https://r2.erweima.ai/mathbot/image/17260342529029oab1v0h.pdf

The document contains exercises for class 2, focusing on functions of two variables. Here's a summary of the exercises:

### Exercises:
1. **Determine the domain of the following functions:**
   - (a) \( f(x, y) = \ln(1 - x^2 - y^2) \)
   - (b) \( f(x, y) = x\sqrt{1 + y^2} \)
   - (c) \( f(x, y) = x\sqrt{1 - y^2} \)
   - (d) \( f(x, y) = \frac{xy^2}{x + y - 1} \)

2. **Determine the level sets of the following functions:**
   - (a) \( f(x, y) = x^2 - y^2 \)
   - (b) \( f(x, y) = \sqrt{x^2 + y^2} \)

3. **Draw the graph of the following functions:**
   - (a) \( f(x, y) = x^2 - y^2 \)
   - (b) \( f(x, y) = \sqrt{x^2 + y^2} \)

Would you like help solving any of these exercises, or do you have any specific questions about them?

### Related Questions:
1. How do you determine the domain of a function involving square roots and logarithms?
2. What are level sets, and how do you find them for a given function?
3. How can you sketch the graph of a function of two variables?
4. What is the significance of the domain in multivariable functions?
5. How do you interpret the level sets geometrically?

### Tip:
For functions involving square roots, ensure the expression inside the square root is non-negative to find the domain.

Determine the domain, level sets, and graph of functions such as f(x, y) = ln(1 - x^2 - y^2), f(x, y) = x√(1 + y^2), and others.

Explore the domain, level sets, and graphs of multivariable functions like f(x, y) = ln(1 - x^2 - y^2) and f(x, y) = x√(1 + y^2). This problem set covers logarithmic functions, square root domains, and 3D graphing, suitable for university-level students.

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