102-2008&2009-3-F10-August2009

102-2008&2009-3-F10-August2009 - Kuwait University...

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Kuwait University Department of Mathematics and Computer Science Math 102 : Calculus II Summer Semester 2009 Final Examination Saturday 15 August 2009 Duration: 120 minutes Total marks: 40 Justify all your answers 1. Let f ( x ) = 3 x 3 ¡ 2 x ¡ 1 ¡ 1, x > 0. (a) Show that f is one-to-one on (0 ; 1 ). [1 mark] (b) Find the domain and range of f ¡ 1 . [1 mark] (c) Find ( f ¡ 1 ) 0 (0). [2 marks] 2. Consider f ( x ) = (1 ¡ x ) ln(2 x +1) . (a) Find the domain of f . [1 mark] (b) Find f 0 (1 = 2). [2 marks] 3. Find the value of a > 0 for which lim x !1 ± ax + 1 ax ¡ 1 x = 9. [4 marks] 4. Evaluate the following integrals. (a) Z ln 1 + p x · dx . [3 marks] (b) Z cos 3 x + 2cot x sin 2 x dx . [3 marks] (c) Z ( x 2 + 4 x + 3) ¡ 3 = 2 dx . [3 marks] 5. Test Z 1 1 1 x ( x 2 + 4) dx for convergence and evaluate it if it is convergent. [5 marks] 6. Consider the curve C parametrized by x = 1 2 ln(1 ¡ t 2 ) and y = arccos t for 0 t 3 = 4. (a) Find the length of C . [3 marks] (b) Find an equation of the tangent line at the point corresponding to t = 1 = p 2. [2 marks] 7. Find the centroid of the region bounded by the curves y = sec 2 x , y = 0, x = 0 and x = …= 4. [5 marks] 8. Find the area inside the polar curve r = 1 ¡ cos ± and outside the polar curve r = cos ± . [5 marks]
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Kuwait University Department of Mathematics and Computer Science Math 102 : Calculus II Summer Semester 2009 Final Examination ANSWERS 1. (a) f 0 ( x ) = 9 x 2 + 2 x ¡ 2 > 0 for x > 0. So f is increasing, and consequently one-to-one on (0 ; 1 ).
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102-2008&2009-3-F10-August2009 - Kuwait University...

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