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Lesson 18: Geometric Representations of Functions
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.
4.
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
6.
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.
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.
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?
17.
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
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