31Bc08sm1a

# 31Bc08sm1a - Mathematics Department UCLA T Richthammer...

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Mathematics Department, UCLA T. Richthammer spring 08, midterm 1 Apr 23, 2008 Midterm 1: Math 31B Calculus, Sec. 2 1. (6 pts) Show that f ( x ) = x 2 +4 x is not 1-1 on R . Find a domain D f (as large as possible) such that f is 1-1 on D f , and determine f - 1 , D f - 1 and R f - 1 . Answer: f ( x ) is not 1-1 because f (0) = f ( - 4) = 0, for example. By drawing the graph of f we ﬁnd D f = [ - 2 , ) to be a domain as desired and R f = [ - 4 , ). Solving y = x 2 + 4 x for x we get x = y + 4 - 2. So f - 1 ( x ) = x + 4 - 2. Furthermore D f - 1 = R f = [ - 4 , ) and R f - 1 = D f = [ - 2 , ). 2. (6 pts) Calculate the derivatives of ( a ) f ( s ) = (1+ s ) 3 1+ s 2 e s , ( b ) g ( x ) = x 1 /x . For (a) use logarithmic diﬀerentiation! Answer: (a) ln f ( s ) = 3 ln(1 + s ) + 1 2 ln(1 + s 2 ) - s , so we get d ds ln f ( s ) = 3 1+ s + s 1+ s 2 - 1, which implies f ± ( s ) = (1+ s ) 3 1+ s 2 e s ( 3 1+ s + s 1+ s 2 - 1). (b) We have

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## This note was uploaded on 05/01/2008 for the course MATH 31B taught by Professor Valdimarsson during the Spring '08 term at UCLA.

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31Bc08sm1a - Mathematics Department UCLA T Richthammer...

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