lesson 10: functions and level sets
DESCRIPTION
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 surfaceTRANSCRIPT
![Page 1: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/1.jpg)
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
![Page 2: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/2.jpg)
Outline
Graphing with traces
Contour plotsTopographic maps
Contour plots of quadric surfacesUtility Functions and indifference curves
Worksheet
![Page 3: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/3.jpg)
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.
![Page 4: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/4.jpg)
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.
![Page 5: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/5.jpg)
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.
![Page 6: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/6.jpg)
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.
![Page 7: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/7.jpg)
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.
![Page 8: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/8.jpg)
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.
![Page 9: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/9.jpg)
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.
![Page 10: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/10.jpg)
Outline
Graphing with traces
Contour plotsTopographic maps
Contour plots of quadric surfacesUtility Functions and indifference curves
Worksheet
![Page 11: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/11.jpg)
UTM 19 327126E 4694768N (NAD27) Somerville, USGS Boston North (MA) Quadrangle
Projection is UTM Zone 19 NAD83 Datum
Enter the topographic map
![Page 12: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/12.jpg)
Outline
Graphing with traces
Contour plotsTopographic maps
Contour plots of quadric surfacesUtility Functions and indifference curves
Worksheet
![Page 13: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/13.jpg)
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
![Page 14: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/14.jpg)
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
![Page 15: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/15.jpg)
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
![Page 16: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/16.jpg)
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
![Page 17: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/17.jpg)
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
![Page 18: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/18.jpg)
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
![Page 19: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/19.jpg)
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
![Page 20: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/20.jpg)
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
![Page 21: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/21.jpg)
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
![Page 22: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/22.jpg)
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
![Page 23: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/23.jpg)
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
![Page 24: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/24.jpg)
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
![Page 25: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/25.jpg)
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
![Page 26: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/26.jpg)
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
![Page 27: Lesson 10: Functions and Level Sets](https://reader035.vdocuments.site/reader035/viewer/2022081401/55615063d8b42adb6b8b51a8/html5/thumbnails/27.jpg)
Outline
Graphing with traces
Contour plotsTopographic maps
Contour plots of quadric surfacesUtility Functions and indifference curves
Worksheet