Math 132 Final Exam Spring 2016 Name ID Number Section Number Section Instructor Day Time MWF 10 10 MWF 9 05 Nikolaou MWF 11 15 Nikolaou MWF 12 20 Zhao Zhao Wen Wen Yaping Lowell MW 2 30 MW 4 00 TuThu 8 30 TuThu 10 00 1 2 3 4 5 6 7 8 Section Instructor Day Time 9 10 11 12 13 15 16 Sunukjian TuThu 11 30 Benincasa TuThu 4 00 MWF 11 15 MWF 12 20 MWF 1 25 TuThu 11 30 TuThu 1 00 Farelli Bates Hart Le Johnson No calculators papers or notes may be used Please don t just give an answer Clearly explain how you get it provid ing appropriate mathematical details An answer with no support ing work will be awarded zero points This is a 2 hour exam Question MC Total Out of 25 6 Out of 20 7 Out of 15 8 Out of 20 9 Out of 20 Total Out of 100 a a a a Grade cid 107 b cid 107 b cid 107 b cid 107 b 1 Mutiple Choice Section Choose the one option that best answers the question There is no partial credit for questions 1 5 1 5 points Find a polar equation for the curve represented by the Carte 2 5 points Which of the following is the radius of convergence for the sian equation x 6 A r 6 tan B r 6 C r 6 cos D r 6 sec power series A 1 5 B 1 C 5 D A 3 x 3 3 x 1 B 1 3 C 3 x 3 3 x 1 D 1 3 cid 88 n 1 1 n xn n25n cid 88 n 1 xn 1 3n 2 3 5 points For which x values does the following series converge 4 5 points Consider the integral ln 2x dx Which of the following is cid 90 true A This integral does not exist B This integral can be found using integration by parts C This integral can be found using u substitution D This integral can be found using trigonmetric substitution 5 5 points Which of the following is a Polar representation of the Carte sian coordinate 2 A cid 17 cid 16 3 1 3 cid 19 cid 19 5 6 5 3 cid 18 cid 18 B 2 C 2 cid 16 D 2 cid 17 6 Please ll in your letter answer for questions 1 5 below 1 2 3 4 5 3 Free Response Portion Show all work for each of the following ques tions Partial credit may be awarded for questions 6 9 6 a 10 points Evaluate the integral cid 90 1 0 x3 1 x4 4 dx 31 20 4 6 b 10 points Represent the following function as a power series Express your answer in summation notation and simplify completely within the summation f x x2 1 4x3 2 cid 88 n 1 1 n 1 4n 1 n x3n 1 5 7 a 10 points Find the Taylor series for the function below Express your answer in summation notation and simplify completely within the sum mation f x ln 1 x Centered at a 1 ln 2 1 n 1 x 1 n 1 2n n cid 88 n 1 6 7 b 5 points The Maclaurin series of tan 1 x Determine the Maclaurin series for cid 88 n 0 1 n x2n 1 2n 1 Express your answer in summation notation and simplify completely within the summation f x 9x tan 1 4x3 9 cid 88 n 0 1 n42n 1 x6n 4 2n 1 7 8 a 10 points Find the radius and interval of convergence of the series Justify any test you use and be sure to verify any necessary conditions cid 88 n 0 1 n 5x n n 2 3 R I 1 5 cid 18 cid 21 1 5 1 5 8 8 b graph below 10 points Consider the polar equation r cos 5 given in the Calculate the area enclosed in one loop of the curve Mathematically justify how you nd the integral bounds 20 9 9 a 10 points Find the exact length of the parametric curve below x t2 y 2t 1 3 2 0 t 4 1 2 1 3 12 10 9 b 10 points Find the equation of the line tangent to the parametric curve given below at the given point Express your answer as y f x x sec t y tan t t 6 y 2x 3 11 This page is intentionally left blank for additional work 12