L11Differentiation

L11Differentiation - Introduction to Microeconomics Lecture...

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Introduction to Microeconomics Lecture 11 Differentiation Simon Cowan
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Outline Finding the Gradient of a function The derivative Stationary Points The Second Derivative Maxima and Minima Convex and Concave Functions Economic Applications For Each Topics
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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 How do we find the gradient when 2 (and 1)? y x mx c m x y x x y = + = = =
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A Non-linear Function
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A Non-linear Function Let’s take an approximation 0 2 (3) (2) 9/ 4 4/ 4 (2.5) (2) 1.25; 1.125 3 2 1 2.5 2 (2.001) (2) 1.00025; lim 1 2.001 2 An approximation from the left gives the same result: (1) (2) 1/ 4 4/ 4 (1.5) (2) 1.5 / 4 4/ 4 0.75; 1 2 1 1.5 2 x y y y y y y y x y y y y ∆ → - - - = = = - - - = = - - - - - = = = - - - 0.875 0.5 = -
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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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A proof for n = 2 2 2 0 0 0 0 2 2 2 2 0 0 0 0 0 ( ) ( ) ( ) 2 ( ) 2 ( ) 2 y x x y x x x x x x x x x x x x x x x x x x + ∆ - + ∆ - = + ∆ + ∆ - ∆ + ∆ = = = + ∆ As x goes to 0 this tends to 2 x 0.
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2 2 when 1: 2 1 2 when 3: 2 3 6 Practise! y
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This note was uploaded on 04/12/2010 for the course ECON DEAM taught by Professor Vines during the Spring '10 term at Oxford University.

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L11Differentiation - Introduction to Microeconomics Lecture...

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