L11Differentiation

L11Differentiation - Introduction to Microeconomics...

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11/11/2007 1 Introduction to Microeconomics Lecture 11 Differentiation Simon Cowan Outline z Finding the Gradient of a function z The derivative z Stationary Points Economic Applications For Each Topics z The Second Derivative z Maxima and Minima z Convex and Concave Functions A Production Function 200 gradient 20 10 Y L Δ == = Δ Another Production Function The gradient of a curve at a particular point is the gradient of the tangent . Finding the Gradient of a Function 2 linear function: ( ) Gradient = slope = yx m x c m =+ () 4 How do we find the gradient when 2 (and 1)? x yx xy = A Non-linear Function
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11/11/2007 2 A Non-linear Function z Let’s take an approximation ( 3 )( 2 )9 / 4 4 / 4 ( 2 . 5 2 ) 1.25; 1.125 32 1 2 . 52 yy y y −− == = 0 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; 12 1 1 . x y x y y Δ→ Δ −Δ = 0.875 0.5 = The Derivative 0 The shorthand for: lim is x yd y x dx dy Δ Δ measures the gradient is the of y dx dy derivative dx Finding the Derivative of the Function y=x n 1 If ,then nn dy y xn x dx A proof for n = 2 22 00 0 0 () ( ) yx x yx x x x xx +Δ − = ΔΔ 2 2 0 0 0 2( ) ) 2 x x x x + Δ+Δ − Δ+Δ =+ Δ As Δ x goes to 0 this tends to 2 x 0. An example 2 2 dy x dx = = when 1: 2 1 2 when 3: 2 3 6 Practise! dy x dx dy dx × = =− = ×− =− More Differentiation Rules The derivative of the function ( ) can be written as ( ) instead of dy y x dx Some rules for differentiation I f () 0 I f I f () () fx a f x fx b gx f x b fx gx hx f x gx hx ′′ ′′′ = ±
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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...

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