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Sketch some of the level curves (contours) or the surface defined by $f$.$$f(x, y)=4 x^{1 / 3} y^{2 / 3}$$

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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04:06

Sketch some of the level c…

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04:42

09:37

01:28

Sketch some level surfaces…

00:47

Describe the level curves …

02:33

Sketch the level curve or …

01:14

02:08

00:44

So if we want to sketch some of the contour lines of this year, one thing that will be really important for us to do is to go ahead and first determine what is going to be the possible outputs or contour lines. Um, as well as what are the possible in points for inputs for X and Y, uh, because once we start moving things around and if we were to just solve for, um, why we might lose some of that information. And actually, when I was trying to double check my answer earlier from what I have got, um, the group or the calculator that I plug this into actually excluded some of the contour lines. Um, and I kind of talk about why, uh, it did that in a moment, But so let's just go ahead and first figure out what is going to be the domain and the range for this. Well, so next to the one third so we could take a cube root of any number, S. O X is just going to be all road numbers. And then why so weaken, square any number and then take the cube root of any number again. So the domain of why is also going to be all real numbers. And over here, the outputs of this should also be all real numbers. So if I said this was Z, then Z should have an output of all real numbers. And the reason why that is, is going to be well, so X to the one third, Um, can now put all real numbers from negative infinity to infinity. And then why, to the two thirds would just be numbers greater than or equal to zero? So if we multiply those together should be all possible road numbers. So that's how we have it. And so this will be important for later on eso. Now, if we were to just try to graft this, though, let's go ahead and try to solve for why? Because that's what we would normally do. Um, so let's go ahead and do why, to the two thirds is equal to So I would divide by four X to the one thirds that would be easy over or X to the one third, um, and then we can go ahead, and I'll first do the cube root on, um each side of this. So this would be why squared is equal to so c cubed over then four cube, which should be 64 then we just have X and then we can go ahead and take the square root on each side so you might end up with just okay, Why is equal to we take the square where 64 which is just eight. So this is 18 and I'll keep this C cube and X here. And the reason why I want to do that is just due to the domain and all that. So now if you were to just leave it like this here, this will be a little bit wrong because why in this case, can Onley ever have an output greater than or equal to zero? Because we have this square root here. So if you're going to go about doing it this way, remember you have to do plus or minus, so you'd have to graft multiple curves in doing this at once. So what I'm going to do instead of having to look at it this way? So let's get rid of that. I'm going to solve for X instead because it will just make things a little bit easier. And we won't have to think about, um whether these numbers can be like output. It is positive or negative or anything like that. So it won't always be the best choice to solve for X, um, or to solve for why say, let's solve for X In this case, eso Over here I would divide by four white to the two thirds, so it's gonna be X to the one third is able to see over four. And then why to the two thirds? And then I would cube each side so I'll just get X is equal to excuse me Z cube over and then 64 And then why squared? And so now this is a little bit easier to graph, but we're just going to have to remember we're now going to be, um, graphing this in terms of why is our inputs and access or outputs? Um, so let's come over here and do that. So the first thing I would want to do is to just checkable what happens when Z is zero. So Z is zero. Well, if I plug that over here That just says X zero. Well, X zero is just this access right here. Um, but notice that if we were to plug Zia zero appear into this original one. Well, not only could x just deserve why could be zero. So sometimes you actually have to go back and look at your original equation to check to see, um, necessarily, like, if you're dividing by zero or something for some value, um, things like this will come up, so it's always good. Once you kind of graft this toe. Just plug in some numbers here for your contours to make sure that you may have not excluded anything. Yeah, So those axes there will be our contour lines when Z is equal to zero. Um, I noticed that even though when I plug zero here, it should be undefined. It shouldn't really matter, because in the original question, I could plug in Syria and zero with no issues. And so that's why this original domain is important. Um, yeah. So we have that, um actually, I probably want to keep this down a little bit, so let's now go ahead and check. Like, actually, I need this equation is Well, so let's check when, like Z is, uh, one. Yeah. So if Z is equal toe one now, that is going to be so X is equal to just 1/64. Why squared? Um so maybe drawing it on here won't be like the best idea. So I'll actually use decimus to graft these, But I'll just go ahead and graph a couple of them or plot a couple of these using just like the numbers so we can get the graphs. And then from there, we can just plug them in later. Yeah, um so I just kind of like give a quick sketch of it over here, and then I'll plug it into desk most to make it look a little bit prettier. Eso this would look something kind of like this because remember, this is X. This is why eso we have, like, this vertical aspecto at zero. Some kind of like that. This would be Z is equal to one. Z is a good one. Now, if we were to do, um let's do like Z is equal to or products Z Yeah is equal to slot. B X is equal to So it be 8/64 which would be won over eight y squared. So it should just be a little Yeah, wider out like this if we were to graph it so that would be Z is equal to. And then it would just kind of keep on going. Uh, let's do a couple of the negative ones. So, like Z is equal to negative one. Well, now that's gonna be X is equal to negative 1/64 y squared. So it's gonna be just like the green line, but now it's just reflected across this access. Um, because now all the X values, we're gonna be out putting negative numbers, so it would look something kind of like this. And then this one also gets reflected over s that Zizi put a negative one. ZZ put a negative one. And then lastly, if we were to do Z is equal to negative, too. Well, that's going to give So X is equal to negative 1/8 y squared. And so then this red line just kind of gets reflected across the Y axis as well. So then this is easy. Is it to negative to then also here is easy to negative two. So not too accurate, cause I don't have, like, any numbers plugged in, but if I didn't, it would just kind of be, uh, really bad for me to draw. So at least this sketch here, I would say it's kind of good enough. And then I'll show you what it looks like. A desk most this. Well, okay, so now here. This is our graph. Um, and I just went ahead and did, like, X is equal to C cubed over 64 y squared. So now I could just kind of move this around. You could see how it gives all the different contour lines for that. And now the thing that I kind of want to point out was if I were to do just Aziz equal to zero once again notice how it on Lee gives us one of the axes as opposed to when we did this over here, we got both of the axes. Um, se I just kind of be careful when you're just using, like, calculators. You should always go back and, like, double check, especially like the weird numbers, like zero of things normally always weird happened for that. Um, that won't do that. But now let's actually just plot a couple of these because I just have this. But it would be nice if I had a couple of the numbers. So, um, we had X 0 to 0. And also wise, you get zero. Just kind of get both of those in there and then let me make this the same color. Is that red one? Okay, so that would be one's easy with zero. And then if I had Z is equal to one, that's the purple. Um, c is equal to its, like that black one. And then we could do the same ones, but with the negatives, negative one. You can see how it's that reflection. Um, going across of like that purple line because if you like, kind of zoom in it maybe a little bit more apparent, that purple on the red or just reflections. And then if we were to do negative to, um you can see how we get just a reflection of that black one. So So I guess you could just do this and then go on here and then right like ZZ 0000 negative one negative to negative one. Negative two. And so on on the other side. Um, but again, I would say, Like what we have over here, at least, is a good kind of just, like sketch of it, though. So, um, depending on how accurate you want it, you could either just usual use this year or use the one that we used in desk mostly then just kind of drawn there. But again, if you're going to use the one from desk most you have to keep in mind that we lose some information when like Z is zero. Because we should have over here. Um, both of the axes being Princz with zero and not just acts is equal to zero.

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