102-2008&2009-1-M10-NOVEMBER2009

102-2008&2009-1-M10-NOVEMBER2009 - Kuwait...

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Unformatted text preview: Kuwait University Math 102 November 9, 2008 Dept. of Math. and Comp. Science First Exam Duration: 90 minutes Calculators and mobile phones are not allowed. Answer all of the following questions. 1. Let f ( x ) = ln ( tan- 1 (2 x- 1) ) + 1. (a) Find the domain of f . (2 pts.) (b) Show that f has an inverse function. (2 pts.) (c) Find f- 1 ( x ) and its domain. (2 pts.) 2. (a) Prove the identity 2sin- 1 ( √ x ) = cos- 1 (1- 2 x ) (0 ≤ x ≤ 1) (3 pts . ) (b) Let a > 1. Solve the following equation for x . log a ( x + 2) = 2ln x ln a (2 pts . ) 3. (a) Use logarithmic differentiation to find y if y = ‡ 1 x- 1 · x tan- 1 x e 3 x √ x 2- 1 ( x > 1) (3 pts . ) (b) Let y = A sinh( mx ) + B cosh( mx ), where A , B and m are constants. Show that y 00 = m 2 y (2 pts . ) 4. Evaluate the following integrals (3 pts. each) (a) Z dx (csc x ) √ 4- cos 2 x (b) Z 1- e- 2 x 2 e- x cosh x dx (c) Z ln x 3 x (4 + 3ln 2 x ) dx Solution key - First midterm - CALCULUS B (Math. 102) 1. (a) D f = { x | tan...
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102-2008&2009-1-M10-NOVEMBER2009 - Kuwait...

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