Your SlideShare is downloading. ×
Lesson 10: Functions and Level Sets
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×

Introducing the official SlideShare app

Stunning, full-screen experience for iPhone and Android

Text the download link to your phone

Standard text messaging rates apply

Lesson 10: Functions and Level Sets

3,509
views

Published on

A contour plot is a nice way to visualize the graph of a function of two variables. If the function is a utility function, this is nothing more than the set of indifference curves. More generally, …

A contour plot is a nice way to visualize the graph of a function of two variables. If the function is a utility function, this is nothing more than the set of indifference curves. More generally, it's like a topographical map of the surface

Published in: Technology

0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
3,509
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
59
Comments
0
Likes
0
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide

Transcript

  • 1. 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
  • 2. Outline Graphing with traces Contour plots Topographic maps Contour plots of quadric surfaces Utility Functions and indifference curves Worksheet
  • 3. Linear Functions The graph of f (x) = mx + b is a line in the plane.
  • 4. 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
  • 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 Solution The graph is a plane.
  • 6. Example Graph z = x 2 + y 2.
  • 7. 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.
  • 8. 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
  • 9. 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.
  • 10. Outline Graphing with traces Contour plots Topographic maps Contour plots of quadric surfaces Utility Functions and indifference curves Worksheet
  • 11. Enter the topographic map
  • 12. Outline Graphing with traces Contour plots Topographic maps Contour plots of quadric surfaces Utility Functions and indifference curves Worksheet
  • 13. A contour plot is a topographic map of a graph Intersect the cone with planes z = c and what do you get?
  • 14. A contour plot is a topographic map of a graph Intersect the cone with planes z = c and what do you get? Circles.
  • 15. 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
  • 16. 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
  • 17. Example Graph z = x 2 + y 2 .
  • 18. Example Graph z = x 2 + y 2 . 3 2 1 0 -1 -2 -3 -3 -2 -1 0 1 2 3
  • 19. 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
  • 20. Example Graph z = x 2 − y 2 .
  • 21. Example Graph z = x 2 − y 2 . 3 2 1 0 -1 -2 -3 -3 -2 -1 0 1 2 3
  • 22. 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
  • 23. Plotting a Cobb-Douglas function Example Plot z = x 1/2 y 1/2 .
  • 24. 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
  • 25. 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
  • 26. 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
  • 27. Outline Graphing with traces Contour plots Topographic maps Contour plots of quadric surfaces Utility Functions and indifference curves Worksheet
  • 28. Worksheet Image: Erick Cifuentes