WEBVTT
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we want to describe The graph of f of X
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is equal to X squared plus six X plus C
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over X as seeing Berries. And we want to
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then Graff several members of the family to illustrate the
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trends that we discover. And in particular, we
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want to determine how the maximum minimum points and points
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and inflection move as C changes. We should also
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identify any kind of transitional values of C average.
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The basic shape of the curve changes. All right
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, So first, maybe what we might do is
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to try toe, get some inspiration for what are
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X intercept should be. So let's go ahead and
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make this into a rational function or at least one
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fracture, I should say, So we can revive
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, says X cubed plus six X squared plus C
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all over X. Now I don't know a good
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way to solve for them, you know you're here
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, so I'm just not going to attempt it.
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Um, so let's move on to the first derivative
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and maybe see if we can get some inspiration for
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that. So they take the derivative of each of
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those terms with meetings powerful. So we get to
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X plus six minus C over X square Because,
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remember, see divided by exes really see times X
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to the negative First power. All right, so
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let's do the same thing. Combine this into one
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fraction. We get to X cube plus six X
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squared minus c all over x weird. Now,
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just like before, I have no idea how to
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solve that Cubic in the numerator, so I'm just
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gonna ignore it. For now, let's move on
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to the second derivative and hopefully get some inspiration from
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this. So take the door of the skin would
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get to plus two C over X cube. Using
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the power rule and combining this into one fractured will
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end up with two x cubed plus two C all
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over. Excuse Now the numerator. This is something
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I know how salt pork. So maybe grow up
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to be able to find some of the collection points
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. So that's going to give us two x cubed
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plus two c is zero. Um, subtract the
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tool over and divide by two would get X cubed
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is eager to negative. See taking the cube root
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of you decide to get X is equal to the
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negative cube root of C, so at least we'll
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know how are inflection Point will change, and we
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know it will be an inflection point. Um,
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since we have a cubic in the numerator we know
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at one point is gonna be causing the one point
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is going to be negative just due to its so
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we at least know that will be our point of
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inflection. So we really didn't get much from this
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. But you might know this if, well,
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let's see equal to zero. This fractional part goes
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away and we're just left with the quadratic X cubed
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plus six sex. So why don't we go ahead
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and look at what happens when C is equal to
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zero and maybe we can get some inspiration from that
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? Well, that tells us after Becks isn't going
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to equal to X squared plus six X Now,
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if we set this here equal to zero, well
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, we could go ahead and factor it, and
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it's gonna get those X X plus six is equal
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to zero. So either X is equal to zero
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. R X is equal to negative six. So
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something we might just assume my happens is we're going
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to have these two ex intercepts and they're just going
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to get shifted, left or right, depending on
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our values first. All right. So now the
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first derivative when c 00 willpower rule game to get
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to expose six almost set this equal to zero.
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So subtract the six over. We get two exes
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, You to 96 to 5. You decide by
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two you would get excessive to me. And since
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the graph of this here is going to look something
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along the lines of this, we will know that
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this will be a minimum, since around exited with
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a negative three of the function is decreasing to elect
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a bit of an increasing. So we might say
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that our minimum for this family should be around exiting
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into negative three and then our second derivative that gives
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us two. And, well, that's strictly greater
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than zero. So this implies is always con que
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up. So something slightly different from what we have
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for the more general case. So it looks like
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we'll have exit except around excessive zero. Next issue
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with negative six. A minimum somewhere around X equals
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negative three and his longest X does not equal to
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zero. Um, we will have our inflection point
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at X is equal to negative que group of sea
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. So now let's go ahead and look at these
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graphs. So at C 00 we end up with
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this transitional of value, as you can see,
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because when cuz good, negative one. And cuz
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Goto one, we end up with a distinct differences
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in grafs and you might notice that at least for
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cuz it's a negative one sees a negative 10.
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We do end up with our two ex intercepts being
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pretty close toe negative six and zero. And you
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might also notice at about ecstasy but a negative three
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, which is about right here we have pretty much
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where minimum should be and also on the other side
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for our minimums. Exes go too negative. Three
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. Looks like we have those minimums there. Um
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, the only difference is four are positive values for
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C. We also have these two maximums and another
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minimum that ends up coming. So we didn't quite
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get everything from making that replacement for season desirable.
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At least it gave us a little bit more information
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also for inflection points so we can see about,
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um, negative keeper of seat. You should have
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our inflection points. So over at C zero.
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Negative one. That should be at one. And
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that looks like our inflection point there at negative 10
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. Well, that should be something about two ends
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. Someone of the earth. Not there. But
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somewhere around here, it looks like we have where
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our point of inflection is first season with negative 10
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. Well, it should be around negative, too
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. So we get our plan into action there and
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seizing the one that should be a negative one.
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And it looks like we do have a corn inflection
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.