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
We need to divide 4,971 by 10 until its a number less then 10.
4,971 / 10 = 497.1
497.1 / 10 = 49.71
49.71 / 10 = 4.971
Now, 4.971 is less then 10, so, we can write this into scientific notation.
4.971 * 10^?
We need to know what power we add to 10 to get 4,971. We can solve for looking at the number of times we divided. As you can see above, we divided 3 times so that will be the power of 10.
Let w represent the width in cm. Then the length is 2w-3, 3 cm less than twice the width. The perimeter is twice the sum of these, so the sum of length and width is half the perimeter:
w + (2w -3) = 18/2
3w -3 = 9 . . . . collect terms
3w = 12 . . . . add 3
w = 4 . . . . divide by 3
2w-3 = 2·4 -3 = 5 . . . . length
The equation is w + (2w -3) = 18/2; the length is 5; the width is 4.
