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

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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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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05:06

Sketch some of the level c…

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04:06

12:10

01:14

Sketch some level surfaces…

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02:18

Sketch the level curve or …

So if we want to get some of the contour lines of this, remember, what we're going to do is essentially just set the sequel to some constant. So let's just go ahead and call it Z, and then we'll just put in different values of Z and then sketch these. Well, now, if you were to look at this, this is actually the equation of a circle, um, centered at the origin 00 with a radius up. So remember this supposed to be our squirt? So this is gonna be Route Z instead. Um, and what that tells us is that our zis that we can plug in so this has to be greater than or equal to zero. Um, because one more plotting these, we can't have, like, a complex output because at least for these were only considering riel valued functions. Eso just something to kind of keep in mind that Z has to be greater than or equal to zero when we're doing this. Um, and at this point, we just plug in some random values for Z and then kind of go from there, so you could pretty much plug in any value of Z, But since we need like this square root here for the radius, Um, I'm just going to plug in some of the easy routes to take, so let's go ahead and do just r is equal to zero to start. I'm sorry. Not r Z is equal to zero. Z is equal to zero. Eso if we do that, that gives us X squared. Plus y squared is equal to zero. Um, well, so this is a circle of radius zero, which essentially means is just a point. So let me come over here on this top corner and kind of do that. So 123123 123 And then I'll do three going down as one. So the season goes zero here is just going to be this singular point. Um, Now let's do something like Z is equal to one. So if we were to do that, though bc squared plus y squared is equal one. And remember, one is really just one square. So this is a circle of radius one. So we just draw a circle ovaries one here and then so versus Z is equal toe one, um, and then, like the next nice circle would be at Z is equal to four. So you could do to if you wanted, but I'm just going to do for just kind of for simplicity sake, and then x squared. Plus y squared is equal to or which is to swirled. So again, radius is two so that we draw this circle. Welcome. Apparently I'm didn't draw this crap too well, because none of these actually look like circles, so it is easy to And then if you do nine, then it would be a raise of three. Um, so it's actually do some of the fractional ones that have nice radio I as well. So let's do like Z is equal to, um 1/4. So that would give us X word. Plus y squared is equal to, um eso won fourth, which is going to be one half square. So our radius here is going to be one half. So this is like one half we would go around like that, and then that is going to be Z is equal to one half on. Then we could also do like, uh, 1/4 I guess, or not 1/4 though. What's the next one? 1/9. So C is equal to one night. So I give us expert Plus Y squared is equal to 1/9 which is going to be one third squared. Um, and so then one third would be around like here or so on the inside that would be Z is equal to one third. So those were just some of the different contours. I guess you could have actually made this little bit bigger, um, or done some of the ones in between the like, nice ones. But these were just some of the lines I decided to do.

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