Math 142 Lecture Notes for Section 8.1
Section 8.1 -
1
Functions of Several Variables
Definition 8.1.1:
An equation of the form z = f (x, y) describes a function of two independent
variables if for each permissible ordered pair (x, y) there is one and only one value
of z determined by f (x, y).
Example 8.1.2:
Given f (x, y) = 2x3 − 3x2 y + 3x − y 2 + 4, find the following:
(a) f (4, 0)
(b) f (1, 2)
Example 8.1.3:
Find the domain of the following functions:
(a) f (x, y) =
p
9 − x2 − y 2
(b) f (x, y) =
x
x+y−4
(c) f (x, y) = ln(x + y)
Math 142 Lecture Notes for Section 8.1
2
Example 8.1.4:
A company makes 100GB and 250GB video game consoles. The weekly demand and
cost equations are p1 = 360 − 6x + y, p2 = 150 + 3x − 5y, C(x, y) = 1000 + 25x + 35y,
with x being the quantity of 100GB consoles demanded and y being the quantity of
250GB consoles demanded. Additionally, p1 is the price of the 100GB console and p2 is
the price of the 250GB console (both in american dollars).
(a) Find the weekly revenue function and R(20, 30).
(b) Find the weekly profit function and P (20, 30).
Definition 8.1.5:
The Cobb-Douglas production function is defined as
f (x, y) = kxm y n
where k, m, and n are positive constants with m + n = 1. Economists use this function
to describe the number of units f (x, y) produced from the utilization of x units of labor
and y units of capital.
Example 8.1.6:
The productivity of a microchip manufacturing facility is given approximately by the
f (x, y) = 20x0.3 y 0.7 with the utilization of x units of labor and y units of capital. If
the company uses 500 units of labor and 1000 units of capital, approximately how many
units will be produced?
Section 8.1 Suggested Homework: 1-25(odd), 39-45(odd)