102-2008&2009-2-M20-May2009

102-2008&2009-2-M20-May2009 - Kuwait University...

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Kuwait University Math 102 May 14, 2009 Dept. of Math. and Comp. Science Second Exam Duration: 90 minutes Calculators and mobile phones are not allowed. Answer all of the following questions. 1. Evaluate the following. [1+3 pts.] (a) lim x →∞ tan - 1 x 1 + x 2 (b) lim x →∞ ± x + 2 x + 3 2 x - 1 2. Evaluate the following. [3.5 pts. each] (a) Z x 2 (9 - x 2 ) 3 / 2 dx (b) Z (sec 6 x ) tan xdx (c) Z sin2 x 5 - cos x dx (d) Z 2 - 3 x 2 (2 x + 1)( x 2 + 2 x + 2) dx (e) Z dx (1 + cos x )sin x (f) Z x (ln x + e x ) dx 1
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Kuwait University Math 102 Date: May 14, 2009 Dept of Math and Comp Sci Solutions to Second Midterm 1. (a) Since lim x →∞ tan - 1 x = π/ 2 and lim x →∞ 1 + x 2 = , it follows that lim x →∞ tan - 1 x 1 + x 2 = 0. (b) lim x →∞ ln( f ( x )) = lim x →∞ (2 x - 1)ln ± x + 2 x + 3 = lim x →∞ ln[( x + 2) / ( x + 3)] (2 x - 1) - 1 which has the indetermi- nate form 0 / 0. Applying L’Hospital’s Rule and simplifying yields lim x →∞ ln( f ( x
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102-2008&2009-2-M20-May2009 - Kuwait University...

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