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 Pn i 1 Pn i 1 i3 Pn n n 1 2 i n n 1 2 i 1 2 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 cos A cos B R 1 2 tan A tan B 1 tan A tan B cos A B cos A B sin A sin B 1 2 cos A B cos A B sin A cos B 1 2 sin A B cos A B sin2 x 12 1 cos 2x cos2 x 21 1 cos 2x sin 2 2 sin cos cos 2 cos2 sin2 csc x dx ln csc x cot x C R sec x dx ln sec x tan x C Math 132 Final Exam 1 Find a power series representation for f x A B C D E X n 1 X n 1 X n 1 X n 1 X Page 2 of 11 2x centered at 0 1 3x 2 2 3n n 1 xn 3n nxn 1 2 3n n 1 xn 2 3n 1 nxn 3n 1 nxn 1 n 1 F G X n 1 X 2 3n 1 xn 1 2 3n x2n n 1 2 Find the third degree Taylor polynomial T3 centered at x 2 for the function f x 1 1 x A f x 1 x x2 x3 B f x 1 3 91 x 2 1 x 27 2 2 1 x 81 3 2 3 1 x 81 2 3 6 x 81 2 3 C f x 1 x 2 x 2 2 x 2 D f x 1 3 19 x 2 1 x 27 2 2 E f x 1 x x2 x3 F f x G f x 1 3 1 3 91 x 2 91 x 2 x 27 1 2 1 3 x 81 x 27 2 2 Math 132 Final Exam 3 Find the Taylor series for f x e3x centered at 1 X 3n e3 x 1 n A n n 0 B C D E X 3n x 1 n n 0 X n 0 X n 0 X n 0 F G n xn n 3n e3 xn n x 1 n 3n X 3e x 1 n n 0 X n 0 n 3n 1 e3 x 1 n n 4 Find the value of the series or conclude that it diverges 2n 2 2 4 6 8 X 1 n 2 2 2 2 2 2n 1 1 3 5 7 n 0 A B C D E F G 2 4 1 2 3 2 2 2 2 4 2 3 2 H The series diverges Page 3 of 11 Math 132 Final Exam Z 5 The indefinite integral arctan x2 dx has a power series centered at 0 that looks like C a1 x a2 x 2 a3 x 3 a4 x 4 a5 x 5 where C is the constant of integration Find a3 a4 A 0 B C E 1 7 1 7 F G 1 3 1 3 D 2 7 2 7 6 Find the interval of convergence of the power series X 1 n x 4 n n 1 2n 1 n 1 A 2 2 B 29 72 C 21 12 D 6 2 E 29 72 F 6 2 G 2 2 H Page 4 of 11 Math 132 Final Exam Page 5 of 11 7 Determine the value of the series n 2 X 2 4 3 n 1 4n n 2 or conclude that it diverges A 0 B C D E F G 152 21 12 7 11 4 5 4 184 21 17 7 H The series diverges 8 Which of the following series converge I X n 2 n 2n 1 n 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 II X 5n 4n2 n 0 III X 2n n 1 n 2 Math 132 Final Exam Page 6 of 11 9 Which of the following alternating series converge conditionally but not absolutely I X n 1 n n 1 n 2 II X 1 n 1 n 2 ln n III X cos n 2n n 1 en n 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 10 Which of the following sequences converge ln n an 2 n 2 4n ln n bn 4 n 1 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 3n2 2n 5 cn 2 6n 4n 10 dn 10n n n10 Math 132 Final Exam 11 Evaluate the following improper integral or conclude that it diverges Z ln x dx x2 2 A 0 B ln 2 2 1 2 C 2 ln 2 2 D E F G ln 2 2 1 2 ln 2 2 ln 2 2 3 1 12 H The integral diverges 12 Evaluate the integral Z arctan x dx A 1 x2 1 B arctan x 2 2 C C C x arctan x 12 ln 1 x2 C D arctan x C E x arctan x ln 1 x2 C F x x2 1 G arctan x 2 2 C 1 x2 1 C Page 7 of 11 Math 132 Final Exam Z 2 3 x2 dx using a left handed Riemann Sum with 4 subdivisions 13 Estimate 0 A 0 26 B 5 31 C 5 26 D 4 31 E 4 26 F 3 31 G 3 31 H 2 Z 0 14 Let F x 1 t2 dt Compute F 0 x x3 2x3 1 x6 6x2 1 9x4 3x2 1 x6 x3 1 9x4 6x2 1 36x4 2x3 1 x6 3x2 A B C D E F G Page 8 of 11 Math 132 Final Exam Page 9 of 11 Z 15 Assume f x is a continuous function satisfying x f t dt x sin x 0 Find f 4 A 0 B 2 C 4 D 2 E 2 F 4 G H It is not possible to find f 4 from the given information 16 Evaluate the definite integral Z 2 0 sin x dx 1 cos2 x A 0 B 1 C D E F 4 2 1 2 3 2 G 2 H Math 132 Final Exam 17 Evaluated the following integral Z 1 x 4 dx 0 x 3 x 2 A ln 3 B 1 ln 2 2 ln 6 C ln 2 ln …
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