Chapter 8 Multivariable Calculus

1. Chapter 8Multivariable Calculus Section 1 Functions of Several Variables

2. 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.

3. 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.

4. 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.

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

6. 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)

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 = 4 + 18 + 24 + 36 + 16 = 98

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

9. 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.

10. 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?

11. 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?

12. z y x 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.

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

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

15. 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.

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

17. 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.

18. 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

19. 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

20. 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.