Polar functions derivatives

WEBVTT
Kind: captions
Language: en

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- [Instructor] What we
have here is the graph

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of R is equal to sin of two
theta in polar coordinates

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and if polar coordinates
look unfamiliar to you

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or if you need to brush up on them

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I encourage you to do a
search for polar coordinates

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in Khan Academy or look at
our pre-calculus section

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but I'll give you a little
bit of a primer here.

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Let's just familiarize ourself

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why this graph looks the way it does.

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So, what we're doing for any point here,

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we could obviously specify these points

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in terms of X and Y coordinates

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but we could also specify
them in terms of an angle

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and a radius so for example,

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this would have some X coordinate

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and some Y coordinate

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or we could draw a line

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from the origin to that
point right over here

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and specify it with some angle theta

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and some R which is the distance

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from the origin to that point.

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And just to familiarize ourselves

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with this curve

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let's just see why it's intuitive.

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So, when theta is zero,
R is going to be zero,

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sin of two times zero is just zero,

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so our R we're just gonna be at the origin

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and then as theta get larger,

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our R gets larger

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and so, we start tracing out

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this pedal of this flower
or clover-looking thing,

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so it starts looking like that

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and we could keep going all the way.

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What happens when theta
is equal to pi over four?

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When theta is equal to pi
over four right over there,

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well, sin of two times pi over four

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is sin of pi over two,

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R is equal to one.

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So, we reach a kind of a maximum R there

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and then and as theta increases,

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our R once again starts to get smaller

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and smaller and smaller.

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Now, we're going to do
this in a calculus context,

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so the first question might be

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well, how do we express
the rate of change of R

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with respect to theta?

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Pause this video and see
if you can figure it out.

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What is R prime of theta?

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Well, there's really nothing new here.

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You just have one variable
as a function of another.

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You just use the chain rule.

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Take the derivative with respect to theta

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right over here.

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So, the derivative of sin of two theta

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with respect to two theta
is going to be cosine

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of two theta

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and then you multiply that,

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times the derivative of two theta

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with respect to theta which is two,

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so we could just say times two here

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or we could write a two out front.

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Alright, that was interesting

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but let's see if we can express this curve

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in terms of Xs and Ys

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and then think about those derivatives.

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So, one primer, a review from pre-calculus

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is that when you wanna go
between the polar world

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and the, I guess you can
say rectangular world,

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you have to remember the transformation

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that Y is equal to R sin of theta

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and that X is equal to R cosine of theta.

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Now, just as a really quick primer,

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why does that make sense?

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Well, let's just take one of
these angle R combinations

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right over here,

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so let's say this is
theta and that is our R.

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Well, the height of that side

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is going to by our Y

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and then the length of this side

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is going to be our X.

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Well, we know from trigonometry

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from our unit circle definition,

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the SOHCAHTOA definition
of our trig functions,

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sin of theta is opposite over hypotenuse,

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sin of theta is equal
to Y over our hypotenuse

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which is R and cosine of theta

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is equal to the adjacent or X over R

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and you just have multiply both sides

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of these equations by R

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to get to what we have right over there

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and once again, if this is going too fast,

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this is a review

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of just polar coordinates
from pre-calculus.

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But now we can use these to express

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purely in terms of theta.

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How do we do that?

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Well, we know that R is
equal to sin of two theta,

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so you just have to replace these Rs

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with sin of two theta.

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So, Y would be equal to sin of two theta,

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sin of two theta times sin of theta,

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times sin of theta

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and X is going to be
equal to sin of two theta,

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sin of two theta

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times cosine of theta,

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times cosine of theta, just like that

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but now we can use these expressions

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to find the rate of change
of Y with respect to theta,

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find a general expression for it.

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Pause the video and
see if you can do that.

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Right, let's work through it together.

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Well, this is once again,

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we're just gonna use our
derivative techniques,

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so I could write Y prime of theta,

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the derivative of Y with respect to theta,

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just gonna use the product
rule right over here,

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derivative of this first expression

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is two cosine of two theta,

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cosine of two theta,

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we've already seen that.

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That's just coming out of the chain rule

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and then times the second expression,

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sin of theta

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and then plus, plus the first expression,

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sin of two theta,

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sin of two theta

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times the derivative of
the second expression,

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derivative of the sin of
theta is cosine of theta.

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Fair enough and we could
do the same thing for X.

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X prime of theta,

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derivative of the first expression,

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it is going to be two
times cosine of two theta,

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two times cosine of two theta,

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times the second expression,

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cosine of theta

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and then you're gonna
have the first expression,

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sin of two theta,

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times the derivative of
the second expression

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which is negative sin of theta,

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negative sin of theta

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and we could use this,

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we could actually
evaluate these at points.

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For example, we could say
well, what's happening

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when theta is equal to pi over four?

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So, when theta's pi over four,

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I'll do that in black right over here.

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We are going to be at this point

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right over there.

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Well, let's evaluate it.

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So, if I were to say Y prime

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of pi over four is equal to, let's see,

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this is going to be equal to two cosine

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of pi over two, two times pi over four,

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times sin of pi over four

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plus sin of two times pi over four

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is sine of pi over two

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times cosine of pi over four.

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Cosine of pi over four.

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What is this going to be equal to?

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Well, cosine of pi over two is zero,

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so if that's zero all of
this stuff's gonna be zero

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and here's sin of pi over two,

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this is one, cosine of pi over four

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is square root of two over two,

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square root of two over two,

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so this is going to be equal
to square root of two over two

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and actually just for the
sake of saving some space

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I'll just write it right over here.

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It's going to be equal to
square root of two over two.

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Well, we could do the
same exercise with X.

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We could say X prime of pi over four.

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Let's see.

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We're still gonna have
two times cosine of two

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times pi over four,

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so that's going to be two
times cosine of pi over two.

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This first part right over
here is gonna look the same,

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so that first term's gonna be zero.

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Then we're gonna have minus,

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so this is all gonna be zero,

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so then we're gonna have minus sin

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of pi over two times pi over four

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is sin of pi over, sin of pi over two,

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and then times sin of pi over four,

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sin of pi over four.

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Now, this is just going to be one

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and so, this is gonna be equal to

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and this is square root
of two over two as well,

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so this is negative square
root of two over two.

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Now let's see why that that makes sense.

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So, let's think about what happens

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as theta increases here.

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If you increase theta a little bit

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from pi over four,

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if you increase it just a little bit,

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your Y coordinate continues to increase,

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so it makes sense you have
a positive slope here.

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But what happens to your X coordinate

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as theta increases a little bit,

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as theta goes from there to there?

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Well, then your X coordinate
starts to decrease

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when theta increases,

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so that's why it makes sense

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that you have a negative rate of change

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right over here.

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Now, the next question that you might say

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is say is well, I wanna
find the rate of change

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of Y with respect to X

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because I want to figure out the slope

00:08:22.188 --> 00:08:24.332
of the tangent line right over there.

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And it looks like it has
a slope of negative one

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but how would we actually calculate it?

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Well, one way to think about it is

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the derivative of Y,

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the derivative Y with respect to X

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is going to be equal
to the derivative of Y

00:08:40.439 --> 00:08:43.772
with respect to theta
over the derivative of X

00:08:43.772 --> 00:08:46.357
with respect to theta.

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And so, at that value,

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so at theta is equal to pi over four,

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this is going to be equal
to positive square root

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of two over two

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over negative square root of two over two,

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negative square root of two over two

00:09:02.963 --> 00:09:06.394
and this all simplifies to
being equal to negative one,

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which makes sense.

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This does look indeed like a tangent line

00:09:10.281 --> 00:09:12.757
that has a slope of negative one.

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So, hopefully this puts it altogether,

00:09:14.349 --> 00:09:16.173
you're feeling a little
bit more comfortable,

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you got a review a little bit

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of the polar coordinates

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but we've augmented that knowledge

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by starting to take some derivatives.


Derivative of __

WEBVTT
Kind: captions
Language: en

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- [Instructor] What we
have right over here

00:00:02.122 --> 00:00:06.236
is the graph of Y is equal to E to the X

00:00:06.236 --> 00:00:08.548
and what we're going to know
by the end of this video

00:00:08.548 --> 00:00:12.966
is one of the most
fascinating ideas in calculus

00:00:12.966 --> 00:00:15.912
and once again it reinforces the idea

00:00:15.912 --> 00:00:19.382
that E is really this
somewhat magical number.

00:00:19.382 --> 00:00:21.996
So we're gonna do a little
bit of an exploration.

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Let's just pick some points on this curve

00:00:24.495 --> 00:00:26.426
of Y is equal to E to the X

00:00:26.426 --> 00:00:29.098
and think about what the
slope of the tangent line is

00:00:29.098 --> 00:00:31.272
or what the derivative looks like

00:00:31.272 --> 00:00:34.534
and so let's say when Y is equal to one

00:00:34.534 --> 00:00:36.074
or when E to the X is equal to one,

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this is the case when X is equal to zero.

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Well, the slope of the tangent line

00:00:40.352 --> 00:00:44.651
looks like it is one,
which is curious because

00:00:44.651 --> 00:00:47.841
that's exactly the value of
the function at that point.

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What about when E to the X is
equal to two right over here?

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Well here, let me do it in another color,

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the slope of the tangent
line sure looks pretty close,

00:01:00.945 --> 00:01:04.215
sure looks pretty close to two.

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What about when E to
the X is equal to 1/2?

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So that's happening right about here.

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Well, it sure looks like the
slope of the tangent line

00:01:15.925 --> 00:01:18.193
is about 1/2.

00:01:18.193 --> 00:01:19.948
We could try what happens when E to the X

00:01:19.948 --> 00:01:21.933
is equal to five?

00:01:21.933 --> 00:01:25.042
Well, the slope of the tangent line here

00:01:25.042 --> 00:01:27.375
sure does look pretty close,

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sure does look pretty close to five

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and so just eyeballing it,

00:01:35.303 --> 00:01:38.387
is it the case that the
slope of the tangent line

00:01:38.387 --> 00:01:42.304
of E to the X is the same
thing, is E to the X?

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And I will tell you and
this is an amazing thing

00:01:45.686 --> 00:01:47.839
that that is indeed true,

00:01:47.839 --> 00:01:51.588
that if I have some function, F of X,

00:01:51.588 --> 00:01:53.838
that is equal to E to the X

00:01:54.773 --> 00:01:56.810
and if I were to take
the derivative of this,

00:01:56.810 --> 00:02:00.192
this is going to be equal
to E to the X as well

00:02:00.192 --> 00:02:01.947
or another way of saying it,

00:02:01.947 --> 00:02:05.780
the derivative with
respect to X of E to the X

00:02:07.069 --> 00:02:08.902
is equal to E to the X

00:02:10.113 --> 00:02:12.647
and that is an amazing thing.

00:02:12.647 --> 00:02:15.389
In previous lessons or courses,

00:02:15.389 --> 00:02:18.191
you've learned about ways to define E

00:02:18.191 --> 00:02:19.453
and this could be a new one.

00:02:19.453 --> 00:02:21.755
E is the number that where
if you take that number

00:02:21.755 --> 00:02:23.553
to the power of X, if
you define a function

00:02:23.553 --> 00:02:25.523
or expression as E to the X,

00:02:25.523 --> 00:02:27.466
it's that number where if you
take the derivative of that

00:02:27.466 --> 00:02:30.299
it's still going to be E to the X.

00:02:32.210 --> 00:02:34.232
And what you're looking here, this curve,

00:02:34.232 --> 00:02:37.107
it's a curve where the
value that's Y value

00:02:37.107 --> 00:02:41.274
at any point is the same as
the slope of the tangent line.

00:02:42.588 --> 00:02:45.030
If that doesn't strike you as mysterious

00:02:45.030 --> 00:02:48.079
and magical and amazing just yet, it will.

00:02:48.079 --> 00:02:50.344
Maybe tonight you'll wake up
in the middle of the night

00:02:50.344 --> 00:02:52.802
and you'll realize just what's going on.

00:02:52.802 --> 00:02:55.494
Now, some of you might be
saying okay, this is cool,

00:02:55.494 --> 00:02:58.693
you're telling me this, but
how do I know it's true?

00:02:58.693 --> 00:03:01.646
In another video, we will do the proof.


Economic models

WEBVTT
Kind: captions
Language: en

00:00:00.443 --> 00:00:04.354
- When you think about what the
field of economics is about,

00:00:04.354 --> 00:00:06.766
it is quite daunting.

00:00:06.766 --> 00:00:10.933
An economy is made up of millions
or even billions of actors

00:00:12.534 --> 00:00:16.392
organized in incredibly complex ways.

00:00:16.392 --> 00:00:19.559
Write down, this is complex real-world

00:00:21.377 --> 00:00:25.015
and each of the actors, human
beings or organizations,

00:00:25.015 --> 00:00:26.594
these are incredibly complex.

00:00:26.594 --> 00:00:27.537
A human brain.

00:00:27.537 --> 00:00:29.700
I can't predict what you're
going to do the next second

00:00:29.700 --> 00:00:31.372
much less what you're going to do

00:00:31.372 --> 00:00:33.134
the next day or the next year

00:00:33.134 --> 00:00:35.217
and imagine trying to make insights

00:00:35.217 --> 00:00:39.266
about what millions or
billions of people will do

00:00:39.266 --> 00:00:42.408
but the field of economics
has borrowed an idea

00:00:42.408 --> 00:00:43.799
from other fields.

00:00:43.799 --> 00:00:46.216
So for example, in chemistry,

00:00:48.508 --> 00:00:52.536
chemists have tried to
understand at a high level,

00:00:52.536 --> 00:00:56.257
well, how do molecules
in a container behave?

00:00:56.257 --> 00:00:59.284
Let's say molecules of gas.

00:00:59.284 --> 00:01:01.615
Well, you could imagine if
you have a container here

00:01:01.615 --> 00:01:04.672
with trillions upon
trillions of molecules,

00:01:04.672 --> 00:01:07.415
this is incredibly complex

00:01:07.415 --> 00:01:10.915
but by making some simplifying assumptions

00:01:12.080 --> 00:01:15.633
about the type of interactions
these particles will have

00:01:15.633 --> 00:01:19.141
or don't have, they
can come up with models

00:01:19.141 --> 00:01:22.652
like the ideal gas law which
you might be familiar with

00:01:22.652 --> 00:01:24.448
or not from your chemistry class

00:01:24.448 --> 00:01:26.818
that relates the pressure to the volume

00:01:26.818 --> 00:01:28.940
to the number of particles you have

00:01:28.940 --> 00:01:31.151
to the actual temperature

00:01:31.151 --> 00:01:32.340
and so this right over here

00:01:32.340 --> 00:01:34.907
where you're taking something
that's hairy and complex

00:01:34.907 --> 00:01:36.747
and making simplifying assumptions

00:01:36.747 --> 00:01:38.320
to help you understand it,

00:01:38.320 --> 00:01:41.403
this thing right over here is a model

00:01:42.831 --> 00:01:44.876
and this is in other fields as well.

00:01:44.876 --> 00:01:47.004
Sometimes, it's not an equation.

00:01:47.004 --> 00:01:50.504
Sometimes, it might be a simpler organism.

00:01:51.352 --> 00:01:53.352
For example, in biology,

00:01:55.180 --> 00:01:59.183
human beings are incredibly
complex organisms

00:01:59.183 --> 00:02:01.878
and not only are they incredibly complex

00:02:01.878 --> 00:02:03.845
but certain forms of experimentation

00:02:03.845 --> 00:02:08.012
would also feel fairly unethical
to our modern moral ethos

00:02:09.073 --> 00:02:11.072
and so what do biologists do?

00:02:11.072 --> 00:02:15.321
Well, they make simplifying
assumptions or they pare down,

00:02:15.321 --> 00:02:18.137
they say, okay, we can't do
that study on human beings

00:02:18.137 --> 00:02:21.528
but maybe we can simplify the problem

00:02:21.528 --> 00:02:23.779
by looking at simpler organisms.

00:02:23.779 --> 00:02:28.227
Maybe you can look at an
individual cell right over here.

00:02:28.227 --> 00:02:31.415
Maybe you can look at
things like fruit flies

00:02:31.415 --> 00:02:35.090
which are famous in the study of genetics.

00:02:35.090 --> 00:02:37.709
Maybe you can even look at
fairly complex organisms.

00:02:37.709 --> 00:02:40.934
Even a mouse is a very complex thing

00:02:40.934 --> 00:02:43.893
but it's still simpler than a human being

00:02:43.893 --> 00:02:45.781
and at least to our modern ethics,

00:02:45.781 --> 00:02:47.780
we're willing to do certain things to mice

00:02:47.780 --> 00:02:50.311
that we aren't willing
to do to human beings

00:02:50.311 --> 00:02:53.215
and so that's why in a biological context,

00:02:53.215 --> 00:02:57.519
you will hear people talk
about things like a mouse model

00:02:57.519 --> 00:02:59.635
where they will test a drug on a mouse

00:02:59.635 --> 00:03:02.488
or try to understand how
something happens in a mouse

00:03:02.488 --> 00:03:05.022
and then say, well, that's
a pretty good indication

00:03:05.022 --> 00:03:07.185
that might be happening to human beings.

00:03:07.185 --> 00:03:09.645
In fact, when they do
drug trials in medicine,

00:03:09.645 --> 00:03:11.154
they often will do it on mice first

00:03:11.154 --> 00:03:13.238
and when they have good
confidence that it works there

00:03:13.238 --> 00:03:14.586
and that it's fairly safe,

00:03:14.586 --> 00:03:17.159
only then will they start
to do the experiments

00:03:17.159 --> 00:03:18.835
on human beings.

00:03:18.835 --> 00:03:20.962
Well, economists are doing the same thing.

00:03:20.962 --> 00:03:24.801
Even before the advent of
computers and computer models,

00:03:24.801 --> 00:03:28.134
economists make simplifying assumptions,

00:03:29.452 --> 00:03:32.282
assumptions like all of
the actors in an economy

00:03:32.282 --> 00:03:36.452
are rational which we already
know is not exactly true.

00:03:36.452 --> 00:03:37.763
I'm not always rational

00:03:37.763 --> 00:03:40.500
and I definitely know people
who aren't always rational.

00:03:40.500 --> 00:03:42.139
They're simplifying assumptions

00:03:42.139 --> 00:03:44.755
that all of the people in an economy

00:03:44.755 --> 00:03:47.162
have the same access to information

00:03:47.162 --> 00:03:49.410
or that they all even
have perfect information

00:03:49.410 --> 00:03:52.843
which we also know isn't
necessarily true in a real economy.

00:03:52.843 --> 00:03:54.151
So depending on the model,

00:03:54.151 --> 00:03:56.235
there are going to be these
simplifying assumptions

00:03:56.235 --> 00:03:59.339
that take this large,
complex real-world thing

00:03:59.339 --> 00:04:01.796
and try to break it down
into simple equations

00:04:01.796 --> 00:04:03.433
or lines or charts.

00:04:03.433 --> 00:04:06.417
We have models early on
in our economic study.

00:04:06.417 --> 00:04:10.597
We will see things like the
production possibility frontier

00:04:10.597 --> 00:04:13.016
where it assumes that
you're only trading off

00:04:13.016 --> 00:04:16.530
between two things and
everything else is equal,

00:04:16.530 --> 00:04:19.780
this notion of ceteris, ceteris paribus

00:04:21.797 --> 00:04:24.691
which means all things equal.

00:04:24.691 --> 00:04:26.633
In a real-world, you're
not gonna be able to say,

00:04:26.633 --> 00:04:28.494
hey, let's just pick
between these two things

00:04:28.494 --> 00:04:30.534
and then hold everything else equal.

00:04:30.534 --> 00:04:32.123
There's hundreds or thousands

00:04:32.123 --> 00:04:34.133
or millions of variables are operating

00:04:34.133 --> 00:04:35.276
but if you wanna make a model,

00:04:35.276 --> 00:04:37.038
maybe we can make these assumptions.

00:04:37.038 --> 00:04:40.386
Same thing with famous price equilibria

00:04:40.386 --> 00:04:42.226
that we're going to study later on

00:04:42.226 --> 00:04:44.844
where you have supply and demand

00:04:44.844 --> 00:04:45.829
and then you have these notions

00:04:45.829 --> 00:04:49.055
of equilibrium prices and quantities.

00:04:49.055 --> 00:04:52.286
These also make similar
types of assumptions

00:04:52.286 --> 00:04:56.036
about rational actors
and perfect information

00:04:57.119 --> 00:05:01.208
and these economic
models can be very useful

00:05:01.208 --> 00:05:03.051
and that's why most of your study

00:05:03.051 --> 00:05:06.481
in a first year economics
course is of these models.

00:05:06.481 --> 00:05:08.109
Now, with that said,

00:05:08.109 --> 00:05:10.480
you should also take
them with a grain of salt

00:05:10.480 --> 00:05:11.949
and you shouldn't just accept them

00:05:11.949 --> 00:05:14.978
as the absolute description of reality.

00:05:14.978 --> 00:05:18.295
In fact, that's when economic
models can get dangerous.

00:05:18.295 --> 00:05:20.163
You always have to be conscientious

00:05:20.163 --> 00:05:22.382
of what are those assumptions you made?

00:05:22.382 --> 00:05:24.752
In fact, Nobel Prizes
have been won in economics

00:05:24.752 --> 00:05:28.022
by revisiting some simplifying assumptions

00:05:28.022 --> 00:05:30.270
and coming up with new models.

00:05:30.270 --> 00:05:33.009
The other difficulty about economics

00:05:33.009 --> 00:05:35.992
is it's hard to test
it in as absolute a way

00:05:35.992 --> 00:05:39.229
definitely as something
like chemistry or physics

00:05:39.229 --> 00:05:40.292
but even in biology

00:05:40.292 --> 00:05:42.870
where you're dealing with
similarly complex systems,

00:05:42.870 --> 00:05:44.546
a human body and an economy,

00:05:44.546 --> 00:05:47.364
these are both extremely complex systems.

00:05:47.364 --> 00:05:48.998
If I wanna see in medicine

00:05:48.998 --> 00:05:50.921
whether a certain medication works,

00:05:50.921 --> 00:05:52.477
I can do a clinical trial.

00:05:52.477 --> 00:05:55.028
I could take hundreds
or thousands of people

00:05:55.028 --> 00:05:58.034
and give maybe half of them the drug

00:05:58.034 --> 00:05:59.195
and I could try to control

00:05:59.195 --> 00:06:01.029
for a bunch of different variables

00:06:01.029 --> 00:06:02.193
but in economics,

00:06:02.193 --> 00:06:05.140
you can't take a thousand
different economies

00:06:05.140 --> 00:06:08.932
that look very similar
in what you think matters

00:06:08.932 --> 00:06:12.166
and then apply some type
of economic prescription

00:06:12.166 --> 00:06:14.572
to half of them and then see what happens,

00:06:14.572 --> 00:06:17.713
to see whether your model is exactly true

00:06:17.713 --> 00:06:19.231
or whether your prescription

00:06:19.231 --> 00:06:22.903
for what makes an economy
grow faster actually works

00:06:22.903 --> 00:06:24.826
and so the big takeaway,

00:06:24.826 --> 00:06:28.668
models are valuable across
the various sciences

00:06:28.668 --> 00:06:30.546
including in economics

00:06:30.546 --> 00:06:33.692
but economics straddles
between a social science

00:06:33.692 --> 00:06:37.499
and the sciences like chemistry or physics

00:06:37.499 --> 00:06:40.031
because you can't run
experiments in the same way

00:06:40.031 --> 00:06:41.951
and we often make simplifying assumptions

00:06:41.951 --> 00:06:44.776
that even though we know
aren't exactly true,

00:06:44.776 --> 00:06:47.142
they're the only way that
we're able to make sense

00:06:47.142 --> 00:06:49.642
of an incredibly complex real-world.


Scarcity

WEBVTT
Kind: captions
Language: en

00:00:00.308 --> 00:00:02.242
- [Instructor] The
entire field of economics

00:00:02.242 --> 00:00:05.151
is based on the idea of scarcity.

00:00:05.151 --> 00:00:06.791
And, arguably, we wouldn't even need

00:00:06.791 --> 00:00:08.603
a field of economics, if there wasn't

00:00:08.603 --> 00:00:11.207
the notion of scarcity in the world.

00:00:11.207 --> 00:00:13.015
So what does scarcity mean?

00:00:13.015 --> 00:00:14.966
Well, think about what does
it mean in everyday life?

00:00:14.966 --> 00:00:16.739
It means that there's not enough

00:00:16.739 --> 00:00:18.547
of something to go around.

00:00:18.547 --> 00:00:20.234
If we're talking about scarce goods,

00:00:20.234 --> 00:00:22.454
scarce services, scarce resources,

00:00:22.454 --> 00:00:24.971
we're talking about
things where if there was

00:00:24.971 --> 00:00:27.365
no cost associated with
them, people would use

00:00:27.365 --> 00:00:31.373
far more of that than
there actually is around.

00:00:31.373 --> 00:00:33.119
And in this video we're gonna think

00:00:33.119 --> 00:00:34.881
about different types
of goods and services

00:00:34.881 --> 00:00:36.862
or just resources and think about

00:00:36.862 --> 00:00:39.387
whether they are scarce or not.

00:00:39.387 --> 00:00:43.410
So a related idea to scarce
resources is it's opposite,

00:00:43.410 --> 00:00:46.307
which is the notion of a free resource.

00:00:46.307 --> 00:00:48.013
So this is something
that, you could argue,

00:00:48.013 --> 00:00:50.074
is infinitely abundant or at least

00:00:50.074 --> 00:00:52.004
in a certain context is so abundant

00:00:52.004 --> 00:00:53.885
that it feels like people can have

00:00:53.885 --> 00:00:55.502
as much of it as they want.

00:00:55.502 --> 00:00:57.268
The more that one person has of it,

00:00:57.268 --> 00:01:00.080
it doesn't take away from someone else.

00:01:00.080 --> 00:01:01.986
And the reason why scarcity is essential

00:01:01.986 --> 00:01:04.954
to economics is because
economics is the study

00:01:04.954 --> 00:01:07.910
of how do you allocate
these scarce resources.

00:01:07.910 --> 00:01:09.785
If there's more demand for it

00:01:09.785 --> 00:01:12.465
than the amount of thing that there is,

00:01:12.465 --> 00:01:14.933
well, who gets it, how
much of it do they get,

00:01:14.933 --> 00:01:16.745
and what do they have to give up

00:01:16.745 --> 00:01:19.832
in exchange to get those scarce resources?

00:01:19.832 --> 00:01:21.453
But for the sake of this video,

00:01:21.453 --> 00:01:23.601
let's just first make sure we understand

00:01:23.601 --> 00:01:25.787
and have a good idea of what resources

00:01:25.787 --> 00:01:29.955
are scarce and which ones aren't, and why.

00:01:29.955 --> 00:01:32.990
So this is a picture of caviar,

00:01:32.990 --> 00:01:34.494
which is essentially fish eggs,

00:01:34.494 --> 00:01:36.587
and it's not easy to get.

00:01:36.587 --> 00:01:38.467
The fish eggs are deep in the water,

00:01:38.467 --> 00:01:40.138
someone has to get to them, and then

00:01:40.138 --> 00:01:42.131
they have to package it in some way,

00:01:42.131 --> 00:01:43.837
and they have to get it to your plate.

00:01:43.837 --> 00:01:45.988
And so, do you think that caviar

00:01:45.988 --> 00:01:49.321
is a scarce resource or a free resource?

00:01:51.002 --> 00:01:52.669
Well, if it was a free
resource, that means

00:01:52.669 --> 00:01:54.956
that we're just swimming in caviar,

00:01:54.956 --> 00:01:56.640
that it's so abundant
that I could just have

00:01:56.640 --> 00:01:59.015
as much caviar as I want and there's still

00:01:59.015 --> 00:02:01.221
as much as you want and that everyone gets

00:02:01.221 --> 00:02:02.950
as much caviar as they like.

00:02:02.950 --> 00:02:04.480
Well, that's clearly not the case.

00:02:04.480 --> 00:02:07.244
Caviar is a scarce resource.

00:02:07.244 --> 00:02:09.390
In fact it is a quite scarce resource,

00:02:09.390 --> 00:02:11.471
and because of that, if you want it,

00:02:11.471 --> 00:02:15.660
you have to give up a good bit to get it.

00:02:15.660 --> 00:02:19.294
This is a picture of some
people working in a factory,

00:02:19.294 --> 00:02:23.427
and the resource that jumps
out here is that of labor.

00:02:23.427 --> 00:02:25.450
And labor's interesting because it's not

00:02:25.450 --> 00:02:27.778
as tangible as something like caviar

00:02:27.778 --> 00:02:29.441
but it is a resource.

00:02:29.441 --> 00:02:32.460
And one could even argue
that caviar on your plate,

00:02:32.460 --> 00:02:35.240
some of its scarcity comes
from the labor involved

00:02:35.240 --> 00:02:37.080
of getting it to your plate.

00:02:37.080 --> 00:02:38.996
But here these are clearly, it looks like

00:02:38.996 --> 00:02:41.379
these are gentlemen who
are putting together

00:02:41.379 --> 00:02:43.004
some type of fabric.

00:02:43.004 --> 00:02:45.776
And so, would you consider labor,

00:02:45.776 --> 00:02:48.029
would you consider that a scarce resource

00:02:48.029 --> 00:02:49.612
or a free resource?

00:02:50.751 --> 00:02:52.911
Well, it would be a
free resource if people

00:02:52.911 --> 00:02:56.922
were willing to just do as
much work for other people,

00:02:56.922 --> 00:02:58.824
actually willing to do an
infinite amount of work

00:02:58.824 --> 00:03:00.933
for other people, which
isn't even humanly possible.

00:03:00.933 --> 00:03:02.956
And even is it was humanly possible,

00:03:02.956 --> 00:03:04.317
people aren't willing to do that.

00:03:04.317 --> 00:03:06.021
They want something in return.

00:03:06.021 --> 00:03:10.276
And so, once again, it
is a scarce resource.

00:03:10.276 --> 00:03:12.871
There's many resources that
are pictured right here.

00:03:12.871 --> 00:03:17.593
You have this beautiful town
next to this alpine lake.

00:03:17.593 --> 00:03:20.966
And so, some clear scarce
resources are here.

00:03:20.966 --> 00:03:23.485
Many people would love to have a view

00:03:23.485 --> 00:03:25.942
like you would get from
this house or hotel

00:03:25.942 --> 00:03:28.199
right over here, but not everyone,

00:03:28.199 --> 00:03:29.926
and many people would love to live there

00:03:29.926 --> 00:03:32.615
because of the view, but
not everyone can live there.

00:03:32.615 --> 00:03:34.978
So that is a scarce resource.

00:03:34.978 --> 00:03:38.372
The water here is an interesting one.

00:03:38.372 --> 00:03:40.532
I can imagine in earlier times,

00:03:40.532 --> 00:03:42.445
if before there was a town here,

00:03:42.445 --> 00:03:43.734
if there was just a primitive village

00:03:43.734 --> 00:03:45.524
living next to this fresh water,

00:03:45.524 --> 00:03:48.215
they would probably view
it as a free resource.

00:03:48.215 --> 00:03:50.339
If someone was thirsty,
they would just go up

00:03:50.339 --> 00:03:52.585
to the lake and they would
just drink from the lake.

00:03:52.585 --> 00:03:55.918
They would not have to give up
anything to drink from that.

00:03:55.918 --> 00:03:58.797
But now the town, it
might be a little bit more

00:03:58.797 --> 00:04:00.469
of a scarce resource.

00:04:00.469 --> 00:04:02.690
They might want to preserve
it for various reasons.

00:04:02.690 --> 00:04:05.940
In order to get the water to your sink

00:04:05.940 --> 00:04:08.491
in your house, there
might be some services

00:04:08.491 --> 00:04:10.227
or goods or labor involved.

00:04:10.227 --> 00:04:11.763
Someone has to set up the pipes,

00:04:11.763 --> 00:04:13.830
maybe it has to be cleaned in some way.

00:04:13.830 --> 00:04:15.341
Well, then it might
become a little bit more

00:04:15.341 --> 00:04:17.533
of a scarce resource.

00:04:17.533 --> 00:04:19.812
Air for most of human history

00:04:19.812 --> 00:04:22.541
has been considered a free resource.

00:04:22.541 --> 00:04:24.553
And even today I'd argue that something

00:04:24.553 --> 00:04:26.484
like oxygen, at least on our planet,

00:04:26.484 --> 00:04:27.970
is considered a free resource.

00:04:27.970 --> 00:04:31.163
When I take a deep
breath (breathes deeply)

00:04:31.163 --> 00:04:33.929
it does not affect your
ability to take a deep breath.

00:04:33.929 --> 00:04:36.840
It does not take oxygen away from you.

00:04:36.840 --> 00:04:39.270
Now is there an infinite amount of oxygen

00:04:39.270 --> 00:04:40.491
in our atmosphere?

00:04:40.491 --> 00:04:43.412
No. But for our purposes it feels

00:04:43.412 --> 00:04:45.245
like there's an infinite amount.

00:04:45.245 --> 00:04:49.312
Now if the photosynthetic
plants were to disappear

00:04:49.312 --> 00:04:51.741
and all of a sudden oxygen
started to get diminished,

00:04:51.741 --> 00:04:53.956
or if we were in a space
station where there isn't

00:04:53.956 --> 00:04:56.403
a seemingly infinite amount of oxygen,

00:04:56.403 --> 00:04:57.988
well then you could imagine a world

00:04:57.988 --> 00:04:59.918
where it could become a scarce resource.

00:04:59.918 --> 00:05:02.133
You can imagine a colony on the Moon

00:05:02.133 --> 00:05:03.941
or on Mars or in the space station

00:05:03.941 --> 00:05:06.298
where it had some type of economic system

00:05:06.298 --> 00:05:09.885
to decide who gets how much oxygen.

00:05:09.885 --> 00:05:11.583
So I will leave you there.

00:05:11.583 --> 00:05:13.233
As already mentioned, scarcity

00:05:13.233 --> 00:05:15.939
is the central idea in all of economics.

00:05:15.939 --> 00:05:17.232
It's the reason why we even need

00:05:17.232 --> 00:05:19.121
a field called economics.

00:05:19.121 --> 00:05:20.823
And as you go forward
in your study of both

00:05:20.823 --> 00:05:24.225
micro and macroeconomics, we'll be looking

00:05:24.225 --> 00:05:27.462
at ways to allocate
these scarce resources.

00:05:27.462 --> 00:05:30.191
We'll try to study what
people have to give up

00:05:30.191 --> 00:05:33.305
in order to have access
to these resources,

00:05:33.305 --> 00:05:35.833
and we will have models that will help us

00:05:35.833 --> 00:05:38.568
understand what are the implications

00:05:38.568 --> 00:05:42.735
for these different methods
of allocating resources.


Analyzing mistakes when finding extrema (example 1)

WEBVTT
Kind: captions
Language: en

00:00:00.457 --> 00:00:01.972
- [Instructor] Pamela was asked to find

00:00:01.972 --> 00:00:03.856
where h of x equals x to the third

00:00:03.856 --> 00:00:08.270
minus six x squared plus 12
x has a relative extremum.

00:00:08.270 --> 00:00:09.784
This is her solution.

00:00:09.784 --> 00:00:12.528
So step one, it looks like she
tried to take the derivative.

00:00:12.528 --> 00:00:15.102
Step two, she tries to find the solution,

00:00:15.102 --> 00:00:17.026
find where the derivative is equal to zero

00:00:17.026 --> 00:00:19.075
and she found that it
happens at x equals two

00:00:19.075 --> 00:00:21.240
so she says that's a critical point.

00:00:21.240 --> 00:00:23.692
In step three, she says,
she makes the conclusion

00:00:23.692 --> 00:00:27.372
that therefore h has a
relative extremum there.

00:00:27.372 --> 00:00:29.460
Is Pamela's work correct?

00:00:29.460 --> 00:00:31.625
If not, what's her mistake?

00:00:31.625 --> 00:00:33.749
So pause this video and try
to work through it yourself

00:00:33.749 --> 00:00:36.240
and see if Pamela's work is correct.

00:00:36.240 --> 00:00:37.261
Alright, well, I'm just gonna try

00:00:37.261 --> 00:00:38.669
to do it again in parallel.

00:00:38.669 --> 00:00:40.293
So first, let me just
take the derivative here.

00:00:40.293 --> 00:00:42.167
So h prime of x.

00:00:42.167 --> 00:00:43.599
Just use the power rule multiple times.

00:00:43.599 --> 00:00:48.010
It's gonna be three x squared
for the x to the third.

00:00:48.010 --> 00:00:52.342
Two times negative six is
negative 12 or minus 12 x

00:00:52.342 --> 00:00:55.701
and then the derivative of 12
x is plus 12 and let's see.

00:00:55.701 --> 00:00:57.975
You can factor out a three here.

00:00:57.975 --> 00:01:02.605
So it's three times x squared
minus four x plus four

00:01:02.605 --> 00:01:06.168
and this part is indeed
equal to x minus two squared.

00:01:06.168 --> 00:01:10.743
So this is equal to three
times x minus two squared.

00:01:10.743 --> 00:01:14.098
So her step one looks right on target.

00:01:14.098 --> 00:01:17.099
Okay, step two, the
solution of h prime of x

00:01:17.099 --> 00:01:20.272
is equal to zero is equal to x equals two.

00:01:20.272 --> 00:01:21.581
Yeah, that works out.

00:01:21.581 --> 00:01:22.848
If you were to say,

00:01:22.848 --> 00:01:26.244
if you were to say three
times x minus two squared

00:01:26.244 --> 00:01:29.075
which is h prime of x,
the first derivative,

00:01:29.075 --> 00:01:30.421
and set that equal to zero,

00:01:30.421 --> 00:01:34.103
this is going to be true
when x is equal to two

00:01:34.103 --> 00:01:36.921
and so any point where
your first derivative

00:01:36.921 --> 00:01:39.861
is equal to zero or it's undefined,

00:01:39.861 --> 00:01:42.231
it is indeed a critical point.

00:01:42.231 --> 00:01:45.217
So this step looks good so far

00:01:45.217 --> 00:01:49.308
and then step three, h
has a relative extremum

00:01:49.308 --> 00:01:50.891
at x equals two.

00:01:51.764 --> 00:01:53.690
Alright, so she made
a big conclusion here.

00:01:53.690 --> 00:01:56.918
She assumed that because
the derivative was zero,

00:01:56.918 --> 00:01:59.164
that we have a relative extremum.

00:01:59.164 --> 00:02:01.908
So let's just see if you can
even just make that conclusion.

00:02:01.908 --> 00:02:04.653
In order to have a relative extremum,

00:02:04.653 --> 00:02:07.142
your curve is gonna
look something like this

00:02:07.142 --> 00:02:09.721
and then you would have a
relative extremum right over here

00:02:09.721 --> 00:02:12.753
and over here, your slope
goes from being positive

00:02:12.753 --> 00:02:16.594
and then it hits zero and
then it goes to being negative

00:02:16.594 --> 00:02:19.172
or you could have a
relative extremum like this.

00:02:19.172 --> 00:02:21.665
This would be a maximum point.

00:02:21.665 --> 00:02:25.415
This would be a minimum
point right over here

00:02:26.301 --> 00:02:28.106
and then in a minimum point,

00:02:28.106 --> 00:02:29.578
your slope is zero right over there

00:02:29.578 --> 00:02:31.464
but right before it,
your slope was negative

00:02:31.464 --> 00:02:33.797
and it goes to being positive

00:02:33.797 --> 00:02:36.501
but you actually have cases
where your derivative,

00:02:36.501 --> 00:02:38.182
your first derivative is zero

00:02:38.182 --> 00:02:40.717
but you don't have an extremum.

00:02:40.717 --> 00:02:45.419
So for example, you could
have a point like this

00:02:45.419 --> 00:02:47.252
where right over here,

00:02:48.203 --> 00:02:52.495
your slope or your derivative
could be equal to zero

00:02:52.495 --> 00:02:54.094
and so your first derivative
would be equal to zero

00:02:54.094 --> 00:02:56.302
but notice, your slope
is positive, it hits zero

00:02:56.302 --> 00:02:59.250
and then it goes back
to being positive again

00:02:59.250 --> 00:03:01.461
and so you can't make the conclusion

00:03:01.461 --> 00:03:03.908
just because your derivative is zero

00:03:03.908 --> 00:03:05.665
that it's definitely an extremum.

00:03:05.665 --> 00:03:06.732
You could say it's a critical point

00:03:06.732 --> 00:03:08.243
and so step two is correct.

00:03:08.243 --> 00:03:10.204
In order to make this conclusion,

00:03:10.204 --> 00:03:12.624
you would have to test what
the derivative is doing

00:03:12.624 --> 00:03:14.862
before that point and after that point

00:03:14.862 --> 00:03:17.599
and verify that it is switching sides

00:03:17.599 --> 00:03:19.028
and we could try to do that.

00:03:19.028 --> 00:03:22.483
So let's make a little table here.

00:03:22.483 --> 00:03:24.347
Make a little table.

00:03:24.347 --> 00:03:25.606
Do it a little bit neater.

00:03:25.606 --> 00:03:29.023
So x, x, h prime of x right over here.

00:03:31.172 --> 00:03:34.799
We know at x equals two,
h prime of two is zero.

00:03:34.799 --> 00:03:36.440
That's our critical point.

00:03:36.440 --> 00:03:37.911
But let's try, I don't know,

00:03:37.911 --> 00:03:40.902
let's see what happens
when x is equal to one

00:03:40.902 --> 00:03:42.618
and then let's see what
happens when x equals three.

00:03:42.618 --> 00:03:45.371
I'm just sampling points
on either side of two

00:03:45.371 --> 00:03:47.907
and let's see, we are going to have,

00:03:47.907 --> 00:03:48.970
when x is equal to one,

00:03:48.970 --> 00:03:53.720
h prime of one is three
times one minus two squared.

00:03:53.720 --> 00:03:55.763
One minus two, negative
one squared is positive one

00:03:55.763 --> 00:03:59.688
times three is still
positive and then three.

00:03:59.688 --> 00:04:02.061
Well, three minus two squared times three.

00:04:02.061 --> 00:04:03.626
That's also gonna be three.

00:04:03.626 --> 00:04:05.478
So this is actually a situation where,

00:04:05.478 --> 00:04:07.743
like I've just drawn it
where our slope is positive

00:04:07.743 --> 00:04:09.359
before we hit the critical point,

00:04:09.359 --> 00:04:13.448
it gets to zero but then it
starts becoming positive again

00:04:13.448 --> 00:04:15.985
and so that's why you
actually have to do this test

00:04:15.985 --> 00:04:18.108
in order to identify
whether it's an extremum.

00:04:18.108 --> 00:04:19.863
It turns out that this is not an extremum.

00:04:19.863 --> 00:04:23.130
This is not a maximum
or a minimum point here.

00:04:23.130 --> 00:04:25.421
So Pamela's work is not correct

00:04:25.421 --> 00:04:27.553
and her mistake is step three.

00:04:27.553 --> 00:04:28.901
In order to make this conclusion,

00:04:28.901 --> 00:04:30.697
you would have to test on either side

00:04:30.697 --> 00:04:31.672
of that critical point.

00:04:31.672 --> 00:04:32.505
Test the first derivative.


Conditions for IVT and EVT: graph

WEBVTT
Kind: captions
Language: en

00:00:00.568 --> 00:00:01.451
- [Narrator] We have the graph

00:00:01.451 --> 00:00:04.366
of y is equal to h of x right over here.

00:00:04.366 --> 00:00:08.806
They ask us does the
intermediate value theorem

00:00:08.806 --> 00:00:12.124
apply to h over the closed interval

00:00:12.124 --> 00:00:13.533
from negative one to four.

00:00:13.533 --> 00:00:17.346
The closed interval from
negative one to four

00:00:17.346 --> 00:00:19.040
right over here.

00:00:19.040 --> 00:00:21.117
We can first remind ourselves what is

00:00:21.117 --> 00:00:22.670
the intermediate value theorem.

00:00:22.670 --> 00:00:23.827
Although you don't have to even know

00:00:23.827 --> 00:00:25.450
what the theorem is actually saying,

00:00:25.450 --> 00:00:26.435
you just have to remember that

00:00:26.435 --> 00:00:28.031
the intermediate value theorem

00:00:28.031 --> 00:00:31.440
only applies over closed intervals

00:00:31.440 --> 00:00:34.542
where the function is continuous.

00:00:34.542 --> 00:00:36.108
You just have to say, you might

00:00:36.108 --> 00:00:37.080
be tempted to say wait, wait

00:00:37.080 --> 00:00:39.416
this function has a point of discontinuity

00:00:39.416 --> 00:00:41.864
but this is outside of that interval.

00:00:41.864 --> 00:00:44.142
The interval we care
about at negative one,

00:00:44.142 --> 00:00:46.392
this is the value of h of x

00:00:47.791 --> 00:00:50.622
and then at four this is the value

00:00:50.622 --> 00:00:53.246
of h of x and you can see over

00:00:53.246 --> 00:00:55.888
that entire interval right over here,

00:00:55.888 --> 00:00:58.628
the function is continuous.

00:00:58.628 --> 00:01:00.526
First, I would definitely say it meets

00:01:00.526 --> 00:01:02.281
the continuity requirement,
it's continuous

00:01:02.281 --> 00:01:04.280
over this closed interval.

00:01:04.280 --> 00:01:06.455
I would say yes, it does,

00:01:06.455 --> 00:01:08.804
the intermediate value theorem does apply.

00:01:08.804 --> 00:01:10.719
Since it does apply, I will be

00:01:10.719 --> 00:01:13.647
able to say that by the
intermediate value theorem

00:01:13.647 --> 00:01:16.388
because we are continuous
over this closed interval,

00:01:16.388 --> 00:01:18.943
the function in this case, h of x

00:01:18.943 --> 00:01:20.958
will take on every value between

00:01:20.958 --> 00:01:24.320
h of negative one and h of four,

00:01:24.320 --> 00:01:25.887
including those values.

00:01:25.887 --> 00:01:27.785
H of negative one, eyeballing it,

00:01:27.785 --> 00:01:29.376
looks like it's negative three.

00:01:29.376 --> 00:01:31.687
H of four looks like it's zero.

00:01:31.687 --> 00:01:34.006
We know by the intermediate value theorem

00:01:34.006 --> 00:01:35.456
that we take on every value

00:01:35.456 --> 00:01:38.123
between negative three and zero.

00:01:39.140 --> 00:01:40.676
You can even see that right over here.

00:01:40.676 --> 00:01:41.867
It actually does take on every one

00:01:41.867 --> 00:01:45.284
of those values in this part of the curve

00:01:48.458 --> 00:01:50.436
right over there.

00:01:50.436 --> 00:01:52.519
Let's do another example.

00:01:53.784 --> 00:01:57.368
Here we're asked does the
extreme value theorem apply

00:01:57.368 --> 00:01:59.047
to g over the closed interval

00:01:59.047 --> 00:02:00.547
from zero to five.

00:02:01.572 --> 00:02:02.832
Once again before we even think about

00:02:02.832 --> 00:02:06.357
what the extreme value theorem says,

00:02:06.357 --> 00:02:07.342
we have to remember that just like

00:02:07.342 --> 00:02:09.228
the intermediate value theorem

00:02:09.228 --> 00:02:11.605
it only applies over closed intervals

00:02:11.605 --> 00:02:13.592
where the function is continuous

00:02:13.592 --> 00:02:15.072
over the entire interval.

00:02:15.072 --> 00:02:16.769
Let's see, is our function continuous

00:02:16.769 --> 00:02:19.492
on the closed interval from zero to five?

00:02:19.492 --> 00:02:20.992
From zero to five.

00:02:23.146 --> 00:02:25.831
No we clearly have a discontinuity

00:02:25.831 --> 00:02:29.265
right over here at x is equal to three.

00:02:29.265 --> 00:02:31.630
First of all, we would say no.

00:02:31.630 --> 00:02:35.254
We can't say that over
this closed interval,

00:02:35.254 --> 00:02:39.113
that we have a well defined
maximum or minimum value.

00:02:39.113 --> 00:02:40.548
This is actually very clear over here

00:02:40.548 --> 00:02:43.157
that you don't have well
defined maximum value,

00:02:43.157 --> 00:02:44.504
we approach infinity over there

00:02:44.504 --> 00:02:46.449
and then we approach
negative infinity over there.

00:02:46.449 --> 00:02:49.016
Extreme value theorem does not apply.

00:02:49.016 --> 00:02:52.235
If instead it had offered
us a closed interval

00:02:52.235 --> 00:02:55.624
where we were continuous,
say they said between

00:02:55.624 --> 00:03:00.354
negative two and zero, let's
say it's that interval.

00:03:00.354 --> 00:03:01.313
It is a different color.

00:03:01.313 --> 00:03:03.851
Say it's between negative two and zero.

00:03:03.851 --> 00:03:05.589
Then the extreme value theorem would apply

00:03:05.589 --> 00:03:08.241
where you would say
okay, over this interval,

00:03:08.241 --> 00:03:11.284
the function has a well
defined minimum value

00:03:11.284 --> 00:03:12.909
and a maximum value.

00:03:12.909 --> 00:03:14.621
In this case it would actually be,

00:03:14.621 --> 00:03:17.289
the maximum would happen
at the right bound

00:03:17.289 --> 00:03:19.925
and the minimum value
happens at the left bound.

00:03:19.925 --> 00:03:21.452
But for this case, the closed interval,

00:03:21.452 --> 00:03:22.714
the one that we started with,

00:03:22.714 --> 00:03:24.932
definitely is not continuous so we do not,

00:03:24.932 --> 00:03:29.557
we would not say that the
extreme value theorem applies.

00:03:29.557 --> 00:03:31.485
Let's do one more example.

00:03:31.485 --> 00:03:32.878
Does the extreme value theorem apply

00:03:32.878 --> 00:03:37.213
to f over the interval from
negative five to negative two?

00:03:37.213 --> 00:03:39.822
Once again, let's just
look at the interval.

00:03:39.822 --> 00:03:43.739
We're going from negative
five to negative two.

00:03:46.522 --> 00:03:48.186
This is our function over that interval

00:03:48.186 --> 00:03:52.423
from x equals negative five
to x equals negative two.

00:03:52.423 --> 00:03:53.917
Even though there is a discontinuity

00:03:53.917 --> 00:03:55.861
in this graph that we
see, it's sitting outside

00:03:55.861 --> 00:03:56.817
of the interval.

00:03:56.817 --> 00:03:59.977
We only care about what's
happening inside of the interval.

00:03:59.977 --> 00:04:02.326
There we actually are continuous.

00:04:02.326 --> 00:04:04.329
We're continuous over
this closed interval.

00:04:04.329 --> 00:04:07.549
Because of that we know that
the extreme value theorem

00:04:07.549 --> 00:04:09.983
would apply, that there is a well defined

00:04:09.983 --> 00:04:13.345
minimum and maximum
value over the interval.

00:04:13.345 --> 00:04:14.623
You can see that the maximum value here

00:04:14.623 --> 00:04:17.090
happens when at f of negative two,

00:04:17.090 --> 00:04:19.409
it's looks like it's around negative 3.5

00:04:19.409 --> 00:04:20.942
or so, or negative 3.6.

00:04:20.942 --> 00:04:22.566
Then the minimum value over this interval

00:04:22.566 --> 00:04:26.643
looks like it happens
at f of negative five.

00:04:26.643 --> 00:04:29.880
It looks like it's pretty
close to negative four

00:04:29.880 --> 00:04:31.428
because it looks like
we're just increasing

00:04:31.428 --> 00:04:34.295
over this entire interval.

00:04:34.295 --> 00:04:37.607
But the important thing here
is to recognize continuity.

00:04:37.607 --> 00:04:38.781
Continuity does not have to apply

00:04:38.781 --> 00:04:42.059
to the entire function or
over its entire domain,

00:04:42.059 --> 00:04:45.944
it just has to be true
over the closed interval

00:04:45.944 --> 00:04:49.479
that we are trying to
apply these theorems to.


Conditions for IVT and EVT: table

WEBVTT
Kind: captions
Language: en

00:00:00.353 --> 00:00:01.756
- [Narrator] We're told this table

00:00:01.756 --> 00:00:03.821
gives a few values of function f.

00:00:03.821 --> 00:00:05.693
It tells us what f of x is equal to.

00:00:05.693 --> 00:00:08.837
That x is equal to two,
three, four and five.

00:00:08.837 --> 00:00:11.693
Which condition would
allow you to conclude

00:00:11.693 --> 00:00:15.319
that there exists a maximum value for f

00:00:15.319 --> 00:00:18.831
over the closed interval from two to four.

00:00:18.831 --> 00:00:22.039
So pause this video and see
if you can figure it out.

00:00:22.039 --> 00:00:23.536
So you might already remember,

00:00:23.536 --> 00:00:27.212
if we're trying to
conclude that there exists,

00:00:27.212 --> 00:00:29.413
anytime people are talking
about, there exists,

00:00:29.413 --> 00:00:30.693
something over an interval,

00:00:30.693 --> 00:00:31.720
we're probably dealing

00:00:31.720 --> 00:00:33.437
with some type of an existence theorem.

00:00:33.437 --> 00:00:36.260
And if we're talking about
the existence of a maximum

00:00:36.260 --> 00:00:40.745
or a minimum value for a
function over an interval.

00:00:40.745 --> 00:00:42.065
That means that we're talking about

00:00:42.065 --> 00:00:43.048
the extreme value theorem.

00:00:43.048 --> 00:00:48.020
Or we are likely talking about
the extreme value theorem

00:00:48.020 --> 00:00:51.419
and in order to apply the
extreme value theorem,

00:00:51.419 --> 00:00:54.188
sometimes abbreviated EVT,

00:00:54.188 --> 00:00:57.234
we need to know that our
function is continuous

00:00:57.234 --> 00:00:59.267
over this closed interval.

00:00:59.267 --> 00:01:00.715
So, in order to do that,

00:01:00.715 --> 00:01:02.483
we need to know that f

00:01:02.483 --> 00:01:06.150
is continuous over our
closed interval here,

00:01:07.496 --> 00:01:08.913
over two to four.

00:01:09.910 --> 00:01:12.918
The closed interval from two to four.

00:01:12.918 --> 00:01:16.446
So let's see which of these
will allow us to conclude this

00:01:16.446 --> 00:01:18.103
because if we're able to conclude this,

00:01:18.103 --> 00:01:19.862
then we're able to apply
the extreme value theorem,

00:01:19.862 --> 00:01:21.470
which tells us that there exists

00:01:21.470 --> 00:01:23.710
a maximum value for f over that interval.

00:01:23.710 --> 00:01:25.174
Alright, choice A says,

00:01:25.174 --> 00:01:29.405
f is increasing in the open
interval from two to three

00:01:29.405 --> 00:01:33.453
and decreasing on the open
interval from three to five.

00:01:33.453 --> 00:01:34.581
Now you can completely,

00:01:34.581 --> 00:01:37.005
this doesn't tell us that
we're definitely continuous

00:01:37.005 --> 00:01:38.213
over that interval,

00:01:38.213 --> 00:01:39.997
over two to four

00:01:39.997 --> 00:01:40.981
so I would rule that out.

00:01:40.981 --> 00:01:42.653
You could definitely
still be discontinuous

00:01:42.653 --> 00:01:44.661
and statement A still being true

00:01:44.661 --> 00:01:46.557
so I'm gonna rule that one out.

00:01:46.557 --> 00:01:50.800
F is continuous over the closed
interval from two to five.

00:01:50.800 --> 00:01:52.318
Well if we're continuous
over the closed interval

00:01:52.318 --> 00:01:53.431
from two to five,

00:01:53.431 --> 00:01:55.833
we're definitely continuous
over a sub-set over that,

00:01:55.833 --> 00:01:58.071
over the closed interval from two to four

00:01:58.071 --> 00:02:01.622
and so if we're able to
make that, if we know that,

00:02:01.622 --> 00:02:04.478
then we can use the extreme value theorem,

00:02:04.478 --> 00:02:06.966
we've met the condition for
the extreme value theorem.

00:02:06.966 --> 00:02:09.152
To say that there exists a maximum value

00:02:09.152 --> 00:02:10.280
for f over that interval.

00:02:10.280 --> 00:02:12.383
So, I like this one.

00:02:12.383 --> 00:02:13.216
Let's just look at this last one.

00:02:13.216 --> 00:02:16.934
F is differentiable over the open interval

00:02:16.934 --> 00:02:18.267
from two to four

00:02:19.672 --> 00:02:22.005
and at x is equal to two.

00:02:23.175 --> 00:02:24.335
So this is close

00:02:24.335 --> 00:02:28.642
because differentiability
does imply continuity.

00:02:28.642 --> 00:02:31.794
But, it isn't telling us what's happening

00:02:31.794 --> 00:02:32.985
at x equals four.

00:02:32.985 --> 00:02:36.330
We could still be
discontinuous at x equals four,

00:02:36.330 --> 00:02:39.402
we're just differentiable
for every point up to four

00:02:39.402 --> 00:02:40.994
because it's talking
about an open interval.

00:02:40.994 --> 00:02:42.034
So this is close.

00:02:42.034 --> 00:02:43.258
If they said differentiable

00:02:43.258 --> 00:02:45.498
over the closed interval from two to four,

00:02:45.498 --> 00:02:47.154
then this would allow us to conclude

00:02:47.154 --> 00:02:49.042
because differentiability
over an interval,

00:02:49.042 --> 00:02:50.666
implies continuity

00:02:50.666 --> 00:02:51.667
but I'm gonna rule this one out

00:02:51.667 --> 00:02:53.650
because it does not ensure continuity

00:02:53.650 --> 00:02:57.643
at our right boundary, at x equals four.

00:02:57.643 --> 00:03:00.842
Let's do another one of these.

00:03:00.842 --> 00:03:03.146
H is a differentiable function.

00:03:03.146 --> 00:03:03.979
And once again,

00:03:03.979 --> 00:03:06.722
they've given us what our
function h is equal to

00:03:06.722 --> 00:03:08.466
at certain sampled x points, x values.

00:03:08.466 --> 00:03:12.633
Clyde was asked whether
there's a solution to h of x

00:03:13.598 --> 00:03:17.384
is equal to negative two on the interval

00:03:17.384 --> 00:03:20.960
from the closed interval
from negative one to three.

00:03:20.960 --> 00:03:22.664
This is his solution

00:03:22.664 --> 00:03:23.632
and then we're asked,

00:03:23.632 --> 00:03:25.288
is Clyde's work correct?

00:03:25.288 --> 00:03:27.392
If not, what is his mistake?

00:03:27.392 --> 00:03:28.840
And then they give us some choices.

00:03:28.840 --> 00:03:31.472
But, before even looking at the choices,

00:03:31.472 --> 00:03:33.504
let us, pause this video

00:03:33.504 --> 00:03:35.816
and see if what Clyde
is saying makes sense

00:03:35.816 --> 00:03:37.888
and if it doesn't make sense.

00:03:37.888 --> 00:03:39.664
Try to pin point where he messed up.

00:03:39.664 --> 00:03:41.304
What step did he make
the wrong conclusion?

00:03:41.304 --> 00:03:43.848
If he made any mistake at all.

00:03:43.848 --> 00:03:46.344
Alright so, the first thing he did

00:03:46.344 --> 00:03:47.177
is he said look,

00:03:47.177 --> 00:03:49.344
negative two, we wanna
see, is there an x value

00:03:49.344 --> 00:03:51.984
where h of x is equal to negative two.

00:03:51.984 --> 00:03:53.696
So first he says look,

00:03:53.696 --> 00:03:55.688
negative two is between h of negative one

00:03:55.688 --> 00:03:57.400
and h of three, so let's see,

00:03:57.400 --> 00:03:59.400
h of negative one is two

00:04:02.304 --> 00:04:04.887
and h of three is negative five

00:04:05.736 --> 00:04:07.448
and he is right,

00:04:07.448 --> 00:04:11.333
that negative three,
or sorry negative two,

00:04:11.333 --> 00:04:12.661
sits right in here

00:04:12.661 --> 00:04:17.472
so negative two is right
between these two values.

00:04:17.472 --> 00:04:19.745
It's right in between h of negative one

00:04:19.745 --> 00:04:20.995
and h of three.

00:04:21.849 --> 00:04:24.017
Since h is differentiable,

00:04:24.017 --> 00:04:27.674
we know it is continuous
on that closed interval.

00:04:27.674 --> 00:04:28.507
That's right.

00:04:28.507 --> 00:04:29.649
We were just talking about that.

00:04:29.649 --> 00:04:32.689
Differentiability implies continuity.

00:04:32.689 --> 00:04:34.881
Continuity does not always
imply differentiability.

00:04:34.881 --> 00:04:37.313
You can't make that assumption

00:04:37.313 --> 00:04:40.065
but differentiability implies continuity.

00:04:40.065 --> 00:04:41.216
If we're differentiable over interval,

00:04:41.216 --> 00:04:43.369
we're definitely continuous
on that interval.

00:04:43.369 --> 00:04:45.082
And so, he's right here.

00:04:45.082 --> 00:04:47.105
That so far step one and step two,

00:04:47.105 --> 00:04:48.297
we can conclude

00:04:48.297 --> 00:04:50.570
that we are continuous
over the closed interval

00:04:50.570 --> 00:04:52.809
from negative one to three

00:04:52.809 --> 00:04:53.729
and then he says,

00:04:53.729 --> 00:04:57.370
the extreme value theorem
guarantees a solution

00:04:57.370 --> 00:05:00.896
to h of x equals negative
two on that closed interval.

00:05:00.896 --> 00:05:03.121
So this feels a little bit fishy.

00:05:03.121 --> 00:05:04.969
Because the extreme value theorem says,

00:05:04.969 --> 00:05:07.393
it says, if we meet our conditions

00:05:07.393 --> 00:05:09.177
and we did for even the
extreme value theorem,

00:05:09.177 --> 00:05:11.129
we're continuous on that closed interval.

00:05:11.129 --> 00:05:12.897
It says that we're gonna
have a well defined

00:05:12.897 --> 00:05:15.105
maximum and minimum value
on that closed interval

00:05:15.105 --> 00:05:17.664
but we don't care about minimum
and maximum values here,

00:05:17.664 --> 00:05:20.353
what we care is that we take on

00:05:20.353 --> 00:05:24.418
an intermediate value in that interval.

00:05:24.418 --> 00:05:26.651
Value in between h of negative one

00:05:26.651 --> 00:05:27.995
and h of three,

00:05:27.995 --> 00:05:31.115
possibly what each of
those boundaries as well.

00:05:31.115 --> 00:05:34.619
And so, he's applying the wrong theorem.

00:05:34.619 --> 00:05:36.843
It should be the
intermediate value theorem,

00:05:36.843 --> 00:05:40.043
I'll just abbreviate that as IVT,

00:05:40.043 --> 00:05:41.539
is what he should be saying.

00:05:41.539 --> 00:05:43.802
The intermediate value theorem,

00:05:43.802 --> 00:05:45.219
because we are continuous
on the closed interval.

00:05:45.219 --> 00:05:48.491
It says that we are going
to take on every value

00:05:48.491 --> 00:05:49.987
between h of negative one

00:05:49.987 --> 00:05:51.526
and h of three

00:05:51.526 --> 00:05:54.236
and negative two is one of those values

00:05:54.236 --> 00:05:55.723
between h of negative one

00:05:55.723 --> 00:05:57.363
and h of three

00:05:57.363 --> 00:05:59.627
and so if he just wrote
intermediate value theorem instead,

00:05:59.627 --> 00:06:01.275
he would've been correct.

00:06:01.275 --> 00:06:04.525
So let's see which choice is consistent

00:06:05.986 --> 00:06:07.506
with what we just wrote.

00:06:07.506 --> 00:06:09.243
He's definitely not correct.

00:06:09.243 --> 00:06:11.874
Step one wasn't incorrect.

00:06:11.874 --> 00:06:15.291
Negative two is between h of negative one

00:06:16.376 --> 00:06:18.307
and h of three.

00:06:18.307 --> 00:06:20.807
Step two was actually correct.

00:06:22.104 --> 00:06:25.362
We know that differentiability
implies continuity.

00:06:25.362 --> 00:06:26.712
Step three is incorrect,

00:06:26.712 --> 00:06:29.462
yes he applied the wrong theorem.


Estimating limits from tables

WEBVTT
Kind: captions
Language: en

00:00:00.554 --> 00:00:03.777
- [Teacher] The function g is
defined over the real numbers.

00:00:03.777 --> 00:00:07.082
This table gives select values of g.

00:00:07.082 --> 00:00:09.368
What is a reasonable
estimate for the limit

00:00:09.368 --> 00:00:12.388
is x approaches five of g of x?

00:00:12.388 --> 00:00:14.306
So pause this video, look at this table.

00:00:14.306 --> 00:00:17.122
It gives us the x values
as we approach five

00:00:17.122 --> 00:00:19.488
from values less than five
and as we approach five

00:00:19.488 --> 00:00:20.753
from values greater than five

00:00:20.753 --> 00:00:24.304
it even tells us what g
of x is at x equals five.

00:00:24.304 --> 00:00:26.638
And so, given that, what
is a reasonable estimate

00:00:26.638 --> 00:00:27.888
for this limit?

00:00:28.843 --> 00:00:30.843
Alright now let's work
through this together.

00:00:30.843 --> 00:00:34.475
So let's think about what g
of x seems to be approaching

00:00:34.475 --> 00:00:38.556
as x approaches five from
values less than five.

00:00:38.556 --> 00:00:41.616
Let's see at four is it 3.37,
4.9? It's a little higher.

00:00:41.616 --> 00:00:43.949
Is it 3.5, 4.99? Is it 3.66?

00:00:45.177 --> 00:00:48.074
4.999; so very close to five.

00:00:48.074 --> 00:00:51.741
We're only a thousandth
away, we're at 3.68.

00:00:53.095 --> 00:00:54.319
But then at five all of a sudden

00:00:54.319 --> 00:00:57.871
it looks like we're
kind of jumping to 6.37.

00:00:57.871 --> 00:01:00.237
And once again, I'm
making an inference here.

00:01:00.237 --> 00:01:01.665
I don't, these are just sample points

00:01:01.665 --> 00:01:02.971
of this function, we don't know exactly

00:01:02.971 --> 00:01:04.889
what the function is.

00:01:04.889 --> 00:01:07.459
But then if we approach five
from values greater than five.

00:01:07.459 --> 00:01:11.042
At six we're at 3.97;
at 5.1 we're at 3.84.

00:01:12.312 --> 00:01:13.729
5.01; 3.7; 5.001;

00:01:14.774 --> 00:01:15.941
we're at 3.68.

00:01:17.059 --> 00:01:19.956
So a thousandth below five
and a thousandth above five

00:01:19.956 --> 00:01:21.955
we're at 3.68, but then at five

00:01:21.955 --> 00:01:24.036
all of sudden we're at 6.37.

00:01:24.036 --> 00:01:26.158
So my most reasonable estimate

00:01:26.158 --> 00:01:30.158
would be, well it look like
we're approaching 3.68.

00:01:30.158 --> 00:01:32.606
When we're approaching
from values less than five.

00:01:32.606 --> 00:01:34.975
And we're approaching 3.68 from values

00:01:34.975 --> 00:01:37.097
as we approach five from
values greater than five.

00:01:37.097 --> 00:01:40.201
It doesn't matter that
the value of five is 6.37.

00:01:40.201 --> 00:01:42.569
The limit would be 3.68 or

00:01:42.569 --> 00:01:46.609
a reasonable estimate for
the limit would be 3.68.

00:01:46.609 --> 00:01:49.059
And this is probably the
most tempting distractor here

00:01:49.059 --> 00:01:51.466
because if you were to
just substitute five;

00:01:51.466 --> 00:01:52.895
what is g of five?

00:01:52.895 --> 00:01:56.283
It tells us 6.37, but the limit does not

00:01:56.283 --> 00:01:59.916
have to be what the actual
function equals at that point.

00:01:59.916 --> 00:02:03.645
Let me draw what this might look like.

00:02:03.645 --> 00:02:07.812
So an example of this, so if
this is five right over here,

00:02:09.092 --> 00:02:11.816
At the point five the value of my function

00:02:11.816 --> 00:02:15.733
is 6.37, so let's say
that this right over here

00:02:17.098 --> 00:02:19.465
is 6.37, so that's the value

00:02:19.465 --> 00:02:22.956
of my function right over there.

00:02:22.956 --> 00:02:25.706
So 6.37, but as we approach five,

00:02:26.608 --> 00:02:29.628
so that's four, actually let
me spread out a little bit.

00:02:29.628 --> 00:02:32.731
This obviously is not drawing to scale.

00:02:32.731 --> 00:02:37.384
But as we approach five,
so if this, that's 6.37;

00:02:37.384 --> 00:02:40.051
then at four, 3.37 is about here

00:02:40.896 --> 00:02:44.079
and it looks like it's approaching 3.68.

00:02:44.079 --> 00:02:46.650
So 3.68, actually let me draw that.

00:02:46.650 --> 00:02:49.400
So 3.68 is gonna be roughly that.

00:02:50.479 --> 00:02:52.979
3.68 is gonna be roughly that.

00:02:53.930 --> 00:02:58.581
So the graph, the graph might
look something like this.

00:02:58.581 --> 00:03:01.440
We can infer it looks like
it's doing something like this.

00:03:01.440 --> 00:03:04.909
Where it's approaching 3.68
from values less than five

00:03:04.909 --> 00:03:08.214
and values greater than
five, but right at five,

00:03:08.214 --> 00:03:09.560
our value is 6.37.

00:03:09.560 --> 00:03:12.212
I don't know for sure if this
is what the graph look like

00:03:12.212 --> 00:03:13.926
once again, we're just
getting some sample points.

00:03:13.926 --> 00:03:14.905
But this would be a reasonable inference.

00:03:14.905 --> 00:03:17.478
And so you can see, our limit.

00:03:17.478 --> 00:03:20.620
We are approaching 3.68,
even though the value

00:03:20.620 --> 00:03:23.008
of the function is something different.


Impact of transforming (scaling and shifting) random variables

WEBVTT
Kind: captions
Language: en

00:00:00.808 --> 00:00:04.111
- [Instructor] Let's say that
we have a random variable x.

00:00:04.111 --> 00:00:05.904
Maybe it represents the height

00:00:05.904 --> 00:00:09.218
of a randomly selected person
walking out of the mall

00:00:09.218 --> 00:00:10.822
or something like that

00:00:10.822 --> 00:00:15.346
and right over here, we have
its probability distribution

00:00:15.346 --> 00:00:18.159
and I've drawn it as a bell curve

00:00:18.159 --> 00:00:20.244
as a normal distribution right over here

00:00:20.244 --> 00:00:21.933
but it could have many other distributions

00:00:21.933 --> 00:00:23.976
but for the visualization sake,

00:00:23.976 --> 00:00:25.852
it's a normal one in this example

00:00:25.852 --> 00:00:29.370
and I've also drawn the
mean of this distribution

00:00:29.370 --> 00:00:30.452
right over here

00:00:30.452 --> 00:00:33.501
and I've also drawn one standard
deviation above the mean

00:00:33.501 --> 00:00:36.684
and one standard deviation below the mean.

00:00:36.684 --> 00:00:38.441
What we're going to do in this video

00:00:38.441 --> 00:00:41.101
is think about how does this distribution

00:00:41.101 --> 00:00:42.814
and in particular, how does the mean

00:00:42.814 --> 00:00:45.471
and the standard deviation get affected

00:00:45.471 --> 00:00:49.100
if we were to add to this random variable

00:00:49.100 --> 00:00:52.651
or if we were to scale
this random variable?

00:00:52.651 --> 00:00:54.979
So let's first think
about what would happen

00:00:54.979 --> 00:00:57.560
if we have another random variable

00:00:57.560 --> 00:01:01.611
which is equal to let's
call this random variable y

00:01:01.611 --> 00:01:06.255
which is equal to whatever
the random variable x is

00:01:06.255 --> 00:01:08.551
and we're going to add a constant.

00:01:08.551 --> 00:01:10.675
So let's say we add,

00:01:10.675 --> 00:01:12.718
so we're gonna add some constant here.

00:01:12.718 --> 00:01:13.779
I'll do a lowercase k.

00:01:13.779 --> 00:01:14.992
This is not a random variable.

00:01:14.992 --> 00:01:15.864
This is a constant.

00:01:15.864 --> 00:01:17.089
It could be the number 10.

00:01:17.089 --> 00:01:19.298
So if these are random heights

00:01:19.298 --> 00:01:21.088
of people walking out of the mall,

00:01:21.088 --> 00:01:22.924
well, you're just gonna add
10 inches to their height

00:01:22.924 --> 00:01:23.757
for some reason.

00:01:23.757 --> 00:01:24.677
Maybe you wanna figure out,

00:01:24.677 --> 00:01:27.657
well, the distribution of
people's heights with helmets on

00:01:27.657 --> 00:01:30.764
or plumed hats or whatever it might be.

00:01:30.764 --> 00:01:32.120
How would that affect,

00:01:32.120 --> 00:01:36.031
how would the mean of y and
the standard deviation of y

00:01:36.031 --> 00:01:37.031
relate to x?

00:01:38.809 --> 00:01:40.571
So we could visualize that.

00:01:40.571 --> 00:01:43.836
So what the distribution
of y would look like.

00:01:43.836 --> 00:01:46.122
So instead of this,

00:01:46.122 --> 00:01:48.008
instead of the center of the distribution,

00:01:48.008 --> 00:01:51.613
instead of the mean here
being right at this point,

00:01:51.613 --> 00:01:53.649
it's going to be shifted up by k.

00:01:53.649 --> 00:01:55.689
In fact, we can shift.

00:01:55.689 --> 00:01:59.617
The entire distribution
would be shifted to the right

00:01:59.617 --> 00:02:01.367
by k in this example.

00:02:02.589 --> 00:02:04.434
Maybe k is quite large.

00:02:04.434 --> 00:02:07.622
Maybe it looks something like that.

00:02:07.622 --> 00:02:11.789
This is my distribution for
my random variable y here

00:02:13.704 --> 00:02:15.594
and you can see that the distribution

00:02:15.594 --> 00:02:18.297
has just shifted to the right by k.

00:02:18.297 --> 00:02:21.945
So we have moved to the right by k.

00:02:21.945 --> 00:02:24.207
We would have moved to
the left if k was negative

00:02:24.207 --> 00:02:25.929
or if we were subtracting k

00:02:25.929 --> 00:02:28.463
and so this clearly changes the mean.

00:02:28.463 --> 00:02:31.942
The mean is going to now be k larger.

00:02:31.942 --> 00:02:33.749
So we can write that down.

00:02:33.749 --> 00:02:38.478
We can say that the mean
of our random variable y

00:02:38.478 --> 00:02:40.645
is equal to the mean of x,

00:02:42.927 --> 00:02:47.094
the mean of x of our
random variable x plus k,

00:02:48.341 --> 00:02:49.569
plus k.

00:02:49.569 --> 00:02:51.204
You see that right over here

00:02:51.204 --> 00:02:53.842
but has the standard deviation changed?

00:02:53.842 --> 00:02:56.291
Well, remember, standard
deviation is a way

00:02:56.291 --> 00:02:59.563
of measuring typical spread from the mean

00:02:59.563 --> 00:03:01.286
and that won't change.

00:03:01.286 --> 00:03:03.703
So for our random variable x,

00:03:04.879 --> 00:03:07.353
this is, this length right over here

00:03:07.353 --> 00:03:09.520
is one standard deviation.

00:03:10.543 --> 00:03:12.323
Well, that's also going to be the same

00:03:12.323 --> 00:03:14.952
as one standard deviation here.

00:03:14.952 --> 00:03:17.206
This is one standard deviation here.

00:03:17.206 --> 00:03:18.442
This is going to be the same

00:03:18.442 --> 00:03:23.250
as our standard deviation
for our random variable y

00:03:23.250 --> 00:03:27.083
and so we can say the
standard deviation of y,

00:03:27.974 --> 00:03:29.155
of our random variable y,

00:03:29.155 --> 00:03:33.322
is equal to the standard deviation
of our random variable x.

00:03:34.529 --> 00:03:37.840
So if you just add to a random variable,

00:03:37.840 --> 00:03:40.787
it would change the mean but
not the standard deviation.

00:03:40.787 --> 00:03:42.343
You see it visually here.

00:03:42.343 --> 00:03:45.084
Now, what if you were to
scale a random variable?

00:03:45.084 --> 00:03:47.374
So what if I have another random variable,

00:03:47.374 --> 00:03:50.043
I don't know, let's call it z

00:03:50.043 --> 00:03:53.932
and let's say z is equal to some constant,

00:03:53.932 --> 00:03:55.682
some constant times x

00:03:57.277 --> 00:03:59.731
and so remember, this isn't,
the k is not a random variable.

00:03:59.731 --> 00:04:00.564
It's just gonna be a number.

00:04:00.564 --> 00:04:02.065
It could be say the number two.

00:04:02.065 --> 00:04:03.973
Well, let's think about what would happen.

00:04:03.973 --> 00:04:08.864
So let me redraw the distribution
for our random variable x.

00:04:08.864 --> 00:04:10.786
So let's see, if k were two,

00:04:10.786 --> 00:04:12.883
what would happen is is
with this distribution

00:04:12.883 --> 00:04:13.904
would be scaled out.

00:04:13.904 --> 00:04:15.594
It would be stretched out by two

00:04:15.594 --> 00:04:17.844
and since the area always has to be one,

00:04:17.844 --> 00:04:19.761
it would actually be flattened down

00:04:19.761 --> 00:04:24.047
by a scale of two as well so
it still has the same area.

00:04:24.047 --> 00:04:27.773
So I can do that with my
little drawing tool here.

00:04:27.773 --> 00:04:31.940
Let me try to, first I'm
going to stretch it out by,

00:04:32.844 --> 00:04:35.918
whoops, first actually
I'll just make it shorter

00:04:35.918 --> 00:04:37.944
by a factor of two

00:04:37.944 --> 00:04:42.640
but more importantly, it is
going to be stretched out

00:04:42.640 --> 00:04:44.438
by a factor of two.

00:04:44.438 --> 00:04:46.466
So let me align the axes here

00:04:46.466 --> 00:04:48.554
so that we can appreciate this.

00:04:48.554 --> 00:04:51.717
So it's going to look something like this.

00:04:51.717 --> 00:04:53.924
It's going to look something like this

00:04:53.924 --> 00:04:56.664
when you scale the random variable.

00:04:56.664 --> 00:04:58.674
This is what the distribution

00:04:58.674 --> 00:05:01.389
of our random variable
z is going to look like.

00:05:01.389 --> 00:05:04.949
I'll do it in the z's
color so that it's clear

00:05:04.949 --> 00:05:06.542
and so you can see two things.

00:05:06.542 --> 00:05:09.608
One, the mean for sure shifted.

00:05:09.608 --> 00:05:12.775
The mean here for sure got pushed out.

00:05:13.735 --> 00:05:16.596
It definitely got scaled up

00:05:16.596 --> 00:05:21.225
but also, we see that the
standard deviations got scaled,

00:05:21.225 --> 00:05:24.785
that the standard deviation
right over here of z,

00:05:24.785 --> 00:05:27.129
that this is a, this has been scaled,

00:05:27.129 --> 00:05:29.212
it actually turns out
that it's been scaled

00:05:29.212 --> 00:05:30.723
by a factor of k.

00:05:30.723 --> 00:05:33.180
So this is going to be equal to k

00:05:33.180 --> 00:05:36.972
times the standard deviation
of our random variable x

00:05:36.972 --> 00:05:40.137
and it turns out that
our mean right over here,

00:05:40.137 --> 00:05:42.426
so let me write that too,

00:05:42.426 --> 00:05:45.772
that our mean of our random variable z

00:05:45.772 --> 00:05:47.612
is going to be equal to,

00:05:47.612 --> 00:05:50.263
that's also going to be scaled up,

00:05:50.263 --> 00:05:52.960
times or it's gonna be k times

00:05:52.960 --> 00:05:55.793
the mean of our random variable x.

00:05:57.251 --> 00:05:59.494
So the big takeaways here,

00:05:59.494 --> 00:06:02.072
if you have one random variable

00:06:02.072 --> 00:06:04.561
that's constructed by adding a constant

00:06:04.561 --> 00:06:05.870
to another random variable,

00:06:05.870 --> 00:06:08.891
it's going to shift the
mean by that constant

00:06:08.891 --> 00:06:11.386
but it's not going to affect
the standard deviation.

00:06:11.386 --> 00:06:12.897
If you try to scale,

00:06:12.897 --> 00:06:16.254
if you multiply one random
variable to get another one

00:06:16.254 --> 00:06:17.548
by some constant

00:06:17.548 --> 00:06:20.714
then that's going to affect
both the standard deviation,

00:06:20.714 --> 00:06:21.982
it's gonna scale that,

00:06:21.982 --> 00:06:24.815
and it's going to affect the mean.


Indefinite integrals: sums & multiples

WEBVTT
Kind: captions
Language: en

00:00:00.603 --> 00:00:02.510
- [Instructor] So we have listed here

00:00:02.510 --> 00:00:06.394
are two significant properties
of indefinite integrals.

00:00:06.394 --> 00:00:07.718
And we will see in the future

00:00:07.718 --> 00:00:09.276
that they are very, very powerful.

00:00:09.276 --> 00:00:12.283
All this is saying is
the indefinite integral

00:00:12.283 --> 00:00:15.218
of the sum of two different functions

00:00:15.218 --> 00:00:19.880
is equal to the sum of
the indefinite integral

00:00:19.880 --> 00:00:21.789
of each of those functions.

00:00:21.789 --> 00:00:25.008
This one right over here
says the indefinite integral

00:00:25.008 --> 00:00:28.065
of a constant, that's not
gonna be a function of x,

00:00:28.065 --> 00:00:31.879
of a constant times f
of x is the same thing

00:00:31.879 --> 00:00:36.046
as the constant times the
indefinite integral of f of x.

00:00:37.072 --> 00:00:38.452
So one way to think about it is

00:00:38.452 --> 00:00:41.178
we took the constant out of the integral,

00:00:41.178 --> 00:00:42.256
which we'll see in the future,

00:00:42.256 --> 00:00:44.777
both of these are very useful techniques.

00:00:44.777 --> 00:00:48.365
Now, if you're satisfied with
them as they are written,

00:00:48.365 --> 00:00:50.110
then that's fine, you can move on.

00:00:50.110 --> 00:00:52.125
If you want a little bit of a proof,

00:00:52.120 --> 00:00:54.710
What I'm going to do
here to give an argument

00:00:54.710 --> 00:00:57.480
for why this is true is use
the derivative properties.

00:00:57.480 --> 00:00:59.330
Take the derivative of both sides

00:00:59.330 --> 00:01:02.730
and see that the equality holds

00:01:02.730 --> 00:01:05.330
once we get rid of the integrals.

00:01:05.480 --> 00:01:07.240
So let's do that.

00:01:07.240 --> 00:01:10.950
Let's take the derivative
with respect to x

00:01:10.950 --> 00:01:13.030
of both sides of this.

00:01:13.030 --> 00:01:15.450
Derivative with respect to x.

00:01:16.710 --> 00:01:18.920
The left side here, well,
this will just become

00:01:18.920 --> 00:01:22.570
whatever's inside the indefinite integral.

00:01:22.570 --> 00:01:24.900
This will just become f of x

00:01:26.060 --> 00:01:26.980
plus g of x

00:01:28.150 --> 00:01:29.150
plus g of x.

00:01:31.700 --> 00:01:33.730
Now what would this become?

00:01:33.730 --> 00:01:36.030
Well, we could just go to
our derivative properties.

00:01:36.030 --> 00:01:38.540
The derivative of the sum of two things,

00:01:38.540 --> 00:01:41.690
that's just the same thing as
the sum of the derivatives.

00:01:41.690 --> 00:01:43.970
So this'll be a little bit lengthy.

00:01:43.970 --> 00:01:46.150
So this is going to be the
derivative with respect to x

00:01:46.150 --> 00:01:50.030
of this first part plus the
derivative with respect to x

00:01:50.030 --> 00:01:51.700
of this second part.

00:01:53.160 --> 00:01:57.000
So this first part is the
integral of f of x dx,

00:01:57.000 --> 00:01:58.230
we're gonna add it.

00:01:58.230 --> 00:02:01.090
And this is the integral of g of x dx.

00:02:01.090 --> 00:02:02.520
So let met write it down.

00:02:02.520 --> 00:02:06.020
This is f of x and then this is g of x.

00:02:07.650 --> 00:02:09.830
Now, what are these things?

00:02:09.830 --> 00:02:12.240
Well, these things, let me
just write this equal sign

00:02:12.240 --> 00:02:13.530
right over here.

00:02:13.530 --> 00:02:14.790
So then this is going to be equal to

00:02:14.790 --> 00:02:16.710
the derivative of this with respect to x

00:02:16.710 --> 00:02:19.130
is just going to be f of x.

00:02:19.130 --> 00:02:21.470
And then the derivative
with respect to here

00:02:21.470 --> 00:02:23.720
is just going to be g of x.

00:02:24.790 --> 00:02:27.450
And this is obviously true.

00:02:27.450 --> 00:02:29.340
So now let's tackle this.

00:02:29.340 --> 00:02:30.970
Well, let's just do the same thing.

00:02:30.970 --> 00:02:34.840
Let's take the derivative of both sides.

00:02:34.840 --> 00:02:38.860
So the derivative with
respect to x of that

00:02:38.860 --> 00:02:43.260
and the derivative with
respect to x of that.

00:02:43.260 --> 00:02:47.430
So the left-hand side will
clearly become c times f of x.

00:02:49.880 --> 00:02:53.230
The right-hand side is going to become,

00:02:53.230 --> 00:02:55.590
well, we know from our
derivative properties,

00:02:55.590 --> 00:02:58.330
the derivative of a
constant times something

00:02:58.330 --> 00:03:02.390
is the same thing as the
constant times the derivative

00:03:02.390 --> 00:03:03.890
of that something.

00:03:04.900 --> 00:03:07.210
So then we have the integral,

00:03:07.210 --> 00:03:10.210
indefinite integral of f of x dx.

00:03:11.380 --> 00:03:13.930
And then this thing is
just going to be f of x.

00:03:13.930 --> 00:03:18.100
So this is all going to be
equal to c times f of x.

00:03:19.810 --> 00:03:24.050
So once again, you can see that
the equality clearly holds.

00:03:24.050 --> 00:03:25.430
So hopefully this makes you feel good

00:03:25.430 --> 00:03:26.710
that those properties are true.

00:03:26.710 --> 00:03:29.810
But the more important thing
is you know when to use it.

00:03:29.810 --> 00:03:34.120
So, for example, if I were
to take the integral of,

00:03:34.120 --> 00:03:36.460
let's say, x squared plus

00:03:38.780 --> 00:03:42.170
cosine of x, the indefinite
integral of that,

00:03:42.170 --> 00:03:44.620
we now know it's going to
be useful in the future.

00:03:44.620 --> 00:03:46.020
Say, well, this is the same thing

00:03:46.020 --> 00:03:47.990
as the integral of x squared dx

00:03:47.990 --> 00:03:52.050
plus the integral of cosine of x dx.

00:03:52.050 --> 00:03:55.720
So this is the same
thing as that plus that.

00:03:57.690 --> 00:04:00.000
Then you can separately evaluate them.

00:04:00.000 --> 00:04:03.330
And this is helpful, because
we know that if we are trying

00:04:03.330 --> 00:04:05.830
to figure out the integral of,

00:04:08.030 --> 00:04:09.610
let's say, pi times

00:04:10.530 --> 00:04:11.860
sine of x dx,

00:04:16.260 --> 00:04:18.140
then we can take this constant out.

00:04:18.140 --> 00:04:20.370
Pi is in no way dependent on x,

00:04:20.370 --> 00:04:22.380
it's just going to stay be equal to pi.

00:04:22.380 --> 00:04:24.580
So we can take it out and
that is going to be equal

00:04:24.580 --> 00:04:27.750
to pi times the integral of sine of x.

00:04:29.900 --> 00:04:31.980
Two very useful properties,

00:04:31.980 --> 00:04:36.140
and hopefully you feel a lot
better about them both now.