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**Partial Differentiation & Application**

Week 9

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**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

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**Functions of Two Variables**

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**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

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**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

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**Function of Two Variables**

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

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**Function of Two Variables**

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

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**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

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**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

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**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

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**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

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**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

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**Graphs of Functions of Two Variables**

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

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**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

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**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

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**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

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**Compute the first partial derivatives**

Example Compute the first partial derivatives Example Partial Derivatives

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**Compute the first partial derivatives**

Example Compute the first partial derivatives Example Partial Derivatives 18

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**Partial Derivatives: Second-Order Partial Derivatives**

27 Partial Derivatives: Second-Order Partial Derivatives

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**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

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**Find the second-order partial derivatives of the function**

Example 29 Partial Derivatives: Second-Order Partial Derivatives

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**Maximum and Minimum of Functions of Several Variables**

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**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

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**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

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**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

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**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

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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

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**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

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

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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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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

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**The solution of the system is**

Maximize Subject to constraint The solution of the system is 48

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**The test in inconclusive.**

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The End 50

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