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.

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

Links & Resources

Exercises

1. Consider $f(x,y)=\ln (y-4x)$.
   1. Evaluate $f(-1,3).$
   2. Find the domain of $f(x,y)$ and sketch the domain of $f(x,y)$ in the $xy$-plane. ​
      Reveal Answer

      1. \(f(-1,3)=\ln (7)\)
      2. Domain: \( \{ (x,y) \mid y-4x>0\}\). See the video for a sketch.

      Watch Video

      
      
      If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing. 
      Functions of Several Variables Exercise V1

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2. Find the domain of $f(x,y)={\displaystyle \frac{\sqrt{9-x^2-y^2}}{x+y}}$ and sketch the domain of $f(x,y)$ in the $xy$-plane.
   Reveal Answer

   Domain: \( \left\{ (x,y) \mid 9-x^2-y^2\geq 0 \; \text{ and } \; x+y\neq 0\right\}\). See the video for a sketch.

   Watch Video

   
   
   If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing. 
   Functions of Several Variables Exercise V2

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

   See the video for the sketch. 

   Watch Video

   
   
   If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing. 
   Functions of Several Variables Exercise V3

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

   See the video for the sketch. 

   Watch Video

   
   
   If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing. 
   Functions of Several Variables Exercise V4

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

   See the video for the sketch.

   Watch Video

   
   
   If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing. 
   Functions of Several Variables Exercise V5

Self-Assessment Questions

1. What are the necessary steps when finding the domain of a surface $z=f(x,y)$?

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2. 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?

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3.  How do the level curves of a surface $z=f(x,y)$ help us visualize, and hence piece together,  the graph of a surface?

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4. What is the relation between a level curve and a horizontal trace?