CONCORDIA UNIVERSITY Department of Mathematics & Statistics Course Number Sections Mathematics 203 All Examination Date Pages Final April 2019 3 Instructors: A. Butaev, C. David, U. Mgbemena, B. Rhodes Course Examiner N. Rossokhata, F.-X. Sanchez, A. Sen A. Atoyan & H. Proppe Special Only calculators approved by the Instructions: Department are allowed Show your work for full marks MARKS [9] 1. (a) Solve for a : 327 + 2 . 3+1 = 42 (b) Given the function f(x) = In(1 + e2x), find the inverse function f-1(x), the range of f(x) and the range of f-1(x). [12] 2. Evaluate the limit if it exists, otherwise explain why the limit does not exist. 1202 - 41 1 + x + 2x3 (a) lim - 2 - 2 202 + 2 - 6 20- - 00 (b) lim (vx2 + 5x + 1+ x) (c) lim In- x (3 + 22 + 202 ) [6] 3. Find (a) all horizontal, and (b) all vertical asymptotes of the function f (20) = 3241 + 2. 4x 4x - 16 [15] 4. Find the derivatives of the following functions: (a) f(x) = 201/2 (Vx- 2-3/2) 230 (b) f ( 2) = In(act . Vx+ 3 ) + Ine? (c) f (20) = 7 arctan (2x) 1 + tan(x) (d) f(x) = sin[vx2 + 1 . cos(e")] (e) f(x) = (1+2.)25 ( use logarithmic differentiation)MATH 203 Final Examination April 2019 Page 2 of 3 [6] 6. [9] 7. (a) Verify that the point (2,1) belongs to the curve dened by the equation my + 2V 3 + y2 = :173 2, and nd the equation of the tangent line to the curve at this point. (b) The length of a rectangle is increasing at the rate of 8 cm/s and its width is increasing at the rate of 5 cm/s. When the length is 20 cm and the width is 12 cm, how fast is the area of the rectangle increasing at that instant? 2 I l (c) Use the 17Hopital's rule to evaluate the lim 6. 23%0 cos(2:z7) 1 Let f(:17)=3lxl32172:1:5 (a) Find the slope m of the secant line joining the points (0, f(0)) and (3, f(3)). (b) Find all points :1: = c (if any) on the interval [0,3] such that f'(c) = m. Consider the function f (:17) = \\/ 251: + l. (a) Use the denition of the derivative to nd the formula for f'(:1:) (b) Write the linearization formula for f at a = 4 (c) Use this linearization to approximate the value of f (3) = W :1: :132:1:+1 (b) A box with a square base is to be constructed with a volume of 54 m3. (a) Find the absolute extrema of f(:1:) = on the interval [0, 3]. The material for the box costs $2/m2, and the material for the top costs $6/m2. Find the dimensions that minimize the cost of the box. (a2 + $2)2 $3 (e) Let f(x) = where a is a real number. Find f\"'(1)