an eCONCORDIA EXAMINATION Department of Mathematics & Statistics Course Number MATH Section 209 Examination EC Date Final Pages August 2015 Instructors 3 Course Examiners Dr. Hal Proppe Dr. Hal Proppe Dr. Fred E. Szabo Dr. Fred E. Szabo Special Instructions Only approved calculators are allowed. . Justify all your answers. Evaluation . MARKS sh is ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co m . This examination counts for 50% towards your final grade 7 x2 . x 4x3 6x2 + 8 (b) Find the value of each of the following: [12] 1. (a) Find lim x2 25 x5 (x 5) x2 3x + a ii. lim where a is a real number. x4 (x 1) i. lim (c) Given lim f (x) = 5 and lim g(x) = 4, find x6 x6 i. lim [3g(x)] x6 Th ii. lim [f (x)]2 x6 iii. lim [g(x) f (x)] x6 f (a + h) f (a) , find the derivative if f (x) = 3+4x3 . h0 h [4] 2. Using the definition of the derivative lim https://www.coursehero.com/file/13349257/Math-209-Final-August-2015pdf/ MATH 209 EC Final Examination August 2015 Page 2 of 3 [17] 3. (a) If f (x) = e 4x17 5x3 , find f 0 (x). (b) If f (x) = (x3 + 2x2 1)(x2 9), find f 0 (x). (c) If f (x) = (7ex + 3)/(x2 4), find f 0 (x). dy 2 =? (d) If y = e(x 7x) , then dx (e) If y = ln(3x3 5)4 , then dy 1 , then =? dx x2 sh is ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co m (f) If y = 5 dy =? dx (g) Find y 0 given xy = e2y + . [8] 4. Does the line tangent to the graph of f (x) = ex at x = 1 pass through the origin? Are there any other lines tangent to the graph of f that pass through the origin? Explain. [5] 5. A cube with a side of 12 centimeters is coated with ice 0.1 centimeters thick. Use differentials to estimate the volume of the ice. [6] 6. The total profit (in dollars) from the sale of x lawn mowers is P (x) = 30x 0.03x2 750, 0 x 1000 (a) Find the average profit per mower if 50 mowers are produced. (b) Find the marginal average profit at a production level of 50 mowers, and interpret the results. Th (c) Use the results from parts (a) and (b) to estimate the average profit per mower if 51 mowers are produced. [5] 7. Suppose a point is moving along the graph of x2 + y 2 = 34. When the point is at (3, 5), its x coordinate is increasing at a rate of 0.6 units per second. How fast is the y coordinate changing at that moment? https://www.coursehero.com/file/13349257/Math-209-Final-August-2015pdf/ MATH 209 EC Final Examination August 2015 Page 3 of 3 [18] 8. Compute the following: R (a) x(3x2 + 5)dx R (b) x2 (2x3 + 7)18 dx 1 xdx +7 (c) R (d) R e35x dx (e) R (f) R (x + 3)4 dx. x2 sh is ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co m x dx. x+7 [8] 9. Evaluate the following integrals [give the exact answer]. (a) R3 (b) R3 2 0 2 ex xdx (t3 + 5)dt [5] 10. A note will pay $40, 000 at maturity 12 years from now. How much should you be willing to pay for the note now if money is worth 2.5%, compounded continuously? [5] 11. Find the area bounded by the graphs of y = x2 3x; y = 0, 2 x 2. [7] 12. Find the interval(s) on which the graph of f (x) = x4 2x3 is concave upward, the interval(s) on which the graph of f is concave downward, and the inflection point(s). Th The present document and the contents thereof are the property and copyright of the professor(s) 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 re-disseminated in any manner or form without the prior written permission of its owner and copyright https://www.coursehero.com/file/13349257/Math-209-Final-August-2015pdf/ Powered by TCPDF (www.tcpdf.org) holder