CONCORDIA UNIVERSITY Department of Mathematics & Statistics Course Number Sections Mathematics 203 All Examination Date Pages Final December 2018 3 Instructors: J. Bernatska, A. Butaev, L. Dube, Course Examiners R. Mearns, E. Mireku, M. White A. Atoyan and H. Proppe Special Only approved calculators are allowed Instructions: Show all your work for full marks. MARKS [11] 1. (a) Solve for x (find the exact value, do not approximate): e 20 = 2ex + 24. (b) Let f(x) = In(4-x2) and g(x) = V1-x. Find fog and determine its domain. (c) Find the inverse function f-1 of f(x) = log2(4 + 2"), and determine the domain and the range of f-1(x). [8] 2. Find the limit if it exists, otherwise explain why it does not exist: Do not use L'Hopital's Rule x - 2| (2 + 3) x - 1 (a) lim 2 2 + 2 - 6 (b ) x- 1 3 - Vx2 + 8 [5] 3. Find all horizontal and vertical asymptotes of the function f(ac) = V16x2 + 1 20 2 2 - 16 x+4 [12] 4. Find the derivatives of the following functions (you don't need to simplify your final answer, but you must show how you calculate it): (a ) f (ac ) = zee" +er ( b ) f (20 ) = " x - arctan(x) 1 + x2 (c) f(x) = cos[x sin(x) + sin(x + cos x)] (d) f(x) = (1 + x2) tan(x) (use logarithmic differentiation) [4] 5. Calculate the second derivative f"(x) of the function f(x) = (x+ a) (x-a)x2 where a is a parameter, and find f"(1).MATH 203 Final Examination December 2018 Page 3 of 3 [14] 10. Given the function f(x) = 2x3 21:2:2 + 36:1: 9. (a) Calculate f' (:17) and use it to determine intervals where the function is increasing, intervals where it is decreasing, and all critical numbers on the :r-axis where f(:1:) has local maximum or local minimum. (b) Calculate f\" (:17) and use it to determine intervals where the function is concave upward, intervals where the function is concave downward, and the inection points (if any). (c) Sketch the graph of the function f(x) using the information obtained above. [5] Bonus Question. Show that the equation arctan(x) = 0 has no solution other 1 + :172 than :1: = 0. (HINT: use Rolle's Theorem.) The present document and the contents thereof are the property and copyright of the professor(s)n 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 redisseminated in any manner or form without the prior written permission of its owner and copyright holder