CONCORDIA UNIVERSITY Department of Mathematics & Statistics Course Number Sections Mathematics 203 All Examination Date Pages Fina December 2019 3 Instructors: A. Danielski, L. Dube, I. Gorelyshev, S. Ky Course Examiners E. Mazzeo, R. Mearns, E. Mireku, J. Num A. Atoyan and H. Proppe Special Only approved calculators are allowed Instructions: Show all your work for full marks. MARKS [13] 1. (a) Solve for a: log2(22 - 4) - 210g2(2 + 2) = -1. (b) Let f(x) = x2 + 2 and g(x) = v2 - x. Find h = fog, and determine the domain and the range of h(x). 3 (c) Given the function f = 64 x+1, find the inverse function f-1, and determine the domain and the range of f-1. [8] 2. Find the limit if it exists, otherwise explain why it does not exist: lim 1202 - 42 - 51 (b) lim Vx2 +8 -3x x-+5 202 - 25 2--1 Vx2 + 15 + 4x [5] 3. Find (a) all horizontal, and (b) all vertical asymptotes of the function f(ac) = 14x + 31 (2202 + 1) (22 - 16) (2x + 1) [12] 4. Find the derivatives of the following functions (for full marks you have to show at least one intermediate step of your calculations): (a) f(x) = In (- e 2 23 Vx+ 3 ( b ) f ( 20 ) = - e- sec x 1 + ex (c) f(x) = Infersina + x sin(e?)] (d) f(x) = (1 + cosx) & cos x [5] 5. Calculate the second derivative f"(x) of the function f(2) = (23/2 + 2-1/2) va arctanx, and find the exact value of f"(1).MATH 203 Final Examination December 2019 Page 3 of 3 [13] 11. Given the function f(x) = ln(2 + :172). (a) Calculate f' (:17) and use it to determine intervals where the function is increasing, intervals where it is decreasing, and the local extrema on the x-axis where f(:1:) has local maximum or local minimum. (b) Calculate f\" (:15) 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 (:17) using the information obtained above. [5] Bonus Question. Is it possible to have a differentiable function f(:1:) such that f(0) = 0, f(2) = 4, and f'(:r)