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MATHEMATICS 184 (Section 922) - FINAL EXAM NAME: STUDENT ID NUMBER: SIGNATURE: INSTRUCTIONS: No notes or books are to be used. Calculators are allowed. No credit will be given for the correct answer without the (correct) accompanying work. Use the back of the pages if you need extra space. Rules Governing Formal Examinations The following are the rules governing formal examinations: 1. Each candidate must be prepared to produce, upon request, a Library/AMS card for identification. 2. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions. 3. No candidate shall be permitted to enter the examination room after the expiration of one-half hour from the scheduled starting time, or to leave during the first half hour of the examination. 4. Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. *Having at the place of writing any books, papers or memoranda, calculators, computers, audio or video cassette players or other memory aid devices, other than those authorized by the examiners. *Speaking or communicating with other candidates. *Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness shall not be received. 5. Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take any examination material from the examination room without permission of the invigilator. 1.(a) Find the Taylor Series about x = 0 for the function f (x) = sin2 (x) Write down at least the first 4 non-zero terms in the series. Hint: Use a trigonometric identity first. [4] (b) Find the Taylor Series about x = 0 for the function f (x) = 1 p 1 + x3 Write down at least the first 4 non-zero terms in the series. [4] 2. Use Newton’s Method with an initial approximation of x0 = −π 2 to find, correct to 5 decimal places, an intersection point of the curves f (x) = 2cos(x) and g(x) = 1 − x [10] 3. Find the points on the curve x2 y 2 = (y + 1)2 (4 − y 2 ) at which the tangent line is horizontal. [10] 4. For the function f (x) = cos2 (x) + 1 (a) Identify the domain and the range. [2] (b) Find the coordinates of the x and y-intercepts. [2] (c) Find all the critical points and corresponding critical values. [4] (d) What are the coordinates of the local maxima and minima? (Indicate which ones are local maxima and which are local minima). Show your work. [3] (e) Find the coordinates of any inflection points. [4] (f) Using the information from parts (a)-(e), sketch a graph of this function. Warning: Your sketch must match the information you determined above. No marks will be awarded for a correct sketch that does not match your results from (a)-(e). [4] 5. Differentiate the following functions: (a) f (x) = x2 sin(x) (b) y = [2] 2x [2] x3 +7 (c) g(x) = arccos(x2 − 1) (Simplify your answer). [3] (d) z = (x5 −6x2 )3 (3x3 +8x−6)4 2 (Simplify your answer). [3] (e) f (p) = p3 ep + pln(p) [3] (f) y = xx [3] 6. Utopia’s national bank offers an annual percentage interest rate of 6%, compounded monthly (12 times a year). The local credit union in competition with the national bank advertises that it will pay an annual percentage interest rate of 5.8% compounded continuously. Determine which (the bank or the credit union) would give you the best return on an investment. [10] 7. The amount of time T , in minutes, it takes me to get to work in the morning varies according to the equation πl T (l) = 45 + l + 15sin( ) −5≤l≤5 4 where l is the number of minutes past 8:30 a.m. that I leave my house. (I think of l as the number of minutes I am running late). (a) According to this equation, how many minutes should I leave after 8:30 a.m. in order to minimize my travel time? (Round your answer to the nearest minute). (b) If I leave the number of minutes after 8:30 a.m. you found in part (a), how long will it take me to get to work? [6] [2] 8. A certain ball has the property that each time it falls from a height h onto a hard, level surface, it rebounds to a height of 0.9h. Suppose the ball is dropped from an initial height of h = 10 meters. (a) Assuming that the ball continues to bounce indefinitely, find the total distance (in meters) it travels vertically. Be sure to count both the distance travelled up and down. [4] (b) If the ball is stopped bouncing after 10 bounces (it is caught at the top of the bounce) what is the total vertical distance the ball has travelled? [3] (c) What would the height of the 11th bounce have been? [3] 9. A reasonably realistic model of a firm’s costs is given by the short-run Cobb-Douglas cost curve 1 C(q) = Kq a + F where a is a positive constant, F is the fixed cost, and K measures the technology available to the firm. (a) Show that C is concave down if a > 1. [4] (b) Assuming that a < 1 and that average cost , C(q) , is minimized q when average cost equals marginal cost, find what value of q minimizes the average cost. [5]