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Use the guidelines of this section to sketch the curve.
$ y = x^4 - 4x $
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Calculus 1 / AB
Calculus 2 / BC
Applications of Differentiation
Summary of Curve Sketching
Missouri State University
Oregon State University
University of Nottingham
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 a graph of curve. Why is it X equal with minus or ex? So in this chapter, they give us this laundry list of steps. We should follow an order to graph something, so it's just kind of go through them step by step. So the first step is they want to figure out what the domain of dysfunction. Well, this is a polynomial, and we know polynomial is have a domain of all real numbers or negative infinity. The next part is one to figure our intercepts, so intercepts. So let's go ahead and find our Why intercept first, just because that's a little bit quicker. So why intercept is when X equals zero? So that would give us why is equitable? Extra support is zero and then four x zero. So we get zero for why interested and now for our X intercept. Remember, he said, why equal does aerial? So you get zero is equal to X to the fourth minus for X. Let's go ahead and pull that or or the X out, and we get X Times X huge minus or so This tells us either X is equal to zero or X visible to the cube root of four. And this year is approximate the one point by. All right. Next, I did not go next. We want to see if there's any symmetry. Intergraph, which is we want to look at of negative X and so doing this year. And so by EPA, Becks, I just mean our original function here. Why? So we can go ahead now and plug in negative X into here and doing that so plugging negative X and X support will just be extra aboard and then plug in negative x in for X with the negative X and that it would become plus four X. Now, this here does not equal to FX, Nor is it negative FX. So what this tells us is no symmetry court, so we got the first little bit done. Now, the next thing they suggest we buy is any acid trips. So we know since this is a polynomial, there should be no acid tips. But we can at least follow the similar steps to figure out it. What kind of in behavior we should have for this. So what we're going to look at is the limit as X approaches Infinity of affects. Well, this is a court IQ function, and we know those kind of look like x squared. So since I was a positive coefficient, we know this is going to go to infinity. And likewise, we know what we have been. Even highest power for a polynomial. We will have the same. So the limit as X approaches negative penny of after Max also goes to the next thing they want. Supply is where this function is increasing and decreasing, so increasing slash decreasing intervals. So that means we want to find why pride. So why prime is going to be so be derivative with respect to X Oh X to the fourth minus war it's and to take the derivative of both of these weeds powerful. So first we will get or X now Q. Because we need instruct all that power and then the derivative before X a jest or so. Let's go ahead and factor that four out. So four X cubed, minus one. So if we were to just kind of look at what X cubed minus one looks like we know it'll look something along the lines of this now it might kind of curve, but at least we know it'll look something like It'll fly an island in, Go back again so well that toeses this here. So why prime will be less than zero win. So let's sexually first step that equals zero to figure out. But that should be so if we said it equals zero that just tell those X should be one So too the left of that point We know the function would be negative. So this will be less than zero when we're on interval. Negative Infinity Q zero. And to remember this here tells us that the function is decreasing and when wide prime is strictly larger than zero. So this should be zero should be infinity too. The value What? So because of that our necks, then it would be one to infinity. And this here is our interval where it is increasing. Oh, so normally you would just set this equal to it are put into a inequality to figure out where it's bigger than equal to zero. But we're just gonna take the cheap way out so you don't have to do more algorithm with you all right. The next step is to find our local Max is slash men's and what we need to find those are critical points and we already found that are critical. Point is exiting the one because we want to find a wide prime is zero and we already know that sex is equal One last step. Now we can use the fact that we know on the interval of negative finito won, the function is decreasing and then it's increasing after. So this tells us we have a local men at exit one. The next step is to look for con cavity and any inflection points we might have. So that means we need to look at the second derivative. So why double price? So we're gonna go ahead and take more X cubed minus four and take the derivative of that that would be derivative with respect to X or excuse me in this war, so would go ahead to use power would take the derivative of X cubes, love the three x squared. We have well X squared and in the derivative of or we'll just be zero. Now notice that this function here, we'll always be greater than or equal to zero. And because of that, we know this wall be con cave up so con a Oh, and also there will be no inflection points Now that we have all this information, this was the last thing they thought was important so we could go ahead and try to grab our option. So let's go ahead and first just locked down our exit. My inner sense. So we have our X intercept being or a wire set being 00 And we have our X intercept 10 and the other one is about one point. Just go ahead and put you grew Oh, four. All right, so we have those intercepts style, So there's no symmetry our ass in tokes we know Well, there were no acid totes, but we do have the end behavior being on both sides, going to infinity. We know at X is equal to one. We have a local men, so now we can just go ahead and connect our graphs and everything. So it's going to come from infinity, and the first point it needs to hit is due. And then, well, we said they were supposed to be a local men at 11 So it needs to curve up again like this, and then the next point need you need to make sure we get would be our other X intercept And then this is just gonna go off until like this. Now it's not drawing very well, but at least you kind of get the idea. That should be more smooth. But unfortunately for me, I was not an artist. Now the only thing I might also say he should find is what value is here. Ah, local men. But I really don't think it's too important since we're just trying to sketch graph with it. But this is all I think we really need to show for our graph. Just maybe draw a little bit.
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