smt - Math 449: Numerical Applied Mathematics Midterm...

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Unformatted text preview: Math 449: Numerical Applied Mathematics Midterm Examination Prof. Wickerhauser 14 October 2009 You may use a calculator, the textbook, your class notes, and the model homework solutions published this semester. Please write your answers in the bluebook. Problem 1 . Express 1 . 01011 01111 110 (base 2) in base 10 notation, giving at least four significant digits. Solution: Compute the answer as 1 + 1 4 + 1 16 + 1 32 + 1 128 + 1 256 + 1 512 + 1 1024 + 1 2048 + 1 4096 = 1 . 359130859375 [Note: This is approximately e/ 2 1 . 3591.] Problem 2 . (a) Find a polynomial p = p ( h ) of minimal degree in h such that ln(1 + h ) = p ( h ) + O ( h 3 ) as h 0. (b) Find > 0 such that | ln(1 + h )- p ( h ) | < . 0005 whenever | h | < . Solution: (a) Taylors theorem gives the formula ln(1+ h ) = h- 1 2 h 2 + 1 3(1+ c ) 3 h 3 for some | c | < | h | , so ln(1 + h ) = p ( h ) + O ( h 3 ) as h 0 where p ( h ) = h- h 2 / 2....
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smt - Math 449: Numerical Applied Mathematics Midterm...

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