M138-FE-W11_solns

M138-FE-W11_solns - Last name (Print): First name (Print):...

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Unformatted text preview: Last name (Print): First name (Print): UW Student ID Number: University of Waterloo Final Examination Math 138 (Calculus II ) Instructor: See table below Section: See table below Date: Ffiday, April 8, 2011 Time: 9:00 am. - 11:30 am. Term: 1111 Duration of exam: 2 1/2 hours Number of exam pages: 11 Exam type: Closed Book (including cover page) Additional material allowed: “Pink tie” calculators Circle your instructor’s name and section number Instructor Section Y.H. Cheng 001 S. Speziale 002 S. Speziale 003 D. Park 004 R. André 005 R. Andre 006 B. Charbonneau 008 P. Smith 009 Instructions 1. Write your name and ID number at the top of this page. Please circle your instructor’s name and your section number up above. 2. Answer the questions in the spaces provided, using the backs of pages for overflow or rough work. ' 3. Show all your work required to obtain your answers. Formulas and identities which may useful for some questions : o sin2a = 2sincxcosa - sinzoz = (1 ——cos 2a)/2 o cosza = (1 +005 2a)/2 o Edgarctanw) = 1+2:2 Math 138 Final Exam, Winter 2011 Page 2 of 11 Name: H [6] 1. Compute the following integrals. °° 2:6 9 ’+ H—t do: “73:41 =— ,7; 1 ZA/Irx’fl/ f 1+7? fl I+x‘ ' / 1 7 _. V&y(/)——«6w(/+{) ~ —«éw(/7‘?’) (b) fxarctan($)dx / $11 )k d 9L “WM—mmww 4v: XJ’WE ’ L V Au 5 a/l-e ‘17“ V 2’7;— i- 3 Al)? “N “MW/f : ch’j/‘Lfll‘flm 3" w 1‘ LL7— ‘3‘)\ f L AL)?” 7' A 1+ 76L -" ' 4X 2: : MC7LL’MA} LI L z/ / 7”" / %M>C 'l' C 7 i mam“ " 1 l ‘ MC“ 2, f '2, vL #1.); 7"‘6 f q/Ic; am» L. .. .o '25.“ +~C_. 7’ ( MCf/Q/IA3K 7’ Math 138 Final Exam, Winter 2011 Page 3 of 11 Name: a [4] 2. Consider the region enclosed by the curve y = cos :5 and y = 0 for O S x S the volume of the solid obtained when this region is rotated about the x— axis. d [4] 3. Find the function y(x) such that ,‘fi + y , = 0 and y(0) 2 e3. Math 138 Final Exam, Winter 2011 Page 4 of 11 Name: [3] 4. Determine whether the following sequence converges or diverges. If the sequence con— verges find its limit. 1 7L {1(2_+6_>;n: 1,2,3,...} 3n 06W KANE) GL4“) ’3 #7” :g w ————«—————-» ire—300 5 a 3 #44 "h 4/] n,“ A z :2..— : .1. v. / .__— __ N) ak. 7’1 "‘ “ N 5 C fiber) 3: 74 3 x— 5 /-AK 00 [7] 5. Suppose the nth partial sum of the series 2 on is given by the expression 7121 n S = 4 ~ n n + 1 (a) Determine if this series converges or diverges. If it diverges, say Why. If it converges to, say S, find S. g: flmfii fng/wéf—L>:4,{::Oi 4, m~3 at) 71—400 )44 l 7/; ' ll 1: I ll w :: 77—4 .... 7‘" 7" 7774‘! 7.. 741*! ~”" __ AL 2 J .— Math 138 Final Exam, Winter 2011 Page 5 of 11 Name: “H— [9] 6. Determine whether the following series converge or diverge. If a series is alternating and convergent state whether the convergence is absolute or conditional. °° (—1)” . °° 1 E N t : A ll—k f t b t th 2 — (a) ( o e ny we nown ac a on e series n can be used.) n=2 n22 97>,th :25 / £4 77 new w 2/ “a!” ’7‘ E I 3% —-9 0 fix» :J chW “6147‘ him” an /\ “:5 23,691)— cflw 4,44. (4 SJ) 7/ «b :71 M 00 ( 1)n+12n O J (7M 2 (b) g n! 7,341' 70 4 M I“) 77 I 523:? H I " M WU")! m gm,“ 9‘2” 77, f .— h~706 22:1 7A“)o¢3 (744,) I, 27/1 / :3 j ”” I} :: o 4 l 74 fl) 6‘3 “4., S/CI’ZJ ‘M C’flxv UM cf 03 oo arctann ’ Wyfi a“ I (c) E n2+1 » Amy/MM“ C/yv‘.f¢/I.!8~W A 77.21 h:' 73~WM 79: 7->/ K“. flA/HFCCCQ mafia/M M ‘ L a”, c: w 1" 74¢ ha” 7:; ha d” 21+: 5 r A “avian/‘24 / “V 7/— “ we/fm fl, — w, A ) z +"_ / P7”), [4] [6] Math 138 Final Exam, Winter 2011 Page 6 of 11 Name: ER 7. Suppose we know that the series 22°21 311—1 converges to some number 8 by the Integral Test. Approximate the value of S with the partial sum 5'2 and give an upper bound for the error. A calculator can be used to simplify your answer, but this is necessary for full points. Hint: Express 2 not absolutely $411 as Haifa—1 and letu=:r. /_ '3 I + 3—,— : i j:vzz.%é\ ”‘ .;L I?» 5"“ // 77‘ "£704: + L. (>0 ,ay 2:.z>~ é hiewbx:fiw~ A ’ 4% an 74%?“ 9 P‘f’h 1490‘“ 2’ M1“ 4 Q.) :3‘x e M Z 1455 aka I)... 2 mace/w» 77* MULqu Z-QDLA : 8. Let 22:0 an be a series of numbers. Prove that if 22:0 Jan] converges then so does 00 n=0 an. é/JA/x 3:0 /QM/ A L L I: 5? 7? cm“ £1.44 72%.Eizwwi—sae 744‘.) 9%. 06 m4 MM] 2.— 2/QM/ 3V1? 4%.” #7 mew EQM Haw) —-———> m gm 5m.“ h: 9 7%. («mafia 3‘1 04) Wl Z,— Qv : L a}, +/4y/ —-—/C?M/) 74:0 74:0 / 1‘ ;(aM+/4M)) _, 3-1/6“; M20 PVT»: --‘ [)1— L m’é— kg ‘72- Math 138 Final Exam, Winter 2011 Page 7 of 11 Name: Rg—E [6] 9. Consider the power series Z(—1)”W(x + 3)". n=1 (a) Find the radius of convergence R of the given power series. 9'2: vé/m all 3 324/0” : him CM+I‘/’/L Cry-H 7‘ ‘70“ MN N 6‘ __, 74“/0L- 7471, convergence .. MW— “ M 0Q 7' FMNM' P" w“ "~ A A , _— 11 am; w /7'/ 2f — as; . £1134 +3) ; ‘__i_.————+—_._-4—~m , é —“3‘i ‘5 —%+I/ 145' Vl/L Wflmwfiw “WW. 33 6A 1.1 a I / :LG‘OQ l 2;)» ——~———>;O@rcluwr ‘ ’3“ ’/~ 77/"1 91900 % M" )4”), L‘ w“! 2’1 " 73914.37“ 51W“ WWW 44L racy/(1 rain/)5 [4] 10. Give a series of numbers whose partial sums can be used to approximate the value of 5 1/10 x the definlte integral /0 1+ dcc. />L/< I 0 /'+‘)L 9” 7 7/0 7/0 / 'lu ab 35 a 5’ I v 3— '” “cl ..__.—-——- 7K :3 DC . ‘J’k :: l 690 K U U /"[-7\) V150 ’00 w M ’2‘: ML? _——. )4 __ :- xy 2 C~I) 3L 33k _ Z 6-))M )L d L 0 “:0 hzd o Math 138 Final Exam, Winter 2011 Page 8 of 11 Name: m [7] 11. (a) Use a well—known trigonometric identity for sinzx to find the Maclaurin series generated by the function f = sin2 (:52) L _ i ‘2, €3,205) :_ /~ deaf”) 2. 9f C 2. '1 ’L )___ [: —@><L) +£39‘4” r"? = "'27" w 4/ WW 1 ‘1‘ ‘1‘ 8 C hf CP [(0 3, s2 2c A— 2 pc— # Q— 3: —— .«LDC + - 3-2: :7! 2 4' 2'55 r 3 8 5 L ’¢ 3 2 5.x"- 2 DC 4. 7, )c/ _ 21x 7 (f/ 4/ 7T— 00 “Jan-Ill")? ' H ‘9 E, < a V (b) Use the series which was computed in part (a) to show that . sin2($2) 111% $2 =0 96 firm-HUGH“ 7‘ ’2 L 2 GDL )/ 06”» w ——- V‘Z/Vw 74-“ an” I 1 ” DC‘H ff“, x470 3‘ 3L As” 00 )flL’y" 4n-‘L ’4. Z >L .. 8/. / 34.40 7‘3, QM)! ;, a , 3 flow [a x. ~ 53/39 + 'J 1': (—0- 4/ 9(-‘)o 7/ :: CD H Math 138 Final Exam, Winter 2011 Page 9 of 11 Name: “mix [8] 12. Suppose the function f (3:) has the power series representation 00 — (_1)nm2n+1 f(x) u go 2n + 1 with interval of convergence [—1, 1]. (a) Determine the value of f (5)(0) for the given function f by using a method of your choice. (Note: f (5)(0) = “the fifth derivative of f evaluated at cc 2 0”.) $2 ("g/fl 98711,] I5 ML Mflé’éamw 3/?PI—4M 2. 5—" (H » Ev 6 (a) : (/52; z 5’ 5/ 74a jg’m , ,L ’57 ‘ '5 ' , 0’1 —\ (9(0) 5 5r, 2 4413 l 1 71 ,. % 5, (b) Using the fundamental theorem of calculus, re-write f (x) as an integral to deter- mine the value of f(1). L 041 1314' _ {4/ m 4’6 : 312194813, Ch: /(9() ‘ 0 0 dt' 0 2774' :_ : (—I/x‘l’ 475 0 WN at 9:0 7— % alt < 0 /- HI] 0 i i 3 3 T C! f Math 138 Final Exam, Winter 2011 Page 10 of 11 Name: MK} [7] 13. Total = 75 pts. (a) Consider the parametric equations Q1 1122," A}.\_ : 5 guys. a,” : 4mg! £17” "/9 dpx/dfi “7 Male "/L/ S/ece 7/5, : fl __g '3 "g: 5L “7' :25 -E : VL 79—} V1, 2 I 1’ J—X‘J'i-KL : ——)- I, 9’ (/L (/7, «1., fix +711 (b) Let C’ be the curve defined by the parametric equations 33:2? = (2/3)?” where O S t S 1. Compute the length of the curve C. ,_ __ Z A o 3 5 I. 2 3 (e) Let C be the curve defined by the parametric equations cc = sint y = sinzt Where 0 S t S Compute the area of the region between C and the m-axis. [K 7/7— / / flpgfi I; /?[O 1//fi)cJé :/ i/S/M2{)<§/wfl c/f D o M Q 3 (g c/ s/Mz‘mz‘dézgggj/ 0 3 O 5 5” r- //M J F .r #7, I 73 3 1 7—— ~ ,4; fl‘ / i L4): ) 1 v)? 2 DC? 1 ’ I U l) x \ ’3‘ /u Math 138 Final Exam, Winter 2011 Page 11 of 11 Name: M This page is for your roughwork. ...
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This note was uploaded on 10/02/2011 for the course MATH 138 taught by Professor Anoymous during the Spring '07 term at Waterloo.

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M138-FE-W11_solns - Last name (Print): First name (Print):...

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