CONCORDIA UNIVERSITY Department of Mathematics & Statistics Course Number Mathematics 205 Examination Date Final April 2017 Instructors: J. Bernatska, A. Boyarsky, L. Colo, S. Link, E. H.-M. Ng Special Only approved calculators are allowed. Instructions: Show all your work for full marks. Sections All Pages 2 Course Examiners A. Atoyan, H. Proppe MARKS [12] 1. (a) Sketch the graph of f (x) = 4 x2 on the interval [1, 2], and approximate the area between the graph and the x-axis on [1, 2] by the left Riemann sum L3 using partitioning of the interval into 3 subintervals of equal length. (b) For the same f (x) = 4 x2 , write in sigma notation the formula for the left Riemann sum Ln with partitioning of the interval [1, 2] into n subintervals R2 of equal length, and calculate 1 f (x) dx as the limit of Ln at n n n P P k 2 = n(n+1)(2n+1) , . NOTE: you may need the formulas k = n(n+1) 2 6 k=1 k=1 (c) Calculate the derivative of the function F (x) = sec(3x) + tan(3x) R 2 et dt 0 (Hint: use the Fundamental Theorem of Calculus and differentiation rules.) [12] 2. Evaluate the following definite integrals (give the exact answers): (a) Z3 x 9 x2 dx Ze (b) 0 [6] ln2 x dx 1 3. Find F (t) such that F 0 (t) = sec4 (t) and F \u0010 \u0011 4 = 0. [10] 4. Calculate the following indefinite integrals: Z (a) (x2 2x) sin(2x) dx (b) Z x2 + 3 dx x2 3x [8] 5. Evaluate the given improper integral or show that it diverges: Z Z1 x 3 (a) dx x2 ex dx (b) 2 x 1 0 0 MATH 205 Final Examination [17] 6. (a) Sketch the curves y = April 2017 Page 2 of 2 2x and y = x and find the area enclosed. (b) Sketch the region enclosed by the parabola x = y 2 + 1 and the line x = 5 and find the volume of the solid obtained by revolution of this region about the line x = 5. (c) Find the average value of the function f (x) = x 1 + 2x on the interval [0, 4]. [9] 7. Find the limit of the sequence {an } or prove that the limit does not exist: (a) an = 3n n 22n (b) an = ln(n3 ) n+1 (c) an = n + 100 n [8] 8. Determine whether the series is divergent or convergent, and if convergent, then absolutely or conditionally : (a) X ln n (1)n n n=2 (b) X n + 100 (1)n+1 100n + 1 n=0 [10] 9. Find the radius and the interval of convergence of the following series (a) X (3x)n 1 n! (b) X (x + 1)3n n=1 n 8n [8] 10. (a) Derive the Maclaurin series of f (x) = x3 ln(1 + 2x2 ) (HINT: start with the series for ln(1 + z) where z = 2x2 ). (b) Use differentiability of power series to find the sum (x 1)n P within its radius of convergence. F (x) = n 1 [5] Bonus Question. A solid is generated by rotating about the x-axis the region enclosed between the curve y = f (x) and x-axis on the interval [0, b], where f is a positive function and x 0. For all values of b 0 the generated solid has the volume b4 . Find the function f . 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