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Sketch the graph of f2+2-4f(x,y) = 9 -x2 _ y2And find the domain.Define the continuity Of a function for tWo variables What 1S the key difference between limit and ...

Question

Sketch the graph of f2+2-4f(x,y) = 9 -x2 _ y2And find the domain.Define the continuity Of a function for tWo variables What 1S the key difference between limit and 2+2-4 continuity-Show that the limit does not exist by considering the limits as (x, y)-(0, 0) along the coordinate es_x +y (xyimoo Zxr+y7

Sketch the graph of f 2+2-4 f(x,y) = 9 -x2 _ y2 And find the domain. Define the continuity Of a function for tWo variables What 1S the key difference between limit and 2+2-4 continuity- Show that the limit does not exist by considering the limits as (x, y)-(0, 0) along the coordinate es_ x +y (xyimoo Zxr+y7



Answers

Sketch the graph of the function with the given rule. Find the domain and range of the function. $$f(x)=9-x^{2}$$

Here. We're going to find and sketch the domain of the function F of x Y equal natural algorithm of nine minus x squared minus nine Y square. So we start by noting that this function has two variable. So the domain is a subsidy of the plane. That it is. The main of F is not by D O M F is a subset of our to our square. That is the it's a subject of the plane. And the second observation we got to make is that the functions if I threw the function natural algorithm, L. M. So we know that the natural logarithms and in fact all logarithms is defined for positive real numbers only because we are working on the real numbers set for the two variables. Then we are talking here about real numbers in general. So in that case we are allowed only two um find the natural reason on real positive real numbers. It means that this argument here get to be a positive quantity. So we can say that the the main of the function is the set points in the plane for which the following inequality is true, That is 9 -6 sq -9. was square is positive because this is the argument. This expression here is the argument of the natural logarithms and the natural algorithm is you find only here is a fine for positive real numbers only. So we got to we know this is the domain but we are going to work a little bit inequality in order to determine better, which said this is So the inequality 9 -6 square -9 was square greater than zero implies that X square Plus nine, Y Square is less than nine. That is because we put these two turns to the right and then right inequality the other way around how we get please from where we can divide both sides by nine. And the sense of inequality remains the same because nine is a positive number. So again, X square plus nine y square over nine Must be less than nine overnight. And that is we can separate the numerator. There's a some we can separate each term of the song of the song through the denominator. So we get X square over nine plus nine. Twice square over nine. Less than one. We simplify the nines here, can we get X square Over nine Plus Why Square? There's someone. And here we identify this equation. In fact, if we replace the inequality by equality, we get the equation X square over nine Plus y squared equals one what? And this equation here, we know that it represents the equation of an ellipse. Yeah. And the Libs This case centered at 00 because we have X zero square and Y -3 square. It's very important that we have some Centre at the Origin 00. And the is equated to one is very important also. And the denominators of the squares affects square. And white square are positive numbers which are the square of other numbers. In this case this is the square of three. And the denominator here is one square of one. So these numbers are the length of the Access of the Ellipse. So in this case we can say that the least center at 00 and mhm has Okay. Yeah. Police is centered Center at 00. And with major axis on the one on the X axis. That is because the creator number, great denominator of the powers of X and Y is in X's square. So the major axis of the ellipse is on the X axis. For the same reason the minor axis of the ellipse is only Y axis. Mhm And we remember that the equations are written Germany this way mm Instead of nine we put three square And we have one which is one square equal one. And we write this way in order to Find the length of the axis that we have three and 1. So the domain of F. Mhm. Okay. Now before writing this, let's see here above again. So we have found that with equality we have the line which is an ellipse. But with the inequality here because for example 00 uh satisfies this inequality. We know that this is the interior of the elites. So the domain of pf equal natural logarithms off nine minus X squared minus nine. Was square is the interior of the A. Lives X square over nine plus y square equal one. Yeah. And the interior means we don't take the ellipse itself. Halley's ellipse itself. He's not included. It's an open set and that's because the inequalities strict. That is. It's less than if we had less than or equal. We will have the elite ellipse included in this case is only the interior of the ellipse. Okay, so you know the major axis is on the X axis, The minor and the Y axis. So we have one 81, 3 and 83. And so we have this type of figure because the vertex of the elites elites are determined by these numbers three and one here. So we have these two vertex here is to the vertex of the lips and take over take some of the minor axis. And then the the main is the interior of the ellipse. Without the ellipse itself. Ellipses not included. Yeah, I stressed the idea. But when we say that the main is the interior of the ellipse, it is clear, it must be clear that the lips is not included. And here's the sketch of the graph it. I'm going to insert now and an image of this. The main a little bit clear which is done with a computer program. Yeah. And here we have he the main as we can see, we have the ellipse as a dot line here, dash line indicating that the ellipse is not in the domain And we have the um vertex The virtus is negative. Three. Here, three here co verdicts The covert this is here one nearly 1. And the interior is shaded in blue in this case two is transparent in order to see the values of the axis. And uh as we can see the major axis of the ellipse is on the Y axis, on the X axis and the minor axis of the ellipse is on the Y axis. And then these shaded blue uh huh region. The interior of the ellipse is the domain of the function F. Come on. Which is represented algebraic plea by this expression here, which is equivalent to this inequality here, which represents interior of the ellipse. Mhm.

All right. So we know for this function that have a natural law and natural logs, our only defined and they are positive. So this has to be greater than zero the argument inside of here. So if we want to rewrite a, we can read it as carrying it out. Whatever is on the inside. So nine minus X squared minus nine y squared has to be greater than zero. So when we continue that we get, we can carry the constant to the other side has to be creator that negative nine can divide everything by the negative. And when we divide by negative, we have to switch the inequality. So we get X squared. Plus nine y squared is less than no. Okay. And the last step is we can recognize this as it's approaching some sort of elliptical form. But the only thing is, we need this nine on the right side to be a what? So how do we change that? We could just abide by night. So we divide by nine. We get X squared over nine plus, why squared over one is less than one. So now it's our elliptical form. So when we graphically understand that it's gonna be in the lips as our domain. All right, So how can we draw this ellipse? Well, we know that it's gonna have a semi major axis or are of three based on the X value and a minor excess of one so strong That and another thing we realized is that we're not gonna have boundary points. So the actual boundary of the Ellipse is not gonna be defined and not gonna be in the domain. So we couldn't recognize that and just start the lips. So this sort of Maine

So when we are looking at this function, which is in ah, except boy is equal to nine minus X squared, minus nine Y squared. First thing that we can do is OK. We can start off, obviously, by just start drawing the graph so X axis will draw. Here you are. Why access here? And we'll draw the FX of why? Or we can just call it to see axis up here. So one thing we can do is look at what it's gonna what This curve is gonna be first when x and wire. About zero when excellently about zero f x y is nine. So we just put that up here, it's gonna be 009 All right? And then how about when X is equal to zero? When X is equal to Syria, we have a Z or FX. ABI is equal to nine minus nine y squared. So then, if we just rearing some variables, we get a Z plus nine y squared is equal to nine. Divided by nine. We get see over nine plus nine y square or just wife square. Does he put one? And if we I think that's the most recon rearrange it to to make it more meaningful. So on this one, we're gonna look at the parabola or whenever we're drawing from the Z ethics. Why why played or the Z wife claim And on the Z Y play and whatever gonna notice? Well, we're gonna notice that if it's on the Z y plane, it's gonna follow some sort of hyperbolic or parabolic tendencies. So that is something to consider. And then also, if we set, why equals zero, we're going to see the same thing. So what we can do then is calculate for intercepts. So our first intercept is if X is equal to zero. So when X equals zero, which were already calculated for we get something along the lines of this. Now, when wise when ZZ put a zero, we know that why is he gonna want So when ex n zero equals zero, which would occur here, Why is you go to one? So it's gonna be like that right here. It was gonna be plus and minus one. Actually, that's something to consider because it's a square root. So then the next thing is what about when xn zero equals zero when X and Z recalled it. Sorry, When? Why in 0.0, what is the Kurgan abuse? Zero equals nine minus X squared. So when Zurich was nine minus X squared negative. X squared equals nine. So it's gonna be a plus minus three. Something along the lines of that. So when we put everything together, we're going to see the curve or are graph look like is something that looks like a cone or a cylinder. Something along the lines of that where we're gonna have a cup, almost an upside down cup. And of course, this extends down beyond. But generally, the shape that we're gonna look for in a photograph is gonna be something along the lines of that and extends on the X X axis two point of three negative three and three and extends in the Y axis. Two point of one negative one, and it Z is gonna be up at night

Here. We want to sketch the graph that has the indicated properties. Um So it's a race previous work. Um And what the graph is going to look like, we know it needs to have The property is that f prime of X is greater than zero given the intervals at kind of X is less than zero given the other intervals. So 0, 3, 8 and 12 are important values. Um What are graph is going to ultimately look like is something like this. Um It looks like this And then at three it's going to start decreasing. Um And then at nine, a little bit before nine right about eight skin to start increasing again until it reaches 12. And then it's going to study off. So that's what our graph needs to look like because in this case it's um greater than zero Between zero and 3 & eight and 12. It's the slope is less than zero in this portion. Um And then the graph is also Going to be concave up from 5 to 9 right here, and then it's going to be concave down um in the other portions that we're worried about right here and right here. So for that reason, we satisfied all the things and then the limit as X goes to infinity is too, which we could shift this ground down further. If we made this too right here, that would be what we need to satisfy the graph.


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