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University of North Texas

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Problem 8 Medium Difficulty

Use the guidelines of this section to sketch the curve.

$ y = (4 - x^2)^5 $

Answer

see solution

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Video Transcript

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