exam 4 spring 2008 solutions

# 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&lt;x&lt;0 i. Let f be the ﬁmction given by ﬁx) a {1 on M1 &lt; x &lt; l and then made if 0 &lt; a: &lt; 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 .) &gt; 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 &quot; § (E) g r a: E (F) 7r g i g; (3} ~27; {2e ggemjee e} “’35 “z 3511?? ,&gt; 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&quot;: &quot;“ “5”” wee/Euro“ w 5} 4 (we? (G) 55g w (H) 3 in .r if (A? ,3 3” {I} e I was if M} &lt; x &lt; O Letf(:z:)m{:rr if0&lt;x&lt;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&gt;wtg a); Xv“ 2 M A .3 W} a V?“ (D). M; .. &quot; {:ﬁéw *3 “£32521 _ afﬁx} “2: EM 4 “W 'wljg ‘ a} _ (E) 0 &gt; 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 &lt; a: &lt; 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 = &quot;g&quot; 3: sinmda: -’ “75,. f z 4 bwwﬁ ) Li]; ) L difﬁwxv‘éwwﬁ W‘s ‘Mté” Ema “3’ B“ Eek fiﬁEﬁiﬁfdw MM fez 4&quot;: m m M: #3... L - m M “A x we?&quot; W 3;. jg f? ’ ‘5 E ewﬁﬂé, 2T L &quot;T .r ‘ &quot;if&quot;; ﬁfZ/Xjﬁm i” it (E) Ba m “ff”er f(x)smﬂ£—\$dx I} - L “Ti: ‘ e vs G CF) Bum-Ei‘ffo f&lt;x&gt;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;»--&gt; W ‘ “MW -- -- .h WMMWW W 1&quot; 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 &lt; x &lt; 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 &quot; i‘; ’ i u‘ ‘ “I ‘ 3 The ﬁmotions y} = :z? and y? = a: are orthogonai on the intervai —1 &lt; a: &lt; 1‘ v E § 3 g f“ g g”; 5 j§j W 2&quot; é? jwsfongXJJWEXAjéd—gx &quot;&quot;g 5“ Main» 5% 5; Consider the following StumI—Liouville problem on the interval} 0 &lt; a: &lt; wmmo .z/(owo yea—&gt;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 &lt;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&quot; ‘ 7 M ﬂ - ﬁlii a} &quot;‘ 5,; &quot;v/ M The set of Legendre polynouﬁals is an orthogona} family on the mayoral M1 q 3; &lt; 1. jam» :3 . (I? 5 . r} 5 fﬁismmoos xdm’ 3: 2L SHI&quot;\$CC}S mix: 3* ggzawgm “ 51&quot; k 7 f. g ‘5 &gt;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&quot;! (In of i :: MM JMK givﬂacmf raft-“454.4291 &amp; 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&quot;;- ' 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 «* &quot;’fo . ‘ 21’ w; i ‘ L “é” » :4&quot; ../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&quot; e a may&quot;? 4' ’&quot; ,4 A i I; W 5 9r i. g fﬁuz. zip/Mr fgmﬁiggzﬁ, a i &lt; E .r n ‘ if v 1’. v ‘ : n; g“ . 5' . , - wage . W . a&quot;: 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&lt;a7&lt;i (8) 19. Letf(x)-{e if\$&gt;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&amp;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.” .&lt;§ ‘jgfjﬁ if ﬁéﬂkwﬁfﬁafﬁ ﬁjmmgﬁée; time A; a 5&quot;” ,. ,2 2?. 2 g m. 2 213‘“ ea} l” W i w &quot;r \&lt; 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,.» &quot;4.3 x \2; ' ». :&quot; (Riem’e fa)?“ a...”fg‘ﬁwewgw‘imuﬁxfi f w“? e W; M“; ewe,» &quot;‘ {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&gt;0 u(m,0) m f(\$) : 103312: + ElﬁsiHSx forO &lt; x &lt; 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} &quot;&quot; were,&quot; wiéég Mega as} if} 5? {:1 3’” L &quot;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. &lt;1) &lt;2) &lt;3) {4) &lt;5) &lt;6) (7) &lt;8) (9) (10) (n) &lt;32) &lt;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&quot; 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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