Unformatted text preview: ξ when x = 0 . 5 and x = 0 . 1. (4) (Cancellation Errors, p52 #12) Near certain values of x , the following functions cannot be accurately computed using the given formula on account of arithmetic cancellations. Identify the values of x where cancellation occurs (e.g., near x = 0 or when x is large and positive). Propose a reformulation that removes the problem (e.g., using Taylor series, rationalization, trigonometric identities, etc.). (a) f ( x ) = 1 + cos x (b) f ( x ) = e − x + sin x − 1 (c) f ( x ) = ln x − ln(1 /x ) (d) f ( x ) = √ x 2 + 1 − √ x 2 + 4 (e) f ( x ) = 1 − 2 sin 2 x (f) f ( x ) = x − sin x (g) f ( x ) = ln x − 1 (5) (Rate of convergence of a sequence, p27 #1) Compute the following limits and determine the corresponding rates of convergence. (a) lim n →∞ n − 1 n 3 +2 (b) lim n →∞ ( √ n + 1 − √ n ) (c) lim n →∞ sin n n 1...
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 Fall '08
 KRASNY
 Math, Taylor Series, threedigit, 1 3K

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