31Bc08sm1a

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

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

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

31Bc08sm1a - Mathematics Department UCLA T Richthammer...

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

View Full Document
Ask a homework question - tutors are online