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Unformatted text preview: April 2008 Mathematics 101 Page 2 of 11 pages Marks [21] 1. ShortAnswer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for correct answers placed in the box, but at most 1 mark will be given for incorrect answers. Unless otherwise stated, simplify your answer as much as possible. (a) Evaluate Z x 3 2 x √ x dx . Answer (b) Evaluate Z π (4 sin θ 3 cos θ ) dθ . You must simplify your answer completely . Answer (c) Express lim n →∞ 1 n n X i =1 1 1 + ( i/n ) 2 as a definite integral. Do not evaluate this integral. Answer Continued on page 3 April 2008 Mathematics 101 Page 3 of 11 pages (d) Write down the Trapezoidal Rule approximation T 3 for Z 4 1 x cos( π/x ) dx . Leave your answer expressed as a sum involving cosines. Answer (e) Let f ( x ) = kx 2 (1 x ) if 0 ≤ x ≤ 1 and f ( x ) = 0 if x < 0 or x > 1. For what value of the positive constant k is f ( x ) a probability density function? Answer (f) Find the first three nonzero terms in the powerseries representation in powers of x (i.e. the Maclaurin series) for Z x t 1 t 8 dt . Answer (g) Let f ( x ) = Z x 3 x √ t sin t dt . Find f ( x ). Answer Continued on page 4 April 2008 Mathematics 101 Page 4 of 11 pages FullSolution Problems. In questions 2–7, justify your answers and show all your work . If a box is provided, write your final answer there. Unless otherwise indicated, simplification of answers is not required....
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 Spring '10
 AKOS
 Math, Maclaurin, Toricelli, University of British Columbia Sessional Examinations

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