please solve and show all work and explain, thank you so much!

CONCORDIA UNIVERSITY Department of Mathematics & Statistics Course Number Sections Mathematics 203 All Examination Date Pages Final December 2022 3 Instructors: A. Atoyan, S. Giovanniello, A. Hindawi, S. Jamali, Course Examiner R. Mearns, U. Mgbemena, A. Panait A. Atoyan Special Only approved calculators are allowed Instructions: Show all your work for full marks. MARKS [13] 1. (a) Solve for a: 2 log2(x - 1) - 10g2(2 + 2) = 2. (b) Let f(x) = 20"-1 and g(x) = v4 -x2. Find h = fog, and determine the domain and the range of h(x). (c) Given the function f(x) = 2et/(1 + e"), find the inverse function f-1, and determine the domain and the range of f-1. 7] 2. Find the limit if it exists : (a) lim 22 - 2x - 3 1 202 - 91 (b) lim (2x - V4x2 - 12x + 1 ) [6] 3. Find (a) all horizontal, and (b) all vertical asymptotes of the function f(2) = V926 + 323 + 323 (220 + 1) 2 (20 + 2 ) [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) = (1+ 2.x) x 1 + tan x (b) f ( xc ) = (e 3 x 3 + 30 ) In v 1 + x (c) f(x) = cos(x2 + cos(x2 + cos(x2))] (d) f(x) = (1 + x2) arctanx [6] 5. Calculate the first derivative f'(x) and the second derivative f"(x) of the function f(x) = x (x + ex) where b is a parameter (i.e. a real number), and then find the exact value of f"(1) as an expression of b.MATH 203 Final Examination December 2022 Page 2 of 3 [4] 6. Consider the following piecewise-dened function: sina17 if x0 (a) For what values of a, if any, is the function f(17) continuous at :1: = 0? (b) Is there a value of the parameter a that makes f differentiable at every :17? 6 7. Consider the function 2 arctan 2:5 . y (a) Find the linearization L(:17) of the function y(:1:) at a = 1/2. (b) Find the differential dy and evaluate it for the values :17 = 0 and d3; = 0.1 . [7] 8. Let f(:17) = x3 21711. (a) Find the slope m of the secant line joining the points (2, f(2)) and (0, f(0)). (b) Find all points a: = c (if any) on the interval [-2,0] such that the rate f'(c) of instantaneous change of f is equal to the slope m of the secant line in (a). [12] 9. (a) Verify that the point (1,2) belongs to the curve dened by the equation y3 :1: y = 2 + 2V3 + x2, and nd an equation of the tangent line to the curve at this point. (b) At 1 PM, ship A is 5 kilometers strictly to the west of ship B. Ship A is sailing west at speed 20 km/hour and ship B is sailing north at 30 km/ hour. How fast (in km / hour) is the distance between ships changing at 3PM? [14] 10. (a) Find the point (:150, go) on the curve y = 2\\/:E that is closest to the point (3,0). (b) A box with a square base is to be constructed with a volume of 50 m3. The material for the bottom and the sides of the box costs $2/m2, and the material for the top costs $5/m2. Find the dimensions that minimize the cost of the box. 2 cos (c) Use l'Hopital's rule to evaluate the lim 5172(93) . 2:40 63: 1