Lesson 10: Functions and Level Sets

Slide 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

Slide 2: Outline Graphing with traces Contour plots Topographic maps Contour plots of quadric surfaces Utility Functions and indifference curves Worksheet

Slide 3: Linear Functions The graph of f (x) = mx + b is a line in the plane.

Slide 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

Slide 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.

Slide 6: Example Graph z = x 2 + y 2.

Slide 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.

Slide 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

Slide 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.

Slide 10: Outline Graphing with traces Contour plots Topographic maps Contour plots of quadric surfaces Utility Functions and indifference curves Worksheet

Slide 11: Enter the topographic map

Slide 12: Outline Graphing with traces Contour plots Topographic maps Contour plots of quadric surfaces Utility Functions and indifference curves Worksheet

Slide 13: A contour plot is a topographic map of a graph Intersect the cone with planes z = c and what do you get?

Slide 14: A contour plot is a topographic map of a graph Intersect the cone with planes z = c and what do you get? Circles.

Slide 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

Slide 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

Slide 17: Example Graph z = x 2 + y 2 .

Slide 18: Example Graph z = x 2 + y 2 . 3 2 1 0 -1 -2 -3 -3 -2 -1 0 1 2 3

Slide 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

Slide 20: Example Graph z = x 2 − y 2 .

Slide 21: Example Graph z = x 2 − y 2 . 3 2 1 0 -1 -2 -3 -3 -2 -1 0 1 2 3

Slide 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

Slide 23: Plotting a Cobb-Douglas function Example Plot z = x 1/2 y 1/2 .

Slide 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

Slide 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

Slide 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

Slide 27: Outline Graphing with traces Contour plots Topographic maps Contour plots of quadric surfaces Utility Functions and indifference curves Worksheet

Slide 28: Worksheet Image: Erick Cifuentes