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# Lesson 10: Functions and Level Sets

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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

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### Lesson 10: Functions and Level Sets

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