2. Contents:
1. Function with two variables
2. First Partial Derivatives
3. Applications of First Partial Derivatives
Cob-Douglas Production Function
Substitute and Complementary Commodities
1. Second Partial Derivatives
2. Application of Second Partial Derivatives
Maxima and Minima of Functions of Several Variables*
Lagrange Multipliers*
*Additional topic
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
Independent variables variable
Function of Two Variables
5. Example Examples of problems with two variables.
a. A company produces two products, A and B.
The joint cost function (in RM) is given by:
C = f ( x, y ) = 0.07 x 2 + 75 x + 85 y + 6000
b. 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 p = 200 − 0.2x − 0.1y
q = 160 − 0.2x − 0.25y
respectively.
Partial Derivatives: Application
6. Example
Let f be the function defined by
2 3
f ( x, y ) = 3 x y − 2 + y .
Find f (0,3) and f (2, −1).
f (0,3) = 3 ( 0 ) (3) − 2 + ( 3)
2 3
= 25
f (2, −1) = 3 ( 2 ) (−1) − 2 + ( −1)
2 3
= −15
Function of Two Variables
7. Example
Let f be the function defined by
x
f ( x ,y ) =
x + y −3
Find f ( 2,3 ),f ( 3,2 )
2
f ( 2,3 ) = =1
2+ 3− 3
3 3
f ( 3,2 ) = =
3+ 2− 3 2
Function of Two Variables
8. Example
Find the domain of each function
a. f ( x, y ) = 3 x − 2 y 2
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.
x
b. g( x, y ) =
2x + y − 3
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. Question
Let f be the function defined by
f ( x, y ) =
(
100 1000 + 0.03 x y 2
)
0 .5
( 5 − 0. 2 y ) 2
Find the domain of the function
g(x, y) is defined as long as ( 5 − 0.2y ) 2
≠0
( 5 − 0.2y ) 2 ≠ 0
5 − 0.2 y ≠ 0
y ≠ 25
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
10. Example 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
1 1 1 3
p = 300 − x − y q = 240 − x − y
4 8 8 8
What is the weekly total revenue function R (x,y)?
R ( x, y ) = xp + yq
1 1 1 3
= x 300 − x − y ÷+ y 240 − x − y ÷
4 8 8 8
1 2 3 2 1
= − x − y − xy + 300 x + 240 y
4 8 4
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
2 y
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
∂f f ( x + h, y ) − f ( x , y )
= lim
∂x h→0 h
∂
provided the limit exits. = f x = f x ( x , y ) = f ( x , y )
∂x
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
∂f f ( x, y + k ) − f ( x , y )
= lim
∂y k →0 k
∂
= f y = f y ( x ,y ) = f ( x ,y )
provided the limit exits. ∂y
Partial Derivatives
16. To get partial derivatives….
To get fx assume y is a constant and
differentiate with respect to x
Example f ( x, y ) = xy 2 + x 2 y
f x ( x, y ) = (1) y 2 + (2 x) y = y 2 + 2 xy
To get fy assume x is a constant and
differentiate with respect to y
Example f ( x, y ) = xy 2 + x 2 y
f y ( x, y ) = x(2 y ) + x 2 (1) = 2 xy + x 2
Partial Derivatives
17. Example Compute the first partial derivatives
f ( x, y ) = 3 x 2 y + x ln y
1
f x = 6 xy + ln y f y = 3x + x ÷
2
y
Example Compute the first partial derivatives
xy 2 + y
g ( x, y ) = e
2 xy 2 + y
g y = ( 2 xy + 1) e xy 2 + y
gx = y e
Partial Derivatives
18. Example Compute the first partial derivatives
f ( x, y ) = 3 x 2 y − 2 + y 3 .
f x = 6 xy f y = 3x 2 + 3y 2
Example Compute the first partial derivatives
f ( x , y ) = 2x + 3y − 4
fx =2 fy =3
Partial Derivatives
19. The Cobb-Douglas Production Function
1−b
f ( x, y ) = ax y b
• 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.
fy is the marginal productivity of capital.
19
Partial Derivatives: Application of First Partial Derivatives
20. Example A certain production function is given by
f ( x, y ) = 28 x y units, when x units of
1/ 4 3/ 4
labor and y units of capital are used. Find the
marginal productivity of capital when labor = 81
units and capital = 256 units. 1/ 4
1/ 4 −1/ 4 x
f y = 21x y = 21 ÷
y
When labor = 81 units and capital = 256 units,
1/ 4
fy 81 3
= 21 ÷ = 21 ÷ = 15.75
256 4
So 15.75 units per unit increase in capital expenditure.
20
Partial Derivatives: Application of First Partial Derivatives
21. Question A certain production function is given by
f ( x, y ) = 28 x y units, when x units of
1/ 4 3/ 4
labor and y units of capital are used. Find the
marginal productivity of labor when labor = 81 units
and capital = 256 units.
1 −3 / 4 3 / 4
f x = 28 x y = 7 x −3 / 4 y 3 / 4
4
When labor = 81 units and capital = 256 units,
f x = 7(81) −3 / 4 (256) 3 / 4 = 49.78
So 49.78 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
∂f ∂g
> 0 and >0
∂q ∂p
Two commodities A and B are complementary
commodities if
∂f ∂g
< 0 and <0
∂q ∂p
23
Partial Derivatives: Application of First Partial Derivatives
24. Example The demand function for two related commodities
are
x = ae q-p
y = be p-q
The marginal demand functions are
δx = - ae q-p δy = be p-q
δp δp
δx = ae q-p δy = - be p-q
δq δq
Because δx/δq > 0 and δy/δp > 0, the two
commodities are substitute commodities.
24
Partial Derivatives: Application of First Partial Derivatives
25. Question
In a survey it was determined that the demand equation for VCRs
is given by
x = f ( p, q ) = 10, 000 − 10 p − e 0.5 q
The demand equation for blank VCR tapes is given by
y = g ( p, q ) = 50, 000 − 4000q − 10 p
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. x = f ( p, q ) = 10, 000 − 10 p − e 0.5 q
y = g ( p, q ) = 50, 000 − 4000q − 10 p
∂x ∂y
= −10 = −10
∂p ∂p
∂x ∂y
= −0.5e 0.5 q = −4000
∂q ∂q
Because δx/δq < 0 and δy/δp < 0, the two commodities are
complementary commodities.
Partial Derivatives: Application of First Partial Derivatives
27. Second-Order Partial Derivatives
∂ f
2
∂ ∂2 f ∂
f xx = 2 = ( f x ) f xy = = ( fx )
∂x ∂x ∂y∂x ∂y
∂ f
2
∂ ∂ f 2
∂
f yy = 2 = ( f y ) f yx = = ( fy )
∂y ∂y ∂x∂y ∂x
27
Partial Derivatives: Second-Order Partial Derivatives
28. Example
Find the second-order partial derivatives of the function
f ( x, y ) = 3 x 2 y + x ln y
1
f x = 6 xy + ln y f y = 3x + x ÷
2
y
x 1 1
f xx = 6 y f yy =− 2 f xy = 6x + f yx = 6x +
y y y
Example Find the second-order partial derivatives of the function
2
a. f ( x, y ) = 3 x − 2 y
fx =3 f y = −4 y
f xx = 0 f yy = −4 f xy = 0 f yx = 0
Partial Derivatives: Second-Order Partial Derivatives
29. Example Find the second-order partial derivatives of the function
xy 2
f ( x, y ) = e
fx =
∂ xy 2
∂x
e ( )2 xy 2
=ye fy =
∂ xy 2
∂y
e ( )
= 2 xye xy 2
f xx = y e 4 xy 2
f yx = 2 ye xy 2
( 1 + xy )2
f xy = 2 ye xy 2 2
(
+ y 2 xye xy 2
) = 2 ye xy 2
(1 + xy )2
f yy = 2 x e ( xy 2
+ye ( xy 2
))
• 2 xy = 2 xe xy 2
(1 + 2 xy )2
29
Partial Derivatives: Second-Order Partial Derivatives
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 f ( x, y ) ≤ f (a, b)
for all (x, y) sufficiently close to (a, b).
f (a, b) is a relative minimum of f if f ( x, y ) ≥ f (a, b)
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. Critical Point of f
A critical point of f is a point (a, b) in the
domain of f such that both
∂f ∂f
( a, b ) = 0 and ( a, b ) = 0
∂x ∂y
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
f x = 0, f y = 0
2. The 2nd Derivative Test: Compute
D( x, y ) = f xx f yy − f xy
2
D ( a, b) f xx (a, b) Interpretation
+ + Relative min. at (a, b)
+ – Relative max. at (a, b)
– Neither max. nor min. at (a, b)
saddle point
0 Test is inconclusive
33
34. Ex. Determine the relative extrema of the function
f ( x, y ) = 2 x − x 2 − y 2
So the only critical
fx = 2 − 2x = 0 f y = −2 y = 0
point is (1, 0).
f xx = f yy = −2, f xy = 0
D(1, 0) = ( −2 ) ( −2 ) − 0 2 = 4 > 0 and f xx ( 1, 0 ) = −2 < 0
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 1 2 3 2 1
R ( x, y ) = − x − y − xy + 300 x + 240 y
4 8 4
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
C ( x, y ) = 180 x + 140 y + 5000
Determine how many assembled units and how many kits
Acrosonic should produce per week to maximize its profit.
35
36. P ( x, y ) = R ( x, y ) − C ( x , y )
1 2 3 2 1
= − x − y − xy + 120 x + 100 y − 5000
4 8 4
1 1
Px = − x − y + 120 = 0K ( 1)
2 4
3 1
Py = − y − x + 100 = 0K ( 2 )
4 4
36
37. Substitute in K ( 1)
y = −2 x + 480
Substitute in K ( 2 )
3 1
Py = − ( −2 x + 480 ) − x + 100 = 0
4 4
= 6 x − 1440 − x + 400 = 0
∴ x = 208
Substitute this value into the equation y = −2 x + 480
∴ y = 64
Therefore, P has the critical point (208,64)
37
38. 1 1 3
Pxx = − Pxy = − Pyy = −
2 4 4
2
1 3 1 5
D ( x, y ) = − ÷ − ÷− − ÷ =
2 4 4 16
Since, D ( 208, 64 ) > 0 and Pxx ( 208, 64 ) < 0 , the
point (208,64) is a relative maximum of P.
38
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
.
( x0 , y0 )
Form the function
F ( x, y , λ ) = f ( x, y ) + λ g ( x, y )
40
41. Then there is a value of λ such that ( x0 , y0 , λ ) is a solution
of the system of equations
∂F ∂f ∂g
= +λ = 0 L ( 1)
∂x ∂x ∂x
∂F ∂f ∂g
= +λ = 0 L( 2)
∂y ∂y ∂y
∂F
= g ( x, y ) = 0 L ( 3)
∂λ
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
z = f ( x, y )
subject to the constraint g ( x, y ) = 0
Step 2: Construct the function F:
F ( x, y , λ ) = f ( x, y ) + λ g ( x, y )
42
43. Step 3: Set up the system of equations
∂F
= 0 L ( 1)
∂x
∂F
= 0 L( 2)
∂y
∂F
= g ( x, y ) = 0 L ( 3 )
∂λ
Step 4: Solve the system of equations for x, y and λ .
Step 5: Test the solution ( x0 , y0 , λ ) to determine
maximum or minimum point.
43
44. 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 z = f ( x, y ) at each solution ( x0 , y0 , λ )
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, λ) = 5x2 + 6y2 - xy + λ(x + 2y - 24)
Fx = δF = 10x - y + λ ; Fxx = 10
δx
Fy = δF = 12y - x + 2λ ; Fyy = 12
δy
Fλ = δF = x + 2y - 24 ; Fxy = -1
δλ
45
46. The critical point,
10x - y + λ = 0
12y - x + 2λ= 0
x + 2y - 24= 0
The solution of the system is x = 6, y = 9, λ = -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
P ( x, y ) = x 2 + 3 xy − 6 y
to maximize profit, how many engines of each type should be
produced if there must be a total of 42 engines produced?
47
48. Maximize z = P ( x, y ) = x + 3xy − 6 y
2
Subject to constraint g ( x, y ) = x + y − 42 = 0
F ( x, y , λ ) = P ( x , y ) + λ g ( x , y )
= x 2 + 3xy − 6 y + λ ( x + y − 42 )
∂F
= 2 x + 3 y + λ = 0 L ( 1) ; Fxx = 2
∂x
∂F
= 3x − 6 + λ = 0 L ( 2 ) ; Fyy = 0
∂y
∂F
= x + y − 42 = 0 L ( 3) ; Fxy = 3
∂λ
The solution of the system is x = 33 y = 9 λ = −93.
48
49. Fxx = 2 > 0
D* = (2)(0) − (3) 2 = −9 < 0
The test in inconclusive.
49
Notes de l'éditeur
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.