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exam 1 fall 2008 solutions

# 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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