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2-Math Review and Elasticities

2-Math Review and Elasticities - Agenda Course Overview...

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8/25/2010 1 Agenda Course Overview Marginal Analysis Elasticities of Demand Unconstrained optimization Iso-surfaces and constrained optimization What To Take Away Microeconomics The Structure of the Course Consumers and Producers Market Interaction Today’s lecture Uncertainty Perfect competition Monopoly and Pricing strategies Competitive Strategy Auctions Information in Markets and Agency Game Theory Introduction to Markets Consumer Theory and Demand Technology and Production
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8/25/2010 2 Agenda Course Overview Marginal Analysis Elasticities of Demand Unconstrained optimization Iso-surfaces and constrained optimization What To Take Away Definition of Derivative Often we are interested in how a given function of interest, say f(x) (which can represent demand, supply, costs, utility, etc…) varies with changes in the underlying independent variable x . One measure of how much a function varies between two points x 1 and x 2 is simply the ratio of the change in the function to the change in the independent variable calculated between those points. If we want to understand the variation of a function at a given point, say x 1 , we could compute the limit of the previous quotient as the point x 2 tends to x 1 . This gives us the derivative f’(x 1 ) of the function f at the point x 1 . 2 1 2 1 ( ) ( ) f x f x x x 1 2 1 2 1 2 1 ( ) ( ) lim 1 x=x x x f x f x df f'(x )= | = dx x x
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8/25/2010 3 Geometric interpretation The derivative gives us the slope of the tangent to the function at a given point We can draw a straight line going through x 1 with slope equal to the quotient . As we move the point x 2 closer to x 1 , this straight line becomes tangent to the function f(x) at the point x 1 . 2 1 2 1 ( ) ( ) f x f x x x x 1 f(x 1 ) x 2 f(x 2 ) x’ 2 f(x’ 2 ) The derivative as a first order approximation To study the behavior of functions it is often convenient to approximate a function by simpler functions. For instance, we can approximate a function by a constant function whose value is the value of the function at a point x 1 , f(x 1 ) . While this approximation is rather accurate for points very close to x 1 , the error that we make with this approximation maybe extremely large as we move away from x 1 . Moving to greater accuracy we could approximate a function by a linear function. In this case if we use the following linear function we can actually get a pretty good approximation. In particular, the maximum possible error that we incur with this approximation increases quadratically as x moves away from the point x 1 . 1 1 1 f(x )+ f'(x )(x- x )
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8/25/2010 4 Some examples The derivative of a constant is the zero function. Linear functions: Power functions: Logarithmic functions: Exponential functions: ( ) d ax a dx 1 ( ) d x x dx 1 (ln ) d x dx x ( ) x x d e e dx Rules of calculus Linearity : The derivative of a sum of functions is the sum of the derivatives.
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