Math 132 Spring 2009 Final Exam NAME STUDENT ID NUMBER This exam contains sixteen questions The first fourteen are multiple choice questions and count for five points each There is no partial credit on these questions so read each question carefully check your arithmetic and make sure that you have marked the answer you intended to mark The last two questions which are each worth fifteen points require written answers and some partial credit might be given However no credit will be given for information that is not germane to the problem at hand Please make sure to write your name and student ID number on the pages that include your answers to the last two questions In fact you will get one point on each of these two questions for writing your name and ID number legibly 1 1 Compute dx x lnx Z a cos ex C b ecos x C c x sin ex C d ln x C e ln x C f 1 ln x C g ln ln x C h 1 x C 2 2 Compute dy dx when y Z x2 0 a cos x b cos x c cos x 1 d x2 cos t e x2 cos x f 2xcos x g cosx C h cos x 1 3 cos t dt 3 Compute Z 0 x x sin dx 2 a 0 b 0 5 c 2 d 4 e f g 2 4 h 1 4 4 Expand a b c d e f g x 4 x 1 2 1 x 1 2 x 1 3 x 1 4 x 1 1 x 1 3 x 1 3 x 1 2 by partial fractions x 3 x 1 2 x 2 x 1 2 x 1 x 1 2 x x 1 2 3 x 1 2 h impossible since undefined at x 1 5 5 Evaluate Z 2 1 dx ln x a Converges by direct comparison with comparison function x12 b Diverges by direct comparison with the comparison function x12 c Converges by direct comparison with the comparison function ex d Diverges by direct comparison with the comparison function ex e Converges by direct comparison with the comparison function x1 f Diverges by direct comparison with the comparison function x1 g Converges by direct comparison with the comparison function e x h Diverges by direct comparison with the comparison function e x 6 6 The region between the curve y 2 x 0 x 1 and the x axis is revolved about the x axis to generate a solid Compute the volume of this solid a 2 b c 2 d 5 2 e 5 f 2 g 2 2 h 16 15 7 7 Suppose dy 1 y dx and y 0 1 Find y 4 a 0 b 2 c 3 d 4 e e4 f ln 4 g 1 4 h 1 4 8 8 Compute the sum of the series X n 1 1 1 cos cos n n 1 a diverges to b diverges to c diverges but not to or d 1 e 1 cos 1 f 0 g 1 h cos 1 9 9 Which of the following three series is convergent A X n 2 3 2 n 1 n B X 3 4 2 n 1 n 2n a A only b B only c C only d A and B only e A and C only f B and C only g all h none 10 C X n 1 n n4 5 2 10 If the Maclaurin series for xex is a 0 b 1 c 1 2 d 1 6 e 1 7 f 1 24 g 1 7 h 1 8 11 P cn xn find c7 11 Estimate Z 1 0 sin x2 dx with an error less than 0 001 a 13 42 b 1 3 c 1 42 d 1 1320 e 1 7 f 2 7 g 1 24 h 5 24 12 12 Compute e i 2 a 0 b 1 c 1 d i e i f e g e h 1 i 13 13 If the Binomial Series for the function f x 1 x2 4 3 is given by P cn xn find c4 a 1 b 0 c 1 d 1 2 e 3 4 f 2 7 g 4 3 h 2 9 14 14 Find the area inside one leaf of the four leaved rose r cos 2 a 8 b 4 c 3 d 2 e f 0 5 g 1 5 h 2 5 15 Name Student ID 15 Find the Taylor Series generated by f x sin x at a 2 Show your work Be sure to include the general term of the series 16 Name Student ID 17 Name Student ID 16 Identify the symmetries of the curve r 1 cos and sketch this curve Show all work 18 Name Student ID 19
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