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Lecture-6-h - Derivatives and Averages Lecture-6...

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1 Derivatives and Averages Lecture-6 Derivatives and Averages Derivative: Rate of change of some quantity – Speed is a rate of change of a distance – Acceleration is a rate of change of speed Average (Mean) – The numerical result obtained by dividing the sum of two or more quantities by the number of quantities
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2 Derivative x x f x f x x x f x f dx df = ¢ = D D - - = D ) ( ) ( ) ( lim 0 dt dv a dt ds v = = speed acceleration Examples 3 4 2 4 2 x x dx dy x x y + = + = x x e x dx dy e x y - - - + = + = ) 1 ( cos sin
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3 Second Derivative xx x f x f dx df = ¢ ¢ = ) ( 2 2 2 3 4 2 12 2 4 2 x dx y d x x dx dy x x y + = + = + = Discrete Derivative ) ( ) ( ) ( lim 0 x f x x x f x f dx df x ¢ = D D - - = D ) ( 1 ) 1 ( ) ( x f x f x f dx df ¢ = - - = ) ( ) 1 ( ) ( x f x f x f dx df ¢ = - - = (Finite Difference)
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4 Discrete Derivative ) ( ) 1 ( ) 1 ( x f x f x f dx df ¢ = - - + = ) ( ) 1 ( ) ( x f x f x f dx df ¢ = + - = ) ( ) 1 ( ) ( x f x f x f dx df ¢ = - - = Left difference Right difference Center difference Example F(x)=10 10 10 10 20 20 20 F’(x)=0 0 0 0 10 0 0 F’’(x)=0 0 0 0 10 -10 0 -1 1 left difference 1 -1 right difference -1 0 1 center difference Left difference
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5 Derivatives in Two Dimensions y y y x f y x f f y f x y x x f y x f f x f y x f y y x x D D - - = = D D - - = = D D ) , ( ) , ( lim
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