102-2008&amp;2009-2-M10-April2009(1)

102-2008&amp;2009-2-M10-April2009(1) - KUWAIT...

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KUWAIT UNIVERSITY April 12, 2009 Math-102 Duration : 90 minutes First Midterm ————————————————————————————————- Calculators and mobile phones are NOT allowed. Answer all of the following questions. 1. (3 pts) Use logarithmic diﬀerentiation to ﬁnd dy dx if y = sin - 1 ( x 2 ) 3 x + 1 ( x + 1) - x cosh x 2. (3 pts) Find the exact value of cos ± 2 tan - 1 ± 1 3 - π 2 3. (3 pts) Show that cosh(ln(tan x )) = csc(2 x ) , for 0 < x < π 2 4. (4+3+3 pts) Evaluate the following integrals (i) Z ± 2 e x 4 + e 2 x + e 2 x 4 + e 2 x dx (ii) Z (coth x ) ln(sinh x ) dx (iii) Z dx 9 x - 4 5. (3 pts) Let g be the inverse function of f and h ( x ) = xg 2 ( x ). If f (1) = 3 and f 0 (1) = 2 then ﬁnd h 0 (3). 6. (3 pts) Solve for x , log 1 e (4 - 3 x ) < 1 1

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Solution key - First midterm - CALCULUS B (Math. 102) 1. ln y = ln(sin - 1 ( x 2 )) + 1 2 ln(3 x + 1) + x ln( x + 1) - lncosh x y 0 = 1 sin - 1 ( x 2 ) 2 x 1 - x 4 + 1 2 · 3 x ln3 3 x + 1 + ln( x + 1) + x x + 1 - sinh x cosh x · y 2. u = tan - 1 ± 1 3 tan u = 1 3 cos u = 3 10 , sin u = 1 10 cos ± 2tan - 1 ± 1 3 - π 2 = cos 2 u - π 2 · = sin(2 u ) = 2sin
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This note was uploaded on 02/23/2010 for the course CAL B 0410102 taught by Professor Deparpment during the Spring '10 term at Kuwait University.

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102-2008&amp;2009-2-M10-April2009(1) - KUWAIT...

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