L'Hopital's Rule to solve for variable

WEBVTT
Kind: captions
Language: en

00:00:00.897 --> 00:00:03.213
- We have an interesting
problem or exercise here.

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Find a such that the
limit as x approaches zero

00:00:07.235 --> 00:00:09.639
of the square route of four plus x

00:00:09.639 --> 00:00:11.093
minus the square route of four

00:00:11.093 --> 00:00:13.792
minus a times x,

00:00:13.792 --> 00:00:17.605
all of that over x, is equal to 3/4.

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And like always, I encourage you

00:00:19.731 --> 00:00:22.610
to pause the video and give a go at it.

00:00:22.610 --> 00:00:24.470
So assuming you have had your go,

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now let's do this together.

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So when you just try to superficially

00:00:28.272 --> 00:00:31.192
evaluate this limit here,
if x approaches zero,

00:00:31.192 --> 00:00:33.102
so if you're just trying to evaluate

00:00:33.102 --> 00:00:35.535
this one x equals zero,
you're going to get...

00:00:35.535 --> 00:00:37.484
let me just try to evaluate the limit.

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As x approaches zero

00:00:39.848 --> 00:00:42.817
of the square route of four plus x

00:00:42.817 --> 00:00:46.667
minus the square route of four minus ax,

00:00:46.667 --> 00:00:49.114
all of that over x.

00:00:49.114 --> 00:00:50.723
Well, this right over here is going to be

00:00:50.723 --> 00:00:52.180
just the principal root of four,

00:00:52.180 --> 00:00:53.602
because four plus zero is four.

00:00:53.602 --> 00:00:54.707
This right over here is just going to be

00:00:54.707 --> 00:00:56.213
the principal root of four,

00:00:56.213 --> 00:00:57.784
because no matter what a is,

00:00:57.784 --> 00:00:59.292
a times zero is going to be zero,

00:00:59.292 --> 00:01:01.280
so you're going to be
left with four minus zero,

00:01:01.280 --> 00:01:03.048
so it's just the principal route of four.

00:01:03.048 --> 00:01:04.313
So you're going to have two.

00:01:04.313 --> 00:01:05.698
This whole thing is going to be two.

00:01:05.698 --> 00:01:07.699
If you just were to substitute x there,

00:01:07.699 --> 00:01:09.714
so this whole thing is two.

00:01:10.262 --> 00:01:12.245
This whole thing right over here

00:01:12.245 --> 00:01:13.574
is going to be two as well.

00:01:13.574 --> 00:01:15.025
You're going to have two minus two,

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and then as x approaches zero,

00:01:16.125 --> 00:01:17.563
this is going to be zero.

00:01:17.563 --> 00:01:19.857
So this looks like we are going...

00:01:20.041 --> 00:01:22.867
we are getting an indeterminate form.

00:01:22.867 --> 00:01:24.973
And when you get to something like this,

00:01:24.973 --> 00:01:28.886
you start to say, "Well,
L'Hopital's rule might apply."

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If I get zero over zero,
or infinity over infinity,

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well, this limit is going
to be the same thing

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as the limit as x approaches zero.

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This is going to be the same thing

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as the limit as x approaches zero

00:01:41.283 --> 00:01:43.126
of the derivative of the numerator

00:01:43.126 --> 00:01:46.157
over the derivative of the denominator.

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So what is the derivative
of the numerator?

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Actually, let me just do the derivative

00:01:49.907 --> 00:01:51.854
of the denominator first,

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because the derivative
of x, with respect...

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Oh, I may have to do that
in a different color.

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The derivative of x with respect to x

00:01:58.262 --> 00:01:59.903
is just going to be one.

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But now let me take the derivative

00:02:00.982 --> 00:02:03.712
of this business up here.

00:02:04.058 --> 00:02:05.692
The derivative...

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The derivative of this with respect to x.

00:02:09.456 --> 00:02:12.759
So this is four plus x to the 1/2 power.

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So this is, the derivative of this part

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is going to be 1/2 times four

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plus x to the negative 1/2 power.

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And so the derivative of
this part right over here...

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Let's see, here the...

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The chain rule applied
here with a derivative

00:02:26.871 --> 00:02:28.923
of four plus x is just one, so we just

00:02:28.923 --> 00:02:30.843
multiply this thing by one.

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But here the chain rule, the derivative of

00:02:32.453 --> 00:02:36.087
four minus ax, with respect
to x, is negative a.

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Now we multiply that,
and we're going to have

00:02:38.089 --> 00:02:39.816
this negative out front,
so this is going to be

00:02:39.816 --> 00:02:41.639
plus a.

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Plus a times...

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times 1/2

00:02:49.075 --> 00:02:52.094
times four minus ax

00:02:52.094 --> 00:02:54.371
to the negative 1/2 power.

00:02:54.371 --> 00:02:57.143
I just used the power
rule and the chain rule

00:02:57.143 --> 00:02:58.938
to take the derivative here.

00:02:58.938 --> 00:03:01.176
And so what is is this going to be?

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Well, this is going to be equal to...

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This is going to be equal
to something over one.

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So we have up here, as x approaches zero,

00:03:09.242 --> 00:03:11.959
this is going to be,
this part, four plus zero

00:03:11.959 --> 00:03:14.660
is just four to the negative 1/2 power.

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Well, that's just going to be 1/2.

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Four to the 1/2 is two,

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four to the negative 1/2 is 1/2.

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And then as x approaches zero here,

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this is going to be four
to the negative 1/2,

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which is once again 1/2.

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So what does this simplify to?

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We have 1/2 times 1/2, which is 1/4.

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That's that there.

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And then over here I have
a times 1/2 times 1/2,

00:03:38.599 --> 00:03:42.720
so that's going to be plus a over four,

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and so this is the same thing as just

00:03:44.123 --> 00:03:47.579
a plus one over four.

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And we say that this
needs to be equal to 3/4.

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This needs to be equal to 3/4.

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That was our original problem.

00:03:54.800 --> 00:03:57.001
So that needs to be equal to 3/4,

00:03:57.001 --> 00:03:58.253
and now it's pretty straightforward

00:03:58.253 --> 00:03:59.619
to figure out what a needs to be.

00:03:59.619 --> 00:04:02.542
A plus one needs to be equal to three,

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or a is equal to 2.

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And we are done.


Tricky L'Hopital's Rule problem

WEBVTT
Kind: captions
Language: en

00:00:00.382 --> 00:00:02.167
- [Voiceover] What I would
like to tackle in this video

00:00:02.167 --> 00:00:04.133
is what I consider to be a particularly

00:00:04.133 --> 00:00:05.800
interesting limits problem.

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Let's say we want to figure out the limit

00:00:07.796 --> 00:00:12.033
as X approaches zero from
the positive direction

00:00:12.033 --> 00:00:14.900
of sine of X.

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This is where it's about
to get interesting.

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Sine of X to the one over

00:00:20.200 --> 00:00:23.166
the natural log of X power

00:00:23.166 --> 00:00:25.766
and I encourage you to pause this video

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and see if you can have a go at it

00:00:27.700 --> 00:00:28.500
fully knowing that this

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is a little bit of a tricky exercise.

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I'm assuming you have attempted.

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Some of you might have
been able to figure out

00:00:34.166 --> 00:00:35.100
on the first pass.

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I will tell you that the first time

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that I encountered something like this,

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I did not figure it out at the first pass

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so definitely do not feel bad

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if you fall into that second category.

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What many of you all probably did

00:00:45.733 --> 00:00:47.166
is you said okay, let me think about it.

00:00:47.166 --> 00:00:48.833
Let me just think about
the components here.

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If I were to think about the limit,

00:00:52.302 --> 00:00:55.400
if I were to think about the
limit as X approaches zero

00:00:55.400 --> 00:00:58.466
from the positive direction of sine of X,

00:00:58.466 --> 00:00:59.733
well that's pretty straightforward.

00:00:59.733 --> 00:01:01.000
That's going to be zero,

00:01:01.000 --> 00:01:03.033
so you could think of like this part of it

00:01:03.033 --> 00:01:05.633
is going to approach
zero but then if you say,

00:01:05.633 --> 00:01:07.566
and you could say, I guess I should say.

00:01:07.566 --> 00:01:09.767
The limit as X approaches zero

00:01:09.767 --> 00:01:12.200
from the positive direction of one over

00:01:12.200 --> 00:01:13.733
natural log of X

00:01:13.733 --> 00:01:15.366
and this is why we have to think about it

00:01:15.366 --> 00:01:16.400
from the positive direction.

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It doesn't make sense to approach it

00:01:17.723 --> 00:01:18.700
from the negative direction.

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You can't take the natural log

00:01:19.733 --> 00:01:21.166
of a negative number.

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That's not in the domain
for the natural log

00:01:23.466 --> 00:01:25.366
but as you get closer and closer to zero

00:01:25.366 --> 00:01:26.866
from the negative direction,

00:01:26.866 --> 00:01:29.000
the natural log of those values,

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you have to raise E

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to more larger and larger negative values.

00:01:32.733 --> 00:01:34.325
This part over here is going to approach

00:01:34.325 --> 00:01:35.799
negative infinity.

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It's going to go to negative infinity.

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One over negative infinity,

00:01:38.566 --> 00:01:40.600
one divided by super large

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or large magnitude negative numbers,

00:01:43.166 --> 00:01:46.033
well, that's just going to approach zero.

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You could say that this right over here

00:01:47.833 --> 00:01:50.566
is also going to be,

00:01:50.566 --> 00:01:52.900
is also going to be equal to zero.

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That doesn't seem to help us much

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because if this thing is going to zero

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and that thing is going to zero,

00:01:59.600 --> 00:02:01.566
it's kind of an implication that well

00:02:01.566 --> 00:02:03.433
maybe this whole thing is going to zero

00:02:03.433 --> 00:02:07.200
to the zero power but we
don't really know what zero,

00:02:07.200 --> 00:02:08.938
let me do the some, those color.

00:02:08.938 --> 00:02:11.033
Zero to the

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zero power but this is one of those great

00:02:14.933 --> 00:02:16.800
fun things to think about in mathematics.

00:02:16.800 --> 00:02:18.800
There's justifications
why this could be zero,

00:02:18.800 --> 00:02:20.633
justifications why this could be one.

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We don't really know what to make of this.

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This isn't really a satisfying answer.

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Something at this point

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might be going into your brain.

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We have this thing that we've been exposed

00:02:31.566 --> 00:02:32.833
to called L'Hopital's rule.

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If you have not been exposed to it,

00:02:33.866 --> 00:02:35.100
I encourage you to watch the video,

00:02:35.100 --> 00:02:37.133
the introductory video
on L'Hopital's rule.

00:02:37.133 --> 00:02:40.368
In L'Hopital's rule, let
me just write it down.

00:02:42.921 --> 00:02:45.133
L'Hopital's rule

00:02:45.133 --> 00:02:47.966
helps us out with situations where

00:02:47.966 --> 00:02:50.500
when we try to superficially
evaluate the limit,

00:02:50.500 --> 00:02:52.033
we get indeterminate forms

00:02:52.033 --> 00:02:53.766
things like zero over zero.

00:02:53.766 --> 00:02:57.133
We get infinity over infinity.

00:02:57.133 --> 00:02:59.733
We get negative infinity
over negative infinity

00:02:59.733 --> 00:03:01.500
and we go into much more
detail into that video.

00:03:01.500 --> 00:03:03.800
It seems this is kind of,

00:03:03.800 --> 00:03:07.366
it feels like we're
getting a zero to a zero.

00:03:07.366 --> 00:03:09.033
It's kind of we're
getting the strange beast

00:03:09.033 --> 00:03:12.333
and at least evokes the
notion of L'Hopital's rule.

00:03:12.333 --> 00:03:15.300
You will not be, as we'll
see in a few seconds,

00:03:15.300 --> 00:03:17.200
you're not wrong to, for that

00:03:17.200 --> 00:03:20.866
L'Hopital's rule neuron to
be triggering in your brain

00:03:20.866 --> 00:03:22.633
although you can't apply it directly

00:03:22.633 --> 00:03:23.600
to this right over here.

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L'Hopital's rule does not apply to

00:03:25.400 --> 00:03:28.500
or directly apply to the
zero to the zero form

00:03:28.500 --> 00:03:31.700
but what we can do is construct a problem

00:03:31.700 --> 00:03:33.466
where L'Hopital's rule will apply

00:03:33.466 --> 00:03:36.300
and then use that to solve, to figure out

00:03:36.300 --> 00:03:38.066
what this is going to be.

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This was essentially the
tricky part of this exercise.

00:03:41.500 --> 00:03:42.433
Well what do I mean?

00:03:42.433 --> 00:03:44.433
Well if we set Y

00:03:44.433 --> 00:03:47.525
equal to Y of

00:03:47.525 --> 00:03:49.033
and maybe let me write it this way.

00:03:49.033 --> 00:03:51.000
If we set, and I'll write Y,

00:03:51.000 --> 00:03:52.392
I could just write Y but I'll say Y

00:03:52.392 --> 00:03:53.566
is clearly a function of X.

00:03:53.566 --> 00:03:55.966
If we say Y of X is going to be sine of X,

00:03:56.842 --> 00:03:59.800
sine of X to the one over

00:03:59.800 --> 00:04:01.733
natural log of X.

00:04:01.733 --> 00:04:03.484
This thing right over
here is essentially saying

00:04:03.484 --> 00:04:05.200
what's the limit as X approaches zero

00:04:05.200 --> 00:04:06.818
from the positive direction of Y

00:04:06.818 --> 00:04:07.766
and once again we don't know.

00:04:07.766 --> 00:04:09.600
Maybe it's zero to the zero

00:04:09.600 --> 00:04:12.166
but we don't know what
zero to zero actually is.

00:04:12.166 --> 00:04:15.066
What we could do, what we could do

00:04:15.066 --> 00:04:16.333
and this is a trick that you see a lot

00:04:16.333 --> 00:04:18.500
and anytime you get kind of
weird things with exponents

00:04:18.500 --> 00:04:20.766
and whether you're doing
limits or derivatives,

00:04:20.766 --> 00:04:23.666
as you'll see it's often times useful

00:04:23.666 --> 00:04:26.133
to take the natural log of both sides.

00:04:26.133 --> 00:04:27.866
Well what happens if
you take the natural log

00:04:27.866 --> 00:04:29.000
of both sides here?

00:04:29.000 --> 00:04:30.800
On the left-hand side

00:04:30.800 --> 00:04:32.461
you're going to have the natural log,

00:04:32.461 --> 00:04:34.767
the natural log, and whenever
I think of natural log

00:04:34.767 --> 00:04:37.627
and E the way I always think about them,

00:04:37.627 --> 00:04:39.666
the color green for some bizarre reason

00:04:39.666 --> 00:04:41.766
but we'll say the natural log of Y

00:04:41.766 --> 00:04:43.733
is equal to.

00:04:43.733 --> 00:04:45.600
If you take the natural log of this thing,

00:04:45.600 --> 00:04:47.266
actually let me just, I
don't want to skip steps here

00:04:47.266 --> 00:04:48.566
because this is interesting.

00:04:48.566 --> 00:04:49.833
This is going to be the natural log

00:04:49.833 --> 00:04:52.700
of all of this business of sine of X,

00:04:52.700 --> 00:04:54.100
let me write this way.

00:04:54.100 --> 00:04:55.666
Sine of X,

00:04:57.050 --> 00:04:59.812
sine, I want to do this
in that orange color.

00:04:59.812 --> 00:05:02.266
The natural log of

00:05:02.266 --> 00:05:05.700
sine of X to the one

00:05:05.700 --> 00:05:07.600
over the natural log of X.

00:05:07.600 --> 00:05:09.866
Well we know from our exponent prior

00:05:09.866 --> 00:05:11.066
our logarithm properties,

00:05:11.066 --> 00:05:13.300
the logarithm of something to a power,

00:05:13.300 --> 00:05:16.033
that's the same thing as the power.

00:05:16.033 --> 00:05:17.933
One over natural log of X

00:05:17.933 --> 00:05:22.366
times the logarithm, this
case the natural logarithm

00:05:22.366 --> 00:05:25.307
of whatever taking the sine of X here.

00:05:27.430 --> 00:05:29.386
Sine of X

00:05:29.386 --> 00:05:33.466
or we could say the natural log of Y.

00:05:33.466 --> 00:05:35.409
Want to keep, stay color consistent

00:05:35.409 --> 00:05:36.766
for at least one more step.

00:05:36.766 --> 00:05:38.966
The natural log of Y is equal to,

00:05:38.966 --> 00:05:40.333
if we just rewrite this

00:05:40.333 --> 00:05:42.900
this is going to be the natural log

00:05:42.900 --> 00:05:45.966
of sine of X,

00:05:46.489 --> 00:05:48.100
the natural log

00:05:48.100 --> 00:05:52.266
of sine of X

00:05:52.266 --> 00:05:56.600
over the natural log of X.

00:05:56.600 --> 00:06:00.066
Well this is all interesting

00:06:00.066 --> 00:06:02.033
but why do we care about this?

00:06:02.033 --> 00:06:03.081
Why did I do this?

00:06:03.081 --> 00:06:05.100
Well instead of thinking
about what is the limit?

00:06:05.100 --> 00:06:07.533
What is the limit as X approaches zero

00:06:07.533 --> 00:06:09.266
from the positive direction of Y?

00:06:09.266 --> 00:06:12.166
Let's think about what
the natural log of Y

00:06:12.166 --> 00:06:14.200
is approaching as we approach,

00:06:14.200 --> 00:06:17.233
as X approaches zero from
the positive direction.

00:06:17.233 --> 00:06:19.333
Let's figure out what the
limit of this expression

00:06:19.333 --> 00:06:22.166
right over here is as X approaches zero

00:06:22.166 --> 00:06:23.256
from the positive direction.

00:06:23.256 --> 00:06:25.073
What is a natural log of Y?

00:06:25.073 --> 00:06:26.600
What is this whole thing?

00:06:26.600 --> 00:06:28.800
Not Y, what is the natural
log of Y approaching?

00:06:28.800 --> 00:06:30.700
Let's think about that scenario.

00:06:30.700 --> 00:06:32.866
Let me write, do this in a new color.

00:06:32.866 --> 00:06:34.633
We want to figure out what is the limit

00:06:34.633 --> 00:06:38.100
as X approaches zero from
the positive direction

00:06:38.100 --> 00:06:40.566
of this business and I'll
just write it in one color.

00:06:40.566 --> 00:06:41.666
The natural log

00:06:41.666 --> 00:06:44.800
of sine of X

00:06:44.800 --> 00:06:47.966
over the natural log of X.

00:06:47.966 --> 00:06:50.633
I don't know, I wrote one time in print,

00:06:50.633 --> 00:06:51.700
one time in cursive.

00:06:51.700 --> 00:06:53.166
I'll just be consistent right over there.

00:06:53.166 --> 00:06:54.933
Now why is this interesting?

00:06:54.933 --> 00:06:56.933
Let's see in the numerator here,

00:06:56.933 --> 00:06:58.689
this thing's going to approach zero,

00:06:58.689 --> 00:07:00.081
natural log of zero
you're going to approach

00:07:00.081 --> 00:07:00.980
negative infinity.

00:07:00.980 --> 00:07:03.033
This thing right over here natural log of,

00:07:03.033 --> 00:07:04.800
as you approach it from
the positive direction.

00:07:04.800 --> 00:07:07.233
Once again you're going to
approach negative infinity.

00:07:07.233 --> 00:07:09.633
This gives you that indeterminate form.

00:07:09.633 --> 00:07:12.233
This is giving you

00:07:12.233 --> 00:07:14.833
that indeterminate form
of negative infinity

00:07:14.833 --> 00:07:16.833
over negative infinity which is neat

00:07:16.833 --> 00:07:20.033
because this triggers or at least tells us

00:07:20.033 --> 00:07:22.966
that L'Hopital's rule
may be appropriate here.

00:07:22.966 --> 00:07:25.966
We could say that this
is going to be equal to

00:07:25.966 --> 00:07:30.200
the limit as X approaches zero

00:07:30.200 --> 00:07:31.400
from the positive direction.

00:07:31.400 --> 00:07:32.867
Let me give myself a little bit more

00:07:32.867 --> 00:07:34.666
real estate to work up with

00:07:34.666 --> 00:07:36.966
and I can take the
derivative of the numerator

00:07:36.966 --> 00:07:38.800
and the derivative of the denominator.

00:07:38.800 --> 00:07:40.866
Derivative of the numerator,

00:07:40.866 --> 00:07:43.366
so the derivative of the numerator,

00:07:43.366 --> 00:07:44.633
I'm going to apply the chain rule here.

00:07:44.633 --> 00:07:47.133
Derivative of sine of X is cosine of X

00:07:47.133 --> 00:07:50.300
and then the derivative,

00:07:50.300 --> 00:07:51.867
the natural log of sine of X with respect

00:07:51.867 --> 00:07:53.866
to sine of X is going to
be one over sine of X.

00:07:53.866 --> 00:07:56.800
This is going to be over sine of X,

00:07:56.800 --> 00:07:58.133
so that's the derivative of the numerator.

00:07:58.133 --> 00:08:01.333
Then the derivative of the denominator

00:08:02.348 --> 00:08:06.033
is just going to be one over X,

00:08:06.879 --> 00:08:08.033
one over X.

00:08:08.033 --> 00:08:10.000
This is all going to be equal to,

00:08:10.738 --> 00:08:12.533
this is equal to the limit

00:08:12.533 --> 00:08:16.600
as X approaches zero from
the positive direction

00:08:16.600 --> 00:08:19.900
of I could write this as cosine of X,

00:08:21.223 --> 00:08:24.584
cosine of X.

00:08:24.584 --> 00:08:25.600
Let's see,

00:08:25.600 --> 00:08:29.033
if I'm dividing by X.

00:08:30.756 --> 00:08:33.700
I'm dividing by X, I am going to get,

00:08:33.700 --> 00:08:37.766
this is going to be X over sine of X,

00:08:38.504 --> 00:08:41.067
X over sine of X.

00:08:41.067 --> 00:08:42.800
When I apply

00:08:42.800 --> 00:08:44.533
and when I try to take the limit here

00:08:44.533 --> 00:08:45.540
I'm going to get a zero,

00:08:45.540 --> 00:08:48.053
once again we got a zero over zero.

00:08:48.053 --> 00:08:50.400
This doesn't feel too satisfying

00:08:50.400 --> 00:08:52.966
but once again this is
where our limit properties

00:08:52.966 --> 00:08:53.733
might be useful.

00:08:53.733 --> 00:08:55.667
As you can tell, this is not the most

00:08:55.667 --> 00:08:56.753
trivial of problems

00:08:56.753 --> 00:08:58.466
but this is going to be the same thing

00:08:58.466 --> 00:09:01.500
and this will take a little
bit of pattern recognition.

00:09:01.500 --> 00:09:03.466
This is the same thing as

00:09:03.466 --> 00:09:05.700
because we know that
the limit of the product

00:09:05.700 --> 00:09:06.500
of two functions

00:09:06.500 --> 00:09:08.036
is equal to the product of their limits,

00:09:08.036 --> 00:09:11.900
this is the same thing as the limit,

00:09:12.576 --> 00:09:15.500
the limit as X approaches zero

00:09:15.500 --> 00:09:17.366
from the positive direction of,

00:09:17.366 --> 00:09:18.833
if we take this part,

00:09:18.833 --> 00:09:20.400
let me do this in a different color.

00:09:20.400 --> 00:09:22.800
If we take this part,

00:09:22.800 --> 00:09:24.233
that's not a different color.

00:09:24.233 --> 00:09:28.033
If we take this part right over here,

00:09:28.033 --> 00:09:33.033
so that's going to be X over sine of X

00:09:33.166 --> 00:09:35.967
and then times the limit.

00:09:35.967 --> 00:09:38.166
Let me put parenthesis here

00:09:38.166 --> 00:09:41.266
times the limit,

00:09:42.004 --> 00:09:43.458
the limit

00:09:44.181 --> 00:09:48.933
as X approaches zero from
the positive direction

00:09:48.933 --> 00:09:50.766
of cosine of X,

00:09:52.166 --> 00:09:55.266
of cosine of X.

00:09:55.266 --> 00:09:57.166
Now this thing right over here

00:09:57.166 --> 00:09:58.240
is pretty straightforward.

00:09:58.240 --> 00:09:59.833
You can just evaluate it at zero,

00:09:59.833 --> 00:10:00.835
you get one.

00:10:00.835 --> 00:10:02.700
This thing right over here is equal to one

00:10:02.700 --> 00:10:04.266
but what's this thing?

00:10:04.266 --> 00:10:05.733
This might ring a bell.

00:10:05.733 --> 00:10:07.500
You might have seen the
limit as X approaches zero,

00:10:07.500 --> 00:10:09.194
of sine of X over X.

00:10:09.194 --> 00:10:11.233
This is just the reciprocal of that.

00:10:11.233 --> 00:10:12.766
This is X over sine of X

00:10:12.766 --> 00:10:14.969
but when you just superficially
try to evaluate it.

00:10:14.969 --> 00:10:17.000
You get zero over zero
so you can then apply

00:10:17.000 --> 00:10:18.833
L'Hopital's rule to this thing.

00:10:18.833 --> 00:10:22.533
Once again this is quite
an interesting scenario

00:10:22.533 --> 00:10:23.833
we find ourselves in.

00:10:23.833 --> 00:10:27.300
This is going to be the
same thing as the limit

00:10:27.300 --> 00:10:30.900
as X approaches zero from
the positive direction.

00:10:30.900 --> 00:10:32.571
Derivative at the top is one,

00:10:32.571 --> 00:10:34.967
derivative at the bottom, is cosine of X.

00:10:34.967 --> 00:10:37.667
Well, this is just going to be one over

00:10:37.667 --> 00:10:39.333
cosine of zero is one,

00:10:39.333 --> 00:10:40.833
so this is just going to be equal to one.

00:10:40.833 --> 00:10:42.666
We got to apply L'Hopital's rule again

00:10:42.666 --> 00:10:46.100
and realize that this limit
is going to be equal to one.

00:10:46.100 --> 00:10:48.233
One times one is one

00:10:48.233 --> 00:10:51.533
so this thing right over
here is equal to one.

00:10:51.533 --> 00:10:53.666
This thing right over here,

00:10:53.666 --> 00:10:57.566
this thing right over here is going to be,

00:10:57.566 --> 00:11:01.033
this thing right over here
is going to approach one

00:11:01.033 --> 00:11:04.366
which tells you that this
thing is approaching one.

00:11:04.366 --> 00:11:05.966
What do we now know?

00:11:05.966 --> 00:11:09.100
We now know and I'll
write it out in language.

00:11:09.100 --> 00:11:12.904
We now know that the limit

00:11:12.904 --> 00:11:16.166
of the natural log of Y,

00:11:16.889 --> 00:11:19.666
the limit of the natural log of Y

00:11:19.666 --> 00:11:23.766
as X approaches zero from
the positive direction

00:11:23.766 --> 00:11:26.666
is equal to one.

00:11:27.834 --> 00:11:30.866
If the natural log of
Y is approaching one,

00:11:30.866 --> 00:11:32.534
what is Y approaching?

00:11:32.534 --> 00:11:35.333
Well in order for the
natural log, so once again

00:11:35.333 --> 00:11:36.676
we just know this thing right over here.

00:11:36.676 --> 00:11:38.133
We know this thing is one

00:11:38.133 --> 00:11:40.100
and this thing is a natural log of Y.

00:11:41.946 --> 00:11:44.033
We now know that this thing,

00:11:44.033 --> 00:11:45.766
the limit as X approaches
zero of this thing

00:11:45.766 --> 00:11:48.433
is one and that's the
same thing as a limit

00:11:48.433 --> 00:11:50.376
as X approaches zero from
the positive direction

00:11:50.376 --> 00:11:52.933
of the natural log of Y.

00:11:52.933 --> 00:11:55.133
These things are
equivalent, is equal to one.

00:11:55.133 --> 00:11:58.200
Well the natural log
of Y is approaching one

00:11:58.200 --> 00:12:02.600
so if the natural log of Y

00:12:02.600 --> 00:12:05.600
is approaching, let me write it this way.

00:12:05.600 --> 00:12:08.433
The natural log of Y is approaching one.

00:12:08.433 --> 00:12:10.500
Well what must Y be approaching?

00:12:10.500 --> 00:12:13.871
Well to get the natural log of something

00:12:13.871 --> 00:12:16.183
that gets you one, well
Y must be approaching E

00:12:16.183 --> 00:12:17.966
because natural log of E is one.

00:12:17.966 --> 00:12:22.866
Then Y must be approaching E

00:12:22.866 --> 00:12:26.233
and we are done because
that's what we cared about.

00:12:26.233 --> 00:12:30.000
We cared about what is
Y, this is Y, remember,

00:12:30.000 --> 00:12:31.233
we defined this whole thing as Y.

00:12:31.233 --> 00:12:33.033
We said what is Y approaching

00:12:33.033 --> 00:12:36.500
as X approaches zero from
the positive direction?

00:12:36.500 --> 00:12:38.666
Well we figured out that
the natural log of Y

00:12:38.666 --> 00:12:41.567
is approaching one as X approaches zero

00:12:41.567 --> 00:12:42.667
from the positive direction.

00:12:42.667 --> 00:12:46.366
That means that Y must be approaching E.

00:12:47.612 --> 00:12:49.633
This tells us that this thing,

00:12:49.633 --> 00:12:51.566
this thing right over here

00:12:51.566 --> 00:12:54.733
is equal to E

00:12:54.733 --> 00:12:56.566
which is somewhat mind blowing.

00:12:56.566 --> 00:12:58.433
We seen other, the E is popping up

00:12:58.433 --> 00:13:00.600
and it's involving sine of X

00:13:00.600 --> 00:13:01.934
and of course natural log of X,

00:13:01.934 --> 00:13:03.633
you expect E to be involved somehow.

00:13:03.633 --> 00:13:08.166
It is a pretty fascinating
problem in my mind.


How to test solutions to equations using substitution

WEBVTT
Kind: captions
Language: en

00:00:00.633 --> 00:00:02.266
- [Voiceover] So we've
got an equation here,

00:00:02.266 --> 00:00:05.433
it says five times x minus three

00:00:05.433 --> 00:00:09.321
is equal to four times x plus three.

00:00:09.321 --> 00:00:10.867
So, what we want to do
is we want to figure out

00:00:10.867 --> 00:00:13.234
an x that satisfies this,
so there's some number

00:00:13.234 --> 00:00:15.233
that if I take five,
multiply it by that number,

00:00:15.233 --> 00:00:17.533
subtract three from it, that's going to be

00:00:17.533 --> 00:00:20.166
the same thing as if I
take four times that number

00:00:20.166 --> 00:00:22.101
and add three to it.

00:00:22.101 --> 00:00:26.402
And, before we go into how to
solve these types of things,

00:00:26.402 --> 00:00:28.433
let's just first see
if we can test whether

00:00:28.433 --> 00:00:32.066
something does satisfy this equation.

00:00:32.066 --> 00:00:34.900
And so, I have three options
here, I have x could be

00:00:34.900 --> 00:00:36.933
equal to five, x could be equal to six,

00:00:36.933 --> 00:00:38.966
and x could be equal to seven.

00:00:38.966 --> 00:00:42.700
And your goal is to pause
this video and figure out

00:00:42.700 --> 00:00:46.400
which of these x's
satisfies this equation,

00:00:46.400 --> 00:00:51.366
which of these values would
make this equation be true.

00:00:51.627 --> 00:00:55.700
So, I'm assuming that you have tried that,

00:00:55.700 --> 00:00:59.033
so let's work through each
of them, step one by one.

00:00:59.033 --> 00:01:01.333
So let's see this first
one: if x is equal to five,

00:01:01.333 --> 00:01:05.933
then in order for this to
be true, five time five,

00:01:05.933 --> 00:01:10.666
right, five times x, so
five times five minus three

00:01:10.666 --> 00:01:13.533
needs to be equal to four
times, everywhere we see

00:01:13.533 --> 00:01:15.066
an x we're going to put a five there,

00:01:15.066 --> 00:01:17.666
four times, actually
let me do it this way.

00:01:17.666 --> 00:01:20.133
Let me just color code it.

00:01:20.133 --> 00:01:21.866
So this is the same thing as saying

00:01:21.866 --> 00:01:26.866
five times five

00:01:28.737 --> 00:01:33.133
minus three, let me do
that in that same color,

00:01:33.300 --> 00:01:38.300
minus three, needs to be
equal to four times five,

00:01:39.712 --> 00:01:44.712
four times five plus three, plus three.

00:01:45.733 --> 00:01:47.500
Color changing is hard.

00:01:47.500 --> 00:01:48.633
Plus three.

00:01:48.633 --> 00:01:49.700
Now, is this true?

00:01:49.700 --> 00:01:52.666
Let's see, five times five
is 25, it's going to be

00:01:52.666 --> 00:01:57.666
25 minus three, needs to
be equal to 20 plus three.

00:01:58.766 --> 00:02:03.334
25 minus three is 22,
needs to be equal to 23.

00:02:03.334 --> 00:02:05.000
No, this is not true.

00:02:05.000 --> 00:02:10.000
So, x does not equal five,
so this is not a solution.

00:02:11.100 --> 00:02:12.833
Let's try x equals six.

00:02:12.833 --> 00:02:15.700
So, once again, we're going
to do five times our x,

00:02:15.700 --> 00:02:17.283
which is going to be six,
actually let me just write it out,

00:02:17.283 --> 00:02:22.283
minus three needs to be
equal to four times our x,

00:02:22.313 --> 00:02:26.309
plus three, and in this case our x is six,

00:02:26.309 --> 00:02:28.033
so it's going to be five
times six minus three

00:02:28.033 --> 00:02:30.466
needs to be equal to four
times six plus three.

00:02:30.912 --> 00:02:32.566
What's five times six?

00:02:32.566 --> 00:02:36.500
Well, it's 30 minus
three, needs to be equal

00:02:36.500 --> 00:02:40.900
to four times six is
24, and then plus three.

00:02:40.900 --> 00:02:43.033
Well, this is true, 30 minus three is 27,

00:02:43.494 --> 00:02:47.600
which is indeed equal to 24
plus 3, it's equal to 27.

00:02:47.600 --> 00:02:51.133
So x equals 6 does satisfy our equation,

00:02:51.133 --> 00:02:54.233
it is a solution, and actually
as we'll see in the future,

00:02:54.233 --> 00:02:57.033
the solution to this
equation right over here.

00:02:57.033 --> 00:02:58.966
X equals six satisfies this.

00:02:58.966 --> 00:03:00.800
Now, just for good
measure, let's just varify

00:03:00.800 --> 00:03:04.033
that x equals seven will not satisfy.

00:03:04.033 --> 00:03:05.633
So I'm going to move this up a little bit.

00:03:05.633 --> 00:03:07.600
So if x is equal to
seven, we're going to get

00:03:07.600 --> 00:03:12.466
five times seven minus
three needs to be equal

00:03:12.466 --> 00:03:16.866
to four time seven plus three.

00:03:16.866 --> 00:03:19.533
And so, we're going to
get, and in all these cases

00:03:19.533 --> 00:03:22.100
we do the multiplication
first, order of operations,

00:03:22.100 --> 00:03:23.733
and it's very clear
when you see it kind of

00:03:23.733 --> 00:03:26.833
in the algebraic notation up
here, so we're going to do

00:03:26.833 --> 00:03:31.833
35 minus three needs to
be equal to 28 plus three,

00:03:32.309 --> 00:03:36.433
35 minus three is 32, 28 plus three is 31,

00:03:36.433 --> 00:03:38.466
these do not equal each other.

00:03:38.466 --> 00:03:43.300
So this is not a solution
to our original equation.


How to test solutions to inequalities

WEBVTT
Kind: captions
Language: en

00:00:00.641 --> 00:00:02.844
- [Voiceover] We have two
inequalities here, the first one

00:00:02.844 --> 00:00:07.116
says that x plus two is
less than or equal to two x.

00:00:07.116 --> 00:00:10.895
This one over here in I guess
this light-purple-mauve color,

00:00:10.895 --> 00:00:15.057
is three x plus four
is greater than five x.

00:00:15.057 --> 00:00:17.826
Over here we have four numbers
and what I want to do in this

00:00:17.826 --> 00:00:21.507
video is test whether
any of these four numbers

00:00:21.507 --> 00:00:24.884
satisfy either of these inequalities.

00:00:24.884 --> 00:00:26.957
I encourage you to pause this video and

00:00:26.957 --> 00:00:29.912
try these numbers out, does
zero satisfy this inequality?

00:00:29.912 --> 00:00:32.353
Does it satisfy this one?
Does one satisfy this one?

00:00:32.353 --> 00:00:33.708
Does it satisfy that one?

00:00:33.708 --> 00:00:35.797
I encourage you to try
these four numbers out

00:00:35.797 --> 00:00:38.949
on these two inequalities.

00:00:38.949 --> 00:00:41.228
Assuming you have tried that, let's work

00:00:41.228 --> 00:00:42.957
through this together.

00:00:42.957 --> 00:00:47.935
Let's say, if we try out
zero on this inequality

00:00:47.935 --> 00:00:50.616
right over here, let's substitute

00:00:51.523 --> 00:00:53.283
x with zero.

00:00:53.283 --> 00:00:57.554
So, we'll have zero plus two needs to

00:00:57.554 --> 00:01:02.329
be less than or equal to two times zero.

00:01:02.329 --> 00:01:04.071
Is that true?

00:01:04.071 --> 00:01:07.235
Well, on the left hand side,
this is two needs to be

00:01:07.235 --> 00:01:09.965
less than or equal to zero.

00:01:09.965 --> 00:01:12.359
Is that true, is two less
than or equal to zero?

00:01:12.359 --> 00:01:14.743
No, two is larger than zero.

00:01:14.743 --> 00:01:18.900
So this is not going to be
true, this does not satisfy

00:01:20.304 --> 00:01:21.822
the left hand side
inequality, let's see if it

00:01:21.822 --> 00:01:26.200
satisfies this inequality over here.

00:01:27.388 --> 00:01:31.936
In order to satisfy it,
three times zero plus four

00:01:31.936 --> 00:01:35.651
needs to be greater than five times zero.

00:01:35.651 --> 00:01:39.495
Well three times zero is just
zero, five times zero is zero.

00:01:39.495 --> 00:01:43.915
So four needs to be greater
than zero, which is true.

00:01:43.915 --> 00:01:47.766
So it does satisfy

00:01:47.995 --> 00:01:50.459
this inequality right over here so zero

00:01:50.459 --> 00:01:53.525
does satisfy this inequality.

00:01:53.525 --> 00:01:55.233
Let's try out one.

00:01:56.902 --> 00:02:01.246
To satisfy this one, one plus
two needs to be less than

00:02:01.246 --> 00:02:03.266
or equal to two.

00:02:03.518 --> 00:02:06.477
One plus two is three, is three
less than or equal than two?

00:02:06.477 --> 00:02:09.200
No, three is larger than two.

00:02:09.576 --> 00:02:13.297
This does not satisfy
the left hand inequality.

00:02:13.297 --> 00:02:16.600
What about the right hand
inequality right over here?

00:02:17.218 --> 00:02:21.556
Three times one plus four
needs to be greater than

00:02:21.556 --> 00:02:24.176
five times one.

00:02:24.176 --> 00:02:27.392
So three times one is three, plus four.

00:02:27.392 --> 00:02:31.926
So seven needs to be greater
than five, well that's true.

00:02:31.926 --> 00:02:35.003
Both zero and one satisfy
three x plus four is greater

00:02:35.003 --> 00:02:38.707
than five x, neither of them
satisfy x plus two is less than

00:02:38.707 --> 00:02:41.502
or equal to two x.

00:02:41.502 --> 00:02:44.262
Now let's go to the two.

00:02:44.262 --> 00:02:46.701
I know it's getting a
little bit unaligned,

00:02:46.701 --> 00:02:49.973
but I'll just do it all in the
same color so you can tell.

00:02:49.973 --> 00:02:54.003
Let's try out two here, two
plus two needs to be less than

00:02:54.003 --> 00:02:56.333
or equal to two times two.

00:02:56.474 --> 00:02:59.744
Four needs to be less
than or equal to four.

00:02:59.744 --> 00:03:01.372
Well four is equal to
four and it just has to be

00:03:01.372 --> 00:03:04.558
less than or equal, so this satisfies.

00:03:04.558 --> 00:03:07.439
This satisfies this inequality.

00:03:07.439 --> 00:03:10.200
What about this purple inequality?

00:03:10.200 --> 00:03:13.687
Let's see, three times
two plus four needs to be

00:03:13.687 --> 00:03:17.064
greater than five times two.

00:03:17.064 --> 00:03:21.265
Three times two is six plus four is ten,

00:03:21.265 --> 00:03:24.266
needs to be greater than 10.

00:03:24.406 --> 00:03:27.090
10 is equal to 10, it's
not greater than 10.

00:03:27.090 --> 00:03:31.394
It does not satisfy this inequality.

00:03:31.394 --> 00:03:34.814
If this was a greater than or equal to

00:03:34.814 --> 00:03:37.211
it would have satisfied but it's not.

00:03:37.211 --> 00:03:40.400
10 is not greater than 10.

00:03:40.528 --> 00:03:42.070
It would satisfy greater than or equal to

00:03:42.070 --> 00:03:43.869
because 10 is equal to 10.

00:03:43.869 --> 00:03:46.104
So two satisfies the left hand one but not

00:03:46.104 --> 00:03:47.833
the right hand one.

00:03:48.091 --> 00:03:50.200
Let's try out five.

00:03:52.486 --> 00:03:56.725
Five plus two needs to
be less than or equal

00:03:56.725 --> 00:03:59.908
to two times five, once
again everywhere we see

00:03:59.908 --> 00:04:02.033
an x, we replace it with a five.

00:04:02.207 --> 00:04:05.499
Seven needs to be less
than or equal to 10.

00:04:05.499 --> 00:04:09.289
Which is absolutely true,
seven is less than 10.

00:04:09.289 --> 00:04:11.662
So it satisfies less than or equal to.

00:04:11.662 --> 00:04:14.397
Five satisfies this
inequality and what you're

00:04:14.397 --> 00:04:16.123
probably noticing now is
that an inequality can

00:04:16.123 --> 00:04:18.180
have many numbers that satisfy.

00:04:18.180 --> 00:04:21.113
In fact they sometimes will
have nothing that satisfies it

00:04:21.113 --> 00:04:23.546
and sometimes they might
have an infinite number of

00:04:23.546 --> 00:04:26.721
numbers that satisfy it and
you see that right over here.

00:04:26.721 --> 00:04:29.025
We're just testing out a few numbers.

00:04:29.025 --> 00:04:31.725
For this left one, zero
and one didn't work,

00:04:31.725 --> 00:04:33.329
two and five did work.

00:04:33.329 --> 00:04:36.352
This right one, zero and
one worked, two didn't work.

00:04:36.352 --> 00:04:38.246
Let's see what five does.

00:04:38.246 --> 00:04:41.616
In order for five to
satisfy it, three times x.

00:04:41.616 --> 00:04:45.716
Now we're gonna try x being five.

00:04:45.716 --> 00:04:49.181
Three times five plus four
needs to be greater than

00:04:49.181 --> 00:04:51.512
five times five.

00:04:51.512 --> 00:04:55.467
Three times five is fifteen,
fifteen plus four is nineteen.

00:04:55.467 --> 00:04:59.704
Nineteen is to be greater
than 25, it is not.

00:04:59.704 --> 00:05:02.603
So five does not satisfy

00:05:02.603 --> 00:05:05.466
this inequality right over here.

00:05:05.466 --> 00:05:08.067
Anyway, hopefully you found that fun.


Two-way frequency tables and Venn diagrams

WEBVTT
Kind: captions
Language: en

00:00:00.394 --> 00:00:04.248
- So what I have here
are 12 pieces of candy.

00:00:04.248 --> 00:00:07.754
The ones that are colored
in brown have chocolate

00:00:07.754 --> 00:00:10.435
on the outside and the ones that have a C

00:00:10.435 --> 00:00:11.809
on them means that they have coconut

00:00:11.809 --> 00:00:12.645
on the inside.

00:00:12.645 --> 00:00:15.802
So for example, this one
over here in the top left,

00:00:15.802 --> 00:00:17.206
it's made out of chocolate on the outside,

00:00:17.206 --> 00:00:19.088
but it doesn't have coconut on the inside.

00:00:19.088 --> 00:00:21.394
While this one right over
here is chocolate on the

00:00:21.394 --> 00:00:24.517
outside and it has coconut on the inside.

00:00:24.517 --> 00:00:27.338
While this one, whoops,
I didn't want to do that,

00:00:27.338 --> 00:00:31.447
while this one right
over here does not have

00:00:31.447 --> 00:00:33.065
chocolate on the outside, but it does have

00:00:33.065 --> 00:00:34.306
coconut on the inside.

00:00:34.306 --> 00:00:36.466
This one right over here has neither

00:00:36.466 --> 00:00:38.427
chocolate nor coconut.

00:00:38.427 --> 00:00:40.538
What I want to think
about is ways to represent

00:00:40.538 --> 00:00:43.326
this information that we are looking at.

00:00:43.326 --> 00:00:46.849
One way to do it is using a Venn diagram.

00:00:46.849 --> 00:00:49.465
So let me draw a Venn diagram.

00:00:49.465 --> 00:00:54.465
So Venn diagram is one
way to represent it.

00:00:55.316 --> 00:00:57.277
The way it's typically
done, the convention,

00:00:57.277 --> 00:00:59.123
is that you would make
a rectangle to represent

00:00:59.123 --> 00:01:00.879
the universe that you
care about, in this case

00:01:00.879 --> 00:01:01.973
it would be all the chocolates.

00:01:01.973 --> 00:01:04.550
All the numbers inside
of this should add up

00:01:04.550 --> 00:01:06.014
to the number of chocolates I have.

00:01:06.014 --> 00:01:07.929
So it should add up to 12.

00:01:07.929 --> 00:01:09.856
So that's our universe right over here.

00:01:09.856 --> 00:01:12.294
Then I'll draw circles
to represent the sets

00:01:12.294 --> 00:01:13.954
that I care about.

00:01:13.954 --> 00:01:15.835
So say for this one I care about the set

00:01:15.835 --> 00:01:18.169
of the things that have chocolate.

00:01:18.169 --> 00:01:20.083
So I'll draw that with a circle.

00:01:20.851 --> 00:01:22.064
You could draw them to scale, but I'm not

00:01:22.064 --> 00:01:23.027
going to draw them to scale.

00:01:23.027 --> 00:01:27.323
So that is my chocolate set, chocolate.

00:01:27.323 --> 00:01:29.646
That is my chocolate set.

00:01:29.646 --> 00:01:32.128
Then I want to have a coconut set.

00:01:32.128 --> 00:01:35.986
So coconut.

00:01:35.986 --> 00:01:37.565
Once again, not drawn to scale.

00:01:37.565 --> 00:01:38.901
I drew them roughly the same size,

00:01:38.901 --> 00:01:41.573
but you can see the
chocolate set is bigger

00:01:41.573 --> 00:01:43.466
than the coconut set in reality.

00:01:43.466 --> 00:01:45.486
Coconut set.

00:01:45.486 --> 00:01:48.156
Now we can fill in the different sections.

00:01:48.156 --> 00:01:51.859
So how many of these things have chocolate

00:01:51.859 --> 00:01:53.542
but no coconut?

00:01:53.542 --> 00:01:58.542
Let's see, we have one,
two, three, four, five,

00:01:59.463 --> 00:02:03.352
six have chocolate but no coconut.

00:02:04.788 --> 00:02:05.969
Let me do that in a different color

00:02:05.969 --> 00:02:07.309
because I think the colors are important.

00:02:07.309 --> 00:02:09.062
So let me do it in green.

00:02:09.062 --> 00:02:14.062
So one, two, three, four, five, and six.

00:02:15.029 --> 00:02:17.757
So this section right over here is six.

00:02:17.757 --> 00:02:18.929
Once again, I'm not
talking about the whole

00:02:18.929 --> 00:02:21.611
brown thing, I'm talking about just this

00:02:21.611 --> 00:02:23.735
area that I've shaded in green.

00:02:23.735 --> 00:02:26.487
Now how many have chocolate and coconut?

00:02:26.487 --> 00:02:29.006
Chocolate and coconut.

00:02:29.006 --> 00:02:33.127
Well that's going to be one, two, three.

00:02:33.127 --> 00:02:35.646
So three of them have
chocolate and coconut.

00:02:35.646 --> 00:02:37.828
Notice that's this section here that's

00:02:37.828 --> 00:02:39.094
in the overlap between.

00:02:39.094 --> 00:02:42.172
Three of them go into both
sets, both categories.

00:02:42.172 --> 00:02:45.224
These three have coconut
and they have chocolate.

00:02:45.224 --> 00:02:46.454
How many total have chocolate?

00:02:46.454 --> 00:02:48.125
Well six plus three, nine.

00:02:48.125 --> 00:02:49.508
How many total have coconut?

00:02:49.508 --> 00:02:50.378
Well we're going to have to figure that

00:02:50.378 --> 00:02:51.708
out in a second.

00:02:51.708 --> 00:02:54.286
So how many have coconut but no chocolate?

00:02:54.286 --> 00:02:58.187
Well there's only one with
coconut and no chocolate.

00:02:58.187 --> 00:02:59.556
So that's that one right over there,

00:02:59.556 --> 00:03:01.493
and that represents this are that I'm

00:03:01.493 --> 00:03:02.727
shading in in white.

00:03:02.727 --> 00:03:04.321
So how many total coconut are there?

00:03:04.321 --> 00:03:06.496
Well one plus three, or
four, and you see that.

00:03:06.496 --> 00:03:08.377
One, two, three, four.

00:03:08.377 --> 00:03:09.805
The last thing we'd want to fill in,

00:03:09.805 --> 00:03:11.477
because notice, six plus three plus one

00:03:11.477 --> 00:03:12.417
only adds up to 10.

00:03:12.417 --> 00:03:13.450
What about the other two?

00:03:13.450 --> 00:03:16.981
Well the other two are
neither chocolate nor coconut.

00:03:16.981 --> 00:03:18.559
Actually, let me color this.

00:03:18.559 --> 00:03:20.382
So that's one, two, these are neither

00:03:20.382 --> 00:03:21.647
chocolate nor coconut.

00:03:21.647 --> 00:03:23.458
I could write these two right over here.

00:03:23.458 --> 00:03:26.082
These are neither chocolate nor coconut.

00:03:26.082 --> 00:03:30.099
So that's one way to
represent the information

00:03:30.099 --> 00:03:32.537
of how many chocolates, how many coconuts,

00:03:32.537 --> 00:03:34.045
and how many chocolate and coconuts,

00:03:34.045 --> 00:03:35.404
and how many neither.

00:03:35.404 --> 00:03:37.018
But there's other ways
that we could do it.

00:03:37.018 --> 00:03:40.199
Another way to do it would
be with a two way table.

00:03:40.199 --> 00:03:45.199
Two way table.

00:03:46.150 --> 00:03:48.546
On one axis, say the vertical axis,

00:03:48.546 --> 00:03:51.136
we could say, let me write this,

00:03:51.136 --> 00:03:56.136
so has chocolate, I'll
write choc for short,

00:03:57.068 --> 00:04:02.068
and then I'll write no
chocolate, choc for short.

00:04:02.187 --> 00:04:05.426
Then over here I could write coconut.

00:04:05.426 --> 00:04:07.214
I want to do that in white.

00:04:07.214 --> 00:04:08.932
I got new tools and sometimes

00:04:08.932 --> 00:04:10.511
the color changing isn't so easy.

00:04:10.511 --> 00:04:13.971
So this is coconut and then over here

00:04:13.971 --> 00:04:16.172
I'll write no coconut.

00:04:21.401 --> 00:04:24.049
Then let me make a little table.

00:04:25.014 --> 00:04:27.615
Make it clear what I'm doing here.

00:04:27.615 --> 00:04:30.970
So a line there and a line there

00:04:30.970 --> 00:04:33.872
and why not add a line over here as well.

00:04:33.872 --> 00:04:36.608
Then I can just fill in
the different things.

00:04:36.608 --> 00:04:39.492
This cell, this square,
this is going to represent

00:04:39.492 --> 00:04:43.182
the number that has coconut and chocolate.

00:04:43.858 --> 00:04:44.846
Well we are already looked into that,

00:04:44.846 --> 00:04:47.574
that's one, two, three, that's these three

00:04:47.574 --> 00:04:48.375
right over here.

00:04:48.375 --> 00:04:50.814
So that's those three right over there.

00:04:50.814 --> 00:04:54.088
This one right over
here, it has chocolate,

00:04:54.088 --> 00:04:55.946
but it doesn't have coconut.

00:04:55.946 --> 00:04:57.282
Well that's this six right over here.

00:04:57.282 --> 00:05:00.405
It has chocolate, but
it doesn't have coconut.

00:05:00.405 --> 00:05:04.723
So let me write this is
that six right over there.

00:05:04.723 --> 00:05:09.067
Then this box would be it has coconut,

00:05:09.067 --> 00:05:10.622
but no chocolate.

00:05:10.622 --> 00:05:11.701
Well how many is that?

00:05:11.701 --> 00:05:14.940
Well coconut no chocolate
that's that one there.

00:05:14.940 --> 00:05:17.019
Then this one is going to be no coconut

00:05:17.019 --> 00:05:18.294
and no chocolate.

00:05:18.294 --> 00:05:19.897
We know what that's going to be.

00:05:19.897 --> 00:05:24.897
No coconut and no chocolate
is going to be two.

00:05:25.493 --> 00:05:26.979
If we wanted to, we could even throw

00:05:26.979 --> 00:05:28.697
in totals over here.

00:05:28.697 --> 00:05:30.707
We could write, actually
let me just do that

00:05:30.707 --> 00:05:31.820
just for fun.

00:05:31.820 --> 00:05:36.487
I could write total, I could write total,

00:05:36.487 --> 00:05:41.119
and if I total it
vertically, so three plus

00:05:41.119 --> 00:05:43.464
one this is four, six plus two is eight.

00:05:43.464 --> 00:05:45.263
So this four is the total number

00:05:45.263 --> 00:05:47.457
that have coconut that has chocolate

00:05:47.457 --> 00:05:48.376
and doesn't have chocolate.

00:05:48.376 --> 00:05:50.210
That's three plus one.

00:05:50.210 --> 00:05:55.210
This eight is the total
that does not have coconut.

00:05:55.643 --> 00:05:56.466
No coconut.

00:05:56.466 --> 00:05:58.733
So the total of no
coconut, and that of course

00:05:58.733 --> 00:06:01.694
is going to be the six plus this two.

00:06:01.694 --> 00:06:03.655
We could total horizontally.

00:06:03.655 --> 00:06:05.757
Three plus six is nine.

00:06:05.757 --> 00:06:07.278
One plus two is three.

00:06:07.278 --> 00:06:08.055
What's this nine?

00:06:08.055 --> 00:06:10.970
That's the total amount of
chocolate, six plus three.

00:06:10.970 --> 00:06:11.631
What's this three?

00:06:11.631 --> 00:06:13.372
This is the total amount no chocolate.

00:06:13.372 --> 00:06:15.497
That's this one plus two.

00:06:15.497 --> 00:06:17.052
Anyway, hopefully you
found that interesting.

00:06:17.052 --> 00:06:19.259
This is just different
ways of representing

00:06:19.259 --> 00:06:21.627
the same information.


Two-way relative frequency tables

WEBVTT
Kind: captions
Language: en

00:00:00.512 --> 00:00:03.634
- [Voiceover] The two-way
frequency table below

00:00:03.634 --> 00:00:06.386
shows data on type of vehicle driven.

00:00:06.386 --> 00:00:08.103
So, this is type of vehicle driven,

00:00:08.103 --> 00:00:11.355
and whether there was an
accident the last year.

00:00:11.355 --> 00:00:13.445
So, whether there was an
accident in the last year,

00:00:13.445 --> 00:00:17.020
for customers of All
American Auto Insurance.

00:00:17.020 --> 00:00:19.319
Complete the following two-way table of

00:00:19.319 --> 00:00:21.455
column relative frequencies.

00:00:21.455 --> 00:00:22.616
So that's what they're talking here,

00:00:22.616 --> 00:00:25.054
this is a two-way table of
column relative frequencies.

00:00:25.054 --> 00:00:28.792
If necessary, round your answers
to the nearest hundredth.

00:00:28.792 --> 00:00:30.209
So, let's see what they're saying.

00:00:30.209 --> 00:00:31.741
They're saying, let's see...

00:00:31.741 --> 00:00:35.841
Of the accidents within
the last year, 28 were

00:00:35.841 --> 00:00:39.055
the people were driving an
SUV, a Sport Utility Vehicle

00:00:39.055 --> 00:00:42.236
and 35 were in a Sports car.

00:00:42.236 --> 00:00:46.625
Of the No accidents in the
last year, 97 were in SUV

00:00:46.625 --> 00:00:48.541
and a 104 were in sports cars.

00:00:48.541 --> 00:00:49.832
Another way you could think of it,

00:00:49.832 --> 00:00:52.824
of the Sport utility
vehicles that were driven

00:00:52.824 --> 00:00:56.400
and the total, let's see it's 28 plus 97

00:00:56.400 --> 00:00:59.952
which is going to be 125.

00:00:59.952 --> 00:01:03.458
Of that 125, 28 had an
accident within the last year

00:01:03.458 --> 00:01:06.194
and 97 did not have an
accident within the last year.

00:01:06.194 --> 00:01:11.177
Similarly, you could say
of the 139 Sports cars

00:01:11.177 --> 00:01:13.257
35 had an accident in the last year,

00:01:13.257 --> 00:01:15.370
104 did not have an
accident in the last year.

00:01:15.370 --> 00:01:18.156
So what they want us to
do is put those relative

00:01:18.156 --> 00:01:20.292
frequencies in here.

00:01:20.292 --> 00:01:22.243
So the way we could think about it.

00:01:22.243 --> 00:01:24.077
One right over here, this represents

00:01:24.077 --> 00:01:26.468
all the Sport utility vehicles.

00:01:26.468 --> 00:01:27.745
So one way you could think about,

00:01:27.745 --> 00:01:30.346
that represents the whole
universe of the Sport Utility

00:01:30.346 --> 00:01:35.022
Vehicles, at least the
universe that this table shows.

00:01:35.022 --> 00:01:38.356
So, that's really representative
of the 28 plus the 97.

00:01:38.356 --> 00:01:40.400
And so, in each of these we want to put

00:01:40.400 --> 00:01:41.604
the relative frequencies.

00:01:41.604 --> 00:01:43.342
So this right over here is going to be

00:01:43.342 --> 00:01:48.311
28 divided by the total.

00:01:48.311 --> 00:01:51.475
So over here is 28,
but we want this number

00:01:51.475 --> 00:01:52.799
to be a fraction of the total.

00:01:52.799 --> 00:01:56.305
Well the fraction of
the total is gonna be,

00:01:56.305 --> 00:02:00.229
28 over 97 plus 28.

00:02:00.229 --> 00:02:03.642
Which of course is going to be 125.

00:02:03.642 --> 00:02:06.173
Actually let me just write
them all like that first.

00:02:06.173 --> 00:02:11.173
This one right over here
is going to be 97 over 125.

00:02:11.304 --> 00:02:13.115
And of course, when you
add this one and this one,

00:02:13.115 --> 00:02:14.782
it should add up to one.

00:02:14.782 --> 00:02:19.570
Likewise, this one's
going to be 35 over 139.

00:02:19.570 --> 00:02:21.231
35 plus 104.

00:02:21.231 --> 00:02:23.550
So, 139.

00:02:23.550 --> 00:02:28.550
And this is going to be
104 over 104 plus 35.

00:02:28.695 --> 00:02:30.692
Which is 139.

00:02:30.692 --> 00:02:33.293
So, let me just calculate
each of them using

00:02:33.293 --> 00:02:34.291
this calculator.

00:02:34.291 --> 00:02:36.590
Let me scroll down a little bit.

00:02:36.590 --> 00:02:41.590
And so, if I do 28 divided by 125,

00:02:43.189 --> 00:02:45.686
I get 0.224.

00:02:45.686 --> 00:02:48.478
They said round your answers
to the nearest hundredth.

00:02:48.478 --> 00:02:51.450
So this is 0.22.

00:02:51.450 --> 00:02:55.065
No accident within the last
year, 97 divided by 125.

00:02:55.065 --> 00:03:00.065
So 97 divided by 125

00:03:00.767 --> 00:03:02.966
is equal to, see here if I rounded

00:03:02.966 --> 00:03:05.010
to the nearest hundredth
I'm gonna round up.

00:03:05.010 --> 00:03:10.010
0.78, so this is 0.78.

00:03:11.395 --> 00:03:13.988
Then, 35 divided by 139.

00:03:13.988 --> 00:03:18.988
35 divided by 139,

00:03:19.267 --> 00:03:23.566
is equal to, round to the
nearest hundredth, 0.25.

00:03:23.701 --> 00:03:25.866
0.25.

00:03:25.985 --> 00:03:28.800
And then 104 divided by 139.

00:03:29.086 --> 00:03:34.086
104 divided by 139, gets me

00:03:35.589 --> 00:03:39.600
if I round to the nearest hundredth, 0.75.

00:03:39.885 --> 00:03:44.885
0.75, then I can check my answer.

00:03:46.061 --> 00:03:47.361
And I got it right.

00:03:47.361 --> 00:03:48.824
But the key thing here is to make sure we

00:03:48.824 --> 00:03:50.078
understand what's going on here.

00:03:50.078 --> 00:03:50.937
So, we could-

00:03:50.937 --> 00:03:53.027
One way to think about this is 22% of the

00:03:53.027 --> 00:03:55.929
Sport utility vehicles had an
accident within the last year.

00:03:55.929 --> 00:03:59.273
Or you could say .22 of them.

00:03:59.273 --> 00:04:02.546
And you could say 78% or 0.78 of the

00:04:02.546 --> 00:04:07.546
Sport utility vehicles had no accidents.

00:04:07.608 --> 00:04:10.186
Likewise, you could say
25% of the Sports cars

00:04:10.186 --> 00:04:12.089
had an accident within the last year.

00:04:12.089 --> 00:04:17.012
And 75% did not have
an accident last year.

00:04:17.012 --> 00:04:18.753
So, it allows you think more in terms of

00:04:18.753 --> 00:04:21.841
the relative frequencies,
the whole, the percentages,

00:04:21.841 --> 00:04:23.374
however you want to think about it.

00:04:23.374 --> 00:04:27.052
While this gives you the actual numbers.


Interpreting two-way tables

WEBVTT
Kind: captions
Language: en

00:00:00.533 --> 00:00:04.500
- The two-way table of column
relative frequencies below

00:00:04.500 --> 00:00:08.600
shows data on gender and voting
preferences during the 2012

00:00:08.600 --> 00:00:10.700
United States presidential election.

00:00:10.700 --> 00:00:12.133
They give us all this data.

00:00:12.133 --> 00:00:14.300
They give us this, as they
say, the two-way table

00:00:14.300 --> 00:00:16.900
of column relative frequencies.

00:00:16.900 --> 00:00:19.567
So for example this column
right over here is Men.

00:00:19.567 --> 00:00:23.100
The column total is 1.00, or
you could say 100 percent.

00:00:23.100 --> 00:00:27.166
And we can see that 0.42
of the Men or 42 percent

00:00:27.166 --> 00:00:29.566
of the Men voted for Obama.

00:00:29.566 --> 00:00:33.833
We can see 52% of the
Men or 0.52 of the Men

00:00:33.833 --> 00:00:36.167
voted for Romney.

00:00:36.167 --> 00:00:40.299
And we can see that the
Other, neither Obama,

00:00:40.299 --> 00:00:45.066
6 percent went for
neither Obama nor Romney.

00:00:45.066 --> 00:00:47.133
And for Women, 52 percent went for Obama,

00:00:47.133 --> 00:00:48.833
43 percent went for Romney and 5 percent

00:00:48.833 --> 00:00:49.900
went for Other.

00:00:49.900 --> 00:00:53.166
And then these, this 52 plus 43 plus 5

00:00:53.166 --> 00:00:56.566
will add up to 100 percent of the women.

00:00:56.566 --> 00:00:59.066
During the 2012 United
States presidential election,

00:00:59.066 --> 00:01:01.833
were male voters more
likely to vote for Romney

00:01:01.833 --> 00:01:03.967
than female voters?

00:01:03.967 --> 00:01:05.233
So let's see.

00:01:05.233 --> 00:01:07.633
If we, there are a
couple of ways you could

00:01:07.633 --> 00:01:08.367
think about it.

00:01:08.367 --> 00:01:09.266
Well, actually, let's go this way.

00:01:09.266 --> 00:01:12.800
Male voters, if you were
a man, 52 percent of them

00:01:12.800 --> 00:01:14.166
voted for Romney.

00:01:14.166 --> 00:01:19.166
While for the Women, 43 percent
of them voted for Romney.

00:01:19.533 --> 00:01:21.866
So a man was more likely.

00:01:21.866 --> 00:01:24.400
If you randomly picked a
man who voted, there was

00:01:24.400 --> 00:01:26.633
a 52 percent chance they voted for Romney,

00:01:26.633 --> 00:01:28.466
while if you randomly picked a woman,

00:01:28.466 --> 00:01:30.833
there was a 43 percent,
of women who voted,

00:01:30.833 --> 00:01:33.867
there was a 43 percent chance
that she voted for Romney.

00:01:33.867 --> 00:01:37.389
So yes, male voters
were more likely to vote

00:01:37.389 --> 00:01:39.033
for Romney than female voters.

00:01:39.033 --> 00:01:40.933
So the answer is Yes.

00:01:40.933 --> 00:01:43.000
And we're done.


Pythagorean theorem in 3D

WEBVTT
Kind: captions
Language: en

00:00:00.243 --> 00:00:01.580
- [Voiceover] So we have an interesting

00:00:01.580 --> 00:00:03.530
shape right over here.

00:00:03.530 --> 00:00:05.931
The base is a rectangular prism.

00:00:05.931 --> 00:00:07.751
And the dimensions of
that rectangular prism,

00:00:07.751 --> 00:00:10.485
it's three units I
guess we could say tall,

00:00:10.485 --> 00:00:14.471
four units wide, and then four units long.

00:00:14.471 --> 00:00:16.909
And then on top of that, on top of that,

00:00:16.909 --> 00:00:19.781
we have what you could
call a right pyramid,

00:00:19.781 --> 00:00:22.277
where the height of this right pyramid,

00:00:22.277 --> 00:00:25.814
so if you start at the center
of its base right over here,

00:00:25.814 --> 00:00:26.876
and you go to the top,

00:00:26.876 --> 00:00:29.996
this height right over here is one unit.

00:00:29.996 --> 00:00:31.784
And this hasn't been
drawn completely to scale,

00:00:31.784 --> 00:00:33.809
and kind of the perspective
skews a little bit.

00:00:33.809 --> 00:00:36.951
But our goal here, our goal here,

00:00:36.951 --> 00:00:40.780
is to figure out what is the length?

00:00:40.780 --> 00:00:45.662
What is the length of one of
these edges right over here?

00:00:45.662 --> 00:00:49.624
So either that one, or
this one right over here.

00:00:49.624 --> 00:00:51.273
What is that length?

00:00:51.273 --> 00:00:55.118
And we will call that,
we will call that x.

00:00:55.118 --> 00:00:57.752
And so I encourage you to pause this video

00:00:57.752 --> 00:00:59.289
and try to think about it on your own.

00:00:59.289 --> 00:01:02.585
Remember, this is a right,
this is a right pyramid.

00:01:02.600 --> 00:01:06.674
So what that tells us
is that this red line,

00:01:06.674 --> 00:01:11.039
that's one unit long, it is perpendicular,

00:01:11.639 --> 00:01:15.702
it is perpendicular to this entire plane.

00:01:15.702 --> 00:01:17.498
It's perpendicular to the top

00:01:17.498 --> 00:01:20.959
of the rectangular prism.

00:01:20.959 --> 00:01:23.860
So with that in mind, I encourage you to

00:01:23.860 --> 00:01:27.263
pause the video and see
if you can figure it out.

00:01:27.263 --> 00:01:29.369
And I will give you a hint.

00:01:29.369 --> 00:01:32.615
You will have to use
the Pythagorean theorum,

00:01:32.615 --> 00:01:35.174
maybe more than once.

00:01:35.174 --> 00:01:39.385
Alright, so I am assuming
you've at least given it a shot.

00:01:39.385 --> 00:01:41.666
So let's work through it together.

00:01:41.666 --> 00:01:46.015
So the key here is to
realize, well okay this point,

00:01:46.015 --> 00:01:48.298
this base right over here,
this point right over here

00:01:48.298 --> 00:01:50.249
it's half way in this direction

00:01:50.249 --> 00:01:51.584
and half way in this direction.

00:01:51.584 --> 00:01:56.321
So we can figure out,
well this entire length

00:01:56.321 --> 00:01:59.318
right over here is length four.

00:01:59.318 --> 00:02:01.324
So half way, this is going to be,

00:02:01.324 --> 00:02:02.286
I'll write it with perspective.

00:02:02.286 --> 00:02:06.082
That's going to be two,
and that's going to be two.

00:02:06.082 --> 00:02:07.610
Just like that.

00:02:07.610 --> 00:02:09.975
And then the other
thing we can figure out,

00:02:09.975 --> 00:02:13.993
we can figure out what
this length is going to be.

00:02:13.993 --> 00:02:16.455
'Cause once again it's
half way in that direction.

00:02:16.455 --> 00:02:20.213
So if this whole thing is two,

00:02:20.213 --> 00:02:21.350
and we see it right over here.

00:02:21.350 --> 00:02:23.146
This is a rectangular
prism, so this length

00:02:23.146 --> 00:02:25.043
is going to be the same
thing as this length.

00:02:25.043 --> 00:02:27.172
So if this whole thing is
two, then each of these,

00:02:27.172 --> 00:02:29.064
this is going to be one,

00:02:29.064 --> 00:02:31.870
and this is going to be
one right over there.

00:02:31.870 --> 00:02:34.165
Well how does that help us?

00:02:34.165 --> 00:02:36.817
Well using that information,
we should be able

00:02:36.817 --> 00:02:40.529
to figure out this length.

00:02:40.990 --> 00:02:42.385
Actually, I'll keep it in this color

00:02:42.385 --> 00:02:44.017
'cause this color's easy to see.

00:02:44.017 --> 00:02:46.766
We should be able to
figure out this length.

00:02:46.878 --> 00:02:48.736
Well why is this length interesting?

00:02:48.736 --> 00:02:50.268
Well if we know that length,

00:02:50.268 --> 00:02:52.799
that length forms a right triangle.

00:02:52.799 --> 00:02:54.618
That length and the one are the two

00:02:54.618 --> 00:02:57.111
non-hypotenuse sides of a right triangle.

00:02:57.111 --> 00:02:59.101
And then the x would be the hypotenuse.

00:02:59.101 --> 00:03:02.240
So we could just apply
the Pythagorean theorem.

00:03:02.240 --> 00:03:05.702
So if we can figure out
this, we can figure out x.

00:03:05.702 --> 00:03:07.142
So let's do one step at a time.

00:03:07.142 --> 00:03:09.326
How do we figure out, how do we figure out

00:03:09.326 --> 00:03:11.735
I dunno, let me call this length.

00:03:11.735 --> 00:03:14.247
How do we figure out length a?

00:03:14.247 --> 00:03:15.593
Well let's just take it out

00:03:15.593 --> 00:03:17.773
and look at it in two dimensions.

00:03:17.773 --> 00:03:19.262
So if we look at it in two dimensions,

00:03:19.262 --> 00:03:22.184
If we look at it in two dimensions,

00:03:22.184 --> 00:03:24.164
it would look something like this.

00:03:24.164 --> 00:03:25.905
So that's our length a.

00:03:25.905 --> 00:03:27.856
We know that this length

00:03:27.856 --> 00:03:30.078
is half of this side right over here,

00:03:30.078 --> 00:03:31.733
so that's going to be one.

00:03:31.733 --> 00:03:33.498
And actually let me do
it in the same colors.

00:03:33.498 --> 00:03:35.423
So this right over here

00:03:35.423 --> 00:03:38.757
is the same thing as this right over here.

00:03:38.757 --> 00:03:40.835
And it is going to be of length one.

00:03:40.835 --> 00:03:44.175
And then this right over here

00:03:45.425 --> 00:03:48.786
is going to be the same
as this right over here,

00:03:48.786 --> 00:03:51.131
which is going to be of length two.

00:03:51.131 --> 00:03:53.891
And so we can just use the
Pythagorean theorem here.

00:03:53.891 --> 00:03:56.076
We know that the hypotenuse squared

00:03:56.076 --> 00:03:59.392
is going to be equal to one squared,

00:03:59.392 --> 00:04:02.367
one squared plus two squared.

00:04:02.590 --> 00:04:05.118
One squared plus two squared,

00:04:05.118 --> 00:04:08.962
which of course, is
equal to one plus four,

00:04:08.962 --> 00:04:10.904
which is equal to five.

00:04:10.904 --> 00:04:12.669
So we could write a...

00:04:12.669 --> 00:04:14.796
Let me do this in the magenta color.

00:04:14.796 --> 00:04:18.690
We can write a squared is equal to five,

00:04:18.690 --> 00:04:20.245
or we could say that a is equal to

00:04:20.245 --> 00:04:22.266
the principal root of five.

00:04:22.266 --> 00:04:24.538
So this length right over here

00:04:24.538 --> 00:04:29.138
is the square root of five,
the principal root of five.

00:04:29.138 --> 00:04:31.770
And now we can use that
information to solve for x.

00:04:31.770 --> 00:04:32.964
So let's take this right triangle.

00:04:32.964 --> 00:04:34.460
And it takes a little bit
of visualization practice

00:04:34.460 --> 00:04:35.419
to visualize this right.

00:04:35.419 --> 00:04:36.620
But notice, this is a right triangle.

00:04:36.620 --> 00:04:40.053
This height right here of
length one is perpendicular

00:04:40.053 --> 00:04:42.405
to this entire plane.

00:04:42.405 --> 00:04:44.202
So let's see if I can draw it.

00:04:44.202 --> 00:04:46.552
So we have this side

00:04:46.605 --> 00:04:49.926
is the square root of five.

00:04:49.926 --> 00:04:51.876
And then we have a height,

00:04:51.876 --> 00:04:53.617
Let me do it, looks like it's in a...

00:04:53.617 --> 00:04:56.430
Well, I'd say we're talkin' about

00:04:56.430 --> 00:05:00.676
this height right over here is length one.

00:05:01.152 --> 00:05:04.879
So I'll draw that, that is of length one.

00:05:04.879 --> 00:05:07.447
And we are trying to figure out x.

00:05:07.447 --> 00:05:11.297
We are trying to figure out x in orange.

00:05:11.820 --> 00:05:15.894
So we're trying to figure out this x.

00:05:15.894 --> 00:05:18.402
And once again, we know
this is a right triangle.

00:05:18.402 --> 00:05:20.915
This is a right triangle, so we can apply

00:05:20.915 --> 00:05:22.835
the Pythagorean theorem again.

00:05:22.835 --> 00:05:27.508
So we will have x squared
is equal to one squared.

00:05:27.508 --> 00:05:28.843
Which is just, I'll just write it

00:05:28.843 --> 00:05:31.769
one squared plus square
root of five squared,

00:05:31.769 --> 00:05:35.250
plus square root of five squared.

00:05:35.250 --> 00:05:38.750
Well, that gets us x squared is equal to

00:05:38.750 --> 00:05:41.644
one plus five, right?

00:05:41.644 --> 00:05:43.076
Square root of five squared is just five.

00:05:43.076 --> 00:05:45.394
So one plus five is equal to six.

00:05:45.394 --> 00:05:49.737
So we get x is equal to
the square root of six.

00:05:49.737 --> 00:05:50.892
And we're done!

00:05:50.892 --> 00:05:54.375
We figured out the length of
this side right over here.


Worked example: cosine function from power series

WEBVTT
Kind: captions
Language: en

00:00:00.141 --> 00:00:02.061
- So I have an infinite series, here.

00:00:02.061 --> 00:00:04.861
The sum from n equals zero
to infinity of negative one

00:00:04.861 --> 00:00:07.921
to the n power times x to the sixth n

00:00:07.921 --> 00:00:11.141
over two n, the whole two n, factorial.

00:00:11.141 --> 00:00:16.141
And my goal in this video is
to evaluate this power series

00:00:17.441 --> 00:00:21.961
when x is equal to the
cubed root of pi over two.

00:00:21.961 --> 00:00:24.361
And I encourage you to pause this video

00:00:24.361 --> 00:00:26.141
and give it a go on your own.

00:00:26.141 --> 00:00:27.621
And I will give you a hint.

00:00:27.621 --> 00:00:29.841
The key to this is to figure out,

00:00:29.841 --> 00:00:32.101
well what function is
this the power series for?

00:00:32.101 --> 00:00:35.041
and then use that
function to evaluate this.

00:00:35.041 --> 00:00:37.001
And there's another clue here, hey,

00:00:37.001 --> 00:00:39.441
this is kind of a
mysterious or a suspicious

00:00:39.441 --> 00:00:41.201
looking number, here, pi over two.

00:00:41.201 --> 00:00:42.841
That looks like something I would use

00:00:42.841 --> 00:00:45.941
a trig function to evaluate.

00:00:45.941 --> 00:00:47.801
That might be a little
bit more straightforward.

00:00:47.801 --> 00:00:50.201
So I'll let you have a go at it.

00:00:51.078 --> 00:00:52.582
So I'm assuming you have tried,

00:00:52.582 --> 00:00:54.202
so let's try to work
through this together.

00:00:54.202 --> 00:00:56.642
And in any of these types of problems

00:00:56.642 --> 00:01:00.202
I like to at least expand
out this power series

00:01:00.202 --> 00:01:02.443
so I get a better sense of what it's like.

00:01:02.828 --> 00:01:04.643
So this right over here,
if I were to expand it out,

00:01:04.643 --> 00:01:06.763
this is going to be
equal to when n is zero,

00:01:06.763 --> 00:01:09.803
this is one, actually,
all of these are one,

00:01:09.803 --> 00:01:11.043
so it's just going to be one.

00:01:11.043 --> 00:01:12.983
When n is one, this is going to be

00:01:12.983 --> 00:01:15.123
negative one, x to the sixth,

00:01:15.123 --> 00:01:18.143
x to the sixth over two factorial.

00:01:18.143 --> 00:01:21.363
when n is two, it's going to be positive.

00:01:21.363 --> 00:01:23.223
negative one squared is positive one,

00:01:23.223 --> 00:01:28.023
times x to the 12th over four factorial.

00:01:28.023 --> 00:01:30.963
and then let's just do one
more: when x is equal to three,

00:01:30.963 --> 00:01:34.364
it's going to be negative x to the 18th.

00:01:35.054 --> 00:01:39.567
x to the 18th over six factorial.

00:01:39.567 --> 00:01:41.367
And you just keep going on and on forever.

00:01:41.367 --> 00:01:45.187
Now, offhand, I don't know a function,

00:01:45.187 --> 00:01:46.707
especially a trigonometric function,

00:01:46.707 --> 00:01:48.187
because that was kind of our clue here,

00:01:48.187 --> 00:01:50.227
this pi over two makes me feel like

00:01:50.227 --> 00:01:53.667
this might be a trigonometric
function right over here

00:01:53.667 --> 00:01:55.847
Nothing jumps out at me offhand,

00:01:55.847 --> 00:01:59.747
but this does look suspiciously familiar.

00:01:59.747 --> 00:02:03.567
This looks awfully close
to the power series,

00:02:03.567 --> 00:02:05.927
or the Maclaurin Series for cosine of x.

00:02:05.927 --> 00:02:08.347
Which we have seen multiple times.

00:02:08.347 --> 00:02:10.387
Let's just remind
ourselves of what that is.

00:02:10.387 --> 00:02:11.607
And if this doesn't look familiar,

00:02:11.607 --> 00:02:14.447
the previous video where
I do the Maclaurin Series

00:02:14.447 --> 00:02:17.367
for cosine of x goes into
detail on how I get this.

00:02:17.367 --> 00:02:22.367
The Maclaurin Series for
cosine of x is equal to...

00:02:22.667 --> 00:02:23.627
So I'll just write a few terms,

00:02:23.627 --> 00:02:25.147
so I'll write approximately equal to

00:02:25.147 --> 00:02:29.567
one minus x squared over two factorial,

00:02:29.567 --> 00:02:34.567
plus x to the fourth over four factorial,

00:02:36.027 --> 00:02:40.847
minus x to the sixth over six factorial,

00:02:40.847 --> 00:02:43.527
and just like that you're
probably seeing the similarities.

00:02:43.527 --> 00:02:45.447
Well the first term is the same,

00:02:45.447 --> 00:02:47.447
the sign negative, positive,
negative, positive,

00:02:47.447 --> 00:02:49.127
negative, positive, negative, positive.

00:02:49.127 --> 00:02:51.887
two factorial, four
factorial, six factorial.

00:02:51.887 --> 00:02:54.667
The difference is the powers,

00:02:54.667 --> 00:02:56.387
the exponents on the x.

00:02:56.387 --> 00:02:58.327
This is x squared, this is x to the sixth.

00:02:58.327 --> 00:03:00.667
This is x to the fourth,
that's x to the 12th.

00:03:00.667 --> 00:03:03.847
This is x to the sixth,
that's x to the 18th.

00:03:05.154 --> 00:03:06.804
Well, what if we...

00:03:06.804 --> 00:03:09.328
I guess, something for
you to think about is,

00:03:09.328 --> 00:03:12.888
well, how can we replace
x with something here?

00:03:12.888 --> 00:03:14.028
Because anything that I change,

00:03:14.028 --> 00:03:15.568
if I take cosine of...

00:03:15.568 --> 00:03:18.028
if I change x to, I don't know, a plus b,

00:03:18.028 --> 00:03:20.308
everywhere we see an x, you'd
replace it with an a plus b.

00:03:20.308 --> 00:03:23.048
Can we put a power of x here,

00:03:23.048 --> 00:03:26.321
so that these things end up like that?

00:03:27.491 --> 00:03:30.742
well, this x to the
sixth is the same thing,

00:03:32.371 --> 00:03:35.892
as x to the third squared.

00:03:35.892 --> 00:03:37.852
That's x to the third squared.

00:03:37.852 --> 00:03:39.672
This right over here, x to the 12th,

00:03:39.672 --> 00:03:42.552
is the same thing as x to the
third to the fourth power.

00:03:42.552 --> 00:03:45.092
This right over here is the same thing as

00:03:45.092 --> 00:03:47.572
x to the third to the sixth power.

00:03:47.572 --> 00:03:51.292
So if we could replace each of
these xs with x to the thirds

00:03:51.292 --> 00:03:54.372
we will get this power series up here.

00:03:54.372 --> 00:03:55.292
Well, how do we do that?

00:03:55.292 --> 00:03:58.052
Well, we would just say the
cosine of x to the third

00:03:58.052 --> 00:04:00.021
and actually let me do
that in a different color.

00:04:01.052 --> 00:04:03.352
So the cosine, and that's
not a different color.

00:04:04.521 --> 00:04:09.072
So the cosine of x to the
third is going to be equal to,

00:04:09.963 --> 00:04:11.399
and once again, everywhere we see an x,

00:04:11.399 --> 00:04:12.536
we replace it with x to the third.

00:04:12.536 --> 00:04:14.936
So it's one minus, and actually,

00:04:14.936 --> 00:04:17.497
I'm just going to put in
parenthesis squared, two factorial.

00:04:17.497 --> 00:04:19.539
I wanted to do that in the green,

00:04:19.539 --> 00:04:21.572
let me do all this in the green.

00:04:21.572 --> 00:04:23.437
Alright, so it's gonna be equal to:

00:04:23.437 --> 00:04:27.617
one minus parenthesis
squared over two factorial

00:04:27.617 --> 00:04:31.917
plus parenthesis to the fourth
power over four factorial.

00:04:31.917 --> 00:04:36.777
minus parenthesis to the
sixth power over 6 factorial.

00:04:36.777 --> 00:04:39.697
And now let me change
back to that mauve color.

00:04:39.697 --> 00:04:42.117
And since I'm taking the
cosine of x to the third,

00:04:42.117 --> 00:04:44.537
well this is gonna be
x to the third squared.

00:04:44.537 --> 00:04:47.157
this is gonna be x to the
third to the fourth power.

00:04:47.157 --> 00:04:49.197
this is gonna be x to the
third to the sixth power.

00:04:49.197 --> 00:04:52.577
Which is exactly what
I have right over here.

00:04:52.577 --> 00:04:55.477
So this right over here
is the power series

00:04:55.477 --> 00:04:57.777
for cosine of x to the third.

00:04:57.777 --> 00:05:00.957
So evaluating this when x is equal to

00:05:00.957 --> 00:05:02.517
the cubed root of pi over two,

00:05:02.517 --> 00:05:04.477
is the same thing as evaluating this

00:05:04.477 --> 00:05:06.857
when x is equal to the
cubed root of pi over two.

00:05:06.857 --> 00:05:08.798
Let me write that down,
because this is interesting.

00:05:08.798 --> 00:05:13.798
So this, so I'll just rewrite
it, from n equals zero

00:05:14.861 --> 00:05:18.360
to infinity of negative one to the n

00:05:18.360 --> 00:05:23.360
of x to the sixth n over two n factorial.

00:05:23.840 --> 00:05:28.120
This is equal to, this is the
power series representation,

00:05:28.120 --> 00:05:32.160
of cosine of x to the third power.

00:05:33.144 --> 00:05:34.160
So if you want to evaluate this

00:05:34.160 --> 00:05:35.680
when x is the cubed root of pi over two,

00:05:35.680 --> 00:05:36.720
we just have to evaluate this

00:05:36.720 --> 00:05:38.240
when x is the cubed root of pi over two.

00:05:38.240 --> 00:05:40.943
And this does suspiciously
work out nicely,

00:05:40.943 --> 00:05:42.903
because if you take the
cube of the cubed root,

00:05:42.903 --> 00:05:44.403
well, good things happen.

00:05:44.403 --> 00:05:49.403
So the cosine of the
cubed root of pi over two

00:05:53.467 --> 00:05:55.663
to the third power, well
that's just the same thing

00:05:55.663 --> 00:05:58.063
as cosine of pi over two,

00:05:58.063 --> 00:06:03.063
which, of course, is equal to zero.

00:06:03.843 --> 00:06:07.843
And we are done.


Worked example: Taylor polynomial of derivative function

WEBVTT
Kind: captions
Language: en

00:00:00.981 --> 00:00:02.200
- So, let's say we've been given all

00:00:02.200 --> 00:00:04.272
this information about the function g

00:00:04.272 --> 00:00:07.036
and it's derivative
evaluated at x equals two.

00:00:07.036 --> 00:00:08.733
We know g of two is equal to three.

00:00:08.733 --> 00:00:10.807
G prime of two is equal to one.

00:00:10.807 --> 00:00:13.911
The second derivative of g
evaluated two is negative one.

00:00:13.911 --> 00:00:18.202
The third derivative of g
evaluated at two is two.

00:00:18.202 --> 00:00:21.811
Given that, what we're
being tasked with is

00:00:21.811 --> 00:00:26.600
we want to use the second
degree Taylor polynomial

00:00:26.600 --> 00:00:31.600
centered at x equals two to
approximate g prime of one.

00:00:33.807 --> 00:00:38.707
Not g of one, g prime of
one and so I encourage you

00:00:38.707 --> 00:00:43.707
to pause this video and try
to think about it on your own.

00:00:44.335 --> 00:00:46.408
I'm assuming you've had a go at it.

00:00:46.408 --> 00:00:50.275
Let's just remind ourselves what a second

00:00:50.275 --> 00:00:52.980
degree Taylor polynomial
centered at x equals two

00:00:52.980 --> 00:00:56.911
would look like for a
general function f of x.

00:00:57.137 --> 00:01:02.137
F of x would approximately be equal to,

00:01:03.592 --> 00:01:08.592
it would be f of two plus f prime of two

00:01:11.568 --> 00:01:16.568
times x minus two plus
f prime prime of two

00:01:19.177 --> 00:01:24.177
times x minus two squared, all
of that over two factorial.

00:01:26.279 --> 00:01:28.309
That would get us to a second degree place

00:01:28.309 --> 00:01:29.730
because it's x minus two squared.

00:01:29.730 --> 00:01:33.602
This is gonna give us a
second degree polynomial.

00:01:33.602 --> 00:01:34.730
This is the general case.

00:01:34.730 --> 00:01:38.811
If we want to find the
approximation for f centered

00:01:38.811 --> 00:01:42.705
at x equals but we're
gonna do it for g prime.

00:01:42.705 --> 00:01:44.412
Let me write this down.

00:01:44.412 --> 00:01:47.943
Alright, so I'll do it in blue.

00:01:48.271 --> 00:01:52.302
So, we have g prime of
x is what we're going

00:01:52.302 --> 00:01:53.913
to try and approximate and then we're

00:01:53.913 --> 00:01:56.772
going to evaluate it at x equals 1.

00:01:56.772 --> 00:02:00.642
G prime of x is going to
be approximately equal

00:02:00.642 --> 00:02:03.674
to, well same thing, it's
going to be the function

00:02:03.674 --> 00:02:05.244
that I'm going to try to approximate

00:02:05.244 --> 00:02:08.976
evaluated at two so g prime of two.

00:02:08.976 --> 00:02:11.079
Notice, so I'm approximate f of x,

00:02:11.079 --> 00:02:12.501
it's that function evaluated at two.

00:02:12.501 --> 00:02:14.306
If I'm approximating g prime of x,

00:02:14.306 --> 00:02:16.974
it's that function evaluated at two.

00:02:16.974 --> 00:02:20.171
Then, plus the first
derivative of this thing

00:02:20.171 --> 00:02:22.280
which is the second derivative of g.

00:02:22.280 --> 00:02:27.009
G prime prime of two times x minus two

00:02:27.009 --> 00:02:29.542
and then plus the second
derivative of the function

00:02:29.542 --> 00:02:32.242
that I'm trying to approximate
but the second derivative

00:02:32.242 --> 00:02:36.173
of g prime is going to be
the third derivative of g.

00:02:36.173 --> 00:02:41.173
It's going to be g prime prime
prime of two times x minus

00:02:42.334 --> 00:02:47.308
two squared, all of
that over two factorial.

00:02:47.308 --> 00:02:50.374
Now, they tell us what these things are.

00:02:52.775 --> 00:02:54.974
Let me use some new colors here.

00:02:54.974 --> 00:02:57.780
They tell us g of two is equal to three.

00:02:57.780 --> 00:02:59.577
This right here, oh, actually no,

00:02:59.577 --> 00:03:01.400
that's not what we're gonna wanna use.

00:03:01.400 --> 00:03:04.005
They tell us we're using g of two.

00:03:04.005 --> 00:03:07.642
They're telling us g prime
of two is equal to one.

00:03:07.642 --> 00:03:11.177
G prime of two, so this right
over here is equal to one.

00:03:11.177 --> 00:03:14.047
G prime prime of two is
equal to negative one.

00:03:14.047 --> 00:03:17.538
This is negative one right over here

00:03:17.538 --> 00:03:19.972
and then finally the third derivative

00:03:19.972 --> 00:03:23.642
of g evaluated at two is two.

00:03:24.670 --> 00:03:26.442
Two over two factorial, two factorials is

00:03:26.442 --> 00:03:28.611
two times one or it's just two.

00:03:28.611 --> 00:03:30.980
So that and that cancel out.

00:03:30.980 --> 00:03:33.478
What are we left with
for our approximation

00:03:33.478 --> 00:03:35.846
our second degree approximation of g

00:03:35.846 --> 00:03:38.311
prime of x centered at x equals two?

00:03:38.311 --> 00:03:43.311
We are left with g prime
of x is approximately

00:03:44.082 --> 00:03:49.082
equal to one minus x minus two.

00:03:53.473 --> 00:03:55.907
One minus x minus two,
I guess I could write

00:03:55.907 --> 00:04:00.907
that as plus two minus
x so plus two minus x.

00:04:02.475 --> 00:04:07.027
The negative of x minus
two is two minus x.

00:04:07.642 --> 00:04:12.169
Plus x minus two squared and obviously I

00:04:12.169 --> 00:04:13.467
could simplify this even more.

00:04:13.467 --> 00:04:18.467
This is three minus x plus x minus

00:04:18.933 --> 00:04:23.080
two squared and now I can evaluate it.

00:04:23.080 --> 00:04:26.280
If I wanna approximate g prime of one,

00:04:26.280 --> 00:04:30.505
I could say g prime of one
is going to be approximately

00:04:30.505 --> 00:04:32.244
equal to, wherever I see the x is here

00:04:32.244 --> 00:04:34.097
I put in a one there, so it's going

00:04:34.097 --> 00:04:39.097
to be three minus one plus
one minus two squared.

00:04:42.133 --> 00:04:44.911
Well, this is going to be two and then one

00:04:44.911 --> 00:04:47.546
minus two is negative one but then

00:04:47.546 --> 00:04:50.175
if I square it I get a positive one.

00:04:50.175 --> 00:04:51.742
So this whole thing is one.

00:04:51.742 --> 00:04:55.280
Two plus one is equal to three.

00:04:55.280 --> 00:04:59.734
Once again, this is an
approximation for g prime of one.

00:04:59.734 --> 00:05:01.138
What did we do here?

00:05:01.138 --> 00:05:02.566
We found the Taylor series.

00:05:02.566 --> 00:05:05.139
The second degree Taylor
series approximation

00:05:05.139 --> 00:05:08.713
for g prime of x centered
around x equals two

00:05:08.713 --> 00:05:11.242
and then we evaluated that approximation

00:05:11.242 --> 00:05:16.242
at x equals one to
approximate g prime of one.

00:05:17.668 --> 00:05:21.240
Anyway, hopefully, you found that fun.