Chapter 8 Multivariable Calculuspmcnamara.weebly.com/uploads/1/3/3/8/13389841/bcb12e_ppt... ·...

Chapter 8

Multivariable Calculus

Section 1 Functions of Several

Variables

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.


Functions of Two or More Independent Variables

Usually we study equations of the form 𝑦 = 𝑓 𝑥 where x is the independent variable and y is the dependent variable.

An equation of the form z = f (x, y) describes a function of two independent variables if for each ordered pair (x, y), there is 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 ordered triple (x, y, z), there is 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.


Evaluating Functions

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

𝐶 5,10 = 1000 + 50 5 + 100(10) = 1000 + 250 + 1000 = 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


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 Compound Interest:

𝐴(𝑃, 𝑟, 𝑡,𝑛) = 𝑃 1 +𝑟𝑛

𝑛𝑛

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

Example: Surface Area


M(x, y) = area of base + 4(area of one side)

𝑀 𝑥,𝑦 = 𝑥2 + 4𝑥𝑦

M 12,5 = 122 + 4(12)(5) = 144 + 240

= 384

You would need (at least) 384 square inches of material to construct the box.

9

Example: IQ


Intelligence Quotient IQ : 𝑄 𝑀,𝐶 =𝑀𝐶∙ 100

M = mental age C = chronological age 𝐹𝐹𝑛𝐹 𝑄(16,17) ≈ 94

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.



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.

nm yxkyxf =),(


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?

7.03.05),( yxyxf =

𝑓 5000, 2000 = 5(5000)0.3(2000)0.7

= 13,163 units of electronics will be produced

13

Example: Business

A small surfboard company produces a standard type of surfboard. If fixed costs are $500 per week and variable costs are $70 per board produced, then the weekly cost function is: 𝐶 𝑥 = 70𝑥 + 500 (x is the number of boards)

The company is expanding its product line to include a high performance competition board. For this board, fixed costs are $200 per week and variable costs are $100 per board produced. The new weekly cost function is:

(x = # standard boards, y = # competition boards) 𝐶 𝑥,𝑦 = 70𝑥 + 500 + 100𝑦 + 200 𝐶 𝑥,𝑦 = 70𝑥 + 100𝑦 + 700 Barnett/Ziegler/Byleen Business Calculus 12e

14

Business (continued) The demand equations for its two types of boards are:

• Price of the standard board: 𝑝 = 210 − 4𝑥 + 𝑦 • Price of the competition board: 𝑞 = 300 + 𝑥 − 12𝑦

In both equations, x is the weekly demand for standard boards and y is the weekly demand for competition boards. 1. Find the weekly revenue R(x,y) function and evaluate

R(20, 10) 2. If the weekly cost function is 𝐶 𝑥,𝑦 = 70𝑥 + 100𝑦 +

700 then find the weekly profit function P(x,y) and evaluate P(20, 10).


15

Business (continued)


𝑝 = 210 − 4𝑥 + 𝑦 𝑞 = 300 + 𝑥 − 12𝑦 𝑅𝑅𝑅𝑅𝑛𝑅𝑅 = 𝑑𝑑𝑑𝑑𝑛𝑑

𝑠𝑛𝑑𝑛𝑑.𝑏𝑏𝑑𝑏𝑑 × 𝑝𝑏𝑝𝑝𝑑𝑠𝑛𝑑𝑛𝑑.𝑏𝑏𝑑𝑏𝑑 +

𝑑𝑑𝑑𝑑𝑛𝑑𝑝𝑏𝑑𝑝.𝑏𝑏𝑑𝑏𝑑 × 𝑝𝑏𝑝𝑝𝑑

𝑝𝑏𝑑𝑝.𝑏𝑏𝑑𝑏𝑑

𝑅 𝑥,𝑦 = 𝑥𝑝 + 𝑦𝑞 = 𝑥 210 − 4𝑥 + 𝑦 + y(300 + 𝑥 − 12𝑦)

= 210𝑥 − 4𝑥2 + 𝑥𝑦 + 300𝑦 + 𝑥𝑦 − 12𝑦2

= 210𝑥 − 4𝑥2 + 2𝑥𝑦 + 300𝑦 − 12𝑦2 𝑅 20,10 = 210 20 − 4 20 2 + 2 20 10 + 300 10 − 12(10)2

= $4,800 is the weekly revenue from 20 standard 10 competition boards

16

Business (continued)


𝐶 𝑥,𝑦 = 70𝑥 + 100𝑦 + 700

𝑅(𝑥,𝑦) = 210𝑥 − 4𝑥2 + 2𝑥𝑦 + 300𝑦 − 12𝑦2

𝑃 𝑥,𝑦 = 𝑅 𝑥,𝑦 − 𝐶(𝑥,𝑦)

= 210𝑥 − 4𝑥2 + 2𝑥𝑦 + 300𝑦 − 12𝑦2 − 70𝑥 − 100𝑦 − 700

= 140𝑥 − 4𝑥2 + 200𝑦 − 12𝑦2 + 2𝑥𝑦 − 700

𝑃 20,10 = $1,700 is the weekly profit from 20 standard and 10 competition boards.

Profit = Revenue - Cost

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.



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.


Three-Dimensional Coordinates (continued)

Find the coordinates of points C and H on the box shown below.

𝐶(2, 4, 0)

𝐻(0, 4, 3)


Three-Dimensional Coordinates (continued)

The point (-3, 5, 2) is graphed below.


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.

23

Homework


Chapter 8 Multivariable Calculuspmcnamara.weebly.com/uploads/1/3/3/8/13389841/bcb12e_ppt... ·...

Documents

Transcript of Chapter 8 Multivariable Calculuspmcnamara.weebly.com/uploads/1/3/3/8/13389841/bcb12e_ppt... ·...