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=e^2x+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= x^0.5y^0.5-10

fx= 0.5x-^0.5(x^0.5)(0.5y)-10       

fx=(y^0.5)(0.5)x-^0.5-0

=0.5x-^0.5 y^0.5                   

y=nx^n-1

When x=25, y=9

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

fy= x^0.5(0.5)y-^0.5

=(0.5)x^0.5y-^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.

Copyright © 2016 Zoë-Marie Beesley

 Licensed under a Creative Commons Attribution 4.0 International License.