
JOURNAL OF RESEARCH IN NATIONAL DEVELOPMENT VOLUME 7 NO 2, DECEMBER, 2009 CALCULUS OF MULTIVARIATE FUNCTIONS: IT’S APPLICATION IN BUSINESS Emmanuel I. Awen
Abstract Introduction y=x3 + 4x2……….. (1) However, there are occasions where the functions contain two or more independent variables, for example, when the cost function of a firm depends on both labour hours and machine hours. A function of differential calculus that has more than one independent variable is known as a multivariate function. According to Budnick (1993), functions which involve more than one independent variable are referred to as multivariate functions, or functions of several variables. A hypothetical multivariate function that has a dependent variable and independent variables is the one that Varberg and Purcell (1992) defined as; if z=f(x,y) we call x and y the independent variables and z the dependent variable. An example of a multivariate function (or a cost function) is as follows: where y = total cost (dependent variable) Types of Multivariate Differentiation
Partial derivative (differentiation) of a multivariate function gives the instantaneous rate of change of the dependent variable with respect to an independent variable while other independent variables are kept constant (Agbadudu, 1987). Partial derivatives of multivariate functions can be determined from formulae for derivatives. This is done by differentiating in the normal
way for one of the independent variables, say x, whilst at the same time treating the other variable, say y, as a constant. For example, let y be a function of two variables x, and x2; that is y=f(x1,x2)………….. (3) ∂y is ∂y = lim f(x1 + ∆x1, x2)  f(x1,x2), ∆x1 0 ∆x1 ∂y Similarly, = lim f(x1, x2 + ∆x1, x2)  f(x1,x2), ∆x2 0 ∆x2 If the limit exists. Note: lim is the short form of limit.
As Agbadudu (1987) puts it, the differentials of the independent variables, say K and L, shall be denoted by dk and dL and defined as dk=∆k and dL=∆L respectively. In total differentiation all the independent variables are differentiated simultaneously. ∂Q ∂Q From the above definition of total deferentiation, the following is the differential formula using quantity Q: dQ= dk + dL …………. (4)
∂z Then dz= dx + dy ………… (5) Higher order partial differentiation ∂ 2z ∂2z and
∂ 2z ∂ 2z or and For a continuous functions, they are equal Maxima and minima for multivariate function without constraints ∂z derivative of the function with respect to x and the partial derivative with respect to y are simultaneously zero. That is or fx(x*=a,y*=b)=0 …………… (6) ∂z and or fy(x*=a,y*=b)=0 …………… (7) The two equations are solved simultaneously for x*=a and y*=b. The values of x and y for which the derivatives are zero are the critical points. The sufficient condition, like the calculus of a single variable, uses the second derivatives. Application of calculus of multivariate functions in business
In business, if a manufacturer determines the particular units of a commodity, say x, that can be sold domestically for say, N 120 per unit, and another unit of a commodity, say y, that can be sold to foreign markets for say, N 200 per unit, then the total revenue obtained from all sales is given by R=120x + 200y ……….. (8) It shows from the above example that revenue R, depends on the prices and quantities of commodities x and y, which are independent of R. Hoffmann etal (2005) has also accepted with the practicality of multivariate functions by saying that in psychology, a person’s intelligence quotient (1Q) is measured by the ratio These are typical; of practical situations in which a quantity of interest depends on the values of two or more variables. other examples include the output of a factory, which may depend on the amount of capital invested in the plant, the size of the labour force, and the cost of raw materials; also the volume of water in a community’s reservoir at any point in time may depend on the amount of rainfall as well as the population of the community. A business organization that is planning for an expansion may depend on increasing their workforce, acquiring more facilities, injecting in more money, and any other factors. (a) Application of partial differentiation
∂u Then and are marginal utility functions. Illustration 1: ∂y Then = 2x1 x2 + 0 + 0 ∂y ∂ x2 = x21 + 2x2 ∂y ∂y =6x21 x22 + x51x3 ∂y and =x5x2 + 6
∂z ∂z Find the partial derivatives of and if z=f(x,y)=tx2y. Solution: f(x+∆x,y) – f(x,y) ∆f = 5(x+∆x)2y – 5(x2y) = 5(x2+2x∆x + ∆x2)y –5x2y = 5x2 + 10x∆x + 5y∆x2 –5x2y = ∆x (10xy + 5y∆x) = = 10xy + 5y∆x ∂z Thus, = lim ∆f(x,y) ∆x 0 ∆x = lim(10xy + ty∆x) ∆x 0 ∂z To find , we have ∆f (x,y) f(x,y+∆x,y) – f(x,y) 0 = 5x2(y+∆y) – 5x2y = 5x2y+5x2∆y – 5x2y = ∂z And = lim ∆f(x,y) ∆y 0 ∆y =lim 5x2 ∆y 0 = 5x2
dq p only depend on its price, but also on the prices of other product and consumer income, that is,
dQ q1=f(P1,P2, ……, Pn, y) ……. (ii) ∂q1 p1 This is the same as in single variable derivatives except that we resort to partial differentiation here. However, this is not the only elasticity we can derive here. This is because the quantity demanded of product one is also linked to the price of products two, three and so on, and to the consumer income. It will be proper to inquire into the effect that changes in those variables would have upon the quantity demanded for product one. ∂ ∂Pj q1 ∂ q1 Elj= …….. (13) This is the measure that will indicate the responsiveness of quantity demanded of q1 to changes in prices of other products. The sign Elj is of significance; if it is positive, it means that product j is a substitute to product one but if it is negative, then product j is complementary to product one. ∂q1 y
……… (14)
Illustration 3: 1 1 1 20 100 (ii) q1= + + + at (10,40,20,350) (iii) q1=p12.5 p20.2 p30.5y Solution: 1 Q1=20020(4)3/4 + 3(10)20 + (1000) 3 ∂Q1 =  (20)P½ =30(4)½ =60 ∂Q1 ∂q1 ∂q1 1 = 3 and = 1j = ∂q1 p1 \ E1= =60 = 4 ∂q1 p2 E12= =3 = ½ ∂q1 p3 E13= =1 = 1/3
1 1000 1 ∂q1 y And Ely= = = (ii) At the given values of p1,p2,p3 and y, q1=20 ∂q1 ∂q1 100 ∂q1 =  = 1, = 1/20, = 1/20 ∂q1 and = 1/50 ∂q1 p1 \ E1= =1 = 1/2 ∂q1 p2 E12= =1/20 = 1/10 ∂q1 p3 E13= =1/20 = 1/20 7 ∂q1 y And Ely= = = ∂q1 y y (iii) Ely= = p12.5 p20.2 p30.5y =1 (b) Application of total differentiation
∂Q ∂Q = 1/4L3/4K3/4, = 3/4L1/4K1/4 ∴ dQ=1/4L3/4K3/4dK + 3/4L1/4K1/4dL. Application of higher order partial differentiation Illustrations 5: Find the first and second partial derivatives of the following functions
fxx=4; fyy=6x10+6y and fxy=fyx=6y
Zxx=4y3; Zyy=12x2y+2x and Marginal utilities are decreasing if their slopes are negative. The slopes of these terms are second partial derivatives of U. (d) Application of maxima and minima for multivariate functions Illustration 7: The cost of construction of a project depends upon the number of skilled
workers x and unskilled workers y. If the cost is given by c(x,y)=40,000+8x372xy+9y2
Solution: Conclusion Single variable functions are those in which one factor depends on just one another factor for a particular result to occur, like total profit being dependent on total cost. However, multivariate functions are those in which one factor depends on two or more other factors for a particular result to occur, like cost of production being dependent on cost of raw materials, cost of labour hours, cost of machines hours, and so on. Multivariate functions can be applied to profit in terms of what profit is dependent upon, as well as cost and total output variables. Multivariate functions are also applicable in business in areas like number of employees selected during interviews and in determination of wages and so on. Multivariate functions can be used in transport businesses by considering so many variables before fixing fares for passengers. It can also be used in hotel business by charging costs that are dependent on many variables. In most organizations, the aforementioned analysis are undertaken using the rule of thumb approach. Such analysis of profit, transport cost, wages determination, and others mentioned, can be determined more scientifically and objectively using calculus of multivariate functions. References Agbadudu, A.B. (1987): Mathematical methods in Business and Economics 1st Ed., Lagos: Lagos University Press. Budnick, F.S. (1993): Applied mathematics for Business, Economic and the Social Sciences 4th Ed., New York: McGrawHill, p. 930. Edwards, C.H., and Penny, D. E. (1994): Calculus with analytic geometry 4th Ed., New Jersey: Prentice Hall. Hoffmann, L.D., Bradley, G.L., and Rosen, K.H. (2005): Applied calculus: for Business, Economics, and the Social and Life Sciences 8th Ed., New York: McGrawHill. Lucey, T. (1996): Quantitative techniques 5th Ed., London: The Tower Building Publishers.

