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University of North Texas



Problem 30 Medium Difficulty

Describe how the graph of $ f $ varies as $ c $ varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when $ c $ changes. You should also identify any transitional values of $ c $ at which basic shape of the curve changes.

$ f(x) = x\sqrt{c^2 - x^2} $


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Video Transcript

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

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