use-the-method-of-the-examples-of-this-section-to-sketch-the-indicated-curves-use-a-calculator-to-53

Question

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

Stephen Hobbs · Accepted Answer

to be able to graft dysfunction. We&#x27;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&#x27;s there. The intercepts. We have an intercept when x zero. Why is zero? So that&#x27;s both the next on the Y intercept symmetry. Um, let&#x27;s see. It looks like we&#x27;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&#x27;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&#x27;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&#x27;ll just take one step to clean this up. We want to find out when it&#x27;s equal to zero or undefined. Um, this is the same thing. Let&#x27;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&#x27;s a few critical points here we&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;ll set up our interval, and then we&#x27;ll analyze it. So let&#x27;s do this. And blue So I&#x27;ve got something. Took the left and negative three in between. Negative three and zero. That&#x27;s something greater than three. So the test values that we could use would be like negative four, for instance. And again, we&#x27;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&#x27;re ready to graph this whole function here because we know, first off, it&#x27;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&#x27;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&#x27;re just gonna make combinations of these things. Well, we know we&#x27;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&#x27;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.

Stephen Hobbs · Answer

to find some of the specs for this function. Here we&#x27;ll start off the domain that includes all real numbers except X cannot equal zero because then we end up with an undefined value. The intercept? Um, it&#x27;s just one x zero, and that will result in why being zero um, symmetry should have some type of odd symmetry here. And Assam totes. Uh, that&#x27;s when we&#x27;ll vertical Assam totes or when you divide by zero. So that happens at X is warm horizontal Assen totes or when you take the limit as X approaches infinity. And that&#x27;s where you look at the ratio of these two in this case, and that would be 1/1. And so the horizontal listen to it is why most one now in terms of decreasing, increasing, Um, let&#x27;s go ahead and take the derivative using the quotient rule. So we start off with low D Hi minus Hi, Do you love all over the bottom square? The X&#x27;s cancel, and so this actually leaves us just with negative one over X minus one square. So, really, the intervals of increasing decreasing just depend on the fact is greater or less than one here. And it turns out that no matter what, the bottom is always positive. And so the tops always negative. So this function is actually just always going to be decreasing, always decreasing. So there&#x27;s no local Max or men or any of that. And then finally, if we want to find out about the con cavity on, we want to take the second derivative. So that&#x27;d be why Double Prime is equal to again. Taking the derivative of this function will do lo de high except the derivative of the top of zero. So that would be zero minus. Hi. De lo would be to X minus one now to the power one times one all over X minus one to the fourth. Let&#x27;s see how this shapes up. So this is going thio equal. Um, double check here. Okay, so this is Yeah, this is just going to give us too X and then minus two all over X minus one. So the fourth, and so in this case, we have some Okay, a possible change in concave ity at X s one. And then if we verify that by plugging values in here, notice if we plug in something less than Well, let&#x27;s start with greater than one. We&#x27;re going to end up with positive values. Actually, let me go ahead and just write. It is a a plus sign instead of, um so positive Con cavity or concave up like that. And then to the left. Negative Sakon cape down. So at this point, we have everything we need. Uh, yeah. So as we begin to plot, one of the first things we could do is just plot, For instance, are vertical aspecto at X s one. We also have a horizontal ASEM toe that why is one so that kind of divides this up a little bit, So it&#x27;s easier to see. Um, we do end up passing through the origin. 00 Here, let&#x27;s graphic in blue And so we&#x27;ll come down con cave down as we knew, um, con cave up and it&#x27;s always decreasing. So no matter where you are, this craft always decreasing. Onda Gennett goes off in both directions to y equals one

Stephen Hobbs · Answer

to find out about this function. We&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;s just going to be positive all the time. So doesn&#x27;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&#x27;s another interval. There&#x27;s gonna be two points that we&#x27;re gonna test when it&#x27;s zero or undefined. And it&#x27;s negative three and positive three. And if we substitute something negative like negative four, we&#x27;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&#x27;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&#x27;s go ahead and put that in there. Okay? We know that we&#x27;re going toe, have a horizontal awesome tau zero, and then that&#x27;s kind of divides it up into our sections. And then we know that we&#x27;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&#x27;s that&#x27;s pretty much it.

Ankit Gupta · Answer

it is given that the question nine. Why is equal Toby Excuse Now we can rewrite it as why is equal Toby ex care upon line Now we need to upload the graph off the given the question by using table values now for different values off X We get different values off way by using the given question and we get to the points which lies on the good off off the ignition. So put X is equal to minus two. When we put X is equal to minus two, we get the value of way is equal to B minus to scare upon nine, which is called a before upon nine. So we get to the point minus 24 upon nine. No put X is equal to minus one when we put X is equal to minus one get the value of y is equal to B minus one scare upon nine, which is equal, Toby one upon nine. So we get to the point minus 11 upon nine. Now put X is equal to zero. When we put X is equal to zero, we get to the value of way is equal to be see those scared upon nine, which is equal to zero. So So we get the 0.0 Now put X is equal to one. When we put X is equal to one, we get the value off way is equal to be one upon nine. So we get to the 0.11 upon nine. Now put X is equal to two. When put X is equal to two, we get the value of why is it called to be too scared upon nine. We get to the very of Why is it called a before upon nine? So we get to the point to for upon nine. Now we&#x27;re going to upload the graph of the given equation by using these table values. So the graph off the given the question will look like. As so, this is a required graph off the given aggression.

Heather Zimmers · Answer

here we have the equation of a circle. And if we write it like this, X minus zero quantity squared. Plus why minus zero quantity squared equals three squared. We will see that the center is 00 and the radius is three. And we can use that information to graph it. So what I&#x27;m gonna do is start by locating the center 00 And then from there I&#x27;m going to go up three units from the centre, right? Three units from the centre left, three units and from the center down. Three units. That gives me four points on my circle. We could certainly do others. And then what we want to do is draw the circle. Of course, it&#x27;s not gonna look perfect, but you just do the best you can, and that&#x27;s the graph.

Problem

Use the method of the examples of this section to…

Problem 17 Easy Difficulty

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

Answer

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Heather Z.

Caleb E.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

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Basic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Grace H.

Heather Z.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus