exam 3 fall 2009 - Engimafiing Mathemaéics {ESE 31’?)...

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Unformatted text preview: Engimafiing Mathemaéics {ESE 31’?) Exam 3 November 4, 2009 "Yhis exam camains five muitipicwhaice prebiems wart}; two points sack five true-fake prebiems wort}; (me: point each, seven shofi-answez problsms wofih one paint 8210;}, and thrae fieeqesyonse proiflams warm 18 points aitogethegr, fer an exam mtai M44} minis. E’gfi i. Mum §.€»ChoéC€ (two poims each} Cicafiy {:21 in {2’33 Gm} 9:: your answer cart“: whiah wacspenfis ta Ehc 0272i}; cerreci :‘e‘spfififi‘c. -. m4 .=. ,, . y . a . H. E. The éifferefi‘iéai farm F * c5? 2 y ewcfx z 63%;; is was: {‘r GU de mi neeé {a} ven‘fy {his} had the: value 6? the fofiowéfig fine integral. «€2.13 E ‘é jam}: glad-Z5 ‘3" x €“"‘"d§’§ @}9 (B) 63 {C} 65 A E mjkfi §}%6 xamfi:%éagzg1 . w»; ‘ <- . 1 £ '_ {A} ? {my} 3 geosmfimuf 2'23; {3 S a g Q7: 8 S L S 4.! ' r - .- ‘: r ‘ r : {E3} r {25: 2;} :2 3305a, Sifi’éfi-f 22;; a} g 2.5 < i g 2 g i 1 fig \ rs ; - ‘7 a {C} r [email protected]} 2 322-05 :5,23:n {m} G g 2: g 2?: 8 g a g i \ ~+r r ~ :3 x ‘ r g {3} z“ {35, 2;; m Laces-ii, mmu, 23 {,5 g u g 25?: i) S a, g :2 ‘ “>5 7 \ 5 . ' ("i t‘ r} {EL} 1* in: v} 2 ya: 09323.3; 31mg, 2.; G g 2; g “7 b g L g E —* ‘x i'.’ i ' "i ‘- A ' {F} r =2); 2 Eqrcesu, 283m 35,1; U 3;: 21. g U S a; 5; i . —-+, .5 ,r . ,. u. g {G} Maggy;:gvceszigvsmmzw 8:113“; figyg if ' W{. 3 fi , , ’ i g ( {H} r gym, z 503 15,631ng 21‘; if} g '22. 27" E S r g i \ —-9- x i { . w, , . {I} r in, if} :: ég-ycegu: 23 sm 1:. i} g” u S .27? E) g 22 g 1 Consider the differentéai eqaation y” + 2x3! 4% 4y 2 G. Since :32; I Q is a raguiar pairs; {haze are pOWE? seréeg selutéens. in the seiutim process, thaw am no pfiiiwof‘f {ems} and fin: rawrrence rfiléitifin cames Gut as ShGWI‘} beiaw (Yam (£0 not {16813 {0 verify :his} Find a Easis Gf scluzécns fer the diff‘eremiai equatian‘ (as you preceed: find 3%? coastanis afi El?) {0 and imiuéifig a5.) 1H -7, I —— m 2 - a germ > )1 a a; _ mm: ” ' W" 2 "15 A / ,3 if‘ ’ 'w _. 14-2 i___ m J : {B} yin—i213 wax figmifamgr;__ { '_ J 512:: “2“ w 992 I wag»: __ w a; A. 1 g i}¥123*21‘*%3*"‘fii- yz:x#$w%rh§fi F . 4 E V {E} 3” flflgfijuffiéxiw Eikfl—Tj-rgx3w " 4; /;3.3: {F} 341:“ *Efi‘rm' y2mwxrgii T.- ‘ r 9 "g = «I I; I a gzflufiaig "“ 2555235“ “Sift,” u?» gags? L13 a; _ g ‘ | [i “J , 4 KH} i w Griff ‘43“ £51533?” «m?- Ealx ‘ a -' 2 :5 r {E} E” 3 9?: flail: w meat— ~ c221“; W“ Egamx‘ * 5%: 3”" ~— ' ‘ 1? i. : = {I} 3} E ‘23!) “ £33 W gfiéj%$* 2655333 wg-r 4. Whaé; series approach? if any, sheuid be used “$0 301% fin: foliowéng déft‘smntiai {aeguafiea aficut 2:1; :- 0'? 4373;” + 29’ y H {3 {A} There is no med {0 use 886% m 503% this {Eiffererz‘iiai equation {'8} The pawe; saz‘ées ma;th shauid be used. {C} The Frobenéus method sheuié be arsed, anti that: wéfi not $36 a ngiurai iagamhm harm. {D} The Fmbenius {named sheaié be useé, and there might be GI“ wifi be a 33mm} logarithm term, S. Accerdésg is she appiica§3§€ canvez‘gems {firemen}, any seariss 33301:? C} in the selu‘imn of {he féiiowing differenfiai aquatics must converge as ieast 0n Whaz intewai? 52;! {F} (8,39; {{3} {s s; {H} a) {sac} As you work shzough she Ems—faiss ems shewamwer praifiems, you may assums that cvsryihéng is “nice.” Specifically? a1} mnsiions are cantinusus with Cantinuoas was} ficfivafivesp ail curves am? surfaces Es: Wiihifi a simpiy—conmcted demaén, and a 2 curves and surfaces as piecswise smooih. 9353“: ii. Yam—Fake {we pom: 632%} Mark “A” (m yoar answer card éé‘éhe statement is Ems; mark “B” éfié is faise. 6. if a “vector {2636 is reta‘iiosaé, than if is m}: consewsiéve. V A , 4 ‘ K , ’ 4 -+ 11 7. L6? 5' be a scam? runways ami is: C %e a cigsed path. Thea éigmd j} a {i 3* x u. 8. The fUfiCfififi f3?) 2 52""? is anaéytic a: my; ‘59}? 3:3 ma} numbers 1:9. 9. yer each eenfiegative imeger n, 3% minimize; ef Legendre's equaiion Bf order n are po§ymmia23 K). Suppese {EM 6 is a sieguéar point efihe (fifiereméai equafien y” {A mix}? «é~ (fl 3 0. Then {he éifferefiéiai equation has at lease one Fmbenéus series: seietian @013? U. Pal“: HI. Sheri Answer (me pail}? each.) The answer its eaefi 9f the fefiewéng is efiher righé: 01‘ wreng: m} war}; is required“, emd n0 partiai credéfi wiié be given. m) Fer preblems E 343, seasider ihe fine integrai 0f 3 veeim" fencfiefi E3 ever a curve C {mm a peim A ta 3 peiet B. Remember that everything %3 “nice” as sieteé Eyefere Ehe iz'uewfaise probéems. Supgese a w _, _ j 33' a d 1' # G. 1:: 1 E. if the afienmtim of ihe curve i5; reverses}, Wig the veiue ofihe fine iniegrai {i} aiways be {he same 83 ii was pm): te {he change {ii} aéways fie ciéfierem:a 0;." {iii} semetémes be {he same am} sometimes; dig/seem? if the curve C 173 parametrized ii: a (Eifferent way (wiéh {he Same erientetionfi wfii the value cf 32:: Eine ietegra} {i} alweys be {he same as it was prim t0 the {:Er‘slange= (ii) aiways be éifi‘erenn or {iii} sometimes be the same arid semefimes differeni‘? i4, 35. 16. i? if a éiifereiii path frem A Ee B is chaser}, wiii {he mine: 0? fine iine integrai {i} aiways Eye the same as it was paler 3:0 the change, (ii) aiways be different, er (iii) semeiimes be the same sad semetimes differeni‘? Wiiat feature of a éiffereriiiai equatien indicates shat it must be seéved using a series metered as sppesed 2e usng the Characteristic equatien? {332$ answer can fie states simpiy am? briefly. if yea don"t mew “whai ii is piesse éon‘i waste yew“ iime making semeiizing up, am} den“; give muiiipie answers? keying that {me Of {hem Wiii “hit? What is she mess imgaeriaiii “re-ainiife exampie (if a difiei‘eiiiiai equatieii which requires ihe pOWB’f series messed to salve ii? 'i‘he powers in a Fm’eeriius series can he say res} numbers, in}: the pawe‘r’s in a power series must be speeifieaiiy what kinés cf sea} numbers? {Your answer must he exactiy correei, neither toe generai nor tee sesfiietecii} Lei Pm. represents: the Legeiidre peiynemiai ef degree n, Wile? is the vaiue of f V31 Pg: @195 2:3}:2’58‘? "J §3a§t EV. Pres Resgaass (pom: “mines 33 Shawn} ¥0§§0w dirsctéens caefifléy? and shew a}? the: steps aeedeé {<5 arrive at year 591133103 {9) '38. L82 {21, m 32f; and Eat C be the curve in {he mfg}in which feiiews; ihfi fine 3; 2 .2: fmm { 13,, 6} EC: {_ V55, xvi-'5}, than foiiews {he cérde «3» 3;? 2 if; camier—cfieckwise fmm {Kg/r21 m {i}? :2), than fefigws {Em yuaxés straight down from it 2} back to O, {3}. Find ’33, M" —+ PW? F the vaiue offiie Eine integral as cfirecteé beiow. C {a} Using Green’s Theerem and yeciafiguiar caerdiaates, set up_ the integrai Gr integrafis which wouid be needed to firm? {he value of the fine integrai. Do {he campEe-{e 36mm so {hat {he vary maxi stag) wmfid be imegration, but {i0 no: do the iategratim Gr finish the probiem. {h‘é Using Gresn‘s- Tfiefifem am} 3033: caerdénaiea set 12;} éhe imegrai 0r éntegmis which woaié be /" needed £0 finé the: vaiae 9f the fine imegrai. ’E‘hen integrate and evaéuate to abtam a finai numer‘émé answsr, simgflifiad sompieéfiy. -—+ {5} 39. Le? F gt m {35 and E61: ihe surface 3 be the Cecilia: parabeieié $2 + 3;: x z, paramee‘izeé by {he fueetiee ?{e, v} z (seen:} i; sing? G§u§23, fifiefgl. {a} Week} it be epgrepfiete is use the Divergence Theerem :0 fine the vaiee of the surface a -—+ inéegrai f( F - 3) 55A? Em cine: werfis, flee; the inergenee Theorem aepiy eere? {Just answer “yes” 01* “no”; you {$0 not neeé :0 give a reasee.) {b} Regareiess of your enswer to par: {3), set 1.1:: {he integrai or integrals which weuié be needeé to find éhe veiue (3f the surface integrai jnfi: E? '2 EMA direcziv (by {he defiméiee). Do the e eempiete setup so {hat the very next stepkwoeid be integration, be: {£0 :19: (ie the integrazien or fifiish the emblem. —* V . - ‘ V e {4) 26. Let F 2 [Qty $2 w ye? egg, let T ee the seixé given eeEQW, and fie: S be the eempieze boundary of :5”. Usng the Divergence Theorem anti sghem'eeé eeerdimtesE set mg {he mieng e? ieiegrais whiee wealé be needed :0 fled {he vaiue of éhe surfeee imegra} [ii .:1"? ° EMA. Be file eempiete sewage C so that the very eext step weulé be iniegratien, but de 3291: ée the ietegmfiee e: fiflish the emblem. {You de mi gee-é {e weny about the issue of eriemafien, since 3 is eréenied outwarég as ii" Should be.) 7‘: $2+y2+zgglee yge z<8 ...
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This note was uploaded on 01/10/2011 for the course ESE 317 taught by Professor Hastings during the Fall '08 term at Washington University in St. Louis.

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exam 3 fall 2009 - Engimafiing Mathemaéics {ESE 31’?)...

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