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exam 1 fall 2008 solutions - Engineering Mathematics(ESE...

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Unformatted text preview: Engineering Mathematics (ESE 31’?) Exam i September £33, 2098 This exam ceeiains five multipiewheice problems worth two paints each, eight Short—answer emblems worth one point each, six true-fake preblerns worth one point each, and four freewesponse emblems werth £6 points airogether, for an exam total of 40 points. There is a table of Laplace transforms on the East page. f’art I. Muitiglefiheice Cieariy circle the oniy correct resyonse. Each is worth two points. 1. Fine the Lapiace transform ofthe function fit} graphed below: 9,. ff? (A) fa C; L J? /‘3 (B) S 3 wfiéxami. $.61?in 1' Ev {lg-r, ; was 1 5 g (C) e 2 (gr) ; “35 “L _. .2, (D) ( ) W ”M {I (E) e“33(;1§+§) % 2. Begin the precess of using Lapiaee transforms to solve the foilowing integrai equation. What will the expression 100k iike immeéiateiy after the Laplace transform of both sides is taken? WI? + 65]? W mp} 6529 fl 1 (3X) YéSj‘j‘“ Wit/{:53 Si j‘i g 25‘;wa m2 3‘ a L {”“““W "W “WM 3.;Wmtflw?““r if? 5: é; } a? “if: i: g} (if, :3 fiéfi _. 3; NEWWWY’wzf 5; a 5,; mwflmwfiwflwww {<3} rfs‘wiwwéxé ; r L. ,2: a z :2 L 5 xgaz—i :1 f g; %f if; L?" 5?; 1L, ,3: I; j , \ 7). E_ .f!, x) 3 7 fl, {3} E £3} ,4 m ‘f {a}; 3 i a} if , x L 51%;); v. 3 $3 § .4, :1, 1 5 3, L i e I g , , a a L ’ x E 19/ 2 1 LL.“ .M ’38 ”E‘s .; i a f “- fl 3: W?’ . a3 ' *MWM f (15:9 M ,. {ME My; 3 SMILE/£5? “ L; .; £15233 “L’ 5., L»; g “’3 we 7"“ =3” g E "ii W E if) ‘r as; ”a” :"W*} age; — H; iffy}; E/ih? W *‘?3ffz}/{g§f g; 7: 9x433} _ :3 L; a {H} w r Find the value if the feiiowing determinant. 3. \{s’ M! g” i , 3M» fag/5.31 aka, u f t 2;“; 12m " i , a E “iii? 2 Z 1... A“ W <W éiiigifé Aw: AM J, x} Q 2 (1) 12 2 m2 _3' 5 4. Find the inverse of the matrix [ 130130 5:41.? W m SW43? 3Mm5mm m _ . w \I; X.) @ mv Sigigéflf 11111:}. 5Mw1w8WTEEEEIJ W W nlwgfifié m M3w45_4 3_1nw1w8m w FliiLMfaiifiL M ) “ \I/ mm / mm 915.3133: www.43w4 I‘m/MIME (H) 5’ Soive the fo-i‘lovong system of equatioes. If there is no solution, select (A). If there are infiniteiy- a: many solutions, select (B). if there is a unique solution 3; , find the sum :t: + 3; wt» 2: among Z xwy+zm—1 choices ((2) through (LE). 3x w y + 22 m 2 Egg-mew serene 5‘7 M g 2 “E 3 (C) m?" E ,3 ~§ :2. 5 '2 j (D) W5 3, e ’2- ~§ ! 5” 3 5»; ,g 3 A (F) M1 333*5‘3 f t o 2. wt : g g (o) 1 Lo 2 W s 5 3 (H) 3 g I 5 - «z t a w 0 Q3+ZW§EQZ i ”E t 5;” (5) "z - e o 2 t (if: {:3 g i 5:} Part II. Short Answer Each of the foiiowing is worth one point. Each answer is either right or wrong: no work is required, and no partial credit wiii be given. 6. Fifi in the {flanks The Laplace transform Es useful for solving differential equations because it .fl, transforms a difierentiai equation into 3.61)- equation. 7. On the axes below, draw the futi-wave rectification of the eine filtration. {a 8. Find 130944206 4:). You do not need to simpfifiz your answer. :? Zég eme%>: 27%?) “Vim-4g} #15 g 3 444533 9‘ F511 in the foilowing 3x 3 matrix so it is skewwsymmetric. 6?» —2 ms’ 2 4:3 1 5 4; 5:3 10. Consider the set V of a1! vectors v m [01,123, 113-, 114} in R4 such that 1.22 3 0 and 114 g 0. This set is go: a vector space. Name one of the ten vector space properties that is not satisfied by V. (You can give the name or the number of the property; whichever you prefer.) 2 , if} cfmemm eefifigfiw’i sémeékfa smfiffiéefifi 41442 {2 fig oefiee mjseocé, a” effweflw 11. Continuing with the set V from problem 10, verify that your chosen property does not hoid by giving a specific exampie fi‘om V of a case in which the property fails. mfié‘émmgeég, [feed 5w;§[;}2}3 :4}: 93-22; 44 4} 24% if exam; m e? 12. The polynomiais 121(3) an x2 + 1 and pets) = x + 5 are eE-ements of the vector space P3. Give an example of a poiynomiai 393(5L‘) in P3 such the: the set {191(4) p230) 133(4)} is Enemy depenéent. WW We WM (if 559 Q44 If); sezég age? J 4msefiéwvgfiie; rng+fi£[x:4xex4é‘ EBA Continuing with the poiynomials pgix) and 372(3?) fiom problem 32, give an example of a poiynomiai 33:43:) in R; such that the set {39; (3:), 392(3), pgfix‘fi is linearly indepemiem. ’3“ E ‘ £3 3 s‘ f ' " J “ 5:“wa {Méwe J‘fifiéés’g :4”; 4 3 "4M” gee r’fififiz‘ Me View» W. ”J - ‘ W I . ,5 ‘ "I '7‘? Part III. True—False Write out the word “true” or “faise” for each of the foiiowing. Each is womb One point. 14. Twe difi‘erent fimctions cannot have the same Lapiace transform f j} jwimqw MjwM agaiggx 42 a); szjifixi W XMMM W1, iaémm ‘f’m MW» %;Lé;gg, fiW 15. If A and B are nonsinguiar nxn matrices, then AB is also a neasingular nxn matnx, and its inverse is A413“1 . .3 3 9. :g «R g ’ £5 M; 53:. MW . MA. fig: yfi Mam jigfiifi ; . f t‘.. K A {:i E 'wgififl’g 3 0 0 O 9 O . 2 1 0 0 O 0 w} 4 1 0 O 0 2 2 7 —3 O 0 1 4 2 5 O 0 6 —1 3 2 l 8 “fig/W gig-3.3;}??? :Lj :1 I: 18. The set V of ail soiutions of the differentiaE equation y” + 5111’ + fig 2 cost is a vector space of dimension two. ”YEW 23> f LM wwwiawmmm . J{"3 3 3 z“ fffiéé$ g5}; Mww MZM 355} 523 gig aim; Eiflggi M‘éggmfi E 3 g; 1.9. The pair 3f” matyices befow spam the meter space 0f 33 2 x 2 diagonai matrices. 1 9 0 0 Wit; gwf x; {YMVEL Wigfim féiM 5‘2: 0 0 0 1 V, , 7154+ ‘ fig?“ “552»; 3 {$3M M £12,; fiMx €3fiM. o/i‘éfiu’ M j}? in J}. g Jfiw g“; W’fiMM “:29"??? J5 Part IV. Free Resganse Foflew directions carefully, and Show aii the steps needed to arrive at your soiution. The paint value for each prohiem is shown to its iefi. (5) 20. Use the Laplace transform £0 find the salutien 0f the foilowing initiai value problem. You do not need to simplify yaur final answer. (In ether words, when you get to the answer 3:0!) 3 ya: do not neeé to put it in any other form.) y”m6y’+25ymé(tm2) y(0)m0y’(0)=0 é‘aiflé‘) M éimkzjiga} 9Z§V€§3 1—” fig” (fiz’éfi 4»— 2§>V5s§ 1 53“"25 \ é Mag r- i z: K3255} $2Mé§+2§ 8 g ,. 4! i w “5;“ 3&4‘ I Oxfdi/Szo figéfiwgv 3: j g({5w3)2+»§23> M ‘1‘ 53 fiifigéé ils’WLLbéK- 5%) {3 :3. 3) 2%.? 3 Li: ZS 2 35%”223 I: Z; 3‘3 $33}; 5%” ’2;> a; 5% M3} Wfigfi“ 53; 4W 2- (4) 21. Use the integral definition of convelufion to find :I:2 :12. You dc not need to Sim 1i I your final P answer. i, 1 am A; g: ' ‘ z 1’ w i akvt Lkfi gigggéf “5:: Fig/32.0}; m: ”" £53 ”a; {gig} waif/fig}? Vi} jag? a; xiii/«iméz’g ”if? 4;. if? E“ 53“? M 3“ {a w“ “:52? g {3) 22. Find £"3(}n(32 +9». m i”/LZ§E+%§E§ : "“ MN (4) 23. Consider the set V of 311 3X2 matrices such that the second raw is twice the first row. This is a vector space. (You do not need to verify this.) (3.) Find a basis for this vector space. fmj 12.. :2) ”a a; g; [a a” {a a _ ’figwfli 2a 2a » t a; 5:15 if? 53 J! u m 5% .1} M Q *2; r} {:3 aaafi; f < {i if)“ [a J {a 5;] {M {31% éarfi/iifiiw g 2, at} i; 15 f} :3: j 5 52‘s If} E g C} 5:} 5 ffléfiej gafi’jiéafijgéaij z... 'a M“ )3 , (b) What is the dimensien of this vectczr space? {I *3 r i g. {'7’ Table of Laplace Transforms - 44444444444) - H1 — _ 4244 44444 44» ‘ ———_ — ...
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