CONCORDIA UNIVERSITY Department of Mathematics & Statistics Course Mathematics Examination Final Instructors: MARKS [9] 1. Sections All Pages 3 Course Examiner A. Atoyan & H. Proppe sh is ar stu ed d vi y re aC s ou ou rc rs e eH w er as o. co m Special Instructions: Number 203 Date April 2014 D. Dryanov, L. Dube, H. Greenspan, M. Hadid, Z. Li Only calculators approved by the Department are allowed Show your work for full marks (a) Let f (x) = ln(1 + x2 ) and g(x) = 4 + x. Find f g and g f and determine the domain of each of these composite functions. (b) Find the range of the function f = e2x + 3, the inverse function f 1 , and the range of f 1 . [12] 2. Evaluate the limits Do not use l'Hpital rule: o x2 x2 + 5x 5 x (a) lim 2 (b) lim (c) lim ln x2 x + x 6 x1 x x1 [6] 3. Calculate both one-sided limits of f (x) = |x2 9| at the point(s) where x+3 the function f is discontinuous. Find the derivatives of the following functions: x7 + x5/2 (a) f (x) = x3 (b) f (x) = ln(x4 x + 3 ) + ln e Th [15] 4. (c) f (x) = arctan(2x) tan(x) (d) f (x) = sin[x2 cos(ex )] (e) f (x) = (1 + 2x)x 2 (use logarithmic dierentiation) https://www.coursehero.com/file/11440385/Math203-FinalExam-Winter-2014/ 1 + x + 2x3 3 + 2x + x3 MATH 203 [15] 5. (a) Final Examination April 2014 Page 2 of 3 Verify that the point (2,1) belongs to the curve dened by the equation xy + 2 3 + y 2 = x3 2, and nd the equation of the tangent line to the curve at this point. (c) [6] 6. Two cars start simultaneously moving away from the intersection of two orthogonal streets at the speeds v1 = 12 m/s going west, and v2 = 16 m/s going north. How fast is the distance between the cars increasing at the instant t = 5 seconds after they start moving from the intersection? sh is ar stu ed d vi y re aC s ou ou rc rs e eH w er as o. co m (b) x2 e 1 Use the l'Hpital's rule to evaluate the lim cos(2x)1 . o x0 Let f (x) = 3 + x + 3 x2 x3 . (a) Find the slope m of the secant line joining the points (0, f (0)) and (3, f (3)). (b) Find all points x = c (if any) on the interval [0,3] such that f (c) = m. [9] 7. Consider the function f (x) = 2x + 1. (a) Use the denition of the derivative to nd the formula for f (x). (b) Write the linearization formula for f at a = 4 (c) Use this linearization to approximate the value of f (3) = x on the interval [0, 3]. x2 x + 1 (b) A box with a square base is to be constructed with a volume of 54 m3 . 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. (a) Find the absolute extrema of f (x) = Th [12] 8. 7 https://www.coursehero.com/file/11440385/Math203-FinalExam-Winter-2014/ MATH 203 Final Examination April 2014 Page 3 of 3 [16] 9. Given the function f (x) = 2x2 x4 . (a) Find the domain of f and check for symmetry. Find asymptotes of f (if any). (b) Calculate f (x) and use it to determine intervals where the function is increasing, intervals where it is decreasing, and the local extrema (if any). sh is ar stu ed d vi y re aC s ou ou rc rs e eH w er as o. co m (c) Calculate f (x) 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). (d) Sketch the graph of the function f (x) using the information obtained above. Th [5] Bonus Question: Let f be a function which is monotonically decreasing (strictly) and dierentiable everywhere on the real axis. Let also g = x2 + 1. Prove that the composite function h = f g has one and only one local extremum and determine whether it corresponds to a maximum or minimum of h(x). https://www.coursehero.com/file/11440385/Math203-FinalExam-Winter-2014/ Powered by TCPDF (www.tcpdf.org)