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Concordia University Department of Mathematics Statistics Number 205 Date June 2015 Section AA Pages 2 Course MATH Examination Final Instructor Dr D Dryanov Instructions Evaluation out of 100 marks Answer the numbered questions The value of each question is indicated in square brackets Show all your steps Only approved calculators may be used This is a closed book exam Any outside materials books notes or recorded materials may not be used 7 marks 1 a Evaluate the de cid 28 nite integral cid 16 cid 90 2 2 cid 112 cid 17 1 4 x2 dx cid 90 1 x F x t cos t2 t2 1 dt by interpreting it in terms of signed area b Use Fundamental Theorem of Calculus Part 1 to evaluate the derivative F cid 48 x of the function and use it to determine whether F x is increasing or decreasing at x 1 10 marks 2 Find the antiderivative F x of the function f x that satis cid 28 es the given condition 15 marks 4 Evaluate the following de cid 28 nite integrals do not approximate give the exact value cid 90 cid 90 2 1 a f x xe x2 F 0 1 2 b f x x2 ln x F 1 8 9 15 marks 3 Find the following inde cid 28 nite integrals a 1 cos x 2 dx dx c x x2 2x 3 dx cid 90 ln x 3 x b cid 90 2 cid 90 cid 90 1 a e1 x x2 dx b 0 x2 cos 2x dx c dx x 2 1 0 10 marks 5 Evaluate the improper integral or show that it diverges cid 90 a 1 dx x ln2 x e b 3 1 x 1 dx cid 90 3 1 14 marks 6 a Sketch the curves y 5x x2 and y x cid 28 nd their points of intersection and then cid 28 nd the area of the region enclosed by the curves b Find the average value of the function f x sin2 x on the interval 0 2 1 cid 88 n 2 cid 88 n 1 1 n 1 n ln n x 1 n n 2n cid 90 x 0 cid 90 0 5 0 e t2 dt F x e t2 dt c Sketch the region bounded by the curves y x2 and x y2 Then cid 28 nd the volume of the solid obtained by rotating this region about the x axis 8 marks 7 Find the limit of the given sequence an n 1 2 or show that it does not exist n 10 n a an b an 1 4n2 n4 1 2n n 5 n 12 marks 8 Determine whether the given series is convergent or divergent and if convergent then is it convergent absolutely or conditionally cid 88 n 1 2 cos n n a b c cid 16 e 3n e 3 n 1 cid 17 cid 88 n 1 9 marks 9 a Find the radius and the interval of convergence of the power series b Derive the MacLaurin series of the function Then use the MacLaurin polynomial of degree 7 for F x in order to approximate the exact value of the de cid 28 nite integral 5 marks Bonus question 1 Show that the area of the region below the graph of f x e and above the x axis from x 0 to x 1 is equal to the area of the region below the graph of g x esin x sin 2x and above the x axis from x 0 to x 2 x 5 marks Bonus question 2 Verify that f x sin 3 to show that 0 sin 3 x dx 1 cid 90 2 1 x is an odd function and use this fact cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 cid 22 The present document and the contents thereof are the property and copyright of 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 2

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