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Unformatted text preview: CS 257 Numerical Methods  Homework 1 August 31, 2006 1. [1pt] Count the number of operations involved in evaluating a polynomial using nested multiplication. Solution: From the pseudocode on page 8, it is clear that there are n multiplications and n additions for a polynomial of degree n . 2. [1pt] Why does the function f ( x ) =  x  not posses a Taylor series at x = 0? Solution: The function does not have a continuous first derivative at 0. 3. [1pt] In the Taylor series for the function 3 x 2 7 + cos x (expanded about x = 0) what is the coefficient of x 2 . Solution: Recall the first few terms of the Taylor series f ( x ) = f (0) + f (0) x + f 00 (0) x 2 2 + . . . and in this case f (0) = 7 + 1 = 6 f (0) = 0 + 0 = 0 f 00 (0) = 6 1 = 5 so the coefficient of x 2 is f 00 (0) / 2! = 5 / 2. 4. [1pt] Using the Alternating Series Theorem, what is the least number of terms required to compute π as 3 . 14 (rounded) using the series π = 4 4 3 + 4 5 4 7 + . . ....
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 Fall '05
 ThomasKerkhoven
 Numerical Analysis, Taylor Series, 1pt, 2pt

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