Section 11.1 Functions and Level Sets Math 21a February 27, 2008 Announcements I Problem Sessions: Monday, 8:30 (Sophie); Thursday, 7:30 (Jeremy); SC 103b I Office hours today 2:30–4pm SC 323. I Mathematica assignment due February 29. Image: Hexod.us
Lesson 10: Functions and Level Sets
May 24, 2015
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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Section 11.1
Functions and Level Sets
Math 21a
February 27, 2008
Announcements
I Problem Sessions: Monday, 8:30 (Sophie); Thursday, 7:30(Jeremy); SC 103b
I Office hours today 2:30–4pm SC 323.
I Mathematica assignment due February 29.
Image: Hexod.us
Outline
Graphing with traces
Contour plotsTopographic maps
Contour plots of quadric surfacesUtility Functions and indifference curves
Worksheet
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
SolutionThe graph is a 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
SolutionThe graph is a 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
SolutionThe graph is a plane.
Example
Graph z =√
x2 + y2.
The traces are the absolute value functions. By staring at it, youcan see z = |r |, so this is just a cone.
-2
0
2
-2
0
2
0
1
2
3
4
-2
0
2
Even this is hard to draw.
Example
Graph z =√
x2 + y2.
The traces are the absolute value functions. By staring at it, youcan see z = |r |, so this is just a cone.
-2
0
2
-2
0
2
0
1
2
3
4
-2
0
2
Even this is hard to draw.
Example
Graph z =√
x2 + y2.
The traces are the absolute value functions. By staring at it, youcan see z = |r |, so this is just a cone.
-2
0
2
-2
0
2
0
1
2
3
4
-2
0
2
Even this is hard to draw.
Example
Graph z =√
x2 + y2.
The traces are the absolute value functions. By staring at it, youcan see z = |r |, so this is just a cone.
-2
0
2
-2
0
2
0
1
2
3
4
-2
0
2
Even this is hard to draw.
Outline
Graphing with traces
Contour plotsTopographic maps
Contour plots of quadric surfacesUtility Functions and indifference curves
Worksheet
UTM 19 327126E 4694768N (NAD27) Somerville, USGS Boston North (MA) Quadrangle
Projection is UTM Zone 19 NAD83 Datum
Enter the topographic map
Outline
Graphing with traces
Contour plotsTopographic maps
Contour plots of quadric surfacesUtility 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?
Circles.A contour plot shows evenly spaced circles.
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
-2
0
2
-2
0
2
0
1
2
3
4
-2
0
2
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
-2
0
2
-2
0
2
0
1
2
3
4
-2
0
2
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
-2
0
2
-2
0
2
0
1
2
3
4
-2
0
2
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
-2
0
2
-2
0
2
0
1
2
3
4
-2
0
2
The paraboloid
Example
Graph z = x2 + y2.
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
-2
0
2
-2
0
2
0
5
10
15
-2
0
2
The paraboloid
Example
Graph z = x2 + y2.
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
-2
0
2
-2
0
2
0
5
10
15
-2
0
2
The paraboloid
Example
Graph z = x2 + y2.
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
-2
0
2
-2
0
2
0
5
10
15
-2
0
2
The hyperbolic paraboloid
Example
Graph z = x2 − y2.
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
-2
0
2
-2
0
2-5
0
5
-2
0
2
The hyperbolic paraboloid
Example
Graph z = x2 − y2.
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
-2
0
2
-2
0
2-5
0
5
-2
0
2
The hyperbolic paraboloid
Example
Graph z = x2 − y2.
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
-2
0
2
-2
0
2-5
0
5
-2
0
2
Plotting a Cobb-Douglas function
Example
Plot z = x1/2y1/2.
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
0
1
2
3 0
1
2
3
0
1
2
3
0
1
2
Plotting a Cobb-Douglas function
Example
Plot z = x1/2y1/2.
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
0
1
2
3 0
1
2
3
0
1
2
3
0
1
2
Plotting a Cobb-Douglas function
Example
Plot z = x1/2y1/2.
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
0
1
2
3 0
1
2
3
0
1
2
3
0
1
2
Utility Functions and indifference curves
I If u is a utility function, a level curve of u is a curve alongwhich all points have the same u value.
I We also know this as an indifference curve
Outline
Graphing with traces
Contour plotsTopographic maps
Contour plots of quadric surfacesUtility Functions and indifference curves
Worksheet
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