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 NotesConcepts
- 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.
1. Consider \(f(x,y)=\ln(y-4x).\)
- Evaluate \(f(-1,3).\)
- Find the domain of \(f(x,y)\) and sketch the domain of \(f(x,y)\) in the \(xy\)-plane.
2. Find the domain of \(f(x,y)=\displaystyle{{\sqrt{9-x^2-y^2}}\over{x+y}}\) 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?