w09exam2sol

w09exam2sol - SOLWONS Math 42 Winter 2009 Seeond Exam —...

Info iconThis preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 12
Background image of page 13
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: SOLWONS Math 42, Winter 2009 Seeond Exam — February 24, 2009 1. (10 points) Determine whether each of the following improper integrals converges. Explain your reasoning completely. co . on (30/ s1n(2:c)d9: =1“ Sosmgbmix + 83mgde ) anrweme/I’cr'vl/y) fiym’i‘walmtejmls —-00 ~00 o OM’HLQ Nat/1+ “4031‘ COHere In Howler ’89:“ M "#9er! 0" 7% W +0 ComWVje" 0'0 N N Bd- OS 5thxe 2 [I'm S Smgx (1K : [MA “Cf-£3] . N906 c2 N900 3” 0 grid Leeaqu {ligand 0/on no‘f “Kf GO /-" C, C I 0% NHL) “SNJVQXDVX diverges {becwgfl ogsmgxah 4/50 oékfies) " (> 1.69” d b — a: 0 \/3_3 This (5 M W T ”+ ex 0 er { r F“ p p 3m! éeiwge X IS fillsczm’ifimws a"; 0 We 3"va be Consdous 070 ‘HLQ ’ch M CIOSQIX WW I i cg It?” 2 ("M Ref—09x a [I'm ax'élilv two“ a We? a I 0140‘ in MFHCUIQP fila‘l COWWQQS . also M0758 {fig—I. 10m 05 (,i l‘ C 76 , r We mg ’\ 6 \4 e Lemuie 6% IS an Bylaws»? ’Ech’i‘mw) WM 771}: Mala/fies 'La '31 a 4+ 0(537; for 0<X<l, ( ( I O‘P Course ) OS £70k Q Cowwvjes hemugg ogéoh 0/063/ l x QM og %C’X COWW es Commas” Ar merOPQr :[méjm/S. {Nita ’flla‘f W 000?"? Law; +0 Comrad‘g #5 value!) Math 42, Winter 2009 Second Exam — February 24, 2009 V 2. (10 points) For this problem, use the following information about any normal (“bell—shaped” or “‘Gaussian”) probability density function f : 1 2 2 0 has the eneral form :1: = 6495'”) /2‘7 f g “ ) am [n+0 = j—oo p+2a o / w .98 —OO f(a:)d$ z .84 Now suppose that the speeds of vehicles on a highway with speed limit 65 mph are normally dis- tributed, with mean 70 mph and standard deviation 5 mph. (a) What is the probability that a randomly chosen vehicle is traveling at avlegal speed (65 or under)? Justify your answer by writing an integral expression that represents this probability and showing how to evaluate this integral. SMCQ/u‘70 and 625) m Lian :70~5 z/Arc 179034441 (12,4 by] EXPRESSch {an} drawn) 05: P 63! r-(x~70)Z/&A52 SW6; 1 i mb{x\<6‘5): S‘s/(2:9 Ax = ‘QXJAX = Amado? ‘0‘” 65 70 .—-oo /ro Value) ne‘fe ‘H’m‘l‘ PmL{XS65) 1‘ [A PrthXZQE) ~ ems laii’erme‘QLi/‘i’y #wgflg l ‘ _ am Lethal axiBNows: ‘ 00 74 PWLGQ 65)”: 65g 4‘600’)‘ 1— 106‘) 0'Y tn“ 0' } ’st , flraw’rlm ) my” z mmgg) w k0?! =, fro“: 95 70.74 /4:70 TS:/u+6‘ ' (b) If the Highway Patrol are instructed to ticket motorists driving 80 mph or more, what percentage of motorists are targeted? Again use an integral to express your answer, and evaluate it with justification. . 00 Nate 41ml 80:70+;1u5 5/A1Qr, Thus; 5mm we neeol ProMx 280) 2 Sumo/X ) 80 ’Hu's Cam Le rem! #16 til/om} reasonwj 1 go = REMIX .33 #093 fig), 110$) 4ff>r07<wdiely PWQM‘H 5p MdOfoS will [we +ar3e+vd7 1Q» +iclte+s Math 42, Winter 2009 Second Exam — February 24, 2009 3. (8 points) (a) Express 1.5342 2 1.53424242 . .. We have [05345 = [053+ 0.005% L93 + 0.0092 + 0,0000%} + 0.000000% +..A .. as a ratio of integers. P. ,.. 5. '53 142 Z12 HZ Smear“ Q ’ 00 '0” we :08 Alli ‘ 40m a~ /I()q) 2 4” f: A + gig COMMJV) /—_‘ ,_______.__~ ~ A mm mm ’00 !*%00 mo {rm/mo) I00 CMoo : I53'?9+‘1'Z. : i578? _ 5053 ' 9900 96200 3300 —-2b —4b —6b ~81) (b) Determine the values of b for which 1 + :5- + 6T + e? + 8—16— + - -- converges, and for those I) find the sum. (Your answer will be an expression in terms of b.) >“2VIL A“ V) 8 n 4351‘ MZO so a J -"1 O( é‘ods ,— Tlm 36mm! ’ferm appears +0 Le ‘HH'S is or jeomalr'rc series inn-Ha Wm! ‘ierm 544‘] and Commn radio F: 1+ converjeg ho 4mg! (an/)1 Irku- E‘e_ €223, Saivinj ’(Z'N‘ L”) we Giff-aim Qflflka €2> £> , fl’#‘{ 9065' COM/err; ’R'LZ 50m )3 2 ,—_4 z ) F r' <L Math 42, Winter 2009 Second Exam -— February 24, 2009 4. (20 points) Determine whether each of the series below converges or diverges. Indicate clearly which tests you use and how you apply them. @)2;:m4—n2+5 W610! (New LIMIth Gmpm'go,‘ 12% erA an: (a 4m: in our'Cnse) fer {‘p HZ!) 41mm 3n?” 7 3012-4) >20) @406 SC BALL {42+ 5 > 3,44,. "2 2 “'2’ 3512A I) 3 Since; gaffer; are POSHL/Vfij ‘HIUS 0!“ 1"”;— is pod-Hm because {+5 Mmemql‘or QMOQ J 3Mq~nz+5 Jehomma’lor aha Posh/7M9 ’Qr‘ n? finenmwhi/e) Ans}? 7O ’lgr m?/ as a 3 ‘ . [I . I’) r 5’) NOW [MA 3.". : Ilm f7... I). L- lrm ‘1 Z HAWK, J“A V14,on gufzmt+5 I new 3n ‘n +5 I :2 [I‘m 14-500 3 A i/yy'L‘tE/fi l 3~0+0 Se 4% LIMP} (emphasny [Est truck/3 (/3 +0 ComeJe ‘vai afler £0444 sews {an (we? 5% l! I. - 3>OJ 09 comer/3e ) 0” ’H’QY diverge ‘ {Ahr is ‘1 P”S€I‘;€$ EVIL/[4. 132' P921 (aka. Tile Harmonic; Series) 50 we know r l Goa dim; a I , J M2365) US é" LE5 fist/v6 Math 42, Winter 2009 Second Exam - February 24, 2009 n20 Z Z , _ f " We have awe;ng 2(2’7)+6f§n 2 ,L+.i'£.+ i. e“) Now/flu ‘H’Ne Semis ) 1 Wm? mt Each TOM/VG series Common ra’DLio [85’s Wilma I [F923; 510%! a" SO Eaclw is 0L conw'jeml Serilei 1:} 43/10va #471 0’0 0° 0% 09 Dc q/l‘n’t‘ggsmfg’é‘ :Q(/’;+%¢fié)cé\ '0 MG q -n:'o 1" ' Q q We Zn 3” J n [75-0 WA {M P0410953 41m HM sen'eg Is @ergn’}, {in 14%) #LC SUM (cm Le (.0me J— Q ‘ Fe) 3 I‘.,/Ll+ ) f’l/G Tm ° [Them CW, a 070 0%?!“ ways 40 $86 ‘IS A CDMWWM‘f 59,111ng iMflLMCQ , J 0M (5M CDM'PU%€ [I'M ZfL< I SU'PFMS‘SPJ LNG, n J any? 50 1‘11 Ral’o 733+ {NH/es ‘Had‘f series ("om/er 63 :( Math 42, Winter 2009 Second Exam w February 24, 2009 _ ((92% Lat “may”, I) w i g\ 4. C \ ‘ QM . E [W 3M, “A, Afngglgn h—apg 3n 1/ [0141)! ~(rm) .n/ 3 Ma I I new 3 . m 50) ’fl'& SUNS” amwpg: g ‘ Math 42, Winter 2009 Second Exam — February 24, 2009 n3 (d) i sin(n) ( n=1 flay/firm 070 14k Seriés are al/ Pos'r/WQ 50 We app/)1 q Cmpariéow 725-15 I! . AOWWV/ 7412 AOWIXPflvas or M 4m ymm‘h a? 7% 1%me SAW/4’ a [40th ’Hm‘f ‘MI‘S Will/ Cowl/1"" Q, Sim X lira/é, jiwe 05 Q (Mud; we tum 1% Jo is 134de mas-Mgr 4/42 Series g of a” 1l€rm5. $0469 O< {SIMM/ </ @ a” n ) M ’F ' r ' > L- orna) MON) We know i?) ng COM/Wye; 51’ch H's/,6) [0*SGr/é5 WNZA f53> flat {Sh/x0); )) nil ? (bovwaeg L7 1422 CWPRNSOA 0s ' ' ' ‘ ' ‘ m0), , 0Q Bm‘ 7%: g {S n; I5 7% Series opaLso/u/e Mal/e; affirms m0 g j 0:) h:/ +503; ABSOIJQ anrjemce Rule) €31???)- m/SO VS. 77:] Math 42, Winter 2009 Second Exam —_ February 24, 2009 00 5. (10 points) Suppose that the series 2 an converges, for positive numbers an. 7121 Decide which of the following series must converge, must diverge, or may either converge or diverge (inconclusive). Circle your answer. You do not need to justify your answers. Kellomeri/q’ham : The above nails (13 ’erf‘ a, 7—“ O“ Inconclusive Diverges ~> Cei‘lhinly Mimi; f 00 %O_ USE/IES‘FFOP Divergence“ Converges Diverges -9 For a” vacmm‘ly lmje n, We,” have angl) 5° 4’th 056020» and We Could apply +h€ Compmismflsi‘ ’fa +42 my} HP 1‘he gene; J Diverges fl) germ-wry Air“ 80." 1. ea; [#0. Use 725%» Divergence, Converges Inconclusive Converges Inconclusive Inconclusive Converges Diverges '9 039, Converjence Rule aw; ‘f/m‘f' 4/, 0),1 Converqu~ 9 1795 Coulo! 3a eH'Ler way. 3 Case. and? we ’91)? fl" £421: is (fillerjen’f. 60+ ancfi I we ~65”! is wavjwfv Converges Diverges Math 42, Winter 2009 Second Exam —~ February 24, 2009 6. (12 points) Find, with complete justification, the interval of convergence of the power series 00 mn n22; nlnn La a - X” w T r a w o. /» ><”*" W h minn- IO 83 . COMP Q Lflnqm :hamoq (“TD/pnfi'“) . X21 I = = “M rim“ 2 pm th'i“ ' e hi im i ——‘ ’1‘ - new I“ (MOIMIMI) 11-90% I “H 140’“) New) ryng $1 : rim—7a» [1‘7" Q H0 2 ‘ 3 . o \ In I ' r - I . +14?“ ’32 Mfiéfewmt [Wt "Ln; [(#3) cam baevaiuflpJ M741 Iflé‘phk Rule/a “m M ’_ hm A ’im 2i! 0: “m “J”- c 5: '4 new IMMI) A 79“)” %+I)) fl View 0 new TLqu L: ("M 1* “19:2:0 ‘ Ma i~ i = IX L 5" 7% WWW “MT” "(gr HXM» {.e, ~(<><< L o9 .,_P‘ _ mfg/L I / A/ Check iegema/Ecwrf— X~~I ymlo’sv‘lp, gene; A “In” ~ gang —— Bing +427] l . , L" 552;; , NCZ 7 New Since nflnn 'Is at Pos'nl‘iVEJ incmsnfl ‘ch‘hom mo n22 +lm+ jmws ’fo 0% as "900) hi ‘m’ow; ‘Hwn‘ L": nil—IE.) O) (i;) EMISLH 1‘2» 0:" mag) and (/71) "M £nfO. heoo Thus) H19, comofi‘Hons mp “Hie Wernij 5mg 155+ me swhswceof Agr €60}! I)" J 50 3+ Converge;a VISSL (X) a . 0C) Hal/((7 Pmdeorn‘F: )(zfi ya‘gu’s file Sepf'fls Q Biz} . Wei” examm€ (xv/474K 541:1 2. ' . a I , , . ~ _ , ms+90%!) SW1 1%; IS posntwe aw! 0160mer Mcipmml 3s Fosfh've m4 mmmgfihj as "01160? alums)“ We, have 0° ‘ I N, i j”: I. 7 “cg/V S X SNiZSW’X MILL?” Vivid] M. .M at” “1.1””! 2 '1‘ ‘ ' a!“ 2 N310 (In an "IV! X552 ‘9 “1‘0an ' sz/c} min/V 0° '2 00) Since jinx arm/«3+0 oo IUOUS zgixaox diverges) mac! so #11 My!" figsf kl’s Us fluff Firm, answer 5 M MLFPWI 04‘ W‘Vevgence is f) ) “3 xaoq. 5 _ é Jr; M5 "S we”. 77 5'2 Math 42, Winter 2009 Second Exam —— February 24, 2009 7. (10 points) Match each function below with its power series, listed'among the choices below. You do not need to justify your answers. (Not all of the series listed have a match.) 3 5 7 2 3 x a: x :1: (a)1—2:c+3x —4$ +-'~‘ (f)§+§+§§+¥+“‘ 1 x m2 x3 32 33 b...____ 2_3_4___5... ()10+100+1000+10000Jr (gm 3”” +213” 3!” + ‘ 2 3 _ 2 4‘6 _E ii: (0)1 ac+33 33+ (h)1 2+3 4t+ 222244266 .1334 325 336 23 25 27 1 x x2 5133 2 _.._3 _5_._7 ‘__,__. n (e) 5’3 3196 +53 7!”: + (J) 10 100+1000 10000+ Function _T Series (choose one of (a) through m “l 9—332 '3: 1 T l 10~—:E ‘h L) 2 ——3:L‘ '— ace ‘ 3 1 l ' (1+ac)2 Q l— J a: :- /t26_3tdt C Key Ohscrvofl‘ronS: ' L_ O L _| " 11K ’firs‘i’iwo ’Rmc’hows May be mwr'rHen as flit/i)... “We Ms 6 I , - "(xi/‘1) I-«(X/m) ‘0 WHY) We Can MawUse, 41w I.o- q Seamejfmc genes ’Qrmulq EFCQTQMQrz'r. I. (ma). r ‘ . I. ,L A a( —' ' ’fiQO‘L turban ,5 Gfif_~ FL): H%(_HX_XZ+X3,H) by “We QaMicSPrkran/fi’. i+x ’ Nde 4M ’Hll. ‘(Z/flC'Ith IS an Mia’eriva‘five figural }‘ ‘H’RN/S Only om four moserlés‘ ,9, ' . a ‘ i 00 4 alxw 5” has ‘HHS same mla’l'lonshlfg. AHcrndeIV) hejm IBM/"fl 66:3 ) ‘n :0 vaS/lHU‘I'mj +7: ~3x @1400 multf/ymj 1:): xi “’0 GHQ)” 7% $9an ‘Haen Wejmhz day} [a Math 42, Winter 2009 Second Exam — February 24, 2009 8. (8 points) Suppose that f is a function with continuous derivatives and f(5) = 3, f’ (5) = —.2, f”(5) = 1, and f’”(5) = —3. (a) Determine the degree—3 Taylor polynomial T3 of f about 5. We meal ’tle Taylor Cozpllca'enfs Cc,J ‘ .2 C3 . (b) Use the Taylor polynomial that you found in part (a) to approximate f (4.9). Express your answer as a number (or sum of numbers): ' 2 2 4'— J. J. J. ,Lo 3 &I0+&I00+&13’Ea (0) Suppose ]f(4) S 2 on the interval [4.9, 5.1]. Use this information to make a statement about the accuracy of the approximation that you found in part ) Byfiylminquy) um. n¢3)a;5 twang #422) we 1m +40% 1“ \_. , M AM; a. . - é 'ZITiX”b/ 2 é-q‘lX’EILf ’Qr a” X in 1” WW“) “0‘” “Mi W A” lt/‘Ifih’GMFUl é 332(ch = I “’L ’ “In I? [0" #120000 I (20000 “01+ 3/ £0”) is “Ceamle ’l’e WILHUEH ’L’“ (Mr/£5 48 0m (I’D/D’UXI‘MGH +0 Math 42, Winter 2009 Second Exam — February 24, 2009 9. (12 points) (a) Compute a power series expansion for cos ac, centered at 0. (Show all of your steps.) ’ We appty ‘Hve, 127/0!‘ Sem'es Tempe ’9» pfx)5m3)< 0+ CFO? M H Thus CZH—aj 0W! gnHAO fir 1420. :0 ’gf’ows POM/effing is . . . 1 — cos ac . . . (b) Fmd a power serres expansmn for / dzz: and determme 113s radrus of convergence. $2 l~cosx -- fid) X1 ‘ has power series I (f 2 L/ c g , f A [AL’e‘L‘J—L X— 2/L X2 XL ( Z q, gffgf~ XL .27.. a I x2 ‘1 3 ‘7‘ W+L~X~+fi __,.~., ‘9 L" G.‘ 3” IO’ Pcosx Nod“) We ’flmfl S QTJV 1405 power sews 3 5 7 C 1A X X - X + X WCQM Wrao’fus Up Coneremc-e VIA 4L0. quo’lésf. I‘D an; ‘ __ Ml 2n+3 ¢ ‘M "f ‘ L: Jim inf im , hm (X21. gar/Jamar)! / W”: “300 an [3-960 (3M3).(gfi+tf)fl (‘UMXm‘H “gag am?) (3:144XQM3YA’MR)! ,. “M [XL 0 3m! i if QM! 7‘“ I ’- A - A '2' , m —_———. = [M 2/“ mm I m3 (am-mM-B) OJ 5w“ new (2»+3)Z{$?n+lf) "3"” (a+%)z(a¢"/) : 0‘ Since L<I war» a” x) ‘Hfle Rana imp/A53 74d 41% Power 5mg; away; gran)“. +1303 ’H’la [7200125 0p COWVErfienoe i3 Z DO I, Math 42, Winter 2009 Second Exam —~ February 24, 2009 as X“ ” " 2M] X ,5 ($1.... h=a @M'O’ {9M3} ) - x20 (d) If you approximate the definite integral fol tag—Sim by taking the partial sum consisting of the first four nonzero terms of the series that you obtained in part (c), what can you say about the accuracy of your approximation? Be as precise as you can, and state your reasoning clearly. I (QM'fO‘fQMfZ)! We MO‘I'Q {3n+|),{&n+.2)‘l is q Pas/'1‘ij incquznj fwdmm 010 mag 4124+ jrows . . , I . WEI/trawl)! large as H400) SO bn~®woafimafl 301481063 fix “Love Ser‘és 3 490,15” ) where We Wrr'Q b” =' ‘Qr n20, nro (i) 13n>0 ) {;/) Ewan «(Z-r n20) MR (5;) km. 5",;0. new 7103 we my app/)1 1% AHe/fiathj Series 755* ml Remam’ér Est/Me ’fo mwc/Ua/a ) mt on!) “V4124 gidfbh Com/€739; 7‘0 a va/ug S (ai’gc‘l' w Area/)1 knew 7Com )l 59+ +174 GFVO!‘ [NM/Mlhdfr) in affirm/waif? 3 (IS/“’3 Weiw’h‘trms ’pmm H‘O’fo {4:3 Sbri’iygwés (Q3':!S”33( {gag {if—OT * fidHLerv/om’g) art/[4003K ‘H’IQ, &€1nibe(£AP“Hiai Sum (consish‘mj UP‘HM 9’3“ ‘QUV'V’OV’Z‘M hm” 5p 7% 5mg from Pant . v ‘ Ar I y ’ Q is [£55 ’qun 44% ’We Value US 077‘» J ’4' '3 “0 “1°19 ‘MQ?’ Jiggm‘f‘fl‘gom ‘MeVMVa/W. ...
View Full Document

This note was uploaded on 01/12/2010 for the course MATH 42 at Stanford.

Page1 / 13

w09exam2sol - SOLWONS Math 42 Winter 2009 Seeond Exam —...

This preview shows document pages 1 - 13. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online