CONCORDIA UNIVERSITY Department of Mathematics & Statistics Course Mathematics Examination Final Instructors: Special Instructions: Number 205 Date April 2016 A. Boyarsky, E. Duma, E. Ghandimi, J. Harnad, K. Lagota, U. Mgbemena Only approved calculators are allowed Show all your work for full marks Sections All Pages 2 Course Examiners A. Atoyan & H. Proppe [MARKS ] [10] 1. (a) Sketch a graph of the function ( 4 x2 2 x 0 f (x) = 2 2|x 2| 0 < x 4 R4 on the interval 2 x 4 and calculate the definite integral f (x) dx 2 in terms of area (do not antidifferentiate). (b) Use the Fundamental Theorem of Calculus to calculate the derivative of Rx2 sin( t) F (x) = e dt, x and determine whether F is increasing or decreasing at x = 1. [6] 2. Find F (x) such that F 0 (x) = x2 + 2x and F (0) = 0. x2 + 4 [11] 3. Find the following indefinite integrals: Z 13 x (a) dx (b) 2 x x6 Z x3/2 ln2 (x) dx [12] 4. Evaluate the following definite integrals (give the exact answers): Z1 (a) 0 2x dx 4x + 1 (b) Z2 4 x2 dx . 1 [8] 5. Evaluate the given improper integral or show that it diverges: Z Z1 dx dx (a) (b) 2 x ln(x ) (1 x)3/4 e 0 MATH 205 Final Examination April 2016 Page 2 of 2 [18] 6. (a) Sketch the curves y = x (3 x2 ) and y = x, and find the area enclosed by the two curves. (HINT: find first the points of intersection of the curves.) (b) Sketch the region enclosed by y = cos(2x) and the x-axis on the interval [0, 2 ], and find the volume of revolution of this region about the axis y = 1. \u0003 \u0002 (c) Find the average value of the function f (x) = sec4 (x) on the interval 4 , 4 . [9] 7. Find the limit of the sequence {an } as or prove that it does not exist: en n3 (1)n n (b) a = (c) an = ln(n + 2n2 ) ln(2n + n2 ). n 2 3n 1 + 4n [12] 8. Determine whether the series is divergent or convergent, and if convergent, then whether absolutely or conditionally convergent: (a) an = (a) X n2/3 1 + 2n n=1 (b) X n+1 (1) n=1 \u0012 \u0013 1 sin n (c) X 2n+1 (1)n+1 n! n=1 [6] 9. Find (a) the radius and (b) the interval of convergence of the series (x + 2)3n P n2 8n n=1 [8] 10. (a) Derive the Maclaurin series of f (x) = x2 e3x . (HINT: start with the series for ez and then let z = 3x). (b) Find the values of x for which the following series converges X (x2 + 1)n n=0 2n+1 and, for these values of x, find the sum of the series as a function of x. Bonus question [5]. Find the values of p (if any) for which the series P 1 is convergent p n=5 n ln n(ln (ln n)) 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