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HW7 assignment

# HW7 assignment - HW#7 EGM6341 Spring 2009 Due Additional...

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HW#7 EGM6341 Spring 2009 Due: 3/26/09 Additional problem (on numerical differentiation): A1. Using Taylor series expansion for f ( x ± 2 h ) & f ( x ± h ) to derive a forth order accurate formula for the 1 st and 2 nd order derivatives, ) ( x f & ) ( x f in terms of f ( x +2 h ), f ( x + h ), f ( x ), f ( x - h ) and f ( x -2 h ). One way you can approach it is as follows: * Expand, for example, f ( x +2 h ) as f ( x +2 h ) = f ( x ) +2 h ) ( x f + 4 ! 2 2 h ) ( x f + 8 3 3! h ) ( x f +16 4 4! h (4) ( ) f x +… * You then do the same for f ( x + h ), f ( x - h ) and f ( x -2 h ). * Now you will have 4 equations and you can solve for 4 unknowns: ) ( x f , ) ( x f , ) ( x f , and (4) ( ) f x by neglecting the higher order terms. Just get the result for ) ( x f & ) ( x f . * Verify your result: consider f(x)= 4 x so that the exact result should be 3 ( ) 4 f x x f = & 2 ( ) 12 f x x f = . Compare your numerical results with the exact results to see if your formulae are accurate to the order you expect. pp. 323-329 #1 Write a program to evaluate ( ) b a I f x dx = using the trapezoidal rule with n subdivisions, calling the result I n to a) dx x I ) exp( 2 1 0 - = ; b) dx x I 5 . 2 1 0 = c)

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