FA07Ma1aSol5

FA07Ma1aSol5 - MA1a HW 5 Solutions November 7, 2007 Problem...

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MA1a HW 5 Solutions November 7, 2007 Problem 1 (p191: 9,10,11) In the following exercises, (a) ﬁnd all points x such that f 0 ( x ) = 0; (b) examine the sign of f 0 and determine those intervals in which f is monotonic; (c) examine the sign of f 00 and determine those intervals in which f 0 is monotonic; (d) make a sketch of the graph of f . In each case, the function is deﬁned for all x for which the given formula for f ( x ) is meaningful. 9. f ( x ) = x/ (1 + x 2 ) . 10. f ( x ) = ( x 2 - 4) / ( x 2 - 9) . 11. f ( x ) = sin 2 x. Solution 1. 9. We compute f 0 ( x ) = (1 - x 2 ) / (1+ x 2 ) 2 and f 00 ( x ) = 2 x ( x 2 - 3) / (1+ x 2 ) 3 . Then f 0 ( x ) = 0 precisely when x = ± 1, and f 0 is continuous, so its sign is constant in the intervals ( -∞ , - 1) , ( - 1 , 1) and (1 , ). In these intervals its sign is ( - ) , (+) and ( - ), respectively, and f is monotonic in the intervals ( -∞ , - 1] , [ - 1 , 1] and [1 , ), because these are the intervals in which either f 0 ( x ) 0 for f 0 ( x ) 0 for all

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This note was uploaded on 02/22/2010 for the course MA 1a taught by Professor Borodin,a during the Fall '08 term at Caltech.

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FA07Ma1aSol5 - MA1a HW 5 Solutions November 7, 2007 Problem...

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