MATH 142 Business Mathematics II

Transcription

1 MATH 142 Business Mathematics II Summer, 2016, WEEK 5 JoungDong Kim Week 5: 8.1, 8.2, 8.3 Chapter 8 Functions of Several Variables Section 8.1 Functions of several Variables Definition. An equation of the form z = f(x, y) describes a function of two independent variables if for each permissible ordered pair (x,y), there is one and only one value of z determined by f(x,y). Ex4) Given f(x,y) = 2x 2 3xy +y 2 4, find the following: a) f(3,0) b) f(1,2) c) f(a,1) 2f(0,b) Ex5) Find the domain of the following functions: a) f(x,y) = 16 x 2 y 2 b) f(x,y) = x x+y 1

2 c) f(x,y) = ln(x y +3) 2

3 Ex6) A company manufactures ten and three speed bicycles. The weekly demand and cost equations are p = 230 9x+y, q = 130+x 4y, C(x,y) = x+30y where $p is the price of a ten speed bicycle, $q is the price of a three speed bicycle, x is the weekly demand for ten speed bicycles, y is the weekly demand for three speed bicycles, and C(x, y) is the cost function. a) Find the weekly revenue function and R(10, 15). b) Find the weekly profit function and P(10,15). 3

4 Definition. The Cobb-Douglas production function is defined as f(x,y) = kx m y n where k, m, and n are positive constants with m+n = 1. Economists use this function to describe the number of units f(x,y) produced from the utilization of x units of labor and y units of capital. Ex7) The productivity of a steel manufacturing company is given approximately by the function f(x,y) = 10x 0.2 y 0.8 with the utilization of x units of labor and y units of capital. If the company uses 3000 units of labor and 1000 units of capital, how many units of steel will be produced? 4

5 Section 8.2 Partial Derivatives Definition. Partial Derivatives If z = f(x,y), then the partial derivative of f with respect to x is f x (x,y) = f x = f f(x+ x,y) f(x,y) (x,y) = lim x x 0 x and the partial derivative of f with respect to y is f y (x,y) = f y = f f(x,y + y) f(x,y) (x,y) = lim y y 0 y Ex8) Find the first-order partial derivatives of the following functions: a) f(x,y) = x 2 y 2 b) f(x,y) = 2x 2 3x 2 y +5y +1 c) f(x,y) = 4x2 y 2 x 2 +2y 2 d) f(x,y) = ln(3x 2 +xy y 8 ) 5

6 Ex9) The productivity of an airplane manufacturing company is given approximately by the Cobb- Douglas production function. f(x,y) = 40x 0.3 y 0.7 a) Find f x (x,y) and f y (x,y) b) If the company is currently using 1500 units of labor and 4500 units of capital, find the marginal productivity of labor (the partial derivative of f with respect to labor) and the marginal productivity of capital (the partial derivative of f with respect to capital) c) For the greatest increase in productivity, should the management of the company encourage increased use of labor or increased use of capital? 6

7 Definition. Second-Order Partial Derivatives Given a function z = f(x, y), we define the following four second-order partial derivatives: f xx (x,y) = f xx = 2 f x = ( ) f 2 x x f yy (x,y) = f yy = 2 f y = ( ) f 2 y y f xy (x,y) = f xy = 2 f y x = y f yx (x,y) = f yx = 2 f x y = x ( ) f x ( ) f y Ex10) Find all second-order partial derivatives of the function, f(x,y) = x3 y 2 7

8 Note. Functions can have more than just one or two independent variables. For instance, a function of three independent variables could be w = f(x,y,z). Here w would have three partial derivatives, one for each independent variable, treating the others as constants. Ex11) Find all of the first-order partial derivatives for the function w(x,y,z) = xe y +ye z. 8

9 Section 8.3 Extrema of Functions of Two Variables Definition. Relative Maximum and Relative Minimum Suppose z = f(x,y) is a function defined on some domain D. We say that f has a relative(local) maximum at (a,b) D if there exists a circle centered at (a,b) and entirely in D such that f(x,y) f(a,b) for all points (x,y) inside this circle. We say that f has a relative(local) minimum at (a,b) D if there exists a circle centered at (a,b) and entirely in D such that f(x,y) f(a,b) for all points (x,y) inside this circle. Definition. A critical point of z = f(x,y) is where both first partial derivatives are zero: and (a,b) is in the domain of f. f x (a,b) = f x (a,b) = 0 and f y(a,b) = f y (a,b) = 0 Definition. Saddle point: Both partial derivatives are zero, but the function has neither a local maximum nor a local minimum. 9

10 Ex12) Classify each labeled point on the graph. Ex13) Determine the critical points of f(x,y) = 2x 2 +3y 2 +2xy +4x 8y +3 10

11 Second Derivative Test for Local Extrema: Let z = f(x,y) be a function of two variables such that f xx (x,y), f yy (x,y), and f xy (x,y) exist for every point inside a circle centered at (a,b). If (a,b) is a critical point (f x (a,b) = 0 and f y (a,b) = 0) we define a number D to be Then, D = f xx (a,b) f yy (a,b) [f xy (a,b)] 2 1. If D(a,b) > 0 and f xx (a,b) < 0 then f has a local maximum at (a,b). 2. If D(a,b) > 0 and f xx (a,b) > 0 then f has a local minimum at (a,b). 3. If D(a,b) < 0 then f has a saddle point at (a,b). 4. If D(a,b) = 0 then no conclusion can be made about f(a,b). Ex14) Find all critical points and determine whether each is a saddle point, local max, or local min. a) f(x,y) = x 2 y 2 +6x+8y 21 11

12 b) f(x,y) = x 3 +y 3 6xy 12