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.

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.

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.

Examples 1. For the cost function C(x, y) = 1,000 + 50x +100y, find C(5, 10).

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)

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

Examples (continued) There are a number of concepts that we are familiar with that can be considered as functions of two or more variables. l w Area of a rectangle: A(l, w) = lw Volume of a rectangular box: V(l, w, h) = lwh h w l

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.

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?

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?

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

Three-Dimensional Coordinates (continued) Locate (3, –1, 2) on the three-dimensional coordinate system. z y x

Three-Dimensional Coordinates (continued) Locate (3, – 1, 2) on the three-dimensional coordinate system. x = 3 x y z z = 2 y = –1

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.

Graphing Surfaces (continued) Some graphing calculators have the ability to graph three-variable functions.

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.

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

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

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.