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Use the guidelines of this section to sketch the curve.
$ y = x(x - 4)^3 $
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Calculus 1 / AB
Calculus 2 / BC
Applications of Differentiation
Summary of Curve Sketching
Missouri State University
University of Nottingham
Idaho State University
In mathematics, the volume of a solid object is the amount of three-dimensional space enclosed by the boundaries of the object. The volume of a solid of revolution (such as a sphere or cylinder) is calculated by multiplying the area of the base by the height of the solid.
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Use the guidelines of this…
we want to sketch the curve. Why is he do X times X minus or cute? Now, in this chapter to give us a laundry list? Oh, steps we should follow in order to graft something. So just go ahead and follow them step by step. So the first thing they suggest that we do is to determine our domain. So we have a polynomial here, factored polynomial, and we know that the domain of the polynomial is just all real numbers or negative infinity. The next thing they say we should find is our intercepts. So are why intercept is going to be when X is equal to zero. So we're gonna have Why is he going to zero zero Bias for and that's just zero are x intersects are going to be when why is equal to zero sort of. What had zero is equal to right. Zero physical too X times x minus for cute and we can use his their product property. The X is equal to zero, and X is equal to the next thing that you suggest we do is to look for symmetry. Now we know that this is not gonna be a periodic function, but we can check to see if of negative X is upper Becks for negative ethics. So plug in end that negative there would get negative X negative X minus four Huge. Now we can factor that negative x out and had become negative. One cube. So negative one cube is just negative one and the negative outfront counsel's weaken X X close or cute. Now, this does not equal to FX, nor that negative the next. So what this tells us is we have no symmetry. The next thing is, we want to look for acid toots. Well, we know this shouldn't have any acid tubs due to it being a partner, but we can use that same logic to look at the end behavior. But let's just go ahead and do that. So the limit as experts, infinity of epidemics. Well, in this case, I mean up about being our function over here. Well, this is essentially why to the fourth power. Um, just because I would be the largest degree and the official would be just one, so we know that even degree polynomial ls with positive politicians go to infinity and we know that the in behavior on both sides of even degree polynomial ls will be the same. So we get that are in behavior will be infinity and infinity. Now, the next two steps I'm gonna go an emergent of one and it's finding our intervals of where the function is increasing. We're decreasing, and also we want to look for max slash mittens. So let's just go ahead and do this on another page toe, save some space. So we have Why is he going to x times X minus four? So we have two functions being multiplied together, So we're gonna need to apply the product. Remember, product will says first function, take the derivative of the second and multiply them together, and then we're gonna add them in the opposite order here. So first we know the derivative of X is just gonna be one. So it's just going to simplify that, and then to take the derivative of x minus for Cuba would have to supply chain. So General says, take the derivative over outside puncture, which in this case is gonna be excused. So that will be powerful. So three X linus uncle minus or squared and then we need to take the derivative of our inside function, which is X minus. Or now the driver of ex my score is going to be one because derivative of X is one derivative 40 not one. And then here we just have X minus four. All right, now, let's go ahead and factor out. Ex. My sport squared and doing that will leave us with three x plus X minus four. So on the inside here we get for X minus four, so we can factor of four hours. Well, four x nice four square and then x minus one. All right, so we have y prime. And if we were to Septus equal to zero, we would get that are critical points. We're going to be X is a good one, and X is equal to or so we'll need these critical points in a minute. We can look at these issues. People being back, sir. Men's all right. So we now know that why prime is equal to or Times X minus four squared and X minus one. All right, and what we want to do to figure out where it's increasing or decreasing. It's the girl worldwide Prime is strictly loved, with zero and strictly less than zero. So for increasing, we want wide prime to be strictly greater than zero. And just to say both some time, I went ahead and did this earlier. Otherwise, there's video a little bit too long. So this function is increasing on 124 and then or two and we can figure out where this function is decreasing by looking at where wide private Shipley lessons there I did once again just for the brevity of the video headed Did this already the negative to one. So let's go ahead and put those critical points we found before down here. So we had that X is equal to one and X is equal to. So now we're gonna imply the first riveted test. All right, so to the left of X is equal to one. We know the function will be decreasing in Texas with one, and then after excessive little one, we know the function is increasing, so it decreases in tow. One increases after, so we know this should be a local men. So I'm saying backs in men, but I should be saying local maximums and now exit before, so it's increasing in before and after four. It's also increasing. So this is not a backs or Ben. It would be a saddle want. All right, so we have our max and our men's tackled. The next thing we want to do is check or contact in and inflection points Okan cavity slashed and collection points. So that means we want to look at Why Double prime and let's just go ahead and go back to that last page of Cigar. What? Why Supply Double Prime is going to equal to well, the derivative of two functions being multiplied together again, Remember, since or is a constant, we can just go ahead and factor that out, since the derivative is linear or squared and X minus one. So we're gonna apply product cool once again. So we're gonna have X minus four. Where times the derivative of X minus one plus. Explain this one time too derivative with respect, X um X minus four squid. So you get explain its or square. So, just like before, the derivative of X plants, one is just want to be, uh, one, and you take the derivative of X minus or square, going out to apply change world and, uh, chain rule. So first used Probable two. That's minus four now for the first power. And then we need to take the derivative of our inside function, which is X minus four. And again, the derivative of X Vice Lord is just going to be one. Or let's go ahead and put this up a little bit. Um, so first, let's go ahead and background ex my sport from this X minus four. So that would be fine. X minus. For there had been a plus two x minus two before X miles more simplify the inside. Here we would end up with three X minus six and like a factor a three out of there to get four times x minus four that a fact about that threes. There should be 12 out front here and exploits for and X minus two. So if we accept this equal to zero, we would get that are points X is equal to and X is he good for? So we could use these in a moment to look for points of an election. So let's go ahead over here and right down with our second orbit of days, which should be, well times X minus four times. That's minus. And to figure out where to calm. Keep up a calm down. We're going to need to look at where Why Double kind is strictly Larkin narrow and strictly lesson. So Khan taped up is when, Why, that will find a strictly larger than Europe. And then, once again, just for the brevity of the problem, I went ahead and solved this already. So this is gonna be evident. Did you union with or to infinity? And over here for down this will be one lie double promise strictly lessons era. And we would be from 2 to 4 Now are two possible points of inflection are gonna be X is equal to and X is equal before So let's go ahead and see so to the left of X is equal to two. We're going to be con cade down to the right of two. We're going to be, uh, to the left of two were actually on this interval here, So we should be Khan paid up and then to the right of two were on the other interval for being down. All right, so this here would tell us we have, ah, inflection point because cavity changes at accessibility. But then at exit Eagle four after four, it will be Kong cape up. So both of these are points of and lecture. So that was everything they suggested we do before we actually start to sketch this. All right, so let's go ahead and first just locked down our interest. Since so we have our Y intercept would be 00 Our X intercept should be zero and executed 41234 So there was no symmetry in the problem. Our Assam totes will no acid trips. But we at least know the in behavior of this craft is going to be going to infinity on each side. We know we're going to have a local men at Exit one. So there's a man around here somewhere and we could figure out whether that needs to be above the X axis or below the ex Access access in a second and at X is equal to so maybe actually above the one I'll quit men and then at two, and or we will have inflection points, so I p Hi, Pete. So look ahead and start with this. So we're going to start from infinity and we're gonna go down until we have our first point. We're just going to be 00 and then at this point, our next important point is our minimum, that ecstasy with one. So we know it's going to curve up something like this here and at exit to we're going to have a change in inflection. So it starts to go from conch Yup, to concrete, down and at exit before we have the same kind of slowly comes in and does something like that. They're so maybe that curve at Exeter before should be a little bit Maur intense, something kind of light This so it's a kind of flatten it out a little bit more, but they used everything over here. We used our inflection points. We can see the con cavity where it's increasing and decreasing. And if I was able to draw a little bit better, that might look a little bit prettier. Um, something else you could do is go in and maybe say what value this occurs that but as far as I'm concerned, since we're just trying to Oh, as well as this inflexion point there. But as far as I'm concerned, just cause we're trying to sketch it, I don't think these numbers matter all that much.
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