Funciones n variables

FUNCTIONS OF N VARIABLES Josefa Ramoni- Giampaolo Orlandoni
PARTIAL DERIVATIVES Continuity is still a condition for differentiability: f(x), x Rn is at a point x=a if it has no jumps or breaks as x approaches a. For functions of one variable, there are only two possible directions from which x approaches a: right and left. Limxa- f(x) = Limxa+ f(x) = f(a) For functions of more than onvariable there exist an infinite number of directions from which a point in Rn can be approached A function f(x), x Rn, is continuous at a point x=a if, in the neighborhood around x=a, the Euclidean distance between f(x) and f(a) tends to zero as x tends to a x-a = (x1,x2)-(a1,a2) = ((x1-a1)2+(x2-a2)2) for R2

The rules of differentiation can also be extended to the functions of n variables,but taking one variable at the time and keeping the others constant: partial derivatives:

For y= f (x1, ..., xn).), y depends on all xi, but the changes with respect to each of the
variables are taken separately: we differentiate f with respect to its ith argument holding all the other arguments fixed. Since the other variables are assumed fixed, there are now only two directions from which toapproach any point: the partial derivative behaves like any other derivative of a function on one variable:

y f ( x1, x2,...,xn)   f 1 = limxi0 xi xi

f ( x1,..., xi  xi,...xn)  f ( x1,...xi,...xn) xi

EXAMPLE:

Y = f(X1,X2) = 100 - 20X12 - 30X22
f2 = - 60X2

f1 = - 40X1

Y/X1 1

Y/X2 1

The fact that each of the n partial derivatives of f exists for all valuesof the domain does not imply that f is differentiable. In fact, it does not even imply that f is continuous. However, if all the partial derivatives of f exist and are continuous functions, then f is differentiable, and in fact its derivative is continuous. A function is continuously differentiable if all its partial derivatives exist and all these partial derivatives are continuous functions.EXAMPLE: Cobb-Douglas production function y=AL0.5K0.5 Marginal product functions: f1=y/L= 0.5 A L0.5-1K0.5= 0.5 AK0.5/L0.5 f2=y/K= 0.5A L0.5K1-0.5 = 0.5A L/K0.5 for A>0

Note: Partial derivatives depend on the level of the other variable: the marginal product function of one input is a function of all inputs:

 


The higher values of capital lead to a bigger increase in output as theresult of an increase in labor The higher values of labor lead to a bigger increase in output as the result of an increase in capital But also, for a fixed amount of capital (labor) available, the marginal product of labor (capital) falls as more labor (capital) is used: Diminishing marginal productivity

EXAMPLE: total revenue function (additively separable function): R(x1,x2) = p1x1 + p2x2f1=R/x1= p1 f2=R/x2= p2

Note: partial derivatives with respect to any of the variables do not depend on any of the other variables

THE CHAIN RULE: Assume y=f(x1(t),x2(t))

f f x1 f x 2 = + t x1 t x 2 t
EXAMPLE: y=3x1 + 5x2 x1=t2 x2=4t3

f = 3 (2t) + 5(12t2) = 6t +60 t2 t

EXAMPLE:

y=A Z-1/r

Z=x1-r + (1-)x2-r

f = -A/r x1-r + (1-)x2-r(-1/r)-1(-rx1-r-1) =Ax1-r-1x1-r + (1-)x2-r(-1/r)-1 x1

f = -A/r x1-r + (1-)x2-r(-1/r)-1(-r(1-)x2-r-1) = A(1-)x2-r-1x1-r + (1-)x2-r(-1/r)-1 x 2

EXAMPLE: the traditional rules of differentiation apply f(x) = x12e3x2+x1x3 + 2x23/x1

f = e3x2+x1x3 (2x1+x12x3) – 2x23/x12 x1

f =? x 2

f =? x 3

FUNCION

PRIMERA DERIVADA

SEGUNDA DERIVADA
2 f  f  f 11  X 1 X 1 =   X 12  D11( f )   2 f f  f 21  X 2 X 1 =  X 2X 1   D21( f )    

 

Y f 1  X 1 = D1( f )

 

Y = f (X1,X2)
2 f f  f 12  X 1 X 2 =  X 1X 2   D12( f )     2 f  f  f 22  X 2 X 2 =   X 22   D22( f )  

 

f 2  Y2 = D2( f ) X

 

FUNCION

PRIMERA DERIVADA
f1 = - 60X1X2 (X2 - 5X1) ( X2 - 10X1)
2 2

SEGUNDA DERIVADA
f11 =...