bcb12e ppt 8 1

Chapter 8

Multivariable

Calculus

Section 1

Learning Objectives for Section 8.1

Functions of Several Variables

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evaluate functions of several variables.

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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

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

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Examples

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,

Barnett/Ziegler/Byleen Business Calculus 12e 7

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

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:

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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.

n m

y

x

k

y

x

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 . 0 3 . 0

5

)

,

(

x

y

x

y

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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?

1087

2000

5000

5

)

2000

,

5000

(

=

⋅

=

f

5

)

,

(

x

y

x

y

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. y

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y z

Three-Dimensional Coordinates

(continued)

Three-Dimensional Coordinates

(continued)

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

y = –1

z = 2

x = 3

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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

Graphing Surfaces

(continued)

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Graphing Surfaces

(continued)

However, many graphing calculators only have the ability to graph two-variable functions.

Graphing Surfaces

in the

x-z

Plane

Here is the cross section of

z = x2 _{+ y}2 _{in the plane y = 0.}

This is a graph of z = x2 + 0.

x z

z

Here is the cross section of

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Graphing Surfaces

in the

y-z

Plane

Here is the cross section of

z = x2 _{+ y}2 _{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 _{+ y}2 _{in the plane y = 2.}

Summary

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