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Unformatted text preview: Homework 10 due Thursday December 16 before 13:30 in corre-sponding group folder on door of oce CM110 1. By Taylor expansion, nd the leading term in the truncation error of the approxi-mation f ( x ) [4 f ( x + h )-3 f ( x )-f ( x + 2 h )] / 2 h. If there is a possible error, of magnitude not exceeding , in each function value, what is the largest possible resulting error in the approximation for f ( x )? How does this compare with the second-order central dierence formula for f ( x )? By Taylor expansion, f ( x + h ) = f ( x ) + hf ( x ) + h 2 2! f 00 ( x ) + h 3 3! f 000 ( x ) + f ( x + 2 h ) = f ( x ) + 2 hf ( x ) + 4 h 2 2! f 00 ( x ) + 8 h 3 3! f 000 ( x ) + . Therefore: 4 f ( x + h )-3 f ( x )-f ( x + 2 h ) = 2 hf ( x )-2 h 3 3 f 000 ( x ) + , i.e. f ( x )-4 f ( x + h )-3 f ( x )-f ( x + 2 h ) 2 h = h 2 3 f 000 ( x ) + ....
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