We show surface graphs and contour plots of functions of two variables

1. Lesson 18 (Section 15.2)
Geometric Representations of Functions of
Several Variables
Math 20
October 31, 2007
Announcements
Problem Set 7 assigned today. Due November 7.
OH: Mondays 1–2, Tuesdays 3–4, Wednesdays 1–3 (SC 323)
Prob. Sess.: Sundays 6–7 (SC B-10), Tuesdays 1–2 (SC 116)

2. Outline
Graphing functions of two variables
Utility Functions and indifference curves

3. Linear Functions
The graph of f (x) = mx + b is a line in the plane.

5. Linear Functions
The graph of f (x) = mx + b is a line in the plane.
Example
Graph the function
f (x, y ) = 2x + 3y + 1

7. Linear Functions
The graph of f (x) = mx + b is a line in the plane.
Example
Graph the function
f (x, y ) = 2x + 3y + 1
Solution
The graph is a plane.

8. Example
x 2 + y 2.
Graph z =

10. Example
x 2 + y 2.
Graph z =
The traces are the absolute value functions. By staring at it, you
can see z = |r |, so this is just a cone.

11. Example
x 2 + y 2.
Graph z =
The traces are the absolute value functions. By staring at it, you
can see z = |r |, so this is just a cone.
4
3
2 2
1
0
0
-2
0
-2
2

12. Example
x 2 + y 2.
Graph z =
The traces are the absolute value functions. By staring at it, you
can see z = |r |, so this is just a cone.
4
3
2 2
1
0
0
-2
0
-2
2
Even this is hard to draw.

13. Enter the topographic map

15. Outline
Graphing functions of two variables
Utility Functions and indifference curves

16. A contour plot is a topographic map of a graph
Intersect the cone with planes z = c and what do you get?

18. A contour plot is a topographic map of a graph
Intersect the cone with planes z = c and what do you get? Circles.

19. A contour plot is a topographic map of a graph
Intersect the cone with planes z = c and what do you get? Circles.
A contour plot shows evenly spaced circles.
3
2
1
0
-1
-2
-3
-1 1
-3 -2 0 2 3

20. A contour plot is a topographic map of a graph
Intersect the cone with planes z = c and what do you get? Circles.
A contour plot shows evenly spaced circles.
3
2
1
0
4
3
2
-1 2
1
0
0
-2 -2
0
-2
-3
2
-1 1
-3 -2 0 2 3

21. Example
Graph z = x 2 + y 2 .

23. Example
Graph z = x 2 + y 2 .
3
2
1
0
-1
-2
-3
-1 1
-3 -2 0 2 3

24. The paraboloid
Example
Graph z = x 2 + y 2 .
3
2
1
0
15
10
-1 2
5
0
0
-2 -2
0
-2
-3
2
-1 1
-3 -2 0 2 3

26. Example
Graph z = x 2 − y 2 .

28. Example
Graph z = x 2 − y 2 .
3
2
1
0
-1
-2
-3
-1 1
-3 -2 0 2 3

29. The hyperbolic paraboloid
Example
Graph z = x 2 − y 2 .
3
2
1
0
5
0
-1 2
-5
0
-2 -2
0
-2
-3
2
-1 1
-3 -2 0 2 3

30. Plotting a Cobb-Douglas function
Example
Plot z = x 1/2 y 1/2 .

31. Plotting a Cobb-Douglas function
Example
Plot z = x 1/2 y 1/2 .
3
2.5
2
1.5
1
0.5
0
1 1.5
0 0.5 2 2.5 3

32. Plotting a Cobb-Douglas function
Example
Plot z = x 1/2 y 1/2 .
3
2.5
2
1.5 3
3
2
1 1
2
0
0
0.5
1 1
2
0
1 1.5
0 0.5 2 2.5 3 30

33. Utility Functions and indifference curves
If u is a utility function, a level curve of u is a curve along
which all points have the same u value.
We also know this as an indifference curve