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102-2008&amp;2009-2-F10-June2009

# 102-2008&amp;2009-2-F10-June2009 - Kuwait University...

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Kuwait University MATH 102 June 14, 2009 Math. Dept. Final Exam Duration: two hours Calculators and mobile telephones are not allowed. Answer the following questions. 1. (4 pts) Let f ( x ) = e 2 x + e x , x 0. (a) Show that f is one-to-one. (b) Find the inverse function f - 1 . (c) Find the domain of f - 1 . 2. (2 pts) Evaluate the limit lim x π 2 - sin - 1 (cos x ) ln(sin x ) 3. (4+4+4 pts) Evaluate the following integrals ( a ) Z sin(3 x ) 2 + 3 cos 2 (3 x ) dx ( b ) Z x p 1 + x dx ( c ) Z e - x 2 ln( e x + 1) dx 4. (3 pts) Determine whether the following improper integral is convergent or divergent, if convergent, find its value Z 2 x ( x 2 - 4) 2 dx 5. (4 pts) Find the centroid of the region bounded by the curves y = x 2 and x = y 2 . 6. (4 pts) Consider the circle r = sin θ. (a) Find the intersection points of the circle and the line θ = π 4 . (b) Find the points on the circle, where the tangent line is parallel to the line y = x . 7. (3 pts) Find the length of the polar curve r = θ 2 , 0 θ π.

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102-2008&amp;2009-2-F10-June2009 - Kuwait University...

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