solve for k.

00:01:39.120 --> 00:01:40.320
And then you have
an expression.

00:01:40.320 --> 00:01:42.100
You have the first
thing they want.

00:01:42.100 --> 00:01:45.060
Find an expression for the
number bacteria after t hours.

00:01:45.060 --> 00:01:48.750
So my question to you is, first
of all, what is I sub zero?

00:01:48.750 --> 00:01:53.030
What is your initial number
of bacterial cells?

00:01:53.030 --> 00:01:54.260
Well they tell us.

00:01:54.260 --> 00:01:55.430
They tell us right here.

00:01:55.430 --> 00:01:58.340
The bacteria culture initially
contains 100 cells.

00:02:01.610 --> 00:02:07.980
So we know that b of 0 when
time equals 0 or 0 hours, that

00:02:07.980 --> 00:02:12.620
is equal to 100 where the
unit is cells, right?

00:02:12.620 --> 00:02:16.320
And so, if we substitute
here t equals 0, this is 1.

00:02:16.320 --> 00:02:19.280
So we know that I naught
is equal to 100.

00:02:19.280 --> 00:02:20.370
Let me do that actually.

00:02:20.370 --> 00:02:25.040
So b of 0 is also equal-- if we
look at it there --I naught

00:02:25.040 --> 00:02:28.730
times e to the what? k times 0
times e to the zero is

00:02:28.730 --> 00:02:29.960
equal to I naught.

00:02:29.960 --> 00:02:33.740
So this tells us that the
bacteria at time 0 is 100.

00:02:33.740 --> 00:02:36.010
And then when we substitute
here, is that the bacteria

00:02:36.010 --> 00:02:37.140
times 0 is equal to I naught.

00:02:37.140 --> 00:02:39.080
So I naught must equal 100.

00:02:39.080 --> 00:02:39.410
OK.

00:02:39.410 --> 00:02:41.460
So we're making some progress.

00:02:41.460 --> 00:02:47.070
So we know that the bacteria
as a function of time is

00:02:47.070 --> 00:02:54.020
equal to I naught, 100,
times e to the kt.

00:02:54.020 --> 00:02:57.800
Now, if we just knew k, then we
would have done this part A,

00:02:57.800 --> 00:03:00.020
find an expression for the
number bacteria after t hours.

00:03:00.020 --> 00:03:02.860
Well how can we figure out k?

00:03:02.860 --> 00:03:04.900
Well they give us
another data point.

00:03:04.900 --> 00:03:08.810
After an hour, the population
has increased to 420.

00:03:08.810 --> 00:03:11.710
So that tells us-- Let me--

00:03:11.710 --> 00:03:26.810
[SIDE COMMENTS]

00:03:26.810 --> 00:03:28.570
Anyway, so what was I doing?

00:03:28.570 --> 00:03:28.800
Oh.

00:03:28.800 --> 00:03:34.550
So this tells us, after
an hour the population

00:03:34.550 --> 00:03:37.190
has increased to 420.

00:03:37.190 --> 00:03:42.070
So that tells us
that b of 1, right?

00:03:42.070 --> 00:03:44.710
After 1 hour, the population
has increased to 420.

00:03:44.710 --> 00:03:48.090
So b of 1 is equal to 420.

00:03:48.090 --> 00:03:52.990
Which is also equal to--
If we put 1 here, that is

00:03:52.990 --> 00:03:58.080
equal to 100 e to the kt.

00:03:58.080 --> 00:04:01.110
Well what's t at b of 1?

00:04:01.110 --> 00:04:01.940
t is 1.

00:04:01.940 --> 00:04:03.750
So e to the k, right?

00:04:03.750 --> 00:04:07.200
So 420 is equal to
100 e to the k.

00:04:07.200 --> 00:04:08.690
And now we can solve for k.

00:04:08.690 --> 00:04:09.490
And let's see what we get.

00:04:09.490 --> 00:04:12.020
So we divide both sides by 100.

00:04:12.020 --> 00:04:12.720
You get 4.2.

00:04:15.360 --> 00:04:17.350
Let me put e to the
k on this side.

00:04:17.350 --> 00:04:19.800
e to the k is equal to 4.2.

00:04:22.890 --> 00:04:25.670
To solve for k, we take the
natural log of both sides.

00:04:25.670 --> 00:04:31.000
And we get k is equal to the
natural log of 4.2, which is

00:04:31.000 --> 00:04:33.440
going to be some
bizarre number.

00:04:33.440 --> 00:04:36.380
And we can-- We'll figure that
out later with a calculator.

00:04:36.380 --> 00:04:39.050
So there we figured out-- Using
the initial condition here,

00:04:39.050 --> 00:04:40.110
we figured out I naught.

00:04:40.110 --> 00:04:43.240
And then using this extra
data point, we solved for k.

00:04:43.240 --> 00:04:48.520
And we got k is equal to the
ln of the natural log of 4.2.

00:04:48.520 --> 00:04:51.390
So what does that leave us?

00:04:51.390 --> 00:04:53.580
So now we know the equation.

00:04:53.580 --> 00:04:56.310
We know k and we know I naught,
so we know that the equation

00:04:56.310 --> 00:05:04.360
is-- So part A, the equation is
b of t, bacteria as a function

00:05:04.360 --> 00:05:07.610
of time is equal to the initial
number of bacteria, 100

00:05:07.610 --> 00:05:10.060
cells, e to the kt.

00:05:10.060 --> 00:05:17.770
And k is ln of 4.2.

00:05:17.770 --> 00:05:19.766
That says 4.2.

00:05:19.766 --> 00:05:23.040
And all of that times t.

00:05:23.040 --> 00:05:23.950
So that's our function.

00:05:23.950 --> 00:05:25.860
I know you can't read that,
but we'll use it again.

00:05:25.860 --> 00:05:27.820
So it should make
a little sense.

00:05:27.820 --> 00:05:33.640
So then they say find-- this
is part B --find the number

00:05:33.640 --> 00:05:36.260
of bacteria after 3 hours.

00:05:36.260 --> 00:05:37.500
So that's pretty
straight forward.

00:05:37.500 --> 00:05:39.100
We have our function now.

00:05:39.100 --> 00:05:40.620
That at any time t, we
can tell you how many

00:05:40.620 --> 00:05:41.370
bacteria there are.

00:05:41.370 --> 00:05:45.970
So let's figure out the number
of bacteria after 3 hours.

00:05:45.970 --> 00:06:04.490
So b of 3 is equal to 100 times
e to the ln of 4.2 times 3.

00:06:04.490 --> 00:06:06.910
And that number we
could calculate if you

00:06:06.910 --> 00:06:07.780
have a calculator.

00:06:07.780 --> 00:06:11.240
e to the ln of-- Let's see.

00:06:11.240 --> 00:06:12.570
That's essentially-- Well
what's this going to be?

00:06:12.570 --> 00:06:13.380
This is the same thing.

00:06:13.380 --> 00:06:15.010
We can actually figure
this out analytically.

00:06:15.010 --> 00:06:22.970
That equals 100 e to the
natural log oh 4.2.

00:06:22.970 --> 00:06:26.920
If you have two things in an
exponent multiplied, that's the

00:06:26.920 --> 00:06:28.770
same thing as taking that and
then raising it to the

00:06:28.770 --> 00:06:29.550
third power, right?

00:06:29.550 --> 00:06:31.273
If we were simplifying, we'd
say oh that's the same thing

00:06:31.273 --> 00:06:32.980
as just multiplying
that times this.

00:06:32.980 --> 00:06:35.330
What's e to the ln of 4.2?

00:06:35.330 --> 00:06:37.730
Well it's 4.2, right?

00:06:37.730 --> 00:06:41.240
Because natural log says well,
what exponent do I have

00:06:41.240 --> 00:06:43.850
to raise e to to get 4.2?

00:06:43.850 --> 00:06:46.720
So if I raise e to that
exponent, I get 4.2.

00:06:46.720 --> 00:06:49.600
So that equals-- I didn't even
have to use a calculator, look

00:06:49.600 --> 00:06:54.330
at that --4.2 to
the third power.

00:06:54.330 --> 00:06:55.400
And I don't know what that is.

00:06:55.400 --> 00:06:58.310
That's going to be probably
70 something and so

00:06:58.310 --> 00:06:59.360
we'll do that later.

00:06:59.360 --> 00:07:00.170
But that's part B.

00:07:00.170 --> 00:07:02.820
That's just plugging
into your calculator.

00:07:02.820 --> 00:07:05.550
That's part B.

00:07:05.550 --> 00:07:07.760
Now what are they asking?

00:07:07.760 --> 00:07:13.800
Find the rate of
growth after 3 hours.

00:07:13.800 --> 00:07:16.370
Let me do that in
a special color.

00:07:16.370 --> 00:07:19.140
So what are they asking us for?

00:07:19.140 --> 00:07:22.200
Well they're essentially asking
us what is the slope of this

00:07:22.200 --> 00:07:25.730
function after 3 hours?

00:07:25.730 --> 00:07:28.660
Or another way you could view
it is what is the derivative

00:07:28.660 --> 00:07:31.080
at time is equal to 3?

00:07:31.080 --> 00:07:31.850
Let's do that.

00:07:31.850 --> 00:07:36.660
Let me erase all this
other stuff I have.

00:07:36.660 --> 00:07:38.440
I think we can erase all of
this because we've written

00:07:38.440 --> 00:07:42.080
down our equation here.

00:07:42.080 --> 00:07:43.380
We can erase that.

00:07:43.380 --> 00:07:45.970
Actually we could even erase
all of this because we've

00:07:45.970 --> 00:07:46.960
just answered that.

00:07:46.960 --> 00:07:50.200
That's just plugging into a
calculator at this point.

00:07:50.200 --> 00:07:52.400
Just want to keep our
original equation.

00:07:52.400 --> 00:07:54.380
So let's do part C.

00:07:54.380 --> 00:07:56.380
Part C, find the rate of
growth after 3 hours.

00:07:56.380 --> 00:07:59.750
So we just have to take the
derivative of the bacteria

00:07:59.750 --> 00:08:01.230
function with respect to time.

00:08:01.230 --> 00:08:03.060
So let's do that.

00:08:03.060 --> 00:08:06.700
So b prime of t.

00:08:06.700 --> 00:08:09.090
It equals what?

00:08:09.090 --> 00:08:12.160
So what's the derivative of--
Let's just do the chain rule.

00:08:12.160 --> 00:08:15.450
What's the derivative of-- so
we can have 100 constant out

00:08:15.450 --> 00:08:19.720
there times the derivative
of this inside function.

00:08:19.720 --> 00:08:21.446
Well, that's just a
constant times t, so

00:08:21.446 --> 00:08:22.450
it's just a constant.

00:08:22.450 --> 00:08:29.053
So times ln of 4.2 times
the derivative of e

00:08:29.053 --> 00:08:29.860
to the whole thing.

00:08:29.860 --> 00:08:32.280
Well, the derivative of e
to the x is e to the x.

00:08:32.280 --> 00:08:34.620
So the chain rule tells us
it's just the derivative

00:08:34.620 --> 00:08:36.860
of this whole expression.

00:08:36.860 --> 00:08:44.570
So times e to the ln 4.2t.

00:08:48.940 --> 00:08:51.500
So that's the derivative
at any time t.

00:08:51.500 --> 00:08:53.390
And then they want to know
what the rate of growth

00:08:53.390 --> 00:08:55.460
is after 3 hours.

00:08:55.460 --> 00:09:06.140
So b prime of 3 is equal
to 100 ln of 4.2.

00:09:06.140 --> 00:09:11.500
All of that times-- Well,
e to the-- As we just

00:09:11.500 --> 00:09:14.360
did this in part B.

00:09:14.360 --> 00:09:19.720
e to the ln 4.2 times 3, that's
the same thing as-- Because

00:09:19.720 --> 00:09:22.110
this expression right
here is just 4.2.

00:09:22.110 --> 00:09:25.310
This is times 4.2 to the third.

00:09:25.310 --> 00:09:28.540
That's just a little bit of
logarithm math we did here.

00:09:28.540 --> 00:09:30.860
And it's the same exact logic.

00:09:30.860 --> 00:09:34.110
All I did is I substituted
3 here and simplified.

00:09:34.110 --> 00:09:35.210
Hopefully that makes
sense to you.

00:09:35.210 --> 00:09:37.140
If it doesn't, you might want
to use your calculator.

00:09:37.140 --> 00:09:40.300
But this all is actually a
pretty important thing to know.

00:09:40.300 --> 00:09:46.690
That e to the ln of x is
just equal to x, right?

00:09:46.690 --> 00:09:49.510
Because ln of x is essentially
saying what exponent do I

00:09:49.510 --> 00:09:51.950
have to raise e to to get x?

00:09:51.950 --> 00:09:55.550
So if I raise e to that
exponent, I get x.

00:09:55.550 --> 00:09:56.710
So that's all I'm saying here.

00:09:56.710 --> 00:10:02.130
That this right here is the
same thing as e to the

00:10:02.130 --> 00:10:06.440
ln 4.2 to the t, right?

00:10:06.440 --> 00:10:08.040
Because if you were to
simplify, you just take

00:10:08.040 --> 00:10:10.430
this exponent and multiply
this and you'd get this.

00:10:10.430 --> 00:10:16.830
And so this is just
4.2 to the t, right?

00:10:16.830 --> 00:10:18.550
Actually that's-- I should
have written our original

00:10:18.550 --> 00:10:19.550
equation as that.

00:10:19.550 --> 00:10:24.630
I could rewrite our original
bacterial function equation as

00:10:24.630 --> 00:10:34.530
100 times-- using that same
thing --times 4.2

00:10:34.530 --> 00:10:35.370
to the t power.

00:10:35.370 --> 00:10:38.320
Actually, that is a simpler
bacterial equation

00:10:38.320 --> 00:10:38.970
than this one.

00:10:38.970 --> 00:10:44.150
And so that would be even a
better answer for part A.

00:10:44.150 --> 00:10:46.440
And then that makes part
B even easier to solve.

00:10:46.440 --> 00:10:48.300
And then part C.

00:10:48.300 --> 00:10:51.020
Well part C is actually-- it
was easier to keep it in this

00:10:51.020 --> 00:10:55.500
form because taking a
derivative of e is easier than

00:10:55.500 --> 00:10:57.010
taking a derivative
of another base.

00:10:57.010 --> 00:10:59.620
And actually we would probably
want to convert backwards.

00:10:59.620 --> 00:11:01.340
So it's good we left it in that
form because it made the

00:11:01.340 --> 00:11:03.120
derivative all that easier.

00:11:03.120 --> 00:11:04.730
But we could rewrite the
derivative as well.

00:11:04.730 --> 00:11:10.580
We could rewrite this
as b prime of t is

00:11:10.580 --> 00:11:17.010
equal to 100 ln 4.2.

00:11:17.010 --> 00:11:22.850
All of that times 4.2
to the t, right?

00:11:22.850 --> 00:11:25.300
I just took a this-- Sorry.

00:11:25.300 --> 00:11:25.430
No.

00:11:25.430 --> 00:11:29.240
I took this and I
replaced it with this.

00:11:29.240 --> 00:11:31.400
Anyway, I got a little bit
messier than I wanted to.

00:11:31.400 --> 00:11:33.500
But then their final
question is when will the

00:11:33.500 --> 00:11:37.366
population reach 10,000?

00:11:37.366 --> 00:11:42.140
So let me see erase what
we did with part C.

00:11:45.360 --> 00:11:54.670
When will the population
reach 10,000?

00:11:54.670 --> 00:11:56.180
Let me write what we
have already done a

00:11:56.180 --> 00:11:58.570
little bit neater.

00:11:58.570 --> 00:12:04.790
We know-- We knew that, part A,
b of t is equal to 100 e to

00:12:04.790 --> 00:12:07.110
the natural log of 4.2t.

00:12:09.860 --> 00:12:14.590
And I said that's the
same thing as 100 times

00:12:14.590 --> 00:12:17.716
4.2 to the t power.

00:12:17.716 --> 00:12:20.370
And they say when will the
population reach 10,000?

00:12:20.370 --> 00:12:22.920
So they're essentially saying
when is-- at what time t

00:12:22.920 --> 00:12:24.490
is b of t equal to 10,000?

00:12:24.490 --> 00:12:37.140
So we can just say 10,000 is
equal to this: 100-- And

00:12:37.140 --> 00:12:39.330
actually, I'm going to use this
one because I want to take the

00:12:39.330 --> 00:12:39.950
natural log of both sides.

00:12:39.950 --> 00:12:41.520
And it just makes
things simpler.

00:12:41.520 --> 00:12:49.020
100e to the ln 4.2t.

00:12:51.740 --> 00:12:53.770
So you could divide
both sides by 100.

00:12:53.770 --> 00:13:05.570
And you get 100 is equal
to e to the ln 4.2t.

00:13:08.280 --> 00:13:10.360
And now take the natural
logs of both sides of

00:13:10.360 --> 00:13:12.460
this equation, right?

00:13:12.460 --> 00:13:14.390
So then we get what?

00:13:14.390 --> 00:13:16.715
We get-- let me draw a
divide-- Well, I'll do

00:13:16.715 --> 00:13:18.320
it in a different color.

00:13:18.320 --> 00:13:20.850
We get-- taking the natural log
of both sides of this --the

00:13:20.850 --> 00:13:27.060
natural log of 100 is equal
to-- Well if you take the

00:13:27.060 --> 00:13:29.680
natural log of e to
something, you're just left

00:13:29.680 --> 00:13:31.330
with the something.

00:13:31.330 --> 00:13:33.920
And you might want to review
all the logarithms if you found

00:13:33.920 --> 00:13:34.720
that a little confusing.

00:13:34.720 --> 00:13:36.660
But that-- If you take
the natural log of this

00:13:36.660 --> 00:13:39.710
side, you're just left
with the exponent.

00:13:39.710 --> 00:13:44.665
So that equals ln of 4.2t.

00:13:44.665 --> 00:13:47.090
And if you want to solve for
t, you just divide both

00:13:47.090 --> 00:13:48.940
sides by ln of 4.2.

00:13:48.940 --> 00:13:53.700
And you get t is equal
to ln of 100 divided by

00:13:53.700 --> 00:13:56.280
the natural log of 4.2.

00:13:56.280 --> 00:13:57.290
And that'll be an hour.

00:13:57.290 --> 00:13:58.260
So how many hours does it take?

00:13:58.260 --> 00:14:00.030
You just plug this in your
calculator and you will get a

00:14:00.030 --> 00:14:02.890
number for the number of hours.

00:14:02.890 --> 00:14:04.580
Hopefully you didn't find
that too confusing.

00:14:04.580 --> 00:14:07.460
And actually since-- Just to
hit another point home, we

00:14:07.460 --> 00:14:08.760
could have used this equation.

00:14:08.760 --> 00:14:10.340
And it's the same thing, right?

00:14:10.340 --> 00:14:11.170
And what would we
would have had?

00:14:11.170 --> 00:14:19.400
We would have had 100 times 4.2
to t power is equal to 10,000.

00:14:19.400 --> 00:14:23.980
Divide both sides by 100
and you get 4.2 to the

00:14:23.980 --> 00:14:27.310
t is equal to 100.

00:14:27.310 --> 00:14:30.950
And so, to solve this-- So
you're essentially saying log

00:14:30.950 --> 00:14:33.650
base-- You would essentially
want to take log base 4.2

00:14:33.650 --> 00:14:34.630
of both sides of this.

00:14:34.630 --> 00:14:42.520
So you would say t is equal
to log base 4.2 of 100.

00:14:42.520 --> 00:14:45.370
And I went over this in the
exponent proper-- in the

00:14:45.370 --> 00:14:47.600
logarithm properties video.

00:14:47.600 --> 00:14:50.360
But it's a pretty useful
thing to know in general.

00:14:50.360 --> 00:14:53.060
How do you figure out-- Your
calculator has two bases.

00:14:53.060 --> 00:14:56.050
It has log base 10 and it
has log base e, which

00:14:56.050 --> 00:14:57.140
is the natural log.

00:14:57.140 --> 00:15:00.210
How do you figure out log base
of another number of 100?

00:15:00.210 --> 00:15:04.200
Well, the simple answer is you
could take natural log of this

00:15:04.200 --> 00:15:06.700
and divide it by the
natural log of this.

00:15:06.700 --> 00:15:10.080
You could also take log base
ten of this and divide it by

00:15:10.080 --> 00:15:13.960
log ten base of this--
log base ten of this.

00:15:13.960 --> 00:15:16.490
Anyway, hopefully I didn't
further confuse you.

00:15:16.490 --> 00:15:19.880
And I will do-- This was
actually exponential growth.

00:15:19.880 --> 00:15:21.730
And you could replace
these words.

00:15:21.730 --> 00:15:24.460
You could say-- Instead of
saying bacteria culture, you

00:15:24.460 --> 00:15:27.680
could say a bank account
initially contains $100 and

00:15:27.680 --> 00:15:31.400
grows at a rate proportional
to its size, which would

00:15:31.400 --> 00:15:32.610
mean compound interest.

00:15:32.610 --> 00:15:35.920
And then you could say, after
an hour the population has

00:15:35.920 --> 00:15:39.480
increased to, whatever, $4.20.

00:15:39.480 --> 00:15:41.500
And then find the expression
for the number of-- So

00:15:41.500 --> 00:15:43.290
essentially, you'd be trying
to figure out the compound

00:15:43.290 --> 00:15:44.290
interest rate and all of that.

00:15:44.290 --> 00:15:46.160
But this is the exact
same phenomenon.

00:15:46.160 --> 00:15:49.020
That the larger you grow,
you're going to grow by an

00:15:49.020 --> 00:15:51.420
absolute value more than that.

00:15:51.420 --> 00:15:53.102
Anyway, I'll do more of these
problems because they can

00:15:53.102 --> 00:15:53.870
be a little confusing.

00:15:53.870 --> 00:15:56.950
And I'll do one on
exponential decay as well.

00:15:56.950 --> 00:15:58.660
See you soon.