Homework_Set_7

# Homework_Set_7 - e x is C 1 and its derivative is e x(g...

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Math 1a B. Simon Fall 2005 HOMEWORK SET 7 DUE AT 1:00 PM ON MONDAY, NOVEMBER 14, 2005 (1) [50 points] The purpose of this problem is to provide an alternate approach to the deFnitions of log( x ) and e x used in class. (a) Prove that the radius of convergence of the power series s n =0 x n n ! is inFnite. DeFne the limit to be e x . (b) DeFne P n ( x ) = n s j =0 x j j ! E n ( x ) = s j = n +1 x j j ! the approximation and error. Prove that if | x | < n , then | E n ( x ) | ≤ | x | n +1 ( n + 1)! p 1 - | x | n P - 1 (c) Use the binomial theorem to prove that | P 2 n ( x + y ) - P 2 n ( x ) P 2 n ( y ) | ≤ e | x | E n ( | y | ) + e | y | E n ( | x | ) (d) Use (b) and (c) to prove that e x + y = e x e y (e) Use the deFnition to prove that, as x 0, e x = 1 + x + O ( x 2 ) and prove that e x is di±erentiable at x = 0 with derivative 1 at x = 0. (f) Use (d) and (e) to prove that

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Unformatted text preview: e x is C 1 and its derivative is e x . (g) Prove that e x is monotone from (-∞ , ∞ ) to all of (0 , ∞ ) and so deFne an inverse function log( x ). (h) Using (f), prove that d dx log( x ) = 1 x . 1 2 HOMEWORK SET 7 DUE AT 1:00 PM ON MONDAY, NOVEMBER 14, 2005 (2) [20 points] Apostol, page 278, problems 3 and 9. Note : In these prob-lems, T n ( f ( x )) denotes the Taylor approximation of order n about x = 0. (3) [10 points] Apostol, page 438, problems 5 and 10. (4) [20 points] Apostol, page 295, problems 1, 4, 8, and 11....
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## This note was uploaded on 03/30/2010 for the course MA 1a taught by Professor Borodin,a during the Fall '08 term at Caltech.

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Homework_Set_7 - e x is C 1 and its derivative is e x(g...

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