Definite integral involving natural log

WEBVTT
Kind: captions
Language: en

00:00:00.402 --> 00:00:01.892
- [Voiceover] Let's now
take the definite integral

00:00:01.892 --> 00:00:03.225
from two to four

00:00:04.109 --> 00:00:06.109
of six plus x squared

00:00:07.868 --> 00:00:10.535
over x to the third power dx.

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At first this might seem pretty daunting.

00:00:14.354 --> 00:00:16.167
I have this rational expression.

00:00:16.167 --> 00:00:19.233
But if we just rewrite this,
it might jump out at you

00:00:19.233 --> 00:00:21.172
how this could be a little bit simpler.

00:00:21.172 --> 00:00:24.781
So this is equal to the
integral from two to four

00:00:24.781 --> 00:00:27.698
of six over x to the third power

00:00:28.653 --> 00:00:32.820
plus x squared over x
to the third power dx.

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I just separated this numerator out.

00:00:35.770 --> 00:00:39.248
I just divided each of those
terms by x to the third power.

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And this I could rewrite.

00:00:40.894 --> 00:00:44.836
This is equal to the
integral from two to four

00:00:44.836 --> 00:00:48.320
of six x to the negative three power,

00:00:48.320 --> 00:00:49.737
that's that first term there.

00:00:49.737 --> 00:00:53.652
And x squared divided by x to the third,

00:00:53.652 --> 00:00:56.394
well that it going to be one over x.

00:00:56.394 --> 00:00:58.477
So plus one over x dx.

00:01:01.336 --> 00:01:03.415
Now this is going to be equal to,

00:01:03.415 --> 00:01:06.263
let's take the antiderivative
of the different parts

00:01:06.263 --> 00:01:09.941
and we're going to evaluate that at four

00:01:09.941 --> 00:01:13.549
and we're going to evaluate that at two.

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And we're going to find
the difference between

00:01:16.160 --> 00:01:19.435
the antiderivative valued
at four and at two.

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Now what is the antiderivative

00:01:21.493 --> 00:01:24.043
of six x to the negative three?

00:01:24.043 --> 00:01:26.151
Well, here, once again we can just use,

00:01:26.151 --> 00:01:29.433
we could use the power rule
for taking the antiderivative

00:01:29.433 --> 00:01:31.903
or it's the reverse of
the derivative power rule.

00:01:31.903 --> 00:01:34.127
We know that if we're taking the integral

00:01:34.127 --> 00:01:35.794
of x to the n dx,

00:01:38.071 --> 00:01:39.469
the antiderivative of that

00:01:39.469 --> 00:01:44.252
is going to be x to the n
plus one over n plus one.

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And if we were just taking
an indefinite integral

00:01:46.110 --> 00:01:47.917
there would be sum pluc C.

00:01:47.917 --> 00:01:49.847
The reason why we don't
put the plus Cs here

00:01:49.847 --> 00:01:54.234
is when you evaluate at
both bounds of integration,

00:01:54.234 --> 00:01:56.623
the C would cancel out
regardless of what it is.

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So we don't really think about the C much

00:01:57.815 --> 00:01:59.783
when we're taking definite integrals.

00:01:59.783 --> 00:02:02.718
But let's apply that to six x
to the negative third power.

00:02:02.718 --> 00:02:04.708
So it's going to be, we're going to take

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x to the negative three plus one.

00:02:08.203 --> 00:02:10.090
So it's x to the negative two.

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And so we're gonna divide
by negative two as well.

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And, of course, we had that
six out front from the get go.

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So that's the antiderivative

00:02:17.538 --> 00:02:19.264
of six x to the negative three power.

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And what's the
antiderivative of one over x?

00:02:21.946 --> 00:02:25.322
You might be tempted to use
the same idea right over here.

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You might be tempted to say, all right,

00:02:26.844 --> 00:02:29.541
well the antiderivative
of x to the negative one,

00:02:29.541 --> 00:02:31.437
which is the same thing as one over x

00:02:31.437 --> 00:02:35.201
would be equal to x to
the negative one plus one

00:02:35.201 --> 00:02:37.597
over negative one plus one.

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But what is negative one plus one?

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It is zero.

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So this doesn't fit this
property right over here.

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But lucky for us, there
is another property.

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And we went the other way

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when we were first taking derivatives

00:02:49.716 --> 00:02:51.499
of natural log functions.

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The antiderivative of one over
x or x to the negative one

00:02:55.482 --> 00:02:57.410
is equal to, sometimes
you'll see it written

00:02:57.410 --> 00:02:59.910
as natural log of x plus C.

00:03:02.359 --> 00:03:05.323
And sometimes and I
actually prefer this one,

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because you could actually
evaluate it for negative values,

00:03:09.202 --> 00:03:11.141
is to say the absolute value,

00:03:11.141 --> 00:03:13.332
the natural log of the
absolute value of x.

00:03:13.332 --> 00:03:15.950
And this is useful,
because this is defined

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for negative values, not
just positive values.

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The natural log of x is only defined

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for positive values of x,

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but when you take the absolute value,

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now it could be negative
or positive values of x.

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And it works, the derivative
of this is indeed one over x.

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Now it's not so relevant here,

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because our bounds of
integration are both positive.

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But if both of our bounds of
integration were negative,

00:03:39.150 --> 00:03:41.384
you could still do this
by just reminding yourself

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that this is a natural log
of absolute value of x.

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So this, we could say
is plus the natural log

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of the absolute value of x.

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It's not a bad habit to do it,

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and if everthing's positive, well,

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the absolute value of x is equal to x.

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And so what is this going to be equal to?

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This is equal to, let's
evaluate everything at four.

00:04:03.176 --> 00:04:04.674
And actually before I
even evaluate it at four,

00:04:04.674 --> 00:04:07.446
what's six divided by negative
two, that's negative three.

00:04:07.446 --> 00:04:10.280
So if we evaluate it at
four, it's going to be

00:04:10.280 --> 00:04:13.030
negative three over four squared.

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Four to the negative two
is one over four squared

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and then plus the natural log of the,

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we could say the absolute value of four,

00:04:24.395 --> 00:04:26.563
but the absolute value
of four is just four.

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So the natural log of four.

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And from that we're going to subtract

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everything evaluated at two.

00:04:34.359 --> 00:04:35.665
So let's do that.

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So if we evaluate it at two,
it's going to be negative three

00:04:41.467 --> 00:04:42.884
over two squared.

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So two to the negative two
is one over two squared,

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over two squared, plus the natural log of,

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the absolute value of positive two is,

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once again, it's just two.

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And so what does this give us?

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So, let's try to simplify it a little bit.

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So this is negative 3/16.

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We'll do that same color.

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So this is going to be equal to negative,

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negative, sorry, not negative 3/16,

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it's gotta be very careful,

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oh, sorry, yes, it is negative 3/16.

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For some reason my brain started thinking

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four to the third power.

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Negative 3/16 plus natural log of four.

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And then this right over here
is negative 3/4, negative 3/4,

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do that same color.

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This right over here is negative 3/4.

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We have this negative sign out front

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that we're going to have to distribute.

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So the negative of negative
3/4 is plus 3/4, plus 3/4,

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and then we're going to subtract,

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remember we're distributing
this negative sign,

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the natural log, the natural log of two.

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And what does this equal to?

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All right, so this is
going to be equal to,

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and I'm now going to
switch to a neutral color.

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So let's add these two terms

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that don't involve a natural log.

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And let's see, if we
have a common denominator

00:06:06.834 --> 00:06:10.032
three over four is the same
thing, that is the same thing

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as we multiply the numerator
and the denominator

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by four, that is 12 over 16.

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And so you have negative 3/16,

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negative 3/16 plus 12/16

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will give you 9/16, 9/16.

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And then we're gonna have the ones

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that do involve the natural log.

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Natural log of four minus
the natural log of two.

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So we could write this plus
the natural log of four

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minus the natural log of two.

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And you might remember from
your logarithm properties

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that this over here,
this is the same thing

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as the natural log of four divided by two,

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this comes straight out of
your logarithm properties.

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And so this is going to
be the natural log of two,

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natural log of two.

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So, we deserve a little
bit of a drum roll now.

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This is all going to be equal to,

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this is going to be
equal to the natural log,

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nine over 16 plus the natural log of two,

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plus the natural log of two.

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And we are done.


Definite integral of radical function

WEBVTT
Kind: captions
Language: en

00:00:00.216 --> 00:00:01.229
- [Voiceover] So, we want to evaluate the

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definite integral from
negative one to eight

00:00:02.763 --> 00:00:05.846
of 12 times the cube root of x dx.

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Let's see, this is going
to be the same thing

00:00:09.102 --> 00:00:10.660
as the definite integral from negative one

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to eight of 12 times, the cube root is

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the same thing as saying
x to the 1/3 power dx

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and so now, if we want to
take the antiderivative

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of the stuff on the inside,

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we're just going to do
essentially the power rule,

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it conduces the power rule of integrals

00:00:28.222 --> 00:00:31.279
or it's the reverse of the
power rule for derivatives,

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where we increase this exponent by one

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and then we divide by
that increased exponent.

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So, this is going to
be equal to 12 times x

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to the 1/3 plus one.

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Let me do that in another color,

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so we can keep track of it.

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X to the 1/3 plus one and
then we're going to divide

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by 1/3 plus one and so,
what's 1/3 plus one,

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well, that's 4/3, 1/3
plus 3/3, that's 4/3.

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So, I could write it this way.

00:01:01.698 --> 00:01:05.000
I could write this x to the 4/3 divided by

00:01:05.000 --> 00:01:08.167
4/3 and this is going to be

00:01:10.566 --> 00:01:14.260
and I'm going to evaluate
this at the bounds.

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So, I'm going to evaluate this at

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and I'll do this in different colors,

00:01:18.217 --> 00:01:19.759
I'm going to evaluate it at eight

00:01:19.759 --> 00:01:22.891
and I'm going to evaluate
it at negative one

00:01:22.891 --> 00:01:25.376
and I'm going to subtract
it and evaluate it

00:01:25.376 --> 00:01:28.807
at negative one from this
expression evaluated at eight,

00:01:28.807 --> 00:01:31.784
and so what is this going to be equal to?

00:01:31.784 --> 00:01:33.773
Well, actually, let me
simplify a little bit more.

00:01:33.773 --> 00:01:36.403
What is 12 divided by 4/3?

00:01:36.403 --> 00:01:38.152
So, 12, I'll do it right, well, I'll

00:01:38.152 --> 00:01:39.004
do it right over here,

00:01:39.004 --> 00:01:43.171
12 over 4/3 is equal to 12 times 3/4,

00:01:45.475 --> 00:01:48.694
which will give you as 12/1 times 3/4,

00:01:48.694 --> 00:01:51.319
12 divided by four is three,

00:01:51.319 --> 00:01:54.487
so this is going to be equal to nine,

00:01:54.487 --> 00:01:56.329
3/4 of 12 is nine.

00:01:56.329 --> 00:01:58.690
So, this, we could rewrite this,

00:01:58.690 --> 00:02:02.857
we could write this as
nine x to the 4/3 power.

00:02:04.171 --> 00:02:06.049
So, if we evaluate it at eight,

00:02:06.049 --> 00:02:09.049
this is going to be nine times eight

00:02:11.074 --> 00:02:14.829
to the 4/3 power and from that,

00:02:14.829 --> 00:02:19.125
we're going to subtract and
evaluate it at negative one.

00:02:19.125 --> 00:02:23.042
So, this is going to be
nine times negative one

00:02:23.896 --> 00:02:25.883
to the 4/3 power.

00:02:25.883 --> 00:02:28.512
So, what is eight to the 4/3 power?

00:02:28.512 --> 00:02:29.523
I'll do it over here.

00:02:29.523 --> 00:02:33.571
So, eight to the 4/3 is equal to eight

00:02:33.571 --> 00:02:36.577
to the 1/3 to the fourth power,

00:02:36.577 --> 00:02:37.954
these are just exponent properties here.

00:02:37.954 --> 00:02:40.537
Eight to the 1/3, the cube root of eight,

00:02:40.537 --> 00:02:42.948
or eight to the 1/3 power, that's two,

00:02:42.948 --> 00:02:44.540
'cause two to the third power is eight

00:02:44.540 --> 00:02:46.140
and two to the fourth power,

00:02:46.140 --> 00:02:48.600
well two to the fourth
power is equal to 16.

00:02:48.600 --> 00:02:52.495
So, eight to the 4/3 is 16
and what's a negative one

00:02:52.495 --> 00:02:53.724
to the 4/3?

00:02:53.724 --> 00:02:57.652
Well, same idea, negative one to the 4/3

00:02:57.652 --> 00:03:00.321
is equal to negative one, there's several

00:03:00.321 --> 00:03:01.154
way you could do it.

00:03:01.154 --> 00:03:02.514
You could say negative one to the fourth

00:03:02.514 --> 00:03:03.580
and the cube root of that,

00:03:03.580 --> 00:03:05.071
or the cube root of negative one

00:03:05.071 --> 00:03:07.140
and then raise that to the
fourth power, either way.

00:03:07.140 --> 00:03:08.616
So, let's do it the first way.

00:03:08.616 --> 00:03:09.832
Negative one to the fourth and then

00:03:09.832 --> 00:03:11.558
take the cube root of that.

00:03:11.558 --> 00:03:13.387
Well, negative one to
the fourth is just one

00:03:13.387 --> 00:03:16.074
and then one to the 1/3 power,

00:03:16.074 --> 00:03:17.772
well that's just going to be equal to one.

00:03:17.772 --> 00:03:19.944
So, what we have here in blue,

00:03:19.944 --> 00:03:22.078
that's just equal to one.

00:03:22.078 --> 00:03:25.911
So, we have nine times
16 minus nine time one,

00:03:27.050 --> 00:03:28.997
well, that's just going
to be nine times 15.

00:03:28.997 --> 00:03:32.081
We have 16/9 and then we're
going take away a nine,

00:03:32.081 --> 00:03:34.907
so that's going to be nine times 15,

00:03:34.907 --> 00:03:35.881
so what is that?

00:03:35.881 --> 00:03:39.130
That is going to be equal to nine times 15

00:03:39.130 --> 00:03:42.213
is 90 plus 45, which is equal to 135,

00:03:45.715 --> 00:03:47.298
135 and we're done.


History of the Democratic Party

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.261
- [Voiceover] Alright,
Kim, we have 216 years

00:00:02.261 --> 00:00:04.170
of Democratic Party history to cover,

00:00:04.170 --> 00:00:06.719
let's cut the pleasantries
and get right to it.

00:00:06.719 --> 00:00:07.897
Who is this man?

00:00:07.897 --> 00:00:09.981
- [Voiceover] That is Thomas Jefferson.

00:00:09.981 --> 00:00:10.818
- [Voiceover] He does not look like

00:00:10.818 --> 00:00:14.802
the baby-faced boy that
he was in this image.

00:00:14.802 --> 00:00:16.230
Is this his presidential portrait?

00:00:16.230 --> 00:00:18.186
- [Voiceover] (laughing)
I believe so, yes.

00:00:18.186 --> 00:00:20.074
So, Thomas Jefferson I think is really

00:00:20.074 --> 00:00:22.994
interesting because he didn't
set out to found a party,

00:00:22.994 --> 00:00:23.827
- [Voiceover] Right.

00:00:23.827 --> 00:00:25.745
- [Voiceover] Although,
he ended up doing so.

00:00:25.745 --> 00:00:28.932
It was more that he had
a competing vision about

00:00:28.932 --> 00:00:31.362
what the United States
should be that was different

00:00:31.362 --> 00:00:34.264
from the vision that was being
promoted by some of the men

00:00:34.264 --> 00:00:36.834
who were in power early
in the nation's history,

00:00:36.834 --> 00:00:40.485
like Washington, Adams,
Alexander Hamilton.

00:00:40.485 --> 00:00:42.157
- [Voiceover] What is the
difference between Jefferson

00:00:42.157 --> 00:00:44.953
and Hamilton's views on governance

00:00:44.953 --> 00:00:48.149
and how did this become
the Democratic Party?

00:00:48.149 --> 00:00:50.788
- [Voiceover] Hamilton had
an idea that he was going

00:00:50.788 --> 00:00:54.154
to try to make the government
a little bit more centrist,

00:00:54.154 --> 00:00:55.906
so a strong central government,

00:00:55.906 --> 00:00:58.752
remember they've just
switched to the constitution

00:00:58.752 --> 00:01:00.598
from the Articles of Confederation.

00:01:00.598 --> 00:01:01.431
- [Voiceover] What year was that?

00:01:01.431 --> 00:01:02.264
- [Voiceover] 1787.

00:01:02.264 --> 00:01:03.099
- [Voiceover] Thank you.

00:01:03.099 --> 00:01:06.612
- [Voiceover] And Hamilton
thinks that there needs

00:01:06.612 --> 00:01:10.226
to be a strong central
government, a national bank,

00:01:10.226 --> 00:01:12.963
a kind of limited democracy, right?

00:01:12.963 --> 00:01:17.130
A democracy that is limited
to more educated, landed men,

00:01:18.208 --> 00:01:21.594
and he wants the United States
to be a lot like England.

00:01:21.594 --> 00:01:22.427
- [Voiceover] And Jefferson,

00:01:22.427 --> 00:01:24.132
and people who felt like Jefferson,

00:01:24.132 --> 00:01:26.714
like James Madison, were
saying, woah, woah, woah,

00:01:26.714 --> 00:01:29.372
this is not what we rebelled
against England for,

00:01:29.372 --> 00:01:31.316
to become England part two.

00:01:31.316 --> 00:01:32.149
- [Voiceover] Right.

00:01:32.149 --> 00:01:33.152
- [Voiceover] Instead what we should have

00:01:33.152 --> 00:01:37.411
is very expanded democracy,
democracy for lots of people,

00:01:37.411 --> 00:01:41.875
doesn't matter their birth,
and much more agrarian,

00:01:41.875 --> 00:01:44.542
state-focused, small government.

00:01:46.026 --> 00:01:46.926
- [Voiceover] And this
is what we still call

00:01:46.926 --> 00:01:48.778
the Jeffersonian Ideal.

00:01:48.778 --> 00:01:49.611
- [Voiceover] Exactly, yeah.

00:01:49.611 --> 00:01:53.778
Small "r" republic citizens
who are mainly farmers.

00:01:57.749 --> 00:02:00.115
Now, of course, the expanded democracy

00:02:00.115 --> 00:02:01.785
that Jefferson's thinking about

00:02:01.785 --> 00:02:04.848
doesn't apply to women
or African Americans,

00:02:04.848 --> 00:02:07.705
Jefferson himself was a slaveholder.

00:02:07.705 --> 00:02:11.673
But he didn't want to limit
governance to only the elite.

00:02:11.673 --> 00:02:14.310
- [Voiceover] So, the
Jeffersonian Ideal is a republic

00:02:14.310 --> 00:02:16.727
for all landholding citizens.

00:02:17.644 --> 00:02:18.477
- [Voiceover] Exactly.

00:02:18.477 --> 00:02:21.500
- [Voiceover] But in 1800 the
idea of what citizen meant

00:02:21.500 --> 00:02:23.836
was very limited and langered.

00:02:23.836 --> 00:02:25.597
- [Voiceover] So this
doesn't really take root

00:02:25.597 --> 00:02:28.764
until the emergence of Andrew Jackson.

00:02:29.761 --> 00:02:33.928
In the late 1820's, he
finally gets elected in 1832.

00:02:34.892 --> 00:02:38.479
So Jackson really pushes
the idea of democracy

00:02:38.479 --> 00:02:41.844
for all white men to
its farthest conclusion,

00:02:41.844 --> 00:02:44.532
which means that in this
time period he expanded

00:02:44.532 --> 00:02:48.844
the franchise to any white man
regardless of his property,

00:02:48.844 --> 00:02:52.748
which means that this
was the most democracy

00:02:52.748 --> 00:02:55.050
in the history of the
world up until this point.

00:02:55.050 --> 00:02:56.035
- [Voiceover] Yeah, but let me point out

00:02:56.035 --> 00:02:58.835
that you did say all white men.

00:02:58.835 --> 00:02:59.668
- [Voiceover] Exactly.

00:02:59.668 --> 00:03:02.668
So, Jackson is unabashedly a racist.

00:03:03.669 --> 00:03:08.353
And his vision of manifest
destiny really involves

00:03:08.353 --> 00:03:10.987
the eradication of all Native Americans.

00:03:10.987 --> 00:03:12.724
- [Voiceover] I mean,
this is a really dark time

00:03:12.724 --> 00:03:15.592
for American Indian policy.

00:03:15.592 --> 00:03:19.624
This is when we see something
called the Indian Removal Act,

00:03:19.624 --> 00:03:22.360
that is literally what it's called.

00:03:22.360 --> 00:03:24.412
Jackson's presidency is also marked

00:03:24.412 --> 00:03:28.579
by the Cherokee Diaspora, the
forced removal of the Cherokee

00:03:29.639 --> 00:03:31.640
along what's called the Trail of Tears.

00:03:31.640 --> 00:03:32.473
- [Voiceover] Yep.

00:03:32.473 --> 00:03:33.306
- [Voiceover] So, okay.

00:03:33.306 --> 00:03:35.678
So, before Jackson, do we really have

00:03:35.678 --> 00:03:38.239
the Democratic Party as an institution?

00:03:38.239 --> 00:03:39.072
- [Voiceover] No.

00:03:39.072 --> 00:03:42.441
Before this we would've called
this the Anti-Federalists

00:03:42.441 --> 00:03:43.990
or the Democratic Republicans.

00:03:43.990 --> 00:03:46.760
- [Voiceover] Okay, so Hamilton
is a Federalist position.

00:03:46.760 --> 00:03:48.035
Jefferson is an Anti-Federalist

00:03:48.035 --> 00:03:49.051
- [Voiceover] Right.

00:03:49.051 --> 00:03:49.884
- [Voiceover] So, okay.

00:03:49.884 --> 00:03:53.191
So we have these kicking
around but they're not parties

00:03:53.191 --> 00:03:55.163
in the same way that
we would see them now,

00:03:55.163 --> 00:03:57.107
they're just sort of policy positions?

00:03:57.107 --> 00:03:58.126
- [Voiceover] Yeah, I would say so.

00:03:58.126 --> 00:04:00.642
Having a mass political party

00:04:00.642 --> 00:04:03.408
was really an invention
of the Jackson era.

00:04:03.408 --> 00:04:07.575
So, one did not run for
president in the early 1800's,

00:04:09.346 --> 00:04:12.163
one stood for president as gentlemen

00:04:12.163 --> 00:04:14.398
around you talked of your virtues.

00:04:14.398 --> 00:04:16.115
It wasn't a campaign.

00:04:16.115 --> 00:04:16.948
- [Voiceover] Oh, so you couldn't

00:04:16.948 --> 00:04:18.416
actually campaign for yourself?

00:04:18.416 --> 00:04:20.432
- [Voiceover] Right,
that was very uncouth.

00:04:20.432 --> 00:04:21.265
- [Voiceover] I see.

00:04:21.265 --> 00:04:24.263
- [Voiceover] So Jackson is
the real antidote to this.

00:04:24.263 --> 00:04:26.832
He creates massive democracy.

00:04:26.832 --> 00:04:30.199
The turnout in this time period
is 80, 90 percent, right?

00:04:30.199 --> 00:04:31.892
So everybody goes to vote.

00:04:31.892 --> 00:04:32.725
- [Voiceover] So, okay.

00:04:32.725 --> 00:04:34.624
So, let's talk about
this political cartoon.

00:04:34.624 --> 00:04:36.809
- [Voiceover] Well, it's
really just making fun

00:04:36.809 --> 00:04:40.357
of Andrew Jackson for
being incredibly stubborn.

00:04:40.357 --> 00:04:43.293
He vetoed everything because the position

00:04:43.293 --> 00:04:46.499
of the Democratic Party,
which is founded with Jackson,

00:04:46.499 --> 00:04:47.863
he vetoes everything because

00:04:47.863 --> 00:04:49.895
he believes in small government.

00:04:49.895 --> 00:04:51.296
So, he thinks that the power

00:04:51.296 --> 00:04:53.815
of the government should
be negative, right?

00:04:53.815 --> 00:04:55.958
The less federal power, the better.

00:04:55.958 --> 00:04:58.378
Which is interesting
for a president, right?

00:04:58.378 --> 00:04:59.211
- [Voiceover] Right.

00:04:59.211 --> 00:05:01.145
- [Voiceover] So it's in this time period,

00:05:01.145 --> 00:05:04.938
from probably this cartoon,
that we get the image

00:05:04.938 --> 00:05:08.538
of the donkey as the iconic
of the Democratic Party.

00:05:08.538 --> 00:05:09.371
- [Voiceover] Cool.

00:05:09.371 --> 00:05:12.868
- [Voiceover] Because Jack-son
many believed was a Jack-ass.

00:05:12.868 --> 00:05:14.825
- [Voiceover] Ah.

00:05:14.825 --> 00:05:16.709
- [Voiceover] So the Democratic Party,

00:05:16.709 --> 00:05:18.488
as a party of all white men,

00:05:18.488 --> 00:05:23.089
ends up having a difficult
relationship with slavery.

00:05:23.089 --> 00:05:27.296
Which is the major issue
of 19th century America.

00:05:27.296 --> 00:05:29.132
And the issue of slavery

00:05:29.132 --> 00:05:32.900
really ends up kinda breaking
the Democratic Party apart

00:05:32.900 --> 00:05:37.650
because in the election of
1860, they actually split

00:05:37.650 --> 00:05:40.852
into the Northern Democrats
and the Southern Democrats

00:05:40.852 --> 00:05:43.430
and run two different
candidates for president.

00:05:43.430 --> 00:05:45.152
- [Voiceover] Woah, why?

00:05:45.152 --> 00:05:47.521
I knew that the Whigs
were split on slavery,

00:05:47.521 --> 00:05:49.004
what split the Democrats on slavery?

00:05:49.004 --> 00:05:51.702
- [Voiceover] Slavery, again.

00:05:51.702 --> 00:05:54.476
The Southern Democrats really supported

00:05:54.476 --> 00:05:56.380
the expansion of slavery everywhere.

00:05:56.380 --> 00:05:58.395
The Northern Democrats were just trying

00:05:58.395 --> 00:06:01.562
to prevent the Union from breaking up.

00:06:01.562 --> 00:06:04.047
- [Voiceover] So they were still
pro-slavery, but pro-Union?

00:06:04.047 --> 00:06:04.880
- [Voiceover] Yes.

00:06:04.880 --> 00:06:06.596
- [Voiceover] And that's why
they only carried Missouri?

00:06:06.596 --> 00:06:08.205
When it says Douglas here,
is that Stephen Douglas?

00:06:08.205 --> 00:06:09.257
- [Voiceover] That is Stephen Douglas,

00:06:09.257 --> 00:06:11.613
famous foe of Abraham Lincoln.

00:06:11.613 --> 00:06:12.446
- [Voiceover] Dang.

00:06:12.446 --> 00:06:14.260
- [Voiceover] And Stephen
Douglas actually died

00:06:14.260 --> 00:06:15.994
I think in 1861.

00:06:15.994 --> 00:06:16.827
- [Voiceover] Woah!

00:06:16.827 --> 00:06:18.492
- [Voiceover] Which was a real shock

00:06:18.492 --> 00:06:20.297
for the Democratic Party because they

00:06:20.297 --> 00:06:23.381
lost their real, major
leader in the North.

00:06:23.381 --> 00:06:26.884
And so they are the minority
party as Abraham Lincoln

00:06:26.884 --> 00:06:30.769
and the first Republican
leads the United States

00:06:30.769 --> 00:06:32.770
to victory in the Civil War.

00:06:32.770 --> 00:06:35.154
And so having been kind
of the opposition party

00:06:35.154 --> 00:06:38.272
against the victorious
party in war means that

00:06:38.272 --> 00:06:42.475
the Democrats are in a fairly
bad political position,

00:06:42.475 --> 00:06:45.308
really for most of the rest
of the 19th century, right?

00:06:45.308 --> 00:06:49.194
'Cause this is the real
ascension era of the Republicans

00:06:49.194 --> 00:06:51.712
as they wave the bloody shirt.

00:06:51.712 --> 00:06:55.031
- [Voiceover] So the Republican party bled

00:06:55.031 --> 00:06:56.261
to save the Union.

00:06:56.261 --> 00:06:57.094
- [Voiceover] Exactly.

00:06:57.094 --> 00:06:58.030
- [Voiceover] Okay.

00:06:58.030 --> 00:06:58.981
- [Voiceover] So that's your bloody shirt.

00:06:58.981 --> 00:07:00.037
- [Voiceover] That's my bloody shirt.

00:07:00.037 --> 00:07:00.870
- [Voiceover] It's amazing.

00:07:00.870 --> 00:07:01.993
- [Voiceover] There it is.

00:07:01.993 --> 00:07:03.060
Little motion blur for you.

00:07:03.060 --> 00:07:03.961
(laughing)

00:07:03.961 --> 00:07:05.142
- [Voiceover] Of waving it, okay.

00:07:05.142 --> 00:07:05.975
I gotcha.

00:07:05.975 --> 00:07:06.836
- [Voiceover] So, okay.

00:07:06.836 --> 00:07:09.101
So this rhetorical strategy of saying,

00:07:09.101 --> 00:07:11.257
hey remember, the
Republicans were the ones

00:07:11.257 --> 00:07:12.691
that saved the Union,

00:07:12.691 --> 00:07:16.732
that basically makes the
Democrats a rump party, right?

00:07:16.732 --> 00:07:17.565
- [Voiceover] Yeah.

00:07:17.565 --> 00:07:18.398
For the rest of the 19th century.

00:07:18.398 --> 00:07:19.410
- [Voiceover] They're in bad shape.

00:07:19.410 --> 00:07:22.034
But they do have a few moments,

00:07:22.034 --> 00:07:25.060
especially in the late 19th century,

00:07:25.060 --> 00:07:27.060
there's a major economic depression.

00:07:27.060 --> 00:07:28.827
And remember, economic
depression is never good

00:07:28.827 --> 00:07:30.480
for the party that's in power, right?

00:07:30.480 --> 00:07:31.313
- [Voiceover] Right.

00:07:31.313 --> 00:07:32.146
Is that here?

00:07:32.146 --> 00:07:34.260
- [Voiceover] So, in the panic of 1893,

00:07:34.260 --> 00:07:37.747
there's just major economic trouble.

00:07:37.747 --> 00:07:39.129
And in this time period,

00:07:39.129 --> 00:07:43.279
this is an important moment
for the Democratic Party.

00:07:43.279 --> 00:07:44.714
- [Voiceover] Alright,
so Kim, who's this fella?

00:07:44.714 --> 00:07:46.549
- [Voiceover] So this is
William Jennings Bryan.

00:07:46.549 --> 00:07:47.382
- [Voiceover] Oh, I know him.

00:07:47.382 --> 00:07:49.714
- [Voiceover] And he's one of those guys

00:07:49.714 --> 00:07:53.704
who ran for president so many
times but never actually won.

00:07:53.704 --> 00:07:54.537
So we don't--

00:07:54.537 --> 00:07:55.370
- [Voiceover] Cross of Gold speech guy.

00:07:55.370 --> 00:07:56.203
- [Voiceover] Exactly, right.

00:07:56.203 --> 00:07:58.002
So he had this idea that the United States

00:07:58.002 --> 00:08:00.128
should move off the gold standard

00:08:00.128 --> 00:08:02.246
and instead allow silver as one of

00:08:02.246 --> 00:08:04.710
the currencies backing the U.S. dollar.

00:08:04.710 --> 00:08:07.444
And the idea behind this was that then

00:08:07.444 --> 00:08:09.596
the money would flood into the economy,

00:08:09.596 --> 00:08:11.149
and people would get wealthier,

00:08:11.149 --> 00:08:12.947
and it would end this depression.

00:08:12.947 --> 00:08:14.149
And that's not how it happened,

00:08:14.149 --> 00:08:16.914
but what it does do is
moves the Democratic Party

00:08:16.914 --> 00:08:19.816
in a more populist direction, taking care

00:08:19.816 --> 00:08:23.983
of the smaller people who
need more help financially.

00:08:25.143 --> 00:08:29.612
Right, so a more
interventionist, economic policy.

00:08:29.612 --> 00:08:31.645
And so they want to kinda start

00:08:31.645 --> 00:08:33.348
helping the little guy with money.

00:08:33.348 --> 00:08:34.448
And you see this kind of in

00:08:34.448 --> 00:08:36.712
what we would call the Progressive Era.

00:08:36.712 --> 00:08:40.947
Which Wilson, who's gonna be
the next Democratic president,

00:08:40.947 --> 00:08:45.785
starting in 1912, he's a big
champion of Progressivism.

00:08:45.785 --> 00:08:49.952
And it's under Wilson's rule
that the 19th Amendment,

00:08:50.934 --> 00:08:53.358
granting women the right
to vote, is passed.

00:08:53.358 --> 00:08:57.670
So this is kind of
anti-corruption, anti-monopoly,

00:08:57.670 --> 00:09:01.020
very pro trying to protect the consumer.

00:09:01.020 --> 00:09:04.458
So this is where the sort
of more liberal aspect

00:09:04.458 --> 00:09:07.776
of Democratic monetary policy comes from.

00:09:07.776 --> 00:09:10.060
I don't wanna give the
impression that Progressivism

00:09:10.060 --> 00:09:13.794
and Wilson were 100 percent good things.

00:09:13.794 --> 00:09:17.247
On one hand, Wilson was the
first Southern president,

00:09:17.247 --> 00:09:18.665
he was a Virginian.

00:09:18.665 --> 00:09:20.648
Since the Civil War, after this era

00:09:20.648 --> 00:09:22.797
of waving the bloody shirt,
and one of the things

00:09:22.797 --> 00:09:26.222
that Wilson does is he
segregates federal jobs.

00:09:26.222 --> 00:09:28.286
So many African Americans who were working

00:09:28.286 --> 00:09:31.450
in the federal government
were forced out under Wilson.

00:09:31.450 --> 00:09:32.417
- [Voiceover] And don't forget that Wilson

00:09:32.417 --> 00:09:36.702
is the president who
screened the pro-KKK film,

00:09:36.702 --> 00:09:40.055
"Birth of a Nation" at
the White House in 1915.

00:09:40.055 --> 00:09:41.329
- [Voiceover] Yeah.

00:09:41.329 --> 00:09:43.260
- [Voiceover] Alright, and
then World War I happened,

00:09:43.260 --> 00:09:45.196
and then the stock market crash happened.

00:09:45.196 --> 00:09:47.050
- [Voiceover] Yeah, and the
stock market crash happened

00:09:47.050 --> 00:09:49.563
under Republican rule, and
it's never a good thing

00:09:49.563 --> 00:09:50.664
when your party's in power,

00:09:50.664 --> 00:09:52.848
and there's a major economic downturn.

00:09:52.848 --> 00:09:53.681
- [Voiceover] Right, and that's during

00:09:53.681 --> 00:09:54.963
the Hoover Administration.

00:09:54.963 --> 00:09:56.484
- [Voiceover] Exactly, before then.

00:09:56.484 --> 00:09:59.502
And so it's after this economic crash

00:09:59.502 --> 00:10:02.783
that Franklin Delano
Roosevelt comes into power.

00:10:02.783 --> 00:10:04.483
And now Roosevelt has this very

00:10:04.483 --> 00:10:07.433
liberal approach to economics, right?

00:10:07.433 --> 00:10:09.987
And this is where sort of contemporary,

00:10:09.987 --> 00:10:12.785
modern American liberalism comes from,

00:10:12.785 --> 00:10:14.840
which is an idea that government

00:10:14.840 --> 00:10:17.558
should regulate the economy.

00:10:17.558 --> 00:10:19.454
You know, they shouldn't
control the economy,

00:10:19.454 --> 00:10:21.296
but they should try to mediate some

00:10:21.296 --> 00:10:24.442
of these major swings that
the economy could make.

00:10:24.442 --> 00:10:26.709
And also that government
has a responsibility

00:10:26.709 --> 00:10:29.360
for the welfare of its people.

00:10:29.360 --> 00:10:32.645
So, FDR implements the New Deal.

00:10:32.645 --> 00:10:36.728
Which is a really sweeping
set of social programs

00:10:38.058 --> 00:10:40.712
designed to try to get people
out of their depression,

00:10:40.712 --> 00:10:42.547
and guarantee a certain standard

00:10:42.547 --> 00:10:44.363
of living for all Americans.

00:10:44.363 --> 00:10:45.445
- [Voiceover] What are
some of those programs

00:10:45.445 --> 00:10:47.563
from the New Deal that
we still have today?

00:10:47.563 --> 00:10:49.352
- [Voiceover] Social Security I think

00:10:49.352 --> 00:10:51.497
would be one of the biggest ones.

00:10:51.497 --> 00:10:55.686
That's a living wage for the
elderly, or the disabled.

00:10:55.686 --> 00:11:00.537
And the FDIC, the Federal
Deposit Insurance Corporation,

00:11:00.537 --> 00:11:03.237
which backs banks, for example.

00:11:03.237 --> 00:11:04.755
But what's interesting about this,

00:11:04.755 --> 00:11:06.953
is that we often think
of the Democratic Party

00:11:06.953 --> 00:11:08.106
and the Republican Party kind of

00:11:08.106 --> 00:11:10.106
having opposite stances on things.

00:11:10.106 --> 00:11:11.906
But the Democratic Party
in this time period

00:11:11.906 --> 00:11:15.124
kinda had a liberal Northern wing,

00:11:15.124 --> 00:11:17.075
and then a Southern conservative wing,

00:11:17.075 --> 00:11:20.051
so there are two kind of
parts of the Democratic Party.

00:11:20.051 --> 00:11:21.434
- [Voiceover] Sure, so
let's look at this map here,

00:11:21.434 --> 00:11:23.303
what is this Solid South thing?

00:11:23.303 --> 00:11:24.519
- [Voiceover] So this is the Solid South,

00:11:24.519 --> 00:11:28.686
and the Solid South is the
Democratic, pro-Jim Crow South.

00:11:30.955 --> 00:11:33.289
So there's the Democratic
Party in the South,

00:11:33.289 --> 00:11:36.593
which is much smaller government,

00:11:36.593 --> 00:11:39.309
really wanted to make sure
that African Americans

00:11:39.309 --> 00:11:42.809
were prevented from having
political power in the South.

00:11:42.809 --> 00:11:46.746
So it's interesting because
African Americans after FDR

00:11:46.746 --> 00:11:49.011
really join the Democratic party,

00:11:49.011 --> 00:11:51.340
but in the South, most of them can't vote.

00:11:51.340 --> 00:11:52.602
- [Voiceover] Right.

00:11:52.602 --> 00:11:54.431
It's a very uncomfortable alliance.

00:11:54.431 --> 00:11:55.863
So, Kim, sorry, I thought that

00:11:55.863 --> 00:11:57.212
this part of the country, the Northeast,

00:11:57.212 --> 00:12:01.126
was still pretty solidly
Republican at this time.

00:12:01.126 --> 00:12:02.915
- [Voiceover] I would say
that FDR puts together

00:12:02.915 --> 00:12:06.665
kind of a New Deal
Coalition of labor unions,

00:12:08.849 --> 00:12:11.198
ethnic and religious minorities,

00:12:11.198 --> 00:12:14.044
sort of the workers of the United States,

00:12:14.044 --> 00:12:18.233
who want a more interventionist
government economy.

00:12:18.233 --> 00:12:19.419
- [Voiceover] Okay.

00:12:19.419 --> 00:12:21.653
- [Voiceover] But don't necessarily want

00:12:21.653 --> 00:12:25.154
the moralizing aspect
that goes along with say,

00:12:25.154 --> 00:12:27.404
the Progressive Era, right?

00:12:28.267 --> 00:12:31.319
The Progressives passed
prohibition, for example, right?

00:12:31.319 --> 00:12:33.969
They thought it was morally bad to drink.

00:12:33.969 --> 00:12:36.152
And one of the first things
FDR does when he gets

00:12:36.152 --> 00:12:39.188
into office is to repeal
prohibition saying,

00:12:39.188 --> 00:12:41.454
you know, what America
needs now is a drink.

00:12:41.454 --> 00:12:42.287
Right?

00:12:42.287 --> 00:12:43.502
So it doesn't have this moral aspect,

00:12:43.502 --> 00:12:45.603
but it does have economic control.

00:12:45.603 --> 00:12:46.436
- [Voiceover] Okay.

00:12:46.436 --> 00:12:48.005
- [Voiceover] But the party
continues to kinda move

00:12:48.005 --> 00:12:52.589
to the left on social issues,
and in the mid 1960's,

00:12:52.589 --> 00:12:57.324
Lyndon Johnson start to pass
civil rights legislation,

00:12:57.324 --> 00:13:00.360
like the Civil Rights Act of 1964,

00:13:00.360 --> 00:13:02.210
and also pretty sweeping social programs,

00:13:02.210 --> 00:13:04.700
which are called the Great Society.

00:13:04.700 --> 00:13:07.390
So these is things like Head Start

00:13:07.390 --> 00:13:09.807
and lots of welfare programs.

00:13:12.124 --> 00:13:13.175
And in this time period,

00:13:13.175 --> 00:13:15.061
along with the Civil
Rights Movement, you know,

00:13:15.061 --> 00:13:19.228
the Democratic Party really
coalesces around this group

00:13:20.428 --> 00:13:24.013
in the North who are
more pro-civil rights.

00:13:24.013 --> 00:13:26.846
So this period from FDR's election

00:13:28.301 --> 00:13:31.349
through Lyndon Johnson's administration

00:13:31.349 --> 00:13:33.792
is really kind of the Democratic heyday

00:13:33.792 --> 00:13:35.446
in the United States.

00:13:35.446 --> 00:13:39.530
Save for Eisenhower, there is
no other Republican president

00:13:39.530 --> 00:13:41.328
that's elected during this time period,

00:13:41.328 --> 00:13:45.495
so this is 30 plus years
of Democratic ascendancy.

00:13:46.433 --> 00:13:49.183
But it kinda comes apart in 1968.

00:13:50.848 --> 00:13:52.386
- [Voiceover] So what happens in 1968?

00:13:52.386 --> 00:13:54.767
- [Voiceover] Well, there's
a lot of social unrest.

00:13:54.767 --> 00:13:57.338
The assassination of
Martin Luther King Jr.,

00:13:57.338 --> 00:14:01.284
the assassination of Robert F. Kennedy,

00:14:01.284 --> 00:14:02.784
many summer riots.

00:14:03.681 --> 00:14:06.885
And many whites in the
South who had, you know,

00:14:06.885 --> 00:14:10.048
followed this Democratic Coalition here,

00:14:10.048 --> 00:14:12.399
began to feel that the
Civil Rights Movement

00:14:12.399 --> 00:14:14.949
and the commitment of the Democratic Party

00:14:14.949 --> 00:14:18.446
to civil rights had
gone too far, basically.

00:14:18.446 --> 00:14:22.045
And there was the problem of Vietnam.

00:14:22.045 --> 00:14:24.165
The Vietnam War, which both parties

00:14:24.165 --> 00:14:27.181
were really strongly
anti-Communist in this time period,

00:14:27.181 --> 00:14:29.821
so many of the sort of young people,

00:14:29.821 --> 00:14:34.202
who might otherwise have
supported the Democratic Party,

00:14:34.202 --> 00:14:36.271
began to feel that the
Democratic Party too

00:14:36.271 --> 00:14:38.822
was part of this war machine that was just

00:14:38.822 --> 00:14:41.843
sending the United States
citizens to fight a war

00:14:41.843 --> 00:14:44.273
that they believed they
had nothing to do with.

00:14:44.273 --> 00:14:46.874
- [Voiceover] So we're seeing
this demographic shift.

00:14:46.874 --> 00:14:47.787
- [Voiceover] Exactly.

00:14:47.787 --> 00:14:51.576
So this Solid South, which
had been voting Democratic

00:14:51.576 --> 00:14:55.917
because they had really
complete political hegemony,

00:14:55.917 --> 00:14:58.907
right, 'cause African
Americans couldn't vote,

00:14:58.907 --> 00:15:01.139
now that African Americans can vote,

00:15:01.139 --> 00:15:02.810
many of those whites end up switching

00:15:02.810 --> 00:15:04.412
to the Republican Party.

00:15:04.412 --> 00:15:05.281
- [Voiceover] Interesting.

00:15:05.281 --> 00:15:07.358
Alright, so Johnson decides
he's not even gonna run again.

00:15:07.358 --> 00:15:08.191
- [Voiceover] Right.

00:15:08.191 --> 00:15:10.793
'Cause he knows he can't
win because the Vietnam War

00:15:10.793 --> 00:15:13.808
has just tarnished him in
the sight of most Americans.

00:15:13.808 --> 00:15:14.641
- [Voiceover] Right.

00:15:14.641 --> 00:15:16.613
- [Voiceover] So, with the
Democrats really split apart,

00:15:16.613 --> 00:15:18.395
in complete disarray, the Republicans

00:15:18.395 --> 00:15:20.296
take back the White House.

00:15:20.296 --> 00:15:21.129
- [Voiceover] So, okay.

00:15:21.129 --> 00:15:23.485
So Kim, that takes us up to about 1968

00:15:23.485 --> 00:15:25.814
and the election of Nixon.

00:15:25.814 --> 00:15:29.100
And there's a lot of history
of the Democratic Party.

00:15:29.100 --> 00:15:30.635
216 years, right?

00:15:30.635 --> 00:15:32.398
- [Voiceover] Yep, it is the oldest,

00:15:32.398 --> 00:15:35.384
voter-based political party in the world.

00:15:35.384 --> 00:15:36.217
- [Voiceover] So, we can't fit

00:15:36.217 --> 00:15:37.725
the entire history into one video.

00:15:37.725 --> 00:15:38.959
- [Voiceover] No, there's too much.

00:15:38.959 --> 00:15:40.344
- [Voiceover] So, next
time when we come back,

00:15:40.344 --> 00:15:41.430
we're gonna talk about

00:15:41.430 --> 00:15:43.566
the new Democrats under President Clinton,

00:15:43.566 --> 00:15:45.547
we're gonna talk about
the Obama Administration,

00:15:45.547 --> 00:15:47.280
and the policy and the plank

00:15:47.280 --> 00:15:49.580
of the Democratic Party going forward.

00:15:49.580 --> 00:15:51.913
- [Voiceover] So stay tuned.


Motion along a curve: finding velocity magnitude

WEBVTT
Kind: captions
Language: en

00:00:00.556 --> 00:00:03.411
- [Voiceover] A particle moves
along the curve xy equals 16,

00:00:03.411 --> 00:00:05.180
so that the y coordinate is increasing,

00:00:05.180 --> 00:00:08.259
we underline this, the y
coordinate is increasing

00:00:08.259 --> 00:00:10.569
at a constant rate of
two units per minute.

00:00:10.569 --> 00:00:13.565
That means that the rate of
change of y with respect to t

00:00:13.565 --> 00:00:15.588
is equal to two.

00:00:15.588 --> 00:00:17.824
What is the magnitude,
in units per minute,

00:00:17.824 --> 00:00:20.550
of the particle's velocity vector when the

00:00:20.550 --> 00:00:23.534
particle is at the point
four comma four, so when

00:00:23.534 --> 00:00:25.617
x is four, and y is four.

00:00:28.121 --> 00:00:30.093
So let's see what's going on.

00:00:30.093 --> 00:00:32.783
So, let's first just
remind ourselves what a

00:00:32.783 --> 00:00:36.321
velocity vector, what the
velocity vector will look like.

00:00:36.321 --> 00:00:40.274
So our velocity is going
to be a function of time.

00:00:40.274 --> 00:00:41.911
And it's going to have two components,

00:00:41.911 --> 00:00:43.904
it's going to be what is
the rate of change in the x

00:00:43.904 --> 00:00:46.435
direction, and the rate of
change in the y direction.

00:00:46.435 --> 00:00:48.670
So the rate of change in
the x direction is going to

00:00:48.670 --> 00:00:52.460
be dx dt, and the rate of
change in the y direction

00:00:52.460 --> 00:00:54.210
is going to be dy dt.

00:00:56.377 --> 00:00:59.545
And they tell us that this
is, that dy dt is a constant

00:00:59.545 --> 00:01:02.057
two units per minute.

00:01:02.057 --> 00:01:03.408
But they're not even
just asking us for just

00:01:03.408 --> 00:01:06.028
the velocity vector for its components,

00:01:06.028 --> 00:01:09.102
they're asking for the magnitude,

00:01:09.102 --> 00:01:12.083
they're asking for the
magnitude of the particle's

00:01:12.083 --> 00:01:13.416
velocity vector.

00:01:14.700 --> 00:01:17.650
Well, if I have some vector,
let me do a little bit of a

00:01:17.650 --> 00:01:21.483
side here, if I have
some vector, let's say a,

00:01:23.332 --> 00:01:27.499
that has components b and
c, well then the magnitude

00:01:29.762 --> 00:01:32.032
of my vector, sometimes you'll
see it written like that,

00:01:32.032 --> 00:01:34.862
sometimes you'll see it
written with double bars

00:01:34.862 --> 00:01:38.407
like that, the magnitude of my vector,

00:01:38.407 --> 00:01:40.517
and this comes straight out
of the Pythagorean theorem,

00:01:40.517 --> 00:01:43.714
this is going to be the
square root of b squared

00:01:43.714 --> 00:01:45.572
plus c squared.

00:01:45.572 --> 00:01:48.004
The square root of the
x component squared plus

00:01:48.004 --> 00:01:49.652
the y component squared.

00:01:49.652 --> 00:01:52.705
So, if we wanted the magnitude
of our velocity vector,

00:01:52.705 --> 00:01:56.705
the magnitude of the
particle's velocity vector,

00:01:57.875 --> 00:02:02.051
well I could write that
as the magnitude of v,

00:02:02.051 --> 00:02:04.066
I could even write it as a function of t,

00:02:04.066 --> 00:02:06.770
it's going to be equal to the square root

00:02:06.770 --> 00:02:10.529
of the x component squared,
so that's the rate of change

00:02:10.529 --> 00:02:14.356
of x with respect to time, squared,

00:02:14.356 --> 00:02:18.356
plus the y component
squared, which in this case

00:02:19.482 --> 00:02:24.075
is the rate of change of y,
with respect to t, squared.

00:02:24.075 --> 00:02:26.931
So how do we figure out
these, how do we figure out

00:02:26.931 --> 00:02:28.908
these two things?

00:02:28.908 --> 00:02:30.975
Well we already know
the rate of change of y

00:02:30.975 --> 00:02:32.745
with respect to t, they say
that's a constant rate of

00:02:32.745 --> 00:02:34.263
two units per minute.

00:02:34.263 --> 00:02:37.166
So we already know that
this is gonna be two,

00:02:37.166 --> 00:02:39.879
or that this whole thing right
over here's gonna be four,

00:02:39.879 --> 00:02:41.056
but how do we figure
out the rate of change

00:02:41.056 --> 00:02:42.993
of x with respect to t?

00:02:42.993 --> 00:02:47.160
Well, we could take our
original equation that describes

00:02:49.396 --> 00:02:52.054
the curve, we could take
the derivative of both sides

00:02:52.054 --> 00:02:54.260
with respect to t, and
then that's going to give

00:02:54.260 --> 00:02:58.912
us an equation that involves
xy and dx dt and dy dt,

00:02:58.912 --> 00:02:59.796
so let's do that.

00:02:59.796 --> 00:03:02.213
So we have xy is equal to 16,

00:03:06.170 --> 00:03:07.815
I'm gonna to take the
derivative with respect to t,

00:03:07.815 --> 00:03:12.148
of both sides, we do that
in a different color,

00:03:12.148 --> 00:03:14.817
just for a little bit of variety,

00:03:14.817 --> 00:03:17.638
so the derivative with respect
to t of the left hand side,

00:03:17.638 --> 00:03:20.862
derivative with respect to
t of the right hand side,

00:03:20.862 --> 00:03:22.589
now the left hand side,
we view this as a product

00:03:22.589 --> 00:03:25.403
of two functions, if we say,
look, x is a function of t,

00:03:25.403 --> 00:03:27.258
and y is also a function
of t, this is we're

00:03:27.258 --> 00:03:28.859
gonna do a little bit of the product rule

00:03:28.859 --> 00:03:30.877
and a little bit of the chain rule here.

00:03:30.877 --> 00:03:32.805
And so this is going to be equal to,

00:03:32.805 --> 00:03:36.722
derivative of the first
function, which is, so,

00:03:39.292 --> 00:03:42.113
we'll first say the derivative
of x with respect to x

00:03:42.113 --> 00:03:45.854
is one, times the derivative
of x with respect to t.

00:03:45.854 --> 00:03:47.702
Remember, taking the derivative
with respect to t, not

00:03:47.702 --> 00:03:51.157
with respect to x, times
the second function,

00:03:51.157 --> 00:03:54.157
so times, times the second function,

00:03:55.695 --> 00:03:59.528
so times y, times y,
plus, the first function,

00:04:02.390 --> 00:04:06.275
which is just x, times the
derivative of the second

00:04:06.275 --> 00:04:08.184
function with respect to t.

00:04:08.184 --> 00:04:12.104
So first what's the derivative
of y with respect to y,

00:04:12.104 --> 00:04:15.004
well that's just one, and
then what's the derivative

00:04:15.004 --> 00:04:18.504
of y with respect to t, well that's dy dt,

00:04:21.962 --> 00:04:25.233
and that is going to be equal to,

00:04:25.233 --> 00:04:28.275
that is going to be equal
to, derivative of a constant

00:04:28.275 --> 00:04:29.358
is just zero.

00:04:30.293 --> 00:04:32.735
So let's see, what does this simplify to,

00:04:32.735 --> 00:04:34.873
this simplifies to, in
fact we don't even have

00:04:34.873 --> 00:04:36.528
to simplify it more,
we can actually plug in

00:04:36.528 --> 00:04:39.526
the values to solve for dx dt.

00:04:39.526 --> 00:04:42.609
We know that dy dt is a constant two,

00:04:43.670 --> 00:04:47.085
and we want the magnitude
of the particle's

00:04:47.085 --> 00:04:48.601
velocity vector when the
particle's at the point

00:04:48.601 --> 00:04:53.062
four comma four, so
when x is equal to four,

00:04:53.062 --> 00:04:57.229
so when x is equal to four,
and y is equal to four,

00:04:58.323 --> 00:05:01.024
and y is equal to four.

00:05:01.024 --> 00:05:03.514
So now, it's a little messy right now,

00:05:03.514 --> 00:05:05.251
but this right here is an
equation we can solve for,

00:05:05.251 --> 00:05:06.473
there's only one unknown here,

00:05:06.473 --> 00:05:08.115
the rate of change of x with respect to t,

00:05:08.115 --> 00:05:12.154
right when we are at the
point four comma four,

00:05:12.154 --> 00:05:13.625
and so if we're able to figure that out,

00:05:13.625 --> 00:05:15.180
we can substitute that
in here and figure out

00:05:15.180 --> 00:05:17.627
the magnitude of our velocity vector.

00:05:17.627 --> 00:05:19.516
So let us write it out,

00:05:19.516 --> 00:05:22.349
so this gives us four, four dx dt,

00:05:24.581 --> 00:05:27.621
plus what is this four times two,

00:05:27.621 --> 00:05:31.574
plus eight is equal to
zero, and so we have

00:05:31.574 --> 00:05:34.741
four dx dt is equal to negative eight,

00:05:36.197 --> 00:05:37.682
just subtracted eight from both sides,

00:05:37.682 --> 00:05:41.463
divide both sides by four, you get dx dt,

00:05:41.463 --> 00:05:44.630
scroll down, is equal to negative two.

00:05:46.221 --> 00:05:48.161
So when all this stuff
is going on the rate of

00:05:48.161 --> 00:05:52.832
change of x with respect
to t is negative two,

00:05:52.832 --> 00:05:57.048
and then you square it, you
get a four, right over here,

00:05:57.048 --> 00:05:59.614
and so the magnitude
of our velocity vector

00:05:59.614 --> 00:06:03.781
is going to be equal to the
square root of four plus four,

00:06:05.595 --> 00:06:09.254
which is equal to eight,
which is the same thing as

00:06:09.254 --> 00:06:12.136
four times two, so this is going to be

00:06:12.136 --> 00:06:15.553
two square root of two, units per minute,

00:06:16.846 --> 00:06:20.763
so that's the magnitude
of the velocity vector.


Mean value theorem example: square root function

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.473
- [Voiceover] Let f of x
be equal to the square root

00:00:02.473 --> 00:00:05.644
of four x minus three,
and let c be the number

00:00:05.644 --> 00:00:08.692
that satisfies the mean
value theorem for f

00:00:08.692 --> 00:00:11.842
on the closed interval
between one and three,

00:00:11.842 --> 00:00:13.526
or one is less than or equal to x

00:00:13.526 --> 00:00:15.553
is less than or equal to three.

00:00:15.553 --> 00:00:16.386
What is c?

00:00:17.401 --> 00:00:20.169
So let's just remind ourselves
what it means for c to be

00:00:20.169 --> 00:00:23.644
the number that satisfies
the mean value theorem for f.

00:00:23.644 --> 00:00:27.171
This means that over this
interval, c is a point,

00:00:27.171 --> 00:00:30.305
x equals c is a point where
the slope of the tangent line

00:00:30.305 --> 00:00:34.873
at x equals c, so I
could write f prime of c,

00:00:34.873 --> 00:00:37.357
so that is the slope of the tangent line

00:00:37.357 --> 00:00:39.783
when x is equal to c, this is equal

00:00:39.783 --> 00:00:42.503
to the slope of the secant line

00:00:42.503 --> 00:00:45.031
that connects these two points.

00:00:45.031 --> 00:00:47.527
So this is going to be equal to,

00:00:47.527 --> 00:00:50.773
see the slope of the
secant line that connects

00:00:50.773 --> 00:00:54.773
the points three, f of
three, and one, f of one.

00:00:56.475 --> 00:01:00.225
So it's going to be f
of three minus f of one

00:01:01.231 --> 00:01:02.981
over three minus one.

00:01:05.005 --> 00:01:08.136
And if you wanted to think
about what this means visually,

00:01:08.136 --> 00:01:10.230
it would look something like this.

00:01:10.230 --> 00:01:12.236
So if this is our x axis

00:01:12.236 --> 00:01:13.986
and this is one, two,

00:01:16.691 --> 00:01:18.876
actually let me spread
it out a little bit more.

00:01:18.876 --> 00:01:20.543
One, two, and three.

00:01:22.493 --> 00:01:26.050
And so you have one, comma,
f of one, right over there,

00:01:26.050 --> 00:01:29.800
so that is at the point
one, comma, f of one,

00:01:30.733 --> 00:01:32.531
and we could evaluate that, actually,

00:01:32.531 --> 00:01:35.172
that's one, comma, one.

00:01:35.172 --> 00:01:38.847
So that's gonna be the
point one, comma, one.

00:01:38.847 --> 00:01:41.044
And then you have the point three, comma,

00:01:41.044 --> 00:01:44.139
let's see, you're gonna
have four times three is 12

00:01:44.139 --> 00:01:48.306
minus three is nine, so it's
gonna be three, comma, three.

00:01:50.226 --> 00:01:54.179
So maybe it's right over
there, three, comma, three,

00:01:54.179 --> 00:01:57.663
and the curve might look
something like this,

00:01:57.663 --> 00:02:00.746
so it might look something like that,

00:02:02.045 --> 00:02:04.693
so if you think about
the slope of the line

00:02:04.693 --> 00:02:07.055
that connects these two points,

00:02:07.055 --> 00:02:10.722
so this line that
connects those two points,

00:02:12.058 --> 00:02:14.199
all the mean value theorem,

00:02:14.199 --> 00:02:15.972
I'm doing a different color.

00:02:15.972 --> 00:02:18.648
All the mean value
theorem tells us is that

00:02:18.648 --> 00:02:21.967
there's a point between
one and three where

00:02:21.967 --> 00:02:25.238
the slope of the tangent line
has the exact same slope.

00:02:25.238 --> 00:02:26.945
So if I were to eyeball it,

00:02:26.945 --> 00:02:28.789
it looks like it's right around there,

00:02:28.789 --> 00:02:31.299
although we are actually
going to solve for it.

00:02:31.299 --> 00:02:34.032
So, some point where the
slope of the tangent line

00:02:34.032 --> 00:02:36.381
is equal to the slope of the line

00:02:36.381 --> 00:02:38.614
that connects these two end points

00:02:38.614 --> 00:02:41.806
and their corresponding function values.

00:02:41.806 --> 00:02:46.180
So that is c, that would
be c right over there.

00:02:46.180 --> 00:02:48.863
So really we just have to solve this.

00:02:48.863 --> 00:02:51.828
So let's first just find
out what f prime of x is,

00:02:51.828 --> 00:02:54.590
and then we could substitute a c in there

00:02:54.590 --> 00:02:57.553
and then we can evaluate
this on the right hand side.

00:02:57.553 --> 00:02:59.886
So I'm gonna rewrite f of x,

00:03:01.522 --> 00:03:04.624
f of x is equal to, and I'm gonna write it

00:03:04.624 --> 00:03:08.869
as four x to the minus
three to the one half power.

00:03:08.869 --> 00:03:10.937
Makes it a little bit more
obvious that we can apply

00:03:10.937 --> 00:03:13.247
the power rule and the chain rule here.

00:03:13.247 --> 00:03:16.759
So f prime of x is going
to be the derivative

00:03:16.759 --> 00:03:19.341
of four x minus three to the one half

00:03:19.341 --> 00:03:21.895
with respect to four x minus three,

00:03:21.895 --> 00:03:25.227
so that is going to be
one half times four x

00:03:25.227 --> 00:03:28.109
minus three to the negative one half,

00:03:28.109 --> 00:03:29.910
and then we're gonna multiply that

00:03:29.910 --> 00:03:31.906
times the derivative of four x

00:03:31.906 --> 00:03:34.112
minus three with respect to x.

00:03:34.112 --> 00:03:37.236
Well, derivative of four x
with respect to x is just four,

00:03:37.236 --> 00:03:39.718
and the derivative of negative
three with respect to x,

00:03:39.718 --> 00:03:41.440
well that's just gonna be zero,

00:03:41.440 --> 00:03:43.214
so the derivative of four x minus three

00:03:43.214 --> 00:03:45.278
with respect to x is four.

00:03:45.278 --> 00:03:46.918
So times four.

00:03:46.918 --> 00:03:48.168
so f prime of x

00:03:50.507 --> 00:03:53.174
is equal to four times one half,

00:03:55.144 --> 00:03:59.311
which is two, over the square
root of four x minus three.

00:04:01.483 --> 00:04:03.763
Four x minus three to the one half

00:04:03.763 --> 00:04:05.786
would just be the square
root of four x minus three,

00:04:05.786 --> 00:04:07.319
but it's the negative one half,

00:04:07.319 --> 00:04:09.590
so we're gonna put it in the
denominator right over here.

00:04:09.590 --> 00:04:12.759
And so f prime of c, we could rewrite this

00:04:12.759 --> 00:04:16.926
as two over the square
root of four c minus three,

00:04:19.663 --> 00:04:22.438
and what is that going to be equal to?

00:04:22.438 --> 00:04:25.437
That is going to be equal to, let's see,

00:04:25.437 --> 00:04:28.512
f of three we already
figured out is three,

00:04:28.512 --> 00:04:31.495
f of one we already figured out is one,

00:04:31.495 --> 00:04:35.181
and so we get three minus
one over three minus one,

00:04:35.181 --> 00:04:38.258
well that's gonna be two over
two, which is equal to one.

00:04:38.258 --> 00:04:41.019
So there's some point
between one and three

00:04:41.019 --> 00:04:43.632
where the derivative at that point,

00:04:43.632 --> 00:04:47.276
the slope of the tangent
line, is equal to one.

00:04:47.276 --> 00:04:50.332
So let's see if we can solve
this thing right over here.

00:04:50.332 --> 00:04:54.416
We can multiply both
sides of this by four c

00:04:54.416 --> 00:04:57.496
by the square root of four c minus three,

00:04:57.496 --> 00:04:59.774
and so then we are going to get

00:04:59.774 --> 00:05:03.941
two is equal to the square
root of four c minus three.

00:05:06.225 --> 00:05:08.310
All I did is multiply both sides of this

00:05:08.310 --> 00:05:10.121
by square root of four c minus three

00:05:10.121 --> 00:05:12.072
to get rid of this in the denominator,

00:05:12.072 --> 00:05:14.075
and so let's see, now we
get rid of the radical,

00:05:14.075 --> 00:05:16.541
we can square both sides, and so

00:05:16.541 --> 00:05:18.540
actually let me just show that,

00:05:18.540 --> 00:05:20.396
so now we can square both sides,

00:05:20.396 --> 00:05:24.229
so we get four is equal
to four c minus three,

00:05:26.516 --> 00:05:30.683
add three to both sides,
seven is equal to four c,

00:05:31.636 --> 00:05:34.553
and then divide both sides by four.

00:05:35.801 --> 00:05:37.944
I'll go right here to do it,

00:05:37.944 --> 00:05:40.485
you're going to get c is
equal to seven fourths.

00:05:40.485 --> 00:05:42.985
C is equal to seven over four,

00:05:44.748 --> 00:05:47.814
which is equal to one and three fourths,

00:05:47.814 --> 00:05:50.314
or we could view this as 1.75.

00:05:51.967 --> 00:05:54.695
So actually, the c value
is a little bit closer,

00:05:54.695 --> 00:05:57.859
I hand drew this, it's closer
to about right over there

00:05:57.859 --> 00:06:01.464
on our diagram, actually
that looks pretty close,

00:06:01.464 --> 00:06:03.168
that actually looks pretty good,

00:06:03.168 --> 00:06:06.222
I just hand drew this curve
so it's definitely not exact.

00:06:06.222 --> 00:06:08.499
But anyway, hopefully that gives you

00:06:08.499 --> 00:06:10.683
a sense of what's going on here.

00:06:10.683 --> 00:06:12.720
We're just saying, hey,
the mean value theorem

00:06:12.720 --> 00:06:14.720
gives us some c where the
slope of the tangent line

00:06:14.720 --> 00:06:16.545
is the same as the slope of the line

00:06:16.545 --> 00:06:20.712
that connects one, f of
one, and three, f of three.


Motion along a curve: finding rate of change

WEBVTT
Kind: captions
Language: en

00:00:00.487 --> 00:00:01.320
- [Voiceover] We're told that a particle

00:00:01.320 --> 00:00:04.913
moves along the curve x squared, y squared

00:00:04.913 --> 00:00:07.321
is equal to 16, so that
the the x-coordinate

00:00:07.321 --> 00:00:09.514
is changing at a constant rate

00:00:09.514 --> 00:00:12.264
of negative two units per minute.

00:00:14.008 --> 00:00:16.683
What is the rate of
change in units per minute

00:00:16.683 --> 00:00:19.023
of the particle's y-coordinate

00:00:19.023 --> 00:00:22.273
when the particle is at the point 1, 4?

00:00:23.250 --> 00:00:26.128
So let's just repeat, or
rewrite, what they told us.

00:00:26.128 --> 00:00:28.568
So the curve is described by x squared,

00:00:28.568 --> 00:00:30.651
y squared is equal to 16.

00:00:31.696 --> 00:00:32.938
They tell us that up there.

00:00:32.938 --> 00:00:34.572
They tell us that the x-coordinate

00:00:34.572 --> 00:00:37.197
is changing at a constant rate.

00:00:37.197 --> 00:00:38.471
We underline that.

00:00:38.471 --> 00:00:39.633
The x-coordinate is changing at

00:00:39.633 --> 00:00:42.931
a constant rate of negative
two units per minute.

00:00:42.931 --> 00:00:45.473
So we could say that dx ...

00:00:45.473 --> 00:00:47.291
I'll write over here
on the right hand side.

00:00:47.291 --> 00:00:48.707
Dx, dt.

00:00:48.707 --> 00:00:50.731
The rate of change of the x-coordinate

00:00:50.731 --> 00:00:53.726
with respect to time is
equal to negative two.

00:00:53.726 --> 00:00:57.643
And they're saying units,
some unit of distance,

00:00:58.775 --> 00:01:01.598
units divided by minute.

00:01:01.598 --> 00:01:03.015
Units per minute.

00:01:04.304 --> 00:01:05.757
What they want us to figure out

00:01:05.757 --> 00:01:07.381
is what is the rate of change

00:01:07.381 --> 00:01:09.679
of the particle's y-coordinate.

00:01:09.679 --> 00:01:10.963
So let me underline that.

00:01:10.963 --> 00:01:15.150
What is the rate of change of
the particle's y-coordinate?

00:01:15.150 --> 00:01:19.789
So what they want us to
find is what is dy, dt.

00:01:19.789 --> 00:01:21.221
What is that equal to?

00:01:21.221 --> 00:01:25.258
And they say when the
particle is at the point 1,4,

00:01:25.258 --> 00:01:29.176
so when x is equal to
one, so x is equal to one.

00:01:29.176 --> 00:01:33.268
And y is equal to four,
y is equal to four.

00:01:33.268 --> 00:01:36.463
So can we set up some
equation that involves

00:01:36.463 --> 00:01:38.773
the rate of change of x with respect to t,

00:01:38.773 --> 00:01:42.019
y with respect to t, x and y?

00:01:42.019 --> 00:01:43.918
Well what if we were
to take the derivative

00:01:43.918 --> 00:01:47.019
of this relation that describes the curve?

00:01:47.019 --> 00:01:48.170
What if we were to take the derivative

00:01:48.170 --> 00:01:50.541
with respect to t on both sides?

00:01:50.541 --> 00:01:52.421
So let me write that down.

00:01:52.421 --> 00:01:53.500
So we're going to take the derivative.

00:01:53.500 --> 00:01:56.083
Actually let me just erase this

00:01:58.458 --> 00:02:01.791
so I have a little bit more space.

00:02:01.791 --> 00:02:03.105
Alright.

00:02:03.105 --> 00:02:06.103
That way I can just add it.

00:02:06.103 --> 00:02:09.443
So let's take the derivative with respect

00:02:09.443 --> 00:02:11.693
to t on both sides of that.

00:02:13.122 --> 00:02:14.384
And if at any point you get inspired

00:02:14.384 --> 00:02:16.685
I encourage you to pause the video

00:02:16.685 --> 00:02:19.076
and try to work through it.

00:02:19.076 --> 00:02:20.625
Well on the left hand side, if we view

00:02:20.625 --> 00:02:24.340
this as a product of two
functions right over here,

00:02:24.340 --> 00:02:28.507
we could take the derivative
of the first function,

00:02:30.095 --> 00:02:32.749
which is going to be the derivative

00:02:32.749 --> 00:02:35.571
of x squared with respect to x.

00:02:35.571 --> 00:02:39.440
So that is 2x, and remember we're not

00:02:39.440 --> 00:02:41.948
just taking the derivative
with respect to x,

00:02:41.948 --> 00:02:43.710
we're taking the derivative
with respect to t,

00:02:43.710 --> 00:02:44.995
so we're going to have
to apply the chain rule.

00:02:44.995 --> 00:02:46.301
So it's going to be the
derivative of x squared

00:02:46.301 --> 00:02:48.325
with respect to x, which is 2x,

00:02:48.325 --> 00:02:51.490
times the derivative
of x with respect to t.

00:02:51.490 --> 00:02:52.823
So times dx, dt.

00:02:55.021 --> 00:02:56.118
And then we're going to multiply

00:02:56.118 --> 00:02:58.508
that times the second function.

00:02:58.508 --> 00:03:00.091
So times y squared.

00:03:01.068 --> 00:03:02.580
Times y squared.

00:03:02.580 --> 00:03:04.141
And then that's going to be plus

00:03:04.141 --> 00:03:08.262
the first function,
which is just x squared

00:03:08.262 --> 00:03:10.211
times the derivative of the second

00:03:10.211 --> 00:03:12.243
function with respect to d.

00:03:12.243 --> 00:03:13.686
So once again we're going
to apply the chain rule.

00:03:13.686 --> 00:03:18.233
The derivative of y-squared
with respect to y is 2y.

00:03:18.233 --> 00:03:19.849
I'm gonna do that in that orange color.

00:03:19.849 --> 00:03:21.349
It is equal to 2y.

00:03:23.891 --> 00:03:27.615
And then times the derivative
of y with respect to t.

00:03:27.615 --> 00:03:28.698
Times dy, dt.

00:03:30.790 --> 00:03:32.985
And then that is going to be equal

00:03:32.985 --> 00:03:36.624
to the derivative with respect to t of 16.

00:03:36.624 --> 00:03:38.404
Well that doesn't change over time,

00:03:38.404 --> 00:03:41.479
so that's just going to be equal to zero.

00:03:41.479 --> 00:03:42.564
So here we have it.

00:03:42.564 --> 00:03:44.255
We need to simplify this a little bit.

00:03:44.255 --> 00:03:46.142
But we have an equation that gives

00:03:46.142 --> 00:03:49.084
our relationship between
x, derivative of x

00:03:49.084 --> 00:03:52.998
with respect to t, y, and
derivative of y with respect to t.

00:03:52.998 --> 00:03:54.220
So actually let me just rewrite

00:03:54.220 --> 00:03:56.314
it one more time so it's
a little bit simplified.

00:03:56.314 --> 00:03:59.564
So this is 2xy squared dx, dt, plus ...

00:04:05.032 --> 00:04:06.702
Actually I don't even
have to rewrite it again.

00:04:06.702 --> 00:04:09.762
All we're trying to do
is solve for dy, dt,

00:04:09.762 --> 00:04:12.055
so let's actually just
substitute the values in.

00:04:12.055 --> 00:04:13.982
So we know we want to figure out

00:04:13.982 --> 00:04:16.104
what's going on when x is equal to one.

00:04:16.104 --> 00:04:20.137
So we know that the x's
here are equal to one.

00:04:20.137 --> 00:04:23.334
This x, x squared, well that's
just going to be one squared,

00:04:23.334 --> 00:04:25.273
so that's just going to be equal to one.

00:04:25.273 --> 00:04:27.321
We know that y is equal to four.

00:04:27.321 --> 00:04:31.488
So this is going to be 16 and
this is going to be eight.

00:04:33.895 --> 00:04:35.603
We know the derivative of x with

00:04:35.603 --> 00:04:37.217
respect to t is negative two.

00:04:37.217 --> 00:04:39.736
They tell us that in
the problem statement.

00:04:39.736 --> 00:04:41.000
Negative two.

00:04:41.000 --> 00:04:43.895
So now this is a good time
to simplify this thing.

00:04:43.895 --> 00:04:46.228
So this will simplify to ...

00:04:47.091 --> 00:04:48.940
Let's see, all of this is going

00:04:48.940 --> 00:04:50.976
to be two times one times negative two,

00:04:50.976 --> 00:04:53.809
so that is negative four times 16.

00:04:55.704 --> 00:04:59.371
So that is negative 64,
and then we have ...

00:05:01.963 --> 00:05:05.064
We'll use a color you can see.

00:05:05.064 --> 00:05:06.516
And then we have all of this.

00:05:06.516 --> 00:05:08.930
Well this is just going to be
one time eight times dy, dt.

00:05:08.930 --> 00:05:11.426
So this going to be eight dy, dt.

00:05:11.426 --> 00:05:14.340
So plus eight times the derivative of y

00:05:14.340 --> 00:05:17.084
with respect to t is equal to zero.

00:05:17.084 --> 00:05:20.203
Add 64 to both sides and we get ...

00:05:20.203 --> 00:05:21.741
I'll switch to a neutral color.

00:05:21.741 --> 00:05:23.523
Eight times the derivative of y

00:05:23.523 --> 00:05:26.672
with respect to t is equal to 64.

00:05:26.672 --> 00:05:30.291
Divide both sides by eight and you get

00:05:30.291 --> 00:05:33.298
the derivative of y with respect to t

00:05:33.298 --> 00:05:37.480
is equal to 64 divided by eight is just 8.

00:05:37.480 --> 00:05:39.726
If you want to look at the units

00:05:39.726 --> 00:05:42.456
it will also be in units per minute.

00:05:42.456 --> 00:05:45.206
Some units of distance per minute.

00:05:47.383 --> 00:05:48.716
And we are done.


Analyzing concavity (algebraic)

WEBVTT
Kind: captions
Language: en

00:00:00.599 --> 00:00:02.635
- [Voiceover] So I have
the function g here.

00:00:02.635 --> 00:00:05.226
It's expressed as a
fourth degree polynomial.

00:00:05.226 --> 00:00:07.957
And I wanna think about the
intervals over which g is

00:00:07.957 --> 00:00:11.098
either concave upwards
or concave downwards.

00:00:11.098 --> 00:00:13.940
Now let's just remind ourselves
what these things look like.

00:00:13.940 --> 00:00:14.857
So concave,

00:00:16.415 --> 00:00:17.665
concave upwards

00:00:19.703 --> 00:00:21.075
is an interval,

00:00:21.075 --> 00:00:23.011
an interval when you're concave upwards

00:00:23.011 --> 00:00:26.584
is an interval over which
the slope is increasing

00:00:26.584 --> 00:00:29.929
and it tends to look like an
upward opening U like that.

00:00:29.929 --> 00:00:34.829
And you can see here that the
slope over here is negative,

00:00:34.829 --> 00:00:38.368
and then as x increases,
it becomes less negative,

00:00:38.368 --> 00:00:40.652
it actually approaches
zero, it becomes zero,

00:00:40.652 --> 00:00:43.334
then it crosses zero and
becomes slightly positive,

00:00:43.334 --> 00:00:45.651
more positive, even more positive.

00:00:45.651 --> 00:00:48.432
So you can see the slope
is constantly increasing.

00:00:48.432 --> 00:00:50.714
And if you think about it
in terms of derivatives,

00:00:50.714 --> 00:00:53.981
it means that you're first
derivative is increasing

00:00:53.981 --> 00:00:55.192
over that interval.

00:00:55.192 --> 00:00:57.218
And in order for your first
derivative to be increasing

00:00:57.218 --> 00:00:59.828
over that interval, your second derivative

00:00:59.828 --> 00:01:02.979
f prime prime of x, actually
let me write it as g,

00:01:02.979 --> 00:01:04.693
because we're using g in this example.

00:01:04.693 --> 00:01:08.002
In order for your first
derivative to be increasing,

00:01:08.002 --> 00:01:09.510
your, let me write this.

00:01:09.510 --> 00:01:10.343
So g,

00:01:11.578 --> 00:01:13.988
so concave upward means

00:01:13.988 --> 00:01:17.274
that your first derivative increasing,

00:01:17.274 --> 00:01:19.019
increasing,

00:01:19.019 --> 00:01:22.850
which means, which means
that your second derivative

00:01:22.850 --> 00:01:25.409
is greater than zero.

00:01:25.409 --> 00:01:28.325
And concave downward is the opposite.

00:01:28.325 --> 00:01:29.742
Concave downward,

00:01:32.728 --> 00:01:33.921
downward,

00:01:33.921 --> 00:01:36.463
is an interval, or you're
gonna be concave downward

00:01:36.463 --> 00:01:40.380
over an interval when
your slope is decreasing.

00:01:41.428 --> 00:01:44.095
So g prime of x is decreasing

00:01:45.913 --> 00:01:49.251
or we can say that our second derivative,

00:01:49.251 --> 00:01:53.333
our second derivative is less than zero.

00:01:53.333 --> 00:01:55.481
And once again, I can draw it on this.

00:01:55.481 --> 00:01:58.902
So, when x is lower, we have a, look,

00:01:58.902 --> 00:02:00.749
or it looks like we have a positive slope,

00:02:00.749 --> 00:02:02.411
then it becomes less positive,

00:02:02.411 --> 00:02:03.549
and then it becomes less positive,

00:02:03.549 --> 00:02:05.020
as it's approaching zero, it becomes zero,

00:02:05.020 --> 00:02:07.723
then it becomes negative,
then even more negative,

00:02:07.723 --> 00:02:09.326
and then even more negative.

00:02:09.326 --> 00:02:11.767
So as you can see, our slope
is constantly decreasing

00:02:11.767 --> 00:02:14.209
as x increases here.

00:02:14.209 --> 00:02:15.787
So in order to think about the intervals

00:02:15.787 --> 00:02:20.008
where g is either concave
upward or concave downward,

00:02:20.008 --> 00:02:23.173
what we need to do is let's
find the second derivative of g,

00:02:23.173 --> 00:02:24.232
and then let's think about the points

00:02:24.232 --> 00:02:27.671
at which the second, where
the second derivative can go

00:02:27.671 --> 00:02:31.624
from being positive to negative
or negative to positive

00:02:31.624 --> 00:02:33.995
and those will be places
where it's either undefined

00:02:33.995 --> 00:02:36.313
or where the second
derivative is equal to zero,

00:02:36.313 --> 00:02:38.303
and let's see what's happening
in the interval between.

00:02:38.303 --> 00:02:39.816
And then we'll know over what intervals

00:02:39.816 --> 00:02:42.197
are we concave upward or concave downward.

00:02:42.197 --> 00:02:43.970
So let's do that.

00:02:43.970 --> 00:02:48.222
So let's take the first
derivative, g prime of x,

00:02:48.222 --> 00:02:49.951
just gonna apply the power rule a lot.

00:02:49.951 --> 00:02:51.290
Four times negative one

00:02:51.290 --> 00:02:55.948
is negative four x to the
third power plus, okay,

00:02:55.948 --> 00:02:57.480
so you're gonna have two times six

00:02:57.480 --> 00:03:02.083
is plus 12x to the first power,
you can just write it as x.

00:03:02.083 --> 00:03:04.266
And then minus two, I could say

00:03:04.266 --> 00:03:06.235
minus two x to the zeroth power,

00:03:06.235 --> 00:03:07.758
but that's just minus two.

00:03:07.758 --> 00:03:09.655
And then the derivative of
negative three, of a constant

00:03:09.655 --> 00:03:10.963
is just zero.

00:03:10.963 --> 00:03:15.046
And now I can take the second
derivative, g prime prime of x

00:03:15.046 --> 00:03:18.026
is going to be equal to
three times negative four

00:03:18.026 --> 00:03:20.026
is negative 12x squared,

00:03:21.695 --> 00:03:24.278
decrement the exponent plus 12.

00:03:25.583 --> 00:03:28.218
And so let's see, where
could this be undefined?

00:03:28.218 --> 00:03:32.132
Well, the second derivative
is just a quadratic expression

00:03:32.132 --> 00:03:34.177
here which would be defined for any x.

00:03:34.177 --> 00:03:36.104
So it's not going to
be undefined anywhere.

00:03:36.104 --> 00:03:39.234
So, interesting points
where we could transition

00:03:39.234 --> 00:03:42.833
from going from a negative to a positive

00:03:42.833 --> 00:03:45.387
or a positive to a
negative, second derivative

00:03:45.387 --> 00:03:47.579
is where this thing
could be equal to zero.

00:03:47.579 --> 00:03:48.621
So let's figure that out.

00:03:48.621 --> 00:03:51.691
So let's figure out where
negative 12x plus 12

00:03:51.691 --> 00:03:53.400
could be equal to zero.

00:03:53.400 --> 00:03:56.084
See, we could subtract 12 from both sides

00:03:56.084 --> 00:04:00.622
and we get negative 12x squared
is equal to negative 12.

00:04:00.622 --> 00:04:02.434
Divide both sides by negative 12,

00:04:02.434 --> 00:04:04.631
you get x squared is equal to one

00:04:04.631 --> 00:04:08.532
or x could be equal to the plus or minus,

00:04:08.532 --> 00:04:10.891
or x could be equal to the plus
or minus square root of one,

00:04:10.891 --> 00:04:12.710
which is, of course, just one.

00:04:12.710 --> 00:04:16.183
So, at the second derivative
at plus or minus one

00:04:16.183 --> 00:04:19.977
is equal to zero, so either
between plus or minus one

00:04:19.977 --> 00:04:22.414
or on either side of
them, we are going to be,

00:04:22.414 --> 00:04:25.463
we could be concave upward
or concave downward.

00:04:25.463 --> 00:04:27.488
So let's think about this.

00:04:27.488 --> 00:04:30.244
And to think about this, I'm
gonna make a number line.

00:04:30.244 --> 00:04:33.761
Let me find a nice, soothing color here.

00:04:33.761 --> 00:04:36.721
All right, that's a nice, soothing color.

00:04:36.721 --> 00:04:39.251
And, let's say, we should
make the number line

00:04:39.251 --> 00:04:40.914
a little bit bigger.

00:04:40.914 --> 00:04:44.914
So, there we go, let's
utilize the screen space.

00:04:45.921 --> 00:04:50.004
And so, if this is zero
and this is negative one,

00:04:50.982 --> 00:04:55.145
this is negative two,
this is positive one,

00:04:55.145 --> 00:04:56.777
this is positive two.

00:04:56.777 --> 00:05:00.782
We know that at x equals negative one

00:05:00.782 --> 00:05:04.553
and x equals one, our second
derivative is equal to zero.

00:05:04.553 --> 00:05:05.862
So let's think about what's happening

00:05:05.862 --> 00:05:08.975
in between those places to
see if our second derivative

00:05:08.975 --> 00:05:11.616
is positive or negative, and
from that we'll be able to say

00:05:11.616 --> 00:05:14.902
where it's concave upward
or concave downward.

00:05:14.902 --> 00:05:19.145
So, on this first
interval right over here,

00:05:19.145 --> 00:05:21.520
so this is the interval from,

00:05:21.520 --> 00:05:24.333
this is the interval where we're going

00:05:24.333 --> 00:05:28.125
from negative infinity to negative one.

00:05:28.125 --> 00:05:30.564
Well, let's just try a
value in that interval

00:05:30.564 --> 00:05:31.825
to see what whether our second derivative

00:05:31.825 --> 00:05:33.191
is positive or negative.

00:05:33.191 --> 00:05:35.727
And let's see, an easy value
there could be negative two,

00:05:35.727 --> 00:05:36.613
it's in that interval,

00:05:36.613 --> 00:05:40.021
so let's take g prime
prime of negative two,

00:05:40.021 --> 00:05:43.967
which is equal to negative 12 times four,

00:05:43.967 --> 00:05:46.049
because negative two
squared is positive four.

00:05:46.049 --> 00:05:48.382
So it's negative 48 plus 12,

00:05:50.210 --> 00:05:52.901
so it's equal to negative 36.

00:05:52.901 --> 00:05:54.751
The important thing to realize, then,

00:05:54.751 --> 00:05:55.988
is well, if over here it's negative,

00:05:55.988 --> 00:05:57.121
then over this whole interval,

00:05:57.121 --> 00:05:59.135
because it's not crossing through zero,

00:05:59.135 --> 00:06:01.262
or it's not discontinuous
at any of these points,

00:06:01.262 --> 00:06:03.000
that's why we picked this interval.

00:06:03.000 --> 00:06:04.204
That over this whole interval,

00:06:04.204 --> 00:06:07.537
g prime prime of x is less than zero,

00:06:08.518 --> 00:06:10.117
which means that over this interval

00:06:10.117 --> 00:06:12.384
we are concave downwards.

00:06:12.384 --> 00:06:14.801
So concave, concave downward,

00:06:17.228 --> 00:06:18.941
concave downward.

00:06:18.941 --> 00:06:23.108
Now let's go to the interval
between negative one and one.

00:06:25.263 --> 00:06:28.981
So this is the open interval
between negative one and one.

00:06:28.981 --> 00:06:30.575
And let's try a value there.

00:06:30.575 --> 00:06:33.089
Let's just try zero
will be easy to compute,

00:06:33.089 --> 00:06:36.517
g prime prime of zero, well,
when x is zero, this is zero,

00:06:36.517 --> 00:06:38.431
so it's just gonna be equal to 12.

00:06:38.431 --> 00:06:41.336
The important thing to realize
is our second derivative here

00:06:41.336 --> 00:06:45.253
is greater than zero, so
we are concave upward,

00:06:46.658 --> 00:06:49.241
concave upward on this interval

00:06:50.327 --> 00:06:51.850
between negative one and one.

00:06:51.850 --> 00:06:55.733
And then finally, let's
look at, let's look at

00:06:55.733 --> 00:06:59.953
the interval where x is greater than one.

00:06:59.953 --> 00:07:03.370
So this is the interval
from one to infinity,

00:07:03.370 --> 00:07:04.723
if we want to do it that way.

00:07:04.723 --> 00:07:06.040
And let's just try a value.

00:07:06.040 --> 00:07:09.164
Let's try g prime prime of two,

00:07:09.164 --> 00:07:10.457
'cause that's in the interval.

00:07:10.457 --> 00:07:12.386
And g prime prime of two
is gonna be the same thing

00:07:12.386 --> 00:07:13.887
as g prime prime of negative two,

00:07:13.887 --> 00:07:15.797
'cause whether you have a
negative two or a positive two,

00:07:15.797 --> 00:07:17.561
you square it becomes four.

00:07:17.561 --> 00:07:19.474
So you're gonna have
four times negative 12,

00:07:19.474 --> 00:07:22.655
which is negative 48 plus
12, which is negative 36.

00:07:22.655 --> 00:07:26.098
So this is negative 36, and
so, once again on this interval

00:07:26.098 --> 00:07:28.931
you are concave, concave downward.

00:07:31.358 --> 00:07:33.479
Now let's, I graphed this ahead of time,

00:07:33.479 --> 00:07:36.817
let's see if what we just
established is actually consistent

00:07:36.817 --> 00:07:40.020
with what the graph actually looks like.

00:07:40.020 --> 00:07:42.123
We were able to come up
with these, these insights

00:07:42.123 --> 00:07:44.718
about the concavity without graphing it.

00:07:44.718 --> 00:07:48.824
But now, it's kind of satisfying
to take a look at a graph

00:07:48.824 --> 00:07:51.184
and actually let me see if I
can match up the intervals.

00:07:51.184 --> 00:07:53.495
Actually this is pretty closely matched,

00:07:53.495 --> 00:07:55.601
all right over here.

00:07:55.601 --> 00:07:56.934
And so, this is,

00:07:58.030 --> 00:08:01.691
actually we can make it a
little bit smaller, all right.

00:08:01.691 --> 00:08:05.345
And so, let me move my boundy box.

00:08:05.345 --> 00:08:08.642
So, I'm saying that I'm concave downward

00:08:08.642 --> 00:08:12.309
between negative infinity,
negative infinity

00:08:14.746 --> 00:08:18.589
all the way until, all the
way until negative one,

00:08:18.589 --> 00:08:21.571
all the way until this
point right over here.

00:08:21.571 --> 00:08:23.027
So, all the way until that point.

00:08:23.027 --> 00:08:24.475
And that looks right.

00:08:24.475 --> 00:08:26.856
It looks like the slope
is constantly decreasing

00:08:26.856 --> 00:08:29.995
all the way until we get
to x equals negative one,

00:08:29.995 --> 00:08:32.507
and then the slope starts increasing.

00:08:32.507 --> 00:08:35.637
The slope starts increasing from there,

00:08:35.637 --> 00:08:37.642
from there all the way,

00:08:37.642 --> 00:08:39.569
and right at x we're transitioning,

00:08:39.569 --> 00:08:42.506
so I'm gonna leave a little,
I won't color in that.

00:08:42.506 --> 00:08:45.401
And so here our slope is increasing,

00:08:45.401 --> 00:08:46.930
let me do that same color.

00:08:46.930 --> 00:08:50.367
Our slope is increasing,
increasing, increasing,

00:08:50.367 --> 00:08:52.527
increasing, increasing all the way

00:08:52.527 --> 00:08:54.132
until we get to x equals one.

00:08:54.132 --> 00:08:56.306
And then our slope
starts decreasing again.

00:08:56.306 --> 00:08:59.200
And we get back into concave downwards.

00:08:59.200 --> 00:09:02.736
Oops, I wanna do that
in that orange color.

00:09:02.736 --> 00:09:05.653
We get back into concave downwards.

00:09:07.178 --> 00:09:08.237
So what we're able to figure out

00:09:08.237 --> 00:09:10.517
by just taking the derivatives
and doing a little algebra,

00:09:10.517 --> 00:09:14.350
we can see quite clearly
looking at the graph.


Second derivative test

WEBVTT
Kind: captions
Language: en

00:00:00.459 --> 00:00:01.605
- [Voiceover] So what I
want to do in this video

00:00:01.605 --> 00:00:05.064
is familiarize ourselves with
the second derivative test

00:00:05.064 --> 00:00:07.735
and before I even get into
the nitty-gritty of it,

00:00:07.735 --> 00:00:09.363
I really just want to
get an intuitive feel

00:00:09.363 --> 00:00:11.660
for what the second
derivative test is telling us.

00:00:11.660 --> 00:00:14.577
So let me just draw some axes here.

00:00:15.452 --> 00:00:20.258
So let's say that's my y-axis,
let's say this is my x-axis

00:00:20.258 --> 00:00:22.822
and let's say I have a function that has

00:00:22.822 --> 00:00:26.072
a relative maximum value at X equals C.

00:00:27.148 --> 00:00:29.257
So let's say we have a situation

00:00:29.257 --> 00:00:31.206
that looks something like that

00:00:31.206 --> 00:00:33.623
and X equals C is right over,

00:00:35.399 --> 00:00:38.336
so that's the point C, F of C.

00:00:38.336 --> 00:00:42.502
So if I can draw a straighter dotted line.

00:00:42.502 --> 00:00:44.919
So that is X being equal to C

00:00:46.667 --> 00:00:51.073
and we visually see that we
have a local maximum point there

00:00:51.073 --> 00:00:53.156
and we can use our calculus tools

00:00:53.156 --> 00:00:54.945
to think about what's going on there.

00:00:54.945 --> 00:00:56.761
Well one thing that we know,

00:00:56.761 --> 00:01:00.179
we know that the slope
of the tangent line,

00:01:00.179 --> 00:01:01.942
at least the way I've
drawn it right over here,

00:01:01.942 --> 00:01:03.624
is equal to zero.

00:01:03.624 --> 00:01:07.374
So we could say F prime
of C is equal to zero

00:01:08.884 --> 00:01:10.300
and the other thing we can see

00:01:10.300 --> 00:01:12.222
is that we are concave downward

00:01:12.222 --> 00:01:15.373
in the neighborhood around X equals C.

00:01:15.373 --> 00:01:18.471
So notice our slope is
constantly decreasing

00:01:18.471 --> 00:01:21.542
and since your slope,
notice it's positive,

00:01:21.542 --> 00:01:23.277
it's less positive, even less positive,

00:01:23.277 --> 00:01:25.227
it goes to zero, then it becomes negative,

00:01:25.227 --> 00:01:27.443
more negative and even more negative.

00:01:27.443 --> 00:01:29.943
So we know that F prime prime,

00:01:31.288 --> 00:01:33.871
we know that F prime prime of C

00:01:36.463 --> 00:01:38.551
is less than zero

00:01:38.551 --> 00:01:42.475
and so I haven't done any
deep mathematical proof here,

00:01:42.475 --> 00:01:46.225
but if I have a critical
point at X equals C,

00:01:48.587 --> 00:01:51.618
so F prime of C is equal to zero,

00:01:51.618 --> 00:01:54.288
and we also see that the
second derivative there

00:01:54.288 --> 00:01:55.650
is less than zero.

00:01:55.650 --> 00:01:56.958
Well intuitively this makes sense

00:01:56.958 --> 00:01:59.094
that we are at a maximum value

00:01:59.094 --> 00:02:00.697
and we could go the other way

00:02:00.697 --> 00:02:04.697
if we are at a local
minimum point at X equals C

00:02:07.212 --> 00:02:09.215
or relative minimum point.

00:02:09.215 --> 00:02:13.382
So our first derivative
should still be equal to zero

00:02:14.235 --> 00:02:16.585
'cause our slope of a
tangent line right over there

00:02:16.585 --> 00:02:17.840
is still zero.

00:02:17.840 --> 00:02:20.590
So F prime of C is equal to zero.

00:02:21.979 --> 00:02:25.290
But in this second situation,
we are concave upwards.

00:02:25.290 --> 00:02:28.922
The slope is constantly increasing.

00:02:28.922 --> 00:02:30.631
We have an upward opening bowl

00:02:30.631 --> 00:02:33.764
and so here we have a
relative minimum value

00:02:33.764 --> 00:02:38.344
or we could say our second
derivative is greater than zero.

00:02:38.344 --> 00:02:40.881
Visually we see it's a
relative minimum value

00:02:40.881 --> 00:02:42.616
and we can tell just
looking at our derivatives,

00:02:42.616 --> 00:02:43.734
at least the way I've drawn it,

00:02:43.734 --> 00:02:45.523
first derivative is equal to zero

00:02:45.523 --> 00:02:47.018
and we are concave upwards.

00:02:47.018 --> 00:02:49.048
Second derivative is greater than zero.

00:02:49.048 --> 00:02:51.451
And so this intuition that
we hopefully just built up

00:02:51.451 --> 00:02:54.388
is what the second
derivative test tells us.

00:02:54.388 --> 00:02:57.379
So it says hey look, if we're
dealing with some function F,

00:02:57.379 --> 00:03:00.183
let's say it's a twice
differentiable function.

00:03:00.183 --> 00:03:02.186
So that means that over some interval.

00:03:02.186 --> 00:03:04.478
So that means that you
could find its first

00:03:04.478 --> 00:03:06.214
and second derivatives are defined

00:03:06.214 --> 00:03:08.270
and so let's say there's
some point, X equals C,

00:03:08.270 --> 00:03:11.421
where its first derivative
is equal to zero,

00:03:11.421 --> 00:03:14.145
so the slope of the tangent
line is equal to zero,

00:03:14.145 --> 00:03:17.029
and the derivative exists
in a neighborhood around C

00:03:17.029 --> 00:03:18.898
and most of the functions we deal with,

00:03:18.898 --> 00:03:19.939
if it's differentiable at C,

00:03:19.939 --> 00:03:23.117
it tends to be differentiable
in the neighborhood around C

00:03:23.117 --> 00:03:25.173
and then we also assume that
second derivative exists

00:03:25.173 --> 00:03:26.695
is twice differentiable.

00:03:26.695 --> 00:03:30.300
Well then we might be
dealing with a maximum point,

00:03:30.300 --> 00:03:32.036
we might be dealing with a minimum point,

00:03:32.036 --> 00:03:34.279
or we might not know
what we're dealing with

00:03:34.279 --> 00:03:38.076
and it might be neither a
minimum or a maximum point.

00:03:38.076 --> 00:03:39.772
But using the second derivative test,

00:03:39.772 --> 00:03:42.068
if we take the second
derivative and if we see that

00:03:42.068 --> 00:03:45.807
the second derivative is
indeed less than zero,

00:03:45.807 --> 00:03:48.584
then we have a relative maximum point.

00:03:48.584 --> 00:03:51.251
Then so this is a situation that

00:03:52.598 --> 00:03:54.321
we started with right up there.

00:03:54.321 --> 00:03:57.926
If our second derivative
is greater than zero,

00:03:57.926 --> 00:03:59.768
then we are in this situation right here,

00:03:59.768 --> 00:04:01.290
we're concave upwards.

00:04:01.290 --> 00:04:03.667
Where the slope is zero,
that's the bottom of the bowl.

00:04:03.667 --> 00:04:06.097
We have a relative minimum point

00:04:06.097 --> 00:04:09.168
and if our second derivative
is zero, it's inconclusive.

00:04:09.168 --> 00:04:12.319
We don't know what is actually
going on at that point.

00:04:12.319 --> 00:04:14.508
We can't make any strong statement.

00:04:14.508 --> 00:04:16.351
So with that out of the way,
let's just do a quick example

00:04:16.351 --> 00:04:18.647
just to see if this has gelled.

00:04:18.647 --> 00:04:22.172
Let's say that I have some
twice differentiable function H

00:04:22.172 --> 00:04:23.481
and let's say that I tell you that

00:04:23.481 --> 00:04:25.814
H of eight is equal to five,

00:04:27.379 --> 00:04:31.546
I tell you that H prime
of eight is equal to zero,

00:04:32.586 --> 00:04:34.883
and I tell you that the second derivative

00:04:34.883 --> 00:04:38.888
at X equals eight is
equal to negative four.

00:04:38.888 --> 00:04:42.493
So given this, can you tell me whether

00:04:42.493 --> 00:04:46.660
the point eight comma five,
so the point eight comma five,

00:04:47.593 --> 00:04:49.676
is it a relative minimum,

00:04:52.953 --> 00:04:55.536
relative minimum, maximum point

00:04:57.653 --> 00:04:59.976
or not enough info?

00:04:59.976 --> 00:05:02.643
Not enough info or inconclusive?

00:05:04.996 --> 00:05:06.011
And like always, pause the video

00:05:06.011 --> 00:05:08.040
and see if you can figure it out.

00:05:08.040 --> 00:05:09.829
Well we're assuming it's
twice differentiable

00:05:09.829 --> 00:05:11.761
and I think it's safe to assume that

00:05:11.761 --> 00:05:13.297
and for the sake of our problem

00:05:13.297 --> 00:05:15.059
we're gonna assume that
the derivative exists

00:05:15.059 --> 00:05:17.489
in a neighborhood around X equals eight.

00:05:17.489 --> 00:05:19.438
So in this example, C is eight.

00:05:19.438 --> 00:05:23.604
So point eight five is
definitely on the curve.

00:05:23.604 --> 00:05:24.992
The derivative is equal to zero.

00:05:24.992 --> 00:05:27.930
So we're dealing potentially
with one of these scenarios

00:05:27.930 --> 00:05:31.508
and our second derivative
is less than zero.

00:05:31.508 --> 00:05:33.537
Second derivative is less than zero.

00:05:33.537 --> 00:05:35.273
So this threw us.

00:05:35.273 --> 00:05:37.570
So the fact that the second derivative,

00:05:37.570 --> 00:05:41.602
so H prime prime of
eight is less than zero,

00:05:41.602 --> 00:05:45.340
tells us that we fall into
this situation right over here.

00:05:45.340 --> 00:05:47.717
So just with the information
they've given us,

00:05:47.717 --> 00:05:50.440
we can say that at the
point eight comma five

00:05:50.440 --> 00:05:53.107
we have a relative maximum value

00:05:54.019 --> 00:05:57.730
or that this is a relative
maximum point for this.

00:05:57.730 --> 00:06:00.774
If somehow they told us the
second derivative was zero,

00:06:00.774 --> 00:06:02.296
then we would say it's inconclusive.

00:06:02.296 --> 00:06:03.819
If they told us and
that's all they told us

00:06:03.819 --> 00:06:04.833
and if they told us the second derivative

00:06:04.833 --> 00:06:07.610
is greater than zero,
then we would be dealing

00:06:07.610 --> 00:06:11.610
with a relative minimum
value at X equals eight.