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.
Copyright © 2016 Zoë-Marie Beesley
Licensed under a Creative Commons Attribution 4.0 International License.