Math 132 Fall 2007 Final Exam 2 3 1 Calculate cos x sin x dx 0 a 1 f 2 3 1 2 3 g 4 b 1 3 3 h 2 c 1 4 4 i 3 d 1 5 1 j 6 e 2 5 t4 2 Let F x dt 1 t3 Calculate the derivative D F 2 of F at 2 x a 4 f 4 b 5 g 5 c 6 h 6 d 7 i 7 e 8 j 8 1 x 3 Calculate x 1 x 2 dx 0 9 a ln 8 7 b ln 6 5 c ln 4 4 d ln 3 9 f ln 5 8 g ln 3 9 h ln 4 16 i ln 3 3 e ln 2 16 j ln 9 1 2 8x 2x 6 4 Calculate dx 1 x 1 x2 0 a 1 ln 2 4 f 4 ln 2 b 1 ln 2 2 g 5 ln 2 c ln 2 d 2 ln 2 e 3 ln 2 h 6 ln 2 i 7 ln 2 j 8 ln 2 e 2 5 Calculate x ln x dx 1 a f 1 3 1 e3 e3 2 3 b g 1 2 e3 1 3 2 e3 1 3 c h 1 e3 2 3 1 2 e3 1 9 d i 2 e3 1 3 1 e3 2 9 e j 1 2 e3 1 3 2 9 e3 1 6 What is the derivative of a ln 2 f 1 ln 2 2 b 1 ln 2 2 g 1 ln 2 x 1 x c with respect to x at x 1 ln 2 h 1 1 ln 2 2 d 1 i 1 2 1 ln 2 2 1 ln 2 4 e ln 2 j 1 4 7 If y 0 0 a sin cos x and dy cos x dx 2 1 y then what is y x cos x b sin sin x c cos 2 f arcsin arcsin x g sin tan x h tan sin x arcsin arctan x d cos sin x 1 i arcsin tan x 2 e arcsin x j 8 Consider the following three statements about a series a with positive terms n 1 n lim a 0 n n I The series converges because a II The series converges because lim n n 1 1 1 b n a III The series converges because lim n and b converges n 1 n n 1 1 a n For each statement determine whether the reasoning is correct or incorrect a I correct II correct b I correct II correct c I correct II incorrect d I correct II incorrect e I incorrect II correct f I incorrect II correct g I incorrect II incorrect h I incorrect II incorrect i Wrong answer j Bonus wrong answer III correct III incorrect III correct III incorrect III correct III incorrect III correct III incorrect 9 Consider the following three statements about a series a with positive terms n 1 n 1 I The series converges because a n 10 n II The series diverges because 1 a n 2 n a III The series converges because lim n n 1 0 a n For each statement determine whether the reasoning is correct C or incorrect F a I C II C III C b I C II C III F c I C II F III C d I C II F III F e I F II C III C f I F II C III F g I F II F III C h I F II F III F i Wrong answer j Bonus wrong answer 10 Consider the three series n5 10n I II n n 0 3 n 0 n and the statements and 1 III n 2 n ln n C The series converges D The series diverges For each series decide which of statements C D is correct a I C II C III C b I C II C III D c I C II D III C d I C II D III D e I D II C III C f I D II C III D g I D II D III C h I D II D III D i Wrong answer j Bonus wrong answer 11 Consider the two series n n n and I 1 1 n n 1 2 1 n n 3 II 4 n 0 3 1 n and the statements AC The series converges absolutely CC The series converges conditionally D The series diverges For each series decide which of statements AC CC D is correct a b c d e f g h i j I AC II AC I AC II CC I AC II D I CC II AC I CC II CC I CC II D I D II AC I D II CC I D II D Wrong answer 12 Consider the two series 1 n I and II n 0 n n 0 10 100 n and the statements C The Ratio Test establishes convergence D The Ratio Test establishes divergence F The Ratio Test is not conclusive Apply the Ratio Test to series I and II and for each decide which of statements C D F is correct a I C II C b I C II D c I C II F d I D II C e I D II D f I D II F g I F II C h I F II D i I F II F j Wrong answer 13 Consider the two series n 3 1 n I n 0 10 100 n2 n3 and the statements and 3 n n II n 1 3n C The Root Test establishes convergence D The Root Test establishes divergence F The Root Test is not conclusive Apply the Root Test to series I and II and for each decide which of statements C D F is correct a I C II C b I C II D c I C II F d I D II C e I D II D f I D II F g I F II C h I F II D i I F II F j Wrong answer 2 1 3 2 x 7 14 Let f x x e What is f 0 72 a 20 f 120 b 40 g 140 c 60 h 160 d 80 i 180 e 100 j 200 5 120 sin 2 x 15 Calculate L lim by finding the Maclaurin series of the numerator x 0 x cos 5 x2 x and the Maclaurin series of the denominator These two Maclaurin series begin with the same degree p monomial In other words for the Maclaurin series for both the numerator and denominator the coefficients of xn are 0 for n p and the coefficients of xp are nonzero What is the value of the product p L a 36 f 96 b 48 g 108 c 60 h 120 d 72 i 132 …
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