{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

exam 3 fall 2009 solutions

# exam 3 fall 2009 solutions - Engineen‘ng Mathematics(ESE...

This preview shows pages 1–7. Sign up to view the full content.

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

View Full Document

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

View Full Document

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Engineen‘ng Mathematics (ESE 31’?) Exam 3 November 4, 28% This exam contains ﬁve maitipieuchoéce probiems worth two paims each, ﬁve true~false probiems worth time point each sever; short-aaswer probiems werth one point each? and three ﬂeeamsponse proklems worth 38 points aiiogeihea for an exam feta} 0MB paiats. Part E. MultipEe—Cheice (two paints each) (313333353 ﬁll £33. {he 0%? 0:: year aﬁSWBi” card which carresponés ti) the 03:11:; curred respaase. I. The diffefaméai form F- ~ :5? : yewgdx 3% x sxydy IS exact. {You {30 not need to verify this.) Find the vaiue of the fellowing line integraé E3 34' j, . {\$3.0}; Ey 5Wd\$ T stydyE f» N 53\$ (A) 0 E1“; E”; r; Ezﬁ'fﬁﬁ- iii; B} e3 ' m “ 346:3 3; “3‘53 3-5133} {Em ””3 ”E13333" E53“? "jg g ij‘ s {C} BE} “1 W J .J i I {it}; __ .. Kit? . elf/W“. j: V‘ ' 53-5 if? (D) 267 "W :: X53: «3» 53 3.3) «53* 3 , ,, 3:333: ' J (33} 2&3“ — 1) 3-»? . 3 3 2 a (3 €?_;?3}:g:} Vin/3‘3} “6:" (F) 38 w; .3; J; .., V {i W EE 5:31} A 1‘ * =5; 3/ ({3} 3(8 3, if g; :13}? j M 5 -3 (H) 566 3 , g {2.} 33} (E) 5{ ) y) i 52,321" .3133; 1.3}; E 623ng ﬂ fa» g . 6‘ # 3 57!;- .' .53 7 g” _ j: . . ﬂ / “Efégéﬁ :3 w {3? 3" r 2: 3,3, 353:"; j 5 {3%)}; {:5} 2. Choose the correci parametrization for the foﬂowing surface. 352 "é" \$5?“ : 41327 G E Z i: 3 ﬁgmﬁb W é’gu. Jami/3,3334; kiwi} \$353335” fzgijffzg . , . 3 ' . E - ”EL/4;“, {A} Efﬁgy-23): Ecosugsmuﬂ-EE G§u§2ﬁ 03135;: 53233313; if {3? {B} Efﬁgy}: EcosmsinaﬁvE 9313:3237 {jg/53\$; {C} ?{u,v§=E2cesu,‘ZsEnu,vE (3333327: {393:1 (D) ?(u,@}=Evcesu,vsénu,2E OgaEZ’K (33133:; {E} ?(u,v}ﬂEvcosu,vs§nu,2E Oﬁ-quﬁ“ (39,31 {31) ”Eff/£3,233}: E2z=cesm2\$sinm3E {Egagihr 033\$: (G) ?{u,v:}:EiscesugvsinuﬂvE {ﬁg-33\$??? ngg% (H) EEK/u ,U}HE’ECOSR,ESEEH 223E (35333327: vagl 2 2 WWW Wm“ “MW 3 m,“ M.“ ‘”:} g , 3’ mm W 3 W 3’ y \- § 3 g: E“) ?('£3, 33:;223300523, 22353311; 32E C3<y<?'3? G_v_1 3 Eif‘XJJQ‘E” EHEH :23. MMWMWM_ .-. M M .nwwmmnn.-- wwwwwww. mm. ‘\ w. €11; W {M £5; T111: Fmbenius 1111263011 sEzéuEd be 11\$€1E 111111 111a": W111 111}: 1312 a 11855121211 Eogaﬁihﬁ 1121111 WMWM7WWMMM “w. ‘ “M’MV’m W—n-w-w—L. 17 7...... (3011311161 1116 d1ffez‘enizai e 113111111 +21 + 41 77 {3 311106 a .7 -{3 1s a 16 Ear 121111 there 2316 if 3 331311161 series 3013:1033. in the soiutien precess, there: are 11:} puEE—sﬁ 1811113, 8.1113 the: recurrence {eiation 130111123 13111 as ShGWE beiow. (1’01: {10 11111 need 11} verify this.) 171.1111 3 E33313 {31" 5131131111115 1‘01 {111-} {11116161111211‘ 1213;113:1011. { As you 13112011126, ﬁnd 3E1 601113131113 11;. 111311} 31111 1112111611115; {1;} F01 m 2 2. a... m 1:} £7.11.-.)- } 2 f5 3- 3 F) “'2 -- '2 . z 7 ~77— 4-... '-\$ ... 77». .... , .. . (A) 911 E. 22: 33: . g}. 3: 3‘ ix {1.} Jig) 2 1 . \$7.}«77-17 :3... . (B) 333 -3 31 y. :1‘ 2:: T “,2“ .: 5.1:. c} r .. . 1 EC) 91:“ “1.332% Ebrw—xwégx‘zwé-“m ? .7»»~7-.........7-~ 7..---- "77- 7 MW .21 g, - m 2:. a... {’13:} / —E--—9:z:3 iﬁﬁw ”.5st .1531: g {1 Q}; m ..... I 3 :12 .3 ‘1‘ 217 s - Mm- wwﬂwwwwwanwm“WW“WWM W-..” - .3..-” W“ .. “ﬂ” 1 j: ...— .2” . -> . 4 :1 5 'I 1, ‘9‘ 35 {"251 {E} 15.: 32x" 3+ 7‘33 ~«—~ gt. 2 ~32 ‘ .ing‘ .. .1 ~ 2 .. .3: 5 f; f _..— t. : 3.-.. g (F) 3! x—Z+§12-~ ygx—3‘+§\$‘w M... -3131; ””3; 61\$... (-1 § .. G :12 1a£—[email protected]£‘—ax§+§¥a>x .‘ggﬁw— .. 73...»: 3.. E ) ’9? 1: 1 11 1 ,5 1. 2 54-3.- - 2.13.1 . 2 1:12. - . f1 '1') . E a; i E {H} 19’ 2 31266113: 6111"- 1" E13132? 1 {3‘9}? ‘1’ Z i z . {E} g7ag+a1\$~2aﬁ1-a \$3M jaw éaix‘j+~- 2 E '> {5} 11mm}. Majzcw gag: gagct"—~ g; :3. :1; .- 5.1., 1 31.; v 512:1: 1.» g. R’. 1 * 131‘; f; if ‘3'; x ”‘9’ if u... 3. _ "1- " W J: W {:3 1'“ ; . M X ”‘3?“ "’ ‘33; ‘1‘" 5% K W 5155.5. 5‘5 ‘”“ 5’1 k ‘E .3 “E“ X W 2 F .1: gj.’ .. , r E \$1..» ,1 1 ; €L1;‘ ” g ﬁfkig 5:33 __ 9;»: i 5%; ”“' j w“ 2;{ ”3" 0,; /- ‘3?“ 3’ ‘5: ”- / - = . ' ,. .-. W, .. .5 E 7 ’3 ~ . 11%.; ”1:: 3371143.! K; ”3 E - E312 - :1: .. 7.2;” «1 “:3 :3: E“ What \$131163 1133;310:3121} if any, shand bi” used 10 soivs 11121: 112% Eowmg 11111121611113} equation 2113:3131: fig : U? gig/“EN—é—‘i’gf—L: 2C? ”a f. . .K_ j} . .J‘ in .. J 1 “J r 5.! {Li 31.43 51g fivWﬁW/i £35331}? ‘ V 3: E M .1 5 = g ‘ ”1” 7M 1.x 3 ”M“ 5.1 .. r: 3 ; Lox 5;” E" :2: 53?: if 1 j” 1:) .3 ”La?” ”W“ 3 H" 3:) f ,1” «54.11.11... W“ ‘3: if; j {3, X ”13:35; 21.?- i am .y‘ ”\$234.3 ‘2'}; .2 f .3 r7 2 .3» .-- ..... 3 1? :1 [3? ”i2; 1;}. 5/23. “WEE 4“ ME 3. at; 6:} .251}. m“ Z «'2’: H" {:5 “’ii“ {" is a; . ‘ 1/ .2333}; 5.3.5.; éZ/mnij (1.1375 3.... 231.714.4163.; 511;. 42.31 ”1&3, :1. j} {A} T133113 ES 1113 11111211 to use 81211131; 113 501.126 this: d1f§12112n113§ 1313113111311. {B} The power 31.1113 111131:th sheuié be 113611. mm_ ,_ m. ._ _..,..,... .m . .7 -. 7..-... 7.-.. .. ... __.......Vw- -.-._.,.......7-.._.~ m7»77~wmuwuw_mw . W». "5 (13} The 1510136111113 metthE shouid 136 used and there 111155111136 01 wiEE be 3 11111111213 ogamhm term SJ: Acceréing 5:0 me apgaiicaizﬂe canvargence theaters. any 3622163 about G in the 3011222612 (2:7 £326 foiigwing {52535552252122.5231 equatim must canverge at 16335021 wha? 222282—2553? , . 2'" , 17—3 _. a” 45‘ “rig-2551" “V 2.23 E5 “‘ G E ) ’ j” 5:} 5.2% £53,. (LA/3‘29? 554MV¢ 5:?afiwé/5'5L} ‘ B 5" H ”‘5 53V 5” { ) \Wataj' - 5 ‘5 ’ .0 ..' ./m {C} 34353:; 5355:5255” 2.5144253252425424 £33145L35’22 .: j: i? .5; .- J fﬁgb} {ME} 3} L} 5.4-“ MW .5. ‘2“ 3 f {E} (9,4,5 W... MM As 35922 work thmugh the 8516553386 and skew-answer pmbEems. you may assume that everything is 55 “125.952. Speciﬁcaﬁy 2212 55221666223 are coniénuous with 00225222220223 partéal derivaééves 2213. CGEWES ané surfaces ﬁe withéna simpiywcennecéed 50227221512 aad 3:1 curves 22326. sufﬁces are: piecewise smooth. Part 33. Trueﬁaisa {53216 2361722: each) Metric “.53.” an yam aﬂSWEﬁ‘ caré if the statement is Ema; mark “3” if it is faise. 6. ’3. 8. Efa. VSCEGY 136362 is roiaéigaai, 2316:: it i3 123': cangewaﬁve. 5 ’ 5": f .5.». . .5’ 5g. ." 25.5”» ”5‘54... 5/ wa¥¥m¢5 42 3.222.253 7‘5; 2/5241: wig 3 35:43.5) 34.55451 544-6552434233 “5’” 55 ‘5’... 45.5: 57/245551. 2.4 515-555;; J "5-. . = .. 5 ‘ 3- ,- 5 ' 5 i'wmm ;' - g 5’ .. f n ' J , . - 25.55.54. 5“” « 54-5 .524. 255.42 525.544.- 42.45 4.55.5235“??? fw’gé’é" {XMV/W'?’ 2555555 5155’ 55555555545» 55555553"- 5 is: f E's a scaiar function. 9.1526. Est C bit a CiOSﬁé 223212. Thea ﬁgmd f) 55 a? E" 2 Q. c j fmmrig3154ka 555554.555 b2 5155555455 35%? 5155/5: 5’ﬁ} 55'4“” ....fm- ‘ V 5.5 :3} 5‘ j . . j 5.412.545 23.3% “jg/km Higgiﬁwxé‘ 5.42 51555593., 2; 5u555gw55545555‘5' ...,4. 5555—554 ‘ 55' ’ 3. . 5; f g N , ._ ﬂ . / 3,5245 .. ,2; Quail/g,“ 3.5:? “fizz.— 455:5, 522.55; 5 Vi. 252.513.5534? §WJL 5M5”? " ‘5 g J , . . prV/ﬁé 3,54; @5363”. 37315: functiém f .2: 5 w e 38 analytic at 3:53 fer a1} 2%} numbers 253;}. .....2. f 555: ﬁpf . .5- 33]. “may... {jﬂié ’3‘?”- r vim VM/wééi v._.« £2.59 (.52., 553... 5-445 \$555M; 52/555255 «525555525 9. Far sac}: nennegati‘v’e integer a: 31% saiuiieﬁs of Lagendre‘s aquatics of oréer n am polynaméalg. 5 . MI»- : z: r“: N é» WW, 33. Suppose that 8 £3 a singﬁiaz min? a?" the. éiffereniiai equatien y” + pkg M?» g; 17:32 2 8. \$2121; the. diffemnzéai equatiﬁa has a? Eaast (me Frebcnius series saiutior: 2355021: U M43252 L"; 35"” «15‘1— /\$4I/?L;;A,aéig&/L glgazi'i’yLM / (Lt/2m}: a ”42 5:7 ’ ,r’ i; ; 1",; r ". ; H, u- A , M A ,4, J»: ,, .. 3 ﬁ : 6 WM WM M 5: MW M MM? Ma .» MM <»‘§2»-i"/’3_~€-<1’£_M,Iﬁnd”£€4,.zg.ug‘rfyli ﬁjﬁ'ﬁ'Z-L? (:1 D Part E}? Short Answer (one win? eacik} The answﬁr is each of the faﬁgwing is eiéher right Or wmng: no work is requiycdt and no partiai credit Wiéi E36 given. 1‘ » § n > ‘ . a; -\ . F0? prebiems ii—E 3y magniﬁer {me Ema mtegral of a vector funcgen F over 5. curve C from a pom: A to a pain? 3. Remembéz' {hat everything is “nice” as staied before ﬁne imeéalse pmb‘iemg. Suppose .-—+ g _, 7 f F . a. 1” ¢ {3. (f I i. if the oﬁemaﬁaéan of tha cum/'6 is reversed} MEI {he mime- of the ﬁne éntegraé (é) aﬁwayss be the 3mm as it was 9330? E0 ths change? (i) always be differena er {iii} sameﬁmes be the same and same‘zimas ﬁif‘femﬁt‘? ' 4 "a, g g M I f f / ‘ M E j Y Z! i w ,_ s, .u ) if i ‘5‘ ,3 5:“1‘: 2:5,.éuyjﬁié/Ti, 9:-» f 3‘ 3’ ”gt A“ 6?;23‘} i Riv:— (4 (w ”5/; MM Ix.) If {Em curve C is paramafr‘ized in a {iif‘fﬁreni way (with ihé: same oriemaétionﬁ wiéi {he vaEue 0? {he ﬁne imegraé {é} ﬁways be the same as ii was prim m the change} (ii) aiways be different, or (iii) sametimea be: the same and sametémes difﬁzrem? kw MAMW/VLMEM W gjgﬁéf/gﬁ’?5i& (\$45; 7 WWWmWmmmmmuh‘MAMmmwmmmmmm“mm“... .m 13. i4. pr”: (.4?! Efa éiffarent gaff: {mm A is B is Gilesen, Wm Ehe vaiae 0f the ﬁrm integral (1} aiwa‘ys be {he same as it was prior t9 ihe change, {ii} alwayg be diﬁ'ereni, 0r {iii} semetimes be the same and semeiimes ﬁiffereni‘? ,1; {Lima} (tjxfﬂ'iérééxfséxéﬁw 2"”fo wwéjf’éué} /M%’%=“M’§’%'é”3 '* fﬁizwxﬁf’ €735,” ffmi: ‘be fiféf. xiiijx/wié’f’? f pafi yrwgmﬂééﬁéidki‘} ”a“ \mw ﬁrm? feaiuz’e 0f 3 {iifferential equation inﬁicams that it mus: be soivaé asmg a seﬁeg mathed as appossé to using “the characteristic equaféan? {The answer can be Sia‘ésci simpiy am? hﬁeﬁy. E‘fyau den‘t know whai: it is, pieasrt: don‘t W333: year {éme making something up, and (ion? give muizipie answers, heping ihaé 6:16 Gfthem wiﬁ “hitfﬁ E4145}? w my, Mfg/75 afﬁx xxx/gm Wat is {he mast imperial}: reaixiife exampis of a éifferemia} equaiien. which requires; the gower series. methoé to saive if? 77 2 f ‘5”?Lfiw pmﬁx/fw ELL x’i vf/Sréﬂwzjéwé / The pawers in a Frebenius seﬁes can E33 any ma} numfasrs, E312? me powers ii} a power? series must be speciﬁcaﬁy what Txinds 0f r632 numbers? {Your answer must be exacﬁy COUGCL wither £00 geﬁemi {301" £00 resmicted.) Let B1453} repreSe-m 5’16 Legendre poimamiai 3f degree 31. What 13 the mine 0f f: P7 @ [355;sz «if? I if ' ”Vii?” {9} Part W, Free Easy-933% (gain: values as ShG‘Wﬁ) Fﬁiiow directiams Gazeﬁﬁiy, ané Show aéi the steps needs-{i {0 arrive at yaw Seiuﬁmz. 18. L6? F ’35 jg} 2 {372, 273515, 2286- Eet C be the curve if}. {he ﬁy—plane which foiiews the ﬁne 3; 2 \$ fmm {i}, 5:}:- t0 {_ Vii V53}, theﬁ faiiews tbs circie \$2 + 3,32 x Q saunter—siecééwiw from { M3. V523- {<3 {i}. 23, than {015.0va the y-axés séraéght dawn "from (G. 23 back t0 {8.83.13iné the vaiue of the has mtegrai 39 E ° 55 r as direczad beiow. {. {a} {b} as ? Using Green‘s Theorem and :‘eﬁangular cocrdinates, 38% rag the integraé or iniegmis which would be needed to fin-ii the vaiue of the ﬁne iniegrai. Do {he camgiete 88195.1}? 30 thai ihe very next £333 weuid be imegratéon, bat do gig d0 the inizegration 0r finish the probiem. § 45% 5 '“ 5“ “a ' 4% , w 5‘- 5% 3 . 5:; it ‘ , , “ g5 .5 ' A f: 6;, , j W; ng( W30? 5;, :53 g” 33: f g‘éwﬁi E .. - 5 Z 1 E2. {3 ,3; £233 3535:4- {Zg “if" v5 5 5 151...; W} ' "’ Using Green‘s Theorem 3315 @0385 caerdinates. sari: up the in‘iegmi or éntegmis which wozﬁé be neeéed is ﬁnd the value of the Ems integral. Thea integraie arid evaiua‘ie to 023mm 3 firm} numericaE answar, simpﬁﬁed compieteiy. (g: :51 {gm-\$5; m A Wei, 2’ 5: :3 j E ’3 g g3, J 5 £2 5:)?” m jﬁ f 3 5:; I/j'? Céfﬁgéifjgg’}; ’Swiéx‘w} Maggi/E {13\$ :9 a 5, '5”? 5*" 5K5” =5“ {\$595555 . WEN-5:7 “j g ) j g 1., ’ )- ~53 5g: 2V5... 3:25:43. “35 gryg 590.35%? :ﬁ-é’t 53’?“ J a,“ M M (5” 5A * % {3:51 A; 3 i 3 EM ,/ 1.?» r’” .5 1: W2“ :1“; f g m 6féjé~ mfg: .- j .3 «55> g .5;- _; w ’ *séj/‘j (5} 39. (4} 222. Let \$222 y, 2,) x :3;, I}, 2:?, 22112.? lei: the smfaee S be the circuiar paraijeieid 2‘2 + 3,22 2: .2, parametrized hythe funcﬁen “gin, 22} :2: 3222:0522, 223322-22, 2:23, {Egagﬁm @ﬁegi. ,._ {a} Weuéd 2: be appmpz‘iate ‘22) use {he Divergeece Them‘em t2) ﬂed the mine of the Surface ~+ integrai If? F ° "#33224? in ether werdg, éees the Dévez‘gence Theerem appiy here? {Jusi anewer “yes” 02' “me“; you ﬁe not need to give a reason} ,, 2 , ,5 - :5 - ,; .-: 23'. 2“ ' ‘ 2/ 3 m ‘ = NM? .r '3; _" ,5 , . , .3 23322-522 2222“}; a..-2‘22L;2={22~é»¢ 3 Magi 22:22:, 2;, 2.2234512: 2-22 ; j F 2.2,”; Regardiess of year answer to part {a}, se: 1233 the iniegrai er integrais which wouid be needeé to ﬁnd the vaéue 0f the surface éntegre? ff? 3 . 713?}23331 direcﬁy (E33; the éeﬁnétéeﬁ}. Be the 5‘ {0} complete set—up so that the very next step wouid be in‘éegratéon, but de 3g; dc {he integraiion or ﬁnish éhe pmbiem. 2‘“ 5222 ' ‘2 2 {a .22 ‘ ,. 2,2 22 . "3» ? EM 3:“ 3,, ﬁézéj'gfh}:? ” i L323”: V?) {:22} 3'} ﬂ; ‘3 Mae ”“3; “”33“ Z 2‘ 2 i: E , 2- M 2 2 3 ._ ‘2 2: j 2‘ 2" 2* ? 2‘ . . . ‘ “‘" w 2 1 2' we ‘ ‘2 222.22 2222.222 2‘2 3 2 3,222 22.22, 2:32 2 .222, 2 _. 3 2 ? 22:33 52:22 ”522222 25% ,3 22‘” ? . 2 ,3: r- was: , v 2 . 322:«23322 2&3 f -2 - .3 --22;-2 2 2‘32 2 22 ,222- 2 {22 2222222222»! ““212 .2222; 1:222 w 5; “£5; * .22.; . ﬁg - 3:: ~ . . . 3 ,_ Lei i”? 2‘: 32g, 22 — 3312,2333, Eet T be the \$0112? gwen eeEGW, and Let S be ﬁne compieie beenéary of T. Usieg {he Divergence Theog'em and wheﬁce? coordinaies, Set 2132 {he iniegrai 02 iniegrais which , *9 weuid be needeé Se ﬁnd the vaiue ef the surface integm? ff (3 F ° "3:32.232. Be 2122‘: compiete \$622123) 3 823 \$3135; the very next step weak? be inéegraéien, bet {i0 332;: {ii} the ietegm‘zion 02‘ finish {he prebiem. {Yen d0 net neeé 20 wow abeut the issee 01302263232202}, since 3 is aﬁented eutwaré, as it 3322222222 ‘36.} 2‘: 22222+235123e 3,222} 2:”; 3“ ,W4 3 M 22> “N ”f j352222222.j?35222§22 ( .2; 2,2 ”‘2’” if {\$222 ‘ ‘ ‘ " ' " M \$2” } ‘ j 0 Egg’j‘ﬁiigéf ﬁjt'iéj ggngéy’én‘ﬁi [gig \$9}: _ 5:. (CL/22,2226“? ...
View Full Document

{[ snackBarMessage ]}

### Page1 / 7

exam 3 fall 2009 solutions - Engineen‘ng Mathematics(ESE...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online