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CONCORDIA UNIVERSITY Department of Mathematics Statistics Course Mathematics Examination Final Instructors 205 Date April 2015 Number Sections All Pages 2 J Brody A Kratsios N Rossokhata Course Examiner C R Norton N Wei A Atoyan Special Instructions Only approved calculators are allowed Show all your work for full marks MARKS 12 1 a Sketch the graph of f x 4 x2 on the interval 2 2 and approximate the area between the graph and the x axis on 2 2 by the right Riemann sum R4 using partitioning of the interval into 4 subintervals of equal length b For the same f x 4 x2 write in sigma notation the formula for the right Riemann sum Rn with partitioning of the interval 2 2 into n subintervals of equal length and calculate cid 82 2 2 f x dx as the limit of Rn at n k2 n n 1 2n 1 k n n 1 n cid 80 n cid 80 NOTE you may need the formulas k 1 2 k 1 c Calculate the derivative of the function F x x3 sin x cid 82 0 6 t2e t2 dt Hint use the Fundamental Theorem of Calculus and di erentiation rules 10 2 Calculate the following inde nite integrals a x2 cos 2x dx b dx 12 3 Evaluate the following de nite integrals give the exact answers 9 x2 dx x a cid 90 3 cid 90 0 6 4 Find F t such that F cid 48 t sin3 t cos5 t and F 8 5 Evaluate the given improper integral or show that it diverges a x2e x3 dx b x x2 1 dx cid 90 0 x2 4 cid 90 x2 2 e cid 90 cid 17 1 0 b ln2 x dx cid 16 1 cid 90 2 0 MATH 205 Final Examination April 2015 Page 2 of 2 17 6 a Sketch the curves y 2x and y x and nd the area enclosed b Sketch the region enclosed by f x sin x and the x axes on the interval 0 and nd the volume of solid of revolution of this region about the axis y 1 c Find the average value of the function f x on the interval 0 4 x 1 2x 9 7 Find the limit of the sequence an or prove that the limit does not exist n 100 b an a an c an n en n 2 n ln n2 n 1 8 8 Determine whether the series is divergent or convergent and if convergent then absolutely or conditionally cid 88 n 2 cid 88 1 cid 88 n 0 cid 88 n 1 a 2 ln n n2 b 1 n 5n 1 1 3 10 9 Find the interval of convergence of the following series a 3x n n b x 1 3n n 8n 8 10 a Derive the Maclaurin series of f x x2 e2x HINT start with the series for ez where z 2x b Use di erentiability of power series to nd the sum F x within its radius of convergence cid 80 1 x 1 n n 5 Bonus question Assume we know that some power series S x about a 1 is convergent at x 2 Can we claim that the series S x is also convergent at x 3 Explain why we can if so or give a counter example if we cannot 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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