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u(x, y)$ is a utility function. Sketch some of its indifference curves.$$u(x, y)=4 x^{3} y$$

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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05:05

u(x, y)$ is a utility func…

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07:22

Graph a typical indifferen…

So if we want to sketch a couple of the indifference curves for this, all we really need to remember is that indifference group is essentially a, um, level curve. But we're going to be restricting toe like the first quadrant because normally we don't have, um, negative goods in most cases. So we're just going to restrict all of our sketches to this first quadrant here, Um, now So let's just go ahead. And, like, say, this is equal to some constant, I'll just call it Z, and then we could solve for y. And it just kind of use this to sketch. So this would be why is equal to, um, Z over four x cubed. And we could just plug in a couple of values for this. So if we have C is equal to one, then we'd have Why is it Goto 1/4 x cute. Um and so let me just go ahead and put some ticks over here so it'd be like 12341234 And so this won't be exactly to accurate, since I'm just trying to get, like, a sketch of it. So if you were to just, like, plug this into a calculator, Um, or a graphing calculator. It should give you a much more accurate, but since I'm doing this by hand, um, it should at least be somewhat representative of it, Right s. So let's go ahead and grab this so it won. The output should be like 1/4. It looks like. And then after that, I mean, it just gets, like, really small really quickly on. Then if I were to plug in, like, one half, that should output, um, one half or at what happens should output to and then, yeah, it just gets, like, really big really fast after this. Uh huh. And now, if we were to do you know, maybe a good one would be like ZZ with four. What could actually over here. I should say this is Z is one. So that would just give us one over x cubed. And then if I were to just plug in some numbers here so it would be like one. Oh, are essentially all these would be just, like four units higher or stretched by a factor of four. So it probably looks something kind of like this, then over here, this to get stretched toe like eight. So actually, that's, like way off the graph. And so then this would be one z is equal to four and then maybe just do like, one more one's easy, but like eight. So that would give us why is equal Thio to over execute. And then again, we're just going to stretch all these by two. So this would go to to and then all of these just kind of come up, But then it gets really close to the bottom like that. Um then, like, way up here and again. Now, if you were to actually plugged this and you'll get a much cleaner sketch of this but this is just me kind of doing this quickly by hand. Um, yeah, so these would just be some of them. So I mean, technically, all of them should something kind of like that a lot smoother, but I can't really draw all that well. So if you were to plug these into like a calculator and then restrict to the first quadrant, though, you would get much cleaner indifference, curves

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