Section 11.1
Functions and Level Sets
Math 21a
February 27, 2008
Announcements
Problem Sessions: Monday, 8:30 (Sophie); Thursday, 7:30
(Jeremy); SC 103b
Office hours today 2:30–4pm SC 323.
Mathematica assignment due February 29.
Image: Hexod.us

Outline
Graphing with traces
Contour plots
Topographic maps
Contour plots of quadric surfaces
Utility Functions and indifference curves
Worksheet

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

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

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.

Example
Graph z = x 2 + y 2.

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

Example
Graph z = x 2 + y 2.
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

Example
Graph z = x 2 + y 2.
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.

Outline
Graphing with traces
Contour plots
Topographic maps
Contour plots of quadric surfaces
Utility Functions and indifference curves
Worksheet

Enter the topographic map

Outline
Graphing with traces
Contour plots
Topographic maps
Contour plots of quadric surfaces
Utility Functions and indifference curves
Worksheet

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

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 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
-3 -2 -1 0 1 2 3

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
-1 2 2
1
0
-2 0
-2
0
-2
-3
-3 -2 -1 0 1 2 3 2

Example
Graph z = x 2 + y 2 .

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

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
-3 -2 -1 0 1 2 3 2

Example
Graph z = x 2 − y 2 .

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

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

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

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

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

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

Outline
Graphing with traces
Contour plots
Topographic maps
Contour plots of quadric surfaces
Utility Functions and indifference curves
Worksheet

Worksheet
Image: Erick Cifuentes