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Find and sketch the domain of $f .$ Then find an equation for the level curve or surface of the function passing through the given point.$$g(x, y, z)=\sum_{n=0}^{\infty} \frac{(x+y)^{n}}{n ! z^{n}}, \quad(\ln 4, \ln 9,2)$$

$$D=\{(x, y, z) | z \neq 0\}$$

Calculus 3

Chapter 14

Partial Derivatives

Section 1

Functions of Several Variables

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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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Find and sketch the domain…

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you're not wrong when we're going to find a sketch the domain of the given function and also find an equation for the level curl at the given coordinates. So to begin, don't just there our function g looks a bit complicated, but if we recall the exponential expansion we're being is equal to this summation and equals, they were to unity. Oh, accident. And over and crying, we can rewrite our equation in this form soon being is equal to being to the X plus y. He's not advising, uh, now that we have the equation and this woman becomes spring clear, that are Dellin is usually all real numbers for the function E. But here, since we're dividing bites and we know that we cannot vehicle to do so what you're doing, we can see that it is all going such that we, uh, not equal to zero. You know, that's a sketch. Did you mean since we're talking about a three dimensional space, we can know that we have boundary planes. X equals zero. My core zero equals zero. The way was that what we have is that our doing is the Internet here. Cream exploitation, peeing except recall that the plane Z equals zero. That's this X Y Keen is not included. That boundary that bound opinion is not included in the domain. Well, the wind, that's everything else. If you know, you may see some light lines that does not include this boundary, please. Okay, now we need to find that you were writing a function here. We have to do. Thanks. Why you and go Teoh into I think that's what wanted interesting. And we're finding the level curve at the point natural level for not for love of nine I'm to. So first we need to be involved with the function at a given point And what that gives them each other. Not for long, for most natural number of line all over too. No. If he used the properties of exponents on properties of logarithms, this would simplify to eat to the natural longer stick which is simply 4 to 6. Uh, So what we have known is an equation e to the X plus y or Z and equal intensity thing within. Simplify here by taking the natural love of both sides, and we have here and that's why over easy is equal to the natural love expect So we have X plus y minus Z times and not for long equal there. This would be the increasing, but it never occurred at the point I want another or that's a love of nine.

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