lesson 18: geometric representations of functions
TRANSCRIPT
![Page 1: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/1.jpg)
Lesson 18 (Section 15.2)Geometric Representations of Functions of
Several Variables
Math 20
October 31, 2007
Announcements
I Problem Set 7 assigned today. Due November 7.
I OH: Mondays 1–2, Tuesdays 3–4, Wednesdays 1–3 (SC 323)
I Prob. Sess.: Sundays 6–7 (SC B-10), Tuesdays 1–2 (SC 116)
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Outline
Graphing functions of two variablesUtility Functions and indifference curves
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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.
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Math 20 - October 31, 2007.GWBWednesday, Oct 31, 2007
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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 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/6.jpg)
Math 20 - October 31, 2007.GWBWednesday, Oct 31, 2007
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![Page 7: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/7.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 8: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/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 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/9.jpg)
Math 20 - October 31, 2007.GWBWednesday, Oct 31, 2007
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![Page 10: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/10.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 11: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/11.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 12: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/12.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 13: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/13.jpg)
Enter the topographic map
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Math 20 - October 31, 2007.GWBWednesday, Oct 31, 2007
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Outline
Graphing functions of two variablesUtility Functions and indifference curves
![Page 16: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/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
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Math 20 - October 31, 2007.GWBWednesday, Oct 31, 2007
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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
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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 20: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/20.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 21: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/21.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
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Math 20 - October 31, 2007.GWBWednesday, Oct 31, 2007
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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 24: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/24.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 25: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/25.jpg)
Math 20 - October 31, 2007.GWBWednesday, Oct 31, 2007
Page7of8
![Page 26: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/26.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
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Math 20 - October 31, 2007.GWBWednesday, Oct 31, 2007
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![Page 28: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/28.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 29: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/29.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 30: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/30.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 31: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/31.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 32: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/32.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 33: Lesson 18: Geometric Representations of Functions](https://reader034.vdocuments.net/reader034/viewer/2022042600/58ee1ccf1a28ab252c8b456b/html5/thumbnails/33.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