Recognizing binomial variables

WEBVTT
Kind: captions
Language: en

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- [Instructor] What we're going to do

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in this video is get
some practice classifying

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whether a random variable
is a binomial variable,

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and we're gonna do it by
looking at a few exercises

00:00:10.120 --> 00:00:11.518
from Khan Academy.

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So this says a manager
oversees 11 female employees

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and nine male employees.

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They need to pick three of these employees

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to go on a business trip,

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so the manager places
all 20 names in a hat

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and chooses at random.

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Let X equal the number of
female employees chosen.

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So they're going to do three trials.

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And on each of those trials,

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you could say success is if
they pick a female employee.

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And then the random variable X is

00:00:38.752 --> 00:00:41.578
the number of females out of those three.

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Is X a binomial variable?

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Why or why not?

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So pause this video,

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and see if you can work
through this on your own.

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All right, now let's
go through each choice.

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Choice A says each trial
isn't being classified

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as a success or failure, so
X is not a binomial variable.

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I disagree with this.

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Each trial is being classified
as a success or failure.

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It's either going to be female or not.

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And since we're counting the
number of female employees,

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if in a trial, we pick a
female, that would be a success,

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so each trial is being classified
as a success or failure.

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So this one over here isn't true.

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There is no fixed number of trials,

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so X is not a binomial variable.

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There is a fixed number of trials.

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They're doing three trials.

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They're picking three names out of a hat.

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The trials are not independent,

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so X is not a binomial variable.

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So this is interesting.

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So, for example, trial one,

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what's the probability of success?

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Well, there are 20 employees,
20 names in the hat,

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and 11 of the outcomes would be success,

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so you have 11/20 probability of success.

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But in trial two,

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what's the probability of success

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given success,

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success in, I'll say trial one, T one?

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Well, if you succeeded in trial one,

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that means that there's now
only 10 female names in the hat

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out of 19.

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And if you don't have
success in trial one,

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then it'll be 11 out of 19.

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So your probability does change
based on previous outcomes.

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And so the trials are not independent,

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and so X is not a binomial variable.

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So this one is true.

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The trials are not independent,

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so that violates that condition

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for being a binomial variable.

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In order to be a binomial variable,

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all your trials have to be
independent of each other.

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And so we'd rule this last one out

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because this last one says that

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X has a binomial distribution
or is or does meet

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all the conditions for
being a binomial variable.

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Let's do another example.

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So here we have different scenarios,

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and I have the conditions

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for a binomial variable
written right over here.

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And so once again, pause the video,

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and look at each of these
scenarios for random variables.

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And look at these conditions,

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and think about whether these

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random variables are binomial or not.

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So let's look at the first one.

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In a game involving a standard
deck of 52 playing cards,

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an individual randomly draws seven cards

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without replacement.

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Let Y be equal to the
number of aces drawn.

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Well, in our introductory
video to binomial variables,

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we talked about, if we're
doing without replacement,

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your probability of getting an ace

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on a given trial,

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where a trial is you're
taking a card out of the deck,

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it's going to be dependent

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on whether you got aces
in previous trials.

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Because if you got an
ace in a previous trial,

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well, that ace,

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then you're gonna have
fewer aces in the deck.

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So the trials, in this
case, are not independent,

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not independent,

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not independent trials.

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Now, on the other hand,

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if on every trial you looked
at whatever card you got

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and put it back in the deck,

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then they would be independent trials.

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The probability of getting an ace

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on each trial would be the same,

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but not when you have without replacement.

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So this is

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not binomial right over here

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because you don't have independent trials.

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The second scenario,

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60% of a certain species of tomato

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live after transplanting
from pot to garden.

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Eli transplants 16 of these tomato plants.

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Assume that the plants live
independently of each other.

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So whether one plant lives isn't dependent

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on whether another plant lives.

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Let T equal the number of
tomato plants that live.

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All right, so let's
look at the conditions.

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The outcome of each
trial can be classified

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as either success or failure.

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So each trial over here is
one of the tomato plants,

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and we have 16 of those trials.

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And success is if the tomato plant lives,

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and failure is if it dies.

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So we have either success or failure.

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Each trial is independent of the others.

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They tell us, the plants live
independently of each other.

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So whether or not a neighboring
plants lives or dies

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doesn't affect whether the
plant next to it lives or dies.

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So each trial is
independent of the others.

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There is a fixed number of trials.

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Yes, we have 16 right over there.

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The probability p of success
on each trial remains constant.

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Well, yeah, according to
at least the scenario,

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they're saying that we have a 60% chance

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for each tomato plant,
which is each trial.

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So it meets all of the
conditions right over here,

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so this one is binomial.

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Now let's look at this third scenario.

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In a game of luck, a turn
consists of a player continuing

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to roll a pair of six-sided die

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until they roll a double,

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two of the same face values.

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Let X equal the number
of rolls in one turn.

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So you're gonna keep rolling
until they roll a double.

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Well, the thing that jumps out at me is

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that you don't have a
fixed number of trials,

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not fixed

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number of trials.

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You could say each trial,
each roll is a trial.

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Success is getting a double,
which has a fixed probability.

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Whether or not you get
a double on each trial

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is gonna be independent
of the previous roll.

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So it meets all the other constraints,

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but it does not meet that
there's a fixed number of trials.

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You're gonna keep, someone,

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there's some chance you might have to roll

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20 times or 200 times

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or who knows however many
times until they roll a double.

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And so this violates there
is a fixed number of trials.

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So since there is not a
fixed number of trials,

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this variable X, not binomial.


Capital vs. consumer goods and economic growth

WEBVTT
Kind: captions
Language: en

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- [Instructor] We've
learned a little bit already

00:00:01.881 --> 00:00:04.618
about how a production
possibilities curve can be used

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to illustrate the concept
of economic growth.

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Let's review the definition
of economic growth.

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Then we're gonna go on to some more depth

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about the tradeoffs
that society faces today

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between the production of
different types of goods

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and how that can affect the level

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of economic growth in the future.

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You'll recall that
economic growth is defined

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as an increase in the ability of a nation

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to produce goods and services over time.

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You've already learned that,

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in a production possibilities curve model,

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economic growth can be
illustrated as a shift outward

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in the nation's PPC over time.

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In the graph on our left here,

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economic growth would be shown

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as a shift from the white curve

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to the pink curve.

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This represents an increase
in the ability of a nation

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to produce both the goods
shown on its PPC over time.

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The question we want to discuss is

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what are the sources of economic growth,

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and how does a nation's decision about

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how to allocate its resources today

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affect its level of potential
output in the future?

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To help us answer this question,

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we're going to put some different goods

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on our production
possibilities curve here.

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And for today's lesson,

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we'll be looking at two types of goods.

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The rubber ducky on the vertical axis

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represents what we're going
to call consumer goods.

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We're gonna define these in just a moment,

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but let's look down at
the horizontal axis first.

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The tractor here, it's a tool.

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This is a good that is used
to produce other goods.

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It's used to produce
agricultural products,

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which could be used to produce all sorts

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of goods a society needs.

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A tractor and any other
tools like tractors

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are what we call capital goods.

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Let's define consumer
goods and capital goods

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before moving on with our analysis.

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Our rubber ducky in our
PPC is a consumer good.

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This is a good intended
to be sold to households.

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Consumer goods are used for consumption,

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and ultimately consumer
goods are disposed of.

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How does this contrast
with capital goods though?

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Capital goods are a bit different.

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Capital goods are any goods

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sold to firms.

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Capital goods, unlike consumer goods,

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are used to produce other goods.

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Therefore, if society
allocates more resources

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towards capital goods today,

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society's essentially choosing

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to produce more goods and
services in the future

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and the tradeoff being less goods

00:02:32.048 --> 00:02:34.228
and services produced today.

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So let's look back at our white PPC here.

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Let's assume that this represents today's

00:02:38.573 --> 00:02:41.631
production possibilities
for a hypothetical country.

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A country can choose to produce
more consumer goods today.

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By doing so, it would
be producing at a point

00:02:48.089 --> 00:02:50.022
such as point A.

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Notice that at point A,

00:02:51.685 --> 00:02:54.038
country is producing more rubber duckies,

00:02:54.038 --> 00:02:56.971
but the opportunity cost
is the capital goods

00:02:56.971 --> 00:03:00.061
of the tools that it could
be producing instead.

00:03:00.061 --> 00:03:02.827
Or society can choose to
allocate its resources

00:03:02.827 --> 00:03:04.809
at a point such as point B.

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Allocating its resources
at point B means society

00:03:08.454 --> 00:03:11.906
is choosing to produce
more capital goods today

00:03:11.906 --> 00:03:16.008
with the opportunity cost
being fewer consumer goods.

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Notice that if a country
were to move from point A

00:03:20.866 --> 00:03:22.158
to point B,

00:03:22.158 --> 00:03:25.526
it would give up all these rubber duckies,

00:03:25.526 --> 00:03:29.091
all the toys, all the consumer goods

00:03:29.091 --> 00:03:30.723
that it could have
produced if it had chosen

00:03:30.723 --> 00:03:32.593
to produce at point A instead.

00:03:32.593 --> 00:03:37.347
So this is the opportunity
cost of producing at point B.

00:03:37.347 --> 00:03:40.464
Now, what if society moved
in the other direction

00:03:40.464 --> 00:03:44.552
and chose to produce more
consumer goods today?

00:03:44.552 --> 00:03:46.797
What if society went from point B

00:03:46.797 --> 00:03:48.295
to point A, in other words?

00:03:48.295 --> 00:03:49.128
Well, in that case,

00:03:49.128 --> 00:03:51.861
the opportunity cost
would be the capital goods

00:03:51.861 --> 00:03:53.472
that it could have enjoyed

00:03:53.472 --> 00:03:56.927
if it had chosen to produce at point B.

00:03:56.927 --> 00:03:57.970
The PPC, once again,

00:03:57.970 --> 00:04:00.623
illustrates this very basic
concept of opportunity cost.

00:04:00.623 --> 00:04:03.947
It shows us what is given up
in order to have anything.

00:04:03.947 --> 00:04:08.414
So let's do some analysis
over here of the consequences

00:04:08.414 --> 00:04:12.491
of this country choosing
to produce at point A or

00:04:12.491 --> 00:04:15.164
choosing to produce at point B.

00:04:15.164 --> 00:04:18.049
What happens if society chooses
to allocate more resources

00:04:18.049 --> 00:04:20.480
towards consumer goods today?

00:04:20.480 --> 00:04:25.022
Essentially, what it's
choosing is current consumption

00:04:25.022 --> 00:04:26.772
of goods and services

00:04:27.895 --> 00:04:28.728
over

00:04:29.955 --> 00:04:31.711
future consumption.

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What do I mean by this?

00:04:37.054 --> 00:04:40.434
Essentially, by producing at point A,

00:04:40.434 --> 00:04:44.455
society is giving up
all those capital goods

00:04:44.455 --> 00:04:45.943
that could have been used

00:04:45.943 --> 00:04:48.914
to produce other goods in the future.

00:04:48.914 --> 00:04:50.393
Therefore, society is choosing

00:04:50.393 --> 00:04:52.718
a higher standard of living today,

00:04:52.718 --> 00:04:54.719
rather than a higher standard of living

00:04:54.719 --> 00:04:56.574
sometime in the future.

00:04:56.574 --> 00:04:58.957
Let's contrast that with point B

00:04:58.957 --> 00:05:01.117
on our production possibilities curve.

00:05:01.117 --> 00:05:03.408
If society chooses to
allocate more resources

00:05:03.408 --> 00:05:05.283
towards capital goods today,

00:05:05.283 --> 00:05:08.450
society is choosing future consumption

00:05:11.775 --> 00:05:12.608
over

00:05:14.205 --> 00:05:15.872
current consumption.

00:05:18.716 --> 00:05:20.340
What do we mean by that?

00:05:20.340 --> 00:05:22.506
By allocating more of its scarce resources

00:05:22.506 --> 00:05:25.991
towards capital goods and tools
and equipment and technology

00:05:25.991 --> 00:05:28.700
that can be used to
produce other goods today,

00:05:28.700 --> 00:05:31.618
society is giving up
the current consumption

00:05:31.618 --> 00:05:33.935
that it could enjoy by producing more toys

00:05:33.935 --> 00:05:35.797
or consumer goods today.

00:05:35.797 --> 00:05:37.271
The benefit though,

00:05:37.271 --> 00:05:40.525
the benefit of producing
more capital goods today

00:05:40.525 --> 00:05:42.108
is likely to be higher rates

00:05:42.108 --> 00:05:44.307
of economic growth in the future.

00:05:44.307 --> 00:05:45.759
Why is that the case?

00:05:45.759 --> 00:05:47.858
Because capital goods are those things

00:05:47.858 --> 00:05:50.079
which are used to produce other goods.

00:05:50.079 --> 00:05:53.057
Capital is a factor of production.

00:05:53.057 --> 00:05:54.793
Consumer goods are not
a factor of production.

00:05:54.793 --> 00:05:56.650
Consumer goods are fun to consume.

00:05:56.650 --> 00:05:57.672
They're fun to enjoy.

00:05:57.672 --> 00:05:59.330
We love playing with our toys,

00:05:59.330 --> 00:06:02.236
but ultimately they get
consumed and thrown away.

00:06:02.236 --> 00:06:04.632
Capital goods aren't quickly disposed of.

00:06:04.632 --> 00:06:06.785
They instead are used in factories.

00:06:06.785 --> 00:06:08.100
They're used in the fields.

00:06:08.100 --> 00:06:09.359
They're used in the mines.

00:06:09.359 --> 00:06:11.124
They're used in all the different ways

00:06:11.124 --> 00:06:13.301
that we produce goods and services

00:06:13.301 --> 00:06:15.575
that society benefits from.

00:06:15.575 --> 00:06:18.697
By allocating more resources
towards capital goods today,

00:06:18.697 --> 00:06:21.782
a country would be
choosing future consumption

00:06:21.782 --> 00:06:23.768
over current consumption.

00:06:23.768 --> 00:06:25.300
Now, to illustrate this concept,

00:06:25.300 --> 00:06:27.975
we can look at some real-world data.

00:06:27.975 --> 00:06:31.243
What I'm about to show you is
some data for two countries

00:06:31.243 --> 00:06:33.369
that shows how those
countries are choosing

00:06:33.369 --> 00:06:35.556
to allocate their resources today.

00:06:35.556 --> 00:06:38.389
We can then compare the
rates of economic growth

00:06:38.389 --> 00:06:42.219
over the last 15 years to show
that one of those countries

00:06:42.219 --> 00:06:46.279
has, in fact, experienced
higher rates of economic growth.

00:06:46.279 --> 00:06:48.611
Let's look at our data first.

00:06:48.611 --> 00:06:52.950
Here we've got the output
data of two countries,

00:06:52.950 --> 00:06:56.676
China on the left and the
United States on the right.

00:06:56.676 --> 00:06:59.067
What can we learn from this information?

00:06:59.067 --> 00:07:00.506
First, let's look at China.

00:07:00.506 --> 00:07:04.554
Notice that, in 2016, the year
for which this data applies,

00:07:04.554 --> 00:07:06.471
China invested 43%

00:07:07.638 --> 00:07:09.838
of its entire GDP.

00:07:09.838 --> 00:07:10.671
What does that mean?

00:07:10.671 --> 00:07:12.757
It means that 43.7 dollars

00:07:12.757 --> 00:07:15.366
out of every 100 dollars spent in China

00:07:15.366 --> 00:07:17.616
was spent on capital goods.

00:07:18.667 --> 00:07:22.608
Compare that to the level
of consumption in China.

00:07:22.608 --> 00:07:23.691
Only 37%

00:07:25.011 --> 00:07:26.538
of China's total spending

00:07:26.538 --> 00:07:29.576
went towards household consumption.

00:07:29.576 --> 00:07:31.879
China is choosing to allocate
more of its resources

00:07:31.879 --> 00:07:33.609
towards the production of capital goods

00:07:33.609 --> 00:07:36.772
than it is towards the
production of consumer goods.

00:07:36.772 --> 00:07:38.635
Let's look over at the United States.

00:07:38.635 --> 00:07:41.797
In the United States in 2016,

00:07:41.797 --> 00:07:43.047
only 15.9%

00:07:44.989 --> 00:07:48.237
of total spending went
towards capital goods.

00:07:48.237 --> 00:07:51.418
On the other hand, 68.6%

00:07:51.418 --> 00:07:54.172
of America's total spending went towards,

00:07:54.172 --> 00:07:56.589
you guessed it, household consumption.

00:07:56.589 --> 00:08:00.485
America has essentially chosen
rubber duckies over tractors

00:08:00.485 --> 00:08:03.830
or consumer goods over capital goods.

00:08:03.830 --> 00:08:05.437
So this is one data point.

00:08:05.437 --> 00:08:07.205
It's not the only thing
that affects the rates

00:08:07.205 --> 00:08:08.784
of economic growth in a country.

00:08:08.784 --> 00:08:11.360
But if we look at the actual
rates of economic growth

00:08:11.360 --> 00:08:15.566
since the year 2000 for
China and the United States,

00:08:15.566 --> 00:08:18.321
an interesting fact emerges.

00:08:18.321 --> 00:08:21.277
China has enjoyed
significantly higher rates

00:08:21.277 --> 00:08:22.924
of economic growth.

00:08:22.924 --> 00:08:24.637
The blue line here

00:08:24.637 --> 00:08:28.908
represents China's economic
growth rates from 2000

00:08:28.908 --> 00:08:29.741
to 2015.

00:08:31.510 --> 00:08:33.383
And we can see that
China has average rates

00:08:33.383 --> 00:08:36.466
of economic growth of between 8%

00:08:38.229 --> 00:08:40.601
and 14% per year.

00:08:40.601 --> 00:08:43.739
That means that every year,
China's economy has produced

00:08:43.739 --> 00:08:48.565
approximately 10% more stuff
than it did the year before.

00:08:48.565 --> 00:08:50.855
Compare that to the United States.

00:08:50.855 --> 00:08:54.296
In the United States, growth
rates have averaged between

00:08:54.296 --> 00:08:58.698
negative 2% and 4%, significantly lower.

00:08:58.698 --> 00:09:00.979
The United States has
achieved economic growth

00:09:00.979 --> 00:09:02.862
for most of the last 15 years,

00:09:02.862 --> 00:09:04.684
but the rate of economic growth

00:09:04.684 --> 00:09:06.326
was quite a bit lower than China.

00:09:06.326 --> 00:09:09.320
China has, of course, a
much larger population.

00:09:09.320 --> 00:09:11.663
Also, China is at a lower level

00:09:11.663 --> 00:09:13.487
of economic development
than the United States,

00:09:13.487 --> 00:09:14.747
so it has more room to grow.

00:09:14.747 --> 00:09:16.709
There are more unused resources in China.

00:09:16.709 --> 00:09:18.492
But that aside,

00:09:18.492 --> 00:09:21.244
China also chooses to
allocate more of its resources

00:09:21.244 --> 00:09:23.497
towards investment in capital goods

00:09:23.497 --> 00:09:26.752
than it does production of
consumer goods for households.

00:09:26.752 --> 00:09:30.893
This fact, going back to our
production possibilities curve,

00:09:30.893 --> 00:09:33.370
supports the idea that by choosing

00:09:33.370 --> 00:09:35.675
to produce more capital goods today,

00:09:35.675 --> 00:09:38.343
a country is essentially
choosing future consumption

00:09:38.343 --> 00:09:39.922
over current consumption.

00:09:39.922 --> 00:09:41.639
Economic growth has many causes.

00:09:41.639 --> 00:09:43.452
Economists have been discussing

00:09:43.452 --> 00:09:46.386
the source of economic growth
for over 200 years now.

00:09:46.386 --> 00:09:48.014
Increases in the population,

00:09:48.014 --> 00:09:50.134
increases in the education level,

00:09:50.134 --> 00:09:52.320
the skill level, the amount of technology,

00:09:52.320 --> 00:09:54.452
and the availability of resources

00:09:54.452 --> 00:09:57.206
are significant sources
of economic growth.

00:09:57.206 --> 00:09:59.114
However, a country's decision

00:09:59.114 --> 00:10:01.353
as to how it allocates its resources today

00:10:01.353 --> 00:10:03.668
also impacts its rate of economic growth.


Comparative advantage - input approach

WEBVTT
Kind: captions
Language: en

00:00:00.485 --> 00:00:01.662
- [Instructor] We just
finished demonstrating

00:00:01.662 --> 00:00:03.307
how to calculate opportunity costs

00:00:03.307 --> 00:00:05.289
and determine who has
the comparative advantage

00:00:05.289 --> 00:00:07.383
in a good's production
using the data provided

00:00:07.383 --> 00:00:10.798
in an output table or a
production possibilities curve.

00:00:10.798 --> 00:00:13.737
In that video we had a table
showing the potential output

00:00:13.737 --> 00:00:14.990
of two goods that two countries

00:00:14.990 --> 00:00:18.283
could produce with a
fixed amount of inputs.

00:00:18.283 --> 00:00:20.444
In this video we're gonna
look at how to calculate

00:00:20.444 --> 00:00:21.691
opportunity cost and determine

00:00:21.691 --> 00:00:22.825
who has a comparative advantage

00:00:22.825 --> 00:00:23.931
of two goods' production

00:00:23.931 --> 00:00:26.864
using the information from
what we call an input table.

00:00:26.864 --> 00:00:27.936
Let's look at this table and discuss

00:00:27.936 --> 00:00:30.209
how it differs from the
information provided

00:00:30.209 --> 00:00:32.161
in the table in the previous video.

00:00:32.161 --> 00:00:33.539
Notice that in this table

00:00:33.539 --> 00:00:35.724
we are given not the number of units

00:00:35.724 --> 00:00:37.875
of output that the two
countries can produce,

00:00:37.875 --> 00:00:39.762
rather we are given the
number of workers needed

00:00:39.762 --> 00:00:42.452
to produce a single unit of output.

00:00:42.452 --> 00:00:44.003
So, the variable in this table

00:00:44.003 --> 00:00:48.170
is the number of inputs needed
to produce 1,000 watermelons

00:00:49.472 --> 00:00:51.624
or a single bicycle.

00:00:51.624 --> 00:00:53.221
With this information we can calculate

00:00:53.221 --> 00:00:55.675
the opportunity costs of
watermelons and bicycles

00:00:55.675 --> 00:00:56.617
in these two countries

00:00:56.617 --> 00:00:58.547
and determine who has
the comparative advantage

00:00:58.547 --> 00:01:00.031
in the two goods' production.

00:01:00.031 --> 00:01:02.928
Let's start with watermelons in country X.

00:01:02.928 --> 00:01:04.430
I find it very helpful to tell a story

00:01:04.430 --> 00:01:06.142
when calculating opportunity cost

00:01:06.142 --> 00:01:08.333
using data from an input table.

00:01:08.333 --> 00:01:11.496
The four workers needed to
produce a single watermelon

00:01:11.496 --> 00:01:14.203
could have produced how many bicycles?

00:01:14.203 --> 00:01:17.595
Four workers could have only produced 4/12

00:01:17.595 --> 00:01:18.595
of a bicycle

00:01:19.657 --> 00:01:21.127
because you would have needed three times

00:01:21.127 --> 00:01:22.632
that many workers in order to produce

00:01:22.632 --> 00:01:23.957
a single bicycle.

00:01:23.957 --> 00:01:26.109
That means that per 1,000 watermelons,

00:01:26.109 --> 00:01:30.669
this country's giving up
only 1/3 of the bicycle.

00:01:30.669 --> 00:01:32.443
Now, I'm gonna go ahead
and convert that once again

00:01:32.443 --> 00:01:35.068
to a decimal to make it easier to compare

00:01:35.068 --> 00:01:37.157
opportunity cost between
the two countries,

00:01:37.157 --> 00:01:41.428
so 1/3 of a bicycle means
that the opportunity cost

00:01:41.428 --> 00:01:45.345
of watermelons is 0.33
bicycles per watermelon.

00:01:46.368 --> 00:01:49.451
Four bicycles, 12 workers are needed.

00:01:50.412 --> 00:01:52.498
The question is how many watermelons

00:01:52.498 --> 00:01:55.169
could those 12 workers have produced?

00:01:55.169 --> 00:01:58.568
The answer to that is 12 divided by four

00:01:58.568 --> 00:02:00.498
because three times as
many workers are needed

00:02:00.498 --> 00:02:01.764
to produce a single bicycle

00:02:01.764 --> 00:02:04.318
as are needed to produce
1,000 watermelons.

00:02:04.318 --> 00:02:06.963
So, the opportunity cost of
bicycles in terms of watermelons

00:02:06.963 --> 00:02:10.782
is what could have been
produced with those 12 workers

00:02:10.782 --> 00:02:12.998
which is three watermelons

00:02:12.998 --> 00:02:15.331
or in fact, 3,000 watermelons,

00:02:15.331 --> 00:02:16.998
so say a 1,000 here.

00:02:19.882 --> 00:02:23.135
3,000 watermelons are given
up for every bicycle produced

00:02:23.135 --> 00:02:25.748
because the 12 workers
needed to produce a bicycle

00:02:25.748 --> 00:02:29.770
could have produced 12
over four watermelons.

00:02:29.770 --> 00:02:31.874
Let's do a similar method to
calculate the opportunity cost

00:02:31.874 --> 00:02:35.291
of watermelons and bicycles in country Y.

00:02:36.220 --> 00:02:39.913
The six workers needed to
produce 1,000 watermelons,

00:02:39.913 --> 00:02:42.734
how many bicycles could
they have produced?

00:02:42.734 --> 00:02:45.057
Well, they would have needed 24 workers

00:02:45.057 --> 00:02:46.848
to produce a single bicycle,

00:02:46.848 --> 00:02:50.515
therefore in country
Y, six over 24 bicycles

00:02:51.532 --> 00:02:55.259
are given up in order to
produce 1,000 watermelons,

00:02:55.259 --> 00:02:59.701
that is 1/4 bicycle per watermelon,

00:02:59.701 --> 00:03:00.920
convert that to a decimal,

00:03:00.920 --> 00:03:03.837
I get 0.25 bicycles per watermelon.

00:03:05.049 --> 00:03:06.848
How about bicycles?

00:03:06.848 --> 00:03:08.387
What's the opportunity cost of bicycles?

00:03:08.387 --> 00:03:12.810
How many watermelons could
those 24 workers have produced?

00:03:12.810 --> 00:03:14.135
Well, only six workers are needed

00:03:14.135 --> 00:03:16.140
to produce 1,000 watermelons,

00:03:16.140 --> 00:03:18.556
so the opportunity cost of each bicycle

00:03:18.556 --> 00:03:23.066
is the 4,000 watermelons
that could have been produced

00:03:23.066 --> 00:03:27.085
using the 24 workers needed
to produce a single bicycle.

00:03:27.085 --> 00:03:29.733
We now have our opportunity costs

00:03:29.733 --> 00:03:32.957
of bicycles and watermelons
in these two countries.

00:03:32.957 --> 00:03:33.790
The next thing we need to do

00:03:33.790 --> 00:03:36.427
is simply determine who has
the lower opportunity cost

00:03:36.427 --> 00:03:38.329
for watermelons and who has
the lower opportunity cost

00:03:38.329 --> 00:03:39.842
for bicycles to determine

00:03:39.842 --> 00:03:41.880
who should specialize
based on the principle

00:03:41.880 --> 00:03:43.319
of comparative advantage.

00:03:43.319 --> 00:03:45.032
Let's start with watermelons.

00:03:45.032 --> 00:03:47.110
You can see right away
that a single watermelon

00:03:47.110 --> 00:03:50.110
costs less of a bicycle in country Y

00:03:51.974 --> 00:03:53.410
than it does in country X.

00:03:53.410 --> 00:03:57.705
Only 1/4 or 0.25 bicycles are given up

00:03:57.705 --> 00:04:00.264
compared to 0.33 in country X.

00:04:00.264 --> 00:04:02.298
That gives country Y a
comparative advantage

00:04:02.298 --> 00:04:04.078
in watermelon production

00:04:04.078 --> 00:04:06.914
and yes, predictably, country X

00:04:06.914 --> 00:04:08.488
therefore has a comparative advantage

00:04:08.488 --> 00:04:10.321
in bicycle production.

00:04:11.202 --> 00:04:13.666
Country X gives up only 3,000 watermelons

00:04:13.666 --> 00:04:15.128
for every bicycle it produces

00:04:15.128 --> 00:04:17.547
whereas country Y gives
up 4,000 watermelons

00:04:17.547 --> 00:04:19.492
for every bicycle it produces.

00:04:19.492 --> 00:04:20.495
Based on these calculations

00:04:20.495 --> 00:04:22.648
we can come to the following conclusions.

00:04:22.648 --> 00:04:26.815
Country X should specialize
in the production of bicycles

00:04:28.762 --> 00:04:30.969
because it has the lower opportunity cost

00:04:30.969 --> 00:04:33.129
compared to country Y.

00:04:33.129 --> 00:04:34.379
Next, country Y

00:04:36.342 --> 00:04:39.390
should specialize in the
production of watermelons

00:04:39.390 --> 00:04:41.066
since it can produce watermelons

00:04:41.066 --> 00:04:45.841
at a lower opportunity cost of
0.25 bicycles per watermelon

00:04:45.841 --> 00:04:49.574
compared to country X which
must give up 0.33 bicycles

00:04:49.574 --> 00:04:51.673
per watermelon, so how
do the two countries

00:04:51.673 --> 00:04:53.552
get the goods that they do not produce?

00:04:53.552 --> 00:04:56.497
Domestically they should
trade with one another

00:04:56.497 --> 00:04:58.387
and through trade both countries

00:04:58.387 --> 00:05:01.386
can enjoy the good that they
are not producing domestically

00:05:01.386 --> 00:05:02.592
at a lower opportunity cost

00:05:02.592 --> 00:05:05.282
than they could have
produced it at domestically.

00:05:05.282 --> 00:05:07.292
To review, the difference
between what we called

00:05:07.292 --> 00:05:10.034
an input table and what
we called an output table

00:05:10.034 --> 00:05:11.345
is that the variable given to us

00:05:11.345 --> 00:05:14.465
is not the number of units of output

00:05:14.465 --> 00:05:15.399
that can be produced,

00:05:15.399 --> 00:05:17.432
rather it is the number of units

00:05:17.432 --> 00:05:20.729
of inputs needed to produce
a single unit of output.

00:05:20.729 --> 00:05:23.391
The input in this table
was the number of workers

00:05:23.391 --> 00:05:25.102
needed to grow 1,000 watermelons

00:05:25.102 --> 00:05:26.740
and the number of
workers needed to produce

00:05:26.740 --> 00:05:27.941
a single bicycle.

00:05:27.941 --> 00:05:29.274
We can calculate opportunity cost

00:05:29.274 --> 00:05:31.337
and determine who has
comparative advantage

00:05:31.337 --> 00:05:35.432
of two goods' productions from
the data in an input table.


Comparative advantage - output approach

WEBVTT
Kind: captions
Language: en

00:00:00.521 --> 00:00:01.639
- [Narrator] In this and the next video

00:00:01.639 --> 00:00:03.699
we're gonna learn how to
calculate opportunity cost

00:00:03.699 --> 00:00:05.962
and determine who has
the comparative advantage

00:00:05.962 --> 00:00:07.894
in the goods production using data

00:00:07.894 --> 00:00:11.493
from both an output
table and an input table.

00:00:11.493 --> 00:00:13.105
If we look at our PPC's in the graph

00:00:13.105 --> 00:00:14.934
on the left, we see the potential outputs

00:00:14.934 --> 00:00:17.759
of two goods, shoes and basketballs

00:00:17.759 --> 00:00:19.750
for two different countries, country A

00:00:19.750 --> 00:00:22.337
in blue and country B in green.

00:00:22.337 --> 00:00:24.721
Notice that the data on this PPC

00:00:24.721 --> 00:00:27.491
is also represented in
the table on the right.

00:00:27.491 --> 00:00:31.183
Country A can produce up to
six pairs of shoes per worker

00:00:31.183 --> 00:00:34.287
or up to eight basketballs per worker.

00:00:34.287 --> 00:00:35.689
Contrast that with country B which

00:00:35.689 --> 00:00:37.301
can produce up to four pairs of shoes

00:00:37.301 --> 00:00:40.682
or four basketballs with a single worker.

00:00:40.682 --> 00:00:43.359
Notice that in this table, the variable,

00:00:43.359 --> 00:00:44.845
the data that appears in the table

00:00:44.845 --> 00:00:46.326
is the output.

00:00:46.326 --> 00:00:48.686
This is the potential
output of the two goods

00:00:48.686 --> 00:00:49.928
that the country can produce with

00:00:49.928 --> 00:00:52.721
a fixed amount of inputs.

00:00:52.721 --> 00:00:56.554
Each worker in other
words, workers are input,

00:00:57.559 --> 00:00:59.771
each worker can produce the numbers

00:00:59.771 --> 00:01:02.969
of basketballs and shoes
indicated in the table.

00:01:02.969 --> 00:01:04.638
To calculate the opportunity cost

00:01:04.638 --> 00:01:07.178
of shoes and basketballs
in these two countries,

00:01:07.178 --> 00:01:09.558
we can use the following method.

00:01:09.558 --> 00:01:12.135
We know that for every six pairs of shoes,

00:01:12.135 --> 00:01:15.136
I'll write six s, country A produces,

00:01:15.136 --> 00:01:17.636
it gives up eight basketballs.

00:01:18.527 --> 00:01:20.219
The resources needed to produce six shoes

00:01:20.219 --> 00:01:22.656
could also produce eight basketballs.

00:01:22.656 --> 00:01:24.325
To find the opportunity cost of shoes

00:01:24.325 --> 00:01:26.461
all I have to do is divide both sides

00:01:26.461 --> 00:01:29.442
of this equation by six and I can see

00:01:29.442 --> 00:01:32.109
that the opportunity
cost of one pair of shoes

00:01:32.109 --> 00:01:36.276
is 8/6 or 4/3 of a basketball.

00:01:38.010 --> 00:01:42.263
Now I like to convert
these fractions to decimals

00:01:42.263 --> 00:01:44.113
because it's easier to compare decimals

00:01:44.113 --> 00:01:46.702
against one another than
it is to compare fractions

00:01:46.702 --> 00:01:49.356
which may possibly have
different denominators.

00:01:49.356 --> 00:01:52.900
I'm gonna go ahead and
convert this 4/3 to a decimal

00:01:52.900 --> 00:01:54.152
and that gives me an opportunity cost

00:01:54.152 --> 00:01:57.235
of 1.33 basketballs per pair of shoes

00:01:58.788 --> 00:02:00.698
produced in country A.

00:02:00.698 --> 00:02:01.849
Let's now calculate the opportunity cost

00:02:01.849 --> 00:02:04.029
of basketballs in terms of shoes.

00:02:04.029 --> 00:02:06.146
Country A can produce
either eight basketballs

00:02:06.146 --> 00:02:07.229
or six shoes.

00:02:08.932 --> 00:02:11.118
To find the cost of
basketballs I just divide

00:02:11.118 --> 00:02:13.588
both sides by eight here.

00:02:13.588 --> 00:02:16.266
I get an opportunity cost of 3/4 of

00:02:16.266 --> 00:02:19.293
a pair of shoes per basketball.

00:02:19.293 --> 00:02:21.296
Again I'm gonna convert this to a decimal

00:02:21.296 --> 00:02:23.238
to make it easier to
compare opportunity costs

00:02:23.238 --> 00:02:25.090
across countries.

00:02:25.090 --> 00:02:28.515
3/4 of a shoe comes out to 0.75 pairs

00:02:28.515 --> 00:02:30.885
of shoes per basketball.

00:02:30.885 --> 00:02:32.611
I now have my opportunity costs

00:02:32.611 --> 00:02:35.588
of shoes and basketballs for country A.

00:02:35.588 --> 00:02:36.486
It's gonna be a lot quicker to calculate

00:02:36.486 --> 00:02:38.452
these opportunity costs for country B

00:02:38.452 --> 00:02:40.855
because country B can
produce either four pairs

00:02:40.855 --> 00:02:42.915
of shoes or four basketballs,

00:02:42.915 --> 00:02:45.843
giving the country A
opportunity cost of shoes

00:02:45.843 --> 00:02:48.843
of one basketball per pair of shoes.

00:02:50.322 --> 00:02:52.748
Likewise for basketballs,
with the resources

00:02:52.748 --> 00:02:54.761
it takes to produce four basketballs,

00:02:54.761 --> 00:02:57.197
the country could have
produce four pairs of shoes.

00:02:57.197 --> 00:02:59.554
Divide both sides by four and I get

00:02:59.554 --> 00:03:02.855
a cost of basketballs of one pair

00:03:02.855 --> 00:03:05.098
of shoes per basketball.

00:03:05.098 --> 00:03:06.822
Now I have all the information I need

00:03:06.822 --> 00:03:09.294
to determine who has the
comparative advantage

00:03:09.294 --> 00:03:11.010
in these two goods.

00:03:11.010 --> 00:03:12.195
With that information I know how

00:03:12.195 --> 00:03:14.088
the country's can specialize and trade

00:03:14.088 --> 00:03:15.783
with one another in a way that benefits

00:03:15.783 --> 00:03:17.839
both countries mutually.

00:03:17.839 --> 00:03:19.311
Let's first look at shoes.

00:03:19.311 --> 00:03:22.048
We can see that country A can produce

00:03:22.048 --> 00:03:26.304
shoes at the opportunity cost
of 1.33 basketballs per shoe

00:03:26.304 --> 00:03:27.489
and country B can produce shoes

00:03:27.489 --> 00:03:31.477
at the opportunity cost of one
basketball per pair of shoes.

00:03:31.477 --> 00:03:33.988
Clearly country B can produce shoes

00:03:33.988 --> 00:03:35.894
at the lower opportunity cost.

00:03:35.894 --> 00:03:37.412
So country B should specialize

00:03:37.412 --> 00:03:39.415
in shoe production since that's what it

00:03:39.415 --> 00:03:41.611
has a comparative advantage in.

00:03:41.611 --> 00:03:43.841
For basketballs country
A can produce basketballs

00:03:43.841 --> 00:03:46.635
at an opportunity cost of 0.75 pairs

00:03:46.635 --> 00:03:48.922
of shoes per basketball, whereas country B

00:03:48.922 --> 00:03:51.652
must give up one pair
of shoes per basketball

00:03:51.652 --> 00:03:54.262
giving country A the comparative advantage

00:03:54.262 --> 00:03:55.684
due to its lower opportunity cost

00:03:55.684 --> 00:03:57.803
of basketball production.

00:03:57.803 --> 00:03:58.858
With this information we can come up

00:03:58.858 --> 00:04:01.330
with some conclusions.

00:04:01.330 --> 00:04:03.655
First, we know that country A

00:04:03.655 --> 00:04:07.238
should specialize in
basketball production,

00:04:09.189 --> 00:04:10.878
because it can produce basketballs

00:04:10.878 --> 00:04:14.626
at a lower opportunity
cost than country B.

00:04:14.626 --> 00:04:18.376
Next we know that country
B should specialize

00:04:19.286 --> 00:04:23.011
in shoe production due to
its lower opportunity cost

00:04:23.011 --> 00:04:25.137
of one basketball per pair of shoes

00:04:25.137 --> 00:04:28.034
compared to 1.33 basketballs
per pair of shoes

00:04:28.034 --> 00:04:29.439
in country A.

00:04:29.439 --> 00:04:31.059
Let's go back over to our PPC on which

00:04:31.059 --> 00:04:32.473
we'll indicate the points at which

00:04:32.473 --> 00:04:35.105
both country's A and B will produce

00:04:35.105 --> 00:04:38.018
based on the principle
of comparative advantage.

00:04:38.018 --> 00:04:40.214
Country A is gonna produce
nothing but basketballs

00:04:40.214 --> 00:04:42.517
due to its lower opportunity cost.

00:04:42.517 --> 00:04:45.678
Putting country A down
here, on its production

00:04:45.678 --> 00:04:48.439
possibilities curve,
country B will then produce

00:04:48.439 --> 00:04:51.025
nothing but shoes due to
its lower opportunity cost

00:04:51.025 --> 00:04:53.960
putting country B at this point

00:04:53.960 --> 00:04:55.959
on this production possibilities curve.

00:04:55.959 --> 00:04:58.386
Now you may be wondering
how does country B

00:04:58.386 --> 00:05:00.078
get basketballs if it's not producing

00:05:00.078 --> 00:05:01.343
any domestically.

00:05:01.343 --> 00:05:03.423
How does country A get
shoes if it's not producing

00:05:03.423 --> 00:05:04.916
any domestically?

00:05:04.916 --> 00:05:07.598
Of course the answer
to that lies in trade.

00:05:07.598 --> 00:05:10.494
By trading with one another both countries

00:05:10.494 --> 00:05:13.718
will be able to consume some combination

00:05:13.718 --> 00:05:16.968
of basketballs and shoes that is beyond

00:05:17.867 --> 00:05:20.820
each country's domestic
production possibilities.

00:05:20.820 --> 00:05:22.205
So these dash lines that I'm drawing now

00:05:22.205 --> 00:05:23.927
represent what we could call

00:05:23.927 --> 00:05:26.828
the trading possibilities curves

00:05:26.828 --> 00:05:28.995
of both country's A and B.

00:05:30.389 --> 00:05:32.460
Through specialization and trade,

00:05:32.460 --> 00:05:35.025
a country can consume at a point beyond

00:05:35.025 --> 00:05:37.775
what would be possible if it were

00:05:38.668 --> 00:05:41.087
to produce everything for itself

00:05:41.087 --> 00:05:43.053
without trading with other countries.

00:05:43.053 --> 00:05:46.353
The two dots here that I'll label X and Y

00:05:46.353 --> 00:05:48.742
represent possible levels of consumption

00:05:48.742 --> 00:05:49.769
which lie beyond what the country

00:05:49.769 --> 00:05:51.104
would have been able to achieve

00:05:51.104 --> 00:05:52.599
if they had chosen to try to produce

00:05:52.599 --> 00:05:54.906
both shoes and basketballs for themselves.

00:05:54.906 --> 00:05:55.739
In the next video we're gonna look

00:05:55.739 --> 00:05:57.589
at a different type of situation

00:05:57.589 --> 00:05:58.808
in which we're given not the number

00:05:58.808 --> 00:06:01.188
of outputs that a country can produce,

00:06:01.188 --> 00:06:02.740
rather the amount of inputs needed

00:06:02.740 --> 00:06:05.594
to produce a single unit
of two different goods.


Intuition for why independence matters for variance of sum

WEBVTT
Kind: captions
Language: en

00:00:01.614 --> 00:00:02.957
- [Narrator] So in previous videos

00:00:02.957 --> 00:00:04.754
we talked about the claim that if

00:00:04.754 --> 00:00:05.671
I have two random variables, x and y,

00:00:05.671 --> 00:00:07.834
that are independent, then the variance

00:00:07.834 --> 00:00:10.675
of the sum of those two random variables

00:00:10.675 --> 00:00:13.070
or the difference of
those two random variables

00:00:13.070 --> 00:00:15.810
is going to be equal to
the sum of the variances.

00:00:15.810 --> 00:00:17.245
So that if you have independent

00:00:17.245 --> 00:00:19.202
random variables, your variation

00:00:19.202 --> 00:00:22.333
is going to increase when you take

00:00:22.333 --> 00:00:24.117
a sum or a difference.

00:00:24.117 --> 00:00:26.234
And we've built a little
bit of intuition there.

00:00:26.234 --> 00:00:27.758
What I wanna talk about in this video,

00:00:27.758 --> 00:00:30.089
it's really about building
even more intuition,

00:00:30.089 --> 00:00:32.324
is get a gut feeling for why this

00:00:32.324 --> 00:00:36.178
independence is important
for making this claim.

00:00:36.178 --> 00:00:37.774
And to get that intuition, let's

00:00:37.774 --> 00:00:39.487
look at two random variables that are

00:00:39.487 --> 00:00:40.922
definitely random variables but that

00:00:40.922 --> 00:00:42.997
are definitely not independent.

00:00:42.997 --> 00:00:46.330
So let's let x is equal to the number

00:00:47.442 --> 00:00:50.692
of hours that the next person you meet,

00:00:54.638 --> 00:00:56.700
so I'll say random person,

00:00:56.700 --> 00:00:59.200
random person slept yesterday.

00:01:05.063 --> 00:01:07.480
And let's say that y is equal

00:01:08.440 --> 00:01:11.107
to the number of hours that same

00:01:13.678 --> 00:01:15.928
person was awake yesterday.

00:01:22.696 --> 00:01:25.145
And appreciate why these are not

00:01:25.145 --> 00:01:27.032
independent random variables.

00:01:27.032 --> 00:01:30.365
One of them is gonna
completely determine the other.

00:01:30.365 --> 00:01:31.262
If I slept eight hours yesterday

00:01:31.262 --> 00:01:33.989
then I would have been awake for 16 hours.

00:01:33.989 --> 00:01:35.251
Or if I slept for 16 hours then I

00:01:35.251 --> 00:01:37.120
would have been awake for eight hours.

00:01:37.120 --> 00:01:40.398
We know that x plus y, even though

00:01:40.398 --> 00:01:41.846
they're random variables, and there

00:01:41.846 --> 00:01:43.631
could be variation in x and there could

00:01:43.631 --> 00:01:44.514
be variation in y.

00:01:44.514 --> 00:01:45.575
But for any given person, remember,

00:01:45.575 --> 00:01:47.473
these are still based on that same person.

00:01:47.473 --> 00:01:51.556
X plus y is always going
to be equal to 24 hours.

00:01:54.620 --> 00:01:58.453
So these are not
independent, not independent.

00:02:00.361 --> 00:02:02.001
If you're given one of the variables

00:02:02.001 --> 00:02:04.947
it would completely determine
what the other variable is.

00:02:04.947 --> 00:02:06.730
The probability of getting a certain

00:02:06.730 --> 00:02:08.600
value for one variable is going to

00:02:08.600 --> 00:02:10.599
be very different, given what value

00:02:10.599 --> 00:02:12.615
you got for the other variable.

00:02:12.615 --> 00:02:14.703
So they're not independent at all.

00:02:14.703 --> 00:02:16.795
So in this situation, if someone said,

00:02:16.795 --> 00:02:19.172
let's just say for the sake of argument,

00:02:19.172 --> 00:02:21.089
that the variance of x,

00:02:22.316 --> 00:02:26.233
the variance of x is
equal to, I don't know,

00:02:27.813 --> 00:02:30.230
let's say it's equal to four,

00:02:31.352 --> 00:02:34.235
the unit's four variance so
it would be squared hours.

00:02:34.235 --> 00:02:36.558
So four hours squared.

00:02:36.558 --> 00:02:38.917
We could say that the standard deviation

00:02:38.917 --> 00:02:42.269
for x in this case would be two hours.

00:02:42.269 --> 00:02:43.805
And let's say that the variance,

00:02:43.805 --> 00:02:46.050
let's say the standard deviation of y

00:02:46.050 --> 00:02:48.300
is also equal to two hours.

00:02:50.884 --> 00:02:53.363
And let's say that the variance of y,

00:02:53.363 --> 00:02:56.103
variance of y, well it would be the square

00:02:56.103 --> 00:02:57.496
of the standard deviation.

00:02:57.496 --> 00:03:00.189
And so it would be four hours,

00:03:00.189 --> 00:03:03.889
four hours squared would be our units.

00:03:03.889 --> 00:03:06.671
So if we just tried to blindly say,

00:03:06.671 --> 00:03:08.632
"Oh, I'm just gonna apply this little

00:03:08.632 --> 00:03:10.644
"expression, this claim we have,"

00:03:10.644 --> 00:03:12.125
without thinking about the independents

00:03:12.125 --> 00:03:14.154
we would try to say,
"Well then, the variance

00:03:14.154 --> 00:03:17.821
"of x plus y, the variance of x plus y

00:03:21.043 --> 00:03:23.404
"must be equal to the
sum of their variances."

00:03:23.404 --> 00:03:25.887
So it would be four plus four.

00:03:25.887 --> 00:03:29.054
So is it equal to eight hours squared?

00:03:32.336 --> 00:03:34.250
Well that doesn't make any sense.

00:03:34.250 --> 00:03:36.964
Because we know that a random variable

00:03:36.964 --> 00:03:39.762
that is equal to x plus y, this is always

00:03:39.762 --> 00:03:41.311
going to be 24 hours.

00:03:41.311 --> 00:03:44.128
In fact, it's not going
to have any variation.

00:03:44.128 --> 00:03:47.099
X plus y is always gonna be 24 hours.

00:03:47.099 --> 00:03:49.405
So for these two random variables,

00:03:49.405 --> 00:03:51.871
because they are so connected.

00:03:51.871 --> 00:03:53.942
They are not independent at all,

00:03:53.942 --> 00:03:56.551
this is actually going to be zero.

00:03:56.551 --> 00:03:58.655
There is zero variance here.

00:03:58.655 --> 00:04:01.279
X plus y is always going to be 24.

00:04:01.279 --> 00:04:04.164
At least on earth where
we have a 24 hour day.

00:04:04.164 --> 00:04:05.408
I guess if someone lived on another

00:04:05.408 --> 00:04:07.414
planet or something it
could be slightly different.

00:04:07.414 --> 00:04:08.369
And we're assuming that we have

00:04:08.369 --> 00:04:10.984
an exactly 24 hour day on earth.

00:04:10.984 --> 00:04:13.201
So this is to give you a gut sense

00:04:13.201 --> 00:04:16.420
of why independence matters
for making this claim.

00:04:16.420 --> 00:04:17.956
And if you have things
that are not independent

00:04:17.956 --> 00:04:19.669
it gives you a good sense for why

00:04:19.669 --> 00:04:22.428
this claim doesn't hold up as much.


Variance of sum and difference of random variables

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:03.511
- So, we've defined two
random variables here.

00:00:03.511 --> 00:00:05.673
The first random variable X

00:00:05.673 --> 00:00:07.367
is the weight of the cereal

00:00:07.367 --> 00:00:11.393
in a random box of our
favorite cereal, Mathies,

00:00:11.393 --> 00:00:14.804
a random closed box of our
favorite cereal, Mathies.

00:00:14.804 --> 00:00:16.599
And we know a few other things about it.

00:00:16.599 --> 00:00:19.849
We know what the expected value of X is,

00:00:19.849 --> 00:00:22.714
it is equal to 16 ounces.

00:00:22.714 --> 00:00:24.907
In fact, they tell it to us on a box,

00:00:24.907 --> 00:00:27.109
they say, you know, net weight, 16 ounces.

00:00:27.109 --> 00:00:28.812
Now, when you see that on a cereal box,

00:00:28.812 --> 00:00:30.224
it doesn't mean that every box

00:00:30.224 --> 00:00:32.774
is going to be exactly 16 ounces.

00:00:32.774 --> 00:00:35.800
Remember you have a discrete
number of these flakes in here,

00:00:35.800 --> 00:00:37.608
they might have slightly
different densities,

00:00:37.608 --> 00:00:38.695
slightly different shapes

00:00:38.695 --> 00:00:41.689
depending how they get
packed into this volume,

00:00:41.689 --> 00:00:43.218
so there is some variation

00:00:43.218 --> 00:00:45.774
which you can measure
with standard deviation.

00:00:45.774 --> 00:00:47.272
So, the standard deviation,

00:00:47.272 --> 00:00:48.816
let's just say for the sake of argument,

00:00:48.816 --> 00:00:51.682
for the random variable X is

00:00:51.682 --> 00:00:55.258
0.8 ounces

00:00:55.258 --> 00:00:56.757
and just to build our intuition

00:00:56.757 --> 00:00:58.053
a little bit later in this video,

00:00:58.053 --> 00:01:00.677
let's say that this,
the random variable X,

00:01:00.677 --> 00:01:04.097
is always stays
constrained within a range,

00:01:04.097 --> 00:01:05.619
that if it goes above a certain weight

00:01:05.619 --> 00:01:06.920
or below a certain weight,

00:01:06.920 --> 00:01:10.254
then the company that produces
it just throws out that box.

00:01:10.254 --> 00:01:12.474
And so, let's say that
our random variable X

00:01:12.474 --> 00:01:16.428
is always greater than
or equal to 15 ounces

00:01:16.428 --> 00:01:19.157
and it is always less than
or equal to 17 ounces,

00:01:19.157 --> 00:01:20.401
just for argument.

00:01:20.401 --> 00:01:22.403
This'll help us build
our intuition later on.

00:01:22.403 --> 00:01:24.147
Now, separately,

00:01:24.147 --> 00:01:25.369
let's consider a bowl,

00:01:25.369 --> 00:01:27.883
we're always gonna consider
the same size bowl,

00:01:27.883 --> 00:01:30.440
let's consider just a four ounce bowl

00:01:30.440 --> 00:01:33.107
because the expected value of Y,

00:01:33.956 --> 00:01:36.100
if you took a random one of these bowls,

00:01:36.100 --> 00:01:37.630
always the same bowl,

00:01:37.630 --> 00:01:39.102
or if you took the same bowl

00:01:39.102 --> 00:01:41.765
and someone filled it with Mathies,

00:01:41.765 --> 00:01:46.052
the expected weight of
the Mathies in that bowl

00:01:46.052 --> 00:01:48.796
is going to be four ounces.

00:01:48.796 --> 00:01:50.058
But once again,

00:01:50.058 --> 00:01:51.951
there's going to be some variation,

00:01:51.951 --> 00:01:54.158
depends who filled it
in, how it packed in,

00:01:54.158 --> 00:01:56.758
did they shake it while
they were filling it?

00:01:56.758 --> 00:01:57.880
There could be all sorts of things

00:01:57.880 --> 00:01:59.556
that could make some variation here.

00:01:59.556 --> 00:02:00.828
And so, for the sake of argument,

00:02:00.828 --> 00:02:02.393
let's say that variation can be measured

00:02:02.393 --> 00:02:05.756
by standard deviation, it's 0.6 ounces.

00:02:05.756 --> 00:02:07.648
And let's say whoever
the bowl fillers are,

00:02:07.648 --> 00:02:08.645
they are also,

00:02:08.645 --> 00:02:11.819
they don't like bowls that
are too heavy or too light,

00:02:11.819 --> 00:02:13.288
and so they'll also throw out bowls,

00:02:13.288 --> 00:02:16.744
so we can say that Y can,

00:02:16.744 --> 00:02:19.042
its maximum value that it'll ever take on

00:02:19.042 --> 00:02:20.613
is five ounces

00:02:20.613 --> 00:02:22.413
and the minimum value that
it could ever take on,

00:02:22.413 --> 00:02:24.984
let's say, it is three ounces.

00:02:24.984 --> 00:02:26.478
So, given all of this information,

00:02:26.478 --> 00:02:28.046
what I wanna do is

00:02:28.046 --> 00:02:31.076
let's just say I take
a random box of Mathies

00:02:31.076 --> 00:02:33.350
and I take a random filled bowl,

00:02:33.350 --> 00:02:36.673
and I wanna think about
the combined weight

00:02:36.673 --> 00:02:39.063
in the closed box and the filled bowl.

00:02:39.063 --> 00:02:41.678
So, what I wanna think about is, really,

00:02:41.678 --> 00:02:42.511
X plus Y.

00:02:43.607 --> 00:02:47.444
I wanna think about the sum
of the random variables.

00:02:47.444 --> 00:02:48.550
So, in previous videos,

00:02:48.550 --> 00:02:51.719
we already know that the
expected value of this

00:02:51.719 --> 00:02:53.955
is just gonna be the sum
of the expected values

00:02:53.955 --> 00:02:56.743
of each of the random variables.

00:02:56.743 --> 00:03:00.275
So, it would be the expected value of X

00:03:00.275 --> 00:03:03.738
plus the expected value of Y,

00:03:03.738 --> 00:03:05.612
and so it'd be 16 plus four ounces,

00:03:05.612 --> 00:03:06.725
in this case,

00:03:06.725 --> 00:03:09.968
this would be equal to 20 ounces.

00:03:09.968 --> 00:03:11.907
But what about the variation?

00:03:11.907 --> 00:03:13.919
Can we just add up the
standard deviations?

00:03:13.919 --> 00:03:16.338
If I wanna figure out
the standard deviation

00:03:16.338 --> 00:03:19.959
of X plus Y, how can I do this?

00:03:19.959 --> 00:03:22.088
Well, it turns out that

00:03:22.088 --> 00:03:23.792
you can't just add up
the standard deviations,

00:03:23.792 --> 00:03:26.232
but you can add up the variances.

00:03:26.232 --> 00:03:30.162
So, it is the case that
the variance of X plus Y

00:03:30.162 --> 00:03:33.049
is equal to the variance of X

00:03:33.049 --> 00:03:36.270
plus the variance of Y.

00:03:36.270 --> 00:03:39.816
And so, this is gonna have
an X right over here, X,

00:03:39.816 --> 00:03:42.938
and then we have plus Y and our Y.

00:03:42.938 --> 00:03:45.208
And actually, both of these assume

00:03:45.208 --> 00:03:46.932
independent random variables.

00:03:46.932 --> 00:03:48.752
So, it assumes,

00:03:48.752 --> 00:03:52.757
assumes X and Y are independent,

00:03:52.757 --> 00:03:54.304
I'm gonna write it in caps.

00:03:54.304 --> 00:03:55.922
In a future video,

00:03:55.922 --> 00:03:58.941
I'm going to give you
hopefully a better intuition

00:03:58.941 --> 00:04:00.717
for why this must be true,

00:04:00.717 --> 00:04:02.243
that they are independent in order to

00:04:02.243 --> 00:04:03.831
make this claim right over here.

00:04:03.831 --> 00:04:05.384
I'm not going to prove it in this video,

00:04:05.384 --> 00:04:08.062
but we could build a
little bit of intuition.

00:04:08.062 --> 00:04:11.046
Here, for each of these random variables,

00:04:11.046 --> 00:04:13.417
we have a range of two ounces over which

00:04:13.417 --> 00:04:14.724
this random variable can take

00:04:14.724 --> 00:04:16.293
and that's true for both of them.

00:04:16.293 --> 00:04:18.293
But what about this sum?

00:04:19.404 --> 00:04:21.199
Well, this sum here

00:04:21.199 --> 00:04:23.577
could get as high as,

00:04:23.577 --> 00:04:25.296
so let me write it this way,

00:04:25.296 --> 00:04:28.971
so X plus Y,

00:04:28.971 --> 00:04:32.636
X plus Y, what's the maximum
value that it could take on?

00:04:32.636 --> 00:04:35.259
Well, if you get a heavy
version of each of these,

00:04:35.259 --> 00:04:38.601
then it's going to be 17 plus five.

00:04:38.601 --> 00:04:40.871
So, this has to be less than 22 ounces,

00:04:40.871 --> 00:04:42.425
that's going to be
greater than or equal to,

00:04:42.425 --> 00:04:44.299
well, what's the lightest
possible scenario?

00:04:44.299 --> 00:04:45.875
Well, if you get a 15 ouncer here

00:04:45.875 --> 00:04:47.154
and you get a three ouncer here,

00:04:47.154 --> 00:04:49.184
and it is 18 ounces.

00:04:49.184 --> 00:04:50.434
And so, notice,

00:04:50.434 --> 00:04:53.975
now, the variation for the sum is larger.

00:04:53.975 --> 00:04:56.797
We have a range that this
thing can take on now

00:04:56.797 --> 00:04:59.426
of four while the range for each of these

00:04:59.426 --> 00:05:00.956
was just two.

00:05:00.956 --> 00:05:02.513
Or another way you could think about it is

00:05:02.513 --> 00:05:05.898
these upper and lower ends of the range

00:05:05.898 --> 00:05:08.959
are further from the mean

00:05:08.959 --> 00:05:11.755
than these upper and
lower ends of the range

00:05:11.755 --> 00:05:14.286
were from their respective means.

00:05:14.286 --> 00:05:15.702
So, hopefully, this gives you an intuition

00:05:15.702 --> 00:05:17.761
for why this makes sense.

00:05:17.761 --> 00:05:19.247
Let me ask you another question:

00:05:19.247 --> 00:05:21.380
what if I were to say

00:05:21.380 --> 00:05:23.305
what about the variance,

00:05:23.305 --> 00:05:25.987
what about the variance

00:05:25.987 --> 00:05:28.595
of X minus Y?

00:05:28.595 --> 00:05:29.428
What would this be?

00:05:29.428 --> 00:05:30.756
Would you subtract the variances

00:05:30.756 --> 00:05:33.054
of each of the random variables here?

00:05:33.054 --> 00:05:35.747
Well, let's just do the
exact same exercise.

00:05:35.747 --> 00:05:39.246
Let's take X minus Y,

00:05:39.246 --> 00:05:42.364
X minus Y and think about it.

00:05:42.364 --> 00:05:44.477
What would be the lowest value

00:05:44.477 --> 00:05:47.060
that X minus Y could take on?

00:05:47.060 --> 00:05:48.248
Well, the lowest values,

00:05:48.248 --> 00:05:50.130
if you have a low X

00:05:50.130 --> 00:05:52.058
and you have a high Y,

00:05:52.058 --> 00:05:54.246
so it'd be 15 minus five,

00:05:54.246 --> 00:05:56.337
so this would be 10 right over here,

00:05:56.337 --> 00:05:57.866
that would be the lowest value

00:05:57.866 --> 00:05:58.699
that you could take on

00:05:58.699 --> 00:06:00.338
and what would be the highest value?

00:06:00.338 --> 00:06:02.726
Well, the highest values,
if you have a high X

00:06:02.726 --> 00:06:03.707
and a low Y,

00:06:03.707 --> 00:06:05.991
so 17 minus three is 14.

00:06:05.991 --> 00:06:06.949
So, notice,

00:06:06.949 --> 00:06:09.553
just as we saw in this case of the sum,

00:06:09.553 --> 00:06:10.788
even in the difference,

00:06:10.788 --> 00:06:13.490
your variability seems to have increased.

00:06:13.490 --> 00:06:15.657
This is still going to be,

00:06:16.687 --> 00:06:19.135
the extremes are still further than

00:06:19.135 --> 00:06:20.927
the mean of the difference,

00:06:20.927 --> 00:06:24.097
the mean of the difference
would be 16 minus four is 12.

00:06:24.097 --> 00:06:27.247
These extreme values are two away from 12.

00:06:27.247 --> 00:06:29.019
And this is just to give us an intuition.

00:06:29.019 --> 00:06:31.162
Once again, it's not a rigorous proof.

00:06:31.162 --> 00:06:33.993
So, it actually turns
out that in either case,

00:06:33.993 --> 00:06:36.108
when you're taking the
variance of X plus Y

00:06:36.108 --> 00:06:37.837
or X minus Y,

00:06:37.837 --> 00:06:39.813
you would sum the variances

00:06:39.813 --> 00:06:42.985
assuming X and Y are
independent variables.

00:06:42.985 --> 00:06:44.070
Now, with that out of the way,

00:06:44.070 --> 00:06:45.962
let's just calculate
the standard deviation

00:06:45.962 --> 00:06:47.785
of X plus Y.

00:06:47.785 --> 00:06:49.503
Well, we know this,

00:06:49.503 --> 00:06:52.185
let me just write it
using this sigma notation,

00:06:52.185 --> 00:06:54.398
so another way of writing
the variance of X plus Y

00:06:54.398 --> 00:06:56.341
is to write the standard deviation

00:06:56.341 --> 00:06:58.103
of X plus Y squared

00:06:58.103 --> 00:07:01.943
and that's going to be
equal to the variance of X

00:07:01.943 --> 00:07:06.422
plus the variance of Y.

00:07:06.422 --> 00:07:08.271
Now, what is the variance of X?

00:07:08.271 --> 00:07:10.597
Well, that's the standard
deviation of X squared,

00:07:10.597 --> 00:07:11.871
0.8 squared,

00:07:11.871 --> 00:07:13.960
this is 0.64,

00:07:13.960 --> 00:07:16.459
0.64.

00:07:16.459 --> 00:07:19.054
The standard deviation of Y is 0.6,

00:07:19.054 --> 00:07:20.836
you square it to get the variance,

00:07:20.836 --> 00:07:22.958
that's 0.36.

00:07:22.958 --> 00:07:24.660
You add these two up

00:07:24.660 --> 00:07:26.955
and you are going to get one.

00:07:26.955 --> 00:07:30.246
So, the variance of the sum is one,

00:07:30.246 --> 00:07:32.516
and then if you take the
square root of both of these,

00:07:32.516 --> 00:07:35.031
you get the standard deviation of the sum

00:07:35.031 --> 00:07:37.692
is also going to be one.

00:07:37.692 --> 00:07:39.095
And that just happened to work out

00:07:39.095 --> 00:07:41.500
because we're dealing with the
scenario where the variance,

00:07:41.500 --> 00:07:44.286
where the square root
of one is, well, one.

00:07:44.286 --> 00:07:46.381
So, this hopefully builds your intuition,

00:07:46.381 --> 00:07:48.677
whether we are adding or subtracting

00:07:48.677 --> 00:07:51.220
to independent random variables.

00:07:51.220 --> 00:07:53.586
The variance of that
sum or the difference,

00:07:53.586 --> 00:07:55.788
the variability will increase.

00:07:55.788 --> 00:07:56.779
In the next video,

00:07:56.779 --> 00:07:58.911
we'll go into some depth talking about

00:07:58.911 --> 00:08:00.607
getting an intuition for why

00:08:00.607 --> 00:08:02.666
independence is an important condition

00:08:02.666 --> 00:08:05.833
for making this statement, this claim.


Polar curve area with calculator

WEBVTT
Kind: captions
Language: en

00:00:01.167 --> 00:00:03.232
- [Instructor] What
we're going to try to do

00:00:03.232 --> 00:00:05.719
is use our powers of calculus to

00:00:05.719 --> 00:00:08.960
find this blue area right over here.

00:00:08.960 --> 00:00:10.677
What this blue area is is the area

00:00:10.677 --> 00:00:12.938
in between successive loops of

00:00:12.938 --> 00:00:14.935
the graph, the polar graph.

00:00:14.935 --> 00:00:18.001
R of theta is equal to
three theta sin theta.

00:00:18.001 --> 00:00:20.290
I'm graphing it in polar coordinates here.

00:00:20.290 --> 00:00:21.744
Just to make sure we understand

00:00:21.744 --> 00:00:24.189
what's going on, this first loop gets

00:00:24.189 --> 00:00:28.106
traced out when theta
goes between zero and pi,

00:00:29.952 --> 00:00:31.868
so this is zero is less than or equal

00:00:31.868 --> 00:00:34.502
to theta is less than or equal to pi.

00:00:34.502 --> 00:00:36.980
That makes sense, when theta is zero,

00:00:36.980 --> 00:00:38.477
this whole thing is zero, but then

00:00:38.477 --> 00:00:42.555
as theta increases, our
R increases as well.

00:00:42.555 --> 00:00:44.762
But then as theta goes back to

00:00:44.762 --> 00:00:46.663
approaching pi, remember sin of pi

00:00:46.663 --> 00:00:48.484
is zero, so then this thing will

00:00:48.484 --> 00:00:50.774
approach zero again, so that makes sense.

00:00:50.774 --> 00:00:53.384
Then as we go from pi to two pi,

00:00:53.384 --> 00:00:55.279
you might expect to see some type

00:00:55.279 --> 00:00:58.022
of a graph down here, but what you need

00:00:58.022 --> 00:01:02.044
to realize is when we
go from pi to two pi,

00:01:02.044 --> 00:01:04.976
sin of those thetas are gonna be negative.

00:01:04.976 --> 00:01:07.328
So what it does is it flips over,

00:01:07.328 --> 00:01:09.756
so instead of having an R right over

00:01:09.756 --> 00:01:11.428
here in the third quadrant, when we're

00:01:11.428 --> 00:01:13.556
a little past theta being a little bit

00:01:13.556 --> 00:01:16.105
more than pi, since R would be negative,

00:01:16.105 --> 00:01:20.301
it flips over and then it
ends up right over there.

00:01:20.301 --> 00:01:22.968
Between theta from pi to two pi,

00:01:24.402 --> 00:01:26.475
we actually trace out this second

00:01:26.475 --> 00:01:29.077
circle right over here, this outer one.

00:01:29.077 --> 00:01:31.320
It makes sense that the radius is

00:01:31.320 --> 00:01:33.251
getting larger because we have

00:01:33.251 --> 00:01:35.335
this three theta right over here.

00:01:35.335 --> 00:01:38.472
As theta is getting larger and larger

00:01:38.472 --> 00:01:40.529
in magnitude, we should say the

00:01:40.529 --> 00:01:43.360
magnitude of our R is getting larger.

00:01:43.360 --> 00:01:45.871
It turns out that the R would have been

00:01:45.871 --> 00:01:47.638
up here if it was positive, but then

00:01:47.638 --> 00:01:49.595
it goes negative, it goes out here,

00:01:49.595 --> 00:01:51.179
but it has a larger and larger

00:01:51.179 --> 00:01:54.093
magnitude because of this three theta.

00:01:54.093 --> 00:01:55.773
With that out of the way, let's

00:01:55.773 --> 00:01:58.976
think about how we find this area.

00:01:58.976 --> 00:02:00.697
In general, if you're looking at

00:02:00.697 --> 00:02:03.000
a polar curve and if you want to find

00:02:03.000 --> 00:02:05.279
an area, the general notion is

00:02:05.279 --> 00:02:08.743
you would go from theta is equal

00:02:08.743 --> 00:02:10.973
to, let's just call it alpha, to

00:02:10.973 --> 00:02:12.804
theta is equal to beta, so this

00:02:12.804 --> 00:02:15.111
would be your bounds on your angle.

00:02:15.111 --> 00:02:16.827
What you do, and we've shown this,

00:02:16.827 --> 00:02:18.608
we've given you the intuition for it

00:02:18.608 --> 00:02:20.843
in previous videos, it would be

00:02:20.843 --> 00:02:24.343
one half times R of theta squared D theta,

00:02:25.616 --> 00:02:27.487
is the general formula for finding

00:02:27.487 --> 00:02:30.359
the area inside of a polar curve.

00:02:30.359 --> 00:02:32.018
But what we're trying to do here is

00:02:32.018 --> 00:02:35.031
find the area between two successive loops

00:02:35.031 --> 00:02:37.037
for this polar curve, where as theta

00:02:37.037 --> 00:02:40.925
goes from zero to two
pi, so how do we do that?

00:02:40.925 --> 00:02:43.825
One way to do that, we can find

00:02:43.825 --> 00:02:46.742
the total area in this larger loop,

00:02:49.561 --> 00:02:51.611
and then subtract from that the

00:02:51.611 --> 00:02:55.790
area in the smaller loop right over here.

00:02:55.790 --> 00:02:57.401
How do we do that?

00:02:57.401 --> 00:02:59.038
At any point, if you get inspired,

00:02:59.038 --> 00:03:00.565
pause the video, it's always good for you

00:03:00.565 --> 00:03:02.240
to try to work it through yourself

00:03:02.240 --> 00:03:04.836
than just passively
watch me work through it.

00:03:04.836 --> 00:03:07.162
Let's do it together, so
what's the area of this

00:03:07.162 --> 00:03:11.081
outer circle if we include
this right over here?

00:03:11.081 --> 00:03:13.061
That's going to be the integral.

00:03:13.061 --> 00:03:14.631
Remember, the outer circle gets traced

00:03:14.631 --> 00:03:17.214
out as we go from pi to two pi,

00:03:18.357 --> 00:03:20.857
and it'd be one half times all

00:03:21.767 --> 00:03:25.168
of this business squared, so three theta

00:03:25.168 --> 00:03:27.668
sin of theta squared, D theta.

00:03:30.837 --> 00:03:33.247
That's the outer circle,
and then from that we

00:03:33.247 --> 00:03:36.682
want to subtract the area
of this inner circle.

00:03:36.682 --> 00:03:39.209
Minus, what are the bounds there?

00:03:39.209 --> 00:03:41.068
We said we'd trace out that inner circle

00:03:41.068 --> 00:03:44.697
as we go from zero to pi, zero to pi,

00:03:44.697 --> 00:03:47.447
one half times three theta sin of

00:03:51.416 --> 00:03:53.333
theta squared, D theta.

00:03:55.304 --> 00:03:57.054
It turns out that these are not

00:03:57.054 --> 00:03:59.524
trivial integrals to solve on your own,

00:03:59.524 --> 00:04:01.270
but we have tools, in fact, if you're

00:04:01.270 --> 00:04:02.945
taking a lot of classes they expect

00:04:02.945 --> 00:04:04.356
you to be able to use tools.

00:04:04.356 --> 00:04:06.130
If you're taking AP Calculus BC,

00:04:06.130 --> 00:04:07.760
they would expect you to be able to

00:04:07.760 --> 00:04:09.214
use a graphing calculator, so that's

00:04:09.214 --> 00:04:10.563
what we're gonna do right now in

00:04:10.563 --> 00:04:12.214
order to calculate this quantity,

00:04:12.214 --> 00:04:13.822
or to calculate this area, so let

00:04:13.822 --> 00:04:17.739
me get out my TI-84
emulator, there we have it.

00:04:19.094 --> 00:04:22.634
Let's see, I want you
to see my keystrokes,

00:04:22.634 --> 00:04:24.446
so let me see if I can remember that.

00:04:24.446 --> 00:04:25.918
The first, well it's the same thing

00:04:25.918 --> 00:04:27.893
over again, it's just different bounds.

00:04:27.893 --> 00:04:29.827
The first time, what you want to do

00:04:29.827 --> 00:04:32.284
is you want to go to the math functions,

00:04:32.284 --> 00:04:33.956
math, and then you see right over

00:04:33.956 --> 00:04:36.584
here this is your definite
integral function.

00:04:36.584 --> 00:04:39.087
I'm just gonna scroll down there,

00:04:39.087 --> 00:04:40.965
to the definite integral and this

00:04:40.965 --> 00:04:43.154
TI-84 Plus CE I guess, writes it in

00:04:43.154 --> 00:04:45.817
nice, definite integral notation.

00:04:45.817 --> 00:04:47.462
Sometimes you'll just see that

00:04:47.462 --> 00:04:48.984
FN int open parentheses and then

00:04:48.984 --> 00:04:51.013
you write the expression, you say

00:04:51.013 --> 00:04:52.640
what your integrating with respect

00:04:52.640 --> 00:04:54.241
to and then you give the bounds,

00:04:54.241 --> 00:04:56.349
but here it's pretty clear.

00:04:56.349 --> 00:04:59.782
Remember that blue
integral right over there,

00:04:59.782 --> 00:05:01.756
I want to go from pi, so I'll do

00:05:01.756 --> 00:05:04.433
second, that's the pi
button right over here,

00:05:04.433 --> 00:05:07.516
pi to two pi, two second pi, and then

00:05:11.549 --> 00:05:15.132
I'm going to have one
divided by two times,

00:05:20.105 --> 00:05:22.354
and I'm just gonna do
it with respect to X,

00:05:22.354 --> 00:05:24.811
it doesn't matter if I do all the thetas,

00:05:24.811 --> 00:05:26.865
instead of thetas I use Xs, so I'm

00:05:26.865 --> 00:05:28.740
gonna have, and I'm just doing that to

00:05:28.740 --> 00:05:31.435
make it easy to type into the calculator.

00:05:31.435 --> 00:05:34.435
This is going to be three X sin of X

00:05:37.982 --> 00:05:41.296
and then I am going to want to square

00:05:41.296 --> 00:05:43.879
this entire thing, so sin of X,

00:05:44.903 --> 00:05:46.926
whoops, I don't want to go there,

00:05:46.926 --> 00:05:50.172
sin of X, I have that doesn't close

00:05:50.172 --> 00:05:52.087
that parentheses yet, okay that

00:05:52.087 --> 00:05:55.453
closes that parentheses, and so I

00:05:55.453 --> 00:05:58.648
want to square that, so squared.

00:05:58.648 --> 00:06:00.172
There you have it, and then I want

00:06:00.172 --> 00:06:03.862
to do DX, so that's that
first blue integral.

00:06:03.862 --> 00:06:07.114
From that, I want to subtract, so I'll

00:06:07.114 --> 00:06:09.220
get my math function again, I can just

00:06:09.220 --> 00:06:11.844
click nine, it does the integral again.

00:06:11.844 --> 00:06:16.796
Now I'm gonna go from zero
to pi of this same thing.

00:06:16.796 --> 00:06:19.879
I'm gonna go from zero to second, pi.

00:06:21.824 --> 00:06:24.771
I'm gonna try to type in that same thing.

00:06:24.771 --> 00:06:27.771
I am doing one divided by two, times

00:06:28.929 --> 00:06:31.762
open parentheses, three X sin of X

00:06:38.864 --> 00:06:41.733
close that, close that, so that's that

00:06:41.733 --> 00:06:45.476
and I'm gonna square it, square it.

00:06:45.476 --> 00:06:48.393
Then I have my DX, so then I do DX,

00:06:49.931 --> 00:06:51.668
and if I typed in everything correctly

00:06:51.668 --> 00:06:55.414
I should just be able to
press enter and get the value.

00:06:55.414 --> 00:06:58.164
I got 139 point, we're just gonna

00:06:59.317 --> 00:07:02.890
round to the nearest hundred, .53.

00:07:02.890 --> 00:07:06.390
This is approximately 139.53 square units.


Justification using second derivative: inflection point

WEBVTT
Kind: captions
Language: en

00:00:00.947 --> 00:00:03.323
- [Instructor] The twice
differentiable function G

00:00:03.323 --> 00:00:07.232
and its second derivative
G prime prime are graphed.

00:00:07.232 --> 00:00:08.788
And you can see it right over here.

00:00:08.788 --> 00:00:10.802
I'm actually working off of the article

00:00:10.802 --> 00:00:12.441
on Khan Academy called

00:00:12.441 --> 00:00:14.942
Justifying Using Second Derivatives.

00:00:14.942 --> 00:00:16.427
So we see our function G.

00:00:16.427 --> 00:00:18.239
And we see not its first derivative

00:00:18.239 --> 00:00:19.726
but its second derivative here

00:00:19.726 --> 00:00:21.089
in this brown color.

00:00:21.089 --> 00:00:23.505
So then, the article goes on to say,

00:00:23.505 --> 00:00:24.938
or the problem goes on to say,

00:00:24.938 --> 00:00:27.362
four students were asked
to give an appropriate

00:00:27.362 --> 00:00:30.497
calculus-based justification for the fact

00:00:30.497 --> 00:00:34.664
that G has an inflection point
at X equals negative two.

00:00:35.863 --> 00:00:38.213
So let's just feel good
that at least intuitively

00:00:38.213 --> 00:00:39.216
it feels right.

00:00:39.216 --> 00:00:41.198
So X equals negative two, remember what

00:00:41.198 --> 00:00:42.414
an inflection point is.

00:00:42.414 --> 00:00:44.387
It's where we're going
from concave downwards

00:00:44.387 --> 00:00:45.597
to concave upwards.

00:00:45.597 --> 00:00:48.306
Or, concave upwards to concave downwards.

00:00:48.306 --> 00:00:50.026
Or another way to think about it

00:00:50.026 --> 00:00:51.551
it's a situation where

00:00:51.551 --> 00:00:54.215
our slope goes from
decreasing to increasing,

00:00:54.215 --> 00:00:56.334
or from increasing to decreasing.

00:00:56.334 --> 00:00:57.970
And when we look at it
over here, it looks like

00:00:57.970 --> 00:01:00.604
our slope is decreasing, it's positive,

00:01:00.604 --> 00:01:02.588
but it's decreasing, it goes to zero.

00:01:02.588 --> 00:01:05.301
Then it keeps decreasing, it becomes

00:01:05.301 --> 00:01:06.514
it's negative now.

00:01:06.514 --> 00:01:08.187
It keeps decreasing until we get to about

00:01:08.187 --> 00:01:09.184
X equals negative two

00:01:09.184 --> 00:01:11.164
and then it seems that it's increasing,

00:01:11.164 --> 00:01:13.410
it's getting less and less
and less and less negative.

00:01:13.410 --> 00:01:15.257
It looks like it's a zero right over here

00:01:15.257 --> 00:01:17.398
then it just keeps
increasing, it gets more

00:01:17.398 --> 00:01:18.774
and more and more positive.

00:01:18.774 --> 00:01:22.166
So it does, indeed, look
like at X equals negative two

00:01:22.166 --> 00:01:24.437
we go from being concave downwards

00:01:24.437 --> 00:01:26.157
to concave upwards.

00:01:26.157 --> 00:01:28.747
Now a calculus based justification

00:01:28.747 --> 00:01:31.394
is we could look at its,
at the second derivative

00:01:31.394 --> 00:01:33.535
and see where the second derivative

00:01:33.535 --> 00:01:35.193
crosses the X-axis.

00:01:35.193 --> 00:01:37.813
Because where the second
derivative is negative

00:01:37.813 --> 00:01:40.167
that means our slope is decreasing

00:01:40.167 --> 00:01:41.952
we are concave downwards.

00:01:41.952 --> 00:01:43.951
And where the second
derivative is positive

00:01:43.951 --> 00:01:46.435
it means our first derivative

00:01:46.435 --> 00:01:49.032
is increasing, our slope
of our original function

00:01:49.032 --> 00:01:50.669
is increasing and we are concave upwards.

00:01:50.669 --> 00:01:53.677
So notice, we do indeed,
the second derivative

00:01:53.677 --> 00:01:55.966
does indeed cross the X-axis

00:01:55.966 --> 00:01:58.351
at X equals negative two.

00:01:58.351 --> 00:02:00.929
It's not enough for it to just be zero

00:02:00.929 --> 00:02:01.923
or touch the X-axis,

00:02:01.923 --> 00:02:05.009
it needs to cross the
X-axis in order for us

00:02:05.009 --> 00:02:06.470
to have an inflection point there.

00:02:06.470 --> 00:02:08.824
So given that, let's look at the students

00:02:08.824 --> 00:02:10.755
in justifications and see

00:02:10.755 --> 00:02:12.972
what we can, if we can kind of play

00:02:12.972 --> 00:02:14.696
put the teacher hat in our mind

00:02:14.696 --> 00:02:16.036
and say what a teacher would say

00:02:16.036 --> 00:02:17.657
for the different justifications.

00:02:17.657 --> 00:02:20.025
So the first one says,
the second derivative of G

00:02:20.025 --> 00:02:23.074
changes signs at X equals negative two.

00:02:23.074 --> 00:02:25.991
Well that's exactly what
we were just talking about.

00:02:25.991 --> 00:02:29.001
If the second derivative changes signs

00:02:29.001 --> 00:02:31.253
in this case, it goes
from negative to positive,

00:02:31.253 --> 00:02:34.020
that means our first derivative went from

00:02:34.020 --> 00:02:35.616
decreasing to increasing.

00:02:35.616 --> 00:02:39.083
Which is indeed, good for saying this is a

00:02:39.083 --> 00:02:41.043
calculus-based justification.

00:02:41.043 --> 00:02:42.851
So, at least for now I'm gonna put

00:02:42.851 --> 00:02:45.182
kudos you are correct there.

00:02:45.182 --> 00:02:47.408
It crosses the X-axis.

00:02:47.408 --> 00:02:49.435
So this is ambiguous.

00:02:49.435 --> 00:02:50.724
What is crossing the X-axis?

00:02:50.724 --> 00:02:51.908
If a student wrote this I'd say,

00:02:51.908 --> 00:02:52.941
what are they talking about, the function,

00:02:52.941 --> 00:02:54.074
are they talking about
the first derivative,

00:02:54.074 --> 00:02:55.687
the second derivative.

00:02:55.687 --> 00:02:58.483
And so I would say, please
use more precise language.

00:02:58.483 --> 00:03:01.234
This cannot be accepted as
a correct justification.

00:03:01.234 --> 00:03:03.203
I will read the other ones.

00:03:03.203 --> 00:03:05.241
The second derivative of G is

00:03:05.241 --> 00:03:07.074
increasing at X equals

00:03:08.354 --> 00:03:09.569
negative two.

00:03:09.569 --> 00:03:11.546
Well no, that doesn't justify

00:03:11.546 --> 00:03:13.230
why you have an inflection point there.

00:03:13.230 --> 00:03:14.524
For example,

00:03:14.524 --> 00:03:17.038
the second derivative is increasing

00:03:17.038 --> 00:03:19.121
at X equals negative 2.5.

00:03:20.238 --> 00:03:22.235
The second derivative is even increasing

00:03:22.235 --> 00:03:23.383
at X equals

00:03:23.383 --> 00:03:24.401
negative one.

00:03:24.401 --> 00:03:25.939
But you don't have an inflection point

00:03:25.939 --> 00:03:27.138
at those places.

00:03:27.138 --> 00:03:29.095
So I would say, this doesn't justify

00:03:29.095 --> 00:03:30.509
why G has an inflection point.

00:03:30.509 --> 00:03:33.596
And then the last student response,

00:03:33.596 --> 00:03:35.544
the graph of G changes concavity

00:03:35.544 --> 00:03:37.304
at X equals negative two.

00:03:37.304 --> 00:03:38.976
That is true

00:03:38.976 --> 00:03:42.192
but that isn't a
calculus-based justification.

00:03:42.192 --> 00:03:45.903
We'd want to use our
second derivative here.


Analyzing related rates problems: equations (trig)

WEBVTT
Kind: captions
Language: en

00:00:00.908 --> 00:00:03.869
- [Instructor] A 20-meter ladder
is leaning against a wall.

00:00:03.869 --> 00:00:07.650
The distance X of T between
the bottom of the ladder

00:00:07.650 --> 00:00:09.989
and the wall is increasing at a rate

00:00:09.989 --> 00:00:12.235
of three meters per minute.

00:00:12.235 --> 00:00:14.818
At a certain instant T sub zero

00:00:15.963 --> 00:00:20.219
the top of the ladder is
a distance Y of T sub zero

00:00:20.219 --> 00:00:23.212
of 15 meters from the ground.

00:00:23.212 --> 00:00:26.440
What is the rate of change of the angle

00:00:26.440 --> 00:00:28.954
theta of T between the ground

00:00:28.954 --> 00:00:31.664
and the ladder at that instant?

00:00:31.664 --> 00:00:34.058
So what I want to do is draw this out.

00:00:34.058 --> 00:00:36.303
And really, the first
step is to think about,

00:00:36.303 --> 00:00:38.223
well, what equation will be helpful

00:00:38.223 --> 00:00:39.603
for us to solve this problem.

00:00:39.603 --> 00:00:41.219
And then we might just go ahead and

00:00:41.219 --> 00:00:42.513
actually solve the problem.

00:00:42.513 --> 00:00:45.154
So a 20-meter ladder is
leaning against a wall.

00:00:45.154 --> 00:00:48.782
So let me draw ourselves a wall here.

00:00:48.782 --> 00:00:50.115
That is my wall.

00:00:51.267 --> 00:00:54.477
Now let me draw our 20-meter ladder.

00:00:54.477 --> 00:00:57.644
So maybe it looks something like that.

00:00:58.639 --> 00:00:59.722
So that is 20

00:01:01.385 --> 00:01:02.567
meters.

00:01:02.567 --> 00:01:05.251
They say the distance X
of T between the bottom

00:01:05.251 --> 00:01:06.759
of the ladder and the wall.

00:01:06.759 --> 00:01:08.744
So, this distance right over here.

00:01:08.744 --> 00:01:09.577
This is

00:01:10.812 --> 00:01:13.208
this distance right over here is

00:01:13.208 --> 00:01:14.041
X of T.

00:01:14.971 --> 00:01:16.898
They say it's increasing at a rate of

00:01:16.898 --> 00:01:18.340
three meters per minute.

00:01:18.340 --> 00:01:20.654
So, we know that we could either say

00:01:20.654 --> 00:01:23.374
X prime of T, which is the same thing

00:01:23.374 --> 00:01:24.207
as DX DT.

00:01:25.323 --> 00:01:28.030
Is equal to three meters,

00:01:28.030 --> 00:01:31.171
I'll write it out because
it's hard if I said M per M

00:01:31.171 --> 00:01:33.133
it might not be that clear.

00:01:33.133 --> 00:01:35.363
Meters per minute, so they give us

00:01:35.363 --> 00:01:36.386
that piece of information.

00:01:36.386 --> 00:01:39.287
So the rate of change of
X, with respect to time,

00:01:39.287 --> 00:01:40.890
they gave us that.

00:01:40.890 --> 00:01:43.318
At a certain sub T sub zero

00:01:43.318 --> 00:01:45.638
the top of the ladder is a distance

00:01:45.638 --> 00:01:46.799
of 15 meters.

00:01:46.799 --> 00:01:49.983
So the top of the ladder, so
let's make this very clear.

00:01:49.983 --> 00:01:52.070
So, this distance

00:01:52.070 --> 00:01:53.113
right

00:01:53.113 --> 00:01:54.560
over here

00:01:54.560 --> 00:01:55.393
is Y of T.

00:01:57.025 --> 00:01:58.253
Y of T.

00:01:58.253 --> 00:02:01.406
And they say, at time T sub zero

00:02:01.406 --> 00:02:03.549
Y of T is 15 meters.

00:02:03.549 --> 00:02:06.772
So let me just write it
in here, Y of T sub zero

00:02:06.772 --> 00:02:08.919
is equal to 15 meters.

00:02:08.919 --> 00:02:12.278
And so, maybe let me write
this right over here.

00:02:12.278 --> 00:02:14.271
This is Y of T sub zero.

00:02:14.271 --> 00:02:16.398
Let's just assume that we're drawing it

00:02:16.398 --> 00:02:18.322
at that moment T sub zero, because I think

00:02:18.322 --> 00:02:19.588
that's going to be important.

00:02:19.588 --> 00:02:22.088
Y of T sub zero is equal to 15

00:02:23.235 --> 00:02:24.415
meters.

00:02:24.415 --> 00:02:26.364
So they want to know,
what is the rate of change

00:02:26.364 --> 00:02:28.995
of the angle theta between
the ground and the ladder.

00:02:28.995 --> 00:02:31.203
And this is saying theta
is also going to change

00:02:31.203 --> 00:02:32.296
with respect to time.

00:02:32.296 --> 00:02:34.114
There's going to be a function of time

00:02:34.114 --> 00:02:36.617
between the ground and the
ladder at that instant.

00:02:36.617 --> 00:02:39.608
So theta, get a new color here.

00:02:39.608 --> 00:02:42.543
Theta is this angle right
over here, this is theta.

00:02:42.543 --> 00:02:44.603
And it's also going to
be a function of time.

00:02:44.603 --> 00:02:46.720
So, what we'll always want to do in these

00:02:46.720 --> 00:02:48.834
related rates problems is we want to

00:02:48.834 --> 00:02:52.344
set up an equation, and
really, an algebraic equation

00:02:52.344 --> 00:02:54.475
maybe a little bit of
trigonometry involved.

00:02:54.475 --> 00:02:56.515
That relates the things
that we care about.

00:02:56.515 --> 00:02:58.130
And then we're likely to have to take

00:02:58.130 --> 00:02:59.850
the derivative of both sides of that

00:02:59.850 --> 00:03:03.015
in order to relate the related rates.

00:03:03.015 --> 00:03:04.202
So let's see.

00:03:04.202 --> 00:03:05.545
We want to know

00:03:05.545 --> 00:03:08.160
we want to know the rate of change

00:03:08.160 --> 00:03:09.868
of the angle between

00:03:09.868 --> 00:03:12.417
the ground and the ladder at that instant.

00:03:12.417 --> 00:03:13.562
So what we need to figure out

00:03:13.562 --> 00:03:18.094
we want to figure out,
theta prime, at T sub zero.

00:03:18.094 --> 00:03:20.811
This is what we want to figure out.

00:03:20.811 --> 00:03:22.953
Now, they've given us
some interesting things,

00:03:22.953 --> 00:03:25.078
they've given us, I
guess, our rate of change

00:03:25.078 --> 00:03:27.459
of X with perspective time is constant

00:03:27.459 --> 00:03:29.714
at three meters per minute.

00:03:29.714 --> 00:03:32.881
And, we know what Y is at that moment.

00:03:33.945 --> 00:03:36.911
So let's see, can we create a relationship

00:03:36.911 --> 00:03:38.883
because they gave us DX DT

00:03:38.883 --> 00:03:40.557
it'll actually be more useful

00:03:40.557 --> 00:03:44.051
to find a relationship
between X and theta,

00:03:44.051 --> 00:03:46.013
and then take the
derivative of both sides.

00:03:46.013 --> 00:03:48.287
And then use this information possibly

00:03:48.287 --> 00:03:50.372
to figure out what the appropriate

00:03:50.372 --> 00:03:52.310
value of X or theta is

00:03:52.310 --> 00:03:53.423
at that moment.

00:03:53.423 --> 00:03:54.438
So let's do that.

00:03:54.438 --> 00:03:57.030
So how does X relate to theta?

00:03:57.030 --> 00:03:58.765
Well, we use a little bit of

00:03:58.765 --> 00:04:00.362
trigonometry right over here.

00:04:00.362 --> 00:04:02.675
If you took the hypotenuse

00:04:02.675 --> 00:04:04.758
times the cosine of theta

00:04:06.493 --> 00:04:08.041
you would get X.

00:04:08.041 --> 00:04:10.741
So let me write this right up here, X of T

00:04:10.741 --> 00:04:12.310
X of T

00:04:12.310 --> 00:04:13.711
is equal to

00:04:13.711 --> 00:04:15.094
the hypotenuse

00:04:15.094 --> 00:04:17.404
20 meters, because that's
the length of the ladder,

00:04:17.404 --> 00:04:18.737
times the cosine

00:04:19.894 --> 00:04:21.227
cosine of theta.

00:04:22.073 --> 00:04:24.304
And I can say the cosine of theta of T

00:04:24.304 --> 00:04:26.087
just to make it clear that this

00:04:26.087 --> 00:04:27.312
is a function of time.

00:04:27.312 --> 00:04:30.114
This comes straight out of trigonometry.

00:04:30.114 --> 00:04:32.787
Actually, our basic trig, trigonometric

00:04:32.787 --> 00:04:34.261
function definitions.

00:04:34.261 --> 00:04:36.964
Now, why is this useful, why
do I think this is useful?

00:04:36.964 --> 00:04:38.864
Well let's think about
what happens when I take

00:04:38.864 --> 00:04:41.668
the derivative of both
sides using the chain rule.

00:04:41.668 --> 00:04:44.444
On the left hand side, I am going to have

00:04:44.444 --> 00:04:45.777
an X prime of T.

00:04:47.615 --> 00:04:49.474
And then that's going to
be equal to, what do I end

00:04:49.474 --> 00:04:51.270
up with on the right hand side?

00:04:51.270 --> 00:04:53.240
Well, using the chain
rule, first, I'll take

00:04:53.240 --> 00:04:55.522
the derivative with respect to theta.

00:04:55.522 --> 00:04:57.133
And so that's just going to be

00:04:57.133 --> 00:04:58.466
negative 20 sine

00:05:01.116 --> 00:05:02.499
of

00:05:02.499 --> 00:05:03.416
theta of T.

00:05:04.286 --> 00:05:06.223
And I need to multiple that

00:05:06.223 --> 00:05:08.140
times theta prime of T.

00:05:09.233 --> 00:05:11.748
So what I can do is, say hey look,

00:05:11.748 --> 00:05:16.052
at T sub zero, I know
what X prime of T is.

00:05:16.052 --> 00:05:19.355
I could try to figure out
what sine of theta of T is

00:05:19.355 --> 00:05:21.845
and then I'll just solve for this

00:05:21.845 --> 00:05:22.845
right over there.

00:05:22.845 --> 00:05:24.140
So let's do that.

00:05:24.140 --> 00:05:25.557
So, at T sub zero

00:05:26.839 --> 00:05:29.214
so at T is equal to

00:05:29.214 --> 00:05:30.131
T sub zero.

00:05:31.091 --> 00:05:33.230
What we're gonna have X prime of T.

00:05:33.230 --> 00:05:35.500
Well that, at every time,
it's three meters per minute

00:05:35.500 --> 00:05:37.974
we'll assume that our rates
are in meters per minute.

00:05:37.974 --> 00:05:39.990
And just our values are in meters

00:05:39.990 --> 00:05:41.403
when we're talking about distance

00:05:41.403 --> 00:05:42.702
and our angles are in radians.

00:05:42.702 --> 00:05:46.191
So, this is going to be equal to three

00:05:46.191 --> 00:05:47.763
is equal to

00:05:47.763 --> 00:05:48.680
negative 20

00:05:50.744 --> 00:05:53.078
times sine of theta of T

00:05:53.078 --> 00:05:55.491
times the derivative of
data with respect to time.

00:05:55.491 --> 00:05:57.707
So, how do we figure
out what sine of theta

00:05:57.707 --> 00:05:59.617
of T is going to be?

00:05:59.617 --> 00:06:03.369
Well, let's just use that
other information they gave us.

00:06:03.369 --> 00:06:05.319
And I'm gonna scroll down a little bit

00:06:05.319 --> 00:06:08.140
to get a little bit more real estate.

00:06:08.140 --> 00:06:11.481
So sine of theta, let
me write it over here.

00:06:11.481 --> 00:06:13.100
Sine

00:06:13.100 --> 00:06:13.933
of theta

00:06:14.775 --> 00:06:17.349
at time T sub not,
that's what we care about

00:06:17.349 --> 00:06:19.043
T is equal to T sub not.

00:06:19.043 --> 00:06:21.602
What's that gonna be?

00:06:21.602 --> 00:06:23.754
Well sine is opposite over hypotenuse.

00:06:23.754 --> 00:06:25.935
So that's gonna be Y

00:06:25.935 --> 00:06:27.498
at T sub not.

00:06:27.498 --> 00:06:29.483
Over our hypotenuse

00:06:29.483 --> 00:06:30.566
of 20 meters.

00:06:32.175 --> 00:06:34.881
Well that's going to be equal to

00:06:34.881 --> 00:06:36.685
that's going to be equal to, they tell us,

00:06:36.685 --> 00:06:39.030
Y of T sub not is 15 meters

00:06:39.030 --> 00:06:40.280
over 20 meters.

00:06:41.376 --> 00:06:44.224
Which is the same thing as three fourths.

00:06:44.224 --> 00:06:46.638
So, by this yellow
information, they actually

00:06:46.638 --> 00:06:48.677
told us that this right over here

00:06:48.677 --> 00:06:51.594
is going to be equal to three fourths.

00:06:51.594 --> 00:06:53.974
So this times three fourths

00:06:53.974 --> 00:06:56.325
times the rate of change

00:06:56.325 --> 00:06:57.158
of theta

00:06:58.210 --> 00:06:59.823
with respect to T.

00:06:59.823 --> 00:07:02.753
And so now we just have to
solve for this and we're done.

00:07:02.753 --> 00:07:04.659
So, this is going to be what's

00:07:04.659 --> 00:07:07.298
negative 20 times three fourths?

00:07:07.298 --> 00:07:08.965
That is negative 15.

00:07:10.510 --> 00:07:12.690
That is negative 15.

00:07:12.690 --> 00:07:15.233
If we divide both sides by negative 15

00:07:15.233 --> 00:07:16.311
we get

00:07:16.311 --> 00:07:18.926
theta prime of T is equal to

00:07:18.926 --> 00:07:20.843
three over negative 15.

00:07:21.833 --> 00:07:23.750
Three over negative 15.

00:07:25.174 --> 00:07:28.466
Which is the same thing as being equal to

00:07:28.466 --> 00:07:30.405
negative one fifth.

00:07:30.405 --> 00:07:32.606
And this, the units here would be in

00:07:32.606 --> 00:07:34.920
radians per minute.

00:07:34.920 --> 00:07:37.277
Because our rates are all in per minute.

00:07:37.277 --> 00:07:40.825
So, if I wanted to I could write radians

00:07:40.825 --> 00:07:41.658
per

00:07:42.792 --> 00:07:43.800
minute.

00:07:43.800 --> 00:07:45.587
Ideally I would write it

00:07:45.587 --> 00:07:46.616
right over here.

00:07:46.616 --> 00:07:47.636
But here you go.

00:07:47.636 --> 00:07:51.239
We were able to figure out and
this is an interesting one.

00:07:51.239 --> 00:07:53.185
Because, they give you
that information on Y

00:07:53.185 --> 00:07:54.573
but really, use that information of Y

00:07:54.573 --> 00:07:55.959
to figure out what sine of theta of T is.

00:07:55.959 --> 00:07:59.976
But the equation you set
up involves X and theta.