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u(x, y)$ is a utility function. Sketch some of its indifference curves.$$u(x, y)=2 x^{1 / 2} y^{1 / 2}$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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Lectures

12:15

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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05:15

u(x, y)$ is a utility func…

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03:52

05:05

02:01

Sketch the shifted exponen…

So if you've done question 34 you've actually already completed this one. Um, because the contour lines, um and the indifference curves eyes going to be the exact same because it's the same function on differences, just like the fancy way of saying contour. So if you've already done number 34 you've essentially already solved this one here. But if you haven't, let's just kind of go through what we did in that one. So, um, one assumption that we will kind of make here that we didn't necessarily make what we were doing 34 was that our input values for X and Y we normally assume to just be positive values. Um, because it's a lot of times doesn't make sense for us to talk about, like a negative. Good. Um, so we would just go ahead and restrict this to the first quadrant here and and then just draw the contour curves from here. But at least from what we did before, this would be exactly the exact same thing other than we're not going to look at zero in this case. Um, yeah. So other than that was, just go ahead and draw some of these, especially first, let's go ahead and set this evil disease. And then we could just move things around and saw for why, um so that would first give us why to the one half is equal to so see over to next to the one half square each side. So why is going to be Z squared over four? Um, just x right. And then remember, since we're just in the first quadrant, all of these inputs, we're just going to be positive numbers that we just go ahead and plug in from here. So let's just put some ticks down. If you want a more accurate graph, you could always just plug this into some graphing calculators. Well, and then just keep it in mind We Onley want in the first quadrant. Um, so first, let's just go ahead and do like Z is equal to one. So doing that we'd get Why is he go to so it will be one square. So be 1/4 X that we just go ahead and graph this, um so at, like an input of 1/4 we should have one, and then at one, it should be like 1/4 and then after that is just going to get, like, really small really quickly on either side like this. So yeah, this will just kind of be like a quick sketch of it. Um, but you could always do a little bit better if you want. Without accurate, This is Yeah, that would be proceeds one. Now, just to make it a little bit bigger. Let's do like, zzz before, so I'll give us why is he going to? So, um, actually, let's do to first. Because since we're squaring Z here, that would be four or four. So just one rex. So then we have one there at, like, one half would output to and like, 14 through output. Or, um And then over here this would just pretty much be around here, here and there that we could connect all these. And then we have Z is equal to two. And then let's just do one more like Z is equal. Thio four. So then that would give why is equal to so four squared over force. They'll just before over X and then we would just pretty much stretch this by four units So it be around here, Um, at one half that would take that to like to. And then this would be, like, one Say, it looks something like this, Um and then see, is it to for? And so if you want, you could doom. Or And I guess technically, all these should be a lot or smooth, but not too good at drawing. Yeah, I would say this is kind of like a good overall kind of Just look at a couple of these, um, indifference curves for this function.

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