lecture11

# lecture11 - Numerical Methods in Engineering ENGR 391...

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Unformatted text preview: Numerical Methods in Engineering ENGR 391 Faculty of Engineering and Computer Sciences Concordia University Numerical differentiation • Know how to compute numerically a derivative of a function What is the velocity at different instance of time? 993 623 383 225 Distance (ft) 13 8 5 3 Time (s) A car traveling along a straight road is clocked at a number of points. The data from the observations are given in the following table: Numerical Differentiation One of the idea would be to approximate f’ ( x ) by the Forward Difference : An alternative would be a Backward Difference : ) ( ) ( ) ( 1 h O h x f ) f(x x f i i i ¡ ¢ c ¡ ) ( ) ( ) ( 1 h O h x f ) f(x x f i i i ¡ ¢ c ¢ Numerical Differentiation ) ( ) ( ) ( ) ( 2 1 h O h x f x f x f i i i ¡ c ¡ ¡ ) ( ) ( ) ( ) ( 2 1 h O h x f x f x f i i i ¡ c ¢ ¢ Assume that y = f ( x ), suppose that we need information about the derivative of f ( x ) £ Taylor’s series expansion x i+1 = x i + h x i-1 = x i- h Averaging the forward and the backward leads to an expression is called Central Difference : ) ( 2 ) ( ) ( 2 1 1 h O h x f ) f(x x f i i i ¡ ¡ c ¡ ¢ ¡ ¢ c c ¢ c ¢ ¢ 2 1 ! 2 ) ( ) ( ) ( ) ( h x f h x f x f x f i i i i ¡ ¢ ¢ c ¢ ¡ ¢ 3 ) 3 ( 1 1 ! 3 ) ( 2 ) ( 2 ) ( ) ( h x f h x f x f x f i i i i » ¼ º « ¬ ª c c ¢ c ¡ ¡ ¡ 2 1 ! 2 ) ( ) ( ) ( )...
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lecture11 - Numerical Methods in Engineering ENGR 391...

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