101-2008&2009-1-F10-January2009

101-2008&2009-1-F10-January2009 - Kuwait University...

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Kuwait University Math 101 Date: January 20, 2009 Dept. of Math. & Comp. Sci. Final Exam Duration: Two Hours Calculators, mobile phones, pagers and all other mobile communication equipments are not allowed Answerthefollowingquestions. Eachquestionweighs 4 points. 1. Evaluatethefollowinglimits,iftheyexist: (a) lim x 2 parenleftbigg 1 x 2 4 x 2 4 parenrightbigg (b) lim θ 0 2sin θ sin2 θ θ 2 2. Let f ( x )= x 2 k x 2 +1 ,if x 0 , x 3 k +1 x 2 +2 ,if x< 0 . (a) Findthevalueof k suchthat f iscontinuousat x =0 . (b) Is f continuousat x =3 ? Justifyyouranswer. 3. Find f (1) , where f ( x )= 1+ x 2 x 3 x 4 +1 . 4. Find all numbers c that satisfy the conclusion of the Mean Value Theorem, of the function f ontheinterval [ 1 , 2] , where f ( x )= x 3 2 x. 5. Showthat: integraldisplay sin 3 x cos xdx = 2 7 (cos x ) 7 / 2 2 3 (cos x ) 3 / 2 + C. 6. Evaluate: 3 integraldisplay 0 3 xdx + 2 integraldisplay 2 x 3 cos xdx. 7. Findthe x -coordinatesofthepointsofinflectionofthecontinuousfunction: f ( x )= x 2 integraldisplay 5 1 3+ t 2 dt + 7 integraldisplay 1 3+ t 2 dt. 8. Showthat: 12 3 integraldisplay 0 x 2 +16 dx 15 .
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