https://r2.erweima.ai/mathbot/image/1725585238487nipg9ec7.jpg

The function provided in the image is:

\[
f(x, y, z) = \arctan\left(\frac{z}{x}\right) + \frac{e^{x+y}}{x^2 + y^2}
\]

It appears to be a multivariable function, dependent on \(x\), \(y\), and \(z\). The first term involves the arctangent of \(\frac{z}{x}\), and the second term is an exponential function divided by the sum of squares of \(x\) and \(y\).

Would you like to analyze specific properties of this function, such as partial derivatives, gradient, or domain? Let me know how you'd like to proceed!

Here are five related questions you might find useful:
1. What is the gradient of this function \(f(x, y, z)\)?
2. How do you compute the Hessian matrix of this function?
3. What is the domain of this function in terms of \(x\), \(y\), and \(z\)?
4. How would you find the critical points of \(f(x, y, z)\)?
5. What is the behavior of this function as \(x\), \(y\), and \(z\) approach infinity?

**Tip**: When computing partial derivatives, remember to apply the chain rule carefully, especially in functions involving compositions like arctangent and exponentials.

What is the function f(x, y, z) = arctan(z/x) + e^(x+y)/(x^2 + y^2)?

Explore the multivariable function f(x, y, z) = arctan(z/x) + e^(x + y)/(x^2 + y^2) with a detailed breakdown of its mathematical properties. Learn about partial derivatives, gradients, and domain analysis of this function. Suitable for students in undergraduate calculus.

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