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Section 9.6: Functions and Surfaces
Practice HW from Stewart Textbook (not to hand in)
p. 683 # 9-13, 19, 20, 23, 24, 25Handout Sheet 1-6, 7-27 odd
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Functions of More Than One Variable
So far most of our experience has been working with functions of one variables.
Some examples are: , , . In this section, we want to examine functions and variables of multivariate equations and functions, like or . We want to look at techniques for obtaining rough sketches of these types of graphs in 3D space.
2)( xxf xxg ln)( xexh )(
663),( yxyxfzyyxfz cos),(
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Cylinders
A cylinder is formed by taking a curve in 3D
space along with lines which intersect the curve
projected in the direction of the coordinate axis
not included in the equation. You can recognize
this by seeing that one of the coordinate axis
variables that are missing.
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Example 1: Make a rough sketch of the equation
.1622 yx
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x
y
z
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Example 2: Make a rough sketch of the equation
.yyxf cos),(
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x
y
z
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Graphing Planes
Recall that the equation of a plane is given by
(note that the variables x, y, and
z). To make a rough sketch of a plane in 3D
space, it is easiest to find the points of
intersection with the coordinate axes.
dczbyax
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Example 3: Make a rough sketch of the equation
.4),( yxf
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x
y
z
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Example 4: Make a rough sketch of the equation
.42 yx
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x
y
z
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Example 5: Make a rough sketch of the equation
.663),( yxyxfz
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x
y
z
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Quadric Surface
Quadric surfaces are the 3D analog of the conic
sections in 2D.
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Conic Sections is 2D
Parabolas
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Ellipse
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Hyperbola
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Types of Quadric Surfaces (summary on p.682 text)
Ellipsoid
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Hyperboloid (One Sheet)
Note: The axis the graph is projected along is the
variable with the negative coefficient.
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Hyperboloid (Two Sheets)
Note: The axis the graph is projected along is the
variable with the positive coefficient.
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Elliptic Cone
Axis of projection is the variable with the negative
coefficient
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Elliptic Paraboloid
Axis of projection is the variable term raised to
the 1st power.
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Hyperbolic Paraboloid
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Note: To sketch a 3D quadric surface, sometimes
it can be useful to set each variable equal to 0
and sketch the corresponding 2D curve. This is
known as a trace.
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Example 6: Identify and make a rough sketch of
the quadric surface . 14
22
2 zy
x
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x
y
z
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Example 7: Identify and make a rough sketch of
the quadric surface . 1449916 222 zyx
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x
y
z
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Example 8: Identify and make a rough sketch of
the quadric surface . 224 zxy
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x
y
z