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Use the guidelines of this section to sketch the curve.

$ y = \frac{x^3}{x - 1} $

see solution

Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Differentiation

Section 5

Summary of Curve Sketching

Derivatives

Differentiation

Volume

Campbell University

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

Lectures

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we want to sketch the curve of Why is it to execute over X minus one? Now, in this tractor, they give us this laundry list of steps which Paula, Let's just go ahead and follow it. So the first thing they tell us we should do is to determine what our domain is going to be. And we know for rational functions, we just need to check to see where our denominator is equal to zero. So we don't want our done. I'm there to be zero in this case, and that would be one exit one. It's our domain is gonna be negative. Entity to one union, one the impending. The next thing they tell us to find it is to look for our interests. So intercepts. So let's look for X intercept first. That's when we set. Why equals zero you get there was he would execute over X minus one. So nothing the denominator. You never make this good to zero. So I just had the numerical zero, which tells us zero is equal X. And since zero is equal to X, that also tells us what are why intercept is going to be. We'll also just B Y 00 So the only intercept we have 00 so we don't need assault for that. The next thing they tell us to look for is symmetry. So rational functions aren't really known for being periodic. But we can check to see if this is even or odd. So would get negative X Q times negative X minus one. Now, negative excuse would just be negative times X cube, so we can factor that out and then distribute it to the denominator. And we get X cubed over X plus one. Now, this here does not equal to FX, nor is it negative FX. So there is no symmetry, at least no symmetry about the why access wore about the origin. The next thing they tell us to look for is Assam tips. So let's go ahead and see. So we're no, we're going to have a vertical ascent over at Exit one. So let's go ahead on and see the behavior as we approach one from the right of at the books. So this is gonna be one from the right, huge over one from the right, minus one now, one from the right is going to be a positive number. So if I cube still going to be possible and if I take one from the right Well, that's something slightly larger than one, and I subtract it from one that's still gonna be positive. So from the right of one, which of the approaching positive infinity And since explain, this one has a odd degree or a degree of one. We know that this vertical Assam tote should have opposite in behavior. So from the left of one that should go too negative. Infinity. All right, now something else we can go ahead and check. Is our in behavior for this as well? So let's see. So the limit it puts us in green, so the limit has X approaches. Infinity of F of X is going to be well X cubed approaches positive energy and X minus one Purchase positivity. So this is going to go to possibility and the limit as ex purchase negative fnd f of X is going to go too well. Negative. Vex goes to negative infinity and negative x my swamp. It's negative infinity. So overall would go to positive. And we know that our horizontal asado is going to go deposit or negativity just due to the fact of O. R. Degree is larger in the new mayor than in the denominator. Now, the next thing they want us to find is our intervals were the function is increasing and decreasing as well as any local. Max is four minutes. So we're gonna need throughout. What? Why Prime is equal to so see that another page? Why is he going to execute over X minus one? So take the story that we're gonna need to apply. Questionable, Questionable says hello. Hi. Minus high d low all over the square of what is below. So we know the derivative of X Cube is going to be three x squared using power rule and the drill Bit of experience. One role during the Texas one derivative negative 10 But that's just one day. Now if you go ahead and do this, algebra here should be left with X squared two X minus three Products Street all over X minus one squared. So let's go ahead. And I should probably do that. Let's find our possible critical value for our critical values so we can find her possible Max's Airmen's. So this is gonna tell us either X is equal to zero or two. X minus three is zero. So that gives us specs. Easy to perhaps so possible. Backs is a men's or a zero ed three house, but so we have X squared U X minus three all over X minus one squared. Now this function will be increasing. Where, Why? Print is strictly larger than zero and already went ahead and solved this beforehand. And this is negative. Infinity to zero union, 0 to 1 union, three halves to a bed and this function will be decreasing. Or were why promise? Strictly less than zero on the last piece of this, which is 123 house. Now let's go ahead and put those values we come before, So he had X x zero X is equal to three hubs. Well, we know to the left of zero, the function is increasing and to the right of zero, it will be increasing until one and then from 1 to 3/2 the function is going to be decreasing and then to the right of three halves, the function is going to be increasing. So this tells us that at X equal dessert we will have a salad point and excessive with three House will be a mogul minimum. Now the last thing they suggest we find is where are function has any inflection points. So we need to find Khan Cappie con on inflection points. So we need to know if I double primates, let's go ahead and find that alibi. So why Double Prime is going to equal to, so we're going to need to use quotient, rule and product cool for this one. So first, let's go ahead and apply the product. Cool. I mean, the questionable and I might be a little bit more space on this, so it's still going to be low. Hi X squared times two X minus three. I actually don't even need to do product because we could distribute that. It's let's do that first, Actually, sufficiency two two x cubed minus three X quick and then minus then in the opposite order to execute minus squared times the derivative. Oh, but we have ended in, um, there's two nice ones and then all over what we have in our denominator squared X minus one, and it was squared before. So now it should be to be power. Now to take the derivative, uh, here, we're gonna need to use power will preach. So it's gonna be six x squared, minus six x and the derivative of exploits. One swear they need to power and changeable. So it's gonna be two times X minus one times the derivative of X minus one. Did it change Will, Which would just be one? And if we go through and simplify all this algebra wound up with two ex over X squared minus three x mostly all over X minus one. Cute. Now we want to set the secret zero so we can find our possible points of inflection that we're going to get ex busy with zero or X squared minus three X plus three zero. So it turns out that this year has no riel solutions. So the only possible point of reflection will have is that X is equal to zero. So let's go ahead and write down our second derivative here, which is going to be two x x squared minus reacts plus three all over X minus one. Cute and again, I just went ahead and it's all for it's gonna become came up. Calm down, look for hand. So conch a boat is going to be where this is strictly larger than there And this happens to be from negative and infinity 20 Union one to infinity and the function is conch aid down when? Why Double prime strictly less than zero all over zero now are possible point of inflection Was that exit so to the left of zero the functions calm keep up and to the right of zero functions concrete down. So this year will be a inflection. And once we did this, it said we can go ahead and actually started graphing. So let's put our intercept first. We have an intercept at only the organ. We have no symmetry. We know are absent toes. So the in behavior is going to be to infinity on each side. And we have vertical awesome totes at Exit one, so X is equal to one. So to be right of this, we should go into infinity and to the left. We should be going too negative Infinity. So at X is equal to three house we know we're gonna have a local men. Let's go ahead. So say this is what happens here. And since we're coming from positive infinity and we have no other Exeter sets or anything like that, we know that our minimum is going to be like that. And we know we have a point of contact bitty at X. So we wanted all of our important pieces. Now we could just go ahead and start connecting lines. So let's go ahead and start to the right of our words on plastic, our bird a classic. So we're starting from positive, Benny, and we're gonna go until we hit our local men. And then it's going to just go up and come next like that and then on the other side. Well, we know that at exit with zero is intercept and we also share the changing cavity. So it should look something like this here, and I'm just gonna reset and hate over they're connected. But this here should be a nice little sketch of our graph. You could possibly go back in and actually say what this minimum is here. But since we're just trying to sketch graph, I think this is sufficient

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