Graphs of functions and level curves.

[Pages:7]Graphs of functions and level curves.

The elliptic parabolloid z x2 y2

ParametricPlot3D Sqrt t Cos u , Sqrt t Sin u , t , u, 0, 2 Pi , t, 0, 4 , PlotPoints 40, AxesLabel

2 y1 0 -1

-2 4

"x", "y", "z" ;

3 z

2

1

0

-2

-1

0

x

1

2

The level curves are circles centered at the origin :

ContourPlot x ^ 2 y ^ 2, x, 2, 2 , y, 2, 2 , PlotPoints 60 ; 2

1

0

-1

-2

-2

-1

0

1

2

2

Graphs and level curves

The elliptic parabolloid z x2 6 y2

ParametricPlot3D Sqrt t Cos u , Sqrt t Sin u Sqrt 6 , t , u, 0, 2 Pi , t, 0, 4 , PlotPoints 40, AxesLabel "x", "y", "z" ;

y 0.5-2 0

-1

-0.5

x 0

1

2

4

3 2z 1 0

Note that now the level curves are ellipses: ContourPlot x ^ 2 6 y ^ 2, x, 4, 4 , y, 3, 3 , PlotPoints 60 ; 3

2

1

0

-1

-2

-3

-4

-2

0

2

4

Graphs and level curves

3

The hiperbolic paraboloid z x2 y2

In[13]:= Plot3D x ^ 2 y ^ 2, x, 1, 1 , y, 1, 1 , BoxRatios Automatic, PlotPoints 30, Mesh False, AxesLabel "x", "y", "z" ;

1 y 0.5 0 -0.5

-1 1

0.5 z

0

-0.5

-1 -1

-0.5

0

x

0.5

1

The level curves are hyperbolas:

ContourPlot x ^ 2 y ^ 2, x, .5, .5 , y, .5, .5 , PlotPoints 60 ;

0.4

0.2

0

-0.2

-0.4 -0.4 -0.2 0

0.2 0.4

4

Graphs and level curves

The function z sin x2 y2

In[14]:= Plot3D Sin x ^ 2 y ^ 2 , x, 2, 2 , y, 2, 2 , BoxRatios Automatic, PlotPoints 30, Mesh False, AxesLabel "x", "y", "z" ;

1

0.5 z

0

-0.5

-1 -2

-1

0

x

1

2 1 0y -1

2 -2

The level curves:

In[5]:= ContourPlot Sin x ^ 2 y ^ 2 , x, 2, 2 , y, 2, 2 , PlotPoints 60 ; 2

1

0

-1

-2

-2

-1

0

1

2

Graphs and level curves

5

The function

sin x2 y2 x2 y2

In[15]:= Plot3D Sin Sqrt x ^ 2 y ^ 2 Sqrt x ^ 2 y ^ 2 , x, 5, 5 , y, 5, 5 , PlotPoints 50, Mesh False, AxesLabel "x", "y", "z" ;

1

z0.5

4

0

2

-4

0y

-2

-2

0

x

2

-4

4

Can you imagine the level curves? Here they are:

In[18]:= ContourPlot Sin Sqrt x ^ 2 y ^ 2 Sqrt x ^ 2 y ^ 2 , x, 3, 3 , y, 3, 3 , PlotPoints 60 ;

3

2

1

0

-1

-2

-3

-3 -2 -1 0

1

2

3

6

Graphs and level curves

The function

Sin x2 y x2 y

In[16]:= Plot3D Sin x ^ 2 y x ^ 2 y , x, 4, 4 ,

y, 4, 4 , BoxRatios Automatic, PlotPoints 50,

ViewPoint

3.753, 3.849, 1.787 , Mesh False, AxesLabel

"x", "y", "z" ;

1

00.5z

4

4

2

2

0

0

y -2 -4 -2 x

What are the level curves in this case?

In[10]:= ContourPlot Sin x ^ 2 y x ^ 2 y , x, 4, 4 , y, 4, 4 , PlotPoints 60 ;

4

2

0

-2

-4

-4

-2

0

2

4

Graphs and level curves

7

The function z xy

In[17]:= Plot3D Abs x y , x, 1, 1 , y, 1, 1 , BoxRatios Automatic, PlotPoints 30, Mesh False, AxesLabel "x", "y", "z" ;

1

0.75 z 0.5

0.25

0 -1

-0.5

0

x

0.5

1 0.5 0y -0.5

1 -1

What are the level curves? They are hyperbolas: In[12]:= ContourPlot Abs x y , x, 1, 1 , y, 1, 1 , PlotPoints 60 ;

1

0.5

0

-0.5

-1

-1

-0.5

0

0.5

1

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