Graphs of functions and level curves.
Ÿ The elliptic parabolloid z = x2 + y2
ParametricPlot3D@8Sqrt@tD Cos@uD, Sqrt@tD Sin@uD, t<,
8u, 0, 2 Pi<, 8t, 0, 4<, PlotPoints -> 40, AxesLabel -> 8"x", "y", "z"<D;
y
2
1
0
-1
-2
4
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, 8x, -2, 2<, 8y, -2, 2<, PlotPoints -> 60D;
2
1
0
-1
-2
-2
-1
0
1
2
2
Graphs and level curves
Ÿ The elliptic parabolloid z = x2 + 6 y2
ParametricPlot3D@8Sqrt@tD Cos@uD, Sqrt@tD Sin@uD  Sqrt@6D, t<,
8u, 0, 2 Pi<, 8t, 0, 4<, PlotPoints -> 40, AxesLabel -> 8"x", "y", "z"<D;
y
0.5-2
0
-0.5
-1
x
0
1
2
4
3
2
z
1
0
Note that now the level curves are ellipses:
ContourPlot@x ^ 2 + 6 * y ^ 2, 8x, -4, 4<, 8y, -3, 3<, PlotPoints -> 60D;
3
2
1
0
-1
-2
-3
-4
-2
0
2
4
3
Graphs and level curves
Ÿ The hiperbolic paraboloid z = x2 - y2
In[13]:= Plot3D@x ^ 2 - y ^ 2, 8x, -1, 1<, 8y, -1, 1<, BoxRatios -> Automatic,
PlotPoints -> 30, Mesh -> False, AxesLabel -> 8"x", "y", "z"<D;
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, 8x, -.5, .5<, 8y, -.5, .5<, PlotPoints -> 60D;
0.4
0.2
0
-0.2
-0.4
-0.4
-0.2
0
0.2
0.4
4
Graphs and level curves
Ÿ The function z = sinHx2 - y2 L
In[14]:= Plot3D@Sin@x ^ 2 - y ^ 2D, 8x, -2, 2<, 8y, -2, 2<, BoxRatios -> Automatic,
PlotPoints -> 30, Mesh -> False, AxesLabel -> 8"x", "y", "z"<D;
1
0.5
z
0
-0.5
-1
-2
2
1
0
-1
-1
0
x
y
1
2 -2
The level curves:
In[5]:= ContourPlot@Sin@x ^ 2 - y ^ 2D, 8x, -2, 2<, 8y, -2, 2<, PlotPoints -> 60D;
2
1
0
-1
-2
-2
-1
0
1
2
5
Graphs and level curves
sinHx2 +y2 L
Ÿ The function €€€€€€€€€€€€€€€€
€€€€€€€€€
x2 +y2
In[15]:= Plot3D@Sin@Sqrt@x ^ 2 + y ^ 2DD  Sqrt@x ^ 2 + y ^ 2D, 8x, -5, 5<,
8y, -5, 5<, PlotPoints -> 50, Mesh -> False, AxesLabel -> 8"x", "y", "z"<D;
1
z0.5
4
2
0
0
-4
-2
0
2
x
y
-2
-4
4
Can you imagine the level curves? Here they are:
In[18]:= ContourPlot@Sin@Sqrt@x ^ 2 + y ^ 2DD  Sqrt@x ^ 2 + y ^ 2D,
8x, -3, 3<, 8y, -3, 3<, PlotPoints -> 60D;
3
2
1
0
-1
-2
-3
-3
-2
-1
0
1
2
3
6
Graphs and level curves
Sin@x2 -yD
Ÿ The function €€€€€€€€€€€€€€€€
€€€€€€€€
x2 -y
In[16]:= Plot3D@Sin@x ^ 2 - yD  Hx ^ 2 - yL, 8x, -4, 4<,
8y, -4, 4<, BoxRatios -> Automatic, PlotPoints -> 50,
ViewPoint -> 8-3.753, -3.849, 1.787<, Mesh -> False, AxesLabel -> 8"x", "y", "z"<D;
1
0.5
0 z
4
4
2
2
0
y
-2
-2
-4
0
x
What are the level curves in this case?
In[10]:= ContourPlot@Sin@x ^ 2 - yD  Hx ^ 2 - yL, 8x, -4, 4<, 8y, -4, 4<, PlotPoints -> 60D;
4
2
0
-2
-4
-4
-2
0
2
4
7
Graphs and level curves
Ÿ The function z = È xy È
In[17]:= Plot3D@Abs@x yD, 8x, -1, 1<, 8y, -1, 1<, BoxRatios -> Automatic,
PlotPoints -> 30, Mesh -> False, AxesLabel -> 8"x", "y", "z"<D;
1
0.75
z
0.5
0.25
0
-1
1
0.5
0 y
-0.5
-0.5
0
x
0.5
1 -1
What are the level curves? They are hyperbolas:
In[12]:= ContourPlot@Abs@x yD, 8x, -1, 1<, 8y, -1, 1<, PlotPoints -> 60D;
1
0.5
0
-0.5
-1
-1
-0.5
0
0.5
1