HW1suggestions - (b) Assume x ∈ Q \ N . Find the...

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Jacobs University, Bremen School of Engineering and Science Prof. Dr. Lars Linsen, Orif Ibrogimov Spring Term 2010 Homework 1 120202: ESM4a - Numerical Methods Homework Problems 1.1. Let f ( x ) = 1+ x + x 2 1 - x + x 2 . (a) Derive the Taylor expansion of the function f at x = 0 up to the fifth term. Find f (4) (0). (b) Use Taylor’s theorem to prove that the inequality 1 + x < e x is valid for all real numbers except x = 0. (c) Prove that Z π 0 e sin 2 x dx > 3 π 2 . (Hint: use (b)) ( 6+4+5 points ) 1.2. Compute the following values with precision ε : (a) cos 9 , ε = 10 - 5 . (b) 5, ε = 10 - 4 . ( 5+5 points ) 1.3. Consider the number representation with respect to basis 10. (a) Is it true that every infinite representation implies irrationality? Justify your answer.
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Unformatted text preview: (b) Assume x ∈ Q \ N . Find the necessary and sufficient condition for which the decimal representation of x is finite. Find the value of x in the following equations: (c) ( 11 · · · 1 | {z } 2009 times , 11 · · · 1 | {z } 2010 times ) 2 = ( x ) 10 . (d) (42) 8 = ( x ) 2 . ( 2+4+2+2 points ) 1.4. (Bonus) Let f ∈ C (2) ([0 , 1]) and f (0) = f (1) = 0. Assume there exists A > 0 such that | f 00 ( x ) | ≤ A for all x ∈ (0 , 1). Prove that | f ( x ) | ≤ A 2 for all x ∈ [0 , 1]. ( 10 points ) Due: 12.02.10, at 3 pm (in the mailbox labeled Linsen in the entrance hall of Res.I )...
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This document was uploaded on 05/18/2010.

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