Mathematics

# Functions of Many Variables

Functions with more than one independent variable.

y=f(x) => one independent variable.

Not just price, but incomes, ect. are independent.

z=f(x, y)     Here, z = dependent, x & y = independent.

Example:

z=100-2x+5y

z=3x2-9y

z=e2x+3y     Expediential fn. Example 1: z=150-2x-3y

X-Intercept  (z=0, y=0)

0= 150-2x-3(0)

=> x=150/2 =75       (75, 0, 0)

Y-Intercept  (x=0, z=0)

0= 150-2(0)-3y

y= 150/3 = 50

Z-Intercept  (x=0, y=0)

z= 150

How dependent variable will a change as a result of the change in independent variables.

Use partial derivatives z=f(x, y)

Terms:   Change: Δ, Delta: δ To what effect a change in x has on z while holding y constant. To what effect a change in y has on z while holding x constant.

Example 1:

z=x3+3x2y2+y3

z=x3+[3y2]x2+[y3]

fx= 3x2+(3y2)(2)x+0

=3x2+6xy2                           Constant Multiplication Remains.

fy= 0+(3x2)(2)y+3y2   ->2 = (3-2)

=6x2y+3y2

fx Slope of fn with respect to x.

fy Slope of fn with respect to y.

Example 2:

z= x0.5y0.5-10

fx= 0.5x-0.5(x0.5)(0.5y)-10

fx=(y0.5)(0.5)x-0.5-0

=0.5x-0.5 y0.5

y=nxn-1

When x=25, y=9

fx=0.5(25)-0.5(9)0.5= 3/10

fy= x0.5(0.5)y-0.5

=(0.5)x0.5y-0.5_____Partial Derivative

Plug in values= (0.5)(25)0.5(9)-0.5 = 5/6

Example 3: z=3x2y3

fx= (3y3)(2)x = 6xy3

fy= (3x2)3y2 = 9x2y2

Example 4: z=5x3-3x2y2+7y5

fx= (15x2)-6xy2

fy= -6x2y+35y4

Example 5: z=(3x+5)(2x+6y)

fx= (3)(2x+6y)+(2)(3x+5)

Take derivative of 1st term with respect to x. Do not differentiate 2nd term of multiplication.

=6+18y+6x+10

=12x+18y+10

fy=(0)(2x+6y)+(6)(3x+5)

=18x+30

Example 6: Q=36KL-2K2-3L2    Production Function fL= 36K-0-6L

=36K-6L

MPK= 36K-6K

Featured image supplied from Unsplash.