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Use the method of the examples of this section to sketch the indicated curves. Use a calculator to check the graph.$$y=\frac{9 x}{9-x^{2}}$$

Calculus 1 / AB

Chapter 24

Applications of the Derivative

Section 6

More on Curve Sketching

Derivatives

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to be able to graft dysfunction. We'll just go through our list here. The domain is all real numbers, except when we divide by zero so X cannot equal positive or negative. Three. That is an indicator that we have vertical aspecto. It's there. The intercepts. We have an intercept when x zero. Why is zero? So that's both the next on the Y intercept symmetry. Um, let's see. It looks like we'll have odd symmetry because we have a nod divided by an even, but should still keep us at a nod. Symmetry and then ask him Totes eso We just said we have to vertical Assam totes here at positive and negative three. And then the horizontal Assam tote is when you take the limit as X goes to infinity, power is bigger in the denominator. So the h A and A B A. If that's funky, um, and the denominator with it blow up before the top does. So the this would be like, well, zero for the horizontal. Then, to find the, um, increasing and decreasing intervals, we'll go ahead and go ahead and take the quotient rule here so low the high through a bit of of the top is just one minus high, do you? Well, prove it over the bottom is two X and then we square the denominator X squared minus nine And then we'll just take one step to clean this up. We want to find out when it's equal to zero or undefined. Um, this is the same thing. Let's see as negative, um, expert minus nine. Sorry. I was just getting a little next up here because of this notation here. There were no parentheses, curls. So all of that over the denominator squared X squared minus nine squared. And there's a few critical points here we've got Well, the top is actually always going to be negative. Okay, there we go. The top will always be negative on the bottom is actually always going to be positive. So based on that, we don't even really need to look any further. This graph is always going to be decreasing because the derivative is always negative. So, in other words, always decreasing the derivatives always negative. So it's always decreasing the con cavity. On the other hand, why double prime? See what this would get us? Um Okay. So low X word. Find this mine squared D hi, Derivative of the top Here is just negative. Two X minus high de lo driven to the bottom. We need chain reward to quantity. X squared minus nine for the power one. I'm untraditional two X No, this looks kind of intimidating all over. X squared, minus nine square. One thing that we could do is notice that each of these has a factor of X squared minus nine. So one of these would go away, one of these one of these, and then from there, I'm just gonna go ahead and jump down to it. So then this will simplify. Do it on the bottom, right? 22 x parentheses. X squared, lost 27 and then all over X squared minus nine. And it's helpful to know what this is when we plug in our values. So there are a few critical points for the second derivative here. One of them is that X zero that would make the whole expression zero or when X is again, positive or negative. Three. So then we'll set up our interval, and then we'll analyze it. So let's do this. And blue So I've got something. Took the left and negative three in between. Negative three and zero. That's something greater than three. So the test values that we could use would be like negative four, for instance. And again, we're plugging into this expression here that would end up giving us a negative positive, a negative and a positive for different test values in between these or outside of these. Um And so now we're ready to graph this whole function here because we know, first off, it's always helpful to know we have the 0.0 so that would be right in the middle. Um, the vertical Assam totes positive and negative three. So these air just points at which we don't have any values, but kind of restrict us. We have, ah, horizontal lots in tow, which again you can pass through, by the way. So it just helps us with the endpoint behavior mostly. And so then we're just gonna make combinations of these things. Well, we know we're always decreasing and then con cape down would look like this decreasing con cave down, decreasing con Keep up would look like this decreasing con cape down again. Would look like that and then decreasing. Can't keep up would look like this. Okay, so we have basically what this graph looks like, and we know that one point the middle. So this is the general shape of the graph.

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