MidtermKey

MidtermKey - University of California, Davis Department of...

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University of California, Davis Department of Applied Science Spring 2004 David M. Rocke Numerical Methods EAD 115 April 27, 2004 Midterm Examination NAME For all problems on this midterm examination, we will use the function f ( x )=2 x 3 - 3 x 2 - 11 x +6 . 1. Compute the 0 through third order Taylor series approximations of f (1) at x = 0 (i.e., the Maclaurin series). Compute the true absolute error E t and the approximate relative error E a in each case, where the latter is obtained from the change in the approximation from one iteration to the next. Solution We have f ( x x 3 - 3 x 2 - 11 x f 0 ( x )=6 x 2 - 6 x - 11 f 0 ( x )=1 2 x - 6 f 0 ( x 2 f (0) = 6 f 0 (0) = - 11 f 0 (0) = - 6 f 0 (0) = 12 The ±rst four approximations of f (1) using the Maclaurin series f ( x )= f (0) + f 0 (0) x + 1 2 f 0 (0) x 2 + 1 6 f 0 (0) x 2 + ··· are ˆ f 0 (1) = 6 ˆ f 1 (1) = 6 - 11(1) = - 5 ˆ f 2 (1) = 6 - 11(1) - (1 / 2)(6)(1) = - 8 ˆ f 3 (1) = 6 - 11(1) - (1 / 2)(6)(1) + (1 / 6)(12)(1) = - 6 The respective true absolute errors are 12, 1, 2, and 0, while the estimated absolute errors of the last three are 11, 3, and 2 (the ±rst one has no estimated error since there is no previous approximation).
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MidtermKey - University of California, Davis Department of...

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