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Lecture 11kg - Introduction to Microeconomics Lecture 11...

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Introduction to Microeconomics Lecture 11 Differentiation Kathryn Graddy
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Outline Finding the Gradient The derivative Stationary Points The Second Derivative Maxima and Minima Convex and Concave Functions Economic Applications For Each Subject!!
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A Production Function 200 gradient 20 10 Y L = = =
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Another Production Function The gradient of a curve at a particular point is the gradient of the tangent .
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Finding the Gradient of a Function 2 linear function: ( ) Gradient = slope = ( ) 4 What do we do? y x mx c m x y x = + =
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A Non-linear Function
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A Non-linear Function We could take an approximation 0 (3) (2) 1.25 3 2 (2.5) (2) 1.125 2.5 2 (2.001) (2) 1.00025 2.001 2 lim x y y y y y y y x ∆ → - = - - = - - = -
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The Derivative 0 The shorthand for: lim is measures the gradient is the of y x y dy x dx dy dx dy derivative dx ∆ →
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Finding the Derivative of the Function y=x n 1 If ,then n n dy y x nx dx - = =
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An example 2 2 when 1: 2 1 2 when 3: 2 3 6 Practice! y x dy x dx dy x dx dy x x dx = = = = × = = - = × - = -
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More Differentiation Rules The derivative of the function ( ) can be written as ( ) instead of dy y x y x dx Some rules for differentiation If ( ) ( ) 0 If ( ) ( ) ( ) ( ) If ( ) ( ) ( ) ( ) ( ) ( ) f x f x f x ag x f x ag x f x g x h x f x g x h x α = = = = = ± = ± We now have a new tool– let’s use it to understand economics!
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