Chapter 8

Multivariable Calculus

Section 1

Functions of Several Variables

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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

l

w

Volume of a rectangular box: V(l, w, h) = lwh

lw

h

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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

where k, m, and n are positive constants with m + n = 1.

nm yxkyxf ),(

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Cobb-Douglas Production Function

The production of an electronics firm is given approximately by the function

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?

7.03.05),( yxyxf

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Cobb-Douglas Production Function

The production of an electronics firm is given approximately by the function

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?

7.03.05),( yxyxf

f (5000,2000) 55000 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 (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.

x

y

z

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x

y

z

Three-Dimensional Coordinates(continued)

Locate (3, –1, 2) on the three-dimensional coordinate system.

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Three-Dimensional Coordinates(continued)

Locate (3, – 1, 2) on the three-dimensional coordinate system.

y = –1z = 2

x = 3

x

y

z

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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

Here is the cross section of z = x2 + y2 in the plane y = 0. This is a graph of z = x2 + 0.

x

z

x

zHere is the cross section of z = x2 + y2 in the plane y = 2. This is a graph of z = x2 + 4.

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Graphing Surfaces in the y-z Plane

Here is the cross section of z = x2 + y2 in the plane x = 0. This is a graph of z = 0 + y2.

y

z

y

z

Here is the cross section of z = x2 + y2 in the plane y = 2. This is a graph of z = 4 + y2.

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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.