math_172_(spring_2009)_exam_2_solutions

math_172_(spring_2009)_exam_2_solutions - Math 172 EXAM 2...

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Unformatted text preview: Math 172 . - EXAM 2 , April 2, 2009 Name: ID#: m Section#: A SHOW APPROPRIATE WORK or EXPLANATION on each problem for full credit. Box or circle your ﬁnal answers Calculators/ note sheets are NOT allowed. Numbers 1n the [ ] indicate What each problem is worth. Check the board for useful info. Known Maclaurin series: Tools 82: tests for Series: = 1 + a: + x2 + 11:3 + . .. Geometric Series Test for Divergence Integral Test p-series Comparison Test Limit Comparison Test Alternating Series Test Ratio Test Root Test life“ 1m; beau Vevlabdeol as K, Teckicqlla', Smc‘e cc #0, K can W'f _ be. zcvo. Hawavw, seHm) [4:0 9w}: era-3’ u‘uc‘l‘ ﬂawlv‘fﬁcopé. \$6 {In ﬂu HM! «ha/75:3, kl’ah been? Constant. 2. [10 each] Using appropriate tests, determine whether each series is convergent or divergent. °° 3 a);n\/n+l " 3 c i»;- Armin-2:, “{ugj (mﬂavi 50“ %5+: y‘Jnf‘h “m -— ”3/,“ “Ni INTI" “EM 0"“ “31 aw. ros‘t+lv'¢*+cmd (may 0W4! MkM-Qw ‘4'5‘1 332’1—1/3. rower,“ [p— —S¢n¢{ col/p22 1/7') {a in, prav‘liou T-‘cs‘i" Z ”F... 41% LEE Us" I LMJ. [M "a“; pd- ‘. 307k ﬁﬁf ”‘4 ”:35; am:- pos'liw’e AM a" M21, and - %}3_ 7"“ ”‘27" 27W“ * -- =7w~ ~ “-209 .1.— h-Oob "in“ (A. “a.” 1’9“?! ' “3/:- SVC—w-(g 3m): 351:3 glue. ’f‘k'uliwi-r; mum and Lucknow Zn?» (mvevys, L.C.-T Sm); {1.4, 1w’bk ("‘23- f” - b) E 1117(1n) W3 (mnav:m 72d . ’“x’ 10.) > 1 tr n 23 5o ‘W h..." > 3 l 9:3 “a“ f'k d‘W; (WVMMIC Qﬂeﬁl. M \$0 duos ;)hc:) ‘43 (Manson-155+ [D‘UEP-é WIL- Mi )5 fusifwc and radiations m XZZ?’ an! Mgw-g inky“! 37'25-(1 ﬂ)”: /"'7Mx,, wthk i; do w(”I X>e . IN’W’. 4 Is alo alum: smu— PM“: I 5 \3 "‘3 I X" 3‘” 7mm in“) f: Z‘L‘i’dx ‘31:: udu a 1M", J0me]: {[Msﬂ ——-:oo “9“” '7” M M?) 7" (3)» ' meWWWWW—rw 2. (continued) C) :(3717: 1)” ~ M95 {200+ Tes'l’: 7m 1/Iant=.’lu’~ "(33,)" -7,“ (3:11)-)m. J1) =J.‘ name "“900 "—99-. Sun}. 41": Jiw't-i‘k (— [, Pod“ Tcd' 547\$ ABSGLLITELY CONVEMENT what '3 7L4 34““ ﬁr 0: yos‘rﬁvcﬁkrmw! smés . I . . °° 3"+1 7. 3. [5] Fmd the sum of the senes Z 4n . 32":{f‘11 :1) Mt, 11:2 7717: F: agent-WM. semis: M71. fird' km 4: ~37: -—- t1 [6 / and comma vm‘fo r: 21: So H; mm is ﬁn‘blem __ 3‘3 33:5! . I—V‘ 1 \ ‘ 3? pa U \ 9‘ x u 4. [10 each] Using appropriate tests, determine Whether each series is absolutely convergent, condi-- tionally convergent, or divergent. (”it/17311 71;; 7; an 414:th Sunk: M11 515% / “ﬂak is draft». n=q » , dccmur‘u) I aw! 77W (315‘): o, {o ”Magma; i: («4W n4» (ag-is) 19:) ’ﬂv Memﬁé‘g Yen}; 73:31. #0.»ve :(agl 7.23,};— = :39; m a”. mmgmt- p-sm‘e; {ppm 1 I 99 co ‘ go] . 2-, a», (MNVW’ bod— 274‘“ ANCW C6 717: Sine! it , ’ I @NDITMNALLY (ONVEPGENT' } ‘ . ‘ r . . p Np’t: Rah» T-esf" wt" 31w. 1:; (1.1%,):— i I {o :31 aMJusnk M ' {Anita 735+ aft! wovlc ken: ”A man au+{( Tm ((#0)an “I. I “ .:‘ I ‘ 4“ M» (MI)! 6N)" ’73M [12* ~ “IN-90° an)”. (Moe. 11m um?” 41, 4M: was is Mrownw (ouvznbENT 6:? m “44“0 723+. 5. [10] Find the interval of convergence for the power series 2 (.73) (x —‘ 4)” . n=¢ ‘ a . q __ an «N ”st. Q—ﬂ‘f‘o TVS'h 71M 1 .131! 1: 7"“ [(2 (X‘J‘fz O. M - “A a“ h.” , n+1 E?>"‘(x~"l)M = 7% AIM!) = zlxvql. so («.me um NH!" h—hu “*1 ' =9 lxwlcf Now ckéd‘ a?" 6M! punk: 4 . ”(a)" 535)“ .. '° .L h' . UAW)“- 3;, sen/m bums Z » Z n DIVEP-éENT' (havwmc) ‘ ' ’ Suit: 5‘ \ﬂ .0 wkwx=7i,éevuésbecams 5% = 23%? (”mum bf, AH. v I ‘ Semis T-cs'i‘. H M €43 ﬂkrvn/ A‘ (mum-yum. is (34-, ‘11.] .6. [10] Find a Maciaurin series representation for / exzdx. Give your answer using 2 notation for .fuil credit. X -— - 1+ 2: 1 ~— X+Lv+ X . c . n, 2:. T! ,L n 2. co 1. no 2M ‘1 X‘ 3C a) X .— " K1. "xﬂ-‘F "" ‘0‘. 6a 8 V: y‘\ :1 Z “I —— 1+ +2! 3., + . O ‘ ' a ., . WWWWWWWWWWWWWWWWWMWMWMWW“NJm 7. [10] Find the Taylor series centered at a z 1 for the function f (:3) = ln(\$). Give your answer using 2 notation, and write out the ﬁrst three non—zero terms of the series. Find: M’“ '5" ‘fm‘ ° ‘ - §o= 7%”? Vg-{(:.)?)(x"9“ I " d I " °° u if ”(2): (:R’z _: :23) =I-a 3 fl,” “Z {2500“ f {”’(u=(—I;(~z\$x”——». f ”70:61)sz = o + i—w/Mg (x-I)" (mm: ear-mania f“’zu=éo(—zJ(-3) : n: ‘77)}: format! B only V‘dl34 19¢ "24/” \$0 ﬂu. M10 km marl- b< 5m,”— ’ﬂi ’° n Could «he do somﬂukj [ﬂu 1‘1”}; 7am: [£61K = (”a”) “X -.= @9960!) dx. . —— C+ OZ ”H . Since km 0’ C 0": wand 91‘“ Oh C” n ' 7am: Tim—«W . 43w —L—-’_x Wright. 8. [5] Using an appropriate series, compute an approximation for cos(0.1) with [error] < .0001. (o\$[x.)- I 2! ;— 4! 7?. f. ‘Fnraux, (olfI an" (.1) ‘ -- §o(s.l="’—-—~+.—_—-___ cf ) ! 7.! 6! + 7k; 7: An al-Irmﬁé, avér’ cc. 5“, M A Ihmﬁit“ {an}; BSH‘MA 721w, if M uy. 7k An? 2 km.“ an “ﬁn-ate, ﬂieWW wt" 13¢ (Minors 7‘11“ Msiz—cﬁﬂe?”¥cm, L331}, \$0; . I . .uaol (as(.l)’«‘-(I-— 31/1 mﬁ. leww'l ¢_ “M . ...
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