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

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 …

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Sketch the graph of the fu…

So if we want to sketch some of the indifference curves for this utility function here, remember, all that is really telling us to do is sketch some of the level curves of this function. Um, but normally will be restricted to the first quadrant. So all of our inputs going to be positive values. Um, So when we're looking over here, if we were to call this like Z, Well, this is just the equation of a line. If we plug in any value of Z, um, and then we just kind of keep in mind that we want to plug in, um, values Act only our larger than one for our Z for this or larger than zero, I should say, Um, so let's first just go ahead and do like Z Is he gonna one just may be a good starting place. So again, we're going to kind of restrict ourselves to this first quadrant. Um, 12341234 So this is our X axis and r y axis. So if we were to do that, we end up the question three x plus, or why is it that one? And so you could go ahead and, like, solve for y and all that. But I'm just going to find the two intercepts and plot those. Or you could also just do this using a graph and get similar urge. Calculate an autograph and you should get something similar. Just remember, we want to kind of restrict to the first quadrant here. Okay, Um, so if we were to go ahead and plug in like X is equal to zero, that would say for why is it Goto one which implies why is equal to 1/4? So that would be summer, maybe around, like here. Then, if we do, why is he go to zero? That would give us three. X is you Go one which implies X is you go to one third, which probably around like they're so actually, this is probably not the best. And we probably want to plug in a larger value for our utility. So let's do like so this is C is gonna one now. Um, I think a good value too. Maybe plug in where we have like a nice one is if we find some common multiple of this. So let's do like a 12. So Z is equal to 12. Then we end up with three X plus four. Why is it 12? So when X is equal to zero, we end up with four. Why is he go to 12? Which implies, Why is he go to three? So it would be like way appear? And then, if why is equal to zero? The window with three X is equal to 12, which would imply X is equal to four. So we'd be over here now. And so this would be our utility when Z is equal to 12. Yes, that seems so big. So now let's find just one more maybe toe plot in the middle, at least for like the graph I have over here. Actually, let me expand this screen and I'll just scoop this over on Ben. I'll do maybe, like Z is equal. Thio nine. Yeah, so that would give us three X Plus four y is equal to nine. And then So when access zero that gives or why is equal to nine. So that would imply Why is he go to nine force, which would be slightly larger than two so around like here and then when why is equal to zero would get three. X is equal to nine, which implies X is equal to three. So that would be around like right here. And so then this would be our utility curve for when Z is equal to not actually. Let's just do one more, maybe to fill this out. Um, So maybe, like, um, four would be a good number in between these, So c is a good whore. So three x plus for why would be four and eso when X is zero That would give us or why is equal to four, which implies, why is just one So we'd be like right here and then when why is equal to zero? That would be three X is equal to four, which implies X is equal to four thirds. That would be slightly larger, so around like right here so that that would be a utility of four eso again. You could just kind of plug this into a calculator to get some. But at least since these airlines air not too hard to kind of graft by hand like this here

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