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# Lesson 18: Geometric Representations of Functions

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

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### Lesson 18: Geometric Representations of Functions

1. 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. 2. Outline Graphing functions of two variables Utility Functions and indiﬀerence curves
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 x 2 + y 2. Graph z =
7. 7. 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.
8. 8. 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
9. 9. 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.
10. 10. Enter the topographic map
11. 11. Outline Graphing functions of two variables Utility Functions and indiﬀerence curves
12. 12. A contour plot is a topographic map of a graph Intersect the cone with planes z = c and what do you get?
13. 13. A contour plot is a topographic map of a graph Intersect the cone with planes z = c and what do you get? Circles.
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. A contour plot shows evenly spaced circles. 3 2 1 0 -1 -2 -3 -1 1 -3 -2 0 2 3
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 4 3 2 -1 2 1 0 0 -2 -2 0 -2 -3 2 -1 1 -3 -2 0 2 3
16. 16. Example Graph z = x 2 + y 2 .
17. 17. Example Graph z = x 2 + y 2 . 3 2 1 0 -1 -2 -3 -1 1 -3 -2 0 2 3
18. 18. 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
19. 19. Example Graph z = x 2 − y 2 .
20. 20. Example Graph z = x 2 − y 2 . 3 2 1 0 -1 -2 -3 -1 1 -3 -2 0 2 3
21. 21. 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
22. 22. Plotting a Cobb-Douglas function Example Plot z = x 1/2 y 1/2 .
23. 23. 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
24. 24. 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
25. 25. 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