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Sections All Pages 2 Course Examiner A Atoyan and H Proppe x2 cid 82 1 x CONCORDIA UNIVERSITY Department of Mathematics Statistics Course Mathematics Examination Final Instructors December 2016 205 Date Number M Bertola J Brody A Conti K Lagota Special Instructions MARKS 10 1 a Sketch the graph of the function Only approved calculators are allowed Show all your work for full marks f x 1 2x if x 1 1 x 1 x x 3 if x 2 if 1 x 2 3 cid 82 0 cid 90 and nd the de nite integral do not antidi erentiate f x dx in terms of area b Use the Fundamental Theorem of Calculus to calculate the derivative of F x 2 t2 dt cid 90 x2 3 x 2 3 2 2 cid 90 x a x2 cos 2x dx 0 16 2 Find the following inde nite integrals a dx b arcsin x dx c 12 3 Evaluate the following de nite integrals give the exact answers cid 90 x3 4 x3 4x dx 4 cid 90 0 16 x2 dx x b cid 17 cid 16 cid 90 2 6 8 4 Find F t such that F cid 48 t sin3 t cos5 t and F 0 5 Evaluate the given improper integral or show that it diverges 1 e cid 90 0 a dx x ln2 x ex ex 4 dx b 0 MATH 205 Final Examination December 2016 Page 2 of 2 3 x 16 6 a Sketch the curves y x and y 4 and nd the area enclosed b Find the volume of a solid obtained by rotating the region bounded by the curve y sin x and the x axis on the interval 0 about the axis y 1 c Find the exact average value of f x tan2 x on the interval 0 4 6 7 Find the limit of the sequence an at n or prove that it does not exist a an 3n 1 2 6n b an ln 1 2n2 2 ln 1 n 12 8 Determine whether the series is divergent or convergent and if convergent whether absolutely or conditionally 1 n 1 n3 n2 cid 88 n 0 3 n 5 en cid 88 n 2 sin n n2 a b c 6 9 Find a the radius of convergence and b the interval convergence of the series x 1 n n 1 2 n cid 19 8 10 a Use the integrability of the power series to express the function F x n t2n 1 dt as an elementary function n 1 cid 88 cid 80 cid 18 cid 80 n 0 x cid 82 0 n 1 i e sum the series for F x within the radius of its convergence b Find the MacLaurin series for the function x e x2 Hint start with the series for ez then replace z by x2 5 Bonus Question A solid is generated by rotating about the x axis the region under the curve y f x where f is a positive function and x 0 The volume generated by the part of the curve from x 0 to x b is b2 for all b 0 Find f x The present document and the contents thereof are the property and copyright of the professor s who prepared this exam at Concordia University No part of the present document may be used for any purpose other than research or teaching purposes at Concordia University Furthermore no part of the present document may be sold reproduced republished or re disseminated in any manner or form without the prior written permission of its owner and copyright holder

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