Lesson 10: Functions and Level Sets

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

Worksheet

Image: Erick Cifuentes