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)

Outline

Graphing functions of two variablesUtility Functions and indifference curves

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.

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.

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.

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.

Math 20 - October 31, 2007.GWBWednesday, Oct 31, 2007

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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.

Enter the topographic map

Math 20 - October 31, 2007.GWBWednesday, Oct 31, 2007

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Outline

Graphing functions of two variablesUtility Functions and indifference curves

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

Math 20 - October 31, 2007.GWBWednesday, Oct 31, 2007

Page5of8

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

Math 20 - October 31, 2007.GWBWednesday, Oct 31, 2007

Page6of8

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

Math 20 - October 31, 2007.GWBWednesday, Oct 31, 2007

Page7of8

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

Math 20 - October 31, 2007.GWBWednesday, Oct 31, 2007

Page8of8

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