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Contours and Domain: For full credit, show all the necessary work with emphasis on the uderlinedparts of the questions: (a) For the following function, draw and dis...

Question

Contours and Domain: For full credit, show all the necessary work with emphasis on the uderlinedparts of the questions: (a) For the following function, draw and discuss the contour map showing several level curves.flwy)=(3 312 - J)V?(b) Find;_sketch_and discuss the domain of the following twO-variable function fly) cos"'(-212 ITy? - 3)

Contours and Domain: For full credit, show all the necessary work with emphasis on the uderlinedparts of the questions: (a) For the following function, draw and discuss the contour map showing several level curves. flwy)=(3 312 - J)V? (b) Find;_sketch_and discuss the domain of the following twO-variable function fly) cos"'(-212 ITy? - 3)



Answers

Draw a contour map of the function showing several level curves.
$$f(x, y)=x^{3}-y$$

This question. Discuss about the heavily function first we consider and why you go to the hip public sign under X. And then the craft of this one it would be is fully Yeah, we have. It looks like some ladies. So look at the ground. We see that and the man in the ranch equal to the ranch and equal to own dream number here and now. The second one, the y e go to the hip public course I am the X. We have the craft. It looks like the farm here when we have every base to studying from one here on then up like that. Look at the craft. We see that and the man will be only a number on the ranch in co. Two from one up to infinity. And now the next function, What you consider will be. Why equal to the public attention of the eggs. Then the craft will have. Here we have It looks like one minus one. And then we have the crab. What looks like this farm and look at the ground. We see that that the man be own real number. But the ranch, it will ego to from minus 1 to 1, Not including the two in Ponta And the next function when you consider with me why you go to the cause, helical Second off the X it will be the inverse off the reciprocal of the symbolic second. So the craft looks like this one. And look at the crime. Proceed under the man he go to the ranch and ICO Jew, Romans, Infinity to the zero union from zero to infinity. And now the next function we want to consider will be Why ICO to the public. Second on the X and the craft we have here, it will. Looks like we have it would be up to the value one. Yeah, And this I look at the graph. We see that that the man, he could only a number. But the rent here in equal to from the zero up to one And the last function will be why you got your hip political tension. I'm the ex and the cramp we have had And being Matt, I have it would it looks like, uh, from minus one. Yeah, till here. And then you go up to I just and this will be. The one is to be minus one and look at the grab a seat under the man you go to from months infinity to zero union from the empty infinity from the rent it would go to from a minus infinity to the Manus one union from one after infinity.

And this problem will be identifying the indicated features of a given function. The first will look at the Dellin, but the natural log. We knew that the argument in here has to be greater than general, So I want to tell the Bender Dellin in the X Squared Plus y squared has to be greater than one okay, and a range of other function is the same. All right. What a regular natural look function with being. And that would be over numbers when regret the level curves. What we get are concentric circles, all centered at the origin, and it's circle would have a radius that is greeted in one. The boundary points of are dealing would be de question ex cleared. That's worth grade equal to one on our domain would be considered open because there are no bundle. Ponte points included into do Mean. Finally, the domain is also considered unbounded because the domain does not lie in a disco tonight radius

For this problem we are given the equations equals six minus two X minus three. Why? And asked to plot the contour plots for the given values of C. Now to begin we want to write C instead of Zeb there. So we have C equals six minus two X minus three Y. Now to make this easier to plot we can rearrange this to get why by itself. So we'll get that. Why is going to equal six plus two X minus? Um uh Yes sorry so it would be C minus six plus two X. All divided by three. So we would then have that, Why will equal in order here just negative six plus two X over three. Then actually we can reduce that down that negative six can go to just be negative too plus 2/3 X. And then we'll have let's see here when it's too we'll have negative 4/3 plus two X. That's +44 minus six will have negative to over three plus two X. That should be two X over three. That's two X over three than we will have negative two plus two X over three. Wait, does that make sense? No. Would be just two X over three. We'll have um positive to over three plus two X over three. And lastly we'll have positive um 4/3 Plus two X over three. So we can see that we'll have a family of curves or not curves really a family of lines each passing through the X. X. Or the Y axis with a different intercept. So let me just establish one of these one. This would actually be the first one. Can see that it has a rise of two over a run of three. It would be along the lines of that. Then I'll extend this outwards. Let's see here right there. Then, since I'm working on a tablet, I can just copy and paste these around. We started negative two. Then the next one would be negative for over three, just a little bit under um negative one. You have negative to over three, just a little bit over naked one And we intercept at zero, Then add to over three, Then at 4/3, something along those lines.

In this problem will be I didn't define indicated features of given function The first look at the do me since we do with the function e we have no restrictions. Hair sore Dellin would be all point in the x went green or the range. First we need to note that the argument negative x squared y squared. The only possible valuably can get here our own less than or equal to zero. And if that is the case, then zero all function people toe one which is the maximum bottling and any of the value will give us a fraction between 011 So here we we can state that the range would be between zero and one where zero is not included. When the great the level curves, What we get in here are concentric circles. I'm based on what we have here. I just know that when C is equal to one, then the level curve with actually the point in the reserve. Otherwise, we have concentric circles centred under origin. The boundary points Um, in this case we have no boundary point because our domain is the entire X Y plane. So therefore, any point would be considered an interior point. And since we have no boundary, um, this can be considered open. But it can also be, um, considered close, since we have no boundary of report can be considered a bunch of point. And finally this Our domain would be considered unbounded. We could It does not lie inside and did okay tonight, radius.


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