MATH REVIEW
Properties of Exponents
1. xa * xb = xa+b Examples: x3 * x2 = x5, x1/2 * x1/3 = x5/6, x3* x−1/2 = x5/2
2. xa/xb = xa−b Examples: x5/x3 = x2, x3/x5 = x−2, x3/x1/2 = x5/2
3. (xa)b = xab Examples: (x3)2 = x6, (x−1/2)7 = x−7/2, (x2/3)5/7 = x10/21
Functions and Graphs
y = f(x) “y is a function of x”
x is the independent variable and y is the dependent variable
1. Constant Function: y = f(x) = 7 (forms a straight line with zero slope)
2. Linear Function: y = f(x) = 2x + 3 (forms a straight line)
y = f(x) = mx + b
b = intercept on the y axis
m = slope of the line = ΔY/ΔX = (y2 - y1)/(x2 - x1)
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3. Quadratic Function: y= f(x) = x2
Derivatives
The derivative of a function at a point serves as a linear approximation of the
function at that point. The derivative of a function at a chosen input value describes
the best linear approximation of the function near that input value. For functions of
a single variable, the derivative at a point equals the slope of the tangent line to the
graph of the function at that point. The derivative shows the rate of change dy/dx
= Δy/Δx = (y2 - y1)/(x2 - x1) when x2 - x1 are very close.
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dy/dx = the derivative of function y = f(x)
The derivative is also noted as df(x)/dx or f’(x).
Function y = f(x)=x2 has a derivative function of dy/dx = f’(x) = 2x. Or you can say
d(x2)/dx = 2x.
p
Power Rule: If y = f(x) = x , then dy/dx = f ′(x) = p x
If y = x3, then dy/dx = 3x2
p−1
Constant Rule: If y = f(x) = C where C is a constant, then dy/dx = f’(x) = 0
If y = f(x) =7, then dy/dx = f’(x) =0.
If y = (3x2 – 5x +15)/p, its derivative is dy/dx = (6x – 5)/p if p is a constant (does
not depend on x).
Quotient Rule: If y = f(x) = A(x)/B(x) then dy/dx = f’(x) = (BA’ – AB’)/B2 where
A’ = dA/dx, B’= dB/dx
Example: y = (5x2 – 3x + 1)/(4x – 7).
dy/dx = [(4x – 7)(10x - 3) - 4(5x2 – 3x + 1)]/(4x – 7)2
= [(40x2 - 82x + 21) – (20x2 - 12x + 4)]/(4x – 7)2
= (20x2 – 70x +17)/(4x – 7)2
Functions of Two Variables:
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Let U(x, y) = x2 y. Then, dU/dx = 2xy, dU/dy = x2
The marginal rate of substitution is
MRS = -(dU/dx)/(dU/dy) = -(2xy)/x2 = -2y/x
MRS – graphically illustrated
y
A
B
0
x
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