Chapter 8
Multivariable
Calculus
Section 1
Functions of Several
Variables
Learning Objectives for Section 8.1
Functions of Several Variables
■ The student will be able to
identify functions of two or
more independent variables.
■ The student will be able to
evaluate functions of several
variables.
■ The student will be able to use
three-dimensional coordinate
systems.
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Functions of Two or More
Independent Variables
An equation of the form z = f (x, y) describes a function of
two independent variables if for each permissible order pair
(x, y) there is one and only one z determined. The variables x
and y are independent variables and z is a dependent
variable.
An equation of the form w = f (x, y, z) describes a function
of three independent variables if for each permissible
ordered triple (x, y, z) there is one and only one w
determined.
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Domain and Range
For a function of two variables z = f (x, y), the set of all
ordered pairs of permissible values of x and y is the domain
of the function, and the set of all corresponding values
f (x, y) is the range of the function.
Unless otherwise stated, we will assume that the domain of a
function specified by an equation of the form z = f (x, y) is
the set of all ordered pairs of real numbers f (x, y) such that
f (x, y) is also a real number.
It should be noted, however, that certain conditions in
practical problems often lead to further restrictions of the
domain of a function.
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Examples
1. For the cost function C(x, y) = 1,000 + 50x +100y,
find C(5, 10).
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Examples
1. For the cost function C(x, y) = 1,000 + 50x +100y,
find C(5, 10).
C(5, 10) = 1,000 + 50 · 5 + 100 · 10 = 2,250
2. For f (x, y, z) = x2 + 3xy + 3xz + 3yz + z2,
find f (2, 3, 4)
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Examples
1. For the cost function C(x, y) = 1,000 + 50 x +100 y,
find C(5, 10).
C(5, 10) = 1,000 + 50 · 5 + 100 · 10 = 2,250
2. For f (x, y, z) = x2 + 3xy + 3xz + 3yz + z2,
find f (2, 3, 4)
f (2, 3, 4) = 22 + 3 · 2 · 3 + 3 · 2 · 4 + 3 · 3 · 4 + 42
= 4 + 18 + 24 + 36 + 16 = 98
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Examples
(continued)
There are a number of concepts that we are familiar
with that can be considered as functions of two or more
variables.
Area of a rectangle: A(l, w) = lw
w
l
Volume of a rectangular box:
V(l, w, h) = lwh
h
w
l
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Examples
(continued)
Economist use the Cobb-Douglas production function to
describe the number of units f (x, y) produced from the
utilization of x units of labor and y units of capital. This
function is of the form
f ( x, y)  k x y
m
n
where k, m, and n are positive constants with
m + n = 1.
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Cobb-Douglas Production
Function
The production of an electronics firm is given
approximately by the function
f ( x, y)  5 x
0.3
y
0.7
with the utilization of x units of labor and y units of capital.
If the company uses 5,000 units of labor and 2,000 units of
capital, how many units of electronics will be produced?
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Cobb-Douglas Production
Function
The production of an electronics firm is given
approximately by the function
f ( x, y)  5 x
0.3
y
0.7
with the utilization of x units of labor and y units of capital.
If the company uses 5,000 units of labor and 2,000 units of
capital, how many units of electronics will be produced?
f (5000,2000)  55000
0.3
2000
0.7
 13,164
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Three-Dimensional Coordinates
A three-dimensional coordinate system is formed by three
mutually perpendicular number lines intersecting at their
origins. In such a system, every ordered triplet of numbers
z
(x, y, z) can be associated with a
unique point, and conversely.
We use a plan such as the one
to the right to display
this system on a plane.
y
x
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Three-Dimensional Coordinates
(continued)
Locate (3, –1, 2) on the three-dimensional coordinate system.
z
y
x
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Three-Dimensional Coordinates
(continued)
Locate (3, – 1, 2) on the three-dimensional coordinate system.
z
z=2
y = –1
y
x=3
x
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Graphing Surfaces
Consider the graph of z = x2 + y2. If we let x = 0, the equation
becomes z = y2, which we know as the standard parabola in the
yz plane. If we let y = 0, the equation becomes z = x2, which we
know as the standard parabola in the xz plane.
The graph of this equation z = x2 + y2 is a parabola rotated
about the z axis. This surface is called a paraboloid.
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Graphing Surfaces
(continued)
Some graphing calculators have the ability to graph
three-variable functions.
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Graphing Surfaces
(continued)
However, many graphing calculators only have the ability
to graph two-variable functions.
With these calculators we can graph cross sections by
planes parallel to the xz plane or the yz plane to gain
insight into the graph of the three-variable function.
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Graphing Surfaces
in the x-z Plane
z
Here is the cross section of
z = x2 + y2 in the plane y = 0.
This is a graph of z = x2 + 0.
x
z
Here is the cross section of
z = x2 + y2 in the plane y = 2.
This is a graph of z = x2 + 4.
x
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Graphing Surfaces
in the y-z Plane
z
Here is the cross section of
z = x2 + y2 in the plane x = 0.
This is a graph of z = 0 + y2.
y
z
Here is the cross section of
z = x2 + y2 in the plane y = 2.
This is a graph of z = 4 + y2.
y
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Summary
■ We defined functions of two or more independent
variables.
■ We saw several examples of these functions including the
Cobb-Douglas Production Function.
■ We defined and used a three-dimensional coordinate
system.
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