WEBVTT
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We want to describe how the ground of the black
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Sea z x times screw that C squared, minus
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expert buried as C Berries. Then we want to
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graph several members of the family to illustrate the trends
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that we discover. And in particular, we want
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to investigate how the maximum minimum points and inflection points
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very see various. And we want to widen by
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any transitional values. Upsy which some basic shape of
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the curve changes. So first, something we might
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want to do is Pharrell what our domain dysfunction is
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going to be, uh, and also identify our
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x intersects. So first, let's just find the
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domain. So what do homicide here? So domain
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? Well, we need to make sure that C
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squared minus X squared is going to be greater.
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Their equal zero. Well, we could go ahead
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and addicts Universum get C squared plus two x square
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. And when we square root each side, we
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end up with C. I should actually put out
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the value of C is less than or equal to
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ex lesson or equal to the negative absolute value of
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C. Since we don't know if she's going to
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be possible negatives Or at least we know our domain
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is going to be those that use their. And
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now let's go ahead and figure out our intercepts so
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we can go ahead and took the secret zero s
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. So that tells us either X is equal to
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zero or the square root of C squared minus X
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squared zero, which will be the same thing,
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is just C squared minus X where is equal to
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zero and following pretty much the same thing. Bella
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toes that X is going to eat in the plus
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or minus absolute value of seats. And since we
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have the absolute value, we could really just right
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is expected to plus or minus. All right,
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so we know our ex intercepts, but we know
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our domain. And one thing we should also know
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this is that C cannot equal to zero for any
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about because if it is, we end up with
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X square root of negative X squared and expert is
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always going to be positive. So I'm always taking
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the negative of a square or I'm always square doing
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something that is negative, which is underfunded. All
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right. So, really, Scott, all of
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the information out of just F of X that we
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can extract. Or at least I think we can
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extract. So let's go to the first trip.
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So prime of X is going to be well,
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we're going to need to use product will take this
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derivative. So taking the dribble of our first function
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extra just in this one. Selfie C squared minus
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X squared plus X times the derivative of square root
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. They're ready to use changeable for that. So
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we still have the X. And then it's going
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to be 1/2 one over the square root of C
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squared minus X squared, and we take the drift
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on the inside, which is going to be well
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, the derivative of C Square's gonna be zero negative
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. X squared becomes, too are negative to X
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Noticed. These twos here can cancel out and those
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exit just become ex squid. So let me go
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under. Relentless c squared my sex where both that's
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where it over see scored. Find sex Bird Square
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boot. Well, let's go ahead and add these
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into a one direction, so I would need to
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multiply the luck in the left by the square root
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of C squared my sex wherein which would just be
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C squared minus expert and the numerator And I almost
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forgot my negative right here. So then negative x
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were all over the square root of C squared minus
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x squared. And then that simplifies to C squared
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minus two x squared over. Understand? C squared
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my sex square square. Now, if we were
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to set this here zero well, we just had
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the numerous equals zero. And that would tell us
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that C square isn't too too. Times X squared
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, square moving divide. My treatments were rooting inside
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to get X is equal to plus or minus C
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squared over too. And if we want, we
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can rewrite this again as just plus or minus c
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over the square root of two. And so this
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here will be within our domain since, um,
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see will be, at least within our domain and
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then dividing and by a deposit number will also keep
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us in our domain. Um, then let's look
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at the second to repeat it. So I'm going
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to use this one here to take the derivative.
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So we're just gonna use pushing rules, remember?
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Questionable says low the eye So we're gonna have C
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squared minus X squared square rooted then the derivative of
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our numerator. So that's just going to be negative
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for X and then minus them in the opposite order
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. So C squared minus X squared times the derivative
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of the square root of See My second word.
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And we already found that from right here. So
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let's just going to be X over this world of
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cease word by sex. Weird, negative. And
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then all over what we have in our denominator squared
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. So just be C squared, minus X squared
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and then everywhere to just go through and do that
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algebra we should quit two x cubed minus three c
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squared that's all over C squared, minus, exploited
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to the green. And then if we said this
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equal to zero But we just let the new Marie
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equals zero So naturally just asking First assault for two
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x cubed physical to three C square. Actually,
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let me go ahead and factor in X out first
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. So x two X squared minus C squared is
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equal to zero. So by the zero product property
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that tells this either exit with zero or this other
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expression is a 02 X squared minus are Let me
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just go ahead of two equals C squared Divide by
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two square group. We get X is equal to
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plus or minus three C squared over two square Reuben
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. And again, let's go ahead and pull that
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sea out without value. But plus or minus makes
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soldiers really matter. So we get plus or minus
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C times the square root of three over two.
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Now notice that the square root of three of the
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two something like this on your side spirit of three
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over two is strictly larger than one. So if
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it's positive, is going to be strictly larger than
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the absolute value of see and if it's negative,
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is going to be strictly less than so. This
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here is actually not in domain. So if we
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have an inflection point, it's only going to be
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at X is equal to zero. All right,
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so I think this is all the information we can
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really pull out from this. Oh, and maybe
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over here I should say that the reason by really
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this, remember, is to find our possible maximums
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and minimums. But so I went ahead and graft
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a handful of these for burying values of seat,
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and so are zeros end up being where we expect
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them to be. So at plus or minus one
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with value proceeds and at zero in each case,
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so at sea is even to 1/2. We get
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it at taking 1/2 half zero season, plus or
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minus one negative+110 And for a season with plus
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or minus five, we get zero and negative thoughts
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. Our maximums and minimums are all pretty much where
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we would expect as well. So if we divide
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one by the square root of two, well,
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that's going to be about 1.8. So that's about
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where that's B and the other one's air, about
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the same as well we have our inflection point at
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X is equal to zero. So it looks like
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everything we have pretty much matches up with what we
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got from the first part.