Homework_Set_7

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online