# Section 1: Functions of Several Variables

### Instructions

- First, you should watch the concepts videos below explaining the topics in the section.
- Second, you should attempt to solve the exercises and then watch the videos explaining the exercises.
- Last, you should attempt to answer the self-assessment questions to determine how well you learned the material.
- When you have finished the material below, you can start on Section 2 or return to the main several variable calculus page.

### Resources

Download Notes### Concepts

- The definition of a function of two variables
- The graph of a function of two variables with domain D and range R
- The level curves of a function of two variables

### Exercises

**Directions:**You should attempt to solve the problems first and then watch the video to see the solution.

- Evaluate \(f(-1,3).\)
- Find the domain of \(f(x,y)\) and sketch the domain of \(f(x,y)\) in the \(xy\)-plane.

3. Sketch the level curves for \(f(x,y)=1-5x+y\) for \(k=1, 0, -1.\)

4. Sketch the level curves for \(f(x,y)=\sqrt{4-x^2-y^2}\) for \(k= 0, 1, 2.\)

5. Sketch the level curve for \(f(x,y)=\sqrt{y^2-x^2}\) for \(k=4.\)

### Self-Assessment Questions

**Directions:**The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.

- What are the necessary steps when finding the domain of a surface \(z=f(x,y)\)?
- When finding the domain of a surface \(z=f(x,y)\), how do we determine the set of all points \((x,y)\) where \(f(x,y)\) is
**not**defined? - How do the level curves of a surface \(z=f(x,y)\) help us visualize, and hence piece together, the graph of a surface?
- What is the relation between a level curve and a horizontal trace?