WEBVTT
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we want to sketch the curve. Why is it
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for my sex where to be the power? So
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in this shocker, they give us this laundry list
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of steps we should follow whenever we want to grab
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something. So we're gonna follow those steps. So
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the first thing they tell us is to determine our
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domain. Well, this here is a somewhat factored
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polynomial. So we know our domain is going to
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be all build numbers. The next thing they tell
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us to do is to find our intercepts. So
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intercepts and let's go ahead and find our why intercept
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first. So why intercept? This is when X
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is equal to zero. So it's gonna be why
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is it too? Or minus zero squared, which
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would just be zero. And so it before to
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the Yeah. All right. Um, so I
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really don't care what this number is. I just
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care that it's possible Negative. So we're just gonna
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leave in this sport, but you can rewrite it
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if you want. Next. Let's go ahead and
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find our ex intercepts. So now this is when
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Why is it zero? So we're going to have
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zero is equal to or minus squared to the fifth
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, Will. This implies that we really just have
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00 to 4 minus X squared, which is X
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is equal to plus or minus two. And the
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last thing is, they want us to find symmetry
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. I'm going with him. Move these over a
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little bit, but I lost my zero along the
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way. Next. So Step three should be symmetry
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, so we don't polynomial. They're not gonna be
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periodic, but we can look to see if it's
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even on persimmon tree about the wire. Ex access
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. I mean, the Y axis were the order
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. So plug in this inn, we're gonna get
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four minus a negative X square to the foot.
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It's wearing the negative extra sees a sex. So
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we get four minus x squared to the phone,
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which is EPA, Becks, and by epa,
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Lex. I mean, the function we're working with
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up here, so this implies that this is an
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even function. So you have symmetry across B Y
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access. The next thing, they want us to
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find his ass. So we know that polynomial should
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not have any acid. Oops, but we can
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go ahead and look at the end behavior with the
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same idea. So the limit as X approaches and
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very of pepper vex Well, this would be really
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so I'm just kind of break. This is for
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very large value is going to be cool, too
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. Negative x to the 10. So we know
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it is even degree polynomial the negative. So efficient
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. So this year should go too negative. Infinity
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. And since it's and even degree polynomial whatever behaviors
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on one side, it'll be on the other.
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So this should also be negative. The next thing
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we want to find is intervals of and I'm gonna
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combine this Step six Awesome. We want to find
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where the function is increasing, slash decreasing and any
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local Max Cashman's that we have. So that means
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we're gonna first need to find what wide prime it's
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. So let's go ahead and do that on another
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page, since this will take a little bit of
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work. So why is equal to or minus X
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squared to the left? Well, the first thing
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I'm gonna do is just factor out a negative from
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here. So I wanna write negative X squared minus
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four to the fifth. Remember, we can factor
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that out since it's an odd power, all right
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? And I just want to do this because it'll
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help me keep things in place when I'm taking derivatives
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and stuff. So the first derivative will be,
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Well, we're gonna need to use the Powerball and
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changeable for this. So first would have squared minus
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four to now the fourth power, and then we
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need to multiply a five out front. Then we
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need to take the derivative over inside function, which
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is going to be X squared minus sport. And
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we know the derivative of this here is just going
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to be two x, so we end up with
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negative 10 x x squared minus four to the fourth
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. Now, we're going to go ahead and set
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up a secret zero because we need to know that
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critical points of this Help us find our local maxes
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and local mittens if we do happen. So this
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would imply that there X is equal to zero or
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X squared. Minus four is a good a zero
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. And once again, this is gonna be X
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is equal to plus or minus two so critical place
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. They're zero and plus a nice to you.
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All right. So now let's go ahead and write
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down. What wide promise. So, Floyd prime
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is going to be negative. Jim Specs, times
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X squared minus four to the fourth power. Now
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we can go ahead and determined where this is going
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to the men that makes you write this right,
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So increasing. Which will be where? Why,
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Prime, it's strictly larger than zero. So I
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just went ahead and already solved for this. Just
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thio cut down the length of the video a little
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bit. That would take too long to do this
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. So we would have that this is strictly grade
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zero on the interval of negative infinity to negative too
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. And then union negative too. 20 And we
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know this is going to be decreasing when Why Prime
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is strictly less than zero. And this would just
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be the rest of the interval. Minus are critical
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points. So it would be zero. Did you
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union to To Infinity. Now let's go Then plop
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down those critical points we have. So we had
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negative to zero and two and we know this will
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be increasing on negative negative too and negative. 2
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to 0. And we know it's going to be
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decreasing on 0 to 2 and two to infinity.
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So that tells us Negative, too, is a
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saddle point by the first derivative test two is also
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a saddle point by the first derivative tests, since
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we do not have a change in the derivatives.
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But we get that zero here, since it's going
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from increasing to decreasing. Should be a local backs
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, all right? And then the last thing they
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suggest we find is the con cavity of this and
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our inflection points. So we'll need to know what
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the second derivative of this function will be. So
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let's go over here and do that also. So
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we found that. Why prime is this right here
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? So we want to take the derivative of that
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and we'll do that by using the product. Cool
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. So why double prime is going to be and
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I'm gonna factor that negative 10 out front here,
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remember, product will says, keep the person you're
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multiplying and then multiply by the derivative of the second
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function. So X squared, minus four forth.
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And then we're going to add to this where they're
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switched so X squared, minus war before and then
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the derivative with respect to X of the other culture
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, which is X. So now we already know
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the derivative of X is just going to be one
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. And to take the derivative of expert minus four
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to the fourth, it'll be just like before where
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we had to use changeable. So the board is
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gonna come out front, X squared minus four and
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ours to the third power. Then you need to
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take the derivative of our inside function so expired minus
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four. And then over here, we'll just have
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X word minus four to the fourth Power. Now
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the derivative of X Square is gonna be to expect
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power will get and then the derivative for just zero
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, since it's a constant. So let's go ahead
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and clean that up a little bit. This could
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be negative. 10 eight x squared, X squared
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, minus or thirds waas x squared minus four to
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the port. So we could go ahead and factor
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out this X squared minus four to the third.
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And doing that will leave us behind with eight x
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squared plus X squared minus four, which could be
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simplified down further. Two nine X squared, minus
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So now if I were just setting this equal to
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zero will notice We have two perfect squares here,
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and we already know that this here will get us
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X is equal to plus applies to. And this
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here will give us X is equal to closer minus
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two. So factor that since they're perfect squares I
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mean the difference of squares, not perfect squares.
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And it should just be plus or minus. Those
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answers are so we'll need to know that there as
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well. All right, so let's go ahead and
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write down our second derivative, so it should be
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negative. Tip X squared by this four cubed and
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nine X squared minus four. Now, just like
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we did before, we want to figure out where
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this would be con cave up, calm down so
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we can figure out our inflection points. So Kong
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Kate up is gonna be when? Why, double
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private? Strictly greater than zero. And then,
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just like before, just for the sake of brevity
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, we will, um, I'll just say where
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this occurs so we don't have to go through all
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the tedious algebra. So this year will be on
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negative too, to negative 2/3 and also 2/3 22
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And then the remaining interval or inter bowls will just
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be where it's Kong came down. So negative.
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Infinity two magnitude union negative 2/3 two, 2/3 and
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two to infinity. Now we can go ahead and
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use this to figure out our once of inflection.
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So before we found negative too negative. 2/3 2/3
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and two. Now we know this is Kong.
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Keep up on negative too, to negative 2/3 and
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2/3 to 2 and it will be conquered down on
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the rest of these intervals. So negative 2 to
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2 has a switching call cavity. This is inflection
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point the same thing for negative 2/3. Same thing
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for 2/3 and same thing for so all for these
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are points of inflection. All right, now that
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we have this done, it tells us we could
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go ahead and finally sketches. The person we should
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probably plot is our intercepts, and we're going to
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have zero negative too. And two, or are
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so 0.0, we're going to settle our why Intercept
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actually was supposed to be up here at or to
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the fifth and then the ex intercepts are negative,
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too. All right, so we have that.
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And by number three, we know this is asymmetric
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function. So if we just plot the left hand
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side of this, then we could just reflect that
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graph across the Y access. All right, so
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on the left, we have are in behavior going
208
00:15:52.759 --> 00:15:54.909 A:middle L:90%
to negative 20 from part for, so we know
209
00:15:54.909 --> 00:16:00.159 A:middle L:90%
we need to start down here. And what other
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00:16:00.159 --> 00:16:06.320 A:middle L:90%
important points do we have? What? Negative,
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too. We have a saddle point. So that
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00:16:07.289 --> 00:16:14.549 A:middle L:90%
means it should fly and helped right here, and
213
00:16:14.549 --> 00:16:18.549 A:middle L:90%
we know it. Zero. It will, um
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00:16:18.399 --> 00:16:21.169 A:middle L:90%
, be a max. And we also have inflection
215
00:16:21.169 --> 00:16:27.049 A:middle L:90%
points that negative 2/3 so negative 2/3 will need an
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00:16:27.059 --> 00:16:33.159 A:middle L:90%
inflection point. So let's go ahead and start wrapping
217
00:16:33.169 --> 00:16:37.039 A:middle L:90%
this now. So you started negative, Penny.
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And the first point of interest is going to be
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negative, too. And like I was saying,
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it needs to flatten out here, since this is
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00:16:42.860 --> 00:16:47.139 A:middle L:90%
a saddle point, and it's also a point of
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00:16:47.149 --> 00:16:49.730 A:middle L:90%
where our cavity will change. And then we hit
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00:16:49.730 --> 00:16:53.429 A:middle L:90%
negative 2/3 and the capacity should start to change as
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00:16:53.440 --> 00:17:03.259 A:middle L:90%
well. We hit our maximum here, zero from
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00:17:03.840 --> 00:17:07.849 A:middle L:90%
park five and six. And now all we need
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00:17:07.849 --> 00:17:10.869 A:middle L:90%
to do is reflect this graph across the Y access
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00:17:11.079 --> 00:17:15.849 A:middle L:90%
due to seven tree. So, bye, really
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00:17:17.539 --> 00:17:22.519 A:middle L:90%
? You can just go ahead and starts the same
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00:17:22.569 --> 00:17:27.180 A:middle L:90%
thing you did before. So it flattens out here
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00:17:27.190 --> 00:17:33.089 A:middle L:90%
and then goes down too. Thank you. So
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00:17:33.089 --> 00:17:33.490 A:middle L:90%
you could go ahead and be a little bit more
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00:17:33.500 --> 00:17:37.599 A:middle L:90%
accurate with this if you want. But since we're
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00:17:37.599 --> 00:17:38.200 A:middle L:90%
just trying to sketch it, I don't think we
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00:17:38.200 --> 00:17:41.299 A:middle L:90%
really need to add all these numbers and stuff.
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00:17:41.440 --> 00:17:42.980 A:middle L:90%
But you could always go back and possibly figure out
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00:17:44.400 --> 00:17:47.619 A:middle L:90%
Like what this actual value here is for the changes
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00:17:47.619 --> 00:17:48.299 A:middle L:90%
on cavity. Same thing on the other side.
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00:17:48.339 --> 00:17:51.740 A:middle L:90%
But I don't think it really matters all that much
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00:17:51.740 --> 00:17:52.190 A:middle L:90%
since we just want to sketch it.