Chapter 3: Functions of Several Variables
Section 3.1: Functions and Their Graphs
Example 3.1.1
Graph the surface determined by z=fx,y=10−3 x2−7 y2.
Solution
Maple Solution - Interactive
Write the rule for f.
Context Panel: Plots≻Plot Builder≻3-D plot
Use the interactive controls to modify the graph as needed.
10−3 x2−7 y2→
A graph of 10−3 x2−7 y2 can also be obtained with the . In the main panel shown in Figure 3.1.1(a) click either Preview or Plot. The resulting graph is shown in Figure 3.1.1(b).
Figure 3.1.1(a) Main panel of the Interactive Plot Builder
Figure 3.1.1(b) Interactive Plot Builder's default graph
Figure 3.1.1(d) is produced by changing the default ranges in the Plot Builder's main panel to −2≤x,y≤2, and clicking the Options button. Figure 3.1.1(c) shows the settings in the Options panel that are responsible for Figure 3.1.1(d). The Range from/to option is set to z∈−5,10, and "z" is added in the Label column. The Style is changed to "surface", Axes are changed to "frame", and Constrained Scaling is selected.
Figure 3.1.1(c) Options panel
Figure 3.1.1(d) Alternate rendering
There is a considerable difference between Figure 3.1.1(b) and Figure 3.1.1(d). Just a few changes in how a surface is rendered can give an entirely different aura to the graph of a surface. The optimal choices for a graph's parameters are a matter of trial and error, and may even depend on the intended purpose of the graph.
An alternative approach would be to launch the Context Panel on the expression 10−3 x2−7 y2 and to select Plots≻Plot Builder.
Maple Solution - Coded
Figure 3.1.1(b) is generated with the following instantiation of the plot3d command.
plot3d10−3 x2−7 y2,x=−5..5,y=−5..5
Figure 3.1.1(d) is generated with the following instantiation of the plot3d command.
plot3d10−3 x2−7 y2,x=−2..2,y=−2..2,view=0..10,axes=frame,style=surface,scaling=constrained,labels=x,y,z
If the "show command" option is selected in the interactive Plot Builder, the following implementation of the plot3d command will be written to a window beneath the graph. It can be copied and pasted in as input to generate the graph programmatically.
plot3d−3*x^2−7*y^2+10, x = −4 .. 4, y = −2 .. 2
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