Math Review

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Economics Undergraduate Board (EUB)
Math Review Autumn 2013

EXPONENT LAWS:

xa+xb=xa+xb (xa)b= xab

xa*xb=xa+b xaxb=xa-b DERIVATIVE NOTATION: Single variable: ddxfx=dfdx=f'(x)= “Taking the derivative of the function f with respect to x” Multiple variables: ∂∂xfx,y=∂f∂x=fx= “Taking the partial derivative of the function f with respect to x” ∂∂yfx,y=∂f∂y=fy = “Taking the partial derivative of the function f with respect to y” DERIVATIVE RULES: Product Rule: ddxfx*gx=f'x*gx+fx*g'(x) Quotient Rule: ddxfxgx=f'x*gx-fx*g'(x)gx2 Chain Rule: ddxfgx=f'x*g'(x) USEFL DERIVATIVES: ddxxn=nxn-1 ddxlnx=1x OTHER USEFUL THINGS: MUx=∂Ux,y∂x=∂U∂x= “Marginal utility with respect to x”: How much more utility you get from obtaining one more unit of x MUy=∂Ux,y∂y=∂U∂y= “Marginal utility with respect to y”: How much more utility you get from obtaining one more unit of y Endogenous Variables: Dependent variable explained by the model Exogenous Variables: Independent variables outside of the model and affect a model without being affected by it; given parameters

Practice Problems →
DERIVATIVES

Single variable:

e.g., given fx=2x0.2+x0.4+ x-0.3 , find f'x

Exercises: for each of the following function, find f'x

(i) fx=3x
(ii) fx=6x19
(iii) fx=x2-2xx7+3
(iv) fx= 2ln⁡(x2)

Multiple variables (partial derivatives) :

e.g., given fx,y=2x+3y+1 , find fx and fy

e.g., given fx,y=6x2y2-7xy-x , find fx and fy

Exercises: for each of the following function, find fx , fy

(i) fx,y=10xy2
(ii) fx,y=5x2-6y+1
(iii) fx,y=20x1/2y1/4
(iv) fx,y=2xlny3-5xy ln⁡(x2) Answers (Single Variable): (i) 13x- 23 (ii) 23x- 89 (iii) x9-2x8+3x2-6x (iv) 4x

Answers (Multiple Variable): (i) fx=10y2, fy=20xy (ii) ) fx=10x2, fy=-6y (iii) fx=10x-12y14 ,…...

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