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Unformatted text preview: Math 151A HW1 Solutions Will Feldman 1. We want to find the third order Taylor polynomial for the function f ( x ) = 1 + x about x = 0. Recall the formula for the Taylor polynomial of order n centered at x is given by P n ( x ) = n X k =0 f ( k ) ( x ) k ! ( x- x ) k . Calculating some derivatives of f you will find that P 3 ( x ) = 1 + x 2- x 2 8 + x 3 16 . Now we want to use this to approximate some values of the square root. Note that y = f ( y- 1), so that (for example) . 5 = f (- . 5) P 3 (- . 5) . 711. The same method applies to the other values of square root. 2. We can calculate the Taylor series for f ( x ) = (1- x )- 1 by calculating that f ( n ) ( x ) = n !(1- x )- ( n +1) . Then we have P n ( x ) = n X k =0 x k . In order to calculate the error of using P n ( x ) to approximate f on x [0 ,. 5] note that we can calculate P n ( x ) exactly since xP n ( x ) = x n +1 + P n ( x )- 1 P n ( x ) = x n +1- 1 x- 1 ....
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- Spring '08