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Fall2011-HW1

# Fall2011-HW1 - expression to determine the numerical value...

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Homework 1 Math 371, Fall 2011 Assigned: Thursday, September 8, 2011 Due: Thursday, September 15, 2011 (1) (Finite precision numbers) The floating point representation of a real number takes the form x = ± (0 .a - 1 a - 2 . . . a - n ) β · β e , where a - 1 = 0 , - M e M . Suppose that β = 2 , n = 4 , and M = 5 . (a) Find the smallest and largest positive numbers that can be represented in this floating point system. Give your answers in decimal form. (b) Find the floating point number in this system that is closest to 2 . (2) (Rounding arithmetic) Use three-digit, decimal rounding arithmetic (i.e., β = 10 and n = 3 ) to compute the following sums. Add the numbers by hand in the specified order. (a) 6 k =1 1 3 k (b) 6 k =1 1 3 7 - k (3) (Taylor series) (a) Calculate the first three terms of the Taylor series for f ( x ) = ln x about x = 2 . Express the coefficients analytically (not as decimals). (b) The first-order Taylor expansion of f ( x ) = exp x about x = 0 with exact remainder is f ( x ) = 1 + x + f ( ξ ) 2! x 2 for some point ξ between zero and x . Find an analytic expression for ξ . Use this expression to determine the numerical value of
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Unformatted text preview: expression to determine the numerical value of 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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