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# Lesson 10: Functions and Level Sets

## by Matthew Leingang, Clinical Associate Professor of Mathematics at New York University on Feb 27, 2008

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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, ...

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 SetsPresentation Transcript

• 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
• Outline Graphing with traces Contour plots Topographic maps Contour plots of quadric surfaces Utility Functions and indiﬀerence curves Worksheet
• Linear Functions The graph of f (x) = mx + b is a line in the plane.
• 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
• 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.
• Example Graph z = x 2 + y 2.
• 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.
• 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
• 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.
• Outline Graphing with traces Contour plots Topographic maps Contour plots of quadric surfaces Utility Functions and indiﬀerence curves Worksheet
• Enter the topographic map
• Outline Graphing with traces Contour plots Topographic maps Contour plots of quadric surfaces Utility Functions and indiﬀerence curves Worksheet
• A contour plot is a topographic map of a graph Intersect the cone with planes z = c and what do you get?
• 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 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
• 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
• Example Graph z = x 2 + y 2 .
• Example Graph z = x 2 + y 2 . 3 2 1 0 -1 -2 -3 -3 -2 -1 0 1 2 3
• 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
• Example Graph z = x 2 − y 2 .
• Example Graph z = x 2 − y 2 . 3 2 1 0 -1 -2 -3 -3 -2 -1 0 1 2 3
• 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
• Plotting a Cobb-Douglas function Example Plot z = x 1/2 y 1/2 .
• 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
• 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
• 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
• Outline Graphing with traces Contour plots Topographic maps Contour plots of quadric surfaces Utility Functions and indiﬀerence curves Worksheet
• Worksheet Image: Erick Cifuentes