7.1 Functions of Two or More Variables

Transcription

1 7.1 Functions of Two or More Variables Hartfield MATH 2040 Unit 5 Page 1 Definition: A function f of two variables is a rule such that each ordered pair (x, y) in the domain of f corresponds to exactly one number f(x,y). Definition: The domain of a function f of two variables is the collection of all ordered pairs for which the function is defined. If the domain is not stated explicitly, then the domain is presumed to be the largest set of ordered pairs for which the function is defined which is called the natural domain. y Ex.: For the function f ( x, y), x 1 A: state the domain. B: evaluate f(3, 16).

2 Functions of Three or More Variables Hartfield MATH 2040 Unit 5 Page 2 The definition of a function of two variables can be extended to three or more variables without significant change in the wording. B: evaluate f(1, 1, 2). Ex.: For the function f ( x, y, z) ln x y, z A: state the domain.

3 Applications of Multi-Variable Functions Hartfield MATH 2040 Unit 5 Page 3 Ex. 1: Suppose that you own a company makes two models of audio speakers: the Ultra Mini and the Big Stack. Your total monthly cost (in dollars) to make x Ultra Minis and y Big Stacks is given by C(x,y) = x + 40y. (Source: Finite Mathematics & Calculus Applied to the Real World (1996) p. 1089, #2) a. Explain the significance in each term of the cost function. b. If your company can manufacture no more than 150 total speakers per day, what is the domain of your cost function? c. How much would it cost to produce 120 Ultra Minis and 30 Big Stacks?

4 Hartfield MATH 2040 Unit 5 Page 4 A function used to model the output of a company or a nation is called a production function. The most famous production function is Cobb-Douglas production function. Cobb-Douglas model: P(L, K) = al b K 1 b, where P is the total production, L is the number of units of labor, K is the number of units of capital, a is a multiplier based on conditions unrelated to labor and capital, and b is the share of contribution by labor. Ex. 2: The Cobb-Douglas production function for an automobile manufacturer is P(L, K) = 10L 0.3 K 0.7. Find and interpret P(240, 100).

5 Hartfield MATH 2040 Unit 5 Page 5 Ex. 3: Determine the amount of material needed to construct a box as shown in the illustration below. Then calculate the amount of material when the length and width of the box are 12 inches and the height is 4 inches. y x z

6 Graphing in Three Dimensions Hartfield MATH 2040 Unit 5 Page 6 To graph a function of two variables, you need a three-dimensional coordinate system. We will not be graphing in three dimensions by hand although you may see some three dimensional graphs through a computer algebra system (CAS).

7 7.2 Partial Derivatives Hartfield MATH 2040 Unit 5 Page 7 With functions of several variables, it is possible to differentiate with respect to one variable while treating the other variable as a constant. This type of differentiation results in partial derivatives. Ex. 1: Find each partial derivative below. a. x 5 2 xy f x, y represents x the derivative of f with respect to x with y held constant. Statement: The notation b. y 5 2 xy The notation f x, y represents y the derivative of f with respect to y with x held constant. c. x 2 y

8 Hartfield MATH 2040 Unit 5 Page 8 Ex. 2: Find each partial derivative below. Ex. 3: Find each partial derivative below. x a. 3x 2 5xy 4 4y 3 2x 6 y a. 3x 2 5xy 4 4y 3 2x 6 x b. 2x 2y e e xlny ylnx b. x y x y y y x

9 Subscript Notation for Partial Derivatives Hartfield MATH 2040 Unit 5 Page 9 With functions of one variable, the shorthand notation of derivatives uses prime notation. For instance, d f x f x. dx With functions of several variables, the concise notation for partial derivatives uses subscript notation. Ex. 1: Find f x, y if 4 2 f( x, y) x y 1 3. x f x, y fx x, y x f x, y fy x, y. y Thus, and

10 Hartfield MATH 2040 Unit 5 Page 10 x, y if Ex. 2: Find f y 2 2 xy Ex. 3: Find x 3 2 f ( x, y). x y f x, y if 2 3 x y f( x, y) ln x y e.

11 Interpretation of Partial Derivatives as Rates of Change and as Marginals Instantaneous rate of change f, x x y of f with respect to x when y is held constant Instantaneous rate of change f, y x y of f with respect to y when x is held constant Hartfield MATH 2040 Unit 5 Page 11 Let C(x,y) be the total cost function for x units of product A and y units of product B. Then Cx Cy Marginal cost function for x, y product A when production of product B is held constant. Marginal cost function for x, y product A when production of product B is held constant.

12 Application Hartfield MATH 2040 Unit 5 Page 12 An electronics company s profit P(x,y) from making x DVD players and y CD players is given below. P(x,y) = 3x² 4xy + 4y² + 80x + 100y (Source: 4 th edition p. 508, modified #46) c. Find the marginal profit function for CD players. a. Find the marginal profit function for DVD players. b. Evaluate your answer to part (a) at x = 200 and y = 300 and interpret your result. d. Evaluate your answer to part (c) at x = 200 and y = 120 and interpret your result.

13 Higher-Order Partial Derivatives Hartfield MATH 2040 Unit 5 Page 13 Just as functions of one variable can be differentiated more than once, functions of several variables can be partially differentiated more than once. a. Differentiate twice with respect to x c. Differentiate first with respect to x and then with respect to y 2 yx f = f xy 2 x 2 f = f xx d. Differentiate first with respect to y and then with respect to x b. Differentiate twice with respect to y 2 xy f = f yx 2 y 2 f = f yy

14 Hartfield MATH 2040 Unit 5 Page 14 Ex.: Find all the second-order partial derivatives of f x y x x y x y 3 2 4,

15 7.3 Optimizing Functions of Two Variables Hartfield MATH 2040 Unit 5 Page 15 Similar to functions of one variable, functions of two variables can only have relative extrema at critical points. Ex.: Find the critical numbers of f. f x, y 3x 2y 2xy 8x 4y 2 2 Definition: A critical point (a, b) of a function f(x, y) satisfies the following pair of conditions: f a, b 0 and x f a, b 0. Relative extreme values can only occur at critical numbers. y

16 Second Derivative Test for Functions of Two Variables: The D-Test To determine if a function of two variables has a relative maximum, relative minimum, or a saddle point at a critical number, we will use this variation on the Second Derivative Test of one variable. Hartfield MATH 2040 Unit 5 Page 16 Ex.: Find the relative extreme values of f. f x, y 3x 2y 2xy 8x 4y 2 2 The D-Test: If (a, b) is a critical number of a function f and if D is defined by 2 D fxx a, b fyy a, b fxy a, b, then f at the point (a, b) has a: i. relative maximum if D > 0 and f a, b < 0 xx ii. relative minimum if D > 0 and f a, b > 0 xx iii. saddle point if D < 0.

17 Hartfield MATH 2040 Unit 5 Page 17 Ex. 2: Find the critical points of f and then find the relative extreme values of f. f x, y x y 3x 6y 3 2

18 Applications of Optimizing Functions of Two Variables Hartfield MATH 2040 Unit 5 Page 18 Ex. 1: A company manufactures products A and B. The price function for product A is p = 16 x (where 0 < x < 16) and the price function for product B is q = 19 1 x (where 0 < y < 38), both in 2 thousands of dollars with x and y representing the amounts of product A and product B, respectively. If the cost function is C(x, y) = 10x + 12y xy + 6 thousand dollars, find the quantities and prices of the two products that maximize profit. Also find the maximum profit. (Source: 4 th edition p. 519, #22)

19 Hartfield MATH 2040 Unit 5 Page 19 Ex. 2: An airline flying to a Midwest destination can sell 20 coach-class tickets per day at a price of $250 and six business-class tickets per day at a price of $750. It finds that for each $10 decrease in the price of the coach ticket it will sell four more per day and for each $50 decrease in the businessclass price it will sell two more per day. What prices should the airline charge for the coach- and business-class tickets to maximize revenue? How many of each type will be sold at these prices? (Source: 4 th edition p. 520, #30)