exam 4 spring 2008 solutions - Engineering Mathematiee(ESE...

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Unformatted text preview: Engineering Mathematiee (ESE 317) Exam 4 ' Aprii 23, 2008 This exam contains seven multipioehoiee probiems worth two points each, i1 true—fake prohiems worth one point each, and some fieenresponse probiems worth 15 points altogether, for an exam total of £38 points. There is a foonuia fist on the final page. Part” I. MultigieAChoice Cieariy circle the oniy correct respooSe. Each is worth two points. 0 ifwl<x<0 i. Let f be the fimction given by fix) a {1 on M1 < x < l and then made if 0 < a: < 1 pefiodie with period 33 m 2. L :1 I; Find the Fourier coefficient of}. (In other woros, find the constant 0.0 in the Fourier sefies for f .) > g ’g‘ O f M f fl ) ‘3 f x E o” a “We (C) 1 a g; Kg) M“ g . $9“ 2y L. M 3 g I (D) 2 i " § (E) g r a: E (F) 7r g i g; (3} ~27; {2e ggemjee e} “’35 “z 3511?? ,> i 2. Let f be as in probiem I. Find the Fourier coefficient (25,. (It: other words, find the constant 335 in the Fourier series-for f.) A 0 9 - “fl z“ X ? _ . o (-) 2 5% '” jg”? fizijfimoefixaix (B) 25% g 4 3 g . g I I r a (C) fig; j; 3% “Rog: 9:33;, “W 4’5 i (D) é m Mé g: E ' ”‘“ ’“’ m5; of?“ g {B M £3 , 2 g a {g} 5% ‘1": "“ “5”” wee/Euro“ w 5} 4 (we? (G) 55g w (H) 3 in .r if (A? ,3 3” {I} e I was if M} < x < O Letf(:z:)m{:rr if0<x<1 w: 0 otherwise n: Then the Fourier integrai of f exists and is giver: by fix) 3 my: need to verify this.) 00 Findthe vaiue ufthe integrai j; d . w 3;. WM: 3 m2 (Yau do (A) “‘5 :35 :1 if sax/3,53,:sz { (B) Mi“ .. 3 y M W! W (C) it fi7f+fi>wtg a); Xv“ 2 M A .3 W} a V?“ (D). M; .. " {:fiéw *3 “£32521 _ affix} “2: EM 4 “W 'wljg ‘ a} _ (E) 0 > X .1 Wm mam; gmw‘ ; m #7:?“ (F) 4 M 5; W cm W 3.. 1 g; 3.} (G) 5 W 73' (H) 3; 7T (1) 5 (J) 00 Classifythe pania: differential equaticn g% m g: + g: + gr:- WWWWNR {MW ‘g. (A) three-dimensianai, ignearg homogew (B) three—dimensional, Einear, nonhemogeflcous (C) {D} {E} (F) {G} (H) (E) {3-} {K} {L} three—dimensional, nonlinear, homogeneous threeuéimeasionai, noniinearg manhomogeneaus feur-«d-imensiefiai, ii-neara hemageaeaus {cur-adimensimai; Imam, acnhomagenews feurfiimeasicmi, neniinear? hamngefieeus foar—éimensieaaij. fieniinear, nanhfimageneaus five—{iimefisi-anaia 5216311 hameganeeus fivendimensianah imam, mnfiomogfiaems fivemdimensimaa maiinear, fiemggefiams fivedimensicmai, mniinear, nenhG-magenawg Suppege the fundamental frequency of a vibrating string on the intervai O < a: < 1 is 88 E112. Ifthe initiai dispiacemeni is éeubied, (for example, changed from um, 0) m fix) m .901 sin #53: to etc, 0) as flat) m .002 sin 7m , what will then be the findamental frequency efthe string? (A) 20 Hz {,1} j, a . y g g i (B) 40H; 1' m (25%.;me W» . . “mung L, I I) I if “we; W 2L4 ' , (D) 160 Hz : (E) 320 32 Suppose that, during the process of solving a partial differential equation, the feflowing point has been reached. u(:1:,0) = 355 sin? m f(x) {ise this expression to find a formula for the censtants En. L , I e I 'K V A e W ’ (A) B“ “w” 2%:le flxfiinflgg‘h Cmmeegie 15;? 1:, érflwé» *Ae L . g r a ‘ (B Bu = "g" 3: sinmda: -’ “75,. f z 4 bwwfi ) Li]; ) L diffiwxv‘éwwfi W‘s ‘Mté” Ema “3’ B“ Eek fifiEfiififdw MM fez 4": m m M: #3... L - m M “A x we?" W 3;. jg f? ’ ‘5 E ewfiflé, 2T L "T .r ‘ "if"; fifZ/Xjfim i” it (E) Ba m “ff”er f(x)smfl£—$dx I} - L “Ti: ‘ e vs G CF) Bum-Ei‘ffo f<x>sinfl££dz w“ z? s ;3*‘ a; éfefii mix, .x «vb 1 E L (G) BR : fife f(x)sinflg—x-dx (H) Bu 2 %L£f(x)sineiedx The selutioas ef the steadyustate iwovdimensienai heat equation ievoive which types of fimctions? (A) cesines ané sieee (B) cesines 313d hyperbelie engines ((3) cosine-,5 anii hyperboiic Sines {D} Sines emf hyperbeiie cesines .4;»--> W ‘ “MW -- -- .h WMMWW W 1" g {E} 3.2: {es mi. hyeerieefie siees (F) hyperbeii‘c memes ami hyperboiie Sines E’art II. TmowFalse Write out the word “true” or “faise” for each of the followiag. Each is worth one point. 8. 10. 11. i2. The foliowing boundary vaiue prohiem on -1 < x < 1 is a StumaLiouville problem. 2:29” + M1 m :52)?! z 0 y(-1) m 0 99(1) = 0 f f ‘ “g - 4 o 1 Mk 35 z ‘J’ééAai-w J}; ‘%L?{,£, aéwfliwr (:1, k E j 3 § j J ,’ w 3‘2, x f " i‘; ’ i u‘ ‘ “I ‘ 3 The fimotions y} = :z? and y? = a: are orthogonai on the intervai —1 < a: < 1‘ v E § 3 g f“ g g”; 5 j§j W 2" é? jwsfongXJJWEXAjéd—gx ""g 5“ Main» 5% 5; Consider the following StumI—Liouville problem on the interval} 0 < a: < wmmo .z/(owo yea—>20 The function y n cos m: is an eigenfunction of this Stum—Liouvifle probiem corresponding to the eigenvame A 3 71:2. g i (E: ‘1 <13 s 9“ t g ‘2 v g” g » “ffiflgfi’fi, if {2w}? ago-fig g:me '“ I (7; M»? , 2%. \E w, / w 3 oog'fi’y w z?” (awwfi: » {is v’ r I 2" ‘ 7 M fl - filii a} "‘ 5,; "v/ M The set of Legendre polynoufials is an orthogona} family on the mayoral M1 q 3; < 1. jam» :3 . (I? 5 . r} 5 ffiismmoos xdm’ 3: 2L SHI"$CC}S mix: 3* ggzawgm “ 51" k 7 f. g ‘5 >35} : $53£2€ mfg ifiiflé 5);?» ,Aéwfioéfiw s 13. Any function f such that f and f’ are piecewise contiguous can be written as a Fourier series g I A , is“ mz 422:, @ZE/mwj M.) v x y i 1% i I ‘ 3- ii: ££®V 3L2} oi gégjéj/fifi 14. If f is an even function which has a Fourier series, then its Fourier series contains no sine terms. 15. The fimction pictured to the right is absoiutely integrable, j» 3 l ; x 2 ‘3' a A . :6. Consider Laplaoe’s equation + g?” m 0. Under the usual assumptioo (With the separation of win . am? vaziables method) that Maggy) 2 F'(x)G(y), computations Show that F(;r:) = 6:83” + (2263 C(y) = ogcos 3531+ again kg; (The preceding statement is two. You do not need to verify it or judge it. The following statement is the one you must judge.) Thus the compiete set of solutions- to Laplaoe‘s equation is given by Mm, y) m (516“ + CgewkxMCBCOS Icy + c4sin kg) . ; , 57 a ,. , 3:: «mM' '{%¢; imam; W“? .; n i f? 3 '2 n: g; I; ’ '2: “@343 @5325) rim-r535 “ff ,i x i I a A _ . g"! (In of i :: MM JMK givflacmf raft-“454.4291 & 1?. If ti; and ii? are solutions of a partied differential equation than any linear combination a 2 am} + (3qu is also a. solution. / g' i Vi 2% g";- ' 3‘ gm E3 Sin, wamfi' fig, fiafiyx mess/w”: 1U§V1€T§13m°éfim i i z i? i f M? - $231. $52 _ r -‘ : .43 . f «* "’fo . ‘ 21’ w; i ‘ L “é” » :4" ../W?’§yr’ «iv/i Mi” Mg «kayak: fwflfiw it, a» L :8. if the fimdamentai frequency of a vibratiog sirng is SE) Hz: that; 230 Hz is om of the overtones, J i s. 3" e a may"? 4' ’" ,4 A i I; W 5 9r i. g ffiuz. zip/Mr fgmfiiggzfi, a i < E .r n ‘ if v 1’. v ‘ : n; g“ . 5' . , - wage . W . a": mo: 44; 4;,“ %“m 44%; 54M fimgw (f g; Part. III. Free Resgonee Foiiow directions earefiziiy. The {Joint vaiue for each probiem is shown to its iefi. $2 if0<a7<i (8) 19. Letf(x)-{e if$>1 (a) Draw the odd extension of f . (b) This function (the ode exterision of f) is absolutely integrabie. (You do eat need to verify this.) Find its Fourier integral (not its Fourier series). Show ail the steps needed to arrive at the correct solution. You (£0 not need to do aoy aigebraic simplifications to your final answer. If Afiw) = 0 or B(w} fl 0, yen may state that without any reason or explanation. .24.; zese’o j 3‘: fifth 5‘ ,- ,ng ‘7 I §.H&fl£, weal/2% a iii/3:35 3“ J; :3? {a}; figmezew iggw} ”' Q; Weieeee‘éize 3-)“ __r W :2, 2 K g E E , w 4.3,” A I.” .<§ ‘jgfjfi if fiéflkwfiffiaffi fijmmgfiée; time A; a 5"” ,. ,2 2?. 2 g m. 2 213‘“ ea} l” W i w "r \< m. W ' :7 a S i . fis'fioélef‘éfp 3? {MQ‘E’ “‘23? if c e1 ‘52“; £3 M“; W 7, CL A “e 3%; Wfiéflw “w {Low-wee §3 i “5?? iv 5W; W 5-5,.» "4.3 x \2; ' ». :" (Riem’e fa)?“ a...”fg‘fiwewgw‘imufixfi f w“? e W; M“; ewe,» "‘ {w W M; féééisszzxaeo “ff/.415; of w W m (1) 20. What is theith ofthe expression as $2: approaches zero? we I .‘ w’gjgé'fig’i’) (6) 21. Find the solution of the foliowing initial boundary vaiue problem a“ m. fig améc‘h? 2263,25) m0 u(27r,i) 30 fort>0 u(m,0) m f($) : 103312: + ElfisiHSx forO < x < 271' (Do not take {he time to work out the solution from the beginning using separation of variabies. Rather, you may work with the soiution form which we have already derived, once you recognize which type of equation this is.) Ree} "" were," wiéég Mega as} if} 5? {:1 3’” L "5; 2??” 6% A 2 m .1 the $335 were “3.: Meg} w Z. ewe m; a 7?.“33 ‘ ax} r“ ; Kg I “fie—wiping?” E g~gfifigg MW 3 iwé aging w ER ' gr?“sz g. 3 9% We. 32- ;» EX». 5 fafiwifi ‘4’ gal? 3“: 8§§§PL 2» if“ glfiifiw‘i é» jgggflg” 2 ¢ w’lw gag )X' 4,, ‘3» 5 ‘ Y Q ‘Mégag; “5” If 3 ., weigrf} M ffifime fl: {3: “*9” m; 55%?1 a}: 5:; {Ti} EXfRA CREDIT You aiready cameo this, point if you me your Fog-mole Lise io ciess on Memos}? and tamed it 3113. <1) <2) <3) {4) <5) <6) (7) <8) (9) (10) (n) <32) <13) Pentium List foes ax d3: = is‘max fsiricmdm m w icosam W E x - xcamxdx ~— gfigcosax + gsmam [xsinaxdx m ggsinax “— $005339 @2532—2 - a3 fxgcosax dzc m iégcosam + ‘F ‘3 . . ‘3“— fxzsmaxdx = Egsmazc w a ‘13 2063113 femcosbx dz: 3 fig (:0be + bsinbx) (11" feflsinbx d2; m e (a sinbx ~— bcosbm) 32—h? fcosi’assdx = $3: + isinzasc ' 2 4a . g M i w m; . fsm axdx m 2:3 4a smch sin 3 sing m awcoflx + y) «E. cos(x ~ y)] 1 cosxcesy : i; [COS($ + 'y) ~§~ 005(3: w 31)} sinmcosy w %[sin(x ~é- y) + sin(m — y)] ...
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