differentiation_discrete_function.pdf

# The acceleration at t16 is given by 16 16 t t v dt d

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, The acceleration at t=16 is given by   16 16 t t v dt d a Given that   5 . 22 10 , 0054606 . 0 13065 . 0 289 . 21 3810 . 4 3 2 t t t t t     t v dt d t a 3 2 0054606 . 0 13065 . 0 289 . 21 3810 . 4 t t t dt d 5 . 22 10 , 016382 . 0 26130 . 0 289 . 21 2 t t t 2 16 016382 . 0 16 26130 . 0 289 . 21 16 a 2 m/s 664 . 29

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15 Lagrange Polynomial n n y x y x , , , , 1 1 th n 1 In this method, given , one can fit a order Lagrangian polynomial given by n i i i n x f x L x f 0 ) ( ) ( ) ( where ‘ n ’ in ) ( x f n stands for the th n order polynomial that approximates the function ) ( x f y given at ) 1 ( n data points as     n n n n y x y x y x y x , , , , ...... , , , , 1 1 1 1 0 0 , and n i j j j i j i x x x x x L 0 ) ( ) ( x L i a weighting function that includes a product of ) 1 ( n terms with terms of i j omitted.
16 Then to find the first derivative, one can differentiate   x f n for other derivatives. For example, the second order Lagrange polynomial passing through     2 2 1 1 0 0 , , , , , y x y x y x is         2 1 2 0 2 1 0 1 2 1 0 1 2 0 0 2 0 1 0 2 1 2 x f x x x x x x x x x f x x x x x x x x x f x x x x x x x x x f Differentiating equation (2) gives once, and so on Lagrange Polynomial Cont.

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17     