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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.$$g(x, y, z)=\int_{x}^{y} \frac{d t}{1+t^{2}}+\int_{0}^{z} \frac{d \theta}{\sqrt{4-\theta^{2}}}, \quad(0,1, \sqrt{3})$$

$$\begin{aligned}&D=\{(x, y, z) |-\infty<x<\infty,-\infty<y<\infty,-1 \leq z \leq 1\}\\&\text { The required level curve is } \tan ^{-1} y-\tan ^{-1} x+\sin ^{-1} \frac{z}{2}=\frac{7 \pi}{12}\end{aligned}$$

Calculus 3

Chapter 14

Partial Derivatives

Section 1

Functions of Several Variables

Campbell University

Harvey Mudd College

Baylor University

University of Nottingham

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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Sketch the region enclosed…

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in this problem, we're going to find and sketch domain of the function and also firing Don't present the level curve under given point so we know that our function looks a bit complicated. But if you remember your integration formula for the inverse, the inverse strict functions then you can use when you're that wonderful in terms on in verse of new over eight is equal to the integral. Do you over its greatness. You scream. We also knew that flying in verse are you over A is equal to the integral game Over the square root of a square blue you square Using these two formulas we can simplify Yeah, thanks As town in birth, this first integral becomes tent in verse Oh, team evaluated from extra y Look, trying anglers of data the value it were evaluated We're to decode here toe work that ankle you want It gives us time in bursts of white My mom found in verse thank nine inverse of the, um an inverse of zero simply there. So the Suki I will function z No. The domain of the inverse standards only numbers but a domain of in verse Sign is from negative. Born to positive, born domain of our function. Gene would be on point X for anything such that near We knew that Oh, is all the numbers from the government and the need to infinity? I think a y what me would be from negative born to one inclusive? I don't mean Org No. When we after gratis what we need to know that we will have This is a three dimensional beast. So our boundary Miss Cates, we have two planes and that is just lean. Where's the vehicle to negative one on the plane where Z is equal to one before I assure you to grab the domain. There's once like correction. I need to meet my highest right in writing. I forgot to add this, but this here should be inverse finals data over to you. Just beat him thing because I always two squared that this would be the over to down here. So here we have the graph of our to me. Right, And that would be everything. In the highlighted area, our boundary points are included. You have a close doing? No. We will move on to finding the level curve at that point number one negative three. First we evaluate oh function either given point and what we have you and on in verse one mind on in versus zero Fine X squared of three number two When we get here to an inverse of oneness fire before this term is zero and buying inverse of skirt or 3/2 Yes, prior, but three. I didn't These that we get every party, We're 12. So the equation of the level curve at a given point would be turn in worse of wine lying found in verse. Thank for buying. Leading by two is equal to They won't pine orderto You know, this is at the point no one good of three.

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