This document provides definitions and examples related to functions of two variables. It begins by defining a function of two variables as a rule that assigns one real number (f(x,y)) to each ordered pair of real numbers (x,y) in some set. It discusses the domain of such functions and provides examples of evaluating specific functions. It also covers partial derivatives, second-order partial derivatives, and the chain rule as they apply to functions of two variables.

1. Chapter Three
Function of Two
Variables

2. Definition
Functions of Two Variables
A function f of the two variables x and y
is a rule that assigns to each ordered pair
(x, y) of real numbers in some set one
and only one real number denoted by
f(x,y).
2

3. Functions of Two Variables
The Domain of a Function
of Two Variables
The domain of the function f(x, y) is the set
of all ordered pairs (x, y) of real numbers
for which f(x, y) can be evaluated.
3

4. Example 1
Functions of Two Variables
For f(x, y) 3x y2 find
a) f(2,3)
b) f(2, 2
2
f 2,3 3 2 3 15
2
f 2, 2 3 2 2 8
4

5. Example 2
Functions of Two Variables
x
For f x, y e ln y . Find the domain
of f and f(0,1)
The domain ( x, y ) / x 0 and y 0
0 0
f (0,1) e ln1 e 0 1
5

6. Example 3
Functions of Two Variables
2
3x 5y
f x, y
x y
a) Find the domain of f.
b) Compute f(1, 2)
6

7. Example 4
Functions of Two Variables
A pharmacy sells two brands of aspirin.
Brand A sells for $1.25 per bottle and
Brand B sells for $1.50 per bottle.
a) What is the revenue function for aspirin?
b) What is the revenue for aspirin if 100
bottles of Brand A and 150 bottles B are
sold?
7

8. Example 4 (Continued)
Functions of Two Variables
a) Let
x the number of bottles of Brand A sold
y the number of bottles of Brand B
sold. Then, the revenue function is
R x, y 1.25x 1.50y
b)
R 100,150 1.25 100 1.50 150
125 225
350
8

9. Functions of Two Variables Cobb-Douglas
Production Functions
Economists use a formula called the Cobb-
Douglas Production Functions to model
the production levels of a company (or a
country). Output Q at a factory is often
regarded as a function of the amount K of
capital investment and the size L of the
labor force. Output functions of the form
1
Q K,L AK L
9

10. Functions of Two Variables Cobb-Douglas Production
Functions (Cont.)
where A and are positive constants and
0 1 have proved to be especially useful
in economic analysis. Such functions are
known as Cobb-Douglas production
function.
10

11. Example 5
Functions of Two Variables
Suppose that the function Q 500x0.3y0.7
represents the number of units produced
by a company with x units of labor and y
units of capital.
a) How many units of a product will be
manufactured if 300 units of labor and
50 units of capital are used?
b) How many units will be produced if
twice the number of units of labor and
capital are used?
11

12. Example 5 (Cont.)
Functions of Two Variables
0.3 0.7
a)Q 300,50 500 300 50
500 5.535 15.462
42,791 units
b) If number of units of labor and capital
are both doubled, then x 2 300 600
and y 2 50 100
0.3 0.7
Q 600,100 500 600 100
500 6.815 25.119
85,592 units
12

13. Definition
Let z f x, y)
Partial Derivatives
a) The first partial derivative of f with
respect to x is
z f x x, y f x, y
fx x, y lim
x x 0 x
b) The first partial derivative of f with
respect to y is:
z f x, y y f x, y
fy x, y lim
y y 0 y
13

14. Computation of Partial Derivatives
oThe function z / x or fx is obtained by
Partial Derivatives
differentiating f with respect to x, treating
y as a constant.
oThe function z / y or fy is obtained by
differentiating f with respect to y, treating
x as a constant.
14

15. Example 1
For the function
Partial Derivatives
f x, y 4x 2 3xy 5y 2
Find z / y and z / x
Treating y as a constant, we obtain
z
8 x 3y
x
Treating x as a constant, we obtain
z
3 x 10 y
y
15

16. Example 2, 3
Find the partial derivatives fx and fy if
Partial Derivatives
2 2 2y
f x, y x 2xy
3x
Find fx (1,2) and fy (1,2) if
f x, y xexy y2
16

17. Example 4
Suppose that the production function
Partial Derivatives
Q(x,y) 2000 x​0.5y0.5 ​is known. Determine
​
the marginal productivity of labor and the
marginal productivity of capital when 16
units of labor and 144 units of capital are
used.
17

18. Example 4 (Cont.)
Partial Derivatives
Q 0.5 1000y 0.5
2000 0.5 x y 0.5 0.5
x x
Q 1000 x 0.5
2000 0.5 x 0.5 y 0.5
y y 0.5
substituting x 16 and y 144, we obtain
Q 1000(144)0.5 1000 12
3000 units
x (16,144) (16)0.5 4
Q 1000(16)0.5 1000 4
333.33 units
y (16,144) (144)0.5 12
18

19. Example 4 (Cont.)
Partial Derivatives
Thus we see that adding one unit of labor
will increase production by about 3000
units and adding one unit of capital will
increase production by about 333 units.
19

20. Example 5
It is estimated that the weekly output at a
Partial Derivatives
certain plant is given by the function
Q(x,y) 1200x 500y x2 y x3 y2 units, where
x is the number of skilled workers and y the
number of unskilled workers employed at the
plant. Currently the work force consists of 30
skilled workers, and 60 unskilled workers.
Use marginal analysis to estimate the
change in the weekly output that will result
from the addition of 1 more skilled worker if
the number of unskilled workers is not
changed. 20

21. Second-Order Partial
Derivatives
Partial Derivatives
If z f(x, y)
the partial derivative of fx with respect to x is
2
z z
fxx fx x or 2
x x x
the partial derivative of fx with respect to y is
2
z z
fxy fx y or
y x y x
21

22. Second-Order Partial
Derivatives (Cont.)
Partial Derivatives
If z f(x, y)
the partial derivative of fy with respect to
x is 2
z z
fyx fy or
x x y x y
the partial derivative of fy with respect to
y is 2
z z
fyy fy or 2
y y y y
22

23. Example 6
Partial Derivatives
Compute the four second-order partial
derivatives of the function
3 2
f x, y xy 5xy 2x 1
23

24. Example 7
Partial Derivatives
Find all four second partial derivatives of
2
f x, y ln x 4y
then find
fxx 2,1 2
24

25. Remark
Partial Derivatives
The two partial derivatives fxy and fyx are
sometimes called the mixed second-
order partial derivatives of f and fxy fyx .
25

26. The Chain Rule; Approximation by the
Chain Rule for Partial
Derivatives
Suppose z is a function of x and y, each of
Total Differential
which is a function of t then z can be
regarded as a function of t and
dz z dx z dy
dt x dt y dt
26

27. The Chain Rule; Approximation by the
Chain Rule for Partial
Derivatives
Remark 1
Total Differential
z dx
rate of change of z with respect
x dt
to t for fixed y.
z dy
rate of change of z with respect
y dt
to t for fixed x.
27

28. The Chain Rule; Approximation by the
Total Differential
dt
dz
Find if z
x 2
3 xy , x
2t 1
Example 1
t ,and y
2
28

29. The Chain Rule; Approximation by the
Example 1
By the chain rule,
Total Differential
dz z dx z dy
2x 3y 2 3x 2t
dt x dt y dt
Which you can rewrite in terms of t
dz
4(2t 1) 6t 2 3(2t 1)(2t ) 18t 2 14t 4
dt
29

30. Example 1
The Chain Rule; Approximation by
By the chain rule,
the Total Differential
dz z dx z dy
2x 3y 2 3x 2t
dt x dt y dt
Which you can rewrite in terms of t
dz
4(2t 1) 6t 2 3(2t 1)(2t ) 18t 2 14t 4
dt
30

31. Example 2
The Chain Rule; Approximation by
A health store carries two kinds of multiple
vitamins, Brand A and Brand B. Sales
the Total Differential
figures indicate that if Brand A is sold for x
dollars per bottle and Brand B for y dollars
per bottle, the demand for Brand A will be
2
Q x, y 300 20x 30 y bottles/month
It is estimated that t months from now the
price of Brand A will be
x 2 0.05t dollars per bottle
31

32. Example 2 (Cont.)
The Chain Rule; Approximation by
It is estimated that t months from now the
price of Brand A will be
the Total Differential
x 2 0.05t dollars per bottle
and the price of Brand B will be
y 2 0.1 t dollars per bottle
At what rate will the demand for Brand A
be changing with respect to time 4 months
from now?
32

33. Example 2 (Cont.)
The Chain Rule; Approximation by
Our goal is to find dQ/dt when t 4. Using
chain rule, we obtain
the Total Differential
dQ Q dx Q dy
dt x dt y dt
12
40 x 0.05 30 0.05t
when t 4, x 2 0.05 4 2.2
and hence,
dQ
40 2.2 0.05 30 0.05 0.5 3.65
dt
33

34. Approximation Formula
Suppose z is a function of x and y. If ∆x
The Total differential
denotes a small change in x and ∆y a
small change in y, the corresponding
change in z is
z z
z x y
x y
34

35. Remark 2
z
x change in z due to the change in
The Total differential
x x
for fixed y.
z
y
y change in z due to the change in y
for
fixed x.
35

36. The Total Differential
If z is a function of x and y, the total
The Total differential
differential of z is
z z
dz x y
x y
36

37. Example 3
At a certain factory, the daily output is
Q=60 K1/2 L2/3 units, where K denotes the
The Total differential
capital investment measured in units of
$1,000 and L the size of the labor force
measured in worker-hours. The current
capital investment is $ 900,000 and 1,000
and labor are used each day. Estimate the
change in output that will result if capital
investment is increased by $1,000 and
labor is increased by 2 worker-hours.
37

38. Example 3
Apply the approximation formula with
K 900, L 1000, ∆K 1 and ∆L 2 to get
The Total differential
Q Q
Q K L
K L
30 K 1/2 L1/3 K 20 K 1/2 L 2/3 L
1 1
30 10 1 20 30 2
30 100
22 units
That is, output will increase by
approximately 22 units. 38

39. percentage change
Approximation of Percentage Change
The percentage change of a quantity
expresses the change in the quantity as a
percentage of its size prior to the change.
In particular,
change in quantity
Percentage change 100
size of quantity
39

40. Approximation of
Percentage Change
Approximation of Percentage Change
Suppose z is a function of x and y. If ∆x
denotes a small change in x and ∆x a
small change in y , the corresponding
percentage change in z is
z z
x y
z x y
% z 100  100
z z
40