8.1 Function of Two Variables
Many functions have several variables.
Ex. There are 3 types of football tickets. Type A costs $50, type B costs $30, and type C costs
$20. If in a match, x tickets of type A, y tickets of type B, and z tickets of type C are sold,
the total income of ticket sale is f(x, y, z) = 50x + 30y + 20z.
Def. A real-valued function of two variables f consists of
1. A domain A consisting of ordered pairs of some real numbers (x, y).
2. A rule that associates with each ordered pair in A with one real number, denoted by
z = f(x, y).
Ex. (Ex 1, p.532) f(x, y) = x + xy + y
w + 2. Compute f(0, 0), f(1, 2), and f(2, 1).
Ex. Find the domain of a. f(x, y) = x
2+y
2
, b. g(x, y) = 2
x−y
, c. h(x, y) = p
1 − x
2 − y
2.
Ex. Find the domain of g(r, s) = √
rs.
The graph of a function z = f(x, y) is the collection of all points {(x, y, f(x, y)) : x, y ∈ A}
in R
3
(Fig 5, p.534; Fig 6, p.535).
The graph of z = f(x, y) is 3 dimensional and it is difficult to draw. So we use level curves.
A level curve is the graph of
c = f(x, y)
on xy-plane for a constant c. By drawing the level curves corresponding to several admissible
values of c, we obtain a contour map. (Fig 7, p.535; Fig 8, p.536)
Ex. Ex 5, p.536.
HW. C8.1: SC1, 2, 3, Ex 31, 33
