Math 132 Final Exam Fall 2016 20 multiple choice questions worth 5 points each Exam is comprehensive No calculators For the multiple choice questions mark your answer on the answer card Useful Formulas cid 80 n cid 80 n i 1 i n n 1 2 cid 16 n n 1 cid 17 2 i 1 i3 2 cid 80 n i 1 i2 n n 1 2n 1 6 sin2 cos2 1 1 tan2 sec2 1 cot2 csc2 sin A B sin A cos B sin B cos A cos A B cos A cos B sin A sin B tan A B tan A tan B 1 tan A tan B sin A sin B 1 2 cos A B cos A B cos A cos B 1 2 cos A B cos A B sin A cos B 1 2 sin A B cos A B sin2 x 1 2 1 cos 2x cos2 x 1 2 1 cos 2x sin 2 2 sin cos cid 82 csc x dx ln csc x cot x C cos 2 cos2 sin2 cid 82 sec x dx ln sec x tan x C Math 132 Final Exam Page 2 of 11 1 Find a power series representation for f x 2x 1 3x 2 centered at 0 n 1 n 1 n 1 cid 88 cid 88 cid 88 cid 88 cid 88 cid 88 cid 88 n 1 n 1 n 1 n 1 A B C D E F G 2 3n n 1 xn 3nnxn 1 2 3n n 1 xn 2 3n 1nxn 3n 1nxn 1 2 3n 1xn 1 2 3nx2n 2 Find the third degree Taylor polynomial T3 centered at x 2 for the function f x 1 1 x 3 1 27 x 2 2 1 A f x 1 x x2 x3 9 x 2 1 B f x 1 C f x 1 x 2 x 2 2 x 2 3 9 x 2 1 D f x 1 E f x 1 x x2 x3 9 x 2 2 F f x 1 27x2 1 9x 1 G f x 1 27 x 2 2 6 81x3 27 x 2 2 1 3 1 3 1 3 1 81 x 2 3 81 x 2 3 81 x 2 3 Math 132 Final Exam Page 3 of 11 3 Find the Taylor series for f x e3x centered at 1 n 0 n 0 n 0 cid 88 cid 88 cid 88 cid 88 cid 88 cid 88 cid 88 n 0 n 0 n 0 n 0 A B C D E F G 3ne3 x 1 n 3n x 1 n n n xn n 3ne3xn n x 1 n 3n 3e x 1 n n 3n 1e3 x 1 n n A 2 4 1 2 B C D 3 2 2 E 2 4 F 2 G 2 2 3 H The series diverges 4 Find the value of the series or conclude that it diverges 1 n cid 0 2 cid 1 2n 2 2n 1 cid 88 n 0 cid 0 cid 1 2 2 1 cid 0 cid 1 4 2 3 cid 0 cid 1 6 2 5 cid 0 cid 1 8 2 7 Math 132 Final Exam Page 4 of 11 5 The inde nite integral arctan x2 dx has a power series centered at 0 that looks like cid 90 C a1x a2x2 a3x3 a4x4 a5x5 where C is the constant of integration Find a3 a4 A 0 B 1 3 1 C 3 D 1 7 1 E 7 F 2 7 2 7 G cid 88 n 1 1 n x 4 n 2n 1 n 1 A 2 2 B 9 2 7 2 C 1 2 1 2 D 6 2 2 7 E 9 2 F 6 2 G 2 2 H 6 Find the interval of convergence of the power series Math 132 Final Exam Page 5 of 11 7 Determine the value of the series cid 18 2n 2 4 3 n 1 cid 19 4n cid 88 n 2 or conclude that it diverges A 0 B 152 21 C 12 7 D 11 4 E 5 4 F 184 21 G 17 7 H The series diverges cid 18 n cid 19 n 2n 1 cid 88 n 2 I A None of them B I only C II only D III only E I and II F I and III G II and III H All of them 8 Which of the following series converge cid 88 n 0 5n 4n2 II III cid 88 n 1 2n n 2 Math 132 Page 6 of 11 9 Which of the following alternating series converge conditionally but not absolutely Final Exam cid 88 n 2 1 n 1 ln n II cid 88 n 1 III cos n 2n en n 10 Which of the following sequences converge 4n ln n n4 1 cn 3n2 2n 5 6n2 4n 10 dn n 10n n10 cid 88 n 2 n 1 n n 1 I A None of them B I only C II only D III only E I and II F I and III G II and III H All of them an bn ln n n2 2 A an only B an bn only C cn dn only D an cn only E an bn dn only F an cn dn only G All of them Math 132 Final Exam Page 7 of 11 11 Evaluate the following improper integral or conclude that it diverges cid 90 2 A 0 ln x x2 dx B ln 2 2 1 2 C 2 ln 2 2 2 ln 2 2 ln 2 2 ln 2 D 2 E 1 F G 3 1 12 H The integral diverges 12 Evaluate the integral cid 90 arctan x dx A 1 x2 1 C arctan x 2 2 2 ln 1 x2 C B C C x arctan x 1 D arctan x C E x arctan x ln 1 x2 C F x x2 1 C arctan x 2 2 G 1 x2 1 C Math 132 Final Exam Page 8 of 11 13 Estimate 3 x2 dx using a left handed Riemann Sum with 4 subdivisions cid 90 2 0 26 5 31 5 26 4 31 4 26 3 31 3 31 2 A 0 B C D E F G H cid 90 0 x3 A 2x3 1 x6 B 6x2 1 9x4 C 3x2 D x3 1 36x4 2x3 F 1 x6 G 3x2 6x2 E 1 x6 1 9x4 14 Let F x 1 t2 dt Compute F cid 48 x Math 132 Final Exam Page 9 of 11 15 Assume f x is a continuous function satisfying x sin x f t dt cid 90 x 0 F 4 G H It is not possible to nd f 4 from the given information Find f 4 A 0 B 2 C 4 D 2 E 2 cid 90 2 A 0 B 1 …