Part 4: Graphs of Functions of Two Variables

The graph of a function z = f(x,y) is also the graph of an equation in three variables and is therefore a surface. Since each pair (x,y) in the domain determines a unique value of z, the graph of a function must satisfy the "vertical line test" already familiar from single-variable calculus. Some of the surfaces we have encountered in the preceding sections are graphs of functions and some are not.

1. Which (if any) of the Cartesian coordinate surfaces x = c, y = c, and z = c are graphs of functions? Explain.
2. What familiar surface is the graph of the function z = x² + y²?
3. Which (if any) of the following quadric surfaces are graphs of functions? Explain.
• Sphere, x² + y² + z² = p²
• Elliptic paraboloid, z = 3x² + y²
• Hyperbolic paraboloid, z = x² - y²
• Hyperboloid of one sheet, x² + y² - z² = 1
• Hyperboloid of two sheets, x² - y² - z² = 1

4. Which (if any) of the following cylinders are graphs of functions? Explain.
   • Circular cylinder, x² + y² =4
   • Circular cylinder, x² + z² =4
   • Parabolic cylinder, y² = z
   • Parabolic cylinder, z² = x
   • Sinusoidal cylinder, y = sin(x)
   • Sinusoidal cylinder, z = sin(x)

From economics we have the important concept of a Cobb-Douglas production function, the simplest example of which is f(x,y) = . In economic terms, the function relates productivity to labor and capital. The graph of this function for 0 < x < 2 and 0 < y < 2 is shown below.

5. Make your own picture of the Cobb-Douglas function f(x,y) =  in your worksheet, and rotate it to view the surface from various angles.
6. A more general form for Cobb-Douglas functions is f(x,y) = x^ay^1-a, where 0 < a < 1. What value of a gives the function in step 5? Vary a between 0 and 1, and redraw the surface. What stays the same when you change a? What changes?

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Copyright CCP and the author(s), 2001-2002