4 - ECMT 1020 Summer School 09 Lecture 4 Partial...

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ECMT 1020 Summer School 09 Lecture 4 – Partial Differentiation
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Review of Functions of One Variable A function may be defined for the variables z and x in the following manner; z = f (x) This suggests that z is in some way determined by the value of x. “f” represents a general transformation function that links z and x Examples: In production, Output (z) may be a function of input (x) or sales (z) may be some function of promotion level (x).
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Derivatives of One Variable Functions If z = f (x) then we can define the derivative dz/dx as lim h 0 (f (x +h) – f (x))/ h The derivative of a function is equivalent to the slope of the function at x. The derivative is useful in identifying turning points of functions (if they exist), related maxima and minima of functions and/or points of inflection.
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Conditions for optima The conditions for finding maxima and minima of functions of one variable are; 1. dz/dx ( also called f x or f ’(x)) = 0 (This is called the first order condition) 2. f “ (x) = d 2 z/dx 2 < 0 (relative max) f “ (x) = d 2 z/dx 2 > 0 (relative min) f “ (x) = d 2 z/dx 2 = 0 (further investigation) These are the second order conditions
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Example Suppose z = 2x 3 - 15x 2 +36x The first derivative dz/dx = 6x 2 – 30x + 36 The second derivative d 2 z/dx 2 = 12x – 30 To find max or min (if they exist) set dz/dx = 0 Thus 6x 2 – 30x + 36 = 0 x = 2 or x =3 Second Order conditions: (d 2 z/dx 2 = 12x – 30) For x = 2 d 2 z/dx 2 = - 6 (rel max) For x = 3 d 2 z/dx 2 = 6 (rel min)
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Functions of More than One Variable There are many situations where more than one independent variable will influence the value of some target variable. Production output (z) is determined by the amounts of inputs. There could be two inputs such as Labour (x) and Capital (y). Sales (z) could be determined by promotional spend (x) but also price (y). We can generalise the functional form to z = f (x, y)
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Derivatives of Functions of more than one variable The situation for derivatives becomes a little more complicated when there is more than one independent variable. We need to distinguish between partial derivatives and total derivatives . Partial derivative : The change in the dependent variable when an independent variable is incrementally changed (provided the other independent variables have been kept constant)
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Partial Derivatives As in the univariate function case there will be first order and second order partial derivatives. For a multivariate function there will be
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This note was uploaded on 02/10/2012 for the course ECON 1002 taught by Professor Markmelatos during the Three '10 term at University of Sydney.

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4 - ECMT 1020 Summer School 09 Lecture 4 Partial...

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