2008_solutions - Name: April 2008 Marks [21] 1....

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Name: April 2008 Mathematics 101 Page 2 of 11 pages Marks [21] 1. Short-Answer 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 diFculty. ±ull 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 x 3 2 x x dx . Answer (b) Evaluate π 0 (4 sin θ 3 cos θ ) . You must simplify your answer completely . Answer (c) Express lim n →∞ 1 n n i =1 1 1 + ( i/n ) 2 as a de²nite integral. Do not evaluate this integral. Answer Continued on page 3
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April 2008 Mathematics 101 Page 3 of 11 pages (d) Write down the Trapezoidal Rule approximation T 3 for 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 ±rst three nonzero terms in the power-series representation in powers of x (i.e. the Maclaurin series) for
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This note was uploaded on 10/26/2010 for the course MATH MATH100 taught by Professor Akos during the Spring '10 term at The University of British Columbia.

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2008_solutions - Name: April 2008 Marks [21] 1....

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