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Sketch some of the level curves (contours) or the surface defined by $f$.$$f(x, y)=2 x^{1 / 2} y^{1 / 2}$$

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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04:42

Sketch some of the level c…

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04:06

12:10

00:43

02:18

Sketch the level curve or …

01:14

Sketch some level surfaces…

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So if we want to sketch a couple these level curves one thing that we should make note of is what is the domain of our X and are wise here as well as our possible output. So, like, if we call this Z, So I'm first actually just going to rewrite this as to root x times route? Why, um is equal to see. And so now we know that we can't plug in negatives into these routes um, sister and even route. So that means X and Y needs to be greater than or equal to zero. And then also, we know the outputs from the absolute about early of square roots are going to be zero or larger. And if I multiply something zero larger by two, that means Z our output here is also going to be larger than or equal to zero. So when I come over here on, I solve for why in the second, we have to keep in mind that all of our outputs for Z have to be greater than or equal to zero, uh, and all of the values for X and why would we go to graft? These also have to be greater than or equal to zero, because otherwise that may kind of get lost when we're doing this. So this is very important, because otherwise you'll get contour curves that you actually don't have, um, or this. So let's go ahead and solve for why Now? Um, so I would divide by Z Or actually, I would divide by two root X, so that would give us route. Why is equal to see over to root X on. Then I would square each side's that would give us why is eager to so z squared over for X like that. And so again, remember, we have to keep in mind that X Y and Z all have to be greater than or equal to zero. Um, so essentially, since X and Y are greater than or equal to zero, we're only going to be working in the first quadrant. So I guess I should have maybe said that here, this is first wandering. We're not going to rock this anywhere else. Um, so let's go ahead and I'll skip this down and then let's scrap this So this we're only working in the first quadrant. That's why I'm when I'm going to actually draw over here. Okay, so now we could just plug in some values. Um, for this. So let's do, um, z is equal to zero. Actually, that we do z is equal to zero. We actually end up with something kind of interesting, because this is going to say, why is equal to, um Well, it would be zero over for X, which is just going to be zero. Um, but that would just be like this line here, So that would be when Z is equal to zero. Um, but something else if we actually keep it in, like this original form appear. And we're looking here, notice that we could also get where X is always equal to zero. So if I were to kind of switch these around, this one could also say, Well, X is equal to 0/4. Why? Which is going to be zero. And so we actually end up with both of these axes here being, um, zero. So just something kind of good toe make note of. And we actually have where this is going to be included, because if we come up here and plug in zero for X and Y Z is still going to be zero. So that's one thing you kind of have to keep in mind when you're trying to just solve for, like, why or X You will have things like this that kind of come up. Um, but other than ZZ, go to zero, I think all of the other ones we don't have to worry about that, though. Yeah, So now let's do like Z is equal one. So that would give us why is equal to 1/4 x eso if we were to go ahead and actually out graft this in blue. So if I plug in one, that would give me, like, 1/4 and then 1/8 Oh, yeah, This is just gonna kind of be pretty small numbers. Um, and then if I were to plug in like one half, that would be a half. If I were to plug in, 18 would be here. And then it just kind of keep on going like that and getting bigger and bigger. I guess you could be a little bit more exact with the values, but I think it will be okay, if we just do this here as of this is going to be easy, As you could have one. Um, if we were to do like Z is equal to that's going to give us why is equal to So it be two squared. So four. So it'll just be one over X. So then that would give us so one here. And then I'd have, like, one half 1/4 18 and it should just be slightly above. And then over here, all these should also be a little bit bigger as well, because that one half it should output to. Yeah, I actually did not draw this well at all, but let me just try to draw a little bit better. Um, se let me erase this. I try to do that again. So yeah, so, at one, it should be about 1/4 which should be around here and eight. And then, Yeah, just I didn't choose to good of numbers. Then one half it should be at a half. So around here, Um, and then at 1/4 it should be at one. And then, yeah, just keep on going up from here like that. Okay, so I think that looks a little bit better when we actually come to graph that. All right. Now, when we come back over here to plot one over X s So we have 11 and then when we do, one half would be at to thing. Yeah, they're just kind of go up from here, like, but then this one at two, it should be at one half, and then it kind of just gets closer and closer to this other one. And so then this is going to be C is equal to two. Um, but anything like the next best one to do when maybe be like Z is equal to four. Uh, that way we can get away from having to plot, like, fractions. Um, if I do, three would be nine force, and I really don't want to do that. So I'll just do cuz you before and so that would be so. Would be 16/4. So it be wise, you go to four over X. So when I plug in one that should have an output of four. Um, than if I plug in to that should have an output of two. If I plug in for that shaving output of one and, you know, look, something kind of like and then you could go ahead and graph some more of these if you want, but I was just kind of stop there. Um, so again, the thing to kind of keep in mind, um, is so when you just have where you can, like, plug numbers and you don't really have the issue of, like, quote unquote dividing by zero or anything like that, you could just wrap them like you normally do. But remember when we had, like, this zzz which zero we ran into, like, this Weird kind of. Well, it could either be where X is always zero wise, always zero. But if we look at it, just those forms, we might not actually see all of that. So sometimes, even when you like sulfur in these, you still need to kind of go back to the original equation. Ask yourself. Okay. Well, what values could these actually beef? And then also any kind of restrictions that you have placed on these values the original function have to also be placed when you're drawing these contour lines because otherwise, um, when you saw for this here, there's no necessarily, um there's not really a restriction placed on, like X and Y and Z and all that other than, like, X can't be zero. But even then, we know X can actually be zero from the original equation. Yeah, you just gonna have to keep stuff like that in mind when you are going about trying to graph thes contour lines because otherwise you will lose out on some information.

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