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

2. 2 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objectives for Section 8.1 Functions of Several Variables ■ The student will be able to: ■ Identify functions of two or more independent variables. ■ Evaluate functions of several variables. ■ Use three-dimensional coordinate systems.

3. 3 Barnett/Ziegler/Byleen Business Calculus 12e Functions of Two or More Independent Variables

4. 4 Barnett/Ziegler/Byleen Business Calculus 12e 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. 5 Barnett/Ziegler/Byleen Business Calculus 12e Evaluating Functions 1. For the cost function C(x, y) = 1,000 + 50x +100y find C(5, 10). 2. For f (x, y, z) = x 2 + 3xy + 3xz + 3yz + z 2 find f (2, 3, 4)

6. 6 Barnett/Ziegler/Byleen Business Calculus 12e Other Functions There are a number of concepts that we are familiar with that can be considered as functions of two or more variables. Perimeter of a rectangle: P(l, w) = 2l + 2w l w Volume of a rectangular prism: V(l, w, h) = lwh l w h

7. 7 Example: Surface Area  A company uses a box with a square base and an open top to hold an assortment of coffee mugs. The dimensions (x by x by y) are in inches. Barnett/Ziegler/Byleen Business Calculus 12e 1.Find the function M(x,y) that represents the total minimum amount of material required to construct one of these boxes. 2.Use your function to evaluate M(12, 5)

8. 8 Example: Surface Area Barnett/Ziegler/Byleen Business Calculus 12e M(x, y) = area of base + 4(area of one side) You would need (at least) 384 square inches of material to construct the box.

9. 9 Example: IQ Barnett/Ziegler/Byleen Business Calculus 12e

10. 10 Economics  In 1928 Charles Cobb and Paul Douglas published a study in which they modeled the growth of the American economy during the period 1899 - 1922.  They considered a simplified view of the economy in which production output is determined by the amount of labor involved and the amount of capital invested.  While there are many other factors affecting economic performance, their model proved to be remarkably accurate. Barnett/Ziegler/Byleen Business Calculus 12e

11. 11 Barnett/Ziegler/Byleen Business Calculus 12e Economics Cobb-Douglas production function: f(x,y) = total production in one year k = productivity factor x = labor input (total number of person-hours worked in a year) y = capital input (monetary value of all equipment, machinery, and buildings) k, m, and n are positive constants with m + n = 1.

12. 12 Barnett/Ziegler/Byleen Business Calculus 12e Example: Economics The productivity 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?

13. 13 Example: Business Barnett/Ziegler/Byleen Business Calculus 12e

14. 14 Business (continued) Barnett/Ziegler/Byleen Business Calculus 12e

15. 15 Business (continued) Barnett/Ziegler/Byleen Business Calculus 12e

16. 16 Business (continued) Barnett/Ziegler/Byleen Business Calculus 12e Profit = Revenue - Cost

17. 17 Graphing in 3D  When we graph y = f(x), we use an x-y plane.  When we graph z = f(x, y) we use an x-y-z plane which is a 3-dimensional coordinate system. Barnett/Ziegler/Byleen Business Calculus 12e

18. 18 Barnett/Ziegler/Byleen Business Calculus 12e 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 triple of numbers (x, y, z) can be associated with a unique point in space.

19. 19 Barnett/Ziegler/Byleen Business Calculus 12e Three-Dimensional Coordinates (continued) Find the coordinates of points C and H on the box shown below.

20. 20 Barnett/Ziegler/Byleen Business Calculus 12e Three-Dimensional Coordinates (continued) The point (-3, 5, 2) is graphed below.

21. 21 Barnett/Ziegler/Byleen Business Calculus 12e Graphing Surfaces Consider the graph of z = x 2 + y 2. If we let x = 0, the equation becomes z = y 2, which we know as the standard parabola in the yz plane. If we let y = 0, the equation becomes z = x 2, which we know as the standard parabola in the xz plane. The graph of this equation z = x 2 + y 2 is a parabola rotated about the z axis. This surface is called a paraboloid.

22. 22 Barnett/Ziegler/Byleen Business Calculus 12e Graphing Surfaces (continued) Some graphing calculators have the ability to graph three-variable functions.

23. 23 Homework Barnett/Ziegler/Byleen Business Calculus 12e