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u(x, y)$ is a utility function. Sketch some of its indifference curves.$$u(x, y)=2 x y^{3}$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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u(x, y)$ is a utility func…

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Graph a typical indifferen…

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For each of the following …

So if we want to sketch some of the indifference courage for this utility function here remember, utility or indifference Curves is really just a fancy way of saying the level curves. So we would like set this equal toe like Z or something and then plug in some values for Z and then just sketch whatever curbs come out from that Now these are going thio kind of require in a lot of cases where X and Y are positive values. Um, because it wouldn't really make sense for us to talk about a negative value for, like, a good in a lot of cases on because of that will normally have to restrict to the first quadrant for this. Um, so let's just go ahead and first solve for why then we can go from there to kind of graph. Or better yet, we could actually solve for X in this case and then graph it that way. So I'll just divide by two and White Cube and then that's going to give us, uh, X is equal to Z over to Why keep the only reason why I want to do this? Because it's a little bit easier to plug in like values of X and y instead, the only thing we have to keep it room keep in mind is that we're plugging in values of why and getting out output of X when we kind of graft this though, um, so yeah, let's just do a couple of the level curves and also show these and desk most as well. Um, because I just kind of do like a quick little sketch of these. So if Z is one, actually, let's do is easy. But to, um Then we would end up with X is equal to so just be won over. Why cute? A. So, if I come over here, just graph some of these. So remember what this is saying is, when Why is one we get an output of one for X? Um, when why is too we get an output of 1/8 so two would be like 1/8. So, like, it's really small, really close like this. And then over here it does essentially the same thing. So, um, we would have just kind of like that and that this is going to be our level curve are different script, I should say when Z is equal to two. Um then if we were to do like Z is equal to maybe like, four, this would give us to over Why, Cube. So essentially all this is going to do is bring up all the values by two or kind of, ah, school. Push it mawr to the right by two. So, like, this value would get brought up a little, brought up a little brought up a little on. Then this one would get brought over here and then just kind of all these get brought over by some amount. So it looks something kind of like this, er, ze, Is it four and then lastly, I don't know, maybe we could do like Z is equal to eight are very audio Z 06 So that would give us three over White Cube. And so again, this is just going to kind of push it slightly more to the right. And so it looks something along these lines here, Um I mean, the only one that I can kind of think of all the time ahead would just be This would be the real X is equal to this, then all the other ones just kind of get pushed over a little bit. Yeah, that this would be a Z 06 So this is kind of like the sketch of them for the level curves on. Then I'll show you how to do this in decimus is Well, um so over here I just went ahead and plugged in what we had for the function first. And then I set X to be greater than because they were just kind of restrict us to this first quadrant. But again, you could just kind of, like, scoop this over and not have any issues. But then, yeah, let's plug in those values that we had. So that Zizi the one we also did Z is equal to two. So I'll just change z to go to to. Then we did. Or and then we also did six. And so if we just get rid of that first one So these would be the ones that we had graft or somewhat graft before, so you could see it's the same kind of idea. Um, yeah, you could probably just, like, take a screenshot of this and then say OK, this is Z is equal. Thio 20 to 4 is easier to six for those indifference curves. Yeah, so it just kind of depends on how accurate you want your graft Actually look.

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