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Use the guidelines of this section to sketch the curve.$$y=\frac{1}{x^{2}-9}$$

$x=\pm 3$

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 4

Curve Sketching

Derivatives

Differentiation

Applications of the Derivative

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to find out about this function. We'll just start down the list here. The domain is going to be all real numbers, except when we divide by zero, which happens that ex cannot equal positive or negative three. So we know that we have either holes or vertical assam totes here. Vertical assam totes on our case. The intercepts. There's actually no intercepts here. The symmetry. Although, I mean, I guess I should back that up just a little bit because of x zero. We do end up with negative one night. So there is Ah, uh, why intercept? Just know X intercepts and symmetry. It looks to be even because of the even Power X squared. So it should be somewhat symmetric with respect to the y axis. Um and then lastly, Assam totes. So that's when the denominator equal zero for vertical assam totes. So we have to vertical assam totes at X is positive or negative three. And then the horizontal ass from tow is when we take the limit as X approaches infinity, um, of this top function and that would just be one over zero. Oh, and then, of course, the one I'm sorry. One over infinity, which is zero. So that would be or why, uh asked him to next. We could move on to increasing decreasing. Find out what? Why prime is and another way to rewrite this function. It's the same thing. Is X squared minus nine to the power and negative one. And this just makes taking the derivative easier. Because then it's negative. One X squared minus nine all to the power Negative too. Rewriting that will get negative one over. Expert Oh, sorry. Multiplied by. Of course, I forgot the derivative of the inside, which is two x So the numerator is negative. Two x all divided by X squared minus nine. All of that squared. This is what the first derivatives equal to double check here. Yeah, that looks good. And then way have three critical points here. X zero x is positive or negative three. So we could then set up our interval intervals of increasing decreasing here Negative three zero positive three and then choosing test values inside and outside of these, if we were to test like negative four, for instance, and plug that in tow derivative. The bottom is always positive, so we can just ignore it. But a negative times a negative is a positive similar to negative two. Negative. Uh, like, let's say we substituted negative one in there. Negative two times negative one is a positive, and then vice versa. Eso we get negatives if we plugged in positives, if that makes sense. So at this point, we do have a local max, and that would be at the point x zero since the graph goes, you know, increasing to decreasing there and plugging zero in. We know that from earlier. That's going to give us a negative one night. Okay. Lastly, we could look at the conch cavity. So, cavity, we want to take the derivative of what we have there and that will give us why. Double prime. So low that d Hi. Thank you. Too. Minus high de lo. It's a little bit longer and a little bit more messy. Blue expert. My miss mom. Times two x all over the denominator. Well, again, we can ignore it because it's just going to be positive all the time. So doesn't affect the sign. Okay. And finally, if we set this equal to zero here, um, there is really just going to be So that's another interval. There's gonna be two points that we're gonna test when it's zero or undefined. And it's negative three and positive three. And if we substitute something negative like negative four, we're gonna get a positive second derivative negative for something like zero and positive for something greater than three when you plug those into this. So therefore, we know that this graph is gonna give up thank you down and can't keep up again. And so now we have everything we need school down. Not that far to be able to graph this. So using all the info that we have, Okay, we know that we have again a vertical ascent owed it positive and negative three. So let's go ahead and put that in there. Okay? We know that we're going toe, have a horizontal awesome tau zero, and then that's kind of divides it up into our sections. And then we know that we're con keep down in the middle and we hit that point, which is zero negative 1/9. So I put it not at not at the origin just well below it. And then we know we have concave up and increasing and so that would be this shape. But then over here we have decreasing and Khan cave up. And so that looks like this. And so this is the general shape of this graph, that's that's pretty much it.

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