1. Partial Differentiation & Application
Week 9

2. Contents: Function with two variables First Partial Derivatives
Applications of First Partial Derivatives
Cob-Douglas Production Function
Substitute and Complementary Commodities
Second Partial Derivatives
Application of Second Partial Derivatives
Maxima and Minima of Functions of Several Variables*
Lagrange Multipliers*
*Additional topic

3. Functions of Two Variables

4. A Function of Two Variables
A real-valued function of two variables, f, consists of:
1. A set A of ordered pairs of real numbers (x, y) called the domain of the function.
2. A rule that associates with each ordered pair in the domain of f one and only one real number, denoted by z = f (x, y).
Dependent variable
Independent variables
Function of Two Variables

5. Partial Derivatives: Application
Example
Examples of problems with two variables.
A company produces two products, A and B. The joint cost function (in RM) is given by:
Country workshop manufactures both furnished and unfurnished furniture for home. The estimated quantities demanded each week of its desks in the finished and unfinished version are x and y units when the corresponding unit prices are
respectively.
Partial Derivatives: Application 5

6. Function of Two Variables
Example
Let f be the function defined by
Find
Function of Two Variables

7. Function of Two Variables
Example
Let f be the function defined by
Find
Function of Two Variables 7

8. Function of Two Variables
Example
Find the domain of each function
Since f (x, y) is defined for all real values of x and y (x and y is linear function), the domain of f is the set of all points (x, y) in the xy – plane.
g(x, y) is defined as long as 2x + y – 3 is not 0. So the domain is the set of all points (x, y) in the xy – plane except those on the line y =–2x + 3.
Function of Two Variables

9. Function of Two Variables
Question
Let f be the function defined by
Find the domain of the function
g(x, y) is defined as long as
So the domain is the set of all points (x, y) in the xy – plane except those on the line y=25
Function of Two Variables 9

10. Function of Two Variables
Acrosonic manufactures a bookshelf loudspeaker system that may be bought fully assemble or in a kit. The demand equations that relate the unit prices, p and q to the quantities demanded weekly, x and y, of the assembled and kit versions of the loudspeaker systems are given by
What is the weekly total revenue function R (x,y)?
Example 10
Function of Two Variables

11. Recall the Graph of Two Variables
Ex. Plot (4, 2)
Ex. Plot (-2, 1)
Ex. Plot (2, -3)
(4, 2)
(-2, 1)
(2, -3)
Function of Two Variables

12. Graphs of Functions of Two Variables
Three-dimensional coordinate system: (x, y, z)
Ex. Plot (2, 5, 4) z 4 y 2 5 x
Function of Two Variables

13. Graphs of Functions of Two Variables
Ex. Graph of f (x, y)= 4 – x2 – y2
Function of Two Variables

14. First Partial Derivatives of f (x, y).
f (x, y) is a function of two variables. The first partial derivative of f with respect to x at a point (x, y) is
provided the limit exits.
Partial Derivatives

15. First Partial Derivatives of f (x, y).
f (x, y) is a function of two variables. The first partial derivative of f with respect to y at a point (x, y) is
provided the limit exits.
Partial Derivatives

16. To get partial derivatives….
To get assume y is a constant and differentiate with respect to x
To get assume x is a constant and differentiate with respect to y
Example
Example
Partial Derivatives

17. Compute the first partial derivatives
Example
Compute the first partial derivatives
Example
Partial Derivatives

18. Compute the first partial derivatives
Example
Compute the first partial derivatives
Example
Partial Derivatives 18

19. The Cobb-Douglas Production Function
a and b are positive constants with 0 < b < 1.
x stands for the money spent on labor, y stands for the cost of capital equipment.
f measures the output of the finished product and is called the production function
fx is the marginal productivity of labor.
Fx measures the rate of change of production with respect to the amount of money expended for labor, with the level of capital expenditure held constant.
Fy measures the rate of change of production with respect to the amount expended on capital, with the level of labor expenditure held constant/fixed.
fy is the marginal productivity of capital. 19
Partial Derivatives: Application of First Partial Derivatives 19

20. Partial Derivatives: Application of First Partial Derivatives
A certain production function is given by
units, when x units of labor and y units of capital are used. Find the marginal productivity of capital when labor = 81 units and capital = 256 units.
Example
When labor = 81 units and capital = 256 units,
So units per unit increase in capital expenditure. 20
Partial Derivatives: Application of First Partial Derivatives

21. Partial Derivatives: Application of First Partial Derivatives
A certain production function is given by
units, when x units of labor and y units of capital are used. Find the marginal productivity of labor when labor = 81 units and capital = 256 units.
Question
When labor = 81 units and capital = 256 units,
So units per unit increase in labor expenditure. 21
Partial Derivatives: Application of First Partial Derivatives

22. Substitute and Complementary Commodities
Suppose the demand equations that relate the quantities demanded, x and y, to the unit prices, p and q, of two commodities, A and B, are given by
x = f(p,q) and y = g(p,q) 22
Partial Derivatives: Application of First Partial Derivatives

23. Substitute and Complementary Commodities
Two commodities A and B are substitute commodities if
Two commodities A and B are complementary commodities if 23
Partial Derivatives: Application of First Partial Derivatives

24. Partial Derivatives: Application of First Partial Derivatives
Example
The demand function for two related commodities are
x = ae q-p
y = be p-q
The marginal demand functions are  
x = - ae q-p y = be p-q
p p
x = ae q-p y = - be p-q
q q
Because x/q > 0 and y/p > 0, the two commodities are substitute commodities. 24
Partial Derivatives: Application of First Partial Derivatives

25. Partial Derivatives: Application of First Partial Derivatives
Question
In a survey it was determined that the demand equation for VCRs is given by
The demand equation for blank VCR tapes is given by
Where p and q denote the unit prices, respectively, and x and y denote the number of VCRs and the number of blank VCR tapes demanded each week. Determine whether these two products are substitute, complementary, or neither. 25
Partial Derivatives: Application of First Partial Derivatives

26. Partial Derivatives: Application of First Partial Derivatives
Because x/q < 0 and y/p < 0, the two commodities are complementary commodities.
Partial Derivatives: Application of First Partial Derivatives

27. Partial Derivatives: Second-Order Partial Derivatives27
Partial Derivatives: Second-Order Partial Derivatives

28. Find the second-order partial derivatives of the function
Example
Find the second-order partial derivatives of the function
Example
Partial Derivatives: Second-Order Partial Derivatives 28

29. Find the second-order partial derivatives of the function
Example 29
Partial Derivatives: Second-Order Partial Derivatives

30. Maximum and Minimum of Functions of Several Variables30

31. Relative Extrema of a Function of Two Variables
Let f be a function defined on a region R containing (a, b).
f (a, b) is a relative maximum of f if
for all (x, y) sufficiently close to (a, b).
f (a, b) is a relative minimum of f if
for all (x, y) sufficiently close to (a, b).
*If the inequalities hold for all (x, y) in the domain of f then the points are absolute extrema. 31
Partial Derivatives: Application of Second Partial Derivatives

32. Partial Derivatives: Application of Second Partial Derivatives
Critical Point of f
A critical point of f is a point (a, b) in the domain of f such that both
or at least one of the partial derivatives does not exist. 32
Partial Derivatives: Application of Second Partial Derivatives

33. Determining Relative Extrema
1. Find all the critical points by solving the system
2. The 2nd Derivative Test: Compute
Interpretation + +
Relative min. at (a, b) – +
Relative max. at (a, b)
Neither max. nor min. at (a, b)  saddle point –
Test is inconclusive 33

34. Ex. Determine the relative extrema of the function
So the only critical point is (1, 0).
and
So f (1,0) = 1 is a relative maximum 34

35. Application
Ex: The total weekly revenue (in dollars) that Acrosonic realizes in producing and selling its bookshelf loudspeaker systems is given by
where x denotes the number of fully assembled units and y denotes the number of kits produced and sold each week. The total weekly cost is given by
Determine how many assembled units and how many kits Acrosonic should produce per week to maximize its profit. 35

36. 36

37. Substitute this value into the equation
Substitute in
Substitute in
Substitute this value into the equation
Therefore, P has the critical point (208,64) 37

38. Since, and , the point (208,64) is a relative maximum of P.38

39. Lagrange Multipliers
Reading: Mizrahi and Sullivan, 8th ed., 2004, Wiley
Chapter:17.5 39

40. Method of Lagrange Multipliers
A method to find the local minimum and maximum of a function with two variables subject to conditions or constraints on the variables involved.
Suppose that, subject to the constraint g(x,y)=0, the function z=f(x,y) has a local maximum or a local minimum at the point .
Form the function
Maximize f(x,y) subject to g(x,y) = 0 40 40

41. provided all the partial derivatives exists.
Then there is a value of such that is a solution of the system of equations
provided all the partial derivatives exists. 41

42. Steps for Using the Method of Lagrange Multipliers
Step 1: Write the function to be maximized (or minimized) and the constraint in the form:
Find the maximum (or minimum) value of
subject to the constraint
Step 2: Construct the function F: 42

43. Step 3: Set up the system of equations
Step 4: Solve the system of equations for x, y and .
Step 5: Test the solution to determine maximum or minimum point. 43

44. If D*  0  Fxx  0  maximum point Fxx  0  minimum point
Find D* = Fxx . Fyy - (Fxy)2
If D*  0  Fxx  0  maximum point
Fxx  0  minimum point
D*  0  Test is inconclusive
Step 6: Evaluate at each solution found in Step 5. 44

45. Example: Find the minimum of f(x,y) = 5x2 + 6y2 - xy
subject to the constraint
x+2y = 24
Solution:
F(x,y, ) = 5x2 + 6y2 - xy + (x + 2y - 24)
Fx = F = 10x - y +  ; Fxx = 10 x
Fy = F = 12y - x + 2 ; Fyy = 12 y
F = F = x + 2y - 24 ; Fxy = -1  45

46. The solution of the system is x = 6, y = 9,  = -51
The critical point,
10x - y +  = 0
12y - x + 2= 0
x + 2y - 24= 0
The solution of the system is x = 6, y = 9,  = -51
D*=(10)(12)-(-1)2=119>0
Fxx = 10>0
We find that f(x,y) has a local minimum at (6,9).
f(x,y) = 5(6)2+6(9)2-6(9)= 720 46

47. Example
A manufacturer produces two types of engines, x units of type I and y units of type II. The joint profit function is given by
to maximize profit, how many engines of each type should be produced if there must be a total of 42 engines produced? 47

48. The solution of the system is
Maximize
Subject to constraint
The solution of the system is 48

49. The test in inconclusive.49

50. The End 50